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Doctoral Thesis

Quantum Networks with Superconducting Circuits

Author(s): Kurpiers, Philipp

Publication Date: 2019

Permanent Link: https://doi.org/10.3929/ethz-b-000378338

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ETH Library D ISS .ETH NO. 26152

QUANTUMNETWORKS WITH SUPERCONDUCTINGCIRCUITS

A thesis submitted to attain the degree of D OCTOROF S CIENCES of ETH Z URICH

(Dr. sc. ETH Zurich)

presented by

Philipp Kurpiers

MSc ETH Physics, ETH Zurich

born on 02.02.1988

citizen of Germany

accepted on the recommendation of

Prof. Dr. Andreas Wallraff

Prof. Dr. David Schuster

2019

Abstract

Our modern information society relies on high-performance local information processing and global information distribution. Complementing today’s information technology with quantum networks promises exponentially faster computation, and more efficient and secure communication. First quantum networks have been investigated during the last decades showing the feasibility to utilize quantum technologies locally and globally. In this thesis, we present the experimental implementation of a local quantum network in a circuit quantum electrodynamic (circuit QED) architecture, which has become one of the leading platforms for quantum information processing. Extend- ing the circuit QED framework by reliable, high-bandwidth quantum communication techniques thus paves the way towards distributed quantum computation. Besides quantum computation, combining quantum communication and our fast quantum- state readout techniques provides a basis for studying non-local physics based on Bell tests in circuit QED. Circuit QED systems with typical transition frequencies in the microwave regime need to be operated at temperatures in the tens of millikelvin range to reduce their thermal population. To build circuit QED-based quantum networks we develop a novel modular design of a cryogenic network. We experimentally investigate two cryogenic prototype systems with a length of 4 meter and 10 meter. Based on these characterizations we perform simulations for systems up to 40 meter. The experimental and simulation results indicate that large-scale cryogenic networks can be realized using our modular design. In addition to cryogenic networks, reliable quantum communication at high transfer rates requires low-loss information transmission between distant quantum systems. For that purpose, we characterize coaxial cables and rectangular microwave waveguides down to millikelvin temperatures using a resonant-cavity technique. With this technique we measure attenuation constants down to 1 decibel/kilometer at the single photon level. These low attenuation constants show the potential to perform quantum communication with microwave photons over several tens of meters. In the next part, we realize a direct quantum channel between two remote circuit QED nodes on separate chips. We utilize a fully microwave controlled interaction between a quantum system and a resonator to emit time-symmetric single itinerant photons from the first node and absorb them at the second node. To characterize the performance of the direct quantum channel we transfer quantum states directly and generate entangled states between the remote nodes. We experimentally investigate two different mappings of local quantum states to the itinerant photons. First, we map local quantum states to the Fock-state-basis of the itinerant photons and realize the first deterministic quantum communication between two remote quantum systems. Second, we show that quantum communication errors can be detected by mapping quantum states to a time-bin superposition of a single photon. To herald quantum communication errors by an ancilla quantum systems in future experiments, we test a novel all-microwave interaction between two quantum systems. Quantum gates based on this all-microwave interaction can also be used in more advanced quantum communication algorithms like quantum teleportation or error-correction schemes like entanglement distillation. For all remote quantum communication experiments, we validate the high level of experimental control by observing an excellent agreement with numerical master equation simulations. In conclusion, we present a framework for modular cryogenic quantum networks based on low-loss direct quantum channels for future quantum information processing applications and fundamental studies of non-local physics.

iv Zusammenfassung

Unsere moderne Informationsgesellschaft basiert auf lokalen Hochleistungscompu- tern und weltweiter Informationsverteilung. Quantennetzwerke, welche die heutige Informationstechnologie ergänzen können, versprechen sowohl exponentiell schnelle- re Berechnungen als auch eine effizientere und sicherere Kommunikation. Während der letzten Jahrzehnte sind erste Quantennetzwerke erforscht worden, die eine zu- künftige Nutzung von Quantentechnologien lokal und weltweit ermöglichen könnten. Die vorliegende Arbeit untersucht die experimentelle Realisierung von lokalen Quan- tennetzwerken in supraleitenden Schaltkreisen, welche sich zu einer der führenden Plattformen für Quanteninformationsverarbeitung entwickelt haben. Wir erweitern supraleitende Schaltkreise um zuverlässige Techniken zur Quantenkommunikation mit hoher Bandbreite und schaffen somit die Grundlage für verteilte Quantencompu- tersysteme. Neben Anwendungen im Rahmen der Quanteninformationsverarbeitung schafft die Kombination von Quantenkommunikation und den von uns entwickelten schnellen Messtechniken von Quantenzuständen die Grundlage, um nicht-lokale physi- kalische Phänomene mit Hilfe bellscher Ungleichungen in supraleitenden Schaltkreisen zu untersuchen. Quantensysteme in supraleitenden Schaltkreisen müssen aufgrund ihrer Übergangs- frequenzen im Mikrowellenbereich auf Temperaturen von einigen zehn Millikelvin ab- gekühlt werden, um ihre thermische Anregung zu reduzieren. Um zukünftig supraleitenden Quantennetzwerke aufzubauen, entwickeln wir ein neues modulares Design für kryogene Netzwerke. Wir untersuchen experimentell zwei kryogene Prototypsysteme mit einer Länge von 4 Meter und 10 Meter und nutzen diese Charakterisierungsmessun- gen, um Systeme bis zu 40 Meter Länge zu simulieren. Die experimentellen Ergebnisse und Simulationen deuten darauf hin, dass gross angelegte kryogene Netzwerke mit unserem modularen Design realisiert werden können. Zusätzlich zu kryogenen Netzwerken benötigt eine zuverlässige Quantenkommu- nikation mit hoher Transferrate eine verlustarme Informationsübertragung zwischen entfernt liegenden Quantensystemen. Wir charakterisieren hierzu Mikrowellenkoaxi- alkabel und Rechteckhohlleiter bis zu Temperaturen im Millikelvin-Bereich mit Hilfe einer Resonanz-Kavität-Messmethode. Mit dieser Technik messen wir Absorptionsko- effizienten kleiner als 1 Dezibel/Kilometer bei einer Signalstärke auf dem Niveau von einzelnen Photonen. Diese niedrigen Absorptionskoeffizienten zeigen das Potential, dass breitbandige Quantenkommunikation mit Mikrowellenphotonen über mehrere zehn Meter durchgeführt werden kann. Im nächsten Teil dieser Arbeit realisieren wir einen direkten Quantenkanal zwischen zwei entfernt liegend, chip-basierten supraleitenden Schaltkreisen. Wir ver- wenden eine vollständig durch Mikrowellensignale kontrollierte Wechselwirkung zwischen einem Quantensystem und einem Resonator, um einzelne, zeitsymmetrische, sich ausbreitenden Mikrowellenphotonen von einem Netzwerkknoten zu emittieren und an einem zweiten zu absorbieren. Um diesen direkten Quantenkanal zu charakterisieren, übermitteln wir Quantenzustände direkt zwischen zwei Netzwerkknoten und erzeugen verschränkte Zustände zwischen diesen. Wir untersuchen experimentell zwei verschiedene Übertragungsverfahren der lokalen Quantenzustände auf die sich ausbreitenden Mikrowellenphotonen. Zum einen übertragen wir lokale Quantenzu- stände auf die Fock-Zustands-Basis der sich ausbreitenden Mikrowellenphotonen und realisieren damit die erste deterministische Quantenkommunikation zwischen zwei entfernt liegenden Quantensystemen. Zum anderen zeigen wir, dass Quantenkommu- nikationsfehler detektiert werden können, indem die lokalen Quantenzustände auf die Superposition von Zeitintervallen eines einzelnen Photons übertragen werden. Um Quantenkommunikationsfehler durch ein Hilfsquantensystem in zukünftigen Experi- menten anzuzeigen, testen wir eine neue, mikrowellen-kontrollierte Wechselwirkung zwischen zwei Quantensystemen. Quantengatter, die auf dieser Wechselwirkung basie- ren, können auch in erweiterten Quantenkommunikationsalgorithmen wie der Quan- tenteleportation oder zur Fehlerkorrektur mit Hilfe der Verschränkungs-Destillation genutzt werden. In allen Quantenkommunikationsexperimenten belegt eine ausge- zeichnete Übereinstimmung mit numerischen Simulationen der Mastergleichung ein hohes Mass an experimenteller Kontrolle unserer Quantensysteme. Zusammenfassend präsentieren wir ein Design für ein modulares kryogenes Quan- tennetzwerk, das auf verlustarmen Quantenkanälen basiert und das zukünftige An- wendungen im Bereich der Quanteninformationsverarbeitung und der Grundlagenfor- schung zu nicht-lokalen physikalischen Phänomenen ermöglicht. Contents

Abstract iii

Zusammenfassung v

1 Introduction to Quantum Networks 1

2 Basics of Quantum Communication in Circuit Quantum Elec- trodynamics 7 2.1 Quantum Information Processing ...... 7 2.1.1 Single Qubits and Qutrits ...... 8 2.1.2 Multiple Quantum Systems ...... 10 2.1.3 Bell Inequalities ...... 11 2.1.4 Experimental Loopholes in Bell Tests ...... 16 2.1.5 Entanglement Witnesses ...... 18 2.2 Circuit Quantum Electrodynamics ...... 21 2.2.1 Basic Circuit Elements ...... 22 2.2.2 Coplanar Waveguide Resonators ...... 25 2.2.3 Josephson Junctions ...... 27 2.2.4 Transmon: a Multi-Level Artiﬁcial Atom ...... 30 2.2.5 Jaynes-Cummings Model ...... 37 2.2.6 Dispersive Readout ...... 39 2.2.7 Purcell Filters ...... 46 2.2.8 Cavity-Assisted Raman Process: F0G1 ...... 51 2.2.9 Sample Fabrication ...... 59

3 Cryogenic Quantum Networks 61 3.1 Basics of Cryogenics ...... 62 3.1.1 Dilution Refrigerators ...... 62 3.1.2 Heat Transfer in Solids ...... 63 3.1.3 Thermal Properties of Copper ...... 65 3.1.4 Thermal Conductivity and Mechanical Stability of Insulators . . 69 3.1.5 Thermal Conductivity of Superconductors ...... 71 3.1.6 Thermal Boundary Resistance ...... 73 3.1.7 Thermal Radiation ...... 74 3.2 Experimental Setup for Circuit QED ...... 78 3.2.1 Passive Heat Load ...... 78 3.2.2 Input Microwave Cabling ...... 79 3.2.3 Cryogenic Detection Lines ...... 83 3.2.4 Room-Temperature Setup ...... 86 3.3 Cryogenic Networks ...... 87 3.3.1 Cryogenic Requirements for a Loophole-Free Bell Test with Superconducting Circuits ...... 87 3.3.2 Modular Design of a Cryogenic Network ...... 89 3.3.3 Characterization of a Cryogenic Prototype System ...... 93 3.3.4 Cooldown of a 10-Meter Cryogenic Network ...... 97 3.3.5 Simulations of a 30-Meter Cryogenic Network ...... 100 3.3.6 Outlook on Cryogenic Networks ...... 102

4 Characterization of Coaxial and Rectangular Microwave-Frequency Waveguides 105 4.1 Propagating Microwave Signals ...... 106 4.2 Overview of Evaluated Coaxial Cables and Rectangular Waveguides . . . 111 4.3 Experimental Setup ...... 112 4.4 Characterization of the External Couplings ...... 114 4.5 Illustration of the Resonant-Cavity Technique ...... 116 4.6 Measurements Results of Attenuation Constants ...... 117 4.7 Conclusion and Outlook on Extended Waveguide Measurements . . . . 125

5 Quantum Communication with Microwave Photons 129 5.1 Literature Overview ...... 131 5.2 Quantum Communication via Direct Quantum Channels ...... 131 5.3 Calibrations of the Excitation Transfer using Time-Symmetric Photons ...... 133 5.4 Emission and Absorption of Microwave Photons ...... 137

viii 5.5 Deterministic Quantum State Transfer ...... 140 5.6 Deterministic Generation of Remote Entanglement ...... 141 5.7 Quantum Communication using Time-Bin Encoding ...... 143 5.8 Time-Bin-Encoded Quantum State Transfer ...... 146 5.9 Time-Bin-Encoded Generation of Remote Entanglement ...... 149 5.10 Photon Loss Model and Outlook on Bell Tests ...... 152 5.11 Two-Qubit Gates based on the FGGE-Transition ...... 158 5.12 Conclusion and Outlook ...... 164

Bibliography 167

List of Publications 221

Acknowledgements 223

Curriculum Vitae 225

Chapter 1

Introduction to Quantum Networks

What I cannot create, I do not understand. Richard Feynman in February 1988

The development of computers and their interconnection via the internet has funda- mentally changed the way we process and transfer information. Today, we are able to run extensive computations on high-performance computers remotely from nearly everywhere on earth and we use secure communication channels to communicate with each other or with our connected devices. Our current information technology is based on hardware components like transistors and lasers, whose developments require the understanding of quantum mechanical laws combined with electrical and optical engineering techniques [Dowling03]. Information itself is however treated classically in these modern devices, meaning that no exclusive quantum phenomena like quantum superposition or entanglement can be utilized for processing or transmitting data. Using quantum systems as information carriers turned out to promise exponentially faster computation compared to classical computers, e.g. as shown in Shor’s quantum algorithm to factor large integers [Shor94] or the quantum algorithm to solve linear systems of equations [Harrow09]. In addition, quantum systems can contribute to simulate complex chemical reactions [Feynman82, Kandala17], to optimize supply chains and ﬁnancial services [Rebentrost18], or to improve machine learning [Lloyd13].A universal quantum computer which can perform all those tasks is still beyond current technology, but controlling up to 100 quantum systems individually seems feasible in the near future. Using these intermediate-scale quantum systems new many-body physics phenomena can be simulated [Preskill18] and next substantial steps towards 2 Chapter 1 Introduction to Quantum Networks

a fault-tolerant quantum computer can be investigated [Devoret13]. Besides quantum computing, the collaboration of information science and quantum physics has led to developments of quantum communication techniques which can be utilized to transmit classical information more efﬁciently, e.g. superdense coding [Bennett92], or in a proven-secure way, e.g. quantum key distribution [Bennett84a, Ekert91, Gisin02]. Combining these quantum communication and computing ideas leads to full quantum networks consisting of multiple interconnected quantum devices. Similar to today’s classical information processing, quantum networks can be categorized into infrastruc- tures which distribute information globally via a quantum internet [Kimble08] and those which process information in local area networks (quantum LAN) [Duan10]. The quantum internet for intercity or intercontinental distances focuses on the transmission of quantum information and distribution of entanglement in environments with high transmission loss and noise. These networks require either indicators which herald the generation of entanglement [Duan10] or quantum repeaters [Briegel98] at intermediate distances. Quantum repeaters correct for emerging transmission errors during the quantum communication [Childress06, Zwerger18] using entanglement swapping [Zukowski93˙ ] and distillation [Deutsch96, Bennett96a] schemes. After entanglement has been established between two remote nodes, quantum teleportation [Bennett93, Bouwmeester97] can be used to transmit quantum states by only sending classical information [Northup14]. This indirect transfer of quantum information can thus tolerate high transmission loss with the cost of lower transmission rates. First long range quantum networks have been experimentally implemented based on optical photons in free space via satellites [Ren17, Yin17, Liao18] or in optical ﬁbers [Chen10, Wengerowsky19]. As in classical intercity or intercontinental communication, transmission of quantum information at optical frequencies is ideal due to the lower loss and smaller depolarization compared to lower frequencies. Local quantum networks are tailored for distributed quantum computation in which quantum states and entanglement are shared between multiple quantum processors [DiVincenzo00]. As for non-distributed quantum computers, these distributed quantum machines can provide exponential speed up compared to classical computing [Cacciapuoti18]. The separate quantum processors consist of many coupled stationary, atomic or solid-state quantum systems and store and process quantum information [Monroe14, Reiserer15]. Subroutines of the full quantum algorithm run on the separate processors [Grover97, Cleve97, Cirac99, Jiang07] which are connected by classical and quantum channels [Yepez01]. The quantum channels of local quantum networks are optimized for high-bandwidth quantum communication and enable error- 3

correction across different nodes to protect quantum information against decoherence [Fowler10, Fujii12, Horsman12, Nickerson15, Kumar19]. These requirements make especially direct quantum channels between two nodes an efficient way for scaling up distributed quantum computers. Direct quantum channels offer the possibility to transmit arbitrary quantum states directly and to generate entanglement deterministically between two nodes using single photons [Cirac97, vanEnk97, Duan10]. First algorithms for distributed quantum computing [Van Meter08, Ying09] and efficient implementations of quantum gates between different nodes [Eisert00, Sarvaghad-Moghaddam18, Cohen18] have been developed. However, the experimental realization of a deterministic quantum communication between separate quantum systems has not been demonstrated and is addressed in this thesis. In future quantum networks a combination of both types of quantum networks can be envisioned [Cacciapuoti18, Caleffi18, Kumar19, Dahlberg19], which will link long distance communication channels with local high-performance quantum computers, similar to our classical infrastructure today. Besides these computing and communication applications, quantum information science has also strongly deepened our understanding of the quantum world by introducing an alternative description of quantum systems. For example, entangled systems provide stronger correlations than any classical system [Einstein35] which can be treated as an information resource. Based on these stronger-than-classical correlations John Bell has shown that the statistical predictions of quantum mechanics are incompatible with any local physical theory [Bell64]. Experimental Bell tests [Aspect82b, Weihs98, Rowe01, Hensen15] have successively closed all loopholes which could have diminished the validity of the measurement results. Bell tests have thus expanded and challenged our understanding of locality and of physical theories in general [Brunner14]. In addition, loophole-free Bell tests provide the basis for device- independent quantum communication protocols since they guarantee that two systems are entangled independent of any implementation details. Device-independent protocols can for example be used in quantum cryptography [Arnon-Friedman18] and randomness amplification [Colbeck12] applications in the future. Quantum information science therefore combines fundamental quantum mechanical research questions and applied computation and communication tasks, which are at the basis of our modern information society. In this thesis, we present our experimental implementation of a local quantum network using circuit quantum electrodynamics (circuit QED) systems. The field of circuit QED studies the interactions of microwave-frequency artificial atoms and resonators 4 Chapter 1 Introduction to Quantum Networks

both being realized in superconducting electrical circuits [Blais04]. The flexibility in designing circuit QED systems [Wallraff04, Koch07a] and controlling them with high-precision [Kelly15, Córcoles15, Rol17] has made circuit QED one of the leading platforms for quantum information processing [Devoret13]. Beyond on-chip quantum information processing, first circuit QED-based quantum networks have been realized. In these quantum networks entanglement has been generated probabilistically between distant systems using a joint measurement [Roch14] or two-photon interference and detection schemes [Narla16]. In this thesis, we extend the circuit QED framework by direct high-bandwidth quantum channels using single itinerant microwave photons as information carriers [Eichler11, Pechal14]. For short-distance quantum communication, itinerant microwave photons are advantages to optical photons, since no low- efficiency, low-bandwidth conversion to optical frequencies [Lecocq16, Han18, Fan18] is required. All circuit QED systems need however to be operated at temperatures in the millikelvin range to reduce their thermal population at their microwave transition frequencies. To built larger-scale quantum networks for distributed quantum computing in circuit QED, we present a novel modular design of a cryogenic network including low-loss superconducting microwave waveguides.

In Chapter 2, we describe basic concepts of quantum information science and Bell tests. We next discuss selected topics of circuit QED which are necessary for the presented experiments in the later chapters. In Chapter 3, we describe the basics of cryogenic systems and of the thermal properties of matter at cryogenic temperatures. Next, a typical setup for circuit QED experiments is explained. Using the cryogenic basics, we introduce a novel design of a modular cryogenic network to host quantum networks. We present experimental tests of two cryogenic prototype systems with a length of 4 meter and 10 meter. The 10 meter-system is, to our knowledge, the ﬁrst cryogenic network which realizes a connection between two cryogenic systems at millikelvin temperatures. Based on our experimental characterization results we simulate cryogenic systems up to 40 meter, which can be used to perform loophole-free Bell tests with superconducting circuits. In Chapter 4, we present our characterization measurements of the loss of microwave waveguides at cryogenic temperatures using a resonant-cavity technique. We review the background of propagating microwave signals, characterize our experimental setup and detail our adaptations of the resonant-cavity technique. Next, we show measurement results of typical, commercial available coaxial cables and rectangular waveguides in a frequency range of 4-13 gigahertz and measure attenuation constants down to 1 decibel/kilometer at the single photon level. In addition, we investigate the power and magnetic ﬁeld dependence of low-loss coaxial cables. 5

Our experimental results illustrate the potential to perform quantum communication with microwave photons over several tens of meters. In Chapter 5, we present our experimental implementations of a direct quantum channel between two remote circuit QED systems on separate chips using a fully microwave controlled interaction. Before discussing our experimental details, we provide a literature overview of remote quantum communication experiments in various physical systems. Next, we explain our experimental procedure to calibrate the direct transfer of single time-symmetric microwave photons. We investigate two quantum communication protocols based on different mappings of our local quantum states to the itinerant photons. First, we map our quantum states to the Fock-state basis of the itinerant photons and realize the ﬁrst deterministic quantum communication between two separated quantum systems. Second, we show that communication errors can be detected by mapping our quantum states to a time-bin superposition of a single photon. We obtain an excellent agreement of our experimental results with numerical master equation simulation and introduce an analytical model to study the effect of photon loss for future Bell tests between remote circuit QED systems. We conclude with a characterization of a novel all-microwave quantum gate which is relevant in the context of heralded quantum communication and future quantum communication algorithms. 6 Chapter 1 Introduction to Quantum Networks Chapter 2

Basics of Quantum Communication in Circuit Quantum Electrodynamics

This chapter provides an introduction to basic concepts of quantum information science in the context of circuit quantum electrodynamics (circuit QED). We ﬁrst introduce the fundamental information unit, a quantum bit or qubit, and multi-qubit systems. Based on these multi-qubit systems, we discuss Bell tests and entanglement witnesses for qubits and higher-dimensional quantum systems. Next, relevant aspects of circuit QED are explained including coplanar waveguide resonators and superconducting qubits. We focus on the transmon qubit as the quantum system used in the experiments of this thesis. We describe the experimental calibration of transmon-qubit control pulses, Jaynes-Cummings type transmon-resonator couplings, and how its states can be read out. Next, an additional ﬁlter circuit is introduced, which allows to optimize the readout parameters and high-bandwidth quantum communication. We then discuss an all-microwave transition between a transmon and a resonator degree of freedom used for quantum communication. We conclude with a description of the fabrication process and an overview of experimental parameters used within this thesis.

2.1 Quantum Information Processing

In this section, we present the basics of quantum information processing using the reference by Michael Nielsen and Isaac Chuang [Nielsen11]. 8 Chapter 2 Basics of Quantum Communication in Circuit Quantum Electrodynamics

2.1.1 Single Qubits and Qutrits

Similar to classical computers, quantum computers are based on basic processing units which are deﬁned, in the simplest case, by two possible states - either g or e in the Dirac notation. In contrast to a classical bit, a quantum bit or qubit can| 〉 be in| 〉 any superposition state of these two states

θ iφ θ ψ = cos g + e sin e (2.2) | 〉 2 | 〉 2 | 〉 with θ and φ corresponding to the inclination and azimuthal angle of a three-dimensional unit sphere - the Bloch sphere (Fig. 2.1a). In the standard representation of the Bloch sphere the states g and e are at 1 along the z-axis, the states + , = | 〉 | 〉 ± | 〉 |−〉 1/p2( g e ) at 1 along the x-axis, and the states i , i = 1/p2( g i e ) at 1 along| 〉 the± | y-axis.〉 ± | 〉 |− 〉 | 〉 ± | 〉 ± Pure quantum states can always be expressed as state vectors. In reality however, quantum systems need to be considered as statistical mixtures of pure states, due to their interaction with the environment. These mixed states can formally be expressed

Figure 2.1 The state = ( g e )/p2 is represented a) as a point in the Bloch sphere, b) by its expectation values|−〉 of| the〉 − Pauli | 〉 matrices, and c) by the real part of its density matrix for the qutrit basis states. 2.1 Quantum Information Processing 9

as density operators or matrices [Gerry05] X ρˆ = pi ψi ψi (2.3) i | 〉〈 | with the sum over all basis sates of the system, for example g and e in the qubit | 〉 | 〉 case, and the probability pi to be in state ψi . We can decompose all qubit density matrices | 〉 1 ρˆ σˆ b σˆ b σˆ b σˆ (2.4) = 2( 1 + x x + y y + z z) using the Bloch vector ~b b , b , b , the identity matrix σˆ , and the Pauli matrices = ( x y z) 1 0 1 0 i 1 0 σˆ x = , σˆ y = , σˆ z = . (2.5) 1 0 i −0 0 1 − The Bloch vector deﬁnes the coordinates of ρˆ within the Bloch sphere [Bertlmann08] and the Pauli matrices form a basis of the two-dimensional Hermitian matrices (Fig. 2.1b).

Besides qubits, higher-dimensional quantum systems can be used as computational primitives or their higher levels can be advantageous for operations on the qubit subspace [Strauch03, Zeytinoglu15˘ , Bækkegaard19] making especially three-dimensional systems (d = 3), so called qutrits, interesting. A pure qutrit state

ψ = α g + β e + γ f (2.6) | 〉 | 〉 | 〉 | 〉 can be deﬁned by the three complex numbers α, β, γ which fulﬁll the normalization 2 2 2 condition α + β + γ = 1 (Fig. 2.1c). Mixed qutrit states can be decomposed using the Gell-Mann| | | | matrices| |

      0 1 0 0 i 0 1 0 0 λˆ   , λˆ i −  , λˆ   , 1 = 1 0 0 2 =  0 0 3 = 0 1 0 0 0 0 0 0 0 0− 0 0     0 0 1 0 0 i λˆ   , λˆ  −  , 4 = 0 0 0 5 = 0 0 0  1 0 0 i 0 0       0 0 0 0 0 0 1 0 0 1 ˆ   ˆ   ˆ   λ6 = 0 0 1 , λ7 = 0 0 i , λ8 = 0 1 0 . (2.7)     3   0 1 0 0 i −0 p 0 0 2 − 10 Chapter 2 Basics of Quantum Communication in Circuit Quantum Electrodynamics

The Gell-Mann matrices are an extension of the Pauli basis for three-dimensional ˆ ge Hermitian matrices: λ1,2,3 correspond to the Pauli matrices σˆ x,y,z in the qubit (ge) ˆ gf ˆ ef ˆ ge ef subspace, λ4,5 to σˆ x,y, λ6,7 to σˆ x,y and λ8 is the diagonal matrix (σˆ z +2σˆ z ).

2.1.2 Multiple Quantum Systems

Combining n, d-dimensional quantum systems leads to an exponential increase of the number of basis states with dimensionality d n of the combined system. Pure two-qubit A B states can thus be expressed as vectors in the combined Hilbert space H = H H A B A B A ⊗ B using the four basis states g, g = g g , g, e = g e , e, g = e g , A B | 〉 | 〉 ⊗ | 〉 | 〉 | 〉 ⊗ | 〉 | 〉 | 〉 ⊗ | 〉 e, e = e e | 〉 | 〉 ⊗ | 〉 X X ψ = αi,j i j = αi,j i, j (2.8) A A | 〉 i=g ,e | 〉 ⊗ | 〉 i=g,e | 〉 B B j g,e j=g ,e = with the complex numbers αi,j which need to fulﬁll the normalization condition as in the single qubit case. Multi-qubit states can be categorized in product or separable A states, which can be written as a tensor product of two-single qubit states ψ = ψ B ⊗ ψ with ψA HA (ψB HB) and non-separable or entangled states [Einstein35, ∈ ∈ Schrödinger35]. An example of a maximally entangled two-qubit state is the Bell state

ψ+ = ( g, e + e, g )/p2 (2.9) | 〉 | 〉 | 〉 which cannot be separated into individual single-qubit states and shows perfect anti- correlations even if both qubits are measured independently. A B In the case of mixed states, we distinguish between a product state ρˆ = ρˆ ρˆ P A B ⊗ and a separable state ρˆ = i piρî ρî , which is a mixture of product states with ⊗ probabilities pi [Gühne09]. Separable states are classically correlated meaning that they can be generated using only local operations and classical communication and are not specific to quantum theory. States are called entangled if their correlations cannot originate from classical procedures only and are thus typical features of quantum theory (see ψ+ ) [Gühne09]. John Bell [Bell64] showed based on the condition of locality that the| non-classical〉 correlations of ψ+ can be utilized to experimen- | 〉 tally [Freedman72, Aspect82b] test local-realistic models against non-local quantum mechanical predictions which is discussed in the next sections. 2.1 Quantum Information Processing 11

2.1.3 Bell Inequalities

The consequences of entangled states for quantum theory and physical theories in general have led to controversial discussions since their discovery [Einstein35, Bohr35, Schrödinger35, Furry36, Bohm57] and are still challenging our understanding of the world [Goldstein11]. In 1964, John Bell proved based on an inequality that the statistical predictions due to the perfect anti-correlations of entangled states are incompatible with any local physical theory [Bell64]. Bell’s theorem has stimulated a wide range of concepts [Brunner14] reaching from fundamental questions about non-locality [Bell04], to applications in quantum information science [Horodecki09], like quantum cryptography [Bennett84b, Ekert91, Deutsch96, Gisin02, Acín06, Arnon-Friedman18, Lucamarini18], measurement based quantum computing [Raussendorf01], quantum teleportation [Bennett93, Bennett96b] or randomness ampliﬁcation [Colbeck12] to mention some examples. In this section, we focus on the concepts needed to perform a Bell test using the CHSH-Bell inequality [Clauser69] based on the review by Nico- las Brunner et al. [Brunner14]. Philosophical aspects related to Bell’s theorem are discussed in a review by Sheldon Goldstein [Goldstein11]. John Clauser, Michael Horne, Abner Shimony, and Richard Holt (CHSH) [Clauser69, Clauser70] generalized Bell’s idea to experimentally realizable conditions by present- ing an inequality which does not depend on perfect correlations which cannot be realized experimentally due to imperfections [Goldstein11, Larsson14]. The CHSH- scenario can be seen as a ’black box’ situation [Brunner14] in which each of two distant observers, Alice and Bob, has access to one box, and the two observers share a communication channel (Fig. 2.2a). Alice (Bob) can ask one of the two possible questions x 0, 1 (y 0, 1) to her box A (B) and she (he) obtains one of the two possible results ∈ ∈ a 1, +1 (b 1, +1). Alice and Bob locally select different questions, measurement∈ − settings, and∈ − obtain measurement results from their ’black boxes’. The results of these experiments are characterized by a joint probability distribution p(ab x y) which attributes the obtained results a and b to the selected questions x and y|. A CHSH- scenario is then characterized by the set p = p(ab x y) of all probabilities [Tsirelson93] which is referred to as a behavior or correlations.| These correlations need to fulﬁll the normalization constraint P p ab x y 1 and CHSH introduced the expression a,b= 1 ( ) = ± |

S = a0, b0 + a0, b1 + a1, b0 a1, b1 (2.10) | 〈 〉 〈 〉 〈 〉 − 〈 〉 | P to characterize this type of scenario. In Eq. (2.10), ax, by = a,b ab p(ab x y) is the expectation value of the product ab for a selected measurement〈 〉 choice (x,|y). 12 Chapter 2 Basics of Quantum Communication in Circuit Quantum Electrodynamics

a) b) 4

NS x y 2 2

 QM communication Bell inequality 2 CHSH,

a b L Alice Bob fully mixed point 0

Figure 2.2 a) Schematic of a typical Bell scenario: Alice and Bob are at two remote places and share a common communication channel. They can ask individual questions x, y to their systems and obtain measurements results a, b. b) Illustration of obtainable CHSH- values S under the no-signaling (NS), quantum-mechanical (QM) or local (L) constraints. The shapes of the constraints are adapted from a set discussion for which we refer to Refs. [Barrett05, Brunner14].

The basic idea of the CHSH and, in general, Bell inequalities is, that different physical models translate into additional constraints on the behavior p and thus result in different bounds on S. In general, three type of correlations can be distinguished:

No-Signaling Constraints

These constraints avoid a direct conflict with relativity by preventing Alice and Bob to use their boxes for instantaneous signaling if they are space-like separated. This can be expressed [Cirel‘son80, Khalfin85, Rastall85, Popescu94] as X p(a x) p(a x y) = p(ab x y) a, x, y | ≡ | b= 1 | ∀ X± p(b y) p(b x y) = p(ab x y) b, x, y (2.11) | ≡ | a= 1 | ∀ ± meaning that the local marginal probability of Alice p(a x) (Bob p(b y)) are independent of Bob’s (Alice’s) measurement settings. Hence, we can| define a set| NS (Fig. 2.2b) including all behaviors which satisfy the no-signaling constraints Eq. (2.11). Popescu and Rohrlich [Popescu94] showed that a behavior p exist which fulfills Eq. (2.11) such that

Sns 4. (2.12) ≤ 2.1 Quantum Information Processing 13

Locality Constraint

Under the locality constraint the assumption is made that a set of past causes λ (often referred to as hidden variables or beables [Bell85]) can be found which affect both measurement results and can fully explain the correlations between a and b. The outcome on one side, however, depends only on λ and the local inputs and is fully independent of the inputs on the other side. The locality constraint can thus be formally written as

p(ab x y, λ) = p(a x, λ)p(b y, λ), (2.13) | | | so that we can explain p(ab x y, λ) by only knowing the local probability functions | p(a x, λ) and p(b y, λ). These probability functions depend only on the locally asked questions| x, y and| the past causes λ, but are not affect by the other party. In a general local theory we can also introduce a probability density p(λ x y) according to which λ are distributed, to take into account that λ are not fully controllable| in the experiment. We can deﬁne the set of local behaviors L as all behaviors which fulﬁll Z p(ab x y, λ) = p(λ)p(a x, λ)p(b y, λ) dλ. (2.14) | | |

Note that we set p(λ x y) = p(λ) which involves the important hypothesis that x and | y can be chosen freely or randomly [Bell04, BBT18]. And as Bell pointed out: if x and y are not free variables, "the apparent free will of experimenters, and any other apparent randomness, would be illusory" [Bell04] p.244. These type of theories are, hence, called ’superdeterministic’ ones which are impossible to test experimentally [Bell85, Colbeck12]. In summary, any local behavior fulﬁlls the no-signaling constraint but not vice versa, so that L NS. Using Eq. (2.14) we obtain a , b R p a b d with the x y = (λ) x λ y λ λ local expectation⊂ values a P a p a x〈, and〉 b 〈P 〉b〈 p b〉 y, and can x λ = a ( λ) y λ = b ( λ) express Eq. (2.10) as 〈 〉 | 〈 〉 |

Z Sl = p(λ) a0 λ b0 λ + a0 λ b1 λ + a1 λ b0 λ a1 λ b1 λ dλ 2. (2.15) 〈 〉 〈 〉 〈 〉 〈 〉 〈 〉 〈 〉 − 〈 〉 〈 〉 ≤

Here, we used a0 λ , a1 λ [ 1, 1] and assumed b0 λ b1 λ 0 without loss of 〈 〉 〈 〉 ∈ − 〈 〉 ≥ 〈 〉 ≥ generality [Brunner14]. 14 Chapter 2 Basics of Quantum Communication in Circuit Quantum Electrodynamics

Quantum-Mechanical Constraint

The set of all quantum-mechanical achievable behaviors QM, can be deﬁned by the constrain

ˆ ˆ p(ab x y) = tr ρÂBMa x Mb y (2.16) | | ⊗ | with the density matrix ρÂB in the Hilbert space HA HB and Alice’s (Bob’s) measure- ˆ ˆ ˆ ⊗ ˆ ment operators Ma x (Mb y ) acting on HA (HB). Ma x and Mb y can be general quantum- | | | | mechanical positive operator-valued measures (POVM) and need to fulfill Mˆ a x 0 P ˆ | ≥ and a Ma x = 1 [Brunner14, Nielsen11]. Here, we consider, without loss of gener- | ality, orthogonal projective measurement operators defined by Mˆ 1 σ~ˆ ax~σ~ˆ ) a x = 2 ( 1 + | and Mˆ 1 σ~ˆ b ~yσ~ˆ) introducing the vector of Pauli matrices σ~ˆ σˆ , σˆ , σˆ b x = 2 ( 1 + = ( x y z) | and the normalized vectors x~ and ~y. Using these orthogonal projective operators we ψ+ obtain ax, by | 〉 = x~ ~y. Next, setting Alice’s measurement choices to x~0 = (1, 0, 0), 〈 〉 x~1 = (0, 1, 0) and Bob’s choices to ~y0 = (1, 1, 0)/p2, ~y1 = (1, 1, 0)/p2, we obtain − the expectation values a0, b0 = a0, b1 = a1, b0 = 1/p2 and a1, b1 = 1/p2. From these expectation〈 values〉 follows〈 〉 〈 〉 〈 〉 −

Sqm 2p2 2.8284 (2.17) ≤ ≈ under the quantum mechanical constraint. With these considerations, it can be eas- P λ λ λ ily shown [Brunner14] that separable states, ρˆAB = λ p (ρˆA ρˆB ), only lead to correlations of the local form Eq. (2.14) (here discrete) ⊗

ˆ ˆ X λ λ λ ˆ ˆ p(ab x y) = tr ρÂBMa x Mb y = tr p (ρÂ ρˆB )(Ma x Mb y) | | ⊗ | λ ⊗ | ⊗ | X λ λ ˆ λ ˆ X λ = p tr ρÂ Ma x tr ρˆB Mb y = p p(a x, λ)p(b y, λ), (2.18) λ | | λ | | and thus, that the observation of non-local correlations requires entanglement. The reversed statement, if entanglement always leads to non-locality and how to quantify non-locality, is only known for pure states [Capasso73, Gisin91, Home91] but an open research question for mixed states [Badziag03, Fonseca15, Lipinska18]. The exemplary measurement settings above are chosen in the x-y plane (equatorial plane) of the Bloch sphere (Fig. 2.3a). In general, in any original plane (defined by its normal vector n~) of the Bloch sphere S = 2p2 can be obtained for the state ψ+ by | 〉 2.1 Quantum Information Processing 15

using the following relation between the measurement settings

π π π x~ R x~ , ~y R x~ , ~y R ~y (2.19) 1 = n~ 2 0 1 = n~ 4 0 2 = n~ −2 0 with the 3D rotation matrix Rn~(θ) around the vector n~. However, for non-maximally entangled states the maximal violation of the CHSH-inequality might be reached for q 2 other measurement settings. For example, states of type γ = γgg g, g + 1 γgg e, e q | 〉2 | 2 〉 − | 〉 with 0 < γgg < 1 show a maximal violation of 2 1 + 4γgg(1 γgg) for x~0 = (0, 0, 1), − x~1 = (1, 0, 0), ~y0 = (sin(φ), 0, cos(φ)) and ~y1 = ( sin(φ), 0, cos(φ)) with φ = q 2 − tan(γgg 1 γgg) [Popescu92]. The maximal violations of Bell inequality has also − been studied for mixed states [Braunstein92], however, mixed entangled states exist which do not violate any Bell inequalities [Werner89, Barrett02] (see Section 2.1.5 and Fig. 2.4). In summary, the violation of a Bell inequality shows that the measured correlations cannot be explained assuming the locality constraint for a speciﬁc tested state [Werner01]. A violation of a Bell inequality also implies that the measured correlations of the real quantum experiment can be distinguished from its attempted simulation

a) b) Classical answer e1 Choice of measurement setting

a b

x2 y1

x1 e2 time, t

y2 entangled x y

distance, l

Figure 2.3 a) Representation of the measurement settings used in a CHSH-Bell test for Alice

(x~i ) and Bob (~yj ). For Bell states the maximal violation is obtained for x~1 (~y1) orthogonal to x~2 (~y2) and ~y1 rotated by π/4 compared to x~1 which is not true for an arbitrary input state (see main text for details). b) Space-time diagram of a CHSH-Bell test. The x-direction shows the distance between Alice and Bob who share an entangled state and the y-direction shows time. The red (blue) colored regions illustrate the future light cone of Alice (Bob). The stars symbolize the moment in time when the measurement settings are chosen (e.g. by random number generators [Hensen15] or humans [BBT18]) and the cross symbolizes the moment when the classical answer is received. 16 Chapter 2 Basics of Quantum Communication in Circuit Quantum Electrodynamics

without specifying the used physical degree of freedom or the type of measurement [Brunner14]. This distinguishability is called device-independence and implies that the violation of a Bell inequality guarantees that the two systems are entangled independent of any implementation details (for experimental considerations see next Section 2.1.4). The violation of a Bell inequality thus is a key requirement for quantum cryptography [Arnon-Friedman18] and randomness ampliﬁcation protocols [Colbeck12] since it allows to test them device-independently. In addition, the strict inclusion L QM NS ⊂ ⊂ has been shown in Refs. [Khalﬁn85, Rastall85, Popescu94] meaning that no-signaling behaviors exist which are not quantum behaviors. However, the strength of non- local correlations in all physical theories is closely related to the uncertainty principle [Heisenberg27] and steering [Schrödinger35, Wiseman07]. This relation implies that a trade-off has to be made between non-locality, uncertainty in a measurement, and which states Bob can prepare given a measurement result of Alice [Wolf09], and shows that quantum mechanics cannot be more non-local without violating the uncertainty principle [Oppenheim10]. For further details on non-local correlations beyond quantum mechanics we refer to Refs. [Popescu14, Carmi18], and conclude with an example of the consequences of these correlations for distributed computing: Gilles Brassard [Brassard06] and Wim van Dam [vanDam13] showed that maximally no-signaling correlations could be used to perform all distributed computations with a trivial amount of communication since the value of any distributed function could be determined by Alice after only one bit of communication from Bob.

2.1.4 Experimental Loopholes in Bell Tests

Bell’s fundamental insight has opened the possibility to experimentally distinguish local and non-local events. However, loophole-free experimental Bell tests [Hensen15, Shalm15, Giustina15, Hensen16, Rosenfeld17, Li18b] require certain experimental settings to avoid that local theories can reproduce the non-local correlations [Larsson14]. In general, we can distinguish between loopholes, which need to be closed by experimental conditions, like the ’detection’ [Pearle70, Rowe01, Ansmann09], ’freedom of choice (free will)’ [Bell04, Yuan15, Rauch18, BBT18], and ’locality’ [Bell64, Aspect82a, Jarrett84, Weihs98, Kent05, Kent18] loopholes, and those which need be taken into account for rigorous statistical analysis of the experimental results, like the ’memory’ loophole [Aspect76, Elkouss16]. For the purpose of this thesis, we focus on the locality loophole, since it is the main challenge to close this loophole in a circuit QED system. In Chapter 3, we therefore introduce our design and show ﬁrst prototype 2.1 Quantum Information Processing 17

tests of a novel cryogenic system, and in Chapter 4, 5 we present low-loss communication channels and remote quantum communication protocols as prerequisites to close the locality loophole. For a detailed description of other loopholes, we recommend Ref. [Larsson14] and the citations mentioned above.

Locality and Freedom-of-Choice Loopholes

In a Bell test which closes the locality loophole, the measurement settings of the two parties need to be chosen faster than a communication between the two sites can happen to exclude that a local realistic theory can explain the results [Bohm57, Bell64]. Before the first experimental tests [Aspect82a, Weihs98, Hensen15], which closed the locality loophole, it was an open question [Bell04] if non-local quantum correlations would disappear under these experimental conditions. Closing the locality loophole, therefore, includes enforcing that no communication between Alice and Bob is possible during the time of an experimental run which starts at the moment when the questions x and y are asked and ends when the answers a and b are obtained, and showing that non-local quantum correlation still exist. From a fundamental point of view, the locality loophole can never be closed completely, since it is impossible to define the start and end time of a measurement run exactly [Brunner14]. The doubts about the start time are closely related to the question whether Alice and Bob can select their questions freely, which results in the unclosable loophole of ’superdetermism’ (see locality constraint). The definition of the end time is linked to our understanding of a quantum mechanical measurement and the exact moment of the ’collapse’ of the wave function which could be used by a local hidden variable theory to mimic quantum correlations. This aspect is summarized in the ’collapse locality’ [Kent05] loophole. These theoretical doubts have motivated serious experimental efforts to close the locality loophole experimentally to the best of our knowledge. In the last decades, various experiments have been realized using a definition of the start point of the experiment so that as little common causes as possible are supposed to be shared with the past. This has, for example, been achieved by triggering quantum random number generators [Weihs98, Scheidl10, Hensen15, Shalm15, Giustina15, Rosenfeld17] to obtain less-predictable inputs as previous attempts. Other experiments use the randomness of cosmic photons [Rauch18, Li18b] to push back the time at which the choice of the measurement setting could have been exploited by any local influences. We contributed to a work which uses human ‘free will’ to generate free variables [BBT18] (without closing the locality loophole). 18 Chapter 2 Basics of Quantum Communication in Circuit Quantum Electrodynamics

In summary, the locality loophole can be closed on experimental grounds by assur- ing that the time when a random number generator is triggered is space-like separated from the point in time when the measurement result is obtained by the other party. A space–time diagram, which illustrates this, is shown in Fig. 2.3b and the experimental timings are discussed in Section 3.3.1.

