HYBRID SUPERCONDUCTING QUANTUM COMPUTING ARCHITECTURES

by

Matthew A. Beck

A dissertation submitted in partial fulﬁllment of the requirements for the degree of

Doctor of Philosophy

(Physics)

at the

UNIVERSITY OF WISCONSIN–MADISON

2018

Date of ﬁnal oral examination: 6/21/18

The dissertation is approved by the following members of the Final Oral Committee: Robert F. McDermott, Professor, Physics Mark Saﬀman, Professor, Physics Mark A. Eriksson, Professor, Physics Maxim G. Vavilov, Professor, Physics Mikhail Kats, Assistant Professor, Electrical and Computer Engineering ProQuest Number:10838041 All rights reserved INFORMATION TO ALL USERS The quality of this reproduction is dependent upon the quality of the copy submitted. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if material had to be removed, a note will indicate the deletion.

ProQuest 10838041 Published by ProQuest LLC ( 2018). Copyright of the Dissertation is held by the Author. All rights reserved. This work is protected against unauthorized copying under Title 17, United States Code Microform Edition © ProQuest LLC. ProQuest LLC. 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, MI 48106 - 1346 © Copyright by Matthew A. Beck 2018 All Rights Reserved i

Abstract

Quantum computing holds the promise to address and solve computational problems that are otherwise intractable on a classical, transistor based machine. While much progress has been made in the last decade towards the realization of a scalable superconducting quantum processor, many questions remain unanswered. The work contained in this thesis addresses two equally important concerns; These are speciﬁcally that of quantum information storage and transfer and the scaling of current qubit control and readout methods. Superconducting quantum processors are exactly what their name implies: processors. While the goal is to eventually build a universal quantum computer, it is not unreasonable to envision near term quantum processors hard wired to perform speciﬁc computational tasks. This idea of compartmentalized quantum processing necessitates that the quantum results of a computation either be stored and/or transferred for latter / further use. A natural candidate to realize such a quantum memory is the neutral Rydberg atom. The hyperﬁne states of cesium atoms exhibit coherence times greater than 1 second while adjacent Rydberg energy levels have electric dipole transitions in the gigahertz regime; These properties make it a suitable candidate to realize a quantum memory and information bus between adjacent superconducting processors yielding an unprecedented ratio of coherence time to gate time. To realize such a computing architecture, the coherent coupling between a single Rydberg atom and superconducting bus resonator must ﬁrst be demonstrated. This ﬁrst half of this thesis details the development of a superconducting interface meant to realize strong coupling to a single Rydberg atom. To date, the experimental liquid Helium 4 K UHV cryostat has been built, characterized, ii

and installed. Superconducting niobium coplanar waveguide (CPW) resonators have been designed and fabricated to facilitate strong coupling to the Rydberg atom through on-chip microwave ﬁeld engineering. Additionally, the CPW resonators have been tailored to achieve quality factors above 104 at 4 K. The project is currently still on-going with single-atom trapping and state characterization near the 4 K chip surface under investigation. The second portion of this thesis details the development of a superconducting single ﬂux quantum (SFQ) pulse generator for transmon qubit control. As the size of superconducting quantum processors scales beyond the level of a few tens of qubits, the control hardware overhead becomes untenable. For current technology based on microwave control pulses generated at room temperature followed by ampliﬁcation and heterodyne detection, the heat load and physical footprint of the required classical hardware preclude brute force scaling to qubit arrays more than ∼ 100. The work contained herein details the development, fabrication, characterization and ﬁnally integration of a dc/SFQ driver with a transmon qubit on a single chip as a ﬁrst step towards an all superconducting digital control scheme of quantum processors. Details of the multi-additive layer processing and fabrication required to realize these devices are discussed in the context of maintaining high (> 10 us) qubit coherent times and small superconducting resonator loss. To date, coherent qubit rotations have been achieved via application of SFQ pulses with pulse to pulse spacing aligned with subharmonics of the qubit frequency. Interleaved randomized benchmarking (RB) of SFQ driven single qubit gates realized are currently at 90% level. Future plans regarding a ﬂip chip / multi-chip module approach to increasing gate ﬁdelities will also be discussed. iii