2.1.5 Entanglement Witnesses

In this section we discuss the veriﬁcation and quantiﬁcation of entanglement for two- qubit and two-qutrit systems. To violate a Bell inequality a certain amount of entanglement is required, as discussed in Section 2.1.3. However, Bell inequalities are non-optimal entanglement witnesses meaning that they do not detect all entangled states [Hyllus05] as illustrated in Fig. 2.4a. General overviews of techniques how to detect entanglement are provided by Steven Enk et al. [vanEnk07], Otfried Gühne and Géza Tóth [Gühne09] and Ryszard Horodecki et al. [Horodecki09]. Here, we focus on entanglement witnesses, which quantify the entanglement of two-qubit states, like the concurrence [Hill97, Wootters98] and techniques to verify entanglement for 2 2 or ×

a) b)  ℬ 2.8 maxC  ent. & no LHV 2.6 minC  ent. & LHV 2.4 CHSH,

seperable 2.2

2 LHV & not QM 0 0.2 0.4 0.6 0.8 1

concurrence,  5 0 5 Figure 2.4 a) Illustration of the difference between entanglement witnesses and Bell inequalities. Correlations can either originate from quantum mechanics or a local hidden variable model (LHV) resulting in four different sets of states: separable, entangled and LHV,entangled and no LHV,LHV and not-quantum-mechanical (not QM). Bell inequalities (B) denote the nonexistence of a local model for the observed correlations, which however require entanglement, while entanglement witnesses (W) distinguished between separable and entangled states. Figure is based on Ref. [Hyllus05]. b) Maximal (red) and minimal (blue) CHSH-value S for a given concurrence C. Figure is based on Ref. [Verstraete02] 2.1 Quantum Information Processing 19

2 3 dimensional systems using the positive partial transpose (PPT) criterion [Peres96] × and the computable cross norm or realignment (CCNR) criterion [Chen03, Rudolph05] for d d 3 3. The concurrence× ≥ × is deﬁned for two-qubit mixed states by

C(ρˆ) = max(0, l1 l2 l3 l4) (2.20) − − − with the decreasingly ordered eigenvalues li of the matrix [Gühne09, Horodecki09]

rÆ Æ ρˆ(σˆ y σˆ y)ρˆ (σˆ y σˆ y) ρˆ. (2.21) ⊗ · ⊗ Equation (2.21) shows the advantage of the concurrence as an analytically computable metric. The concurrence is 0 for separable and 1 for maximally entangled states. Frank Verstraete and Michael Wolf [Verstraete02] derived general bounds on S given a concurrence C (Fig. 2.4b). They use the Horodecki lemma [Horodecki95] to show that 2 a CHSH-value between SmaxC(C) = 2p1 + C and SminC(C) = 2p2C can be obtained given a concurrence C. The exact S obtained for a given concurrence strongly depends on the type of errors which lead to the reduction of the concurrence. For example, states of the form   0 0 0 0 1 0 1 γ 0     (2.22) 2 0 γ 1 0 0 0 0 0

p 2 which are affected only by pure dephasing show C = γ and S = 2 1 + γ = SmaxC.

Amplitude damping, however, leads to the minimal possible violation SminC. A general characterization and optimal local ﬁltering are discussed in Refs. [Verstraete01, Verstraete02].

Further commonly used entanglement witnesses are the entanglement formation E [Bennett96c] and the negativity N [Vidal02]. E can be related to the concurrence for 1 p 2 two-qubit systems as E(C) = h( 2 (1 + 1 C (ρˆ))) with the binary entropy function − h(x) = x ln(x) (1 x) ln(1 x) [Gühne09]. N is deﬁned as the violation of the positive− partial transpose− − (PPT)− criterion

ρˆTA 1 N ρˆ tr . (2.23) ( ) = || ||2 −

In Eq. (2.23), ... tr indicates the trace norm and || || 20 Chapter 2 Basics of Quantum Communication in Circuit Quantum Electrodynamics

X X TA ρˆ = pji,lm i j l m (2.24) i,j l,m | 〉〈 | ⊗ | 〉〈 |

P P is the partial transpose of ρ = i,j l,m pi j,lm i j l m . The PPT criterion [Peres96, | 〉〈 |⊗| 〉〈 | T Horodecki96] states that for bipartite 2 2 and 2 3 systems the positivity of ρˆ A is necessary and sufﬁcient for ρˆ to be separable.× Similar× to S, bounds on the negativity can be obtained for a given concurrence in the case of 2 2 systems [Verstraete01] p 2 2 × between NmaxC(C) = C and NminC(C) = (1 C) + C (1 C). − − −

For higher dimensional systems, e.g. bipartite 3 3, the computable cross norm × or realignment criterion (CCNR criterion) [Chen03, Rudolph05, Horodecki06] is a strong criterion for entanglement detection since it detects many higher-dimensional entangled states [Gühne09]. However, it is not sufﬁcient for 2 2 systems and therefore, × complements the PPT criterion [Rudolph03, Gittsovich08]. The CCNR criterion is based on realignment of elements of a density matrix and is a permutation criterion as the PPT. We deﬁne the CCNR criterion using the Schmidt decomposition [Nielsen11] in the operator space of the density matrix

X ˆA ˆB ρˆ = µk Gk Gk (2.25) k ⊗

Â ˆB with the non-negative, real coefficients µk and orthonormal bases Gk and Gk of the A B observables on H and H [Gühne09]. For example, the identity with the (properly normalized) Pauli or Gell-Mann matrices form an orthonormal basis in the qubit or P qutrit case. The CCNR criterion states that a state ρˆ must be entangled if k µk > 1 [Chen03, Rudolph05] making the CCNR criterion a sufficient but not necessary condition for entanglement. The CCNR criterion can therefore be seen as a measure of the correlations between local orthogonal observables [Shang18]. Several nonlinear extensions of the CCNR criterion exist, e.g the covariance matrix criterion, which also detects entangled sates for which the CCNR criterion fails [Gittsovich08]. Furthermore, criteria have been investigated which are based on a single generalized measurement, known as symmetric informationally complete positive operator-valued measures (SIC POVM) [Chen15, Xi16, Shang18]. These SIC POVM criteria provide stronger entanglement detection [Shang18] since they can detect more entangled states than the CCNR criterion, e.g. bound entangled states [Gühne09]. 2.2 Circuit Quantum Electrodynamics 21

2.2 Circuit Quantum Electrodynamics

In the previous sections, we have discussed qubits and qutrits mainly as independent, conceptual objects which can ’somehow’ interact with each other or be controlled from the outside. A powerful framework which describes the interaction of real qubits, like atoms, with light is given by cavity quantum electrodynamics (cavity QED) [Haroche89]. In cavity QED systems, atoms are coupled to quantized modes of the electromagnetic field (photons) inside a cavity. This coupling can be utilized to control and readout the state of atoms [Boozer06] and the cavity field [Sayrin11], let multiple atoms interact with each other [Casabone13, Ritter12] or with photons [Chen13, Reiserer14, Hacker19]. Furthermore, this coupling can protect the qubit state against further interactions with the environment and therefore mitigate the effect of decoherence [Haroche06]. In this section, we introduce the field of circuit QED [Blais04], as an electrical-circuit equivalent cavity QED framework, and discuss selected topics of circuit QED which are relevant for remote quantum communication and loophole-free Bell tests. The field of circuit QED studies the interactions of artificial atoms with microwave-frequency resonators both being realized by superconducting electrical circuits [Blais04]. These electrical circuits consist of a large number of individual atoms and electrons and can range in size up to several millimeters making circuit QED systems macroscopic in comparison to typical cavity QED experiments with single (Rydberg) atoms or atomic clouds in optical cavities. Circuit QED systems can, however, be well described by collective degrees of freedom since all Cooper pairs in a superconductor [Bardeen57] form a quantum-mechanical, macroscopic state which is well-modeled by a single wave function [Tinkham15]. In circuit QED systems, atomic-level structures are mimicked using non-linear elements which are based on the Josephson effect [Josephson62, Josephson74] in our experimental implementations. First quantum phenomena in such systems have been observed in the 1980s [Voss81, Devoret85, Martinis85] based on which superconducting artificial atoms have been developed in the following decades [Nakamura99, Vion02, Yu02, Martinis02]. The flexibility in designing the properties of these artificial atoms [Koch07a, Manucharyan09, Yan16a] and their couplings to resonators [Wallraff04, Majer07, McKay15] in combination with controlling them with high-precision [Kelly15, Córcoles15, Rol17] has made circuit QED one of the leading platforms to study quantum physics [Devoret13]. For example, circuit QED experiments have been realized which investigate quantum optical phenomena [Schuster07, Lang13, Sundaresan15], violate Bell’s inequality [Ansmann09, Dewes12, Zhong19], start implementing quantum processing units 22 Chapter 2 Basics of Quantum Communication in Circuit Quantum Electrodynamics

[DiCarlo09, Steffen13b, Vlastakis13, Kelly15, Naik17, Wang18], perform quantum simulations of interacting bosons [Houck12, Eichler15, Underwood16, Ma19], fermions [Barends15], spin systems [Salathé15], molecules [O’Malley16, Kandala17], and bi- ological systems [Potoˇcnik18] or realize quantum communication between remote systems [Roch14, Narla16, Axline18, Campagne-Ibarcq18, Kurpiers18, Leung19]. In the Sections 2.2.1 to 2.2.9, we especially focus on main aspects in relation to generate remote entanglement in chip-based circuit QED systems, and read out qubit states rapidly to make non-local experiments possible. We start by introducing basic circuit elements, like coplanar waveguide resonators, and the transmon as a multi-level superconducting artiﬁcial atom. We discuss the calibration routines of transmon control pulses, and investigate its Jaynes-Cummings-type coupling to resonators in the dispersive regime. Based on these basics, we detail dispersive readout and introduce an additional ﬁlter circuit to optimize parameters for rapid readout. Next, we discuss an all-microwave cavity-assisted Raman transition [Pechal14, Zeytinoglu15˘ ] to realize a tunable atom-photon coupling for high-bandwidth quantum communication. We conclude with a summary of the fabrication process and an overview of the sample parameters used in the experiments of this thesis.

2.2.1 Basic Circuit Elements

In general, electrical circuits can be model as networks [Devoret97, Burkard04a, Vool17] consisting of branches b which connect two nodes n0 and n1. Two linear circuit- elements - capacitor, inductor - and a non-linear inductive element - Josephson element - are the basic elements of all dissipation-less circuits. We first use a lumped-element description in which we consider the physical dimension of the elements to be small compared to typical wavelengths of circuit QED experiments of around 20 mm (frequencies around 6 GHz) so that the voltage and current do not vary along elements. Lumped-element circuits are, for example, used in our experiments for testing certain fabrication processes (Fig. 2.5a) or building quantum-limited parametric amplifiers [Yurke06, Castellanos-Beltran07, Eichler14a] (Fig. 2.7c). The electric circuit elements of each branch can be described using the branch voltage V which is the voltage drop across the branch b and branch current I which denotes the current flowing along b which are in general time dependent. V and I are related to the conjugate variables flux Φ and charge Q as V (t) = Φ˙, I(t) = Q˙ which we use to define the constitutive relations for the individual elements. In a purely capacitive element, V is only a function of Q, which results for a linear capacitor in V = 2.2 Circuit Quantum Electrodynamics 23

2 Q/C with capacitance C and an energy Ecap = Q /2C stored in the electric field of the capacitor. Equivalently, a purely inductive element is defined by its linear inductance 2 L as I = Φ/L and an energy Eind = Φ /2L stored in the magnetic field around the inductor. A Josephson element is a dissipation-less, nonlinear inductive element based on tunneling of Cooper pairs between two superconductors [Tinkham15] through an insulating layer forming a potential barrier and is described by the first Josephson relation [Josephson62]

Φ I = I0 sin . (2.26) Φ0

The critical current I0 sets the maximal dissipation-less current through the element above which it behaves like a normal resistor. Brian Josephson discovered that I0 is a d macroscopic parameter and proportional to e− with the thickness of the insulating layer d and the area of the Josephson element (Fig. 2.5b). Φ0 = ħh/2e is the reduced magnetic ﬂux quantum and provides the scale of the non-linearity [Vool17]. The energy of Josephson element is EJE = Φ0 I0 cos(Φ/Φ0) = EJ cos(Φ/Φ0) with the Josephson − − energy EJ. Furthermore, the second Josephson relation

˙ ˙ V = Φ = Φ0φ (2.27) relates the phase difference φ of the superconductors across the junction to the ﬂux so that I = I0 sin(φ).

a) b)

Figure 2.5 a) Circuit diagram and micrograph of an LC circuit (by P. Zhou, J.-C. Besse). b) Circuit representation and scanning electron microscope (SEM) micrograph of a Joseph- son junction (by T. Walter). 24 Chapter 2 Basics of Quantum Communication in Circuit Quantum Electrodynamics

With these basic building blocks a wide range of dissipation-less circuits can be modeled and systematic procedures have been developed to ﬁnd the Lagrangian and Hamil- tonian functions of these circuits based on which the circuits can be quantized following standard canonical quantization (see for example [Gupta77, Yurke84, Devoret97, Burkard04b, Nigg12, Girvin14, Ulrich16]). In Sections 2.2.1 to 2.2.8, we focus on the discussion of Hamiltonians of some basic multi-element circuits which results in the Hamiltonian describing the presented experiments. We start with discussing the Hamiltonian of a simple parallel LC-circuit (Fig. 2.5a)

Q2 Φ2 Q2 C H ω2Φ2. (2.28) LC = 2C + 2L = 2C + 2 r

HLC is equal to the sum of Ecap and Eind since Q and Φ are conjugate variables in the sense of momentum and position of mechanical harmonic oscillators and the resonance 1 frequency is given by ωr . We can next replace the classical variables by operators = pLC ˆ with the commutator relation [Φˆ,Q] = iħh. Introducing the annihilation aˆ and creation operator aˆ† of a single excitation, which can be interpreted as a photon due to the ˆ † ˆ underlying electromagnetic field [Eichler13], and we obtain Φ = Φzpf (aˆ + aˆ ), Q = Ç Ç iQ aˆ aˆ† . Φ ħhZ0 and Q ħh are the zero-point fluctuations (standard zpf ( ) zpf = 2 zpf = 2Z0 deviation)− − of the flux and charge of the LC-oscillator’s ground state with the impedance q L Z0 = C [Devoret97]. The Hamiltonian of a LC-circuit becomes

ˆ † HLC = ħhωr (aˆ aˆ + 1/2) (2.29)

† since [aˆ, aˆ ] = 1. To justify that LC-oscillators can be treated quantum mechanically they have to be operated at temperatures T at which kB T ħhωr so that the thermal ﬂuctuation energy is small compare to the energy quantum. For typical frequencies around 6 GHz this corresponds to T 300 mK. The experimental details how these temperatures are reached are discussed in Chapter 3. In addition, the energy relaxation rate κr of the mode has to be small compared to ωr meaning that the internal loss of the circuit and its coupling to the environment need to be reasonably small [Devoret97, Vool17] which we discuss in the next Section 2.2.2. Even if the mentioned requirements are met, the average quantities of the LC-circuits, as a harmonic oscillator, can always be described by classical equations using the correspondence principle [Glauber63]. We thus discuss the importance of non-linear Josephson elements to study direct quantum phenomena by building superconducting artiﬁcial atoms in Section 2.2.3 and Section 2.2.4. 2.2 Circuit Quantum Electrodynamics 25

2.2.2 Coplanar Waveguide Resonators

In this section, we introduce a type of distributed linear resonator, the coplanar waveguide (CPW) resonator, which are commonly used in circuit QED experiments. Using these CPW resonator we also discuss the effects of external coupling and internal loss. CPWs are one type of planar transmission lines on top of a substrate. CPWs consist of a thin center conductor which is separated from two grounded conductors by a slot in the metal. CPWs are well described using transmission line theory [Collin91, Pozar12] and can be modeled as a series of inﬁnitesimal lumped-element circuits [Blais04]. CPWs support the propagation of quasi transverse electromagnetic (TEM) waves since the phase velocity vph is the speed of light in vacuum c = 1/pε0µ0 in the region above the conductors and c/pεr in the substrate with the vacuum and relative permittivity ε0 and εr and the vacuum permeability µ0. This can be combined in a relative effective permittivity εeff which is a function of ε0 and εr and the geometry of the CPW so that vph = c/pεeff. Due to these well-deﬁned waves CPWs are typically described as planar coaxial cables and are widely used to route microwave signals on planar, chip-based circuits [Pozar12].

Using the lumped-element circuit model, we can deﬁne a self-inductance Ll and p capacitance Cl per unit length for the CPW and obtain a phase velocity vph = 1/ Ll Cl p and a characteristic impedance Z0 = Ll /Cl which is usually chosen to be 50 Ω. Cl depends only on the geometry and εeff of the CPW [Gevorgian95, Garg13], while

Ll can in addition be dependent on the kinetic inductance of the superconductor [Watanabe94].

λ/2 Resonators

Using CPWs, quasi-one-dimensional transmission-line λ/2 resonators can be deﬁned by opening the center conductor at two positions (Fig. 2.6c). These open ends im- pose voltage antinodes at boundary conditions and can be modeled as capacitors with capacitance Cc at these points. The λ/2 resonator has resonance frequencies

(n + 1) c 1 νn = ωn/2π = (2.30) pεeff 2l with the integer mode number n = 0, 1, 2, ... and the length of the resonator (distance between the two capacitors) l making them the microwave counterpart to optical Fabry-Perot cavities.

We can furthermore add a small internal loss per unit length αatt along the CPW 26 Chapter 2 Basics of Quantum Communication in Circuit Quantum Electrodynamics

to our model, so that αattl 1. Under this assumption, the λ/2 resonator can be well described as a parallel LCR-oscillator [Göppl08, Pozar12] for frequencies close to νn 2 2 with lumped-element equivalent Ln = 2Ll l/n π , C = Cl l/2 and R = Z0/αattl. Due to the external capacitive coupling and the internal loss, the nth mode of the resonator n n n has a frequency width κr /2π = κint/2π + κext/2π. We deﬁne the loaded quality factor th n n of the n mode as QL = ωn/κr and obtain the general relation

1 1 1 n = n + n . (2.31) QL Qint Qext

For λ/2 resonators we can express

n ωnCl Z0 (n + 1)π Qint = = 2αatt 2αattl ω C l 1 Qn n l 1 (2.32) ext = 2 2 2 + 4 ωnCc Z0

n n showing that Qint is independent of l while Qext l for a ﬁxed frequency ωn. Note that in Eq. (2.32) we assumed an equal coupling at the input and output port, which can be disadvantageous for the signal-to-noise ratio in typical transmission measurements, and have neglected a shift of the resonance frequencies due to this coupling and the internal loss, which is valid if C Cc and αattl 1 [Göppl08].

λ/4 Resonators

Opening the center conductor at one position and shorting it to the grounded conductors at a distance l creates a λ/4 resonator imposing a voltage antinode at the input capacitor and a voltage node at the shorted end (Fig. 2.6a,b). λ/4 resonators have resonance frequencies at (2m + 1) c 1 νm = (2.33) pεeff 4l with mode number m = 0, 1, 2, 3... in the limit of small input coupling. A second port can be added to a λ/4 resonator by adding a T-junction at distance x from the shorted end to the center conductor which results in a galvanic coupling to the external circuit. This resonator can be modeled as three CPW sections: CPW1 has a length l x between the input capacitor to the T-junction, CPW2 is a shorted stub of length x between− the T-junction to the shorted end of the λ/4 resonator, and CPW3 is a semi-inﬁnite CPW from the T-junction towards the output port. This model holds under the assumptions that the impedance at the T-junction is Z0 and that the ground is 2.2 Circuit Quantum Electrodynamics 27

d) b) l/4

c) l/2

Figure 2.6 (a) Micrograph and b) schematic circuit diagram of a λ/4 coplanar waveguide resonator. The input is colored blue and the output dark gray (by T. Walter). c) Schematic circuit diagram of a λ/2 resonator. d) Air-bridge across a coplanar waveguide to connected the ground planes (by J.-C. Besse).

connected across the junction using, for example, airbridges (Fig. 2.6d) [Steffen13a]. The transmission (ABCD) matrix method can be used to characterize the frequency response of the system [Pozar12, Dellsperger12] which shows that the coupling to the output port is set by the position of the T-junction. This can be qualitatively understood as κout being proportional to the fraction of the total voltage at the position of the T- junction [Jeffrey14]. We use the ﬁrst mode of galvanically coupled λ/4 resonators as the basic linear resonators in the circuit QED experiments within this thesis and approximate them as single mode resonators due to the large frequency spacing between the ﬁrst and higher modes.

2.2.3 Josephson Junctions

Superconducting qubits are typically based on Josephson (tunnel) junctions which consist of a Josephson element in parallel to a capacitance CJ due to the ﬁnite distance between the two superconducting electrodes (Fig. 2.5a). In this section we discuss Josephson junctions and so-called superconducting quantum interference devices (SQUIDs) which consist of two Josephson junctions in parallel before we introduce superconducting qubits in the next section. 28 Chapter 2 Basics of Quantum Communication in Circuit Quantum Electrodynamics

a) b) c)

......

Figure 2.7 Overview of different Josephson junction based circuits. We show the circuit diagrams and micrographs of a) a superconducting quantum interference device (SQUID), b) a transmon qubit (see Section 2.2.4), and c) a lumped-element Josephson parametric ampliﬁer (JPA) [Eichler14a]. Fabricated and pictures taken by T. Walter, J.-C. Besse, S. Gasparinetti, M. Oppliger.

The Hamiltonian of the a Josephson junction is given by [Büttiker87]

Q2 Φ H E cos 4E n2 E cos φ (2.34) JJ = J = CJ J ( ) 2CJ − Φ0 − introducing the variable n = Q/2e which describes the number of Cooper pairs having tunneled through the junction and the charging energy E e2/2C of the junction. CJ = J Similar to the LC-circuit, n and φ are conjugate variables since the Hamilton equations of motion are ħhdn/d t = ∂ H/∂ φ, ħhdφ/d t = ∂ H/∂ n. We quantize the system by − ˆ − ˆ introducing the operators nˆ and φ which obey the commutator relation [nˆ, φ] = i and obtain Hˆ 4E nˆ2 E cos φˆ Pechal16a . JJ = CJ J ( ) [ ] Josephson junctions− can also be embedded in a larger electrical circuits. For example, a Josephson junction can be shunted by an additional capacitance Cs which can be integrated into the Hamiltonian by replacing E E e2/2C with the total CJ C = σ → capacitance Cσ (Fig. 2.7b) or by an additional inductance modeled by adding a term 2 2 2 ˆ proportional to Φ0/2Lφ = EL/2φ to HJJ. In more advanced circuits, the effective charging energy can be obtained from a full capacitive network analysis as, for example, detailed in [Koch07a]. 2.2 Circuit Quantum Electrodynamics 29

Superconducting quantum interference devices (SQUIDs) are formed by two Joseph- son junctions in parallel. SQUIDs are widely used in circuit QED systems to make the transition frequency of superconducting qubits tunable by an external magnetic ﬂux

Φext (Fig. 2.7a). The inductive part of the SQUID Hamiltonian is

H E cos φ E cos φ (2.35) squid = J1 ( 1) J2 ( 2) − − with the Josephson energies E and E of the two junctions and the phases φ and J1 J2 1 φ2 across them. Since the two Josephson junctions form a loop the relation φ1 =

Φext/Φ0 φ2 between the two phases and the external ﬂux is required from Faraday’s law. Using− this relation we obtain

v Φext t Φext H E cos 1 d2 tan2 cos φ (2.36) squid = JΣ + ( ) − 2Φ0 2Φ0 with E E E , d E E /E and the new effective phase difference φ JΣ = J1 + J2 = ( J2 J1 ) JΣ = − (φ1 +φ2)/2 arctan(d tan(Φext/2Φ0)) [Vool17]. For a constant external magnetic ﬂux, Eq. (2.36) simpliﬁes− to

v Φext t Φext E E cos 1 d2 tan2 (2.37) J JΣ + → 2Φ0 2Φ0 or in the case of two symmetric junctions (d=0)

Φ E E cos ext (2.38) J JΣ → 2Φ0 showing that a SQUID can be modeled as a single Josephson junction with a ﬂux tunable Josephson energy EJ(Φext).

Furthermore, combining Josephson junctions or SQUIDs to an array of length 2 M (Fig. 2.7c) reduces the non-linearity M − [Eichler14b] which can be used to implement low-loss superinductors [Masluk12] or Josephson parametric ampliﬁers (JPA) [Castellanos-Beltran08, Eichler14a, Planat19] since the bifurcation power also increases quadratically with M. 30 Chapter 2 Basics of Quantum Communication in Circuit Quantum Electrodynamics

2.2.4 Transmon: a Multi-Level Artificial Atom

In this section, we focus on the transmon (Fig. 2.7b) as the multi-level artiﬁcial atom used in our circuit QED experiments. The transmon is an advancement of a Cooper pair box [Nakamura99] and consists of a Josephson junction or SQUID shunted by a large capacitance so that EJ EC and EL = 0. The Hamiltonian of the transmon is given by Hˆ replacing E E e2/2C with the total capacitance C Koch07a . JJ CJ C = σ σ [ ] → In addition, we introduce an effective offset charge ng, which is controllable by an external capacitively-coupled gate voltage ng = CgVg/2e, and obtain ˆ 2 ˆ Htransmon = 4EC(nˆ ng) EJ cos(φ) (2.39) − − For an exact solution, the Hamiltonian can be written in the basis φ of the phase ˆ | 〉 operator φ [Devoret03, Koch07a] ψ(φ) = φ ψ 〈 | 〉

d 2 4EC( i ng) EJ cos(φ) ψ(φ) = Eψψ(φ) (2.40) − dφ − − with the periodic boundary condition ψ(φ) = ψ(φ + 2π).

a) c) 15 01 1.5 10 01 / E

5 α 1

energy, E / 0 b) 3 01 0.5 2

1 anharmonicity, -0.1 energy, E / 0 -1 -0.5 0 0.5 1 0 20 40 60 80 100 energy, E /E gate charge, ng J C

Figure 2.8 Eigenenergies of the lowest four eigenstates compared to the eigenenergy of

the lowest level E01 as a function of the offset charge ng in a) the Cooper pair box regime with EJ/EC = 1 and b) the transmon regime with EJ/EC = 75 [Koch07b] similar as used in experiments within this thesis (see Tab. 2.1). c) Anharmonicity α compared to E01 versus EJ/EC. The horizontal lines indicate α/E01 for EJ/EC and the minimal α/E01 for → ∞ EJ/EC 17.5. ≈ 2.2 Circuit Quantum Electrodynamics 31

2i z ng Using f (z) = e− ψ(2z) we obtain

Eψ EJ f 00(z) + + cos(2z) f (z) = 0 (2.41) EC EC which is the canonical form for Mathieu’s differential equation [Meixner54] and leads to the eigenenergies and eigenfunctions

E n E a E /2E (2.42) m( g) = C 2(ng+k(m,ng))( J C) − with the Mathieu’s characteristic value ar (x) and a function sorting the eigenvalues k(m, ng) [Koch07a]. In the limit of the Cooper pair box (EJ EC) [Bouchiat98] EC dominates the eigenenergies but for ng mod 1 = 1/2 the degeneracy is lifted by EJ resulting in an effective two-level system which is however sensitive to charge noise

(Fig. 2.8a). In the transmon limit (EJ EC) the sensitivity to charge noise is exponentially suppressed (Fig. 2.8b) with the cost of a reduction of the anharmonicity

α = (E1,2 E0,1)/ħh < 0 with a weak power law (Fig. 2.8c). We deﬁne the absolute ε − and relative εrel charge dispersion between two energy levels as

∆Emax ∆Emin ħhε ħhεrel = − = (2.43) ∆Emean ∆Emean with the maximal ∆Emax, minimal ∆Emin and mean ∆Emean energy difference between the two levels (Fig. 2.9a) [Pechal16a].

The charge dispersion of level m can also be approximated for EJ/EC 1 using asymptotics of the Mathieu’s characteristic value [Koch07a]

4m+5 v m 3 2 t 2 E 2 + 4 m J 8EJ/EC εm ( 1) EC e−p (2.44) ≈ − m! π 2EC

8EJ/EC showing the exponential decrease e−p with increasing EJ/EC but also the 4m+5 exponential increase 2 with the level m. Therefore, signiﬁcantly larger EJ/EC ratios (smaller anharmonicities) have to be used if the transmons is operated as a qutrit in comparison to a qubit (Fig. 2.9c). For example, using a f01 = E01/h = 6.3 GHz (see Tab. 2.1) the absolute value of the anharmonicity α has to be reduced from 335 MHz | | to 275 MHz if a frequency splitting of 1.6 kHz (1/ε 100 µs) due to charge dispersion should be kept constant for the ef-transition compared≈ to the ge-transition.

In the transmon Hamiltonian Eq. (2.39) EJ cos(φ) can be interpreted as a strong potential in the EJ/EC 1 limit. This justiﬁes the use of a perturbation approach, in 32 Chapter 2 Basics of Quantum Communication in Circuit Quantum Electrodynamics

a) 1 rel ϵ 10-3 h↔i c) energy, EJ/EC f↔h 130 95 75 60 50 45 10-6 e↔f 106

g↔e [ μ s ] rel. dispersion, -9 10 104 b) 2.2 h↔i 100 1.8 g↔e f↔h 1 1.4 e↔f e↔f f↔h g↔e 10-2 1 inv. dispersion, 1 / ϵ 〈 j + 1 | n 〉 / g e 0.6 220 260 300 340 380 420

0.2 h↔g anharmonicity, |α| [MHz] coupling, 0 20 40 60 80 100

energy, EJ/EC

Figure 2.9 a) Relative charge dispersion εrel and b) relative coupling strength j + 1 n j / e n g versus EJ/EC between the lowest ﬁve consecutive eigenstates. c) Inverse charge〈 | dispersion| 〉 〈 | | 〉 1/ε versus anharmonicity α for the transmon parameters speciﬁed in Tab. 2.1 and used in the remote communication| | experiments presented in Chapter 5. The vertical and horizontal lines indicate the experimentally measured values of α and the corresponding inverse charge dispersion of e f . | | | 〉 ↔ | 〉

2 4 which the cos(φ) 1 φ /2! + φ /4! + ... is expanded for small φ to ﬁnite order. This perturbation approach≈ − removes the periodicity of the system allowing us to drop ˆ the offset charge ng, and we obtain an harmonic oscillator Hamiltonian H0 = 4EC(nˆ 2 EJ ˆ2 ˆ ˆ4 − ng) EJ + 2 φ which is perturbed by V = EJ(φ /24 + ...) [Pechal16a]. We rewrite −p4 † † − ˆ p4 † nˆ = i EJ/32EC (ˆc ˆc) = i n0 (ˆc ˆc) and φ = 2EC/EJ (ˆc + ˆc) with the typical annihilation and creation− operators ˆc−and ˆc† of the harmonic oscillator approximating the transmon. With these operators we obtain

1 ˆ p † H0 = 8EJ EC ˆc ˆc + EJ (2.45) 2 −

† and with [ˆc, ˆc ] = 1 up to ﬁrst order

4 EC † Vˆ ˆc + ˆc . (2.46) ≈ −12 2.2 Circuit Quantum Electrodynamics 33

Using perturbation theory [Sakurai10] the perturbed eigenenergies are

E E p ˆ p C C 2 Ek = ħhωk = 8EJ EC k + k V k = 8EJ EC k k (2.47) 〈 | | 〉 − 2 − 2 introducing the eigenvectors k of nˆ which correspond to eigenvalue k and dropping | 〉 constant energy shifts to set E0 = 0. In this perturbation approach we obtain the anharmonicity ħhα = (E2 2E1) = EC and the transition frequency between level k − − and k + 1

p ωk,k+1 = (ωk+1 ωk) = 8EJ EC EC /ħh k EC/ħh = ωge + k α. (2.48) − − − The perturbed eigenstates up to ﬁrst order in V

p ψk nˆ ψk+1 n0 k + 1 (2.50) | 〈 | | 〉 | ≈ with the same scaling as in the Fock basis (Fig. 2.9b) and

EJ/EC →∞ ψk nˆ ψk+j 0 (2.51) | 〈 | | 0 〉 | → with j0 > 1. If higher order terms of the cos(φ) expansion are included, all states with odd j are coupled [Khezri16]. The corresponding matrix elements decrease | | exponentially with j making the above approximation also valid in the large EJ/EC | | limit [Pechal16a]. Using the ﬁrst-order approximation, we introduce new operators which contain only nearest-neighbor elements of n

1 X ˆb ψ nˆ ψ ψ ψ (2.52) = n k k+1 k k+1 0 k 〈 | | 〉 | 〉〈 |

ˆ † ˆ and approximate nˆ in0 (b b). The Hamiltonian of the transmon is therefore given ≈ − 34 Chapter 2 Basics of Quantum Communication in Circuit Quantum Electrodynamics

E 4 E ˆ 0 p ˆ †ˆ C ˆ † ˆ ˆ †ˆ C ˆ †ˆ †ˆˆ Htransmon = 8EJ EC b b b + b = ħhωge b b b b bb (2.53) − 12 − 2 for which we again neglected terms which couple non-nearest-neighbors and constant energy shifts in the last expression. Alternatively, we can express

ˆ 0 X Htransmon = ħh ωk ψk ψk (2.54) k | 〉〈 | in an explicit diagonal form using the perturbed eigenstates ψk . Equation (2.54) | 〉 illustrates the typical labeling of the transmon states as g = ψ0 , e = ψ1 , f = | 〉 | 〉 | 〉 | 〉 | 〉 ψ2 , h = ψ3 , ... . | 〉 | 〉 | 〉 Besides the transmon and Cooper pair box, qubits have been realized in various superconducting circuits. These superconducting qubits can be grouped according to their most relevant degree of freedom into charge qubits, ﬂux qubits, and phase qubits. We present an overview of different types of superconducting qubits based on their energy ratios EL/EJ, EC/EL [Vool17, Wendin17] in Fig 2.10.

a) b) 0.5 flux qubit 0.4 rf-squid

J J flux noise

/ E c-shunt / E L 0.3 L

0.2 critical current noise energy, E energy, E 0.1 fluxonium phase qubit charge noise Cooper pair box transmon 0 10-2 1 102 104 106 energy, EJ/EC energy, EJ/EC

Figure 2.10 a) Schematic categorization of Josephson-junction-based superconducting

qubits using their EJ/EC and EL/EJ ratios. Typical ratios for ﬂux qubits (red), phase qubits (dark blue), charge qubits (light blue) are shown. Figure and data are based on Ref. [Wendin17]. b) Sketch of noise types which are dominating the qubit decoherence in dependence of the EJ/EC and EL/EJ ratios. Figure is based on Ref. [Vool17]. 2.2 Circuit Quantum Electrodynamics 35

Transmon-Pulse Calibration and Benchmarking

Transitions between transmon states can be driven by a time-dependent microwave

ﬁeld of envelope Ω(t) = ΩI(t) cos(ωd t) + ΩQ(t) sin(ωd t) where ωd is the drive frequency, and ΩI(t) and ΩQ(t) are the two quadrature amplitudes which can be individ- ˆ ˆ 0 ually modulated. Adding this drive results in the Hamiltonian Htransmon = Htransmon + ˆ ˆ 0 ˆ ˆ † Hd = Htransmon + Ω(t)(b + b ). This control allows to prepare any superposition state of the two involved states. Since the transmon is a weakly anharmonic system, we k a use derivative removal by adiabatic gate (DRAG) pulses Rij (between level i and j around axis k and angle a), which reduce leakage to other levels, e.g. the ef-transition if one intends to drive only ge-transition, and phases errors [Motzoi09, Gambetta11]. In circuit QED experiments typically Gaussian shaped DRAG pulses with a duration of

10-30 ns are used for Rge and Ref [Chen16]. We ﬁx the pulse duration in our experiments and sweep the two quadrature amplitudes to obtain Rabi oscillations [Rabi37] (Fig. 2.11a). Using these Rabi oscillations we extract the amplitudes for rotation angles of π and π/2 pulses. To calibrate the drive frequencies to be on resonant with the transmon transition frequencies we use π/2 Ramsey experiments [Ramsey50, Wallraff05] (Fig. 2.11b) which consist of two Rij pulses separated by a variable time delay τram. From these Ramsey fringes on the R ge- and ef-transition we obtain νge, α and the Ramsey coherence times T2ij. Gaussian shaped DRAG pulses require two effective scaling parameters to reduce leakage and phase errors. The ﬁrst parameter αq introduces a weight between the two quadrature amplitudes and the second a constant detuning compared to the transition frequency

[Gambetta11, Chen16]. In our experiments, we only calibrate αq using the pulse y π/2y π y π/2x π y π/2 x π sequences of Rij Rij , Rij Rij , Rij − Rij (Fig. 2.11c). Our calibration sequence mainly minimizes phase errors. Leakage to other levels can be calibrated using a n π sequence of 2 Rij pulses and optimize both DRAG parameters [Chen16].

We extract the energy relaxation time T1ij by sweeping a time-delay between a π Rij (Fig. 2.11d) and a readout pulse. Spin-echo coherence times T2ij are measured by π/2 π sweeping the delay between two Rij intersected with a dynamical decoupling Rij in the middle of the delay (Fig. 2.11e) [Berger15a, Berger15b]. In addition, we characterize the single qubit gates using randomized benchmarking (RB) techniques [Chow09] (Fig. 2.11f). For details on the pulse calibration and RB we refer to Refs. [Baur12a, Steffen13a, Ciorciaro15, Haberthür15, Storz16, Balasiu17]. 36 Chapter 2 Basics of Quantum Communication in Circuit Quantum Electrodynamics

A a) Rge b) π/2 π/2 Rge τram Rge meas

0 20 0 20 80 100

pulse duration, τ [ns] pulse duration, τ [ns] 1 1 e e 0.8 0.8 0.6 0.6 0.4 0.4

population, P 0.2 population, P 0.2

| e 〉 0 | e 〉 0 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 5 6 7 8

drive amplitude, Age [norm.] delay, τram [μs] x π/2 x,y,-y π c) Rge Rge d)

1 s 1 e 0.8 0.6 0.8 0.4

population, P 0.2 0.6 | e 〉

0 sequence fidelity, F 0.5 -2.5 -1.5 -0.5 0.5 0 50 100 150 200 250

DRAG weight, αq sequence length, ncl π π π τ Rge R ef Rge π/2 t2 e) f) Ref τt1

s 1 Pg s 1

0.8 Pe 0.8

0.6 Pf 0.6 0.4 0.4 0.2 0.2

state population, P 0 state population, P 0 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 10

delay, τt1 [μs] delay, τt2 [μs]

Figure 2.11 Pulse schemes and typical experimental results of DRAG pulse calibration measurements for the g e (ge) and e f (ef) transition: a) Rabi oscillations of ge, b) Ramsey fringes| of〉 ge, ↔ c) | 〉 DRAG scale| parameter〉 ↔ | 〉 of ge, d) randomized benchmarking of

ge,e) energy relaxation time T1ef of ef, f) and spin-echo coherence time T2ef of ef. See main text for details. 2.2 Circuit Quantum Electrodynamics 37

2.2.5 Jaynes-Cummings Model

After introducing the individual circuit elements we next couple individual elements to obtain a circuit QED system. We introduce a generalized Jaynes-Cummings Hamil- tonian which describes a coupled transmon-resonator system as a coupling of a multilevel quantum system with a single mode of the resonator (Fig. 2.12). The Jaynes- Cummings Hamiltonian of a qubit coupled to a cavity consists of a bare qubit Hamilto- ˆ ˆ ˆ nian Hqb, a bare resonator Hamiltonian Hr and an interaction Hamiltonian Hint

ω ˆ ˆ ˆ ˆ † ge † HJC,qb = Hr + Hqb + Hint = ħhωraˆ aˆ + ħh σˆ z + ħhg(aˆ + aˆ)(σˆ + + σˆ ) (2.55) 2 − with the ladder operators of the qubit σˆ = 1/2(σˆ x iσˆ y) and the coupling constant ± ˆ ˆ 0 (†)∓ g. Replacing the Hqb with Htransmon and σˆ with b we obtain ± E ˆ † ˆ †ˆ C ˆ †ˆ †ˆˆ † ˆ † ˆ HJC = ħhωraˆ aˆ + ħhωge b b b b bb + ħhg(aˆ + aˆ)(b + b). (2.56) − 2 The coupling constant g depends in general on the physical implementation of the resonator and transmon. In our experiments, we couple the transmon capacitively 0 to a λ/4 resonator at its voltage anti-node and thus, ﬁnd that g VrmsβC with the 0 root-mean-square voltage of the vacuum ﬂuctuations of the resonator Vrms and to the parameter βC which depends on the geometries and capacitance of the circuit [Koch07a, †ˆ † Pechal16a]. Next, we perform a rotating wave approximation (RWA) neglecting aˆ b and ab terms, which couple levels with large energy differences. The RWA is valid if g/(ωge + ωr) 1 (circa 1/50 in our experimental setup, see Tab. 2.1) and we obtain E ˆ † ˆ †ˆ C ˆ †ˆ †ˆˆ †ˆ ˆ † HJC = ħhωraˆ aˆ + ħhωge b b ħh b b bb + ħhg(aˆ b + aˆb ). (2.57) − 2 Using Eq. (2.54) we obtain

ˆ † X X † HJC = ħhωraˆ aˆ + ħh ωk ψk ψk + ħh gk,k+1(aˆ ψk ψk+1 + h.c.) (2.58) k | 〉〈 | k | 〉〈 |

ˆ with gk,k+1 = ψk b ψk+1 g pk + 1 g. 〈 | | 〉 ≈ 38 Chapter 2 Basics of Quantum Communication in Circuit Quantum Electrodynamics

Dispersive Regime

In circuit QED experiments, in which quantum information is stored and processed in transmons and the transmission line resonators are used as a quantum buses or for readout, it is advantageous to operate the system in the dispersive regime. In this regime the direct energy exchange between the transmon and the resonator is suppressed due to a large detuning of the transmon and resonator transition frequencies compared to g (Fig. 2.12). If we want to operate the transmon as a qubit or qutrit especially g/ ω0,1 ωr = g/ ∆qr 1 and g/ ω1,2 ωr 1 have to hold. Furthermore, the number| − of excitations| | in| the resonator| n has− to| be small compared to a critical photon 2 ∆qr number ncrit = 4g2 (see Section 2.2.6) since the transition matrix elements scale with † ˆ † pn. In the dispersive regime, ħhg(aˆ b + ab ) can be treated as a small perturbation, and the general Jaynes-Cummings Hamiltonian can be diagonalized using the unitary transformation [Koch07a]

X gk,k 1 Uˆ exp Aˆ exp + aˆ ψ ψ h.c. . (2.59) = = ω ω ( k+1 k ) k k,k+1 r | 〉〈 | − − ˆ ˆ ˆ † ˆ ˆ ˆ We calculate UHJCU H + [A, H] + ..., neglect terms which couple levels with large ≈

qb r

Figure 2.12 Schematic energy level diagram of the eigenstates of a transmon-resonator

system in the dispersive regime. ωge and α specify the transmon transition frequencies and ωr the resonator frequency. The black arrows show Jaynes-Cummings-type nearest- neighbor couplings and the light gray arrows couplings neglected in the rotating wave approximation. Figure is based on Ref. [Pechal16a]. 2.2 Circuit Quantum Electrodynamics 39

energy differences and keep terms up to second order in g to obtain the dispersive Hamiltonian

ˆ X † X Hdisp = ħh ωr (χk+1 χk) ψk ψk aˆ aˆ + ħh ωk + χk ψk ψk (2.60) − k − | 〉〈 | k | 〉〈 | with the dispersive shift [Blais04, Koch07a]

2 2 2 gk 1,k g k g k χk = − = (2.61) ωk 1,k ωr ≈ ωge + (k 1)α ωr ∆qr + (k 1)α − − − − − ˆ and χ0 0 [Pechal11]. To next order, we obtain the approximations of gef e b f g ≡ ˆ ≈ 〈 | | 〉 ≈ p2g(1 + α/2ωge) and gfh f b h g p3g(1 + α/ωge) [Sete15]. The dispersive shift renormalizes the transmon≈ 〈 | frequencies| 〉 ≈ and resonator frequencies by introducing a transmon-state dependent shift of the resonator frequency and by adding an ac Stark shift to the transmon transition frequencies. These dispersive transmon- state-dependent resonator shifts are used for quantum non-demolition (QND) readout [Blais04, Wallraff05, Bianchetti10b] which we discuss in the next section. In addition, the dependence of the resonator frequency on the transmon-resonator detuning ∆qr can be utilized to tune the resonator to the desired frequency [Dickel18, Kurpiers18]. The ac Stark shift [Schuster05] can be used to relate the drive power of the resonator to the intra-resonator ﬁeld in units of photons [Schuster07] and thus provides an absolute measure of the signal power at the position of the resonator.

2.2.6 Dispersive Readout

The state of a transmon in the dispersive regime is typically read out by applying a gated microwave tone at the readout resonator and monitoring the state-dependent resonator response using room-temperature electronics. Details of the experimental setup are described in Section 3.2.3. The dependence of the resonator response on the transmon state is caused by the dispersive coupling of the transmon and resonator (Fig. 2.13). In this section, we ﬁrst introduce the dispersive readout and discuss its power dependence. Second, we introduce average readout schemes including a discussion on different reference traces. Third, we present single-shot readout schemes: we introduce the basic ideas for fast, high ﬁdelity single-shot qubit-readout, and show how single-shot readout of qutrit-states can be utilized for readout characterization. According to Eqs. (2.60) and (2.61), we obtain the effective dispersive shifts on 40 Chapter 2 Basics of Quantum Communication in Circuit Quantum Electrodynamics

the resonator between the states g , e , and f | 〉 | 〉 | 〉 e g 2 ω|r 〉 ω|r 〉 χ2 g α χge = − = χ1 = 2 − 2 ∆qr ∆qr + α f g 2 ω|r 〉 ω|r 〉 g α(2∆qr + α) χgf = − = 2 ∆qr (∆qr + α)(∆qr + 2α) χgf 2∆qr + α α = 2 3 (2.62) χge ∆qr + 2α ≈ − ∆qr

g e g f g with ω|r 〉 = ωr χ1, ω|r 〉 = ω|r 〉 + 2χge or ω|r 〉 = ω|r 〉 + 2χgf. The effective dispersive − shiftsχge and χgf vanish in the linear oscillator limit α = 0 showing that the dispersive readout depends on the non-linearity of the transmon. However, χge is substantially 2 smaller than for a two-level system g /∆qr [Blais04]. To read out qubit or qutrit g g states, the resonator is typically probed at a frequency in the interval [ω|r 〉 2χgf, ωr| 〉], − and the resonator response is integrated over time (Fig. 2.14b) [Blais04, Gambetta07, Bianchetti10a]. The time dynamics of the dispersive readout are set by the effective dispersive shift χi,j and the external energy-relaxation rate of the resonator κ [Gambetta08] and can be modeled using Cavity-Bloch equations [Bianchetti09, Li18a]. The optimal signal-to-noise ratio (SNR) in the steady-state response is obtained for a χi,j/κ = 0.5 [Gambetta08] while χi,j/κ = 0.1-0.5 optimize the SNR in the transient regime [Walter14]. In Section 2.2.7 we discuss the effects of external coupling in detail and

a c 1 transmon ) ) ] 0.8

0.6 in1 norm. | [ l/4 out2

21 0.4 S | b 0.2 0/ R ge 2cge ) π 0 meas, |S | 21 4.67 4.71 4.754 4.8 4.84 drive frequency, f GHz [ ] Figure 2.13 a) Circuit diagram of a filtered resonator-transmon system (gray, orange) based on λ/4 resonators in which the filter is driven (blue). For details on the (Purcell) filter we refer to Section 2.2.6. b) Pulse scheme and c) pulsed transmission spectrum measurements of the resonator-transmon system for the transmon in state g (blue) and e (red) applying 0 π | 〉 | 〉 no pulse Rge or Rge. The solid lines are fit to the data using the steady-state model Eq. (2.81) discussed in the main text . 2.2 Circuit Quantum Electrodynamics 41

introduce ﬁlters, which allow to set these optimal ratios to realize fast qubit readout. In the next paragraph, we focus on the power dependence of the dispersive readout based on which we obtained circuit parameters for fast qubit readout.

Power Dependence of the Dispersive Readout

For a qubit coupled dispersively to a resonator an increasing number of photons in the resonator introduces transitions between the qubit levels, also know as coherent mixing of the dressed states of the Jaynes-Cummings Hamiltonian [Blais04, Boissonneault08, Khezri16]. The probability of these transitions is proportional to v 2 1 1 t n + 1 µt(n) = sin tan− (2.63) 2 ncrit

2 ∆qr α with the critical photon number ncrit 2 in the transmon case. This = 4g = 4χge(1+α/∆qr) shows that n ncrit is required to stay in the dispersive limit and in addition ncrit 1. The second condition must hold since for n = ncrit we obtain µt(ncrit ) 15% → ∞ ≈ [Blais04], but µt(ncrit 0) = 1/2. Since ncrit is proportional to 1/χge an upper limit on → χge is given by ncrit 1 to stay in the dispersive regime, which led us choose ncrit 14 ≈ and used n/ncrit 5 in our readout experiments [Walter17]. Maxime Boissonneault ≈ et al. [Boissonneault08] used third-order corrections to the dispersive approximation to show that photons in the cavity induce incoherent relaxation and excitations at rate

n γ S( ∆qr) . (2.64) ↓↑ ± ncrit

S( ∆qr) is the power spectral density of a classical noise parameter fφ(t) which leads ± ˆ to qubit dephasing, described by the Hamiltonian Hφ fφ(t)σˆ z. The incoherent rates γ limit the QND character of the dispersive readout with increasing n and thus, reduce ↓↑ the achievable SNR which is proportional to the readout ﬁdelity Fr = 1 P(e g) − | − P(g e) with the probability P(k l) that a qubit prepared in state k is measured in state | | l [Gambetta07]. Daniel Slichter et al. [Slichter12] measured that S( ∆qr) shows the typical 1/f magnetic ﬂux noise dependence observed in SQUID-based± devices

[Wellstood87, Koch07b]. This dependence makes large ∆qr advantageous to keep γ small. Detailed discussions on resonator-photon introduced± qubit-transitions can ↓↑ also be found in Refs. [Boissonneault09, Boissonneault10, Boissonneault12, Sank16].

Based on these consideration we chose ∆qr/2π 1.55 GHz and g/2π 0.208 GHz for ≈ ≈ ncrit 14 resulting in a dispersive shift χge/2π 7.9MHz and refer to Ref. [Walter17] ≈ ≈ − 42 Chapter 2 Basics of Quantum Communication in Circuit Quantum Electrodynamics

for experimental details.

In addition, the dispersive shift χi,j(n) is also dependent on the number of photons in the resonator, e.g.

dχge(n) 9 α χge(0) (2.65) dn ≈ −8 ∆qr ncrit assuming α ∆qr and including four transmon levels [Sete15]. The decrease of χge with increasing| | |n also| leads to a saturation of the SNR unlike predicted by the linear approximation which shows a linear increase of the SNR with n [Boissonneault08, Boissonneault09]. Following the same argument a critical photon number for the ef-transition can ∆ α 2 be approximated by nef ( qr+ ) relevant for single-shot readout of qutrit states crit 4(p2g)2 ef ≈ [Sete15] and ncrit ncrit/6 with our experimental settings. ≈ Average Dispersive Readout

For average readout, the population of an arbitrary transmon state is extracted from a large number of individual measurements. For each of these measurements, we describe the recorded in-phase (I) and quadrature (Q) components of the measurement pulse as a readout trace S(t). These individual readout traces are ﬁrst averaged Sarb(t) and the population Pi are obtained by comparing Sarb(t) to known, averaged 〈 〉 〈 〉 references traces, e.g Sg(t) , Se(t) , Sf(t) for a qutrit readout. We obtain 〈 〉 〈 〉 〈 〉

Sarb(t) = Pg Sg(t) + Pe Se(t) + Pf Sf(t) , (2.66) 〈 〉 〈 〉 〈 〉 〈 〉 assuming no populations in other transmon states. Experimentally, S(t) is a 2m- dimensional vector with the number of time points m and a factor 2 due〈 to〉 concate- nating the I and Q quadratures. The reference traces form a d 1 dimensional plane − for a d-level system. In the absence of noise Sarb(t) always lies within the plane 〈 〉 spanned by the reference traces, however in reality Sarb(t) can be outside this plane and needs to be projected into this plane to obtain the〈 populations.〉 This projection can be performed by a weighted integration over the discrete time trace Sarb(t) leading to 〈 〉 an optimal population extraction for known reference traces [Dahlberg16, Pechal16a]. Preparing eigenstates of the transmon as reference traces is, however, not directly possible due to thermal population of the transmon states. Several approaches have been investigated to extract population in average readout mitigating the effect of thermal population. First, the transmon states can be actively cooled by performing a 2.2 Circuit Quantum Electrodynamics 43

reset to the ground state [Valenzuela06, Ristè12b, Campagne-Ibarcq13, Geerlings13, Egger18, Magnard18, Salathé18b] making the reference traces approximately the eigenstates up to rethermalization. Second, so-called cavity-Bloch equations can be used [Bianchetti09, Li18a] which approximate the dynamics of the resonator response based on resonator and transmon parameters making the thermal populations indepen- th dent ﬁt parameters. Third, the thermal population Pi can be measured [Geerlings13] and modiﬁed reference traces are assumed for the population extraction [Pechal16a]

th th th th th Sg (t) = (1 Pe Pf ) Sg(t) + Pe Se(t) + Pf Sf(t) 〈 〉 − − 〈 〉 〈 〉 〈 〉 th th th th th Se (t) = Pe Sg(t) + (1 Pe Pf ) Se(t) + Pf Sf(t) 〈 th 〉 th 〈 〉 th− − 〈 th〉 th th〈 〉 Sf (t) = Pe Sg(t) + Pf Se(t) + (1 Pe Pf ) Sf(t) (2.67) 〈 〉 〈 〉 〈 〉 − − 〈 〉 We use the ﬁrst (cooling) and third (modiﬁed reference traces) approach in our circut QED experiment (see Section 5.3 and Section 5.12).

Fast Single-Shot Qubit Readout

The state of a transmon can also be inferred in every single repetition of the experiment if the SNR is sufficiently high [Gambetta07, Gambetta08]. Single-shot readout is, for example, essential for error-correction algorithms based on feedback [Ristè12a, Salathé18b], deterministic generation of entanglement based on measurement [Ristè13], deterministic quantum teleportation [Barrett04, Steffen13b] and loophole-free Bell tests [Hensen15, Rosenfeld17] to close the detection loophole. During the presented thesis, we have optimized qubit dispersive single-shot readout in terms of measurement time and fidelity to be applicable for loophole-free Bell test with superconducting circuits. For details, we refer to our work [Walter17], and further references [Mallet09, Vijay11, Walter14]. The main results of our work are summarized in Fig. 2.14 illustrating the time response of our Purcell-filtered transmon-resonator system in single-shot and on average (Fig. 2.14b) and showing that a readout fidelity Fr 98.4% can be ∼ reached within τ = 56 ns (Fig. 2.14c) using dispersive readout (see Section 3.3.1 for timing considerations of a loophole-free Bell test with superconducting circuits). In addition, the observed overlap error (eO) decreases exponentially with time and the readout fidelity is limited by qubit energy relaxation for τ > 64 ns with T1ge = 7.7 µs (see Tab. 2.1). The limitation due to qubit energy relaxation illustrates the need of fast single-shot readout to achieve high-fidelity readout also in other circuit QED experiments. 44 Chapter 2 Basics of Quantum Communication in Circuit Quantum Electrodynamics

0/ a) R ge pre π measurement meas. 250 100 -18 0 time, t [ns] - - b) 0.75

0.00

τ intraresonator field, Q(t) 0.75 - 50 0 50 100 150 200 - time, t [ns]

c) 0 10 1- r ee ℱ eg

r e0 10 - 1

10 - 2 meas. error, e=1- ℱ 10 - 3

10 - 4 0 20 40 60 80 100 120 140 Integration Time, τ [ns] Figure 2.14 a) Scheme of pulses applied to the resonator and transmon drive line. The resonator pulses include a pre-selection measurement pulse to select thermally excited states 0 π before the actual qubit measurements. We use no pulse Rge and a DRAG pulse Rge to prepare the qubit in state g and e . b) Typical single-shot (dots) and averaged (solid lines) measurement dynamics| with〉 transmon| 〉 in state g (blue) and e (red). The red and blue shaded regions depict the respective standard deviations| 〉 for 60000| 〉 measurement trajectories. The black dashed lines show the theoretically expected dynamics using independently extracted sample parameters and Eq. (2.77). The typical integration time τ 56 ns is indicated by ≈ the gray region. (c) Measurement errors e = 1 Fr with readout ﬁdelity Fr as a function of integration time τ. e is obtained from single-shot− experiments at optimal drive strength which results in approximately n = 2.5 photons in the readout resonator. The black circles show the direct measurement results. The open circles indicate the measurements errors on the excited (red) and ground state (blue), and overlap errors (green) between them due to imperfect signal-to-noise ratio. These measurement errors are extracted from Gaussian ﬁts to the excited and ground state measurement results. Figure is based on Ref. [Walter17]. 2.2 Circuit Quantum Electrodynamics 45

Qutrit-State Single-Shot Readout for Population Extraction

Single-shot readout can also be used to better characterize the measurement errors of qutrit readout (see Chapter 5) which is detailed in this paragraph. For the readout characterization, we prepare the transmon in either state g , e and f using DRAG pulses, and record the single-shot traces. We use two mode-matched| 〉 | 〉 weight| 〉 functions R w1(t) and w2(t) [Walter17] to obtain the integrated quadratures u = S(t)w1(t) d t R and v = S(t)w2(t) d t for each traces. The integrated traces form three Gaussian shaped clusters in this u-v plane (Fig. 2.15) corresponding to the Gaussian probability distributions of the trace when the qutrit is prepared in one of the three states. We model the probability distribution x~ = (u, v) as a sum of three Gaussian distributions with density X As 1 x~ µ σ 1 x~ µ f x~ e 2 ( s)> − ( s) (2.68) ( ) = p − − · · − s 2π σ | | and ﬁt parameters As, µs and σ. Based on the obtained ﬁt parameters we divide the u-v plane into three regions used to assign the result of the readout of the qutrit state

(Fig. 2.15). If an integrated trace is in the region labeled s0 we assign it state s0. We count the number of traces prepared in state s and assigned value s0, and normalize | 〉 i the recorded counts. We summarize those normalized counts in a vector N | 〉 for each prepared state i and we estimate the assignment probabilities matrix Rss P s s 0 = ( 0 ) = g e f | 〉 | | 〉 (N | 〉, N | 〉, N | 〉). Experimentally, the measurement power and signal integration time 1 can be optimized in order to minimize the measurement error probability 6 I R 1, || − || i.e. the sum of all measurement misidentiﬁcations (the off-diagonal elements of Rss ). 0 Exemplary measurement results are shown in Fig. 2.15. The probability

X Ms P s ρˆ P s s ρss (2.69) 0 = ( 0 ) = ( 0 ) | s || 〉 · to assign value s0 to a single shot measurement of a qutrit in state ρˆ can be expressed as M = R ~ρdiag with the vector ~ρdiag which consists of the diagonal elements of ρˆ. The · assignment probabilities M can in a ﬁrst approach be estimated by ~ρdiag = M. This approach is sensitive to measurement errors, but independent of state preparation errors. 1 Setting ~ρdiag = R− M effectively accounts for single-shot readout error. However, R needs to be estimated· precisely making it sensitive to state-preparation error. Due to our small transmon reset inﬁdelities of approximately 0.2% [Magnard18], and single qubit gate errors of 0.6% (measured with randomized benchmarking, see Fig. 2.11f) state preparation errors are expected to be lower than readout errors. 46 Chapter 2 Basics of Quantum Communication in Circuit Quantum Electrodynamics

Figure 2.15 Scatter plot of the measured integrated quadrature values u, v to characterize qutrit single-shot readout. Measurement results using the sample Node A (a-c) and B (d-e) (see Tab. 2.1) when prepared in state g (blue), e (red), f (green), respectively. We plot the first 1000 of the total 25000| repetitions〉 | for〉 each state| 〉 preparation experiment. Based on all measurement results we estimate the assignment probabilities of identifying the prepared states as the measured states (show as numbers in the corresponding region). For example, the numbers in the region g indicate correct identification in (a) and (d), the others misidentifications. The dashed lines are the qutrit state discrimination thresholds used to obtain these assignment probabilities. Figure is based on Ref. [Kurpiers18].

The described formalism to correct for single-shot readout errors and extract the state populations can also be extended to two-qutrit systems. For two-qutrit states we obtain the assignment probability matrix R P s s s s P s s sAsB,s s = ( A0 B0 A B ) = ( A0 A ) A0 B0 | | 〉 | | 〉 · P(sB0 sB ) using the outer product of the single-qutrit assignment probability matrices. | | 〉

2.2.7 Purcell Filters

For transmon readout and remote quantum communication between separate chips resonators need to be coupled to external circuits. This external coupling, however, opens a energy relaxation channel for the transmon via the resonator, known as Purcell effect [Purcell46], which is suppressed in the dispersive regime compared to the resonant case but cannot be neglected for experiments in which the state of a qubit should be readout fast (see Section 2.2.6 and Ref. [Walter17]) or a high-bandwidth quan- 2.2 Circuit Quantum Electrodynamics 47

tum communication is desired (see Chapter 5 and Ref. [Kurpiers18]). We therefore introduce filter circuits to mitigate this energy relaxation channel. Before introducing these filtered systems, we first consider a circuit QED system in which a transmon is coupled to a resonator which is directly coupled to an external circuit. We do a semi-classical analysis of this system in which the intra-resonator field

αr is considered classically. For an input-drive at frequency ωd and amplitude εd the evolution of αr in a frame rotating at ωd is given by

κr κr α˙r = i∆rdαr αr ipκinεd = i∆rdαr αr iγin (2.70) − − 2 − − − 2 −

i with the resonator-drive detuning ∆rd = ω|r 〉 ωd, and the input (κin), output (κout), − and total energy relaxation rate κr = κout +κin assuming no internal loss. The evolution of αr in Eq. (2.70) is derived based on the input-output formalism of a driven damped harmonic oscillator in the classical (coherent) limit [Gardiner85]. The input-output formalism also relates the outgoing ﬁeld amplitude γout = pκoutαr to the intra-resonator 2 ﬁeld. The normalization should be chosen such that αr = n is the average number | | of photons in the cavity. In steady state, α˙r = 0, we obtain

pκr γout = γin (2.71) i∆rd + κr/2 with the typical Lorentzian line shape of a harmonic oscillator measured in transmission. To realize a nearly directional transmission measurement, the resonator is weakly coupled at the input port and strongly to the output with typical ratios of κin/κout ≈ 1/100 resulting in κr κout. This asymmetric coupling lets most of the intra-resonator ≈ field αr leak through the output while a sufficiently high field amplitude can be used at the input port to keep αr constant. The input-to-output coupling ratio cannot be chosen arbitrarily large, since a direct coupling between the input and output port, e.g. via box-modes in the sample holder around the chip, scales also with the input-amplitude. This directly transferred field can introduce undesired interference effects or saturate the amplifiers in the output line.

Purcell Eect

k 1,k To estimate the energy relaxation rate γPurcell− from transmon level k to k 1 via the resonator to the environment we treat the external coupling as a perturbation,− and use 48 Chapter 2 Basics of Quantum Communication in Circuit Quantum Electrodynamics

Fermi’s golden rule to obtain

2 gk 1,k γk 1,k κ (2.72) Purcell− = r − 2 (ωk 1,k ωr) − − in the absence of additional photons in the resonator (n = 0) and based on the single ge g2 mode approximation of the resonator (Fig. 2.16) Koch07a . γ κr 2 is to ﬁrst [ ] Purcell = ∆qr order independent of α and is therefore equivalent to the energy relaxation of two cou- ge χge pled linear systems. Using Eq. (2.62) we can also approximate γPurcell κr α showing ge ≈ that γPurcell cannot be kept small while κr and χge are small, since α is limited by charge dispersion for transmons which is relevant for fast qubit readout (see Section 2.2.6) and high-bandwidth quantum communication (see Section 2.2.8). Moreover, additional photons in the resonator affect the energy relaxation rate as

2 2 2 2 g 3g g g (3∆qr + 4α) γge n κ 1 6n 2n , (2.73) Purcell( ) r 2 2 2 + 2 ≈ ∆qr − ∆qr − ∆qr ∆qr(∆qr + α)

ge expressed here in an approximation up to third-order [Sete15]. γPurcell(n) is lowered by 3g2 ge 3α a constant term ∆2 compared to γPurcell but increases with increasing n for ∆qr > 2 . qr −

Purcell Filters

Alternatively, an impedance calculation can be utilized to derive the Purcell relaxation rate in electrical circuits

env γPurcell(ω) Re(Z (ω)) (2.74)

env where Z is the impedance of the dissipative environment at frequency ω [Nigg12, Solgun14]. This method takes the full mode spectrum of the environment into account and can predict the relaxation rate of transmons over a larger frequency ranges [Allenspach15], for example, if a transmon is placed at a frequency in the center between two resonator modes [Houck08]. In addition, the impedance description illustrates the usefulness of impedance engineering techniques [Pozar12] to decrease ge γPurcell significantly at ωge while keeping the resonator frequency and bandwidth nearly unaffected. Various filter circuits have been proposed and realized in the recent years which are based on band-stop filters at ωge [Reed10, Bronn15a], band-pass filters at ωr [Jeffrey14, Sete15] or filters integrated at the output coupling [Bronn15b]. For all 2.2 Circuit Quantum Electrodynamics 49

these implementations we obtain

g2 γge ω κ ω , (2.75) filter( ) r( ) 2 ≈ ∆qr meaning that the transmon-resonator coupling g remains unaffected by the additional

ﬁlter circuit but κr(ω) becomes frequency dependent. In our experiments [Walter17, Magnard18, Kurpiers18], we have chosen to use additional broadband resonators as band-pass ﬁlters (Fig. 2.13a) on our readout and quantum communication resonators to obtain high κr and χge simultaneously. For example, the relaxation rate using our experimental parameters is suppressed by the factor

ge γ 0.27 µs 1 f Purcell (2.76) filter = ge 733 µs 2720 γfilter ≈ ≈ for our readout resonator (Tab. 2.1) and by ffilter 1/1950 for our quantum commu- ≈ nication resonators due to the use of Purcell filters. ffilter illustrate the relevance of Purcell filters in our experiments and we derive the most relevant quantities based on Ref. [Sete15] in which they investigated a notch-type filter system in detail. The- ses systems are also shown to be advantages for the detection of individual itinerant microwave photons [Besse18] and to be scalable to multi-qubit architectures with multiplexed readout [Jeffrey14, Kelly15, Heinsoo18].

Expanding the input-output formalism to a system in which a resonator at frequency

ωr is coupled by J to a filter-resonator with intra-filter field βf, frequency ωf and external coupling κf J leads to

α˙r = i∆rdαr iJβf − − ˙ κf βf = i∆fdβf iJ ∗αr βf iγin (2.77) − − − 2 − with ∆fd = ωf ωd, assuming that the ﬁlter-resonator is driven with γin = pκinεd and asymmetrically− coupled to the input and output line. In the quasi-steady state 1 approximation, in which we neglect the fast oscillations on the time scale of κf− and ˙ set βf = 0, we obtain

eff κr J α˙r = i∆e rdαr i αr γin (2.78) − − 2 − κf/2 + i∆fd

eff with a shifted frequency of the resonator ∆e rd = (∆rd ∆fdκr /κf) due to the coupling. − 50 Chapter 2 Basics of Quantum Communication in Circuit Quantum Electrodynamics

This equation has the same form as for a single resonator Eq. (2.70) and we model the system as a single (Purcell-ﬁltered) resonator with an effective energy relaxation rate

4 J 2 1 κeff (2.79) r = | | 2 κf 1 + (2∆fd/κf)

eff which is drive-frequency dependent by ∆fd. Evaluating κr at ωr and ωge we obtain

1 2∆ /κ 2 κ 2 f + ( fd f) f (2.80) filter = 2 1 + (2∆qf/κf) ≈ 2∆qf with ∆qf = ωge ωf and assuming ∆fd κf and ∆qf κf in the last relation − | | | | (Fig. 2.16). ffilter shows a quadratic suppression of the Purcell energy relaxation rate in the ratio of filter bandwidth to qubit-filter detuning. Furthermore, we obtain for the transmitted field of the notch-type system (Fig. 2.13c) in the steady state

κ 2∆ nt p f rd γ γin. (2.81) out κ /2 i∆ eff f + fd κr + 2i∆e rd

nt Another important property for readout of the notch-type system (γout) compared

730 μs

[ μ s ] 1000

1ge 100

4 10 Δ

1 0.3 μs 0.1 Δ2 0.01 energy relaxation time, T

3.5 4. 4.5 5. 5.5 6. 6.5

qubit frequency, νqb [GHz]

Figure 2.16 Energy relaxation time T1ge of the ge-transition for the sample parameters specified in Tab. 2.1 with (blue) and without (red) including the Purcell filter. The black dashed line is a fit ∆4 to in the Purcell-filtered case. The vertical line indicates approximately the qubit transition frequency at 6.3 GHz. 2.2 Circuit Quantum Electrodynamics 51

tr to a system which is driven in transmission (γout) via the inner resonator is a reduction of the output power. We deﬁne the output power ratio

nt 2 γ ∆ κ 2 Γ nt out rd f (2.82) tr = | tr |2 2 γout ≈ 2J | | nt assuming ∆rd κf and ∆fd = 0 and obtain Γtr 1/7 for ∆rd χge using our experimental| parameters| (Tab. 2.1). This makes the notch-filtered≈ setup≈ advantageous compared to a transmission-type setup since the requirements on the dynamic range of the first amplifier are weakened [Eichler14b]. However, considering the full dynamics of the system, the indirect drive via the Purcell-filter adds a latency to the raising edge of the readout-resonator field which can partially be mitigated by using shaped resonator drive pulses [Govenius12, Kurpiers13, Jeffrey14, McClure16, Walter17]. eff In conclusion, Purcell filters are a key tool to make the choice of κr practically ge eff independent of χge while keeping γPurcell high. Therefore, the relevant ratios χge/κr 0.1 0.5 for qubit readout can be chosen optimally for large parameter ranges. How-≈ ever,− this is only valid in the low excitation regime since the dispersive approximation png/ ∆qr 1 still has to hold as the transmon-resonator coupling is not altered by the| Purcell-filter.| In addition, Purcell filters are an excellent extension for single- photon quantum communication to obtain a high bandwidth which we discuss in the next section.

2.2.8 Cavity-Assisted Raman Process: F0G1

As mentioned in Section 2.2.7, coupling to external circuits is also required for quantum communication between remote circuit QED systems since photon wave packets can be exchanged via microwave transmission lines. At the emitter and receiver a driven Jaynes-Cummings-type system (Section 2.2.5) can be utilized to transfer the quantum state between the transmon and photons in the resonator [Cirac97]. In this section, we discuss an implementation based on a second-order cavity-assisted Raman process [Zeytinoglu15˘ ] which realizes an amplitude and phase tunable coupling between the f , 0 g, 1 (f0g1) transition with the Fock states of the resonator 0 or | 〉 ↔ | 〉 | 〉 1 [Gasparinetti16]. We follow a calculation by Baptiste Royer (also summarized in | 〉 supplementary material of Ref. [Kurpiers18]) to obtain the effective coupling strength g˜f0g1 between these states. Note that similar transitions can also be used to realize two-qubit gates in a transmon-resonator-transmon system [Egger19] or in a system of two capacitively coupled transmons (see Section 5.11). 52 Chapter 2 Basics of Quantum Communication in Circuit Quantum Electrodynamics

Tunable Transmon-Resonator Coupling

We start with the Jaynes-Cummings Hamiltonian in the rotating wave approximation, ˆ ˆ ˆ † Eq. (2.57), and add a transmon-drive term Hd = ħhΩf0g1 cos(ωd t + φf0g1)(b + b )

E Hˆ /h ω aˆ†aˆ ω ˆb†ˆb C ˆb†ˆb†ˆbˆb g aˆ†ˆb aˆˆb† Ω cos ω t φ ˆb ˆb† dJC ħ = r + ge 2 + ( + )+ f0g1 ( d + f0g1)( + ) − (2.83) with EC = EC/ħh for this derivation. We move to a frame rotating at ωd and obtain

EC Ωf0g1 Hˆ /h ∆ aˆ†aˆ ∆ ˆb†ˆb ˆb†ˆb†ˆbˆb g aˆ†ˆb aˆˆb† ˆb eiφf0g1 ˆb† e iφf0g1 dJC ħ = r + qb 2 + ( + ) + 2 ( + − ) − (2.84) with ∆r = ωr ωd and ∆qb = ωge ωd in the rotating frame, neglecting fast oscillating − − ˆ ˆ terms at 2ωd. Second, we perform the displacement operations aˆ aˆ γa, b b γb which results in → − → −

E ˆ † 2ˆ †ˆ C ˆ †ˆ †ˆˆ †ˆ ˆ † HdJC/ħh = ∆raˆ aˆ + ∆qb 2EC γb b b b b bb + g aˆ b + aˆb − | | − 2 Ωf0g1 ˆ † iφf0g1 2 † + b e− + ∆qbγb gγa + EC γb γb + h.c. + aˆ ∆rγa gγb + h.c. 2 − | | − − ˆ †ˆ †ˆ ˆ †ˆˆ 1 2 ˆˆ 2ˆ †ˆ † + EC γb b b b + γb∗ b bb (γb∗ bb + γb b b ) . (2.85) − 2 The displacement operations allow us to set the linear drive terms to zero

Ωf0g1 iφf0g1 2 0 = e− + ∆qbγb gγa + EC γb γb (2.86) 2 − | | 0 = ∆rγa gγb. (2.87) − − and we obtain

iφf0g1 1 Ωf0g1 e− γb (2.88) ≈ 2 ∆qb

2 for a small displacement γb 1 (linear displacement regime) and assuming g /(∆r∆qb) ˆ ˆ 1. Next, we use a Bogoliubov transformation aˆ cos(Λ)aˆ + sin(Λ)b, b sin(Λ)aˆ + ˆ → → − cos(Λ)b resulting in 2.2 Circuit Quantum Electrodynamics 53

ˆ HdJC/ħh ˜ K˜ 1 ˜ † ˜ ˆ †ˆ αˆ †ˆ †ˆˆ † † † ˆ †ˆ ˆ †ˆ † †ˆˆ = ∆raˆ aˆ + ∆qb b b + b b bb + aˆ aˆ aâˆ + 2χãˆ aˆb b g˜f0g1aˆb b + g˜f0g1∗ aˆ bb 2 2 − p2 2 2 2 †ˆ ˆ † + g cos (Λ) g sin (Λ) + (∆r ∆qb + 2EC γb ) cos(Λ) sin(Λ) aˆ b + aˆb − − | | 3 †ˆ †ˆˆ ˆ †ˆ †ˆ 3 † † ˆ † ˆ † + EC cos (Λ) sin(Λ) aˆ b bb + aˆb b b + EC cos(Λ) sin (Λ) aˆ aˆ aˆb + aˆ aâˆb E C 2 2 † †ˆˆ ˆ †ˆ † cos (Λ) sin (Λ) aˆ aˆ bb + aâˆb b − 2 3 ˆ †ˆ †ˆ ˆ †ˆˆ 3 † † † + EC cos (Λ) γb b b b + γb∗ b bb EC sin (Λ) γbaˆ aˆ aˆ + γb∗ aˆ aâˆ − 2 † ˆ † ˆ ˆ †ˆ † + 2EC cos (Λ) sin(Λ) aˆ aˆ(γb b + γb∗ b) b b(γbaˆ + γb∗ aˆ) . (2.89) − This transformation is similar to the dispersive transformation defined in Eq. (2.59). ˜ 2 2 2 We obtain a resonator-drive detuning ∆r = ∆r cos (Λ) + (∆qb 2EC γb ) sin (Λ) − | | − g sin(2Λ) which is shifted compared to the bare detuning and depends on ∆qb, EC and ˜ 2 2 g. Similarly we obtain for the qubit-drive detuning ∆qb = (∆qb 2EC γb ) cos (Λ) + 2 − | | ∆r sin (Λ) + g sin(2Λ). Furthermore, we define the qubit anharmonicity under this 4 ˜ transformation α˜ = EC cos (Λ), the transmon-induced resonator anharmonicity K = 4 − 2 2 EC sin (Λ), the dispersive shift χ˜ = EC cos (Λ) sin (Λ), and the f0g1-coupling rate − 2 − g˜f0g1 = p2ECγb cos (Λ) sin(Λ). To cancel the linear coupling terms we set

2 2 2 g cos (Λ) g sin (Λ) + (∆r ∆qb + 2EC γb ) cos(Λ) sin(Λ) = 0 (2.90) − − | | and obtain

1 g Λ tan 1 2 . (2.91) = − 2 2 ∆qr 2EC γb − | | We approximate

g Λ (2.92) ≈ ∆qr assuming that the dispersive parameter g/∆qr 1, the relative anharmonicity EC/∆qr 1 and the displacement γb 1 are small, and we operate the system in the linear dis- 2 EC g placement regime. In this limit, we obtain also α˜ EC and χ˜ − 2 , which is ∆qr ≈ − 4 ≈ ˜ g the approximation of χge for EC/∆qr 1, and K/EC ∆4 1. Using the unitary ≈ qr ˆ ˜ † ˜ ˆ †ˆ α˜ ˆ †ˆ †ˆˆ transformation U = exp(i t(∆raˆ aˆ + ∆qb b b + 2 b b bb)) we obtain the resonance 54 Chapter 2 Basics of Quantum Communication in Circuit Quantum Electrodynamics

frequency of the f0g1-transition

˜ ˜ 2∆qb ∆r α˜ = 0 (2.93) − − and the drive frequency up to ﬁrst order

2 2 f0g1 g g 2 ωd = 2 ωge + EC ωr 4EC γb (2.94) ∆ − − − ∆qr − | |

f0g1 showing an ac Stark shift quadratic in γb Ωf0g1. A drive at ωd therefore induces an effective coupling up to ﬁrst order | |

αg χge iφf0g1 iφf0g1 g˜f0g1 Ωf0g1 e− = Ωf0g1 e− (2.95) ≈ p2∆qr(∆qr + α) p2g between the states f , 0 and g, 1 [Pechal14] which is directly tunable in phase and | 〉 | 〉 amplitude by the drive tone. Similar as discussed in Section 2.2.5 for χge, we obtain for a linear system g˜f0g1(α 0) = 0, which can be seen as a destructive interfer- → ence of the two paths between f , 0 and g, 1 . In the other limit, g˜f0g1(α ) = iφf0g1 | 〉 | 〉 → ∞ Ωf0g1 e− g/∆qr is equal to the effective coupling of an ideal Λ-system with only one possible transition path between the two states [Zeytinoglu15˘ ]. Therefore, max- imizing α would also maximize g˜f0g1, making superconducting circuits with a larger anharmonicity like capacitively shunted (c-shunt) ﬂux qubits [Yan16b] interesting systems to study this transition. Finally, we obtain the Hamiltonian

˜ f0g1 αˆ †ˆ αˆ †ˆ †ˆˆ K † † Hˆ /ħh = b b + b b bb + aˆ aˆ aˆaˆ − 2 2 2 † ˆ †ˆ 1 ˆ †ˆ † †ˆˆ + 2χ˜aˆ aˆb b (g˜f0g1aˆb b + g˜f0g1∗ aˆ bb) (2.96) − p2 modeling the introduced coupling of the f0g1-transition in a frame rotating at ∆˜ r for ˜ the resonator and ∆qb + α/2 for the transmon.

To second order in Ωf0g1/∆qr the ﬁdelity of a SWAP operation between the f , 0 | 〉 and g, 1 can be approximated by [Zeytinoglu15˘ ] | 〉 Ω /2 2 2Ω /2 2 3Ω /2 2 f0g1 f0g1 p f0g1 p f0g1 F˜ 1 + + (2.97) ≈ − ∆qr + α ∆qr ∆qr α − ˜f0g1 illustrating that ∆qr α should be chosen large compared to Ωf0g1 to obtain a high F . In our Purcell-ﬁltered± transmon resonator system, we chose α/2π 0.265 GHz to min- ≈ 2.2 Circuit Quantum Electrodynamics 55

imize charge dispersion on the f -level (see Fig. 2.9), ∆qr/2π = 2 GHz, and g/2π = − 0.335 GHz to obtain g˜f0g1 0.0132 Ωf0g1. Experimentally, we obtain g˜f0g1 /2π | | ≈ | | ≈ 6 MHz for Ωf0g1/2π 450 MHz without measuring signiﬁcant deviations from the ≈ f0g1 linear approximation of g˜f0g1 or the quadratic approximation of ωd with increasing Ωf0g1 (see Fig. 5.3 or Ref. [Kurpiers18]).

Emission of Shaped Photon Wave Packets

Due to the amplitude and phase tunable coupling between the transmon degree of freedom and the resonator, a constant coupling of the resonator to the external circuit A κout is sufﬁcient to emit a photon wave packet with wave function f (t) into a transmission line at node A. We use the phase tunability to compensate for the drive-induced ac Stark shift and therefore, we stay resonate with the f0g1-transition and emit photons with constant phase. The experimental calibration is described in Section 5.3 and a theoretical derivation of the used shape of f (t) and g˜f0g1(t) is given in Appendix E of Marek Pechal’s PhD thesis [Pechal16a] and is summarized in this section. The state of the propagating photon wave packet can be written as

Z † 1TL = f (t) aˆout(t) 0 d t (2.98) | 〉 | 〉

† Æ A † with the creation operator aôut(t) = κoutaˆ (t) describing a propagating photon in † † † time interval (t, t + d t). aôut(t) is normalized such that [aôut(t), aôut(t0)] = δ(t t0). − To obtain a high absorption probability of the emitted photon, f (t) has to be chosen such that it can be reversed by the receiver B [Cirac97]

B A fin (t) = fout( t). (2.99) −

Since the ﬁeld amplitude inside a resonator cannot fall off faster than exp( κr t/2), the shortest physically possible, reversible photon shape can have an exponential− rising B A edge κin of the receiving node B and an exponential falling edge κout assuming B B A A B A κin κr and κout κr . Choosing κin = κout = κr is therefore ideal for a symmetric photon≈ wave function.≈ We choose a symmetric, analytic function with an exponentially rising and falling edge to approximate the shortest possible photon shape

q p κeff f (t) = p (2.100) 2 cosh(κeff t/2) 56 Chapter 2 Basics of Quantum Communication in Circuit Quantum Electrodynamics

p p with an effective photon bandwidth κeff κr. Introducing κeff decouples the photon bandwidth from the real resonator bandwidth≤ and provides the option to adjust experi- B A A mental deviations between κin and κout. κout should therefore be designed higher than the speciﬁed maximal obtainable g˜f0g1 to avoid limitations of the photon bandwidth due to the resonator bandwidth.

To calculate the time dependence of the effective coupling g˜f0g1(t) which is needed to emit a photon with the desired wave function, we model the system with a non- Hermitian Hamiltonian acting on the states f , 0 and g, 1 | 〉 | 〉 0 g˜ ˆ f0g1 f0g1 Heff = . (2.101) g˜f0g1∗ iκr/2 − ˆ f0g1 The non-Hermitian term iκr/2 of Heff traps the system in the dark state g, 0 af- − | 〉 ter the photon emission. To obtain the time evolution of a wave function ψ(t) = | 〉 cos(Θ) f , 0 + sin(Θ) g, 1 the non-unitary Schrödinger equation has to be solved and we obtain| 〉 | 〉 p p p p κefft κefft κeff 1 e + (1 + e )κr/κeff g˜f0g1(t) = p − . (2.102) Ç κp t p κp t 4 cosh(κeff t/2) eff eff (1 + e )κr/κeff e − p For κeff = κr this equation simpliﬁes to

κr g˜f0g1(t) = (2.103) 2 cosh(κr t/2) illustrating the strong relation between the tunable coupling used for the photon emission and the wave function of the propagating photon Eq. (2.100). Note that this type of analysis is not speciﬁc to our choice of states and could be used in any other Λ-type system, for example in another circuit QED implementation [Campagne-Ibarcq18].

Coupling of Two Remote Circuit QED Systems

Two remote systems which are described by Hˆ f0g1 can be coupled via a transmission line. Bisecting this transmission line with a circulator avoids hybridization of the resonators of system A and B since signals can only pass in one direction. The circulator makes the systems more independent from each other, but it is not necessary if a perfect absorption of the propagating photon is realized since nothing is reﬂected in this case [Cirac97]. In our experimental implementation, we chose to use a circulator since it also offers the possibility to study the output ﬁeld of both resonators via its third 2.2 Circuit Quantum Electrodynamics 57

port. We model the loss between the emitter and receiver by adding a beam splitter which routes the lost signal towards an additional, imaginary port l and use a master equation approach which describes the dynamics of an open quantum system B driven by a source quantum system A [Gardiner93, Carmichael93, Gardiner04]. Using input- output theory we obtain the relations

out in q A q A aÂ = aÂ + κr aÂ = κr aÂ out in Æ B aˆB = aˆB + κr aˆB (2.104)

in out for the input field aî and output field aî of the resonators of system A and B, and assuming the vacuum state at the input of system A. The dynamics of aÂ and aˆB are described using the Heisenberg equations

κA ˙ ˆ f0g1 r aÂ = i[HA , aÂ] aÂ − 2 κB ˙ ˆ f0g1 r Æ B in aˆB = i[HB , aˆB] aˆB κr aˆB . (2.105) − 2 − The directional connection due to the circulator of the two system results in

q aˆin aˆout t A aˆ B = pηAB A ( τ) = κr ηAB A (2.106) − with the photon-transmission probability ηAB and a constant delay τ due to the propagation time between the two samples. τ can be eliminated by a time-shift transformation on the second system [Cirac97] which we performed on the last relation. We use the time-shifted variables in the further derivation without changing notation. Introducing the circulator and the imaginary beam splitter to model loss leads to

out q A p ˆ Æ B q A Æ B aˆB = κr ηAB aÂ + 1 ηAB lin + κr aˆB = κr ηAB aÂ + κr aˆB q − q ˆl A aˆ ˆl A aˆ out = κr (1 ηAB) A pηAB in = κr (1 ηAB) A (2.107) − − − ˆ assuming the input lin at the imaginary port of the beam splitter to be in the vacuum out state. aˆB is therefore a superposition of the output fields of both resonators which can be used to measure the emitted fields of the resonators of system A and B. Using Eq. (2.107) we model the dynamics of the cascaded system, described by the density 58 Chapter 2 Basics of Quantum Communication in Circuit Quantum Electrodynamics

matrix ρˆAB, by the master equation [Lindblad76, Gerry05]

˙ ˆ f0g1 ˆ f0g1 ˆ out ˆ ρÂB = i[HA + HB + Hc, ρÂB] + D(aˆB ) ρÂB + D(lout) ρÂB. (2.108) − ˆ ˆ ˆ ˆ † ˆ † ˆ Hc is an effective coupling term and D(O) = O O O O, /2 denotes the dissi- • • − { •} pation super-operator. Calculating the evolution of the expectation values of aÂ and aˆB using the master equation Eq. (2.108) and comparing it to the evolution obtained from the Heisenberg equations Eq. (2.105) we obtain [Cirac97]

ÆκAκB ˆ r r † † Hc = +ipηAB aˆAaˆB aˆAaˆB (2.109) 2 −

κAκB showing an effective direct coupling with coupling constant η p r r . For η 0 p AB 2 AB = we thus obtain two independent systems and two independently decaying modes aÂ and aˆB. For an ideal connection, ηAB = 1, we obtain Æ κAκB ˙ ˆ f0g1 ˆ f0g1 r r † † q A Æ B ρÂB = i HA + HB + i aÂaˆB aÂaˆB , ρÂB + D( κr aÂ + κr aˆB) ρÂB. − 2 − (2.110)

Howard Carmichael [Carmichael93] showed that Eq. (2.110) can be rewritten using an ensemble of pure states ψ in a quantum trajectory approach instead of ρˆAB. In his approach the dynamics of| ψ〉 are described by the Schrödinger equation with the non-Hermitian Hamiltonian | 〉

i ˆ ˆ f0g1 ˆ f0g1 A † B † q A B † Hid/ħh = HA + HB (κr aÂaÂ + κr aˆBaˆB + 2 κr κr aÂaˆB). (2.111) − 2 ˆ Hid illustrates the directionality due to the circulator since only an interaction term † aÂaˆB appears, which creates a photon at system B while annihilation one at system A.

In our numerical simulation of a photon transfer between system A and B we i additionally added internal energy relaxation rates of the resonators κint, and energy i i i i i i i relaxation γ1ge = 1/T1ge, γ1e f = 1/T1e f and dephasing γφge = 1/2T1ge 1/T2ge, i i i − γφe f = 1/2T1e f 1/T2e f of the ge- and ef-transitions and obtain the master equation − 2.2 Circuit Quantum Electrodynamics 59

˙ ˆ f0g1 ˆ f0g1 ˆ ρÂB = i[HA + HB + Hc, ρÂB] − out ˆ + D(aˆB ) ρÂB + D(lout) ρÂB X i i i + κintD(aî) ρÂB + γ1geD( g e i) ρÂB + γ1e f D( e f i) ρÂB i=A,B | 〉〈 | | 〉〈 | i i + γφgeD( e e i g g i) ρÂB + γφe f D( f f i e e i) ρÂB . (2.112) | 〉〈 | − | 〉〈 | | 〉〈 | − | 〉〈 |

2.2.9 Sample Fabrication

Our circuit QED devices, depicted in Fig. 2.17, are fabricated at the cleanroom facilities ETH Zurich FIRST-Lab and at Binnig and Rohrer Nanotechnology Center (BRNC) at IBM Zurich campus Rüschlikon. We start with a 500 µm thick, C-plane sapphire wafer (2-inch) on which a 150 nm niobium layer is sputtered on one side. The coplanar waveguide resonators and 50Ω feed lines are patterned into niobium using photolithography and reactive ion etch (RIE) techniques [Göppl09]. The Al/Ti airbridges (Fig. 2.6d) are fabricated in two additional photolithography steps with the ﬁrst deﬁning the contact area to the niobium ground plane and the second the bridge outline followed by a

a) b)

chip transfer

transmon

2 um

readout

Figure 2.17 a) False-colored micrograph and b) simpliﬁed circuit of a sample of the same design as used in the remote communication experiments (Chapter 5). The sample consists of a remote communication (transfer) circuit (yellow), a readout circuit (gray), and a transmon (orange) which is capacitively coupled to the two resonator systems. The input lines of the transmon and readout circuit are marked in blue, and the input to the transfer circuit is an auxiliary line to perform resonator spectroscopy in transmission on the transfer circuit. The inset in (a) shows a scanning electron microscope (SEM) micrograph of the asymmetric SQUID with a designed ratio of 1 : 5 between the areas of the Josephson junctions. The output transmission lines are galvanically coupled to the corresponding circuit. Figure is based on Ref. [Kurpiers18]. 60 Chapter 2 Basics of Quantum Communication in Circuit Quantum Electrodynamics

quantity, symbol (unit) sample fast readout Node A Node B readout resonator frequency, νR (GHz) 4.762 4.787 4.780 readout Purcell filter frequency, νRpf (GHz) 4.754 4.778 4.780 readout resonator bandwidth, κR/2π (MHz) 37.5 12.6 27.1 readout circuit dispersive shift, χR/2π (MHz) 7.9 5.8 11.6 transfer resonator frequency, νT (GHz) 8.400 8.400 transfer Purcell filter frequency, νTpf (GHz) 8.426 8.415 transfer resonator bandwidth, κT /2π (MHz) 7.4-10.4 12.6-13.5 transfer circuit dispersive shift, χT /2π (MHz) 6.3 4.7 qubit transition frequency, νge (GHz) 6.316 6.343 6.096 transmon anharmonicity, α (MHz) -340 -265 -308 Josephson to charging energy ratio, EJ/EC 61 94 68 f0g1-transition frequency, νf0g1 (GHz) 4.022 3.485 max max. eff. f0g1-coupling, g˜f0g1/2π (MHz) 6.0 6.7 energy relaxation time on ge, T1ge (µs) 7.7 5.0 0.4 4.5 0.2 energy relaxation time on ef, T1ef (µs) 4.4 2.2 ± 0.6 1.5 ± 0.2 R ± ± coherence time on ge, T2ge (µs) 2.6 3.4-7.7 2.6-5.7 R coherence time on ef, T2ef (µs) 2.1-3.2 0.9-2.3 e -level population at thermal equilibrium, nth (%) 0.4 18 13 | 〉 Table 2.1 Summary of device parameters used in our single-shot readout characterization experiments (sample fast readout), Section 2.2.6 and Ref. [Walter17], and remote communication experiments (sample Node A and B), Chapter 5 and Refs. [Kurpiers18, Kurpiers19]. series of evaporation steps [Puebla-Hellmann13, Steffen13a]. Next, the wafer is cut with a dicing saw into individual devices of 4 7 µm size. × The Al/AlOx/Al Josephson junctions and the transmon capacitor pads are written with electron-beam lithography and shadow evaporated aluminum with lift-off. In this process, we deposit two layers of aluminum under different evaporation angles separated by a controlled oxidation [Fink10, Oppliger17]. The thickness of the aluminum-oxide layer, which is defined by the oxidation time, and the area of the

Josephson junctions define EJ. We choose an area ration of 1/5 for the asymmetric SQUIDs used in our samples. Next, the sample is glued to a gold-plated, copper printed circuit board (PCB) and the niobium ground plane and each feed line are connected to the PCB using Al wire bonds. On the PCB, 50Ω coplanar waveguides are patterned which extend the feed-lines to SubMiniature version P (SMP) connectors to which standard microwave coaxial cables can be connected. The PCB with the chip is placed in a copper sample holder and thermalized to the base temperature stage of our dilution refrigerators (see Chapter 3). Miniature superconducting wire coils are mounted at the bottom of the sample holder underneath the chip to tune the transmon frequencies by applying a controlled, static magnetic flux bias [Fink10, Fankhauser16, Hutchings17]. The samples are shielded against stray magnetic fields using two high permeability

® shields made from CryoPerm [Schär17]. In Tab. 2.1 we summarize the characteristic device parameters. Chapter 3

Cryogenic Quantum Networks

At any stage in the development of quantum technologies, there will be a largest size attainable for the state space of individual quantum processing units, and it will be possible to surpass this size by linking such units together into a fully quantum network.