Acknowledgments

I would like to take just a few lines to give thanks to some notable people without whom this thesis and the work contained herein would not be possible. First and foremost, I would like to extend my deepest gratitude to my advisor Dr. Robert F. McDermott. It has been with Robert’s guidance that I have learned how to be a better scientist. To my graduate student forefathers in the group, Dr. David Hover, Dr. Steven Sendelbach, Dr. Yuanfeng Gao and Dr. Guilhem Ribeill, I would like to take a sentence or two and extend many thanks to you all for your patience and guidance during my ﬁrst few years in the group. The debt I owe you I hopefully can repay through forwarding these lessons to younger students. With regards to the hybrid atom–CPW project, I would like to thank the tireless eﬀorts of Jonathan Pritchard, Joshua Isaacs and P.I. Mark Saﬀman of the atomic team. You three have been a pleasure to work with. This acknowledgment section (nor this thesis) could not have been completed without a mention of the Edward M. Leonard, Jr. The past few years working side by side in the lab and clean room have been the most enjoyable professional experience of my life. Your tireless eﬀorts in always striving for cleaner samples, cleaner measurements and cleaner code have been an inspiration. Your leadership in the lab has be a shining example to all in just how much one person can achieve. I, as does the entire lab, owe you a great deal of gratitude. I also would like to take a moment and thank the cowboys of the McDermott Family Research Ranch, Alex, J.P., Nathan, Naveen, Ted, Brad, Ryan, and Joey. Thank you all for iv the many riveting discussions and helping to maintain a scalable lab. Raising a child through undergraduate and graduate study could not have been possible without the endearing love and support of my parents. I wish to give many thanks to my mother who has been a bedrock of patience and understanding and to my father who showed me what it means (and takes) to be a good man and father. And ﬁnally, Alyssa and Zoe. Alyssa, you have been the love of my life since we were kids. Thank you for everything you do everyday and showing me what happiness truly is. Zoe, watching you grow up into the wonderful young adult you are has been a privilege. I can only hope to teach you the number of lessons you have taught me. With that, this is for you, kid. v

To my family. vi

Contents

Contents ...... vi

List of Tables ...... ix

List of Figures ...... x

1 Introduction ...... 1 1.1PhysicalInformation...... 2 1.2QuantumBits...... 4 1.2.1 Qubit Metrics ...... 5 1.3 Physical Qubit Realizations ...... 7 1.3.1 NeutralAtoms...... 8 1.3.2 Superconducting Qubits ...... 9 1.3.3 Current Hybrid Quantum Computing Approaches ...... 12

2 Josephson Physics and Superconducting Circuit Fundamentals ... 14 2.1BasicPhenomena...... 15 2.1.1 RCSJModel...... 17 2.1.2 Superconducting Quantum Interference Devices ...... 20 2.2QuantizationOfElectricalCircuits...... 22 2.3 Superconducting Qubit Zoo ...... 26 2.3.1 Phase qubit ...... 26 2.3.2 Flux Qubit ...... 27 2.3.3 Transmon...... 27

3 Circuit Quantum Electrodynamics ...... 29 3.1 Cavity–Qubit Hamiltonian ...... 30 3.1.1 Resonant Jaynes–Cummings Hamiltonian ...... 33 3.1.2 Dispersive Jaynes-Cummings Hamiltonian ...... 34 vii

3.2 Qubit-Cavity Bringup ...... 37 3.2.1 Coupling The Qubit–Cavity System to the Outside World . . . 37 3.2.2 Qubit Signs of Life ...... 41 3.2.3 Finding the Qubit Frequency ...... 45 3.2.4 Orthogonal Qubit Axis Control ...... 45 3.3 Qubit and Qubit Gate Benchmarking ...... 49 3.3.1 Characterizing and Calibrating the Qubit States ...... 51 3.3.2 Randomized Benchmarking of Qubit Gates ...... 53

4 Hybrid Atom–CPW cQED Experiment ...... 57 4.1ByTheNumbers...... 57 4.2 Experimental Apparatus ...... 61 4.2.1 CryostatPerformance...... 67 4.3 Superconducting CPW Resonator Design ...... 68 4.3.1 EngineeringtheQualityFactor...... 71 4.3.2 MeasuringtheQualityFactor...... 78 4.3.3 Engineering the atom-CPW coupling strength ...... 81 4.3.4 StrayElectricFieldCompensation...... 91 4.4CurrentStatus...... 93

5 Single Flux Quantum Qubit Control ...... 94 5.1 Superconducting Supercomputing with Single Flux Quanta ...... 95 5.1.1 Rapid Single Flux Quantum ...... 98 5.1.2 EnergyEﬃcientSFQ...... 98 5.1.3 ReciprocalQuantumLogic...... 98 5.2 SFQ Control of a Transmon Qubit ...... 99 5.3BridgingtheQuantum–ClassicalDivide...... 103 5.3.1 SFQDriverDesign...... 104 5.3.2 SFQ–Qubit Circuit Layout ...... 111 5.3.3 Fabrication...... 113 5.4Measurement...... 119 5.4.1 LHe SFQ Circuit Pre-screening ...... 121 5.4.2 100 mK Quantum Circuit Pre-screening ...... 121 5.4.3 Initial SFQ–Qubit Bringup ...... 124 5.4.4 Establishing Orthogonal Axis Control ...... 133 5.4.5 Randomized Benchmarking of SFQ–Qubit Gates ...... 137 5.4.6 SFQ Generated Quasiparticle Studies ...... 140 viii