H. Jeff Kimble in The quantum internet, p. 1023

As mentioned in the previous chapter, circuit QED systems which operate at gigahertz- frequencies need to be cooled to temperatures far below T ħhωr/kB 300 mK so that the thermal ﬂuctuation energy is small compared to the energy quantum.≈ In this chapter, we ﬁrst focus on cryogenic systems for circuit QED experiments, explain relevant thermal and cryogenic properties, and the microwave cabling between the room-temperature setup and the circuit QED systems at millikelvin temperatures via coaxial lines.

Beyond these basic aspects of our experimental setup, we discuss cryogenic networks which can host quantum networks based on circuit QED systems. We investigate the feasibility of performing a loophole-free Bell test in these systems and introduce our modular design of a cryogenic link which connects two cryostats at millikelvin temperatures. We analyze cooldowns of two prototype cryogenic systems - the ﬁrst system has a length of 4 m and uses one cryostat, the second system connects two cryostats at a distance of 10 m. Based on these experimental results, we make predictions how cryogenics networks can be scaled. 62 Chapter 3 Cryogenic Quantum Networks

3.1 Basics of Cryogenics

State-of-the-art cryostats for quantum information experiments use cryogen-free dilution refrigerator systems in which heat is transferred by conduction towards a cryogenic cooling unit. Individual temperature stages in these dilution refrigerators are isolated from each other by low-conductive solid support structures. In the ﬁrst part of this section, we introduce the working principle and characteristic cooling powers of state-of-the-art dilution refrigerators. In the second part, we discuss the basics of heat transfer by conduction and by thermal radiation complemented with experimental data. In addition, heat is transferred by convection of the residual gas, whose contribution to the total heat transfer can however be considered small at typical pressures reached in dilution refrigerators (see Ref. [Parma14] for details).

3.1.1 Dilution Refrigerators

We use cryogen-free dilution refrigerators from BlueFors Cryogenics Oy Ltd. (BF-LD250 and BF-LD400) which consist of several temperature stages (Fig. 3.1a) at approximately 295 K (RT), 35-50 K (50K), 2.8-4 K (4K), 0.8-1.1 K (Still), 0.1 K (cold plate, CP) and 0.01 K (base temperature, BT) with the abbreviation in∼ brackets used in the rest of∼ this thesis. At the RT-stage a common o-ring-sealed vacuum can provides a high vacuum environment for the whole dilution refrigerator. Within each temperature stage high thermal conductive metals are used and different stages are connected by low conductive support posts. Radiation-shields, which enclose the lower temperature stages, are installed at the 50K-, 4K- and Still-stage, and an optional shield at the BT-stage can be used to better protect the experimental space from thermal radiation. Cryogen-free dilution refrigerators are based on two cooling mechanisms: ﬁrst, the 50K- and 4K-stages are cooled using a two-stage pulse tube cryocooler (PT) [Radebaugh00] which provides cooling powers of 40-55 W at 45 K on the ﬁrst stage and 1.5-2 W at 4.2 K (Cryomech Inc. PT415 or PT420) (Fig. 3.1b). The PT also pre-cools the lower stages during the cooldown of the dilution refrigerator to temperatures around the boiling point of liquid helium-4 (TlHe4 = 4.2 K) by using heat switches for establishing thermal contact between the 4K- and Still-stage and between the Still- and BT-stage. After pre- cooling the heat switches are turned off and the second cooling mechanism is activated: 3 4 The Still- and BT-stage are cooled based on pumping on an He/ He mixture inside a 3 4 3 dilution unit [Pobell06]. The He/ He mixture shows a diluted He and a concentrated 3He phase for temperatures below 0.8 K and energy is required to move 3He from the concentrated to the diluted phase.∼ This mechanism typically provides cooling 3.1 Basics of Cryogenics 63

a b 125W 100W ] 300 75W K

) ) [ 250 4K 200 50W

150 25W 100 0W 50 250W 200W 4K stage, T 0 Still 0W 50W100W 150W 50 100 150 200 250 300

c temperature 50K stage, T50K K d CP ] ) 30 ] 5000 [ ] ) W

mW 1000 [ [μ 20 BT

Still 100  BT  10 10 power, Q

power, Q 0 1 500 700 900 1100 10 20 50 100 250

temp, TStill mK temp, TBT mK [ ] [ ] Figure 3.1 a) Photograph of the three lowest temperature stages of a dilution refrigerator (DR). The most upper plate is the Still-stage, the one in the middle to the cold plate (CP) and the lowest one to the base temperature stage (BT). In addition, the dilution refrigerator hosts the setup of the remote communication experiments (see Chapter 5). On the photograph microwave cables, two copper sample holders, and a magnetic shield (CryoPerm®) hosting a JPA are visible. b) Capacity curve (blue) of the pulse tube cryocooler (Cryomech Inc. PT415) between approximately 30 300 K on the 50K-stage and 4 300 K on the 4K-stage. The capacity curve shows the relation− of the temperature of the two− stages to the cooling power at each stage. The data are provided by the manufacturer Cryomech Inc. The red line shows the measured temperatures of the 50K- and 4K-plate during a cooldown of our cryogenic prototype system discussed in Section 3.3. Cooling power of the Still- (c) and BT-stage (d) versus temperature measured in a BlueFors BF-LD400 system. powers of approximately 20 µW at 20 mK on the BT-stage, of 200 µW at 140 mK on the CP-stage, and of 40 mW at 1.2 K on the Still-stage (Fig. 3.1c) [Krinner19].

3.1.2 Heat Transfer in Solids

Thermal conduction in solids is described by Fourier’s law

˙ q~ = k(T) ~ T, (3.1) − ∇ which states that the heat ﬂux density q~˙ is proportional to the gradient of the temperature T and the temperature-dependent thermal conductivity k(T) [Bergman11]. The thermal conductivity of matter k(T) is a transport property which is characterized by 64 Chapter 3 Cryogenic Quantum Networks

the heat transfer mediated by electrons and phonons. These conduction electrons and phonons are scattered by other electrons, phonons, lattice defects or impurities. This scattering results in a diffuse heat transfer process, which is discussed in Section 3.1.3 for metals, in Section 3.1.4 for insulators and in Section 3.1.5 for superconductors. In typical cryogenic setups, a 1-D approximation can be used to calculate the heat transfer through elongated solids. Integrating Fourier’s law in this 1-D approximation gives

T A Z h Q˙ k T dT (3.2) = l ( ) − Tc with the heat ﬂux Q and a constant cross-section A along length l between a low

Tc and high Th temperature. Equation (3.2) can, for example, be used to calculate the heat load on a cold stage due to thermal conduction in support posts connecting it to a stage at higher temperature. The thermal conductivity of a material can be obtained by measuring Tc and Th and a heater installed at the position of Th so that ˙ Q = Pheater neglecting other types of heat sources, e.g. heater wiring which is discussed in Section 3.1.3 and Section 3.1.4. In another typical example for cryostat design, a tube of perimeter U, length l and wall thickness d is cooled at one position to temperature

Tc and heated along l by a uniform heat rate per area ψ, which can be expressed as

U(l z)ψ dz = Q˙(z) dz = Udk(T) dT. (3.3) −

For a temperature independent thermal conductivity k(T) = k we obtain the temperature at position z ψl ψ 2 ∆T(z) = T(z) Tc = z z (3.4) − dk − 2dk ψ 2 showing a quadratic dependence on the distance, e.g. ∆T(l) = 2dk l . Furthermore, the wall thickness d is the main design parameter to achieve a desired temperature difference ∆T(l). Materials with a high thermal conductivity k have to be selected to keep ∆T(l) small. This example is especially relevant for the 50K-stage of a dilution refrigerator, since it is heated uniformly by room-temperature (TRT = 295 K) radiation 4 ψ TRT and k(T) k for metals between 100 K and TRT (see Fig. 3.2). ≈

Time-dependent heat transfer in solids is modeled by the heat equation, which can be derived from Fourier’s law and the ﬁrst law of thermodynamics,

˙ ˙ ρ(T)ch(T)T (k(T) T) = Ψ (3.5) − ∇ ∇ 3.1 Basics of Cryogenics 65

with the density ρ(T), the speciﬁc heat capacity ch, and internal heat per unit volume ˙ Ψ [Cannon84]. The volumetric heat capacity ρ(T)ch(T) measures the temperature- dependent amount of heat which is stored in an material per volume and per ∆T. For a constant thermal conductivity k(T) k we obtain ≈ k T ˙ ( ) 2 ˙ 2 ˙ T T + Ψ = αd(T) T + Ψ. (3.6) ≈ ρ(T)ch(T)∇ ∇ introducing the thermal diffusivity αd(T) which is a measure of the diffusion rate of heat in solids.

3.1.3 Thermal Properties of Copper

Copper is the typical metal used at temperatures < 4 K due to its high thermal conductivity, e.g. kCu(4.2 K)/kAl(4.2 K) 24 (Fig. 3.2b), and since it does not become superconducting (see Section 3.1.5).≈ In the thermal conductivity comparison, we assumed a residual resistivity ratio for Cooper RRRCu = 200 and 1100 aluminum alloy according to Aluminum Association and refer to later in this Section 3.1.3 for details. For temperatures > 4 K copper is also preferable compared to aluminum if high thermal performance is required, since the ratios of the mean thermal diffusivities R 50 K Cu R 50 K Al R 295 K Cu R 295 K Al 4 K αd dT/ 4 K αd dT 3.2 and 50 K αd dT/ 50 K αd dT 1.25 are advantageous. However, disadvantages≈ of copper compared to aluminum≈ are its high density

ρCu/ρAl 3.3 which leads to higher demands on the support structures, its high ≈ reduction of thermal conductivity by deformation and impurities [Dies67], and the requirement for vacuum annealing to reach high thermal conductivities at low temperatures [Simon92]. For a detailed comparison of aluminum and copper we also refer to Refs. [Woodcraft05a, Woodcraft05b]. In our cryogenic application (see Section 3.3), the highest priority is a high thermal diffusivity over full temperature range 0.01 300 K which led us chose copper on all temperature stage of the cryostat (Fig. 3.2).− This section is therefore focused on copper. However, the general statements are also valid for other metals. In metals, k(T) is dominated by electron conduction since the velocity of the conduction electrons is much larger than the sound velocity of the phonons [Pobell06]. At high temperatures scattering of electrons at thermally excited phonons is the limiting factor of conduction resulting in an approximately constant thermal conductivity kCu(295 K)/kCu(100 K) 0.85. In contrast, at low temperatures the number of phonons is small and scattering≈ at lattice defects or impurities dominates which leads to k(T) = k0 T linearly decreasing with T for T < 5 K. Between these two limits, 66 Chapter 3 Cryogenic Quantum Networks

a) b) 5000 5000 RRRCu ] ] 280 - 1 1000 m - 1 m 140 - 1

- 1 100

1000 80 [ WK Cur100 Al1100 k [ WK 10 Cur300 Al6061 500 c) ]

- 1 -1 Cur100

s 10

2 Al1100 A A -2 100 7 CuOF CuOF [ m 10

NA A d

Cu Cu α 50 OFE OFE 10-3 NA A th. conductivity, k CuETP CuETP diff, 10-4 6 10 15 20 25 35 5 10 20 50 100 300

temperature, T [K] temperature, T [K]

Figure 3.2 a) Measured thermal conductivity k versus temperature T (dots) for Cu-OF, Cu- OFE, and Cu-ETP copper alloys (see text for details) which are annealed (A) or non-annealed (NA). The dashed lines are fits to the data using the model discussed in Ref. [Simon92]. The numbers indicate 4 fitted RRRCu. Comparison of thermal conductivities (b) and thermal diffusivity αd (c) of copper with an RRR of 100 (and 300 in b) and aluminum alloys 1100 (and 6061 in b) in a temperature range between 4 300 K using the National Institute of Standards and Technology (NIST) database [Veres−, Johnson61, Simon92, Touloukian65, Marquardt00, Bradley13]. the thermal conductivity of metals typically exhibits a maximum. In general, the scale of this maximum, at which temperature it occurs, and the thermal conductivity at low temperatures depend strongly on the purity of the metal and the lattice defects introduced during the manufacturing process. For example, we observe from the mea- 6 surements of two Cu-OF samples that a mass fraction of > 20 10− of phosphorus · in copper [Dies67] can reduce kCu for temperatures below 10 K by factors of 35 50 (Fig. 3.2a, red). In addition, cold hardening and deformation introduce lattice defects− which can be recrystallized by special heat treatment, e.g. vacuum annealing, leading to a significant increase of the thermal conductivity at low temperatures [Pobell06]. For example, we observe an increase of the thermal conductivity due to annealing for a Cu-OFE (blue Fig. 3.2a) and a Cu-ETP sample (gray Fig. 3.2a). At low temperatures (impurity scattering) and high temperatures (phonon scattering) the electrical conductivity σe and thermal conductivity are related by the 2 Wiedemann-Franz law k = LTσe with the Lorenz number L = (πkB/e) /3. Due to the strong dependence of the electrical and thermal conductivity at low temperatures on the purity and cold-working/heat treatment, the residual resistivity ratio (RRR)

RRR = σe(TlHe4)/σe(TRT) k(TlHe4)/k(TRT) TRT/TlHe4 (3.7) ≈ 3.1 Basics of Cryogenics 67

4 of the conductivities at the boiling point of liquid HeTlHe4 = 4.2 K and TRT became the standard parameter to characterize the electrical and thermal conductivity of metals (Fig. 3.2a,b). For example, Simon et al. [Simon92] reported an effective fit function for the thermal conductivity of copper between approximately 1 1000 K using the RRR as the only fit parameter. Their study is based on 22 references covering− a RRR range of 19 to 1800 and a temperature range 0.2 1250 K. They conclude that the thermal conductivity of all data could be fitted up to −15% using their fit function. Similar results have ± been found in studies performed at CERN [Verweij07, Manfreda11], are summarized in Brookhaven National Laboratory [Jensen80], and are used in material libraries of ® finite-element simulation software like COMSOL Multiphysics [COMSOL4.312]. The RRR is also used throughout this thesis to parametrize the thermal conductivity of copper (Fig. 3.2a,b). Cu 2 1 The thermal diffusivity of copper αd (T) (γ + β T )− increases at cryogenic temperatures (Fig. 3.2c) since the density of copper is nearly temperature independent, e.g. ρCu(TRT)/ρAl(4.2 K) 0.99 and the specific heat is a sum of electronic Tand 3 ≈ 3 phononic T contributions ch = γT + β T [Pobell06].

Characterization of High-Thermal-Conductivity Materials

To measure the thermal conductivity of highly conductive materials such as copper, we use a typical measurement setup as described for example in Ref. [Ekin06] and sketched in Fig. 3.3. The sample is placed inside a vacuum-tight sample holder which is operated inside a dipper-type (dip stick) cryostat. The dipper-type cryostat consists 4 of a Dewar ﬁlled either with He or liquid nitrogen (boiling point TLN = 77 K) and allows for measurements between approximately 5-30 K and 80-120 K. The vacuum- tight dip stick consists of a stainless tube, in which the sensor and heater wiring is

to RT T1a T2a T3a

Tcold sample heater stainless steel

tube T1b T2b T3b

vacuum-tight sample holder

Figure 3.3 Sketch of the experimental setup to measure high-thermal-conductivity materials in a dipper-type cryostat. The setup consists of a vacuum-tight sample holder made out of a stainless-steel tube and a brass can. The sample is thermalized at position labeled

Tcold and heat can be applied using a heater. Temperatures are measured at three positions along the sample using two temperature sensors at each position. 68 Chapter 3 Cryogenic Quantum Networks

guided, and a brass sample holder at the cold end. We use stainless steel and brass due to their simple machinability, mechanical stability and low thermal conductivity. Between the vacuum-sealed connector at room-temperature and the thermalization point we use constantan looms for temperature-sensor wiring due to their low thermal conductivity. All temperature-sensor wires are thermalized at the lower temperature stage 50-150 mm before the temperature sensor making the heat transfer in the wiring small compared to the one in highly conductive sample. We measure the temperature of the sample at three positions and use two resistive temperature sensors (Lakeshore Cernox® CX-1010) at each position to check for consistency across the piece and between the sensors. These temperature sensors have a sensitivity of 32 0.5 Ω/K for a temperature range of 4.2-77 K according to the manufacturer and are thus− optimized for the thermal conductivity measurements in liquid helium. We use an ac resistance bridge (Lakeshore Model 370 or 372) to measure the sensor resistance via four point measurements. We install a heater (Lakeshore HTR-25-100 ) at one end of the sample and thermalize the other end via a brass block to the cryogenic liquid. The heater is based on a nickel-chromium resistance wire and shows a nearly constant resistance with temperature. We choose copper wires (Lakeshore WHD-30-100) for heater wiring due to their high fuse current of 8.8 A and a small cross-section compared to the sample ( 1/100). We also thermalize these wires approximately 150 mm before the heater. For∼ high conductance materials cooling due to thermal radiation can be neglected ˙ ˙ 5 (Qcond/Qrad 10− , see Section 3.1.7) and good vacuum is obtained by pre-pumping ≈ 4 the dip stick at room-temperature to pressures < 10− mbar. Due to the speciﬁc material choice of the wiring and negligible lateral heat transfer via radiation and convection, Eq. (3.2) can be directly used to obtain k(T) (Fig. 3.2a).

Copper Alloys used in the Presented Cryogenic Network

As detailed in this section and observed in our thermal conductivity measurements, a high-purity copper alloy with clear speciﬁcations of phosphorus is required to obtain high thermal conductivities at cryogenic temperatures. We have chosen Cu-OFE (European Standard Number CW009A) from Aurubis with Cu min 99.99%, and phos- 6 phorus P < 3 10− for all radiation-shields within our cryogenic setup (for details see Section 3.3).· For ﬂexible elements (braids), we use Cu-ETP (CW004A) provided by Druseidt with Cu min 99.90% at 50K-stage and Cu-OF (CW008A) from Tranect with Cu min 99.95% for all other stages. Cu-OF is also used for the shield and plates 3.1 Basics of Cryogenics 69

inside the dilution refrigerator. For future cryogenic network applications, Cu-ETP1 (Cu-NOSV) by Aurubis could be an alternative copper alloy since the manufacture spec- ETP1 iﬁes RRRCu 400 and Cu-ETP1 is for example used at the Cryogenic Underground ≥ Observatory for Rare Events (CUORE) [D’Addabbo18].

3.1.4 Thermal Conductivity and Mechanical Stability of Insulators

In insulators, heat is mostly conducted by phonons. The thermal conductivity k(T) is dominated by phonon-phonon scattering for high temperatures, and by phonon- impurity scattering for low temperatures. In general, k(T) decreases with a weak power law in T at low temperatures. The exact temperature scaling is, however, strongly dependent on the frequency dependence of the involved phonons and the crystal structure of the particular insulator [Pobell06]. The thermal diffusivity αd of insulators is however approximately constant at cryogenic temperatures since the specific heat is also dominated by excitations of phonons leading to k ch. Typically insulating materials [Runyan08, Woodcraft09, Kramer14] used in cryostats are stainless steel (e.g. type 304 [Touloukian65, Mann77] by American Iron and Steel Institute), polymers (PEEK), Polytetrafluoroethylene (PTFE, Teflon), polyamides (Ny- lon [Veres, Johnson61]), polyimides (Vespel), carbon fibers, epoxy glass fibre composites (Vetronite G10), aramid fibers (Kevlar [Ventura00, Ventura09]), glass-ceramics (Macor [Rutt02, Runyan08]) and epoxy-based glues (Stycast [Rondeaux02]). Due to rapid progress of 3D printing technologies over the last decades, special printing-compatible composites have been developed, e.g. Somos® Perform or Accura® Bluestone which have been investigated as cryogenic support structures at CERN [Fessia13] and we discuss our implementation in Section 3.3.2. We have measured the thermal conductivity of Bluestone for which, to the best of our knowledge, no thermal conductivity measurements have been published (Fig. 3.4). We use a power law fit ( T 2.3) in the temperature range below 3.7 K and a polynomial fit to the logarithmic data between 3.7 K and TRT as used in the NIST database [Veres] (Fig. 3.4b blue line). An important property which benchmarks insulating materials for support structures is the yield strength to thermal conductivity ratio rsk(T) = σs(T)/k(T). The yield strength is the point in a stress-strain curve at which a material starts deform- ing non-linearly (elastic limit). rsk for composites and plastics can be larger by fac- PEEK 304ss tors of 10 100 compared to stainless steel. For example, at TRT rsk /rsk 30 − Kevlar 304ss ≈ [Woodcraft09, Touloukian65, Mann77], at T = 1 K rsk /rsk 200 [Ventura00, Macor 304ss ≈ Ventura09], or at T = 0.5 K rsk /rsk 20 [Woodcraft09]. ≈ 70 Chapter 3 Cryogenic Quantum Networks

Measurements of Low-Thermal-Conductivity Materials

To measure the thermal conductivity of insulating materials below 4 K we have built a measurement setup at the BT-stage of a dilution refrigerator (Fig. 3.4a). With this setup we have measured thermal conductivities between 0.6-9 K by operating the cryostat either by only using the pulse tube cooler to cool the system or as a full dilution refrigerator (Section 3.1.1). For low thermal conductivity materials it is critical to use low-conductive wiring since wires can lead to a signiﬁcant heat transfer in parallel to the sample. At T < 10 K niobium–titanium (NbTi) wires are ideal since NbTi becomes superconducting around 10 K leading to a low thermal (Section 3.1.5) and high electrical conductivity. We use NbTi twisted pairs with PTFE core insulation for the temperature sensor and heater wiring. Due to the low thermal conductivity these NbTi wires should be thermalized at several points to ensure that the whole wire becomes superconducting. We use resistive temperature sensors (Lakeshore Ruthenium Oxide Rox™ RX-102B) with temperature range of use 0.01-40 K according to the manufacturer and heaters based on a nickel-chromium resistive wire (Lakeshore HTR-50). To obtain thermal conductivity measurements between 80-295 K we, in addition,

a b 10 ) ] ) 1 - m 1

- 1 WK [ 0.1

2 10 Bluestone - Nylon 10 3 Macor - PEEK th. conductivity, k 304 steel 10 4 - 0.5 1 5 10 50100 300

temperature, T K [ ] Figure 3.4 a) Photograph of the setup to measure thermal conductivities of insulating materials at the BT-stage of a dilution refrigerator. The white part shows a support post of our cryogenic link (see Section 3.3) clamped to the BT-plate. The Cu-OF block on top hosts a temperature sensor (Lakeshore RX-102B) and a heater (Lakeshore HTR-50). b) Measured (dots) thermal conductivities of Bluestone (blue), Nylon (dark red), carbon-ﬁber-reinforced Nylon (light red), and Macor (bright) using the setup shown in (a). The solid lines of 304 stainless steel, PEEK, Macor and Nylon are obtained from Refs. [Veres, Johnson61, Touloukian65, Mann77, Rutt02, Runyan08, Woodcraft09, Kramer14]. The blue solid line is a ﬁt to our thermal conductivity measurements of Bluestone (see main text for details). 3.1 Basics of Cryogenics 71

installed a measurement setup below the 50K-stage of the dilution refrigerator. Similar to the low temperature measurements, we have used low thermal conductivity wiring for the heater (Lakeshore phosphor-bronze WDT-32-25) thermalized approximately ˙wire ˙ins 400 mm apart from the heater to obtain Qcond/Qcond 1/100 and phosphor-bronze ≈ ˙wire ˙ins wiring (Lakeshore 36 AWG) for the temperature sensors (Qcond/Qcond 1/200). We use the same type of heaters and resistive temperature sensors (Lakeshore≈ Cernox® CX-1080 sensors) with a sensitivity of 480-0.5 Ω/K for a temperature range of 20-300 K. In these high temperature measurements, thermal radiation (see Section 3.1.7) is the ˙ ˙ dominant mechanism of cooling. We estimate Qrad/Qcond 5 for measurements with ≈ the temperature at the position of heater around TRT. We thus use an effective model instead of Eq. (3.2) and finite element simulations (COMSOL Multiphysics® ) to obtain ˙ ˙ k. In the effective model we assumed Qcond and Qrad to be independent heat transfer ˙ ˙ ˙ A 4 4 channels and obtained Qtot = Qcond + Qrad = l kf(Th Tc) + µf(Th Tc ). We fitted − − this model to our data assuming k(T) = k to be constant and using kf and µf as fit parameters (Fig. 3.4b).

3.1.5 Thermal Conductivity of Superconductors

To transfer information at cryogenic temperature over distances on the laboratory scale the use of superconducting microwave waveguides is advantageous (see Chapter 4 and Ref. [Kurpiers17]). In this section we discuss the thermalization of superconductors theoretically and based on measurements of a test sample using a similar measurement setup as for insulating materials (Fig. 3.5a). We focus especially on aluminum as a typical material used in commercially available microwave waveguides (see Sec- tion 4.7), but the general conclusions also hold for other conventional superconductors, e.g. Niobium [Mamyia74]. In superconducting metals the number of single electrons decreases exponentially with temperature exp( 1.76Tcrit/T) below a critical temperature Tcrit by forming − Cooper-pairs [Tinkham15]. However, Cooper-pairs do not carry heat since they are in the same low energy state with zero entropy [Pobell06]. Thus, the thermal conductivity of a superconductor is proportional to the product of the thermal conductivity of the ’normal-metal’ single electrons ( T) and their temperature-dependent number

1.76Tcrit/T ks(T) T e− . (3.8) ks(T) becomes smaller than the phononic thermal conductivity for T Tcrit/10 showing that the total thermal conductivity of superconducting metals approaches≤ those 72 Chapter 3 Cryogenic Quantum Networks

a c offset in Q: 0.008 W ) ) 1000 μ Al 4 inches 100 6061 Cu thermalization 10 ]

W 1 [μ

Q 0.100

0.010 b linear fit ) 0.001 ref. Al1100

80 100 150 200 300 400 temperature, T mK [ ] Figure 3.5 a) Photograph of the setup to measure the temperature distribution versus applied heat of a 4 in aluminum rectangular-waveguide WR90Al (see Section 4.2) which is directly screwed to the BT-stage of a dilution refrigerator. b) Photograph showing the thermalization of the WR90Al using copper braids. For technical details we refer to Ref. [Meinhardt18]. c) Measurement results (dots), linear fits (solid lines) and reference data [Phillips59, Pobell06] (dashed line) of the WR90Al heat flow measurements. We first screwed the WR90Al directly to the BT-plate (a, red) and second, used the copper braid thermalization (b, blue). A offset heat load of Qoffset = 8 nW is extracted from the fits and the lowest observed temperatures (squares) are plotted at Qoffset for illustration. of insulators, e.g kAl(80 mK) kPTFE(80 mK) [O’Hara74, Sahling81, Wasserbäch01, ≈ Al Woodcraft05b, Pobell06, Veres] with Tcrit 1.2 K. This low thermal conductivity has to be taken into account for the thermalization≈ of superconducting parts (Fig. 3.5a,b) or by using non-superconducting coatings. In addition, superconductors need to be shielded against external heat loads (e.g. radiation) to avoid large temperature differences below Tcrit/10. The specific heat due to phonon excitations is not influenced by the superconducting phase transition and is T 3 as in the normal state. However, the superconducting state offers an additional degree of freedom for the electrons which leads to an increase of the electronic specific heat at Tcrit. For T < Tcrit the electronic specific heat decreases exponentially as described for the thermal conductivity [Yaqub60, Pobell06]. For alu- Al,s minum the total heat capacity ch (T) increases by approximate a factor of 2.5 at Tcrit [Phillips59, Berg68, Sahling01, Pobell06] leading to an increase in the thermal diffusiv- Al,s Al,s Al,s ity αd (Tcrit). Between Tcrit and Tcrit/10 we obtain αd (Tcrit/10)/αd (Tcrit) 1/5 in- ≈ dicating a lower heat transfer rate in the superconducting state [Sahling81, Sahling01]. 3.1 Basics of Cryogenics 73

3.1.6 Thermal Boundary Resistance

In the previous sections, we discussed the thermal conductance of bulk solids. In cryogenic systems, however, several solid parts are attached to each other to obtain the designed geometry which leads to interfaces between the parts. In this section, we focus especially on metal-metal interfaces. At these interfaces the thermal contact conductance Kc (units W/K) decreases with decreasing temperature according to a power law x Kc T (3.9) with x between 0.75 to 2.5 [Salerno97]. Note that 1/Kc is the thermal boundary resistance Rc. Kc typically starts limiting the heat transfer around T < 20 K [Pobell06]. For

b Cu Cu: 3400 N ) Cu/A Cu:~ 3400 N Cu/Cu:/ 200~ N a ] 10 Cu/Cu:~350 N K ) / Cu/A Cu:~ 200 N W [ 5 Cu/A/Cu:~350 N

F c / / ~

heater T1a T2a 1 Cu100 2 0.5 A 330 mm l =0.5 m T 1b = 4K plate T2b th. conductanc e, K

0.1

5 10 20 30 temperature, T K [ ] Figure 3.6 a) Sketch of the experimental setup for the thermal boundary conductance measurements. The brown parts indicate an adapter plate (lighter) and a counterpart 2 (darker) which we pressed together at surface of A = 330mm using either brass screws (applied force F 3400 N) or a carbon-fiber rod connected to a pneumatic cylinder at the RT-stage to set a≈ defined force up to approximately 350 N. The adapter plate is screwed to the 4K-stage of a dilution refrigerator. We used heaters based on a nickel-chromium resistive wire (Lakeshore HTR-50) and resistive temperature sensors (Lakeshore Cernox® 1010) at the indicated positions. b) Measured thermal boundary conductances (dots) versus mean temperature for copper/copper interfaces without (Cu/Cu) and with (Cu/A/Cu) coating the surfaces with low-temperature grease Apiezon N. The solid lines are power-law fits T x to the data with extracted x between 1.25 to 1.6. The dashed line shows the thermal conductance of a copper rod with RRRCu = 100, cross-section A, and length l = 0.5 m illustrating the magnitude of the contact conductance. 74 Chapter 3 Cryogenic Quantum Networks

2 example, we measured contact conductance across a S = 330 mm surface between two copper parts of 0.1-1 W/K at around 5 K (Fig. 3.6) which is comparable to the rod conductance of a bulk copper rod of kCu (5 K) 0.5 W/K with the same S, l = 0.5 m, ≈ RRR = 100. Furthermore, Kc increases with applied force since an increasing force leads to a larger, effective surface area for the heat transfer. The exact scaling with temperature and applied force, and the order of magnitude of Kc depend on the used metals and the surface finishing. In general, a high thermal conductance across a contact can be achieved by increasing the effective contact area at cryogenic temperatures. This involves two aspects: First, the different thermal expansion coefficient of metals and screw materials have to be considered. For example, brass screws are ideal to press two cooper parts since the thermal expansion coefficient of brass is more negative than that of copper; however aluminum compresses more than brass making more sophisticated designs necessary to thermalize aluminum at cryogenic temperatures [Pobell06, Meinhardt18]. Second, the effective area can be increased by using conforming coating which fills the micro-pores of the surfaces. For example, coating surfaces with low-temperature grease, e.g. Apiezon N, offers high conductance for aluminum-aluminum, copper-copper interfaces for temperatures below 20 K and makes the contact conductance less dependent on the applied force (see Fig. 3.6b and Ref. [Salerno97]).

3.1.7 Thermal Radiation

In addition to conduction through solids, heat is also transferred via thermal radiation. Thermal radiation is a significant source of heat transfer for temperatures > 50 K since it scales with T 4 and can also lead to significant temperature differences in low conductive materials at lower temperatures. In this section, we introduce how we model thermal radiation in our cryostat design and discuss improvements using multilayer-insulation (MLI) to reduce radiation load based on Ref. [Parma14]. All surfaces at finite temperature emit energy which is in general a function of the wavelength. We first use the common model of an idealized black-diffusive body which absorbs all incoming radiation and emits it uniformly in all directions. Using this black-diffusive body model, the total emitted power per unit area is obtained by integrating the intensity of the emitted radiation I(ν), which is given by Planck’s law, over the whole frequency spectrum

Z ∞ 4 φb = I(ν) dν = σSB T (3.10) 0 3.1 Basics of Cryogenics 75

8 2 4 with the Stefan-Boltzmann constant σSB = 5.67 10− W m− K− . Real surfaces, however, emit only a fraction of the black body radiation· and are characterized by the emissivity ε(T) = φ(T)/φb(T) 1 which is equal to the absorptivity of the body a. We obtain in this so-called ideal≤ gray body model

Φ φ ε T σ T 4 (3.11) = S = ( ) SB with the total emitted power Φ and surface area of the body S. In the following we assume opaque bodies in our models which do not transmit radiation and obtain a + r = ε + r = 1 with the diffusive reﬂectivity r.

The emissivity ε(T) is, in general, a temperature dependent material parameter which is also strongly dependent on the surface treatment. For example, cleaned metal surfaces show low emissivities: copper εCu(TRT) 0.02-0.1, εCu(80 K) 0.06, ≈ ≈ εCu(4 K) 0.02 or aluminum εAl(TRT) 0.02-0.15, εAl(80 K) 0.08 [Estalote77, ≈ ≈ ≈ Parma14]. The range at TRT indicates the importance of surface treatment and the purity of the material. The speciﬁed minimum values are obtained from 99.999% pure copper and aluminum samples directly after mechanical and electrolytic polish- ing [Estalote77]. The maximum values and the values at cryogenic temperatures correspond to typical values observed at CERN [Parma14]. Metal oxides or insulators have typically high emissivities, e.g. copper oxides εCuOx(TRT) 0.6, oil paints ≈ εop(TRT) 0.9. The high ε of oxides shows that oxidizing metals are not ideal for long-term≈ reliable, low-emissivity surfaces and should be covered by e.g. multi-layer insulation if low ε is essential for a cryogenic system. Due to the high ε of insulators any oil, grease or non-metalized tape should be avoided on surfaces which are exposed to radiation and properly cleaned on low emissivity surfaces.

Next, we discuss the heat exchange between two surfaces, which is the typical conﬁguration in cryostats, based on the radiosity method. The radiosity method can also be generalized to multiple surfaces and provides analytical solutions for simple geometries. An adaptation of this method is also used in ﬁnite element solvers to simulate heat transfer via conduction and radiation in arbitrary geometries simultaneously [COMSOL4.312]. The radiosity, which is the radiated power from a body, is given by

4 4 J = rG + εσSB T = (1 ε)G + εσSB T (3.12) − with the irradiation G and the reﬂected power rG. The irradiation on surface Si is 76 Chapter 3 Cryogenic Quantum Networks

given by 2 X Si Gi = Sj Fji Jj (3.13) j=1 with the geometric view factors F . The view factor F F is deﬁned as the ji ji = Sj Si fraction of radiation leaving surface j and reaching surface i. The→ view factors obey P the reciprocity relation Si Fi j = Sj Fji and j Fi j = 1. Using the reciprocity relation we obtain 2 X Gi = Fi j Jj. (3.14) j=1

Furthermore, for all convex surfaces Fii = 0 holds, since no radiation emit from a convex surface will reach it again. For two surfaces we obtain the two equations

4 J1 = (1 ε1)(F11J1 + F12J2) + ε1σSB T1 − 4 J2 = (1 ε2)(F21J1 + F22J2) + ε2σSB T2 (3.15) − and the total net power exchange

Φ21 = F21S2J2 F12S1J1 = F12S1(J2 J1) (3.16) − − using the reciprocity relation. For a convex surface S1 enclosed by a second surface S2 we obtain 4 4 4 4 σSB(T2 T1 ) σSB(T2 T1 ) Φ21 = S1 − = S1 − (3.17) 1 F 1 1 1 S1 1 1 ε1 + 21( ε2 ) ε + S ( ε ) − 1 2 2 − assuming two concentric inﬁnite cylinders with F S1 in the last relation. Equa- 21 = S2 tion (3.17) shows that the emissivities of both surfaces affect the net heat Φ21 with

1/ε2 being scaled by F21 < 1. Assuming a dilution refrigerator with an aluminum vacuum can and cooper radiation-shields with Si/Sj = ri/rj = 3/4, we obtain φRT,50K = 5-20 W/m2 for the minimum and maximum emissivity values speciﬁed above. For 2 surface areas ( 1.5 m ) of typical dilution refrigerators, φRT,50K is comparable to the cooling power∼ on the 50K-stage showing the importance of using low emissivity sur- 2 faces or multi-layer insulation on this stage. We obtain φ50K,4K = 0.004-0.01 W/m for max cooper radiation-shields and T50K = 50 K using the minimum and maximum emissivity max values speciﬁed above. However, if the 50K-stage is at temperatures T50K 80 K, e.g. for a larger dilution refrigerator, and the copper radiations shields are oxidized,≈ we 80K 2 estimate φ50K,4K 0.3 W/m with ε 0.2, which is comparable to the cooling power ≈ ≈ 3.1 Basics of Cryogenics 77

of the 4K-stage for surface areas on the order of 1 m2. 4K-surfaces should thus be covered by multi- or single-layer insulation to avoid performance degradation over time in large cryogenic systems as for example, implemented at CERN [Parma14]. 4 2 3 2 φ4K,Still = 0.2-0.5 10− mW/m and φStill,BT 0.6-2 10− µW/m are negligible compared to the respective· cooling power. However,≈ our experiments· with superconductors

(Fig. 3.5c) show that φStill,BT on the order of nW leads to a signiﬁcant temperature sc sc difference of ∆TBT = Tsc TBT = 70-80mK. This large ∆TBT illustrates the requirement of a radiation shield at the− BT-stage if bulky superconductors are to reach temperatures close to TBT.

Multi-Layer-Insulation

Multi-layer-insulation (MLI) consists of thin reflective layers, e.g. double-side alu- minized polyester foil, interleaved with insulating spacers, e.g. polyester or glass fibre foils. MLI in cryogenic setups has been studied extensively over the last decades with respect to the radiative and conductive heat transfer rate in MLI [Mikhalchenko85, Krishnaprakas00], the heat flux between TRT and TlN in dependence of the number of layers [Shu86, Bapat90a, Bapat90b, Hofmann06], geometry of the wrapped cryostats [Ohmori05], and the consequences of degraded vacuum [Sun09] have been investigated. Here, we present an effective engineering-model of MLI [Parma14] and report typical heat flux values and design criteria for MLI which we use in the design of a cryogenic system for cryogenic networks (Section 3.3). Heat in MLI is transferred by thermal conduction and radiation across the layers

cc Ts + Tout cr 4 4 q˙MLI = Tout Ts + Tout Ts (3.18) N + 1 2 − N + 1 − with the temperature of the cooled shield Ts and of hottest outermost MLI layer Tout, and two effective parameters cc and cr. Depending on the structure of the MLI the radiative or conductive transfer dominates. In detail, the heat transfer is dominated by radiation for a low density of layers and by conduction for a high density with an optimal density of 15-25 layers/cm [Bapat90a]. In typical cryogenic setups [Parma14] 20-40 layers of MLI are used between the RT- and 50K-stage. In addition, it is recommend by MLI manufactures, e.g. RUAG or Hutchinson, to use several MLI blankets with 10-15 layers each to obtain a good coverage of the surface and to reduce edge effects by covering the edges of lower layers. According to Ref. [Parma14] heat ﬂuxes on the 50K-stage 2 of φRT,50K = 1-2 W/m can be obtained, which we also veriﬁed experimentally in our cl 2 cryogenic system (φRT,50K = φMLI 1.5 W/m , see Section 3.3.4). ≈ 78 Chapter 3 Cryogenic Quantum Networks

3.2 Experimental Setup for Circuit QED

To operate circuit QED systems at the base temperature (BT) stage of a dilution refrigerator we need to control and readout the state of quantum systems with electronics at higher temperature stages while protecting them against environmental noise. We use a room-temperature microwave setup in our experiments, which implies that input signals to and output signals from the quantum systems need to be transferred between different temperature stages. Realizing these microwave cables requires similar considerations as discussed in the previous sections for heat transfer in solids and radiation protection. In addition, we consider active heat load due to dissipated power of the incoming signals in the design of these microwave lines. We first focus on the used materials and the resulting passive heat load on each temperature stage including our direct-current (dc) wiring. Next, we discuss the tradeoff between thermalizing radiation in the microwave cables via attenuators below the single photon level at the BT-stage and significant active heat load on any temperature stage. We continue with describing our cryogenic detection lines and focus on the detection efficiency of a chain of amplifiers. We conclude with explaining our room temperature setup. We discuss the engineering of cryogenic setups for circuit QED in our publication [Krinner19] including an outlook on scaling to hundreds of transmon qubits.

3.2.1 Passive Heat Load

Support structures of the cryogenic systems and microwave coaxial cables, which consist of insulating materials, are required between the different temperature stage ( Section 3.1.4). We use semi-rigid coaxial cables with stainless-steel outer and center conductor with a solid PTFE (sPTFE) dielectric (Carlisle UT85-SS-SS (CC085SS)) and an outer diameter 2.2 mm (0.085 in) if low signal loss in the cable is not required. CC085SS are therefore used for all input lines between the RT- and BT-stages and for output lines between the 4K- and RT-stages. The attenuation constant of CC085SS is 8-10 dB/m at TRT,TlN and TlHe4 for a frequency of 6 GHz according to our measurements (Section 4.6). Based on the thermal conductivity of stainless-steel and PTFE we estimate that a CC085SS has little heat load contribution to the temperature stages if appropriately designed [Krinner19]. To transmit signals with low loss between different stages below the 4K-stage we use microwave cables with niobium-titanium (Nb-47 weight percent (wt%) Ti) outer and center conductors and low density PTFE (ldPTFE) dielectrics (Keycom Corporation 0.085" NbTi-NbTi (CC085NbTi)). These CC085NbTi provide low attenuation of microwave signals < 0.02 dB/m (see Fig. 4.9) and similar 3.2 Experimental Setup for Circuit QED 79

heat loads as CC085SS [Krinner19]. To transmit signals with low loss at the same temperature stage, e.g. at BT-stage, we use coaxial cables with a copper outer conductor, a silver plated copper clad steel (SPCW) (per the standard specification for silver-coated, copper-clad steel wire (ASTM B-501)) center conductor, and a sPTFE dielectric. For example, Carlisle UT85-TP (CC085Cu) with an outer conductor diameter of 2.2 mm (0.085 in) are typically used due to their convenience for installation, e.g a minimum inside bend radius of 1.3 mm compared to 30 mm for CC085NbTi. We use direct-current (dc) wiring to bias our cryogenic amplifiers at the 4K-stage (Fig. 3.7) and to supply a magnetic flux to the SQUID-based transmon qubits by bi- asing superconducting coils. We use 90 µm-diameter copper twisted pairs in woven looms between the RT- and 4K-stage for the amplifier bias which lead to heat loads of approximately 0.1% at the 50K-stage and 3% at the 4K-stage compared to the cooling powers. Theses copper twisted pairs could be replaced by 36 AWG phosphor-bronze wires which would reduce thermal heat loads to < 0.1% compared to the cooling powers. The superconducting coils are fed via 100 µm-diameter Beryllium-Copper (BeCu) twisted pairs in woven looms between the RT- and 4K-stage (heat loads: 0.02% at 50K- and 0.03% at 4K-stage), and NbTi twisted pairs in individually CuNi shields between the 4K- and BT-stage (heat loads: 0.002% at Still-, 0.02% at CP-, and 0.004% at BT- stage). We filter all wires at RT with L-type low pass filters (Tusonix 4601-050LF) with 15 dB attenuation at 30 kHz [Huthmacher13, Lang17]. The coil wiring is further filtered− at the 4K-plate using a 2nd-order RC-filter (Aivon Therma-uD-25G) with a cut-off frequency of 16 kHz and 16 dB attenuation at 30 kHz. All dc wires are thermalized at each temperature stage− by clamping them between to copper plates.

3.2.2 Input Microwave Cabling

In this section, we focus on microwave input lines which are used to realize single- qubit gates between transmon states, all-microwave two-qubit and transmon-resonator gates and to apply drive tones at resonators inputs, e.g. for transmon qubit readout. These input lines have typically a large bandwidth of 3-12 GHz. In the design of the attenuation along these drive lines two competing aspects need to be considered: Enough power needs to be available at the sample to perform the desired operations on transmons and resonators, while thermal radiation propagating along these cables needs to be cooled, to reduce qubit dephasing because of thermal photons in the resonators [Gambetta06, Clerk07, Sears12, Yan16a, Yeh17] and the creation of quasiparticles which can lead to energy relaxation [Catelani11, Serniak18]. 80 Chapter 3 Cryogenic Quantum Networks

-10 LO -3 -3 IF νqt -20 -10 -10 -10 ADCs & FPGA AWG -3 -10 -10 ν -10 f0g1 -3 -3 -3 -10 -10 -10

MWG RT 50K BeCu

CC85SS CC85SS 4K -20 -20 -20 -20 -20 HEMT HEMT Still CC85SS CC85SS CC85SS CC85SS CC85SS NbTi CC85NbTi CC85NbTi CP -20 -20 -20 -20 -20

CC85NbTi CC85NbTi BT -3 -3 -20 -20 -20

cQED photon res. transmon

JPA

-20 ... readout resonator signal generator IQ mixer directional coupler -20 attenuator amplitude and amplifier circulator filter phase control dc source isolator switch dc block

Figure 3.7 Schematic of the setup used for our circuit QED experiments (e.g. Chapter 5). The blue shaded elements indicate the input lines, the yellow shaded ones the detection lines. The dashed lines mark the stages of the dilution refrigerator. At the input lines, the trans-

mon (νqt) and f0g1-transition (νf0g1) drives are combined. We use either single side-band modulation with IQ-mixers driven by a local oscillator (LO) and an arbitrary waveform generator (AWG), or alternatively, directly synthesize drive-pulses by a high-bandwidth AWG (not shown). The input lines are thermalized on each temperature stage and are attenuated at the 4K-, CP- and BT-stage. The frequencies of the transmon and Josephson parametric amplifier (JPA) can be tuned using superconducting coils. All detection lines consist of isolators and a bandpass-filter at the BT-stage, another isolator at the CP-stage, a cryogenic amplifier (HEMT), RT amplifiers and filters, and an analog down-conversion to an intermediate frequency of 250 MHz. The down-converted signal is lowpass-filtered, digitized using an analog-to-digital converter (ADC) and recorded using a field-programmable gate array (FPGA). The JPA is pumped by a LO and the reflected pump signal is canceled at a directional coupler using amplitude and phase controlled destructive interference. 3.2 Experimental Setup for Circuit QED 81

To determine transmon qubit states using dispersive readout schemes, we drive the Purcell-ﬁlter input via a microwave drive line and estimate the steady-state input power at the sample level based on Eq. (2.78) to be approximately Pres 100 dBm. π ≈ − To excite transmons with a Gaussian-shaped DRAG Rge pulse of width 3.3 ns and total pulse duration of 20 ns a maximal power of

max lim 2 Pπ ħhωge T1 ΩRabi/4 74 dBm (3.19) ≈ ≈ − is required at ωge/2π = 6.3 GHz [Krinner19]. ΩRabi is the Rabi rate and we assumed the energy relaxation time limit due to the capacitive coupling of the transmon to the lim drive line T1 300 µs [Carmichael02, Pechal16a]. According to our experiments ≈ (see Ref. [Magnard18]) and in line with perturbation theory calculations and master equation simulations [Zeytinoglu15˘ ], driving the f 0 g1 transmon-resonator | 〉max ↔ | 〉 transition (see Section 2.2.8) requires approximately Pf0g1 48 dBm = 16 nW at the max max ≈ − sample. Based on Pres, Pπ and Pf0g1 a higher attenuation could be used in resonator drive lines than in transmon drive lines. In our experimental setup (Fig. 3.7) however we chose to use the same attenuation to keep the number of different input line conﬁgurations small. Based on these active heat load considerations, we estimate a max maximal attenuation aBT 30 dB at the BT-stage in the input drive lines to keep ≈ − max the dissipated power small compared to the cooling power. In the estimate of aBT we assume a duty cycle of 10% obtained by actively reseting the qubit [Magnard18].

For modeling the thermal radiation propagating along coaxial cables, we consider a quasi one-dimensional model since their diameter is less than the wavelength. The photon ﬂux spectral density

1 S T, ω n T, ω (3.20) ( ) = ħhω = BE ( ) e kB T 1 − of blackbody radiation in such one-dimensional cables is equal to the Bose-Einstein distribution [Eichler13, Norris17] in units of photons per hertz frequency interval per second which is also referred to as Johnson-Nyquist noise in electronics [Johnson28, Nyquist28]. nBE (T, ω) can also be understood as the temperature-dependent photon occupation number of a mode at frequency ω. At room-temperature nBE (TRT, ωge) 940 shows the need to attenuate thermal radiation also in the microwave cables. An≈ attenuator at stage i can be modeled as a beam splitter which attenuates the incoming radiation from stage i 1 by a, e.g. a = 0.01 ( 20 dB) ,and adds 1 a blackbody − − − 82 Chapter 3 Cryogenic Quantum Networks

att radiation at its temperature Ti

att ni(ω) = ai ni 1(ω) + (1 ai)nBE(Ti , ω) (3.21) − − with the photon occupation number ni at stage i. Based on these thermal considerations all attenuation should be placed at the BT-stage to minimize nBT(ω) due to att the added blackbody at Ti . For example, to obtain a photon occupation number 3 nBT(ωge) < 0.3 10− (corresponding to Teff = 40 mK) an attenuation aBT 65 dBm · max ≈ − is required, 3 orders of magnitude above aBT obtained from heat load considerations. Therefore, cascaded attenuators are necessary to keep the dissipation at the individual temperature stage small compared to the cooling power [Krinner19]. For high attenuations a 1 the ratio r(Ti 1, Ti) = nBE(Ti, ωge)/nBE(Ti 1, ωge) between the incoming blackbody radiation from− stage i 1 and the blackbody− radiation of stage i is − a good design value for the attenuation ai of stage i. Choosing ai r(Ti 1, Ti) results in attenuating the incoming radiation to a similar level as the blackbody≈ − radiation at stage i. We obtain r(TRT, T40 K) 9 dB, r(T40 K, T3 K) 11 dB, r(T3 K, T1 K) 5 dB, ≈ − ≈ − ≈ − r(T1 K, T100 mK) 18 dB and r(T100 mK, T40 mK) 21 dB showing the need to install 20 dB at the CP-≈ − and BT-stage. Adding attenuation≈ − at the Still-stage is not necessary due− to the small temperature difference to the 4K-stage. We also chose to combine the attenuation of the 50K- and 4K-stages by placing 20 dB at the 4K-stage to simplify the cabling (Fig. 3.7). In addition, we describe the attenuation− in coaxial cables by a continuous beam splitter model [Norris17] which adds approximately 10 dB of attenuation for a system (BF-LD-400) of distance 1.1 m between the RT- and− BT-plate. ∼ In conclusion, we use approximately 70 dB attenuation in our transmon and resonator drive lines distributed over the individual− temperature stages. We thermalize the attenuators and the CC085SS wiring with custom-made Cu-OF adapters to their respective temperature stage. However, it is still under research how well attenuators thermalize the center conductor of the coaxial cables through the resistive divider realized by the attenuators, and if other attenuator designs can lead to improvements [Yeh17]. In addition, commercial attenuators are typically only specified up to a certain frequency, e.g. XMA-2082 to 18 GHz. For circuit QED experiments, however, radiation above the superconducting band gap, e.g. 82 GHz for aluminum, is relevant due to the creation of quasi-particles which can lead to energy relaxation of superconducting qubits [Catelani11, Córcoles11, Pop14]. Therefore, infrared-photon blocking filters (e.g. based on Eccosorb®), which have high attenuation coefficient in the relevant frequency band, can be installed in the microwave input lines [Barends11, Norris17]. 3.2 Experimental Setup for Circuit QED 83

3.2.3 Cryogenic Detection Lines

In this section we discuss the cryogenic detection circuits of our experimental setup and focus on the detection efficiency of a chain of amplifiers. In cryogenic output lines, signals are routed and amplified between the quantum system and the room- temperature amplifiers and digitizers (Section 3.2.4). We design our output lines (Fig. 3.7) especially for fast and high-fidelity single-shot qubit readout [Walter17] and for the detection of the temporal shape of emitted photons from transmon-resonator systems [Eichler13, Pechal14, Pechal16a]. In our qubit-readout detection lines, we use near quantum-limited Josephson parametric amplifiers (JPAs) [Yurke96, Clerk10, Eichler14a] at the BT-stage because of their excellent noise performance as the first amplifier of the detection chain. We typically obtain a 3 dB bandwidth of 18-32 MHz for a gain of gjpa = 19-26 dB (Fig. 3.7) [Walter17, Kurpiers18]. We operate our lumped-element amplifiers [Eichler14a] in reflection by inserting an isolator and a circulator ( 17 dB isolation each) before the amplifier in the output line. In addition, we install a− 20 dB directional coupler at the input of the JPA (Fig. 3.7). Using this directional coupler− we apply a microwave pump tone at the JPA input to achieve parametric amplification [Eichler13]. The second coupling-port of the directional coupler is used to interferometrically cancel the pump tone reflected from the JPA. The cancellation of the reflected pump tone is required to reduce leakage to the sample which can lead to dephasing of the transmon [Ithier05] and to avoid saturating the next amplifiers in the output line in forward direction. For cancellation and pump input lines, we use 20 dB attenuation at the 4K- and CP-stage, and 3 dB at BT-stage, which sums to 63−dB including the attenuation in the directional− coupler. However, the 99% of the− pump input signal are not dissipated in the directional coupler but routed towards the second coupling-port and are mainly absorbed in the attenuators installed at higher temperature stages. We add an additional isolator after the circulator used for the JPA at the BT-stage followed by a reflective bandpass filter to suppress noise outside the pass band of 4-8 GHz. Installing a third isolator at the CP-stage results in a total isolation of 51 dB against noise propagating backwards from the detection chain towards the sample.− At the 4K-stage, we use high- electron mobility transistor amplifiers (HEMT) amplifiers [Pospieszalski88, Lang17] with approximate gain of 45 dB, bandwidth of 4-8 GHz and added noise photon number noise nhemt = nBE(Themt , ω) 6 according to the manufacturer. ≈ In the photon detection lines, we install three isolators at the BT-stage and one at the CP-stage resulting in a total isolation of 68 dB. We also place a reflective bandpass − 84 Chapter 3 Cryogenic Quantum Networks

ﬁlter with a pass band of 8-12 GHz at the BT-stage. At the 4K-stage, we use a HEMT with 40 dB gain, a bandwidth of 1-12 GHz and nhemt 9. In all output lines, we use CC085Cu (Tab. 4.1) at the BT-stage, CC085NbTi between≈ the BT-CP and CP-4K-stages, and CC085SS between the 4K-RT-stages. All mentioned components are thermalized with custom made Cu-OF adapters to their respective temperature stage.

Cascaded Amplifiers and Detection Eiciency

Our output lines can be modeled as a cascaded system of ampliﬁers and attenuators using Friis’s formula [Pozar12]. Similar as discussed in the Section 3.2.2 for attenuators, we obtain for an ampliﬁer with gain Gamp and noise number namp

out in s = Gamps (3.22) out in n = Gamp(n + namp) (3.23) for the signal sin(out) and noise nin(out) at the input (output). To illustrate the total detection efﬁciency as the input-to-output signal-to-noise-ratio (SNR) of a qubit readout detection line for circuit QED experiments, we consider a quantum-limited JPA with th added thermal noise njpa followed by a conventional low noise semiconductor ampli-

ﬁer (HEMT). In addition, we consider attenuation between the sample and JPA asj at sj jh temperature T and between the ampliﬁers ajh at an effective temperature T . Using Friis’s formula, we obtain the total noise photon number of the cascaded system

1 a out sj th nhemt jh jh nhemt = GjpaajhGhemt nsig + (1 asj)nBE(T , ω) + njpa + + − nBE(Teff, ω) . − ajhGjpa ajhGjpa (3.24)

out Equation 3.24 shows that nhemt is proportional to the total gain of the ampliﬁer chain

GjpaajhGhemt and that the noise of the second amplifier nhemt and the added thermal jh noise between the amplifiers (1 ajh)nBE(Teff) are divided by the effective gain of the − first amplifier ajhGjpa. Next, we make assumptions of low temperatures, high gains and low losses: First, we assume that the temperature of the JPA and the connection th sj between the sample and the JPA are at BT (njpa 1, nBE(T , ω) 1). Second, we assume that the gain of the JPA Gjpa is sufficiently large (typically 20 dB) and that ∼ the attenuation between the JPA and the HEMT ajh is sufficiently small, to obtain jh (1 ajh)nBE(Teff, ω)/(ajhGjpa) 1. Under these assumptions, we obtain − 3.2 Experimental Setup for Circuit QED 85

out nhemt nhemt GjpaajhGhemtnsig 1 + (3.25) ≈ ajhGjpansig and for the total detection efﬁciency

out out SNR s nsig 1 ηamp hemt hemt a η η (3.26) tot = out = out sj nhemt = loss amp SNR n ssig 1 sample hemt ≈ + ajh Gjpa nsig with ηloss = asj and ηamp characterizing the detection efficiency of the amplifier chain. Equation (3.26) illustrates especially the importance of a low attenuation between the sample and JPA asj and a large effective JPA gain ajhGjpa, to obtain a high total amp detection efficiency ηtot . Using the JPA as a phase-insensitive amplifier, which amplifies both field quadratures (2Q) simultaneously, adds at least vacuum noise to the signal, to avoid that the bosonic commutation relation is violated [Clerk10, Eichler13]. We assume vacuum noise on the signal and account for the added vacuum noise by the amplifier 1 1 with nsig = 2 + 2 = 1. For a phase-insensitive amplification, we thus obtain the total detection efficiency of the amplifier chain

2Q 2Q 1 η = ηlossη = ηloss n . (3.27) tot amp 1 hemt + ajh Gjpa

If the JPA is used as a phase-sensitive amplifier, we ideally add no additional noise in ps the amplified quadrature (1Q), and obtain twice the amplitude gain, i.e. Gjpa = 4Gjpa in power [Clerk10, Eichler13]. Assuming the signal only containing vacuum noise 1 1 in the amplified quadrature nsig = 2 /2 = 4 and accounting for noise in the single quadrature only nhemt nhemt/2, we obtain →

1Q 1Q 1 η = ηlossη = ηloss n . (3.28) tot amp 1 hemt + 2ajh Gjpa

The full detection chains (Fig. 3.7) have additional amplifiers at room-temperature amp which can be added to ηtot by replacing nhemt nhemt + nRTamp/ahrGhemt + ... with- amp → out changing the structure of ηtot . These room-temperature amplifiers do not con- amp tribute significantly to ηtot in detection lines since nRTamp/GJPAajhGHEMT ahr is typically small. However in the photon detection lines which do not contain a JPA, noise low-noise room-temperature amplifiers are required (TRTamp 100 K) [Krinner19] to keep their contribution small. Details on the experimental≈ techniques which are typically used to measure ηloss and ηamp can be found in Refs. [Eichler13, Walter17, 86 Chapter 3 Cryogenic Quantum Networks

Bultink18, Heinsoo18, Li19]. We obtain total measurement efﬁciencies for a single amp amp quadrature ηtot,1Q 0.66-0.75 [Walter17] and ηtot,2Q 0.6 [Kurpiers18] for our full ≈ ≈ amp detection chains above other reported measurement efﬁciencies of ηtot,2Q 0.17-0.52 ≈ [Roch14, Jeffrey14, Bultink18, Heinsoo18, Kono18].

3.2.4 Room-Temperature Setup

Similar to the cryogenic microwave cabling the room-temperature setup can also be split into an input control section and an output detection part (Fig. 3.7). For the control of the circuit QED systems we generate continuous and rectangular envelope pulses using microwave signal generators (MWG, Rohde&Schwarz SGS100A). These MWGs output signals up to 12 GHz and we synchronize several instruments using a 1 GHz reference signal to obtain a high phase-stability. To generate arbitrary temporal pulse-shapes we use arbitrary waveform generators (AWG). We either up convert signals from an AWG (Tektronix AWG5014) with a typical bandwidth of 480 MHz (sample rate 1.2 GS/s) using single side-band modulation with in-phase (I) and quadrature (Q) mixers driven by a MWG (local oscillator, LO), or we directly synthesize signals using an AWG (Tektronix AWG70002) with a high bandwidth of 10 GHz (sample rate 25 GS/s). We use a microwave switch to bypass the IQ-mixer for continuous-tone measurements and mixer calibration. Details on the implementation of the up-conversion circuit and IQ-mixer calibration are described in Refs. [Baur12b, Abadal14, Berger15a, Businger15]. We use room-temperature amplifiers with a gain of 24 dB over a bandwidth 0.7 18 GHz to obtain a sufficient drive power for the f0g1-transition (Fig. 3.7). The superconducting− coils are biased with battery-based voltage sources isolated from all ground-references to obtain a low-noise output (Stanford Research Systems SIM928). On the detection part, we amplify and down-convert the measurement signal to obtain the voltage amplitudes and frequencies which are easily digitized using an analog-to-digital converter (ADC). For this purpose, we use two high-frequency, low- noise amplifiers with 30 dB gain in the range from 4-12 GHz and noise temperatures of 75 K and 230 K. Next, we down-convert the signal to an intermediate frequency (IF) of 250 MHz using a mixer (Fig. 3.7). The down-converted signal is further amplified with a low-frequency amplifier with 24 dB gain. For better impedance matching we use attenuators between the amplifiers and the mixer. Low- and high-pass filters are installed to suppress noise at frequencies outside the detection band and to avoid back-folding of these frequency components [Eichler13, Businger15]. The amplified 3.3 Cryogenic Networks 87

and down-converted signal is digitized by an ADC (abaco systems FMC110) with a sampling rate of 1 GS/s and a maximal voltage range of 1 V. The digitized signal is processed by a ﬁeld-programmable gate array (FPGA, Xilinx± Virtex 6) by which the data are digitally down-converted to dc and the corresponding I and Q quadrature values of the signal are recorded using custom-made ﬁrmware [Lang14, Salathé18b, Salathé18a, Reuer18]. In summary, we realize a heterodyne detection chain with a two- step down-conversion [Bozyigit11, Lang14]. Our experimental setup is controlled and the FPGA-processed data saved with a custom-made software (SweepSpot, National Instruments LabVIEW based) [Storz16]. Furthermore, we automated our single- and two-qubit gate calibrations, the active reset of our transmons [Magnard18] and our remote quantum communication experiments using a software (QubitCalib) which combines SweepSpot with the generation of the experimental sequences and analysis (Wolfram Research Mathematica based) [Landig13, Heinsoo13, Storz16].

3.3 Cryogenic Networks

To host quantum networks for distributed quantum computation and to perform loophole- free Bell tests with superconducting circuits, advanced cryogenic networks have to be developed and characterized. In these cryogenic networks quantum systems can be located at cryogenic nodes which are interconnected by cryogenic links. In this section, we discuss our modular design of a cryogenic network, which can be operated at millikelvin temperatures, and present experimental characterizations of two prototype systems with lengths of 4 m and 10 m. We start by detailing the distance requirements to perform a loophole-free Bell test with superconducting circuits.

3.3.1 Cryogenic Requirements for a Loophole-Free Bell Test with Superconducting Circuits

In this section we discuss the timing requirements and the corresponding distance to close the locality loophole in experiments with superconducting circuits. We also specify our thermal design goals along the cryogenic link. Closing the locality loophole requires that the two parties choose free measurement settings and complete the measurements of the qubits faster than a potential communication at the speed of light can occur between the two sites (Section 2.1.4 ). To obtain freely chosen measurement settings we plan to use a quantum random number generator (RNG), which requires approximately 15 ns to generate fresh random 88 Chapter 3 Cryogenic Quantum Networks

a) b) distance, lBell [m] nB 0.010 0 10 20 30 0.001 ϕ RNG Rge out ADC 10-4 sw in sig. int. 10-5 WR90Al

noise photons, n CC85NbTi 0 20 40 60 80 100 40 80 120 160 200 240 time, t [ns] temperature, T [mK]

Figure 3.8 a) Sketch of the estimated time t and corresponding distance lBell required to perform a loophole-free Bell test with superconducting circuits (see Section 2.1.4). We assign a time of 15 ns for the random number generation (RNG), 6 ns to operate the microwave φ switch (sw), 10 ns for Rge to set the measurement basis, 2.8 ns to the input cabling (in) signal propagation delay, 9.6 ns to the output cabling delay including ampliﬁcation and down conversion (out), 50 ns to the signal integration of the qubit readout (sig. int.), and 7 ns to digitization of the signal on the ADC. The estimates are based on Refs. [Schär17, Walter17]. b) Thermal noise photon number nnB at node B versus constant waveguide temperature T for a 30 m waveguide. We assume the ﬁeld to be thermalized by an isolator at node A

to 20 mK, a signal frequency of 8.35 GHz, and an attenuation constant αatt of 0.008 dB/m (0.024 dB/m) for WR90Al (CC85NbTi) (see Chapter 4).

numbers [Abellán15] (Fig. 3.8a). The selection of the measurement basis is performed by triggering a fast microwave switch using the output of the RNG. The switch has a φ switching time of approximately 6 ns to apply or not apply a Rge gate to change the measurement basis (minimal pulse duration 10 ns [Chen16]). We optimized the cabling of the dilution refrigerator [Schär17] to minimize the propagation delay of the input pulses, which need to propagate to the sample, and the output signals, which propagate back to RT (total delay 12.4 ns). To distinguish the qubit state with high fidelity we require approximately 50 ns integration time (see Fig. 2.14 and Ref. [Walter17]). We account for 7 ns digitization time in the ADC and define the moment at which the signal is digitized as the end of our qubit state measurement. In total, we obtain a delay of approximately 100 ns, corresponding to a propagation distance at the speed of light of lBell 30 m, between the RNG at node A and the FPGA at node B (and vice versa). Based on≈ these estimates, we plan to realize a system with 30 m (Fig. 3.13) between the centers of two dilution refrigerators which results in lBell 33 m [Schär17] and leaves a margin of approximately 10 ns. ≈ In addition, sufficiently low temperatures of the transmon qubits at the nodes and the microwave waveguide, used as the quantum channel along the cryogenic link, are required to keep their thermal exciations small. To obtain low temperatures the 3.3 Cryogenic Networks 89

additional heat load from the cryogenic link needs to be small and we set the design node goals of a maximal temperature of TBT = 20 mK at the nodes of the BT-stage and max node of TBT = 100 mK along the cryogenic link. TBT can lead to thermal exciations of 9 the transmons < 2 10− and we also thermalize the ﬁeld of the waveguide at node A using an isolator. We· estimate the thermal excitation of the ﬁeld at node B based on a continuous beam splitter model as

Z l αatt l nA αattz nnB = e− nin + α e nBE(T(z), ω) dz (3.29) 0 with the attenuation constant αatt, and the length of the waveguide l [Norris17]. Due nA node to the thermalization by an isolator we assume nin = nBE(TBT , ωt) and for simplicity, a constant waveguide temperature T(z) = T. In our experimental implementation, photons are transmitted at a frequency of ωt/2π 8.35 GHz which is close to lowest recommended frequency of a WR90 (see Section≈ 4.1). If we use a 30 m WR90Al at max WR90Al TBT we obtain nnB 0.1% (Fig. 3.8b) for αatt = 0.008 dB/m (see Chapter 4 and ≈ CC085NbTi Ref. [Kurpiers17]). Installing a CC085NbTi leads to nnB 0.3% with αatt = 0.024 dB/m. ≈ In conclusion, we set the design goals for a loophole-free Bell test with superconducting circuits to build a 30 m cryogenic network with a maximal temperature of 20 mK at the nodes of the BT-stage and of 100 mK along the cryogenic link. For practical use, we target a cooldown time to BT of less than 7 d. The design and experimental tests towards achieving this goal are discussed in Sections 3.3.2 to 3.3.6. In Section 3.3.5, we also investigate a 40 m cryogenic network for more advanced non-local protocols, e.g. quantum cryptography [Arnon-Friedman18] or randomness ampliﬁcation [Colbeck12], which might require additional gates.

3.3.2 Modular Design of a Cryogenic Network

In our design of a cryogenic network, the cooling units are located at the nodes, which also host circuit QED systems (Fig. 3.9). The nodes are connected by cryogenic links and can, in general, be located on a two-dimensional grid. Low-loss microwave waveguides are placed at the BT-stage of the cryogenic link to execute quantum communication protocols between the different nodes over distances of tens of meters without signiﬁ- cant loss (see Chapter 4). To explore the feasibility of these types of modular cryogenic networks we focus on a 1D version and also study technically realizable distances between the nodes. We use a modular design of the cryogenic network since it has 90 Chapter 3 Cryogenic Quantum Networks

a) b) cooling unit

mA mA mL

mB mB cooling unit

Figure 3.9 a) Sketch of a 2-dimensional cryogenic network for which each cooling unit (blue) hosts a quantum node. For the research work described in this thesis, we designed and realized two nodes with one link. The brown dashed box indicates a unit cell of the network, which is schematically shown in (b). A unit cell consists of a cryogen-free cryostat (cooling unit), which can host a dilution unit and up to 4 adapter modules (mA) (two shown). Unit cells are interconnected using a self-supporting link module (mL) and two flexible thermal interconnects (mB). The distance between two cooling units can be up to 10 m with the approach discussed here. ∼ the advantage, that all basic modules can be tested individually and good predictions on larger systems can be made based on measurements of the individual modules. In addition, a modular design allows for efficient scaling, since existing components can be reused while keeping the flexibility to incorporate additional specialized modules if new demands occur. Furthermore, individual sections of the cryogenic network can be exchanged or maintained without opening the full system. Our design is based on commercial available cryogen-free dilution refrigerators with customized radiation- shields. Together with BlueFors Cryogenics we designed flanges at the side of the vacuum can, and the 50K-, 4K- and Still-radiation-shields to which the cryogenic links is connected. For a BT-radiation-shield along the cryogenic link we place an adapter below the BT-plate (see Section 3.1.5). Flanges at the bottom of the dilution refrigerators are installed to have access to the BT-stage while the cryogenic links are connected, e.g. to replace the circuit QED samples (Fig. 3.9). Due to the large number of critical connections to operate a dilution refrigerator, e.g. for pumping at the vacuum can, or to run the pulse tube coolers and the dilution cycle, we require a design of the cryogenic network which keeps the position of the cooling units fixed. In addition, no forces due to thermal contraction have to appear on the cold stages of the dilution refrigerators during the operation. To fulfill these criteria and to be able to assemble the cryogenic network without moving dilution refrigerators, we based the cryogenic links on three types of modules. Module mA is fixed to a node using the side-port flanges, it consists of self-supporting shields and it serves as an adapter module between the node and the cryogenic link. If different types 3.3 Cryogenic Networks 91

of cooling units are used in the cryogenic network only the flange connections of the adapter module have to be adapted without affecting other modules. The cryogenic link consists of a module mL, which is also made using self-supporting shields, and a module out of flexible cooper braids (mB), which thermally connects an adapter module with a link module or two link modules. The 50K-, 4K-, Still- and BT-radiation-shields of adapter and link modules are made of Cu-OFE with a thickness of d50K = 3 mm, d4K = 2 mm, dStill = 1 mm, dBT = 1 mm 2 2 2 (cross-sectional area A50K = 2030 mm , A4K = 1030 mm , AStill = 370 mm , ABT = 200 mm2) chosen based on the heat transfer calculations (see Section 3.1 and Sec- tion 3.3.5). In our final design, we use octagonal, bent radiation shield due to more reliable production compared to round welded shields, which were used in a first design (Fig. 3.10d). The 50K-radiation-shields of the cooling units and the cryogenic link are covered with MLI to reduce the cooldown times and the steady-state temperatures. We use staggered ends of the radiation-shields on adapter and link modules (Fig. 3.10d). The O-ring sealed vacuum cans (VC) of adapter and link modules are sl made of a main tube (Al, dVC = 4.5 mm) and a sleeve (Al, dVC = 10 mm). This sleeve together with the staggered ends make the assembly of braid modules from the BT- to RT-stage convenient. The braid-connections of mB at the 50K-stage consist of seven Cu-ETP braids (to- b 2 b tal cross-section A50K = 1995 mm ), at the 4K-stage of seven Cu-OF braids (A4K = 2 b 2 1050 mm ), at the Still-stage of eight Cu-OF braids (AStill = 400 mm ), at the BT-stage b 2 of five Cu-OF braids (ABT = 250 mm ). We design the cross-section of the 50K- and 4K-braids to be approximately equal to the radiation shield cross-section and about 9% (25%) larger for the Still-braids (BT-braids). To realize a high thermal contact conductance (see Section 3.1.6) we use three M4 brass screws on the 4K-stage and four M3 brass screws together with Apiezon N grease on the Still- and BT-stage to mount every single braid of mB to the radiation shield of the adapter and link modules. Note however, that special care has to be taken during assembly and disassembly that surfaces, which are exposed to radiation, are not covered by Apiezon N grease (Section 3.1.7). The lengths of all braids are designed to compensate the thermal contraction of the adapter and link module shields during the cooldown and manufacturing tolerances, and to ensure that no forces are applied to the cold stages of the dilution refrigerators. We use radiation-shields at the braid module, which are fixed at one end to be thermalized but freely moveable at the other end to avoid stress because of thermal contraction. These moveable shields protect inner temperature stages from higher thermal radiation along the whole connection. In combination with BT-shields at the 92 Chapter 3 Cryogenic Quantum Networks

mA mB mL

cooling unit b)

mL mA mB

c) d)

Figure 3.10 Schematics of a prototype system, which contains all distinct modules of the cryogenic network. We show a section view a) along the cryogenic connection, b) 3D rep- resentations, and c) a cross-sectional view at the cryogenic link. d) Images of the prototype during assembly. The left image shows a ﬁrst version with round radiation-shields and the two right images the braid module and the support posts in our ﬁnal octagonal version.

nodes, we realize a radiation tight system at each temperature stage in our design.

The inner radiation-shields are separated by insulating support posts from the outer ones to avoid thermal contacts between shields in the adapter and link modules (see Fig. 3.10c,d). We tested different materials of these support posts (Fig. 3.5) and optimized the design in terms of mechanical stability and thermal conductance. We decided for support posts made of 3D-printed composites (Accura® Bluestone, Sauber Aerodynamik AG) between all temperature stages due to its low thermal conductivity 3.3 Cryogenic Networks 93

over the required temperature range (Section 3.1.4) and its offset yield point Rp0.2 100 MPa measured at the department of materials science at ETH Zurich. The support≈ posts are only fixed at one radiation shield to avoid stress on the posts due to differential thermal contraction between the different temperatures. We place one support post in direction of gravity and two rotated by 45◦ at two position along the link axis in the link modules and at one position in the± adapter modules (Fig. 3.10a,b). The choice of the transversal positions allows to assemble the radiation-shields of the link module from outside to inside by moving the shields along the link axis. In addition, it keeps the radiation-shields aligned to each other during operation of the cryogenic system. Our design of the cryogenic link is thus based on radiation-shields guided along the link axis which are only fixed at the cryogenic nodes but can move along the link axis during the operation and the thermal contraction is compensated by the braids of mB realizing a cryogenic network without moveable cooling units. The microwave waveguides are installed within the BT-radiation-shield and are constrained only at center between two dilution refrigerators requiring that their thermal contraction is compensated by flexible coaxial cables at the dilution refrigerators. Along the rest of the cryogenic link, they are placed on holders which allow them to move along the link axis. The waveguides are thermalized every 250 mm using copper braids to compensate differential thermal contraction between the BT-shields and the waveguides.

3.3.3 Characterization of a Cryogenic Prototype System

Steady-State Analysis

To test the thermal performance, the mechanical stability and the assembly of our cryogenic link design, we build a prototype cryogenic system which contains all modules: dilution refrigerator - adapter module - braid module - link module (Fig. 3.10). This prototype system is used to make predictions for extended cryogenics networks towards a loophole-free Bell test. We performed three fully-functional cooldowns of such a prototype system (Fig. 3.11a) and analyzed thermal properties of the system using the techniques described in Section 3.1.2. We obtained temperatures at the end of the endmL link module of the 50K-stage of the prototype T50K = 90-100K (∆T50K = 37-49 K). Based on these temperature measurements we extracted radiation loads of 6-10 W/m2 using a phosphoric-acid based cleaning solution and electropolishing to remove oxides from the copper surfaces before the cooldown but without MLI. On the 4K-stage we measure a temperature difference between the 4K-plate and the end of link module of 94 Chapter 3 Cryogenic Quantum Networks

a length, l m b ) 0 1 2 [ ] 3 4 ) T50K [K] ] 100 K [ CD1 70 CD2 50K T 40 0 1 2 3 CD34

] 6 K

[ 100 5 4K

T 4 c T50K plate

] ) 0.8 280 K T end mN [ 100 50K 0.7 240 T50K start mL ] Still K [ T 0.6 T end mL 0 1 2 3 4 200 50K T plate ] 40 4K 160

mK 300 [ 20

BT 120

T 0 ]

temperature, T 80 120 CD1 mK [ 100 CD2 40 CD3 WR 80 T 0 plate end mA start mL end mL 0 2 4 6 pos Temperature sensor time, t d d [ ] ) e50K, 110 K n50K,stst e50K,stst 50K ~ n4K,50 K e4K,50 K 4K e4K,qstst nBT,50 K eBT,50 mK BT eBT,50 K eBT,qstst BT,min 0 1 2 3 4 5 6 7 8 9 cooldown time, t d Figure 3.11 a) Measured steady-state temperatures (dots)[ ] versus position of the temperature sensor for three prototype cooldowns (CD) indicated in different colors. The data of the BT-stage of cooldown CD3 shows temperature ranges due to limited calibration of the RX-102B temperature sensors. The black lines show the expected BT-distributions of cooldown CD2 and CD3 based on the measurements of the thermal conductivity of copper, of the thermal contact conductances, and of the heat loads of the support-posts. The waveguide temperatures are measured approximately 120 mm from the next thermalization point (Fig. 3.5b). b) Illustration of a finite element simulation (COMSOL Multiphysics®) of the temperature distribution (surface color) and heat flux (black arrows) along the 50K shield in cooldown CD2. c) Measured 50K cooldown dynamics (dots) and time-dependent simulation results (solid lines) using the temperature dependent cooling power of the first pulse tube cooler stage (see e.g. Fig. 3.1b) as the boundary condition. d) Overview of the cooldown dynamics of three prototype cooldowns (indicated by vertical bars) with zero corresponding to the start time of the pulse tube cooler. We show times at which the nodes (n) and the end of the prototype shields (e) on the 50K-, 4K- and BT-stage reach characteristic temperatures (e.g. 110 K, 50 K or 50 mK), a quasi steady state (qstst) temperature, or their minimal (min) temperature.∼ For example, ’eBT, 50 mK’ indicates when the end of the link module of the BT-stage reaches 50 mK which we define as the cooldown time. 3.3 Cryogenic Networks 95

∆T4K = 1-1.8 K dominated by radiation load from the 50K-shields and heat load due to support posts. For the Still-stage we obtain ∆TStill = 70mK, which we attribute to heat load due to support posts and contact resistances between the individual elements. In cooldown (CD) 1 and CD2 we observe a ∆TBT = 25mK, which we could improve to

∆TBT < 5mK in cooldown CD3 by covering all holes in all radiation shields either with 1 layer of aluminum foil or with metalized tape. At the BT-stage we performed measurements of the thermal conductivity of the radiation-shields, and of the thermal contact conductances between the BT-plate and the adapter module (two contacts) and between adapter module and braid module (assumed to be equal to braid module - link module) by applying heat at the end of link module (see Section 3.1.3 and Section 3.1.6). We used Fourier’s law (Section 3.1.2) and approximated the radiation-shields as 1-D objects to analyze these measurements.

We obtained RRRCu = 130 in cooldown CD1 and CD2 and RRRCu = 220 in cooldown CD3, all in line with independent measurements of test-samples at 4 K using the setup described in Section 3.1.3. We extracted thermal contact conductances between BT- BTpmA plate and adapter module of Kc = 5-9 µW/mK at T 100 mK which follow a x ≈ power law (Kc T , Eq. (3.9)) with x 1.2, and between adapter module and braid mAmB ≈ module Kc = 3-4 µW/mK at 100 mK scaling with x 1.5. Based on these thermal measurements and independent measurements of the≈ heat load due to the support posts using the setup discussed in Section 3.1.4, we can predict the BT-distribution (shown for cooldown CD2 and CD3 in Fig. 3.11a) using Fourier’s law and can make predictions for longer systems (see Section 3.3.5). max On the WR90Al we measured temperatures TWR 120 mK in cooldown CD2 in a not radiation tight BT-shield. We showed that the waveguide≈ temperature can be improved in the cooldown of the 10 m cryogenic network discussed in Section 3.3.4.

Dynamic of Cooldowns

The cooldown times of our prototype systems were limited by the cooldown of the 50K-stage (Fig. 3.10c,d). The end of BT-stage reached a quasi-steady-state of T 6-7 K in circa 2.5 d and we had to wait 1.7 d to start operating the dilution unit due≈ to the high 50K-stage temperature. We observed∼ that we can start operating the dilution unit for temperatures between 100-120 K at the end of the link module of the 50K-stage. For higher 50K-stage temperatures the 4K-plate temperature is too high to condense 3 4 the He/ He mixture due to radiation loads. We reached 50 mK at the end of the link nodes module of the BT-stage (TBT 40 mK) only 4.1 h after we have started to successfully ≈ 96 Chapter 3 Cryogenic Quantum Networks

3 4 condense the He/ He mixture. The short cooldown times for temperatures below 50 K can be explained by the high thermal diffusivity αd(T) in this temperature range (Fig. 3.2). The 50K-plate reached a steady-state temperature after 7.1 d and the end of the link module of the 50K-stage after 8.3 d.

Based on these observations, we focused on simulating the 50K cooldown dynamics of the prototype system. Using these time-dependent simulations we also analyzed the effect of different parameters on the cooldown time and made predictions for larger systems (Section 3.3.5). We modeled the 50K-geometry in a finite element simulation software (COMSOL Multiphysics®) and used the independently measured thermal conductivities of the radiation-shields and braids, to fix all thermal conductances keeping the radiation load as the only free parameter in the model. By performing steady-state simulations of the 50K-stage we extracted the radiation load of the individual sections to arrive at the same temperature distribution as measured in the prototype cooldowns. We assumed that the radiation load follows the Stefan–Boltzmann law of a gray body 4 4 µradσSB(TRT T ), Eq. (3.10), with an effective parameter µrad, which we varied in the simulations− (Fig. 3.11b). We obtained a heat flow of approximately 50 W at the position of the pulse tube cooler from these steady-state simulations. A load of 50 W is in agreement with the estimates from our measurement of the capacity curve of the pulse tube cooler. To simulate the cooldown dynamics and to be able to make accurate simulations for larger cryogenic systems, we extracted the temperature-dependent heat flow at the position of the pulse tube cooler, which corresponds to the temperature- dependent cooling power P50K(T). We used the prototype measurement results and the capacity curve provided by Cryomech Inc. (Fig. 3.1b) to obtain P50K(T). Combining the simulation results and the capacity curve was required, since the temperatures of the Cryomech capacity curve are measured at the pulse tube heads while we measured the temperature on the 50K-plate. In a first simulation run, we used the 50K-plate temperatures T50K as the boundary condition in the simulations. We ran time-dependent simulations which reproduced the measured temperature distribution of the full prototype system using only the parameters extracted from independent measurements and the steady state simulations. We concluded from this first simulation run that the cooldown dynamics are well described by our heat flow model. We next extracted the heat flow versus temperature at the pulse tube cooler position of the prototype from the simulation results. We combined the obtained heat flows with the Cryomech capacity to obtain P50K(T) over a temperature range between 32-295 K. We verified the extracted P50K(T) by simulating the dynamics of the cooldowns of the prototype system setting the heat flow at the position of the pulse tube cooler as the boundary 3.3 Cryogenic Networks 97

condition (Fig. 3.11c). We obtain good agreements between these simulations and the measured temperatures and therefore used P50K(T) in all further time-dependent simulations. In summary, we veriﬁed with our prototype cryogenic system that the assembly of the modular design to the cryogenic node and along the link from the BT- to the RT-stage is possible and that all modules are mechanically stable during the cooldown and warm-up. Thermally, we observed temperatures < 20 mK along a distance of 4 m, and that the system can be cooled to BT in around 5 d (Fig. 3.11d). We performed numerical simulations which describe the 50K cooldown dynamics accurately and can be used to make predictions for large cryogenic systems.

3.3.4 Cooldown of a 10-Meter Cryogenic Network

To test our full modular design, including the compensation of the thermal contraction along the cryogenic link, we assembled a cryogenic system, which connects two dilution refrigerators over a distance of 10 m consisting of two adapter modules and three link modules (Fig. 3.12a). We installed MLI on node A and along the cryogenic system up to braid module 4. Cooling the 10 m cryogenic system down, we observed a well- working system as predicted based on the prototype measurements. We measured temperatures at the nodes and along the cryogenic link on all temperature stages and on all modules during the cooldown (Fig. 3.12b,c). In steady-state (Fig. 3.12b), we c,stst obtained a temperature at the 50K-center of T50K = 71 K and a temperature difference to the 50K-plates of ∆T50K = 26 2 K. Based on the measured temperature differences, ± cl 2 we extracted a total heat flux from radiation and support posts of φtot 2.4 W/m along the cryogenic link. Using numerical simulations of the support posts we≈ estimated a heat load of 0.25 W per support post including radiation load. We thus obtained a cl 2 heat flux due to radiation φMLI 1.5 W/m assuming the load of the support posts to be distributed over the surface area≈ of the cryogenic link. At node B and adapter module 2 2, which were not covered by MLI, we measured φtotnB 11 W/m . At node A and 2 ≈ adapter module 1 we obtained φtotnA 13 W/m which we attributed to an improper ≈ MLI installation. From the 4K-plates to the center of the link we measured ∆T4K = 0.8 nA/nB ± 0.2 K. On the Still-stage we stabilized the temperature at the nodes to TStill = 850 mK to operate the dilution units at a fixed 3He flow rate and thus approximately constant cooling power. We measured ∆TStill = 30 mK between the plates and the center. We nA/nB chose TStill to reduce the heat load from support posts to the BT-stage which we ˙ estimated to be Qsup < 0.1µW per support post based on an independent measurement 98 Chapter 3 Cryogenic Quantum Networks a nA nB ) mA1 mL1 mL2 mL3 mA2

mB1 mB2 mB3 mB4 b length, l m ) 0 1 2 3 4 5 [ ]6 7 8 9 10 0.88 ] 70 K [ 55 50K

T 40 0.88 ]

K 4.5 [ 4 4K

T 3.5

] 0.88 K [ 0.88 Still

T 0.85

] 20 mK [ 15

BT 10

T 0 3 5 6 8 9 10 0.88

] 0.88 80 mK [ 70 WR

T 60 plateA end mA1 end mL1 mid mL2 st mL3 st mA2 plateB pos Temperature sensor

c c50K,130 K n50K,stst c50K,stst ) 50K n4K,50 K c4K,50 K c4K,qstst 4K nBT,50 K cBT,50 mK cBT,min BT cBT,50 K★ cBT,qstst 0 1 2 3 4 5 6 7 8 cooldown time, t d Figure 3.12 a) Sketch of a 10 m cryogenic system with[ a] dilution refrigerator at each end. b) Measured steady-state temperatures of the 10 m cryogenic system at the indicated positions. For the 50K-, 4K-, and BT-stage we show the observed minimal temperatures, the Still-plate temperatures are stabilized at 0.85 K (see text). The temperatures of the waveguide close to the middle of the second link module ’mid mL2’ and start of the second adapter module ’st mA2’ (’plate B’) are measured 120 mm (50 mm) from the next thermalization point. The black line shows the BT-distribution estimated using the parameters of prototype CD3 (Fig. 3.11a). c) Overview of the cooldown times of the 10 m cryogenic system with zero corresponding to the start time of the pulse tube coolers. We show times at which the nodes (n) and the center of the 10 m cryogenic system (c) on the 50K-, 4K- and BT-stage reach characteristic temperatures (e.g. 130 K, 50 K or 50 mK), a quasi steady state (qstst) temperature, or their minimal (min) temperature. For example, ’cBT, 50 mK’ indicates when the center of BT-stage reaches 50 mK which we deﬁne as the cooldown time. 3.3 Cryogenic Networks 99

(Fig. 3.4). We obtained ∆TBT = 11 2 mK and a maximum BT in the center of the max ± link of TBT = 21 mK. Based on BT capacity curves (e.g. Fig. 3.1) we estimate a ˙ total heat load of QBT < 3µW on each node. Using the thermal conductivities and contact conductances as in prototype cooldown CD3 and the independently measured heat loads of the support posts, we predict the BT-distribution along the cryogenic link (Fig. 3.12b). We measured a maximal steady-state temperature of WR90Al of max TWR = 75 mK approximately 120 mm from the next thermalization point. We attribute the lower waveguide temperature compared to the prototype cooldown CD2 to a more light tight BT-radiation-shield. However, the low thermal conductivity of aluminum below Tcrit/10 (see Section 3.1.5) results still in significant temperature differences max compared to the BT-shields. Assuming TWR over the full length of the waveguide, we 4 estimate a thermal excitation of the field at node B of nnB 0.8 10− . ≈ · Due to the instillation of MLI along the cryogenic link in the 10 m cryogenic system the cooldown dynamics of the 50K- and the lower temperature-stages were similar (Fig. 3.12c). The center of the BT-stage reached its quasi-steady-state temperature nearly at the same time as the center of the 50K-stage reached 130 K (3.2 d, Fig. 3.12c), 3 4 and we started condensing the He/ He mixture after 3.4 d. 3.8 h after condensing started the system reached temperatures < 40 mK at both nodes and < 50 mK along the whole cryogenic link. The 10 m cryogenic system thus cooled down 15.4 h faster than the fastest prototype cooldown (CD3) despite its additional length of 1.25 m per dilution refrigerator. Similarly, the center of the 50K-stage also reached its∼ steady-state temperature approximately 15 h faster than the prototype cooldowns CD1 and CD3 showing the importance of a small total heat flux φtot for the cooldown time.

In conclusion, the cooldown of the 10 m cryogenic system supported the results of the prototype system, and showed that the modular design can be assembled and aligned between two ﬁxed dilution refrigerators. All parts of the 10 m cryogenic system were mechanically stable during cooldown and warmup and the thermal contraction of the radiation-shields was compensated by the braids of mB. The aluminum rectangular waveguide was thermalized to TWR < 75 mK showing that our design is suitable to build cryogenic networks at millikelvin temperatures over distances of ten meters. The performance of the system can be improved further by improving the MLI at the nodes, and by increasing contact conductances between at the sideports of the cryogenic nodes at the 4K-, Still- and BT-stage. Furthermore, the long-term stability over several cooldowns in sequence without opening the cryogenic link needs to be tested. 100 Chapter 3 Cryogenic Quantum Networks

3.3.5 Simulations of a 30-Meter Cryogenic Network

Based on the prototype and the 10 m system cooldowns we perform simulations of the obtainable cooldown times and steady-sate temperatures of a 30 m cryogenic system to perform a loophole-free Bell test. We study a system based on 4 pulse tube coolers (PTs) at the cryogenic nodes (4PT-30m) which are spaced by 10 m (Fig. 3.13a) and based on 3 pulse tube coolers (3PT-30m, nodes spaced by 15 m). We show the cooldown dynamics of 4PT-30m in Fig. 3.13b for which we assume a shield thickness d50K = 3 mm, RRRCu = 150. For simplicity, we assume the same total heat ﬂux φtot(T50K(t)) = 4 4 µradσSB(TRT T50K(t) ) on all surfaces as sum of the heat ﬂux from radiation and − 2 support posts (see Section 3.3.4). We choose µrad so that φtot = 3 W/m for T50K TRT. Based on these simulations, we obtain a cooldown time between 2.5-3.5 d, and estimate that the 50K-plate reaches a steady-state temperature in approximately 4 d while the center between node 1-2 and 2-3 require circa 5.5 d (Fig. 3.13) to reach a steady-state temperature of around 60 K. The asymmetry of the temperatures at the center between node 1-2 and 2-3 results from a higher heat load on the pulse tube cooler at node 2 compared to node 1 in the PT4-30m design.

Parameter Sweep

We use cooldown-dynamic simulations and sweep (Fig. 3.13c) the radiative heat load

φR on the 50K-stage, the thickness (cross-section) of the 50K-shields and their thermal conductivity (RRRCu). As observed experimentally, we obtain a strong dependence of the cooldown time on the total heat ﬂux at the 50K-stage and the distance between the pulse tube coolers. 2 2 3PT-30m requires φtot < 3.5 W/m and PT4-30m φR < 6 W/m to reach temperatures at the center between the nodes of 110-130 K. In addition, the cooldown time of the 2 PT3-30m system is 1.7 times longer for φtot = 0 W/m compared to the PT4-30m and 2 increases to a factor of 2.4 for φtot = 3 W/m . Based on these simulation results, we recommend a maximal distance of 10 m between two pulse tube coolers, which should 3 4 result in circa 3 d until the He/ He mixture can be condensed and in circa 7 d until the whole system is in steady-state. If additional length is required, 12.5 m between two pulse tube coolers can be tested or the 4PT system can be extended by 5 m at both ends (4PT-40m). The 5 m length increases can be obtained by using an adapter module combined with a link module and an additional module which provides space for circuit QED experiments. 4PT-40m distributes the heat load more equally on the pulse tube coolers and the centers between the nodes reach a steady-state temperature 3.3 Cryogenic Networks 101

a n1 n2 n3 n4 d n1 n2 n3 n4 n1-n2 n2-n3 ) T50K T50K )

30 m 40 m b e 300 300

] T plate n1 ] T plate n1 ) 50K ) 50K K K [ 250 [ 250 T50K mid n1 n2 T50K mid n1 n2 200 - 200 - T50K mid n2 n3 T50K mid n2 n3 150 - 150 - 100 100 50 50 temperature, T temperature, T 0 0 0 2 4 6 8 0 2 4 6 8 time, t d time, t d c [ ] [] ) ] d PT4 30m [ 12 -

CD PT3 30m - 8

cooldown time, T 0 0 2 4 6 2 3 4 5 6 0 50 100 150 200 250 300 2 tot. heat flux, tot W m thickness, d50K mm th.cond. Cu, RRRCu ϕ [ / ] [ ] Figure 3.13 a) Sketch of a cryogenic system of length 30 m with a pulse tube cooler every 10 m (PT4-30m). Only node 1 and 4 host dilution units. b) Cooldown dynamics of the 50K-stage of the PT4-30m system. The temperature of the ﬁrst node (n1), the temperature in the center between the two ﬁrst nodes (mid n1-n2) and between node 2 and 3 (mid n2-n3) are shown versus time. The dashed lines indicate 110 K and 130 K corresponding to the expected temperature range to start operating the dilution units. c) Simulation of

the 50K cooldown times versus absorbed radiative power per unit area φR (left), shield thickness d50K (center), and thermal conductivity RRRCu (right) (see text for details). On the left and center panels we also show the simulated cooldown time of a 30m-system using two pulse tube coolers at the ends and a third pulse tube cooler in the middle (PT3-30m). d) Sketch and e) cooldown dynamics of a 40 m cryogenic system extended by 5 m at both ends compared to PT4-30m.

3 4 in approximately 7.5 d (Fig. 3.13e). We estimate a time of 3.5 d until the He/ He mixture can be condensed in PT4-40m. We study the dependence of the cool down time on the thickness of the 50K-shields for the PT4-30m and 3PT-30m (Fig. 3.13c, center). We ﬁnd an optimum between 2 2-3 mm for a φtot = 3 W/m and RRRCu = 150. For smaller thicknesses than < 1.5 mm, the cryogenic system does not reach steady-state temperatures < 130 K. For larger thicknesses more heat energy needs to be removed from the shields, making the cooldown time limited by the temperature-dependent pulse tube cooling power and 102 Chapter 3 Cryogenic Quantum Networks

not by the heat transfer along the shields. We ﬁnd a weak dependence of the cooldown time on the thermal conductivity of the copper for typical RRRCu > 50 (Fig. 3.13c, right and Section 3.1.3). For example, we obtain a decrease of the cooldown time by only

6% between an RRRCu of 300 to 50. However, if e.g. phosphorus impurities decrease the thermal conductivity signiﬁcantly (RRRCu 10, Fig. 3.2) we observe a decrease ≈ of the cooldown time by 26% between an RRRCu of 50 to 10. This dependence of the cooldown time is mainly explained by the dependence of the thermal conductivity of copper on the RRRCu above 100 K: Between an RRRCu of 300 to 50 the thermal conductivity decrease by 6% between TRT and 100 K, however between an RRRCu of 50 to 10 the conductivity decrease by 25% over the same temperature range. The thermal measurements on the Still- and BT-stage of our prototype and the 10 m cryogenic systems show that no dilution units are required at node 2 and 3 of 30 m system since cooldown times below 4 K are expected to be small compared to the overall cooldown time of the cryogenic network. We estimate a temperature of 13 mK at the BT-plates and a maximal temperature of 50 mK along the BT-shields, taking∼ into account an increased temperature of the Still-stage.∼

3.3.6 Outlook on Cryogenic Networks

We have designed and realized a modular cryogenic network based on stationary cryostats. The repeated operation of the prototype and the 10 m cryogenic systems shows the practicability of our design in terms of assembly, mechanical stability, and thermal performance. These tests also show that a 30 m cryogenic network can be realized with the goal to perform a loophole-free Bell test with superconducting circuits. Based on the BT-distribution of the 10 m cryogenic system we estimate that 50 m max two dilution refrigerator systems can reach TBT < 100 mK and nnB < 0.1% assuming αatt = 0.008 dB/m [Kurpiers17]. These estimates suggest that inter-building or intra-city quantum communication at microwave frequencies are feasible with state- of-the-art dilution refrigerators and the described cryogenic network design. Our design is also adaptable to build cryogenic networks for distributed quantum computing. In these networks standard-size dilution refrigerators can be replaced by larger-scale dilution refrigerators, e.g BlueFors XLD or Oxford Instruments TritonXL, which can host up to several hundreds physical qubits, forming intermediate scale quantum processors [Krinner19]. These processors can be interconnect by several quantum channels hosted in a single cryogenic link and create a large-scale distributed quantum computer similar in design to modern classical supercomputers. To increase 3.3 Cryogenic Networks 103

the density of quantum nodes new modules, which provide more experimental space between nodes, can be integrated into to our cryogenic network. More flexibility during operation can be obtained by using superconducting heat switches at the BT-stage to thermally isolate specific sections temporarily from the rest of the network. In addition, hybrid quantum networks can be realized with the modular design described here by combining quantum nodes for specific tasks, for example, quantum memories based on spin qubits [Veldhorst14, Landig18], which might require magnetic fields can be installed at specialized nodes. Another possibility are cryogenic nodes which host quantum systems for optical-to-microwave-conversion [Balram16, Tsuchimoto17, Higginbotham18, Forsch19]. These cryogenic nodes ideally have high cooling power and are thermally isolated from the rest of the cryogenic network, since optical-to-microwave-conversion is expected to lead to significant heat dissipation [Pechal17]. Cryogenic networks can thus help combining local quantum networks for quantum processing at microwave frequencies and a long distance optical quantum internet [Cacciapuoti18, Caleffi18]. In conclusion, we demonstrated the feasibility to build and operate cryogenic networks on the scale of 10 meters based on two-cryostats and discussed the extensibility to larger systems. In the following chapters we discuss our experimental progress towards realizing quantum networks with superconducting circuits. 104 Chapter 3 Cryogenic Quantum Networks Chapter 4

Characterization of Coaxial and Rectangular Microwave-Frequency Waveguides

In addition to cryogenic networks, low-loss microwave waveguides are required for a reliable quantum communication at high transfer rates between distant circuit QED systems. In this chapter, which is based on our publication [Kurpiers17], we present our results of attenuation constant measurements of commercially-available coaxial and rectangular microwave-frequency waveguides down to millikelvin temperatures and single photon levels. We use a resonant-cavity technique to obtain the attenuation constants from measurements of the internal quality factors versus frequency. In addition, we extract the loss tangent and relative permittivity of commonly used dielectric waveguide materials by comparing our results with established loss models. Our precise measurements of attenuation constants also enable more accurate estimates of the signal levels at the input of cryogenic devices (Section 3.2.2) and can be used to better evaluate the heat load induced by signal dissipation [Krinner19]. In previous studies, attenuation constants have been measured for different types of superconducting coaxial cables down to 4 K by impedance-matched measurements [McCaa69, Ekstrom71, Chiba73, Giordano75, Mazuer78, Peterson89, Kushino08] [Kushino13]. In those works, the attenuation constants are typically evaluated from measurements of the transmission spectra of the waveguides. These transmission spectra are subsequently corrected for the attenuation in the interconnecting cables from room temperature to the cold stage in reference measurements. In these studies lengths of the low-loss superconducting coaxial cables between 20 m and 400 m were 106 Chapter 4 Characterization of Coaxial and Rectangular Microwave-Frequency Waveguides

used for the measurements in order to be dominated by the loss of the tested cable. In this chapter, we study the loss of coaxial cables and rectangular waveguides using a resonant-cavity technique from which we extract attenuation constants at RT, around 4 K and at millikelvin temperatures. By utilizing higher-order modes of these resonators we measure the frequency dependence of the attenuation for a frequency range between 3.5 and 12.8 GHz at cryogenic temperatures, limited by the bandwidth of our detection chain. The following chapter is organized as follows: We ﬁrst review the background of propagating microwave signals and present an overview of the investigated coaxial cables and rectangular waveguides. Next ,we describe the experimental setup, and characterization measurements of the external coupling to the waveguides under test. We illustrate our resonant-cavity technique based on measurements of coaxial cables, and show its adaptation to rectangular waveguides. Based on these measurement results we extract frequency-dependent attenuation constants, and also investigate their dependence on the input power and ambient magnetic ﬁelds. We conclude with measurements of a 3.65 m long rectangular waveguide inside the cryogenic prototype system described in Section 3.3.3.

4.1 Propagating Microwave Signals

Microwave signals are typically transmitted over distances above the chip scale using coaxial or rectangular waveguides. In this section, we describe the theoretical aspects of the propagation in these low-loss waveguides.

Low-Loss Coaxial Cables

The complex propagation constant of coplanar waveguides and coaxial lines

Æ γ = αatt + iβ = (Rl + iωLl )(Gl + iωCl ) (4.1) can be expressed using the equivalent circuit parameters with the self-inductance Ll , the capacitance Cl , the series resistance Rl and shunt conductance Gl per unit length. Based on Ref. [Pozar12], we obtain for a coaxial lines with the radius of the center conductor a and the inner radius of the outer conductor b

a b µ 2πε ε 1 R (ν) R (ν) 2πωε ε tan δ L 0 ln b/a, C 0 r , R s s , G 0 r l = 2π l = ln b/a l = 2π a + b l = ln b/a (4.2) 4.1 Propagating Microwave Signals 107

with the vacuum and relative permittivity ε0 and εr, the vacuum permeability µ0, a k frequency dependent surface resistance of the center and outer conductor Rs (ν), and the frequency independent loss tangent of the dielectric material tan δ. The factors nc 1/a, 1/b and ln(b/a) are deﬁned by the geometry of the coaxial line. Rs (ν) of a normal conductor is proportional to pν and to the direct current (dc) conductivity 1/pσ sc [Pozar12]. In superconductors, Rs (ν) is expected to show a quadratic dependence according to the theory of the high-frequency dissipation [Tinkham04, Mattis58, Kose89, Gao08].

For small conductor loss Rl ωLl and dielectric loss Gl ωCl we approximate p i Rl γ iω Ll Cl 1 + Gl Z0 (4.3) ≈ − 2 Z0 and obtain for the imaginary part (Fig. 4.1a)

ω ε β ωpL C p r . (4.4) = l l = c

c d αatt can be written as a sum of conductor loss αatt and dielectric loss αatt

ε Ra Rb πν ε c d 1 Rl p r s s p r αatt = αatt + αatt = + Gl Z0 = + + tan δ 2 Z0 2µ0c ln b/a a b c 1/a + 1/b pεr πpεr = Rs(ν) + ν tan δ, (4.5) ln(b/a) 2µ0c c

a b introducing an effective surface resistance Rs = Rs = Rs to characterize the total conductor loss of the center and outer conductors in the last relation (Fig. 4.1b).

As described in Section 2.2.2 for coplanar waveguide resonators, the internal quality factor of a λ/2 resonators Eq. (2.32)

n ωnCl Z0 (n + 1)π Qint = = (4.6) 2αatt 2αattl is inversely proportional to αatt [Pozar12]. Using Eq. (4.5), we obtain

1 Qint(νn) = (4.7) 1/a+1/b Rs(νn) tan δ 2πµ0ln(b/a) νn + which is independent of εr. We made use of Eq. (4.7) to determine αatt from measurements of Qint, which is detailed in Section 4.5. 108 Chapter 4 Characterization of Coaxial and Rectangular Microwave-Frequency Waveguides

Rectangular Microwave Waveguides and 3D Cavities

Besides coaxial cables, rectangular waveguides are typically used to transmit signals over large distances if low loss is required. In this section, we describe the basic char- acteristics of rectangular microwave waveguides and 3D cavities. Hollow microwave waveguides consist of a single electric conductor making only the propagation of transverse electric (TE) and transverse magnetic (TM) modes possible [Collin91, Pozar12]. The dimensions of the waveguide, which set the boundary conditions for the wave equation, deﬁne the cutoff frequencies of these TE and TM modes below which their propagation is exponentially suppressed. For rectangular waveguides (WR) in vacuum, TE and TM modes have cutoff frequencies

v c u m2 n2 ν ν t (4.8) TEmn = TMmn = 2 + 2 2 s1 s2 with the dimensions of the waveguide rectangular to the propagation direction s1, s2 (convention: s1 > s2), and mode numbers m, n [Collin91, Pozar12]. The cutoff frequency of the ﬁrst mode TE10 is c ν . (4.9) TE10 = 2s1

For TM00, TM01, TM10 modes the electric and magnetic ﬁelds are zero due to the rectangular boundary conditions leading to a cutoff frequency of the lowest order r TM mode ν c 1 1 which is higher than ν . TE is thus the mode 11 TM11 = 2 2 + 2 TE10 10 s1 s2 which is typically used for transmission in rectangular waveguides. TE and TM show a non-linear dispersion relation with the frequency dependence of the propagation constant

2πÇ β ν2 ν2 . (4.10) TEmn = TE c − mn To avoid strong dispersion of signals transmitted in waveguides and excitations of higher modes a frequency band between approximately 1.3ν and 0.95ν is rec- TE10 TE20 ommended (Fig. 4.1a). Rectangular waveguide are categorized according their recommended frequency bands by Electronic Industries Alliance (EIA) with typical ratios of approximately s1 : s2 = 2 : 1. Attenuation in waveguides in vacuum is dominated by conductor loss which can be modeled by a frequency dependent surface resistance Rs(ν). The lost power per unit 4.1 Propagating Microwave Signals 109

a) b) 400 2.0 ] - 1 ] 1 - 300 TE10 1.5 [ m [ dB m β att

200 TE20 α 1.0 TEM

100 0.5 propagation, attenuation, 0 TM11 0.0 6 8 10 12 14 16 18 6 8 10 12 14 16 18

frequency, f [GHz] frequency, f [GHz]

Figure 4.1 a) Propagation constant β and b) normalized attenuation constant αatt versus frequency for the transverse electromagnetic (TEM) mode of a coaxial cable of type CC085 (see Tab. 4.1), and the lowest two transverse electric modes (TE10, TE20) and the lowest transverse magnetic mode (TM11) of a rectangular waveguide of type WR90 (see Tab. 4.1). The dashed vertical lines indicate the recommended frequency band for the WR90. All TEM attenuation constants are normalized to obtain αatt (10 GHZ) = 1 dB/m indicated by the sc 2 cross and we assume a surface resistance Rs (ν) ν characteristic for superconductors [Tinkham04].

R 2 length Pl = Rs(ν)/2 C Is dl is obtained by integrating the surface current Is along a contour C which encloses| | the inside perimeter of the waveguide walls. The attenuation

TE10 per unit length αatt = Pl /2P10 is the ratio of the lost power per unit length and the power ﬂow of the TE10 resulting in

2 3 2 TE 20 R ν s2c + 2s ν α 10 ν s( ) 1 dB/m (4.11) att ( ) = 2 q [ ] ln 10 s s2µ0c 2 2 2 ( ) 1 ν 4ν s1 c − with the conversion from neper to decibel 20/ ln(10) (Fig. 4.1b) [Pozar12]. To study the attenuation in low-loss waveguides, we use 3D waveguide resonators (3D rectangular cavities). These 3D cavities are typically built using closed waveguide sections which form short circuits at both ends since open waveguide section radiate to the environment. Furthermore, 3D cavities are also used for circuit QED experiments (see for example Refs. [Abdumalikov13, Ristè13, Vlastakis13, Jerger16b, Rosenblum18]). Their TE and TM modes satisfy the same boundary conditions on the sidewalls than in the waveguide case and in addition also the length of the cavity l is constrained resulting in 110 Chapter 4 Characterization of Coaxial and Rectangular Microwave-Frequency Waveguides

v c u m2 n2 k2 ν t (4.12) TEmnk = 2 + 2 + 2 2 s1 s2 l with a third mode number k. Using the typical convention l > s1 > s2 the TE101 is the dominant cavity mode, similar to the lowest λ/2 transmission line resonator mode.

The internal quality factor of TE10k modes assuming only losses in the conducting walls Q 4πν W /P is obtained by the ratio of the stored electric energy W and TE10k = TEmnk e c e the power lost in the walls P Pozar12 . Q is found to be inversely proportional c [ ] TE10k to Rs

3 3 2s1s2lπµ0νTE 1 Q ν mnk . (4.13) TE10k ( TEmnk ) = R ν 2 3 2 s( TEmnk ) c s2(l s1) + 2s1(2s2 + l)νTE − mnk Rectangular waveguides and 3D cavities can be coupled by probe and loop antennas or by small apertures in a common wall. Probe antenna couplings are typically used in right-angle waveguide to coaxial-line adapters. At these adapters the outer conductor of the coaxial-line is shorted to the waveguide and the center conductor is extended wc by a length dc inside the waveguide at the center of longer side s1. The waveguide is wc shorted at a distance dw from the coaxial line to obtain a one direction radiation of wc wc the antenna coupling. dc , dw and the radius of the center conductor are optimized to obtain a high transmission coefﬁcient over a certain frequency band [Collin91]. We use right-angle waveguide to coaxial-line adapters to couple WR90 waveguides to coaxial cables in our experiments and studied them for a direct coupling between 3D cavities and waveguides [Schwegler18]. Probe antennas are also used to couple 3D rectangular cavities to coaxial feedlines [Paik11]. Loop antennas are used for inline waveguide to coaxial-line adapters or to couple to circular cavities [Reagor13].

Small aperture couplings are used to couple two waveguide sections, or a waveguide and a 3D cavity (see Refs. [Ruffieux14, Barthélemy14, Kurpiers17]). Aperture couplings can be modeled as small radiating electric and magnetic dipoles [Bethe44, Collin91] and can be understood as an inductive coupling in an equivalent circuit model [Pozar12, Park09]. We find that the coupling strength scales exponentially with the radius of the aperture and the wall thickness [Barthélemy14] making aperture 3 5 coupling easily tunable over several order of magnitudes between Qext 10 -10 (see Section 4.4). However, if stronger couplings are required antenna couplings≈ should be favored to avoid large apertures which change the mode structure of the waveguide or 3D cavity significantly [Schwegler18]. 4.2 Overview of Evaluated Coaxial Cables and Rectangular Waveguides 111

4.2 Overview of Evaluated Coaxial Cables and Rectangular Waveguides

We evaluate the attenuation constants of coaxial and rectangular waveguides made by a number of different manufacturers and from a range of materials (Tab. 4.1). We characterize 2.2 mm (0.085 in) diameter coaxial cables with niobium-titanium (Nb-47 weight percent (wt%) Ti) or Niobium outer and center conductors. For both cables the manufacturer Keycom Corporation [Key16] uses a low density polytetraﬂuorethylen (ldPTFE) dielectric. We also measure a coaxial cable with an aluminum outer conductor of diameter 3.6 mm (0.141 in) and a silver plated copper wire (SPC) center conductor (per the standard speciﬁcation for silver-coated soft or annealed copper wire (ASTM

ID CC085NbTi CC085Nb CC141Al CC085Cu dim. [mm (in)] 2.2 (0.085) 2.2 (0.085) 3.6 (0.141) 2.2 (0.085) conductor NbTi/NbTi Nb/Nb Al/SPC Cu/SPCW dielectric ldPTFE ldPTFE ldPTFE sPTFE length [mm (in)] 110 110 900 120 TBT [mK] 120 50 60 15 T4K [K] 4.0 4.0 4.0 4.1 ν range [GHz] 4.2-12.7 4.1-12.5 3.5-12.7 3.6-12.7 nBT 0.1-2 0.3-10 1-2 1-3 n4K 0.4-5 0.2-4 8-16 4-9

ID WR90Alc WR90Al WR90CuSn CC085SS dim. [mm (in)] WR90 WR90 WR90 2.2 (0.085) conductor coated Al Al Cu-Sn SS/SS dielectric vacuum vacuum vacuum sPTFE length [mm (in)] 304.8 (12) 304.8 (12) 304.8 (12) 120 TBT [mK] 60 25 50 — T4K [K] 4.3 4.0 4.0 4.2 ν range [GHz] 7.9-12.3 7.7-12.8 7.7-12.8 0.9-13.1 nBT 0.2-1 1-4 1-3 — 1 n4K 2-10 3-12 5-20 > 10 0

Table 4.1 Summary of waveguide and measurement parameters. The indicated dimension (dim.) speciﬁes the outer diameter of the coaxial cables and the EIA type of the rectangular waveguides. The conductor and dielectric materials are speciﬁed as well as the length of the resonant section employed for the measurements. The temperature T measured at the waveguide is indicated (see Section 4.3). The average photon number n on resonance is shown for the investigated frequency (ν) range. CC085SS is measured inside a dipper-type

cryostat at RT,TLN and TlHe4. 112 Chapter 4 Characterization of Coaxial and Rectangular Microwave-Frequency Waveguides

B-298)). The manufacturer Micro-Coax, Inc. [Mic16] uses ldPTFE dielectric for this cable. As a reference, we analyze a standard copper outer conductor, silver plated copper clad steel (SPCW) center conductor (per the standard speciﬁcation for silver- coated, copper-clad steel wire (ASTM B-501)) coaxial cable with a solid PTFE (sPTFE) dielectric and an outer conductor diameter of 2.2 mm (0.085 in) manufactured by Micro-Coax, Inc. We investigate rectangular waveguides of type WR90 by Electronic

Industries Alliance (EIA) standard with inner dimensions of s1 = 22.86 mm, s2 = 10.16 mm (s1 = 0.900 in, s2 = 0.400 in) with a recommended frequency band of 8.2 to 12.4 GHz. Three different conductor materials are characterized: aluminum 6061 with chromate conversion coating per MIL-C-5541E, aluminum 6061 without further surface treatment and oxygen-free copper with tin (Sn wt% > 99.99%) plating of thickness 5-10 µm on the inner surface. All three rectangular waveguides are manufactured by Penn Engineering Components, Inc. [Pen16]. In addition, we evaluate the loss of a 2.2 mm (0.085 in) diameter stainless steel outer and center conductor coaxial cable with a sPTFE dielectric (CC085SS) manufactured by Micro-Coax, Inc. We measure the

CC085SS inside a dipper-type cryostat at RT, approximately TLN = 77 K and TlHe4 = 4.2 K, and discuss it independently of all other cable types.

4.3 Experimental Setup

We construct resonators from coaxial cables and rectangular waveguides as shown in the photographs and schematics of Fig. 4.2a,b. For the coaxial cables we use subminiature version A (SMA) panel mount connectors (TE Connectivity Ltd. 1052639-1). We remove the outer conductor and dielectric material of the coaxial cable at both ends to realize a capacitive coupling between the center conductor of the cable and the connector. We choose a coupling capacitance to obtain largely undercoupled resonators (see Section 4.5). At RT we use a vector network analyzer (VNA) and a through-open-short-match (TOSM) calibration to set the measurement reference planes to the input of the coupling ports (Fig. 4.2c). We adjust the input and output coupling to be approximately equal. As discussed in Section 3.2, for measurements in a dilution refrigerator the microwave signal propagates through a chain of attenuators of 20 dB at the 4K-, CP- and BT-stage before entering the waveguide (Fig. 4.2d). The output signal is routed through an isolator with a frequency range of 4-12 GHz and an isolation larger than 20 dB, and a HEMT ampliﬁer with a bandwidth of 1-12 GHz, a gain of 40 dB and a noise temperature of 5 K, as speciﬁed by the manufacturer. After room temperature 4.3 Experimental Setup 113

a) d) FPGA

RT 50K

CC85SS 4K -20 HEMT 1-12 GHz Still b) c) CC85SS CP

dc VNA -20 CC85NbTi

BT -20

ra TOSM calibration ta

Figure 4.2 a) Photographs and b) schematics of a capacitively coupled coaxial cable and an aperture coupled rectangular waveguide (Section 4.4). c) Schematic of the room temperature setup using a VNA. A through-open-short-match (TOSM) calibration is used to account for loss and phase offsets in the interconnecting cables. d) Schematic of the FPGA-based microwave setup used for the measurements in a dilution refrigerator. amplification and demodulation, the signal is digitized and the amplitude is averaged using a field programmable gate array (FPGA) with a custom firmware (Fig. 4.2d) [Salathé18a]. The waveguides are characterized at nominal temperatures of around 4 K and at millikelvin temperature using a dilution refrigerator. We thermally anchor the waveguides to the base plate of the cryostat using Cu-OF copper braids and clamps. The actual waveguide temperatures are extracted from a measurement of the resistance of a calibrated RX-102B sensor mounted at the center of the coaxial cables or at the end of the rectangular waveguides (Tab. 4.1). 114 Chapter 4 Characterization of Coaxial and Rectangular Microwave-Frequency Waveguides

As discussed in Section 3.1.2, it was essential for the measurements at BT to care- fully anchor all superconducting waveguide elements at multiple points to assure best possible thermalization. The measured temperatures (Tab. 4.1) are found to be significantly higher than typical BTs discussed in Section 3.1.1. We attribute the incomplete thermalization of the superconducting waveguides to the small thermal conductivity of the employed materials below their critical temperature Tc (see Section 3.1.5). We note that when using only a minimal set of anchoring points, we observed even higher temperatures.

4.4 Characterization of the External Couplings

We discuss two test setups in this section in which we have measured the dependence of the capacitive coupling on the separation dc for the coaxial cables (Fig. 4.2b), and the dependence of the aperture coupling on the radius of the circular aperture ra and coupling wall thickness ta for the waveguides (Fig. 4.2b). For the coaxial cables test setup, we control the distance between the input coupler and the center conductor of a coaxial line CC085Cu by clamping the CC085Cu to an adjustable translation stage and measure the transmission spectrum at several distances

(dots in Fig. 4.3a) using the RT-setup. We extract the coupling capacitance Cc from ﬁts of the measured transmission spectra to an ABCD transmission matrix model. We ﬁnd

Cc to decrease exponentially with the separation dc (solid line in Fig. 4.3a).

a) b) 500 16 ta=1.0 mm 100 ta=1.5 mm

8 ]

3 50 [ 10 [ fF ] 4 c 10 C ext

2 Q 5

1 1

0 0.4 0.8 1.2 2 2.5 3 3.5 4

seperation, dc [mm] aperture radius, ra [mm]

Figure 4.3 a) Coupling capacitance Cc (dots) obtained from an ABCD transmission matrix model fit to the measured spectrum versus separation dc between the center conductors of CC085Cu. The solid line is an exponential fit to the data. b) Measured Qext of the TE101 mode of a 3D cavity aperture coupled to a rectangular waveguide. The solid line is obtained from a finite element simulation. 4.4 Characterization of the External Couplings 115

For the waveguide test setup, we determine the external quality factor Qext of an aperture coupled 3D cavity with resonance frequency ν101 = 8.690 GHz in dependence on the coupling wall thickness ta and the radius of the circular aperture ra. We obtain good agreement between the measured Qext and finite-element simulation results using ® COMSOL Multiphysics software (Fig. 4.3b and Ref. [Barthélemy14]). By changing the geometry of the couplers, the coupling of both coaxial cables and rectangular waveguides can be tuned by orders of magnitude which is sufficient to fulfill the condition Qext Qint to extract Qint precisely as described in the following.

a) -40 c) -60 [ dB ] 2 |

21 -80 b) | S

-100

-120 transmission,

-140

0 2 4 6 8 10 12 14

frequency, ν [GHz] mode number, n mode number, n b) c) 5 6 7 8 9 10 62 63 64 65 66 67 -40 Δν= 0.127GHz [ dB ] -80 2 | 21

| S -120

0.7 0.9 1.1 1.3 7.9 8.1 8.3 8.5

frequency, ν [GHz] frequency, ν [GHz]

2 Figure 4.4 a) Transmission coefﬁcient S21 versus frequency ν for CC141Al at RT. The | | dashed line indicates the frequency-dependent insertion loss on resonance IL(νn) = 2 2 10 log10 S21(νn) dB. S21 of b) the ﬁrst 6 measured modes and c) 6 modes around 8− GHz as indicated| | by the| dashed| boxes in (a). 116 Chapter 4 Characterization of Coaxial and Rectangular Microwave-Frequency Waveguides

4.5 Illustration of the Resonant-Cavity Technique

To illustrate the resonant-cavity technique for extracting the attenuation constant of a waveguide we discuss a calibrated S-parameter measurement at RT for the coaxial 2 line CC141Al. The measured transmission spectrum S21(ν) exemplifies the periodic structure of higher-order modes for mode numbers n between| | 5 and 109 (Fig. 4.4). We extract the resonance frequency νn and the external and internal quality factor, Qext and Qint, for each mode n by simultaneously fitting the real and imaginary part of the complex transmission coefficient of a weakly coupled parallel RLC circuit [Petersan98] to the data in a finite bandwidth around each νn (Fig. 4.5)

1 iφ S21(ν) = ( + X )e . (4.14) 1 + Qext/Qint + 2iQext(ν/νn 1) − In Eq. (4.14), X is a complex constant which accounts for impedance mismatches in the SMA panel mount connectors and eiφ is a rotation of the data relative to the measurement plane [Leong02, Probst15]. 2 We observe a decreasing insertion loss on resonance IL(νn) = 10 log10 S21(νn) dB − | |

νres= 3.312 GHz 7.773 GHz 11.469 GHz

-50 10.2 MHz [ dB ] 2

 -60

21 8.1 MHz  S -70 5.2 MHz -80 transmission, -90

-25 0 25 -25 0 25 -25 0 25 frequency, ν [MHz]

2 Figure 4.5 Absolute value squared of the measured transmission S21 (dots) versus frequency ν at the indicated resonances for CC141Al (RT) and modes| | numbers n = 26, 61, 90 . The arrows indicate the full width at half maximum from which we extracted { } Qint. νres is the center frequency of the resonance. The line is the absolute value squared of the simultaneous ﬁt of the real and imaginary part of the S21 scattering parameter. 4.6 Measurements Results of Attenuation Constants 117

(dashed line in Fig. 4.4a) with increasing frequency due to the increase of the effective capacitive coupling strength. We chose IL(νn) > 40 dB to ensure the largely undercoupled regime (Qext Qint) over the entire frequency range. In this regime, Qint is well approximated by the loaded quality factor QL according to 1/Qint = 1/QL 1/Qext 1/QL. − ≈ In our experiments we assure that Qext > 10Qint for all frequencies and temperatures.

Under this condition it is sufﬁcient to extract QL for each mode n from

max Sn S21(ν) = + C1 + C2ν (4.15) Æ 2 2 | | 1 + 4(ν/νn 1) QL − max neglecting the speciﬁc value of the insertion loss (Sn is a free scaling factor). C1, C2 account for a constant offset and a linear frequency dependence in the background 3 [Petersan98] most relevant for measurements of low quality factors (< 10 ) resonances.

4.6 Measurements Results of Attenuation Constants

Coaxial Cables

To determine the frequency dependence of the attenuation constant αatt(ν) of the coaxial line we analyze its measured resonance frequencies νn and quality factors Qint in dependence on the mode number n (see Section 2.2.2 and Section 4.1). We use Eq. (2.30)

(n + 1)c 1 νn = ωn/2π = pεeff 2l to extract εr, and Eq. (4.7)

1 Qint(νn) = (4.16) 1/a+1/b Rs(νn) tan δ 2πµ0ln(b/a) νn + to obtain Rs(ν) and tan δ. The measured external quality factors at RT (Fig. 4.6b and Fig. 4.7b) are in good agreement with the ones expected for a capacitively coupled transmission line (see

Eq. (2.32)) using the capacitance C, the load impedance Z0 and the coupling capacitance Cc as ﬁt parameters. An interpolation of the Qext measurements is used in the 4K- and BT-measurements to estimate the average number of photons stored in the waveguide on resonance at each mode n (see Tab. 4.1 and Fig. 4.11). 118 Chapter 4 Characterization of Coaxial and Rectangular Microwave-Frequency Waveguides

The frequency dependence of the measured quality factors for CC141Al shows the expected pν scaling considering an effective conductivity of the outer and center conductor following the skin effect model of a normal conductor (Fig. 4.6a). This c suggests that αatt is mainly limited by the normal conducting SPC center conductor. 3 The dielectric loss limit of Qint is determined to be approximately 15 10 at BT, see Tab. 4.2. × Following the same measurement procedure, we extract the quality factor of low- loss superconducting cables (e.g. Fig. 4.7 for CC085NbTi). The measured internal quality factors of CC085NbTi at 4K and BT, ranging from 12 103 to 92 103, decrease approximately ν2 (solid line) with a small deviation× at higher frequencies.×

frequency, ν [GHz] a) 2 4 6 8 10 12 100 RT 4K

10 BT ] 3 [ 10

int 1 Q

0.1

b) 5000 ] 3 1000 [ 10 500 ext Q 100 50. 2 4 6 8 10 12

frequency, ν [GHz]

Figure 4.6 a) Measured internal Qint and b) external quality factors Qext of coaxial line CC141Al versus frequency ν extracted from spectra measured at RT (blue dots), at 4K (green dots) and BT (orange dots). The black lines are ﬁts to the loss model discussed in the main text. 4.6 Measurements Results of Attenuation Constants 119

sc p We obtain a better ﬁt assuming a power law dependence of Rs (ν) ν with an exponent p 2.7 0.3 at 4K and p 3.4 0.5 at BT. This peculiar frequency dependence is not≈ explained± by the theory≈ of high-frequency± dissipation in superconductors [Tinkham04, Mattis58, Kose89, Gao08]. Measuring CC085Nb leads to similar results as for CC085NbTi (Fig. 4.9). We also observe a power law dependence with p 3.2 0.4 and p 3.3 0.3 at 4K and BT. Using the same measurement techniques we≈ compare± the dielectric≈ ± and conductor properties of these low-loss coaxial cables with those of CC085Cu for which me measured attenuation ranging from 0.30 dB/m to 0.75 dB/m (Fig. 4.9).

We extract the relative permittivities εr from Eq. (2.30) and the loss tangent tan δ

frequency, ν [GHz] a) 2 4 6 8 10 12 100 RT 4K

10 BT ] 3 [ 10

int 1 Q

0.1

b) 5000 ] 3 1000 [ 10 500 ext Q 100 50. 2 4 6 8 10 12

frequency, ν [GHz]

Figure 4.7 a) Measured internal Qint and b) external quality factors Qext of coaxial line CC085NbTi versus frequency ν extracted from spectra at RT (blue dots), at 4K (bright dots) and BT (red dots). The solid (dashed) black lines are ﬁts to the loss model which assumes sc 2 sc p Rs (ν) ν (Rs (ν) ν ). 120 Chapter 4 Characterization of Coaxial and Rectangular Microwave-Frequency Waveguides

T/parameter Micro-Coax Keycom Micro-Coax LD PTFE ldPTFE sPTFE

εr RT 1.70 0.01 1.72 0.06 1.98 0.07 4K 1.70 ± 0.01 1.72 ± 0.06 2.01 ± 0.07 BT 1.70 ± 0.01 1.72 ± 0.06 2.01 ± 0.07 5 ± ± ± tan δ [ 10− ] RT× 9 1 25 4 4K 8.5± 0.2 0.8 0.2 22 ± 2 BT 6.6 ± 0.2 0.7 ± 0.1 19 ± 4 ± ± ± Table 4.2 Overview of the extracted relative permittivities εr and loss tangents tan δ of the tested dielectric materials. The methods used for the extraction of these parameters are discussed in the main text. from ﬁtting Eq. (4.7) to the measured Qint(ν) for each coaxial cable (Tab. 4.2). The values of Micro-Coax ldPTFE are determined from CC141Al measurements, of Micro- Coax sPTFE from CC085Cu and of Keycom ldPTFE from CC085NbTi and CC085Nb sc measurements. We extract tan δ of the Keycom ldPTFE from the ﬁt assuming Rs (ν) p ν at 4K and BT. Due to the low internal quality factors Qint < 100 limited by the low RT conductivity (measured σ 8 105 S/m for NbTi and Nb) we are unable to extract these quantities at RT. At cryogenic≈ × temperature we observe that tanδ of the ldPTFE of

Micro-Coax and Keycom differ by a factor of 10. εr is found to be 1.7 for ldPTFE and 2 for sPTFE and is nearly temperature∼ independent. ∼ ∼

Rectangular Waveguides

We performed similar measurements with three different rectangular waveguides of type WR90 (Tab. 4.1 and Fig. 4.8). We use an aperture coupling approach by installing two Aluminum 1100 plates (thickness ta = 3 mm) at both ends with a circular aperture (radius ra = 5.3 mm for WR90Alc and ra = 4.1 mm for WR90Al and WR90CuSn) in the center (Fig. 4.2) resulting in inductively coupled rectangular 3D cavities (Section 4.1).

The coupling strength depends on ra and ta of the aperture plates (Fig. 4.3). We perform ﬁnite element simulation to estimate the coupling (see Sec 4.4) and determine the attenuation constant of the rectangular waveguides by a measurement of its internal quality factor. For rectangular waveguide cavities the frequencies of the transverse electric modes

TE10k are given by Eq. (4.12) and the frequency dependent internal quality factor in Eq. (4.13). We invert Eq. (4.13) and extract the surface resistance Rs(νk) from a measurement of Q ν which we use to calculate the attenuation constant of the TE10k ( k) 4.6 Measurements Results of Attenuation Constants 121

a) -50 c) -60 [ dB ] 2 | -70 b) 21 | S -80

-90

transmission, -100

-110 7 8 9 10 11 12 13 14

frequency, ν [GHz] mode number, n mode number, n b) c) 10 11 12 18 19 20

Δν= Δν= 0.303GHz 0.398GHz [ dB ] 2

| -80 21 | S

8.2 8.4 8.6 8.8 11 11.2 11.4 11.6 11.8

frequency, ν [GHz] frequency, ν [GHz] 2 Figure 4.8 a) Transmission coefficient S21 versus frequency ν for WR90Al at RT. The first five modes are below the noise level| due| to the diverging attenuation (see Section 2 2.2.2). S21 of b) modes 10-12 and c) 18-20 as indicated by the dashed boxes in (a). The increasing| ∆| ν illustrates the non-linear dispersion of rectangular waveguides.

TE10 mode of a rectangular waveguide using Eq. (4.11). With this method we extract the attenuation constant of the rectangular waveguides, ranging from 0.06 dB/m to 0.17 dB/m at 4K and 0.007 dB/m to 0.02 dB/m at BT, and determine the frequency dependence of the internal quality factor. For the rectangular waveguides in the normal state at 4K we ﬁnd good agreement to the the- nc oretical model by considering the normal state surface resistance Rs (ν) pν. At sc 2 BT in the superconducting state a surface resistance Rs (ν) ν approximates the data (Fig. 4.9). Note that the frequency dependence cannot be extracted with high

TE10 accuracy for superconducting rectangular waveguides, since αatt (ν) diverges towards the cutoff frequency νco = c/2s1 (Section 4.1). 122 Chapter 4 Characterization of Coaxial and Rectangular Microwave-Frequency Waveguides

a) frequency, ν [GHz] 2 4 6 8 10 12

0.500

0.100 [ dB / m ] 0.050 att α CC85Cu WR90Al 0.010 CC141Al WR90Alc 0.005 CC85NbTi WR90CuSn attenuation, CC85Nb 0.001

b) 0.500

0.100 [ dB / m ] 0.050 att α

0.010 0.005 attenuation,

0.001

2 4 6 8 10 12

frequency, ν [GHz]

Figure 4.9 Attenuation constant αatt (dots) extracted from measurements at frequency ν for CC085Cu, CC141Al, CC085NbTi, CC085Nb, WR90Alc, WR90Al and WR90CuSn. We show measurement results a) at approximately 4 K and b) at BT. The attenuation constant

αatt is plotted in dB/m in a logarithmic scale to cover the full range of measured losses. The solid lines are calculated from the ﬁts to Qint measurements using the same model as for the data. 4.6 Measurements Results of Attenuation Constants 123

16 RT 14 77K 12 4K [ dB / m ] 10 att α 8 6 4 attenuation, 2 0 0 2 4 6 8 10 12 14

frequency, ν [GHz]

Figure 4.10 Measured attenuation constant αatt (dots) versus frequency ν for a CC085SS at RT, at TLN, and at TlHe4. The black lines are calculated from ﬁts to the measured Qint using the model of Eq. (4.7) as described in the main text.

Attenuation Constant of CC085SS

In addition, we evaluate the loss of CC085SS which are typically used in cryogenic applications where loss is of little concern (see Section 3.2). We place the CC085SS inside a dipper-type cryostat and use CC085Cu interconnecting cables and 10 dB attenuation at the couplers for better impedance matching. Using the same measurement− techniques as for the other coaxial cables we obtain the expected frequency dependent attenuation pν of a normal conductor decreasing with temperature (Fig. 4.10). For example, we extract attenuation constants at 6 GHz of 9.7 dB/m (RT), 8.7 dB/m (TLN) and 8.2 dB/m (TlHe4).

Power Dependence of the Attenuation Constants

We ﬁnd the measured attenuation constant of CC085NbTi to be independent of the input power in a range from 140 to 80 dBm corresponding to an average photon number on resonance inside− the resonator− of 105 to 0.2 and an average resonator 2 1/2 6 4 voltage V of approximately 10− to 10− V (Fig. 4.11). Our voltage range is comparable to that of Ref. [Martinis05] in which a clear power dependence of the loss tangent of SiO2 and SiNx dielectric materials is observed. This power dependence is explained by the loss due to the coupling of microscopic two level systems (TLS) to 124 Chapter 4 Characterization of Coaxial and Rectangular Microwave-Frequency Waveguides

avg. number of photons in resonator,- n

2× 10-1 100 101 102 103 104 105 0.1

0.08

[ dB / m ] 4.2 GHz

att 0.06 6.4 GHz α 8.5 GHz 0.04 10.6 GHz 12.7 GHz 0.02 attenuation, 0 -140 -130 -120 -110 -100 -90 -80

input power, Pin [dBm]

Figure 4.11 Attenuation constant αatt (dots) versus input power at the waveguide Pin measured for frequencies between 4.2 GHz and 12.7 GHz. The top axis indicates the average number of photons on resonance n. the electromagnetic ﬁeld within the resonator [Goetz16, Romanenko17]

0 δTLS δTLS(Pr) = p (4.17) 1 + Pr/Pc

0 with the low power TLS loss δTLS and a characteristic power Pc depending on the dielectric material. Following this model we conclude that the ﬁeld inside the dielectric material of the coaxial cable (ldPTFE) does not saturate the individual TLS in the measured power range [Von Schickfus77, Lindström09], since no power dependence of the attenuation of the coaxial cable is observed.

Dependence on Ambient Magnetic Field

We compare the extracted attenuation constants of CC085NbTi cables within (length 110 mm) and without (length 810 mm) a cryoperm magnetic shield otherwise using the same measurement setup as described in Section 4.3. We find no significant effect on the attenuation constants at 4K and BT (Fig. 4.12). Since we expect the internal loss to be the sum of the individual loss contributions we argue that the measured attenuation constants are not limited by an ambient magnetic field which is assumed to be dominated by the isolators installed at the BT-stage of the dilution refrigerator (see Fig. 3.7). 4.7 Conclusion and Outlook on Extended Waveguide Measurements 125

0.100

0.050 [ dB / m ] att α 4K cryoperm 4K 0.010 BT cryoperm attenuation, 0.005 BT

4 6 8 10 12

frequency, ν [GHz]

Figure 4.12 Comparison of the attenuation constants measured at 4K and BT with and without cryoperm magnetic shielding for CC085NbTi. The solid lines are extracted from ﬁts to the measured quality factors according to the model of Eq. (4.7).

4.7 Conclusion and Outlook on Extended Waveguide Mea- surements

We present measurements of the attenuation constant of commonly used, commercially available low-loss coaxial cables and rectangular waveguides down to millikelvin temperatures in a frequency range between 3.5 and 12.8 GHz using a resonant-cavity technique. In this method, we employ weak couplings to the waveguides resulting in resonant standing waves and measure their quality factors. We extract the loss tangent and relative permittivity of different dielectric materials by comparing our measurement results to established loss models. The frequency dependence of the internal quality factors of the normal conducting waveguides are well described by these loss models, while the tested CC085NbTi and CC085Nb show small deviations from the predictions for the high-frequency dissipation in superconductors [Mattis58, Tinkham04]. These small deviations could be investigated in further studies using detection lines with an increased bandwidth. In addition, the coupling mechanism of the coaxial cable and waveguide measurements could be studied more systematically to exactly understand their impact of the extracted attenuations constants. Based on our measurement results (Fig. 4.9b) we conclude to use rectangular waveguides for ﬁrst long distances tests in our cryogenic prototype system with a 126 Chapter 4 Characterization of Coaxial and Rectangular Microwave-Frequency Waveguides

length of 4 m (Fig. 3.10). We decide for WR90Al waveguides since they show the best microwave performance. In addition, WR90Al do not have any additional coating compared to WR90Alc which could decrease the thermal contact conductance at the thermalization clamps (Section 3.1.6), and WR90Al are available by default up to 10 m from Penn Engineering which is not guaranteed for the WR90CuSn since the tin coating is custom-made for our test. We install a WR90Al waveguide of a total length of 3.65 m, consisting of two sections of length 1.15 m and 2.5 m, inside our cryogenic prototype system. We use the same measurement procedure as described 3 in Section 4.6 and obtain attenuation constants below 10− dB/m = 1 dB/km (Qint > 6 2 10 ) [Meinhardt18]. These measured attenuation constants are signiﬁcantly lower ·

a) 12 inches 3.65 m 1×106

5 int 5×10 Q

1×105

0.010 [ dB / m ] 0.005 att α

0.001 attenuation, 5.×10-4 8 10 12

frequency, ν [GHz]

Figure 4.13 a) Measured internal quality factor Qint and b) attenuation constant αatt versus frequency ν for WR90Al waveguides of length 12 inches (same data as in Fig. 4.9b, blue) and 3.65 m (gray). The measurements are performed at BT with measured temperatures of

80 130 mK at the waveguide and for nBT 1. The 3.65 m-waveguide consist of a 1.15 m and− a 2.5 m section [Meinhardt18]. ≈ 4.7 Conclusion and Outlook on Extended Waveguide Measurements 127

than the ones measured on the 12 inches WR90Al section (Fig. 4.13). To investigate the cause and the spread of αatt further measurements need to be performed on these 6 extended waveguides. However, Qint > 5 10 have been reported for aluminum-6061 · 3D-cavities [Reagor13] suggesting that even lower attenuation constants might be achievable with improved surface treatment of rectangular waveguides. Using the measurement results of the 3.65 m WR90Al waveguide, we estimate that microwave signals can be transmitted on a single photon level up to tens of meters with minimal loss. For example, we ﬁnd that more than 99 % of an incident signal can be transmitted over distances of approximately 70 m. 128 Chapter 4 Characterization of Coaxial and Rectangular Microwave-Frequency Waveguides Chapter 5

Quantum Communication with Microwave Photons

Quantum communication techniques are essential to build quantum networks for distributed quantum computing and long-distance quantum information transfer, as discussed in the introduction. In this chapter, we start with an overview of experimentally realized quantum communication protocols in various physical systems in the optical and microwave regime. In the further sections, we focus on quantum communication with single shaped microwave photons via a direct quantum channel. We discuss our experimental implementation based on a cavity assisted Raman-type transition f , 0 g, 1 (see Section 2.2.8), and describe the calibration routines to emit and |absorb〉 ↔ single | 〉 photons with high probability. We present our experimental results of a deterministic quantum state transfer and generation of entanglement between two distant circuit QED systems, which we compare to master equation simulations. For the deterministic transfer we map our local qubits states to flying qubits, which are in a superposition of the vacuum state 0 and the single photon Fock state 1 (Fock- state encoding). In addition, we present| 〉 a protocol in which the flying qubits| 〉 are encoded as a time-bin superposition of single photons to detect errors during the quantum communication (time-bin encoding). Having presented the experimental details, we discuss the expected violation of the CHSH-inequality based on a effective loss model and master equation simulations. For future quantum communication protocols we show first experimental results of a novel all-microwave two-qubit gate based on the f , g g, e transition between two transmons and conclude with an outlook. The| remote〉 ↔ | quantum〉 communication sections are based on our publications [Kurpiers18, Kurpiers19]. 130 Chapter 5 Quantum Communication with Microwave Photons

ion vs ae rec sa nv sc qd sab a 104 Hz ] ) [ 102

ent 1 Γ 10 2 -4 rate, 10 - b 2.8 ) S 2.6 2.4 2.2 CHSH, 2.

c 풞 1. ) 0.8 0.6 0.4 0.2

concurence, 0.

d ℱ 1. ) 0.9 0.8 0.7 0.6 0.5 state fidelity, ( 2014 ) ( 2011 ) ( 2007 ) ( 2005 ) ( 2009 ) ( 2018 ) ( 2016 ) ( 2015 ) ( 2012 ) ( 2019 ) ( 2016 ) ( 2018 ) ( 2007 ) ( 2014 ) ( 2006 ) ( 2008 ) ( 2012 ) ( 2018 ) ( 2013 ) ( 2017 ) ( 2018 ) ( 2013 ) ( 2012 ) ( 2011 ) ( 2008 ) ( 2007 ) ( 2017 ) ( 2018 ) ( 2014 ) Lee et al. Pfaff et al. Roch et al. Yuan et al. Ritter et al. Narla et al. Chou et al. Chou et al. Hucul et al. Dickel et al. Axline et al. Delteil et al. Leung et al. Ibarcq et al. Laurat et al. Maunz et al. Lettner et al. Stockill et al. Usmani et al. Hensen et al. Bernien et al. Kurpiers et al. Slodicka et al. Hofmann et al. Moehring et al. Rosenfeld et al. Humphreys et al. Matsukevich et al. Matsukevich et al. Campagne -

Figure 5.1 Overview of remote entanglement experiments. a) Entanglement generation

rate Γent, b) CHSH-value S, c) concurrence C and d) entangled state ﬁdelity Fs. The experiments are grouped by physical system: atomic ensembles (ae) [Chou05, Matsukevich06, Chou07, Laurat07, Yuan08], trapped ions (ion) [Moehring07, Matsukevich08, Maunz09, Slodiˇcka13, Hucul14], single atom Bose-Einstein-condensate (sab) [Lettner11], vibrational state of diamonds (vs) [Lee11], rare-earth doped crystals (rec) [Usmani12], single atoms (sa) [Hofmann12, Ritter12], nitrogen-vacancy (nv) centers [Bernien13, Pfaff14, Hensen15], superconducting circuits (sc) [Roch14, Narla16, Dickel18, Campagne-Ibarcq18, Axline18, Kurpiers18, Leung19] and quantum dots (qd) [Delteil16, Stockill17]. The colors indicate different implementations: probabilistic unheralded (red), probabilistic heralded (blue), guaranteeing a deterministic delivery of an entangled state at a pre-speciﬁed time (yellow), and fully deterministic (green). The plot markers indicate different schemes to realize the remote interaction: measurement-induced (triangle), single- (cross) or two- photon (squares) interference and detection, direct transfer (diamond), direct transfer with shaped photons (circles). The lines in (c) are bounds [Verstraete02] on the concurrence C calculated from measured CHSH-values S (see Section 2.1.5). 5.1 Literature Overview 131

5.1 Literature Overview

In Fig. 5.1 we present an overview of remote entanglement experiments and compare relevant experimental parameters. The remote entanglement experiments have been performed in various physical systems using optical photons (with atomic ensembles [Julsgaard01, Chou05, Matsukevich06, Chou07, Laurat07, Yuan08], trapped ions [Moehring07, Matsukevich08, Maunz09, Slodiˇcka13, Hucul14], single atom - Bose Ein- stein condensate [Lettner11], vibrational state of diamonds [Lee11], rare-earth doped crystals [Usmani12], single atoms [Hofmann12, Ritter12], nitrogen-vacancy centers [Bernien13, Pfaff14, Hensen15], quantum dots [Delteil16, Stockill17]), or microwave photons (superconducting circuits [Roch14, Narla16, Dickel18, Axline18] [Campagne-Ibarcq18, Kurpiers18, Leung19, Kurpiers19]). In these experiments quantum communication schemes have been implemented which generate remote entanglement probabilistically based on a joint measurement which projects the remote systems on an entangled state, based on the scattering of a single photon or two photons from the two remote systems which is/are interfered on a beam splitter and detected af- terwards, and based on a direct transfer of photons between the two systems. These probabilistic protocols can be implemented using heralding techniques which allow one to re-transmit the quantum information in case an error during the protocol is detected. Heralding protocols can also provide deterministic remote entanglement at predetermined times [Humphreys18]. A fully deterministic quantum communication can be achieved using a direct transfer of photons with a time-reversible shape (see Section 2.2.8) which is detailed in the rest of this chapter.

5.2 Quantum Communication via Direct Quantum Channels

In our adaptation (Fig. 5.2) of the protocol proposed by Ignacio Cirac (see Ref. [Cirac97] and Section 2.2.8), each quantum node consists of a transmon dispersively coupled to two coplanar microwave resonators, analogous to an atom coupled to two cavity modes. For details on the chip see Tab. 2.1 and Fig. 2.17. One resonator is dedicated to dispersive transmon readout and the second one to excitation transfer. The transfer resonators of the two nodes are tuned to have matching frequencies νT 8.400 GHz. ≈ All resonators are coupled to dedicated Purcell ﬁlters [Reed10, Jeffrey14]. An external coaxial line of 0.9 m length, bisected with a circulator, connects the transfer circuits 132 Chapter 5 Quantum Communication with Microwave Photons

g(t) eff g(- t)

b) chip A transfer chip B fAWG transmon 0.4 m 0.4 m

readout FPGA

Figure 5.2 a) Quantum optical schematic of a deterministic unidirectional entanglement protocol between two cavity QED nodes of a quantum network. At the first node, a three level system is prepared in its second excited state f (gray half-circle) and coherently | 〉 driven (g˜f0g1(t), blue arrow) to g (blue half-circle) creating the transfer cavity field 1 (light yellow). The cavity field| couples〉 into the directional quantum channel with rate| 〉 p κT as a single photon wavepacket with an effective bandwidth κeff (yellow hyperbolic secant shape). In the second quantum node, the time reversed drive g˜f0g1( t) transfers the excitation from g to f in the presence of the transferred photon field−1 . Finally, the protocol is completed| 〉 with| 〉 a transfer pulse between f and e (red half-circle)| 〉 to return to the qubit subspace. Additionally, each three level system| 〉 is| coupled〉 to a readout cavity (gray). b) Implementation of the system depicted in (a) in a planar, chip-based, circuit QED architecture (Fig. 2.17). A directional quantum channel between the two nodes is realized using two semi-rigid coaxial cables and a circulator which connect the output ports of the transfer circuit Purcell filters. of both chips. With this setup, photons are routed from node A to B, and from node B to a detection line [Huthmacher14]. As detailed in Section 2.2.8 we generate a controllable light-matter interaction, by applying a coherent microwave tone to the transmon which induces an effective interaction g˜f0g1(t) with tunable amplitude and phase between states f , 0 and g, 1 (f0g1) [Pechal14, Zeytinoglu15˘ ]. The two lowest energy eigenstates g| , e〉 of the| transmon〉 form the qubit subspace and the second excited state f is used| 〉 | as〉 an auxiliary level to control the matter-light interaction in our experiment.| 〉 The f0g1-interaction swaps an excitation from the transmon to the transfer resonator, which then couples to a mode propagating towards node B. By controlling g˜f0g1(t), we shape the itinerant photon to have a time-symmetric envelope. By inducing the reverse process g, 1 f , 0 with the time reversed amplitude and | 〉 ↔ | 〉 phase profile of g˜f0g1(t), we absorb the itinerant photon in the transmon at node B. Ideally, this procedure returns all photonic modes to their vacuum state. 5.3 Calibrations of the Excitation Transfer using Time-Symmetric Photons 133

5.3 Calibrations of the Excitation Transfer using Time-Symmetric Photons

We induce a phase and amplitude tunable coupling g˜f0g1(t), Eq. (2.95), between states f , 0 and g, 1 by applying a microwave tone to the transmon with drive amplitude | 〉 | 〉 Ωf0g1 at the resonance frequency of the f0g1-transition νf0g1 (see Section 2.2.8 for theoretical details). In this section, we discuss the calibration routines to emit and absorb time-symmetric single photons with constant phase. We present the calibration of the drive-induced ac Stark shift νf0g1(Ωf0g1), of the drive-strength dependent coupling rate g˜f0g1(Ωf0g1), and of the timing calibration between the two nodes.

Ac Stark Calibration

0 2 As shown in Eq. (2.94) the f0g1-transition frequency νf0g1(Ωf0g1) = νf0g1 µacΩf0g1 obtains a drive-induced quadratic ac Stark shift in a first order approximation.− To 0 calibrate ∆f0g1(Ωf0g1) = νf0g1(Ωf0g1) νf0g1 we first prepare the transmon in state f by − π| 〉 initializing it in the ground state [Magnard18, Egger18] and use a sequence of Rge and π Ref. Next, we apply a flattop f0g1-pulse with a Gaussian-shaped rise and fall edge Rf0g1 d (Fig. 5.3a). We set different pulse amplitudes Ωf0g1 and vary the drive frequency νf0g1 for each Ωf0g1 to obtain the ac-Stark-shifted resonance frequency of the f0g1-transition

(Fig. 5.3b). We use a pulse duration τf0g1 which is inversely proportional to Ωf0g1 to keep the area below the f0g1-pulse approximately constant. For the first calibration round, we choose τf0g1 based on the expected g˜f0g1 and update τf0g1 in the following rounds using the calibration of g˜f0g1 (see coupling rate calibration). We extract ground d state population Pg for each Ωf0g1 and νf0g1 (Fig. 5.3b), and fit a Gaussian function to the Pg to obtain νf0g1(Ωf0g1) from the center of the Gaussian fit. Plotting ∆f0g1(Ωf0g1) versus Ωf0g1 we observe quadratic ac Stark shifts over the range of the applied drive amplitudes (Fig. 5.3c).

Coupling Rate Calibration

To calibrate the dependence of g˜f0g1 on Ωf0g1 we perform a Rabi-type measurements. Similar to the ac Stark calibration, we prepare the transmon in f and use the same | 〉 ﬂattop f0g1-pulse with a Gaussian-shaped rise and fall edge. To obtain g˜f0g1(Ωf0g1) we set Ωf0g1 and sweep τf0g1 for each Ωf0g1 (Fig. 5.3d). We use the results of the ac

Stark shift calibration to ﬁx the frequency νf0g1(Ωf0g1) to be on resonance with the f0g1-transition. We extract Pg, Pe, Pf and obtain damped Rabi oscillations due to the 134 Chapter 5 Quantum Communication with Microwave Photons

τf0g1 a) 1 Rπ Rπ R ν d) 1 Rπ Rπ R 0.60.8 ge ef f0g1 f0g1 0.60.8 0.60.8 ge ef f0g1 0.60.8 qt0.4 A 0.4 0.4 0.4 0.20 00.2 0.20 00.2 0 30 60 90 0 30 60 90

pulse duration, τ [ns] pulse duration, τ [ns] b) 1 e) 1 s s 0.8 0.8 P 0.6 0.6 g Pe 0.4 0.4 Pf 0.2 0.2 state populations, P state populations, P 0 0 4010 4015 4020 4025 4030 0 20 40 60 80 100 120 140 d d c) f0g1 frequency, νf0g1 [MHz] f) f0g1 pulse duration, τf0g1 [ns]

0 6 [ MHz ]

-3 [ MHz ] / 2 π 4 / 2 π g -6 ˜ f0g1 Δ -9 2 -12 A

eff. coupling, B -15 0 ac Stark shift, 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1

drive amplitude, Ωf0g1 [norm.] drive amplitude, Ωf0g1 [norm.]

Figure 5.3 Pulse scheme of a) the ac Stark and d) Rabi rate calibration of the f , 0 to g, 1 |d 〉 | 〉 transition. b) Measured population Pg (dots) versus f0g1 drive frequency νf0g1 obtained from a pulsed spectroscopy measurement of the ac Stark shifted f0g1-transition frequency

and Gaussian ﬁt (solid line) to the data. c) ac Stark shifts ∆f0g1/2π versus drive ampli- A tude Ωf0g1 for node A and B. ∆f0g1/2π = 0 MHz corresponds to νf0g1 = 4.022 GHz and B νf0g1 = 3.485 GHz (see Tab. 2.1). Each measurement point (dots) is extracted based on a spectroscopic measurement similar to (b - indicated by cross). The solid lines are a quadratic

fit to the data. e) Transmon state populations versus pulse duration τf0g1 (dots), and fit of the damped Rabi model (solid line) explained in the main text. f) Rabi rate g˜f0g1/2π versus Ωf0g1 for node A and B. Each measurement point (dots) is obtained from a Rabi-type measurement similar to (e - indicated by cross). The solid lines are linear fits to the data. 5.3 Calibrations of the Excitation Transfer using Time-Symmetric Photons 135

spontaneous energy relaxation of the resonator ﬁeld to the output line, g1 g0 | 〉 → | 〉 (Fig. 5.3e). We ﬁt Pf [Magnard18] using

ff(τf0g1) = µa pf (τf0g1 τ0) + µb (5.1) − with µa,b to account for potential state preparation and measurement (SPAM) errors and τ0 to account for the Gaussian-shaped rise and fall edge of the ﬂattop pulse. We extend the non-Hermitian Hamiltonian Eq. (2.101), which acts on the states f , 0 and g, 1 | 〉 | 〉 iγ /2 g˜ ˆ f0g1 1ef f0g1 Heff = − (5.2) g˜f0g1∗ iκT/2 − by considering energy relaxation of the f level γ1ef and obtain | 〉 κ γ τ 2 ( T+ 1e f ) ΩRabiτ κT γ1ef ΩRabiτ p e 2 cosh sinh (5.3) f(τ) = − + − 2 2ΩRabi 2

Ç 2 2 with ΩRabi = 4g˜f0g1 + (κT γ1ef) /4. For g˜f0g1 > (κT γ1ef)/4, ΩRabi is purely imag- − − − inary and pf(τ) shows oscillation, else ΩRabi is real and pf(τ) is not oscillating, since the system is overdamped. We ﬁt the measurements results for all Ωf0g1 simultaneously i i using same κT, µa,b, τ0 and g˜f0g1 for each Ωf0g1 as ﬁt parameters. For the drive amplitudes used in our experiments we obtain a linear relation between g˜f0g1 and Ωf0g1

(Fig. 5.3f) showing that larger g˜f0g1 can be obtained in our setup before observing non-linear deviations [Zeytinoglu15˘ ].

Emission of Time-Symmetric Photons

1 q p p To emit photons with a time-symmetric shape 2 κeff sech κeff t/2 , Eq. (2.100), and a constant phase [Pechal14, Pechal16a], we calculate the time-dependent drive amplitude Ωf0g1(t) based on the previous calibration to obtain the effective coupling Eq. (2.102)

p p p p κefft κefft κeff 1 e + (1 + e )κT/κeff g˜f0g1(t) = p − (5.4) Ç κp t p κp t 4 cosh(κeff t/2) eff eff (1 + e )κT/κeff e − p with an effective photon bandwidth κeff κT. Using the g˜f0g1-calibration we relate ≤ the drive-amplitude Ωf0g1 to g˜f0g1 by ﬁtting Ωf0g1 versus g˜f0g1 with a linear ﬁt function f x c x. To set the drive-frequency to be resonant with the f0g1-transition g˜f0g1ToΩ( ) = 1 136 Chapter 5 Quantum Communication with Microwave Photons

for all Ωf0g1, we use the ac Stark calibration. We ﬁt ∆f0g1 versus g˜f0g1 with a quadratic ﬁt function f x c x 2. We obtain the time-dependent phase of the f0g1-pulse g˜f0g1Toν( ) = 2 by integrating the differential equation φ˙ t f g˜ t and obtain the f0g1( ) = g˜f0g1Toν( f0g1( )) experimental pulse shape

Ω t f g˜ t eiφf0g1(t). (5.5) f0g1( ) = g˜f0g1ToΩ( ( ))

To control the f0g1-pulse independently of the ge- and ef-drive, we used a different RT pulse synthesizer which we combined before entering the dilution refrigerator (Fig. 3.7). We adjusted the output voltage of the AWG, the gain of the RT ampliﬁer max A and the attenuation of the f0g1-drive line to obtain approximately g˜f0g1/2 κT which p A B ∼ limited our photon bandwidth κeff = κT < κT. For the absorption process of the photon B p we time reverse the f0g1-drive at node B (Fig. 5.4a) which is asymmetric since κT > κeff (see Eq. (5.4)).

Two-Node Timing Calibration

In addition, to the single-node f0g1-calibrations we optimize the state transfer probability by calibration the timing between the two nodes. To calibrate the timing between the nodes, we start by preparing transmon A in state f . Next, we emit time-symmetric photons from node A using procedure described in| this〉 section, and absorb the photons at node B by time-reversing the emission pulse of node B. We calibrate the timing between node A and B by varying the pulse start time of the absorption pulse at node B (Fig. 5.4a). We extract the population of transmon B and observe a maximum of

Pf which we define as the optimal time delay between node A and B (Fig. 5.4b). To obtain a higher resolution and mitigate noise we use a quadratic fit to Pf to find the optimal time delay. In conclusion, we have implemented an efficient calibration routine to emit and absorb microwave photons in circuit QED systems. We have automated this calibration routine using our QubitCalib software framework [Heinsoo13, Storz16] making it easily useable and extendable for future experiments (see Section 5.11). The calibration routine could be extended by DRAG techniques [Motzoi09, Gambetta11] or by numerically-optimized photon shapes which can minimize driving unwanted transitions and can improve the photon transmission fidelity. These calibration techniques become especially relevant if multiple transmons are used at each node [Schär19] or higher drive amplitudes. Furthermore, asymmetric photon shapes can be investigated to optimize the photon duration, and the emission of photons which are detuned from 5.4 Emission and Absorption of Microwave Photons 137

Figure 5.4 a) Pulse scheme of the timing calibration between node A and B. b) Transmon

state populations of node B versus delay of the pulse start time at node B ∆tf0g1 (dots) and a quadratic ﬁt to Pf. the resonance frequency of the resonator can be studied to lower the requirement of a small frequency detuning of the resonators at the two nodes [Marxer19].

5.4 Emission and Absorption of Microwave Photons

To characterize the excitation transfer, we start by initializing the transmon at the receiving node B in state f . Next, we induce the effective coupling g˜f0g1(t) with τ | 〉 Rf0g1 to emit a symmetric photon (Fig. 5.5a). Here, and in all following measurements of the characterization of quantum communication protocols, the population of the transmon states are extracted using single-shot readout with a correction to account for measurement errors (see Section 2.2.6). The populations of the three lowest levels of the transmon Pg, Pe and Pf are measured immediately after truncating the emission τ pulse Rf0g1 at time τ (see Fig. 5.5b). In this way, we observe that the transmon smoothly evolves from f to g during the emission process. At the end of the protocol the | 〉 | 〉 emitting transmon reaches a ground state population Pg = 95.8% characterizing the limitation of the emission efﬁciency. 138 Chapter 5 Quantum Communication with Microwave Photons

Figure 5.5 The transmons at a) node B and d) node A are prepared in the state f . We | 〉 characterize (dots) the time dependence (τ) of the qutrit populations Pg,e,f (b, e) while driving the f , 0 to g, 1 transition (f0g1). The phase (white-blue shading) of the f0g1 drive is modulated| to〉 compensate| 〉 the drive-induced quadratic ac Stark shift (see Section 5.3). The 2 mean ﬁeld amplitude squared aout(τ) of the traveling photons emitted from c) node B and f) node A is obtained for the| 〈 emitted〉 | photon state ( 0 + 1 )/p2. The effective photon eff | 〉 eff| 〉 bandwidths are adjusted to be κp,A/2π = 10.4 MHz and κp,B/2π = 10.6 MHz. The solid lines in (b, c, e, f, h, i) and i) are results of master equation simulations (see Section 2.2.8 for

details). The time dependence of Ps when executing the excitation transfer protocol (pulse shapes in dashed rectangle in g) from qubit A to qubit B (h) are extracted simultaneously 2 with the amplitude of the emitted ﬁeld from node A. i) shows the residual aout(τ) (light yellow x50) during the absorption process. | 〈 〉 |

To verify that the emitted photon envelope has the targeted shape and bandwidth eff κp,B/2π = 10.6 MHz, we repeat the emission protocol with an initial transmon state ( g + f )/p2 and measure the averaged electric field amplitude aôut(t) of the emit- | 〉 | 〉 〈 〉 ted photon state ( 0 + 1 )/p2 using heterodyne detection [Bozyigit11] (Fig. 5.5c). | 〉 | 〉 We prepare this photon state because of its non-zero average electric field [Pechal14]. Repeating the emission protocol from node A leads to similar dynamics of the transmon population (see Fig. 5.5e). We adjust the amplitude and phase of the transfer pulse (Fig. 5.5d) so that the photons emitted from each node A and B have similar p A effective bandwidth κeff κT . In Fig. 5.5f, the envelope of the photon emitted from node A is slightly distorted≈ by the reflection at node B as determined by the response 5.4 Emission and Absorption of Microwave Photons 139

function of its transfer resonator, which is fully captured by our theoretical model.

R 2 The detected integrated power aôut(t) d t of the photon emitted from node |〈 〉| A is lAB = 23.0 0.5% lower than that emitted from node B. lAB = 1 ηAB is a measure of the photon± loss between node A and B since we detect the field− of each of the emitted photons in the same detection line by making use of the circulator between the two nodes (Fig. 5.2). The path traveled by the two emitted photons towards the detector differs only by the length of the waveguide separating the two samples from each other. We evaluate the ratio of the integrated power of the detected R A 2 R B 2 fields aôut(t) d t/ aôut(t) d t to extract the photon loss lAB between node A and B.|〈 In addition,〉| we|〈 estimate〉| the photon loss between node A and B based on the specifications of the individual elements connecting the nodes: two printed circuit boards including connectors (each 2.5 1%), two coaxial cables of length ± 0.4 m (each 4.0 0.1%) [Kurpiers17] and a microwave circulator (13 2% according to manufacturer).± With these parameters we estimate an accumulated± photon loss between node A and B of 24 3% in good agreement with the measured value of ± lAB = 23.0 0.5%. ± To characterize the absorption of the single time-symmetric photon emitted from node A at the receiving node by time-reversing the emission pulse of node B (Fig. 5.5a and g), we measure the population of transmon B during the process. For this characterization, we apply a π-pulse to transmon B to map f back to the qubit subspace before performing the readout. We observe the population| 〉 of e to smoothly rise and sat | 〉 saturate at Pe = 67.5 % (Fig. 5.5h). This saturation level reflects the efficiency of the protocol for the transfer of a single photon, which is executed in a pulse sequence of 180 ns duration (Fig. 5.5g). From the ratio of the integrated power of the emitted photon in the absence (Fig. 5.5i) or presence (Fig. 5.5f) of the absorption pulse, the absorption efficiency is determined to reach 98.1 0.1%. We estimate that the finite truncation of the f0g1-pulses contributes with 1±.1% to the inefficiency and leakage to the second, weakly-coupled input port of the∼ Purcell filter with 0.8% (Fig. 2.17a). ∼ The results of master equation simulations of the excitation transfer (solid lines in Fig. 5.5), using parameters extracted from independent measurements (see Sec- tion 2.2.8), display excellent agreement with the measured data. This demonstrates a high level of control over the emission and absorption processes and an accurate understanding of the experimental imperfections dominated by qutrit decoherence and photon loss Eq. (2.112). 140 Chapter 5 Quantum Communication with Microwave Photons

5.5 Deterministic Quantum State Transfer

We demonstrate the use of the presented protocol to deterministically transfer an arbitrary qubit state via a direct quantum channel from node A to node B at a rate of 50 kHz. π This is realized by preparing the receiving transmon B in state g , applying a Ref pulse | 〉 to the sending transmon A, followed by the emission/absorption pulse and ﬁnally a ro- π tation Ref on transmon B. We characterize the quantum state transfer by reconstructing its process matrix χdet with quantum process tomography [Chuang97] (Fig. 5.6b,c). For that purpose, we prepare each of the six mutually unbiased qubit basis-states g , | 〉 e , ( g + e )/p2, ( g +i e )/p2, ( g e )/p2, ( g i e )/p2 (see Ref. [vanEnk07] | 〉 | 〉 | 〉 | 〉 | 〉 | 〉−|x 0〉 x π/2| x〉− π/| 2〉 y π/2 y π/2 x π and Section 2.1.1) using the rotations Rge, Rge , R−ge , Rge , R−ge and Rge at node A, transfer them to node B, and reconstruct the transferred state using quantum state tomography (QST). QST of a single qutrit [Bianchetti10b, Bianchetti10a] is performed by measuring the qutrit state population with the single-shot readout method x 0 x π/2 described in Section 2.2.6, after applying the following tomography gates: Rge, Rge ,

Figure 5.6 a) Pulse scheme used to characterize the qubit state transfer between the two ζ φ nodes. We prepare six mutually unbiased input states at node A (denoted by Rge). b) Ex- perimentally obtain process matrix (absolute value shown as colored bars). The gray and red wire frames show the ideal and the master equation simulation of the absolute values of the process matrix, respectively. c) Bloch sphere representation of the quantum states transfer of the six ideal input states states at node A (shown as red dots on the upper sphere) and the corresponding, experimentally obtained output states at node B (shown as green dots on the lower sphere). 5.6 Deterministic Generation of Remote Entanglement 141

y π/2 x π x π/2 y π/2 x π x π/2 x π y π/2 x π x π Rge , Rge, Ref , Ref , ( Rge Ref ), ( Rge Ref ) and ( Rge Ref). The elements of the density matrix are then reconstructed with a maximum-likelihood method [Smolin12], assuming ideal tomography gates. We obtain the process matrix through linear inver- sion, from these density matrices [Steffen13a]. Performing QST of the qutrit subspace is required to characterize residual population in f after the qubit state transfer, which | 〉 π is mainly caused by energy relaxation from f level in combination with a Ref swapping e with f populations. The obtained| density〉 matrices have a non-unit trace in the qubit| 〉 subspace| 〉 and so does the qubit state transfer process matrix. Consequences of that observation are discussed in Section 5.6 on the example of the Bell state density det matrix. We determine a process ﬁdelity of Fp = tr(χdetχideal) = 80.02 0.07%, well above the limit of 1/2 that can be achieved using local gates and classical± communication only. The process matrix χsim calculated with the master equation simulations agrees very well with the data (absolute values shown as red wire frames in Fig. 5.6b).

This is supported by the small trace distance [Nielsen11] tr χdet χsim /2 = 0.015, which ideally is 0 for identical process matrices and 1 for orthogonal| − ones.|

5.6 Deterministic Generation of Remote Entanglement

We use a small modiﬁcation of the excitation transfer protocol to deterministically generate two-qubit remote entangled states between nodes A and B (Fig. 5.8a). The protocol starts by preparing transmons A and B in states ( e + f )/p2 and g , re- | 〉 | 〉 | 〉 π spectively, and applying the emission/absorption pulses followed by a rotation Ref on transmon B to generate the entangled Bell state ψ+ = ( e, g + g, e )/p2. Al- ternatively, a remote entangled state can be generated| 〉 by preparing| 〉 | the〉 transmon π/2 at node A in f , 0 , swapping half of the population to g, 1 (Rf0g1) and using the same g, 1 | f , 0〉 absorption pulse at node B as actually| realized〉 in our experiment. |π/2 〉 ↔ | 〉 The Rf0g1 can be used to decrease the emission time, however, the absorption process requires the same time. Thus, the full alternative protocol is not shorter than our implementation. As leakage to the f level at both nodes leads to errors in the two-qubit density | 〉 det matrix reconstruction, we extract the full two-qutrit density matrix ρˆ3 3 from QST experiments (Fig. 5.7). Therefore, we extend the QST procedure described⊗ in Sec- tion 5.5 to two-qutrit density matrices. We perform two local tomography gates, from the 81 pairs of gates that can be formed from the single-qutrit QST gates, on transmons A and B, and extract the state populations using individual two-qutrit single shot measurements. We observe a residual population of 3.5% of the f level of the | 〉 142 Chapter 5 Quantum Communication with Microwave Photons

Figure 5.7 Two-qutrit state tomography: a) real and b) imaginary part of the density ˆ matrix and c) expectation values of the Gell-Mann operators λk (see Section 2.1.1). The ideal Bell state ψ+ is depicted with gray wire frames. The numerical master equation simulation is depicted| 〉 in red wire frames. The trace distance between the measurement and the simulation is 0.107. transmons after the entanglement protocol showing that the entangled state can not be described rigorously by a two-qubit density matrix. We verify the three-level bipartite entanglement, by using the computable cross norm or realignment (CCNR) criterion (see Section 2.1.5). We obtain ccnr = 1.612 0.003 with the measured entangled state det ± ρˆ3 3, witnessing unambiguously the existence of entanglement of the prepared state. ⊗ det For illustration purposes, we display the two-qubit density matrix ρˆm (Fig. 5.8b,c), det which consists of the two-qubit elements of ρˆ3 3. This choice of reduction from a two- ⊗ det + det + qutrit to a two-qubit density matrix conserves the state fidelity Fs = ψ ρˆm ψ = + det + det 〈 | | 〉 ψ ρˆ3 3 ψ [Nielsen11], however, ρˆm has a non-unit trace. In addition, this reduc- 〈 | ⊗ | 〉 tion method gives a conservative estimate of the concurrence C(ρˆm), compared to a det projection of ρˆ3 3 on the set of physical two-qubit density matrices. We find a state det ⊗ + det + fidelity Fs = ψ ρˆm ψ = 78.9 0.1% compared to the ideal Bell state, and a 〈 det| | 〉 ± concurrence C(ρˆm ) = 0.747 0.004. ± 5.7 Quantum Communication using Time-Bin Encoding 143

Figure 5.8 a) Pulse scheme to generate deterministic remote entanglement between nodes A and B. b) Expectation values of two-qubit Pauli operators and c) reconstructed density matrix after execution of the remote entanglement protocol. The colored bars indicate the measurement results, the ideal expectation values for the Bell state ψ+ = ( e, g + g, e )/p2 are shown in gray wire frames and the results of a master equation| 〉 simulation| 〉 in| red.〉

The density matrix ρˆsim obtained from the master equation simulations of the entanglement protocol (red wireframe in Fig. 5.8) is in excellent agreement with the det experimental results, displaying a small trace distance tr ρˆm ρˆsim /2 = 0.024. We decompose the infidelity into approximately 10.5% from photon− loss (see also Sec- tion 5.10), 9.5% from finite transmon coherence times, and 1% from photon absorption inefficiency.

5.7 Quantum Communication using Time-Bin Encoding

In this section, we extend the deterministic quantum state transfer scheme to a protocol which encodes the quantum information to be transmitted as a time-bin superposition of a single photon, making it suitable for heralded quantum communication in a direct quantum channel. The time-bin encoding transfer protocol relies on encoding the transmitted quantum information in a suitably chosen subspace S such that any error, which may be encountered during transmission, causes the system to leave this subspace. On the receiving end, a measurement which determines whether the system is 144 Chapter 5 Quantum Communication with Microwave Photons

in S but does not distinguish between individual states within S, can be used to detect if an error occurred. Crucially, when the transfer is successful, this protocol does not disturb the transmitted quantum information. The single photon Fock-state encoding used in Section 5.5 is not suitable to detect errors due to photon loss because the error does not cause a transition out of the encoding subspace 0 , 1 . {| 〉 | 〉} As in the deterministic case, the stationary quantum nodes are transmons coupled to coplanar waveguide resonators. The two lowest energy eigenstates of the transmon, g and e , form the qubit subspace and the second excited state, f , is used to detect potential| 〉 | 〉 errors. A small adaptation of the protocol also allows for| 〉 remote, heralded entanglement generation (see Section 5.9). The process for transferring quantum information stored in the transmon into a time-bin superposition state consists of the steps illustrated in Fig. 5.9a: The transmon qubit at node A is initially prepared in a superposition of its ground and ﬁrst excited state, α g + β e , and the resonator in its vacuum state 0 . Next, two pulses are | 〉 | 〉 | 〉 applied to transform this superposition into α e + β f . Then, a f0g1-pulse induces the transition from f , 0 to g, 1 , which is followed| 〉 | by〉 spontaneous emission of a | 〉 | 〉

a) node A: sender b) node B: receiver

Figure 5.9 Schematic representation of the pulse sequence implementing the time-bin encoding process at a) the sender and b) the receiver. In the level diagrams, the gray vertical lines represent Hamiltonian matrix elements due to microwave drive between transmon levels while the diagonal ones show the transmon-resonator coupling (Fig. 2.12). a) The qubit state is initially stored as a superposition of g, 0 (blue dot) and e, 0 (red dot). The ﬁrst two pulses map this state to a superposition| of〉 e, 0 , f , 0 and| the〉 third pulse transfers f , 0 into g, 0 while emitting a photon in time bin| a〉(see| red〉 symbol for photon in mode a| ) using〉 the| f0g1-technique.〉 Then, after e, 0 is swapped to f , 0 , the last pulse again transfers f , 0 into g, 0 and emits a photon| in time〉 bin b. b) Reversing| 〉 the protocol we reabsorb the| photon,〉 mapping| 〉 the time-bin superposition back onto a superposition of transmon states. 5.7 Quantum Communication using Time-Bin Encoding 145

photon from the resonator. The shape of the f0g1-pulse is chosen such that the photon is emitted into a time-symmetric mode centered around time ta. After this ﬁrst step, the system is in the state α e, 0 0 + β g, 0 1a , where 0 and 1a denote the vacuum state of the waveguide| 〉 and⊗ | 〉 the single-photon| 〉 ⊗ | 〉 state in| 〉 the time-bin| 〉 mode a. Next, the population from state e is swapped into f and the photon emission process is repeated, this time to create a| single〉 photon in a| time-bin〉 mode b centered around time t b. The resulting state of the system is g, 0 (α 1b + β 1a ). | 〉 ⊗ | 〉 | 〉 By reversing both f0g1-pulses in the time-bin encoding scheme (Fig. 5.9b) an incoming single photon in the time-bin superposition state α 1b + β 1a will cause the receiving transmon-resonator system, initialized in g, 0 , to| be〉 driven| 〉 to the state | 〉 α e, 0 +β g, 0 as the photon is absorbed. Thus, this protocol transfers the qubit state encoded| 〉 as| a superposition〉 of g and e from transmon A to transmon B. In short, the sequence is | 〉 | 〉

α g A + β e A g B | 〉 | 〉 ⊗ | 〉 → g A α 1b + β 1a g B | 〉 ⊗ | 〉 | 〉 ⊗ | 〉 → g A α g B + β e B (5.6) | 〉 ⊗ | 〉 | 〉 where we have omitted the states of the resonators and the propagating ﬁeld whenever they are in their respective vacuum states.

An important property of this time-bin encoding transfer protocol is its ability to detect photon loss in the communication channel. Indeed, if a photon is lost or not absorbed by the receiver, system B receives a vacuum state at its input instead of the desired single-photon state. This means that both absorption pulse sequences will leave transmon B in its ground state g which will be subsequently mapped into f by the final three pulses. The successful| 〉 quantum state transfer can thus be heralded| 〉 by performing a quantum non-demolition measurement on the transmon which distinguishes between f and the subspace spanned by g , e , but does not measure within this subspace.| Such〉 a binary measurement of a qutrit| 〉 | state〉 can, for example, be realized by suppressing the measurement-induced dephasing in the ge-subspace using parametric amplification and feedback [deLange14] or by engineering the dispersive shifts of two transmon states on the readout resonator to be equal [Jerger16a]. Fur- thermore, the f state excitation can be swapped to an ancilla transmon Ba which | 〉 can be read out independently of transmon B in single-shot [Heinsoo18]. The swap operation can be performed using established flux-based pulses [DiCarlo09, Baur12a] 146 Chapter 5 Quantum Communication with Microwave Photons

or an all-microwave gate driving the f , g into g , e (fgge) transition. Using the B Ba B Ba fgge-transition has the advantage that| it does〉 not| require〉 any additional hardware at RT since the drive line of transmon B can be used to swap the excitation to the qubit subspace of Ba combined with multiplexed-readout techniques [Heinsoo18]. First measurement results on a two-qubit test device are discussed in Section 5.12 illustrating the feasibility of this approach. In addition, the time-bin encoding protocol also detects failures of the state transfer due to energy relaxation at certain times during the protocol, e.g. if no photon is emitted from A due to energy relaxation to g before the ﬁrst time bin. We discuss the detection of qutrit energy relaxation based| on〉 quantum trajectories after the experimental results have been presented.

5.8 Time-Bin-Encoded Quantum State Transfer

We implement the time-bin encoding protocol in a nearly identical setup as was used for the deterministic quantum communication (Fig. 5.2 and Tab. 2.1), up to an exchange of the cryogenic coaxial circulator (Raditek RADC-8-12-Cryo) in the connection between the two samples with a rectangular waveguide isolator (RADI-8.3-8.4-Cryo-WR90) which affects the bandwidth of the transfer resonators due to its different impedance

(see range of κT in Tab. 2.1). We use the waveguide isolator due to an expected lower loss of 0.2 dB compared to the coaxial circulator of 0.6 dB according to the manufacturer.

However, our measurements indicate that the loss between node A and B, lAB, increased from approximately 23% to 28%. The increased loss and the different impedance due to the waveguide isolator might be explainable with our transfer resonator frequency of 8.4 GHz, which is at the edge of the speciﬁed frequency range of 8.3-8.4 GHz, and needs to be further investigated. To characterize the time-bin encoding protocol, we perform qutrit single-shot readout and use quantum process tomography, instead of the binary measurement at transmon B or a swap of the f state population to an ancilla transmon. We ini- | 〉 tialize both transmons in their ground states [Magnard18, Egger18] and subsequently prepare the qubit at node A in one of the six mutually unbiased qubit basis states (Fig. 5.6b). We then run the time-bin encoding and reabsorption protocol (Fig. 5.9) and implement quantum state tomography at node B for all six input states (Sec- tion 5.5). Directly after the tomography pulses, we read out the g , e and f states of transmon B with single-shot readout. Based on these single-shot| 〉 | 〉 measurements,| 〉 we can postselect experimental runs in which transmon B was not measured in the qst f state keeping on average Psuc = 64.6% of the data, and transferring qubit states | 〉 5.8 Time-Bin-Encoded Quantum State Transfer 147

Figure 5.10 a) Pulse scheme used to characterize the time-bin encoding protocol to transfer ζ φ qubit states between two distant nodes using quantum process tomography (QPT). Rij label (DRAG) microwave pulses. b) Real part of the qutrit density matrix ρˆcor for the input state = ( g e )/p2 after the state transfer protocol reconstructed using measurement- error|−〉 correction| 〉−| as〉 for the presented deterministic data. The magnitude of each imaginary pr part is < 0.017. c) Projection of ρˆcor (b) onto the ge-subspace ρˆcor performed numerically pr which we use to reconstruct the process matrix χcor (absolute value shown in (d)). The colored bars show the measurement results, the gray wire frames the ideal density or process matrix, respectively. The results of numerical master equation simulations are depicted as red wire frames.

qst at a rate Γqst/2π = PsucΓexp/2π 32.3 kHz with the repetition rate Γexp = 50 kHz of the experiment. Using the post-selected≈ data, we reconstruct the normalized density matrices ρˆps of the qubit output state at node B using the maximum-likelihood approach and not correcting for readout errors. In this post-selected approach, we obtain the process matrix χps of the quantum state transfer and compute an aver- ps aged state ﬁdelity of Fs = avg( ψin ρˆps ψin ) = 88.2 0.2% and a process ﬁdelity of 〈 | | 〉 ± 148 Chapter 5 Quantum Communication with Microwave Photons

ps Fp = tr(χpsχideal) = 82.3 0.2% relative to the ideal input states ψin and the ideal identity process, respectively.± | 〉 Detecting errors to herald successful transfers would require a measurement that discriminates between f and the subspace spanned by g , e , but does not measure within this subspace, as| described〉 in Section 5.7. To benchmark| 〉 | 〉 only the encoding part of the time-bin encoding protocol we can assume an ideal heralding measurement. We therefore reconstruct all six qutrit density matrices ρˆcor of the output state at node B using the same dataset as for post-selection but correct for measurement errors in the qutrit subspace (see Sec 2.2.6). These qutrit density matrices have a significant average population of level f of 39.1% indicating the detection of errors after the | 〉 qst time-bin encoding protocol, which is compatible with 1 Psuc of the post-selected analysis (Fig. 5.10b). Next, we project these density matrices− numerically onto the pr qubit ge-subspace ρˆcor (Fig. 5.10c), simulating an ideal error detection and reconstruct pr the process matrix χcor of the quantum state transfer (Fig. 5.10d). In this way, we find cor pr an average state fidelity of Fs = avg( ψin ρˆcor ψin ) = 93.5 0.1% and a process cor pr 〈 | | 〉 ± fidelity of Fp = tr(χcorχideal) = 90.3 0.2% based on these measurement-corrected matrices. This analysis allows us also± to compare the time-bin encoding protocol directly to the deterministic quantum state transfer (Section 5.5), in which we obtained det Fp = 80.02 0.07%. This comparison clearly shows the advantage of time-bin encoding to reduce± the effect of photon loss, when assuming perfect readout. In ps det addition, we demonstrated with the post-selected experiments that Fp > Fp with our current readout fidelities. Such direct comparison of fidelities, however, should be done with caution because it depends on the loss of the quantum channel. We expect the fidelity of the deterministic protocol to decrease linearly with loss, while the fidelity of the time-bin encoding protocol remains constant, with a linear decrease of its success probability. The time-bin encoding protocol thus complements deterministic protocols to perform heralded quantum communication in direct quantum channels. In addition, we analyze the sources of infidelity by performing numerical master equation simulations (MES) of the time-bin encoding protocol which we compare to the measurement-error corrected density and process matrices. We find excellent agreement with the experimental results, indicated by a small trace distance pr tr χcor χsim /2 = 0.03. The MES results indicate that approximately 5.5% of the infidelity can− be attributed to f e and e g energy relaxation at both transmons during the protocol. Pure| qutrit〉 → dephasing| 〉 | 〉 →explains | 〉 the remaining infidelity. 5.9 Time-Bin-Encoded Generation of Remote Entanglement 149

5.9 Time-Bin-Encoded Generation of Remote Entanglement

As in the deterministic case, we use a simple modification of the state-transfer protocol to generate entanglement between the distant nodes (Fig. 5.11a). Both transmon- resonator systems are first initialized in their ground states. The first two pulses of the remote-entanglement protocol prepare transmon A in an equal superposition state 1/p2( e + f ), followed by a pulse sequence which entangles the transmon state g and e| with〉 | the〉 time-bin qubit and maps the state of the time-bin qubit to transmon| 〉 B. This| 〉 process can be summarized as

1 ( e A + f A g B p2 | 〉 | 〉 ⊗ | 〉 → 1 g A 1a + e A 1b g B p2 | 〉 ⊗ | 〉 | 〉 ⊗ | 〉 ⊗ | 〉 → 1 g A e B + e A g B (5.7) p2 | 〉 ⊗ | 〉 | 〉 ⊗ | 〉 and, in case of an error, transmon B ends up in state f B (see also Section 5.10). | 〉 We perform post-selected experiments of the entanglement-generation protocol by selecting only experimental runs in which neither qutrit is measured in the f state using individual single-shot readout on both transmons. Under this condition,| 〉 we re- ent ps + + tain Psuc 61.5% of the data and obtain a Bell state fidelity of Fs = ψ ρˆps ψ = 82.3 0.4%≈ compared to an ideal Bell state ψ+ (Fig. 5.11e). In these〈 post-selected| | 〉 ± | 〉 ent experiments, we generate entangled states at rate Γent/2π = Psuc Γexp/2π 30.8 kHz. To benchmark the time-bin encoding entanglement protocol, we use full≈ two-qutrit state tomography of the transmons in which we correct for measurement errors with the same data set as for post-selection. The reconstructed density matrix (Fig. 5.11b,c) displays a high population of the gA, fB , eA, fB states, Pg f = 16.0% and Pe f = 21.4%, and small population of f , g ,| f , e 〉 |and f〉 , f ,P P 2.7%, which in- A B A B A B i= g,e,f f i = dicates that photon loss| is a significant〉 | 〉 source| of〉 error.{ We} project onto the qubit ge-subspace numerically and obtain a two-qubit density matrix, Fig. 5.11d, showing cor + pr + cor a fidelity of Fs = ψ ρˆcor ψ = 92.4 0.4%. Fs is the fidelity we obtain with an ideal, unit fidelity qubit〈 | readout.| 〉 For finite± fidelity in distinguishing between the pair of states g , e and the state f , as presented in Section 2.2.6, we expect a false positive heralding| 〉 of| 〉 the entanglement| 〉 protocol to occur with a probability of approximately 5%, and thus, an overall state fidelity of approximately 87%. Comparing these results det to the fully deterministic case (Section 5.6), Fs 79%, shows the potential of the proposed time-bin encoding protocol to generate remote≈ entanglement independent of 150 Chapter 5 Quantum Communication with Microwave Photons

Figure 5.11 a) Pulse scheme for generating remote entanglement between nodes A and B (see text for details). b) Real and c) imaginary part of the two-qutrit density matrix

ρˆcor reconstructed after execution of the time-bin remote entanglement protocol using measurement-error correction as in the deterministic case. The colored bars indicate the + measurement results, the ideal expectation values for the Bell state ψ = ( gA, eB + | 〉 | 〉 eA, gB )/p2 are shown as gray wire frames and the results of a master equation simulation | 〉 as red wire frames. d) Numerical projection of the ρˆcor onto the ge-subspace to obtain pr the two-qubit density matrix ρˆcor which is in excellent agreement with our MES (trace distance of 0.028). e) Post-selected two-qubit density matrix ρˆps without measurement- error correction. photon loss. Using a MES we attribute approximately 6.5% of the inﬁdelity to energy relaxation and the rest to dephasing. In order to investigate the robustness of the time-bin encoding protocol with respect to qutrit energy relaxation, we simulate a ﬁctitious experiment with the same device parameters in which we monitor quantum jumps from f e and from e g | 〉 → | 〉 | 〉 → | 〉 on qutrits A and B. We perform Ntraj = 2000 simulation runs, and record the time and 5.9 Time-Bin-Encoded Generation of Remote Entanglement 151

Figure 5.12 a) Pulse scheme for generating remote entanglement between node A and B. Probability that a quantum jump from f e or e g of the qutrit at b) node A or c) node B is detected during the time-bin| 〉 → remote | 〉 entanglement| 〉 → | 〉 protocol. The data are based on 2000 stochastic trajectories of a numerical master equation simulation.

type of occurred jumps and the qutrit states after the time-bin encoding protocol for each trajectory [Breuer02]. Based on these data we compute the probability that the error is detected at the end of the time-bin protocol, if qutrit B ends in the f state. Fig. 5.12b,c show this probability as a function of the time at which the jump| occurred〉 for the four energy relaxation processes, f n e n and e n g n, n = A, B . For simplicity, Fig. 5.12b,c only show the cases| 〉 where→ | 〉 a single| 〉 energy→ | 〉 relaxation{ } event occurred. Summing over all trajectories where one or more jumps are present, we can estimate the probability of an undetected energy relaxation event,

1 X P 1 Tr f f ρˆ t , (5.8) undet = N B jump( f) traj jump − | 〉〈 | 152 Chapter 5 Quantum Communication with Microwave Photons

where ρˆjump(tf) is the state at the end of the protocol for a trajectory in which a jump occurred. For the entanglement generation protocol and our experimental parameters, we ﬁnd that the probability of an undetected energy relaxation event is Pundet = 5.5% which corresponds to Pundet/Pjump = 35.7% of trajectories where a jump occurred.

Three types of energy relaxation events contribute the most to Pundet, as shown in

Fig. 5.12b,c. At node A, f A e A jumps (light blue points) in the ﬁrst time bin are generally not detected| since〉 → this | 〉 results in a photon being emitted in the second time bin, with qutrit B ending in e . However, if there is also a photon loss event in the communication channel in the| 〉 second time bin, then qutrit B ends in f and the combination of the two errors is detected. When the communication channel| 〉 is perfect, this type of energy relaxation event is not detected. Second, e A g A jumps (dark blue points) in the second time bin are generally not detected and| 〉 → there | 〉 are two ways they can occur. If there are no other errors in the protocol, then qutrit B ends in g and the error is not detected. However, a e A g A jump can also occur if there| 〉 is a photon loss event in the ﬁrst time bin.| In〉 that→ | situation,〉 qutrit B ends f | 〉 and the error is detected. At node B, f B e B jumps (beige points) in the second time bin are not detected since qutrit| B〉 ends→| in〉 g . In contrast to energy relaxation, pure dephasing does not lead to direct changes| in〉 qutrit populations. Consequently, the time-bin encoding does not allow the heralding of phase errors. In summary, we

ﬁnd that 1 Pundet/Pjump = 64.3% of all energy relaxation events during the time- bin encoding− protocol are detected. However, due to the additional time needed for performing the time-bin encoding protocol relative to the direct Fock-state encoding this protocol is affected more by pure qutrit dephasing.

5.10 Photon Loss Model and Outlook on Bell Tests

’Remember that the disagreement between locality and quantum mechanics is large - up to a factor of p2 in a certain sense. So some hand trembling [of the experimenter] can be tolerated without much change in the conclusion.’, John Bell in Speakable and unspeakable in quantum mechanics, p. 102

Having presented our experimental results of quantum communication in a direct quantum channel between two remote nodes, we discuss how much ’hand trembling’ can be accepted in our speciﬁc physical implementation for future Bell tests. There- fore, we introduce an analytical photon loss model, which we use to derive optimal measurement settings for Bell tests in the deterministic case, and to show that the 5.10 Photon Loss Model and Outlook on Bell Tests 153

entanglement generation rate depends linearly on loss for time-bin encoding. Our photon loss model is based on the so-called Stinespring dilation of quantum channels [Preskill15], in which an interaction with an ancillary system (environment) is introduced and is traced out at the end. This representation is equivalent to the Kraus representation [Nielsen11, Preskill15]. In addition, we compare the photon loss model to the master-equation simulations Eq. (2.112), and study the qutrit energy relaxation and dephasing during the quantum communication protocols. First, we discuss the effect of an imperfect readout on the CHSH-value valid for both quantum communication protocols.

Imperfect Qubit Readout

Alice’s and Bob’s measurements of the qubit states can be imperfect which can be quantiﬁed by the readout ﬁdelity Fr = 1 P(e g) P(g e) = 1 2ε assuming equal − | − | − readout errors ε = P(e g) = P(g e). We obtain the imperfect single qubit measurement | | operators [Steffen13a]

1 ˆ er ε ˆ ε ˆ M 1 n = − M 1 n + M 1 n (5.9) ± | 2 ± | 2 ∓ | with the ideal measurement operator Mˆ 1 σˆ an~σ~ˆ . Assuming equal readout a n = 2 1 + ﬁdelities for both qubits, we obtain |

2 Sr = 2p2Fr (5.10) explained by the independence of Alice’s and Bob’s measurements, which results in an minimum readout ﬁdelity of Fr,min = 84.1% to obtain S > 2 (Fig. 5.13b).

Deterministic Remote Entanglement Generation

In the ideal deterministic entanglement generation sequence, we obtain

1 1 gA, 1 + eA, 0 gB gA, eB + eA, gB 0 (5.11) p2 | 〉 | 〉 ⊗ | 〉 → p2 | 〉 | 〉 ⊗ | 〉 with the Fock states of the photon 0 and 1 . The propagation through a lossy channel can be modeled by loss of the photon| 〉 to the| 〉 environment E (equivalent to amplitude 154 Chapter 5 Quantum Communication with Microwave Photons

damping [Preskill15, Nielsen11])

0 0 0 E | 〉 → | 〉 ⊗ | 〉 p 1 pηAB 1 0 E + 1 ηAB 0 1 E (5.12) | 〉 → | 〉 ⊗ | 〉 − | 〉 ⊗ | 〉 with the transmission probability ηAB as introduced in Eq. (2.106). We obtain for the transfer process including photon loss

1 gA, 1 + eA, 0 gB p2 | 〉 | 〉 ⊗ | 〉 1 p pηAB gA, 1 0 E + 1 ηAB gA, 0 1 E + eA, 0 0 E gB . (5.13) →p2 | 〉 | 〉 − | 〉 | 〉 | 〉 | 〉 ⊗ | 〉

Tracing out the environment by using the partial trace [Nielsen11], results in the density operator

1 (1 ηAB) gA, 0 gA, 0 + ηAB gA, 1 gA, 1 2 − | 〉〈 | | 〉〈 | + eA, 0 eA, 0 + pηAB( eA, 0 gA, 1 + h.c.) gB gB (5.14) | 〉〈 | | 〉〈 | ⊗ | 〉〈 | which leads to the density matrix in the basis of Alice’s and Bob’s qubits after the absorption of the photon by Bob

  1 ηAB 0 0 0 − 1  0 η η 0 ρˆ  AB p AB  (5.15) ηAB = 2    0 pηAB 1 0 0 0 0 0

Using ρˆ , we obtain the state ﬁdelity ηAB

1 FηAB 1 η 2 (5.16) s = 4( + p AB) compared to ψ+ , and a concurrence for ρˆ ηAB | 〉 C ρˆ η (5.17) ( ηAB ) = p AB originating from the off-diagonal η terms. C ρˆ is 0 if η 0 showing that p AB ( ηAB ) AB = ρˆ is entangled for 0 < η 1. ηAB AB ≤ As mentioned in Section 2.1.3, the choice of measurement operators to calculate 5.10 Photon Loss Model and Outlook on Bell Tests 155

S inﬂuences its dependence on ηAB (Fig. 5.13a). We obtain the strongest dependence on ηAB for measurement settings in the x-z (equivalent to y-z) plane. For the ideal + directions of ψ , x~1 = (1, 0, 0), x~2 = (0, 0, 1), and ~y1,2 = (1, 0, 1)/p2, we obtain | 〉 ± 1 + pηAB Sxz p2 ηAB ηAB 2p2 ηAB . (5.18) = (p + ) = p 2

Optimizing the measurement settings in the x-z plane results in

v t1 + ηAB Sxz,opt 2p2 ηAB (5.19) = p 2

p for ~y1,2 = (1, 0, pηAB)/ 1 + ηAB. The weakest dependence on ηAB is obtained for settings in the x-y± plane

Sxy = 2p2 pηAB (5.20)

+ along the ideal directions of ψ . Sxy is also the optimal dependence of S on ηAB which we veriﬁed numerically| (gray〉 dots in Fig. 5.13a) and prove analytically in the following. In addition, we obtain excellent agreement between our master equation ψ+ simulations and our photon loss model for S, C and Fs (Fig. 5.14a).

For the analytical proof that Sxy shows the optimal dependence of S on ηAB, we use the lemma by Ryszard Horodecki et al. [Horodecki95] and Frank Verstraete and Michael Wolf Verstraete02 . For the proof ρˆ needs to written in the basis of the [ ] ηAB identity matrix σˆ and the Pauli matrices ρˆ P b σˆ σˆ . For b , 1 ηAB = i,j=1,x,y,z i,j i j i>1,j>1 we obtain the 3x3 dimensional block ⊗   pηAB 0 0   Bρˆ 0 ηAB 0 (5.21) ηAB =  p  0 0 ηAB − which is diagonal for ρˆ . Using the Horodecki lemma that the maximal achievable S ηAB is given

q 2 2 2 e1 + e2 (5.22) with e1 and e2 being the largest two singular values of B ˆ , results in ρηAB

Smax = 2p2pηAB = Sxy (5.23) 156 Chapter 5 Quantum Communication with Microwave Photons

since pηAB ηAB for 0 ηAB 1. Theses dependencies of S on the measurement ≥ − ≤ ≤ settings can also be understood qualitatively: The linear dependence of Sxz on ηAB originates from the diagonal elements of ρˆ , and the η dependence of the S ηAB p AB xz and Sxy from the off-diagonal terms.

min In conclusion, we obtain a minimum transmission probability ηAB = 50% corre- ψ+ sponding to a state ﬁdelity Fs,min| 〉 = 72.9% for Sx y to obtain S > 2 using our photon loss model for deterministic remote communication. Furthermore, combining readout errors and photon loss we obtain

S F2 xy, r = 2p2pηAB r (5.24) and correspondingly S 2 η η F2 (Fig. 5.13b). xz, r = p (p AB + AB) r

ψ+ ψ+ a) state fidelity, ℱs b) state fidelity, ℱs 0.73 0.8 0.85 0.9 0.95 1. 0.73 0.8 0.85 0.9 0.95 1. 1 2.8 x-y x-z x-z, opt. r 2.6 ℱ 0.95   ξ num. CHSH,  2.4 0.9 2. 2.5 CHSH, 2.1 2.6 2.2 2.2 2.7

readout fidelity, 2.3 2.8 0.85 2.4 2 x-y x-z 0.5 0.6 0.7 0.8 0.9 1 0.5 0.6 0.7 0.8 0.9 1

transmission probability, ηAB transmission probability, ηAB

Figure 5.13 a) CHSH-value S versus transmission probability ηAB and corresponding state ψ+ ﬁdelity Fs| 〉 for different measurement settings using the deterministic remote commu- + nication protocol. The blue line shows the maximal obtainable Sxy = Smax for the ψ + | 〉 measurements settings in the x-y plane, the yellow and brown lines show Sxz for ψ and optimized settings in the x-z plane. The dots are obtained from a numerical optimization| 〉 of S for arbitrary measurement settings. b) Achievable CHSH-values S in dependence on ψ+ + ηAB (Fs| 〉) and the readout ﬁdelity Fr for ψ for measurement settings in the x-y (solid lines) and x-z (dashed) plane. | 〉 5.10 Photon Loss Model and Outlook on Bell Tests 157

Time-Bin Encoding Entanglement Generation

For heralded remote quantum communication, photon loss lowers the transmission probability but does not affect the state ﬁdelity, which we discuss in the following based on the time-bin encoding protocol. We assume the same transmission probability ηAB in the ﬁrst and second photonic modes (label 1,2), and that the two photonic modes are reabsorbed by Bob as

01, 02 fB | 〉 → | 〉 11, 02 eB | 〉 → | 〉 01, 12 gB . (5.25) | 〉 → | 〉 We obtain for the lossy transfer process using time-bin encoding

1 gA, 11, 02 + eA, 01, 12 gB p2 | 〉 | 〉 ⊗ | 〉 1 p pηAB gA, 11, 02 01, 02 E + 1 ηAB gA, 01, 02 11, 02 E + → p2 | 〉 | 〉 − | 〉 | 〉 p pηAB eA, 01, 12 01, 02 E + 1 ηAB eA, 01, 02 01, 12 E gB . (5.26) | 〉 | 〉 − | 〉 | 〉 ⊗ | 〉 Applying the absorption scheme of Eq. (5.25) and tracing out the environment leads to the density operator in the basis of Alice and Bob

1 ηAB( gA, eB + eA, gB )( gA, eB + eA, gB )+ 2 | 〉 | 〉 〈 | 〈 | (1 ηAB)( gA, fB gA, fB + eA, fB eA, fB ) . (5.27) − | 〉〈 | | 〉〈 | Performing a binary measurement on Bob’s qutrit which distinguishes between f and the qubit ge-subspace, but does not measure within ge, realizes a photon loss detection| 〉 + and we obtain ψ with probability ηAB. | 〉 Qutrit Decoherence

To study imperfection due to decoherence of the qutrits, we assume the energy relaxation (T1ge, T1ef) and dephasing (T2ge, T2ef) times of both qutrits to be equal and sweep T1ge, T1ef, T2ge and T2ef independently from each other by setting the times ψ+ which are not swept to inﬁnity (Fig. 5.14b). We observe that the state ﬁdelity Fs is ψ+ least affected by T1ef. In addition, Fs scales similarly with T1ef for the deterministic and time-bin encoding protocols due to the detected energy relaxation errors during 158 Chapter 5 Quantum Communication with Microwave Photons

a b 1 ) ) 풮 2.5 0.95 ψ+ s

2 ℱ CHSH

, 0.9 풞 1.5 det tb Tge 0.85 1 conc ge

, 1 T2

ψ+ s Tef

state fidelity, 1

ℱ 0.8 0.5 ef T2 fid 0 0.75 0 0.2 0.4 0.6 0.8 1 0.5 1 5 10 50

transmission probability, AB time, t s η [μ ] Figure 5.14 a) Comparison of the master equation simulation (MES) results (dots) and ψ+ the analytical model (solid lines) on the Bell-state ﬁdelity Fs (blue), concurrence C (light gray) and CHSH-value S versus transmission probability ηAB. b) MES results of the Bell- ψ+ state ﬁdelity Fs sweeping either the energy relaxation (T1) or the coherence (T2) time of the ge- or ef-transition. The MES are performed for the deterministic (det) or time-bin encoding (tb) remote entanglement protocol. The lines are linear interpolations of the MES results.

time-bin encoding (see Section 5.9). This similar scaling is also observed for T1ge. The ψ+ dephasing times reduce Fs stronger than the energy relaxation times, and stronger in the time-bin encoding case compared to the deterministic case due to the different protocol duration. In summary, we estimate that state ﬁdelities of 98.3% in the deterministic case and 98.1% in the heralded case can be obtained for decoherence times of 30µs on the ge-transitions and of 15µs on the ef-transitions assuming no photon loss and perfect heralding.

5.11 Two-Qubit Gates based on the FGGE-Transition

Similarly as discussed previously for the f0g1-transition of a transmon-resonator system (Sections 2.2.8 and 5.3), a microwave drive at the f , g g, e (fgge) transition frequency of a transmon-transmon system results in| an effective〉 ↔ | coupling〉 between these levels. Using this fgge-interaction we show that excitations can be swapped between the transmons. This excitation swap can be utilized to herald a communication error in our time-bin encoding protocol (see Section 5.7). In addition, we present ﬁrst characterizations of an all-microwave two-qubit gate based on the fgge-interaction, allowing for combined local and remote quantum information processing. In this sec- 5.11 Two-Qubit Gates based on the FGGE-Transition 159

tion, we introduce the effective fgge-coupling, and discuss our experimental calibration routines and characterization results.

Eective FGGE-Coupling

As explained for the transmon-resonator system in Section 2.2.8 we start with the transmon-transmon Hamiltonian [Royer18]

E E ˆ ˆ †ˆ C1 ˆ †ˆ †ˆ ˆ ˆ †ˆ C2 ˆ †ˆ †ˆ ˆ HdJC/ħh = ωqb1 b1 b1 b1 b1 b1 b1 + ωqb2 b2 b2 b2 b2 b2 b2 − 2 − 2 ˆ †ˆ ˆ ˆ † ˆ ˆ † + J(b1 b2 + b1 b2) + Ωfgge(t)(b1 + b1) (5.28)

ˆ ˆ with the annihilation operators b1, b2, the ge-transition frequencies ωqb1, ωqb2, the charging energy EC1, EC2 of transmon 1 and 2, and the transmon-transmon coupling J, similar to Eq. (2.83). We assume ωqb1 > ωqb2 leading to a stronger effective coupling if a microwave drive Ωfgge(t) is applied at transmon 1. Next, performing a displacement and Bogoliubov transformation and neglecting higher order terms as in Eqs. (2.84) to

(2.92), we obtain for EC1 EC2 α, and small transmon-transmon coupling J/ ωqb2 ≈ ≈ | − ωqb1 = J/∆qq 1 an effective coupling | Jα g˜fgge Ωfgge(t) . (5.29) ≈ p2∆qq(∆qq α) − as in the transmon-resonator case.

Chip Design and Parameters

To experimentally study the fgge-interaction, we design and fabricate a test chip with two capacitively coupled transmons (Fig. 5.15a). The transmons have transition fre- 1 2 quencies and anharmonicities νge = 6.184 GHz, α1 = 260 MHz, νge = 5.198 GHz, − α2 = 277 MHz, and a transmon-transmon coupling J/2π = 40 MHz. We obtain a − 2 dispersive zz-coupling χzz = 4J α/((∆qq + α)(∆qq α)) 2π 1.9 MHz assuming both transmons have the same anharmonicity α. We fabricate− ≈ transmon 1 with an asymmetric SQUID with designed ratio 1 : 3, and transmon 2 with a single Josephson-junction. We extract J from a vacuum-Rabi-type measurement [Hofheinz08] by tuning the ge- transition of transmon 1 in resonance with the ge-transition transmon 2 using a superconducting wire coil mounted below the sample. We also use this coil to tune transmon 1 1 to νge. We perform dispersive readout of the transmon states using two resonators at 1 2 1 νr = 7.486 GHz and νr = 7.178 GHz. We extract energy relaxation times T1ge = 7.5 µs, 160 Chapter 5 Quantum Communication with Microwave Photons

Figure 5.15 a) False-color micrograph of the two-qubit sample used to characterize the

f , g g, e interaction (by S. Krinner). b) Measured (dots) ac Stark shifts ∆ac versus | 〉 ↔ | 〉 drive amplitude Ωfgge extracted using pulsed spectroscopy (for details see Section 5.3). ∆ac/2π = 0 MHz corresponds to νfgge = 6.908 GHz. The solid line is a quadratic ﬁt to the n ﬁrst 6 data points and the cross indicates the normalized drive amplitude of Ωfgge = 0.9 used in the further measurements. c) Rabi-type oscillations sweeping the pulse duration

τfgge. The transmon populations Ps (dots) are extracted using average readout accounting for independently measured thermal populations (see Section 2.2.6 for details). The solid

lines are sinusoidal ﬁts to the data. d) Pg sweeping τfgge and the frequency detuning δfgge. 0 MHz indicates the resonance frequency extracted from the ac Stark calibration (b) and the data shown in (c) corresponds to the dashed line.

1 2 2 1 T1ef = 2.4 µs, T1ge = 6.7 µs and T1ef = 1.5 µs, and coherence times T2ge = 6.7 µs, 1 2 2 T2ef = 4.4 µs, T2ge = 4.3 µs and T2ef = 3.0 µs using measurements schemes discussed 1 1 in Section 2.2.4. We observe a thermal population of nth,ge 17.3%, nth,ef 0.4%, and 1 1 ≈ ≈ nth,ge 4.5%, nth,ef 0.03% based on Rabi-type measurements [Geerlings13] which we use≈ for the population≈ extraction in averaged readout (see Section 2.2.6).

FGGE-Pulse Calibration

As discussed for the f0g1-calibration (Section 5.3), we use pulsed spectroscopy on transmon 1 to extract the ac Stark shift of the fgge-transition versus drive amplitude 0 Ωfgge (Fig. 5.15b) and obtain νfgge = 6.908 GHz in the limit of small drives. We observe 5.11 Two-Qubit Gates based on the FGGE-Transition 161

ac Stark shifts up to 100 MHz and a deviation from the quadratic approximation for n normalized drive amplitudes Ωfgge > 0.6 (corresponding to Ωfgge/2π 360 MHz, see ≈ g˜fgge calibration in the next paragraph). We observe Rabi-type oscillations between f and g of transmon 1 sweeping the | 〉 | 〉 pulse duration τfgge of a ﬂattop pulse with a Gaussian-shaped rise and fall edge on resonance with the fgge-transition utilizing the ac Stark calibration (Fig. 5.15c). For a n normalized Ωfgge = 0.9, we obtain a swap of the f population to g in 40 ns corre- | 〉 | 〉 sponding to g˜fgge/2π 5.6 MHz. Using Eq. (5.29) we estimate g˜fgge 0.0103Ωfgge and ≈ n ≈ obtain Ωfgge/2π 550 MHz for Ωfgge = 0.9. Due to the signiﬁcant thermal population we observe a reduced≈ contrast of the Rabi-type oscillations. Sweeping the pulse duration τfgge and the detuning δfgge compared to the resonance frequency obtained from the ac Stark calibration, we obtain the characteristic chevron patterns (Fig. 5.15d). The excitation swap from the f population of transmon 1 to e -population of transmon 2 shows that the fgge-interaction| 〉 can be used to herald communication| 〉 errors during the time-bin encoding protocol as described in Section 5.7.

CNOT-Gate Calibration

In addition, we show that a controlled-not (cnot) two-qubit gate can be performed using the fgge-interaction. The cnot gate can be used to generate Bell states and it forms together with single qubit gates a universal set of quantum gates. Our implementation of the cnot gate is based on a controlled-phase (cphase) gate and single π/2 qubit Rge rotations [Nielsen11]. The cphase gate requires that the e, e state obtains | 〉 a conditional phase of φee = π while φge = φeg = 0. In addition, all excitations need to be transferred back to their original state at the end of the gate (Fig. 5.16c).

To calibrate the cphase gate, we ﬁrst extract the pulse durations τfgge which maximizes the f population after a full oscillation for a given detuning δfgge (Fig. 5.16a) based on the| 〉 chevron-pattern measurements. Next, we perform two-qubit phase calibrations using Ramsey-type experiments (for details see Ref. [Baur12a]). We calibrate the conditional phase φc (Fig. 5.16b) by preparing transmon 1 either in g or e

2π π/2 | 〉φ2 π/| 2〉 and apply Rfgge for various detunings δfgge. On transmon 2 we apply Rge and Rge 2π before and after Rfgge and sweep the phase φ2 of the second pulse. We ﬁt sinusoidal functions to the observed Ramsey-fringes and obtain φc as the difference between transmon 1 prepared in g and e (Fig. 5.16b). With this calibration scheme we | 〉 | 〉 obtain a conditional phase φc = π on state g, e for δfgge = 0 since φc = φfgge + φzz is | 〉 6 accumulated because of the fgge-drive φfgge and the dispersive two-qubit interaction 162 Chapter 5 Quantum Communication with Microwave Photons

Figure 5.16 a) Measurements of the pulse duration τfgge, which maximizes the f popu- | 〉 lation of transmon 1 after a full oscillation, and b) obtained conditional phase φc versus detuning δfgge of the fgge-pulse. The solid line in (a) is a quadratic ﬁt to the data and a linear ﬁt in (b). c) Schematic energy level diagram of the two-transmon system with an illustration of fgge-drive based cphase gate. d) Real part of the measured quantum process matrix (solid bars) of a cnot-gate based on the fgge-interaction assuming thermally excited input states. The ideal gate operation is shown as gray wire frames and the results of MES in red wireframe. The magnitude of the imaginary part of each element of the density matrix is < 0.057.

φzz. This conditional phase can be mapped to the e, e state to obtain φee = π by using | 〉 a (virtual) σz-gate on transmon 2. Due to this σz-gate the phases of the g, e and | 〉 e, e states acquire a minus sign which is canceled on g, e due to the acquired φc = π and| 〉 thus only remains on e, e . We ﬁt a linear function| to〉 the extracted conditional phases versus detuning and| obtain〉 the detuning and pulse duration which realize a cphase gate (Fig. 5.16a,b). In addition, we calibrate the additional single qubit phases a a φge and φeg (see also Ref. [Baur12a]) obtained during the fgge-interaction and the

σz-gate on transmon 2 by performing Ramsey-type experiments on transmon 1 and 2 with and without applying the fgge-drive. We correct these single qubit phases by using virtual-σz-gates on both transmons. As detailed for the quantum state transfer for a single qubit, we perform two- 5.11 Two-Qubit Gates based on the FGGE-Transition 163

qubit process tomography of the cnot gate [Baur12a, Steffen13a] assuming thermally excited input states and extracting the populations of both qutrits with average readout taking the thermal excitation of the reference traces into account (Fig. 5.16d). We thus obtain the process matrix which maps our thermally excited input states to the measured output states and obtain a process fidelity of approximately 98% compared to an ideal cnot gate (gray wireframe in Fig. 5.16d). This fidelity is in agreement with the fidelity of 97% obtained from MES simulations (red wireframe in Fig. 5.16d) [Royer18] using the same device parameters. The experimental results and the master equation simulations indicate that high-fidelity two-qubit gates can be realized using the fgge-interaction for gate times below 90 ns.

Note however, that quantum process tomography is not resilient to state preparation and measurement (spam) errors and further characterizations need to be performed to validate our results. Performing the fgge-based gate with transmons with a low thermal population, utilizing transmon-state reset protocols [Magnard18, Egger18], implementing a higher phase stability to minimize the magnitude of the imaginary part of the quantum process matrix, and using more advanced gate characterization techniques, e.g. randomize benchmarking [Kelly14, Haberthür15] or gate set tomography [Greenbaum15, Dehollain16, Blume-Kohout17] should lead to a quantitative more reliable characterization and should allow to distinguish coherent and incoherent errors. In our implementation, we obtain an on/off ratio of the gate times off ron = toff/ton g˜fgge/χzz = Ωfgge(∆qq α)/(4p2J∆qq) 3 with our current device ≈ | | − off ≈ parameters. According to our MES we expected that ron > 10 can be achievable with optimized device parameters by reducing the the transmon-transmon coupling J and adapting ∆qq to keep the gate time constant.

In conclusion, we have performed first characterization experiments which show that the fgge-interaction is suitable for heralded quantum communication by swapping the f -state population of transmon B to the e -state population of an ancilla transmon| in〉 circa 40 ns with high-fidelity. Performing| 〉 fast, high-fidelity qubit readout on the ancilla transmon [Walter17] should realize the heralding in less than 100 ns. Our first characterization experiments of the cnot-gate indicate that also multiple-qubit (> 2) quantum communication protocols like quantum teleportation [Bennett93, Steffen13b] and error-correction schemes like entanglement distillation [Bennett96c, Oppliger17] or the quantum parity code [Munro12, Borregaard19] can be implemented using fgge-based two-qubit gates. 164 Chapter 5 Quantum Communication with Microwave Photons

5.12 Conclusion and Outlook

Quantum networks across remote nodes are based on reliable, high-bandwidth quantum communication, which has been investigated in various physical systems over the last decades. In this chapter we have presented a direct quantum channel between remote circuit QED nodes on separate chips. The direct quantum channel is established by using time-symmetric itinerant single photons as information carriers. To emit and absorb these itinerant photons with our transmon systems we use an all- microwave cavity-assisted Raman process at the f0g1-transition. The photon-transfer process between the two nodes is performed with a probability of 98.1 0.1%. This all-microwave interaction has the advantage that it only requires the same± drive lines as for single qubit gates. Furthermore, we realize a high bandwidth photon-transfer, which is straightforward to implement and to calibrate with typical circuit QED systems. These advantages make quantum communication based on the f0g1-interaction interesting for future large-scale quantum computing architectures. Using our direct quantum channel, we have experimentally investigated two different mappings of our local qubit states to itinerant photons. First, we use a Fock-state encoding realizing a deterministic quantum communication between two quantum systems. We have achieved a qubit-state-transfer process fidelity of 80.02 0.07% and generated Bell states ψ+ with a fidelity of 78.9 0.1%. Our results are± within the first which have realized| a〉 deterministic quantum communication± between two remote quantum systems. In our experiments, we have demonstrated a high remote quantum communication rate of 50 kHz making the Fock-state encoding particularly interesting for quantum communication in low-loss quantum channels. Our remote communication rate was mainly limited by the experimental repetition rate which can be set closer to the protocol duration in future experiments to increases the rate to approximately 1 MHz. The loss in quantum channels between two circuit QED system can be reduced compared to our on-chip implementation by designing specialized quantum nodes in 3D-cavities which directly couple to the microwave waveguides. We have designed a prototype of these specialized nodes based on which we estimate the loss to be < 1% in the circuit QED system to waveguide coupling (current on-chip implementation: 4.5%) [Schwegler18]. In addition, the isolator (insertion loss: 13 2% ) in our ∼ ± quantum channel can be replaced by a low-loss superconducting switch [Pechal16b] which still allows for thermalization of the waveguide field, or by tunable couplers at both waveguide ends [Bienfait19, Zhong19]. In the second encoding, we map our qubit-states to a time-bin superposition of a single photon and utilize our transmons 5.12 Conclusion and Outlook 165

as qutrits. The time-bin encoding also allows for a direct quantum communication but in addition provides the possibility to detect communication errors making the quantum communication rate linearly proportional to the transmission probability. Us- ing the time-bin encoding we show that the qubit-state-transfer process fidelity can be improved to 92.4 0.4% and the Bell-state fidelity to 90.3 0.2% assuming an ideal error detection.± The time-bin encoding thus provides the± option for heralded quantum communication in lossy, direct quantum channels. In addition, we have characterized the feasibility of an all-microwave heralded quantum communication via ancilla qubits across remote nodes by investigating the fgge-based two-qubit interaction. This fgge-interaction has also the potential to be used for two-qubit gates in multi-qubit quantum algorithms. Our first experimental results indicate that fast excitations swaps (< 50 ns) and high-fidelity (> 95%) two-qubit gates can be performed using this microwave-controlled interaction. The possibility to achieve higher on/off ratios (> 10) of two-quit gates needs, however, to be investigated in further experiments. For all remote communication experiments, we observe an excellent agreement of our experimental results with numerical master equation simulations (MES) indicating the high level of control of our experimental system. Using these MES for the Fock-state encoding, we decompose the Bell-state infidelity into approximately 10.5% from photon loss, 9.5% from finite transmon coherence times and 1% from photon absorption inefficiency for our experimental setup. For the time-bin encoding, we conclude from the MES that all photon loss is detected in our scheme and that the Bell-state infidelity is explained by the finite transmon coherence times. Using time-bin encoding, we also obtain from the MES that circa 2/3 of communication errors due to energy relaxation of the transmon states are detected, since these decay errors can lead to the same final qutrit state as photon loss. Besides investigating our experimental results, the MES allow to make predictions on future remote communication experiments including future Bell tests. For example, we estimate based on the MES that state fidelities of 98.3% using the Fock-state encoding and 98.1% using the time-bin encoding can be obtained for decoherence times of 30 µs on the ge-transitions and of 15 µs on the ef-transitions. In theses estimations we assume no photon loss and perfect heralding but otherwise identical circuit parameters to our current experiments. Improving our circuit QED systems to especially obtain higher photon-bandwidths, we estimate that our Fock-state encoding protocol can be performed in 100 ns (currently 180 ns), and the time-bin encoding protocol in 230 ns. These short protocol durations and high fidelities show that our remote communication 166 Chapter 5 Quantum Communication with Microwave Photons

schemes can be integrated in larger distributed quantum computing algorithms without performance degradation nor time overhead, e.g. state-of-the-art qubit readout times circa 60-300 ns [Ristè12b, Jeffrey14, Walter17, Heinsoo18, Dassonneville19] and two- qubit gate duration circa 40-160 ns [Barends14, Sheldon16, Andersen19, Rol19]. In addition to the MES, we introduced an analytical photon loss model to study the measurement settings for future loophole-free Bell tests showing the importance to optimize those settings dependent on the dominating errors. In conclusion, the presented protocols can serve as a framework for high-ﬁdelity, high-bandwidth quantum communication with superconducting circuits in direct quantum channels. Combined with our cryogenic network systems and low-loss microwave waveguides our quantum communication protocols thus pave the way towards large- scale quantum networks for future distributed quantum computation and non-local experiments beyond loophole-free Bell tests across multiple cryogenic nodes. Bibliography

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List of Publications

1.∗ P. Kurpiers, M. Pechal, B. Royer, P. Magnard, T. Walter, J. Heinsoo, Y. Salathé, A. Akin, S. Storz, J.-C. Besse, S. Gasparinetti, A. Blais, and A. Wallraff. "Quantum Communication with Time-Bin Encoded Microwave Photons." Phys. Rev. Appl. 12, 044067 (2019)

2.∗ P. Kurpiers, P. Magnard, T. Walter, B. Royer, M. Pechal, J. Heinsoo, Y. Salathé, A. Akin, S. Storz, J.-C. Besse, S. Gasparinetti, A. Blais, and A. Wallraff. "Deterministic Quantum State Transfer and Remote Entanglement using Microwave Photons." Nature 558, 264-267 (2018).

3.∗ T. Walter, P. Kurpiers, S. Gasparinetti, P. Magnard, A. Potocnik, Y. Salathé, M. Pechal, M. Mondal, M. Oppliger, C. Eichler, and A. Wallraff. "Rapid, High-Fidelity, Single-Shot Dispersive Readout of Superconducting Qubits." Phys. Rev. Appl. 7, 054020 (2017)

4.∗ P. Kurpiers, T. Walter, P. Magnard, Y. Salathé, and A. Wallraff. "Characterizing the attenuation of coaxial and rectangular microwave-frequency waveguides at cryogenic temperatures." EPJ Quant. Technol. 4, 8 (2017).

5. S. Krinner, S. Storz, P. Kurpiers, P. Magnard, J. Heinsoo, R. Keller, J. Luetolf, C. Eichler, and A. Wallraff. "Engineering cryogenic setups for 100-qubit scale superconducting circuit systems." EPJ Quant. Technol. 6, 2 (2019).

6. P. Magnard, P. Kurpiers, B. Royer, T. Walter, J. Besse, S. Gasparinetti, M. Pechal, J. Heinsoo, S. Storz, A. Blais, and A. Wallraff. "Fast and Unconditional All-Microwave Reset of a Superconducting Qubit." Phys. Rev. Lett. 121, 060502 (2018).

7. The BIG Bell Test Collaboration including J. Heinsoo, P. Kurpiers, Y. Salathé, C. Kraglund Andersen, A. Beckert, S. Krinner, P. Magnard, M. Oppliger, T. Walter, S. Gasparinetti, C. Eichler, A. Wallraff. "Challenging local realism with human choices." Nature 557, 212–216 (2018).

8. J.-C. Besse, S. Gasparinetti, M. C. Collodo, T. Walter, P. Kurpiers, M. Pechal, C. Eichler, and A. Wallraff. "Single-Shot Quantum Non-Demolition Detection of Individual Itinerant Microwave Photons." Phys. Rev. X 8, 021003 (2018). 9. Y. Salathé, P. Kurpiers, T. Karg, C. Lang, C K. Andersen, A. Akin, S. Krinner, C. Eichler, and A. Wallraff. "Low-Latency Digital Signal Processing for Feedback and Feedforward in Quantum Computing and Communication." Phys. Rev. Appl. 9, 034011 (2018).

10. C. Eichler, J. Mlynek, J. Butscher, P. Kurpiers, K. Hammerer, T. J. Osborne, and A. Wallraff. "Exploring Interacting Quantum Many-Body Systems by Experimentally Creating Continuous Matrix Product States in Superconducting Circuits." Phys. Rev. X 5, 041044 (2015).

11. S. Berger, M. Pechal, P. Kurpiers, A. A. Abdumalikov, C. Eichler, J. A. Mlynek, A. Shnirman, Y. Gefen, A. Wallraff, and S. Filipp. "Measurement of geometric dephasing using a superconducting qubit." Nat. Commun. 6, 8757 (2015).

12. Y. Salathé, M. Mondal, M. Oppliger, J. Heinsoo, P. Kurpiers, A. Potoˇcnik, A. Mezzacapo, U. Las Heras, L. Lamata, E. Solano, S. Filipp, and A. Wallraff. "Digital Quantum Simulation of Spin Models with Circuit Quantum Electrodynamics." Phys. Rev. X 5, 021027 (2015).

13. L. Steffen, Y. Salathé, M. Oppliger, P. Kurpiers, M. Baur, C. Lang, C. Eichler, G. Puebla- Hellmann, A. Fedorov, and A. Wallraff. "Deterministic quantum teleportation with feedforward in a solid state system." Nature 500, 319-322 (2013).

First and co-ﬁrst author papers Acknowledgements

During my time at the Quantum Device Lab I conducted research with many wonderful and brilliant people, who have been an incredible source of inspiration and motivation. Without them, I would not have accomplished the projects presented in this thesis and I want to deeply thank all of them for their assistance. First, I would like to express my deep gratitude to my supervisor Andreas Wallraff for providing me a clear guideline of my Ph.D. within the SuperQuNet project. I especially appreciated his continuous support, useful critiques, and encouragement in difficult periods. The opportunity to perform my Ph.D. in his group has substantially enriched my understanding of research and excellence. I would also like to extend my gratitude to David Schuster for taking the time to be my co-examiner, his insightful ideas and our interesting conversations. My special thanks go to the permanent members of the SuperQuNet team: Theo Walter, Janis Lütolf, Paul Magnard, Simon Storz and Melvin Frey. Theo, I am so thank- ful for having experienced your passion about research and life, and the moments we shared and enjoyed together. I sincerely wish that you would still be with us. Janis, thank you for teaching me engineering, all your management and technical support, and for all the exciting free time activities. Paul, thanks for all your scientific input, theory backup, so many fruitful discussions, and especially for supporting another during the last years. Simon, thanks for your fast integration in our team, for your feedback to my thesis, and for the fun of fixing software bugs together. Melvin, thanks for always finding realizable solution to our ideas, and your great technical help. To mention just a few aspects of all your superb assistance and team spirit. It was a highly beneficial, supportive, and motivate experience to work with you in a team. I am also very grateful to all SuperQuNet short-time members: Adrian Beckert, Kevin Reuer, Hangxi Li, Josua Schär, Fabian Marxer, Silvia Ruffieux, Maud Barthélemy, Johannes Fankhauser, Noam Gallmann, Moritz Businger, Lukas Lang, Grégoire Noel, Nicholas Meinhardt, Nick Schwegler, Elias Portolés Marin, Adrian Schwarzer, Raphael Keller, Yannick Hmina, Jonas Künzli, Reto Schlatter and Alain Fauquex. All of them contributed to various aspects of our project which enhanced so many important details and it was a pleasure for me to work together with all of you. In addition, I would like to extend my sincere thanks to our theory collaborators, Alexandre Blais and Baptiste Royer, for their continuous assistance, their constructive and highly valuable inputs, and the many nice and interesting discussions we had and ideas we realized together. Making ice-cream, measuring, skiing, discussing, biking, soldering, playing soccer, making new designs, walking, administrative support, playing board games, fabricat- ing devices, going to rice up, installing cryostats, programming, having whiskey with dry ice,... with all members of the Quantum Device Lab made me enjoy my time in the lab nearly every day - all of you made the difference for me! Since my thesis was somehow related to distance in many respects, I decided to sort everybody by distance from my preferred office spot: Yves Salathé, Arjan van Loo, Johannes Herrmann, Anna Stockklauser, Jean-Claude Besse, Jonas Butscher, Samuel Haberthür, Sebastian Krinner, Robin Buijs, Simon Berger, Simone Gasparinetti, Lukas Huthmacher, Stephan Allenspach, Marek Pechal, Graham Norris, Christian Kraglund Andersen, Mathias Stam- meier, Sébastien Garcia, Tobias Thiele, Gabriela Strahm, Cindy Berchtold, Stefan Filipp, Francesca Bay, Johannes Heinsoo, Markus Oppliger, Lars Steffen, Mintu Mondal, Ste- fania Lazar,˘ Ants Remm, Jonas Mlynek, Christian Lang, Lukas Heinzle, Taro Spirig, Sina Zeytinoglu, Michele Collodo, Abdufarrukh Abdumalikov, Milan Allan, Pasquale Scarlino, David van Woerkom, Anton Potoˇcnik, Florian Lüthi, Antonio Rubio Abadal, Christopher Eichler, Mihai Gabureac and Abdulkadir Akin. Thank you all for supporting me and encouraging me in so many different ways and making the lab a very enjoyable place to be. Finally, I would like to thank my friends outside the lab for all the common activities and especially my family: From the bottom of my heart I thank my wife Christina for her enormous understanding, her patience and tender distractions, and my parents, Christine and Manfred Kurpiers, my brother, Markus, and grandparents, Albertine and Hans Eder, for their immense support during my whole life and interest in my work. Curriculum Vitae

Name Philipp Michael Kurpiers Date of birth 02.02.1988 Citizenship Germany

Education 2013 – 2019 Ph.D. studies, ETH Zürich 2011 – 2013 Master in Physics, ETH Zürich 2008 – 2011 Bachelor in Physics, Ludwig-Maximilians-University, Munich, Germany 1998 – 2007 Abitur, Karl-Ritter-von-Frisch-Gymnasium, Moosburg, Germany Professional experience 2013 – 2019 Research assistant and teaching assistant, Department of Physics, ETH Zürich 2012 Internship, GSI Helmholtz Centre for Heavy Ion Research, Darmstadt, Germany 2010 – 2011 Teaching assistant, Department of Physics, Ludwig-Maximilians-University, Munich, Germany

Zürich, November 2019