5.5NextGenerationDesignandOutlook...... 148 5.5.1 Flip Chip Design ...... 148

A Josephson Energy of a dc SQUID ...... 151

B Preserving High Quality Nb ...... 154

References ...... 158 ix

List of Tables

3.1 Single Qubit Cliﬀords ...... 54

4.1 Bell State density matrix Lindblad superoperators ...... 61 4.2Cryostatmaterialleak-uprates...... 65 4.3NbPlasmaEtchRecipeParameters...... 78 4.4COMSOLSimulatedElectricFieldCompensationParameters...... 92

5.1 dc/SFQ Circuit Inductances ...... 108 5.2ChipFabricationOverview...... 114 5.3 SFQ gate ﬁdelities determined with interleave randomized benchmarking for the n =3andn = 41 subharmonic...... 141 x

List of Figures

1-1 Bloch Sphere ...... 6 ∗ 1-2 Qubit T1 / T2 ...... 7 1-3CesiumLevelDiagram...... 10 1-4 Superconducting Z-Trap ...... 13

2-1 Josephson Junction ...... 16 2-2TiltedWashboardPotential...... 18 2-3JosephsonIVCurves...... 21

3-1cQEDCircuitDiagram...... 31 3-2 Qubit-Cavity Avoided Level Crossing ...... 34 3-3 Inductive / Capactive Resonator Coupling ...... 38 3-4 Measured Resonator–Qubit Spectrum vs. Readout Power ...... 42 3-5 Resonator Spectroscopy Vs. Qubit Flux-Bias Current ...... 43 3-6 Qubit Spectroscopy Data ...... 46 3-7 Dilution Refrigerator Wiring Schematic ...... 50 3-8 Qubit I/π IQ Blob Distribution ...... 51 3-9RandomizedBenchmarkingofaMicrowaveX/2Gate...... 55

4-1ControlledPhaseGateforAtom-CPW...... 58 4-2Bellstatepreparationﬁdelity...... 62 4-3CustomUHVCryostat...... 64 4-4 Cryostat Cold Finger Vibration Measurement ...... 69 4-5 Bell State Preparation Fidelity Vs. Temperature ...... 70 4-6 Complex Conductivity Vs. Temperature ...... 74 4-7 CPW Kinetic Inductance ...... 77 4-8PredictedCPWQualityFactor...... 79 4-9 Multiplexed Quarter-wave Resonators ...... 80 4-10MeasuredCPWQualityFactorVs.CPWGeometry...... 82 xi

4-11 CPW E FieldProﬁle...... 83 4-12PlatedCPWVoltageAntinodeFieldProﬁle...... 85 4-13 Process for electroplating Cu on Nb ...... 87 4-14Atom–Resonatormicrographanddata...... 90 4-15 Sample Mount / DC Compensation Pins ...... 92 4-16 Image of Cs MOT trapped directly beneath superconducting sample ...... 93

5-1JJSFQPhaseEvolution...... 96 5-2BasicSFQcircuits...... 97 5-3 SFQ Driven Qubit Theory ...... 101 5-4 Qubit State Leakage ...... 102 5-5dc/SFQDriverOperation...... 105 5-6dc/SFQPhase...... 107 5-7 dc/SFQ driver WRSpice simulations ...... 109 5-8 dc/SFQ driver WRSpice simulations ...... 112 5-9 Fully Fabricated SFQ–Qubit Sample ...... 120 5-104.2KMeasurementIVMeasurement...... 122 5-11 Bonded Sample / ADR Prescreen Wiring Diagram ...... 123 5-12Hystereticdc/SFQDriveratT=100mK...... 125 5-13ThermallyCycledShuntResistor...... 125 5-14 Initial SFQ–Qubit DR Wiring Diagram ...... 127 5-15 Heavily processed qubit lifetimes ...... 128 5-16 Heavily processed qubit lifetimes ...... 130 5-17 Qubit Subharmonic Forest / Digital Rabi Oscillation ...... 132 5-18 Wiring diagram with mixing on SFQ drive line ...... 134 5-19 Orthogonal SFQ Axis Control ...... 135 5-20 Orthogonal SFQ Control Data ...... 136 5-21 Application and Inverse of single qubit SFQ–Cliﬀords ...... 138 5-22 Qubit T1 with and without background SFQ driving ...... 139 5-23RandomizedbenchmarkingofSFQdrivengates...... 140

5-24 Quasiparticle T1 poisoning ...... 143 5-25QuasiparticleRelaxation...... 144 5-26 Quasiparticle induced qubit frequency shifts ...... 145

5-27 dωq Vs.Γ ...... 147

5-28 dωqb Vs.Γ...... 150

B-1AlFabricationSurfaceandx–raySpectrogram...... 155 xii

B-2NoProtectionLayerSurfaceandx–raySpectrogram...... 157 1

Chapter 1

Introduction

There’s plenty of room at the bottom. — R. P. Feynman (1959)

The meteoric rise in computing power seen in the last four decades has been accompanied by an ever shrinking physical footprint in the size of the transistor. Moore’s law, named for Intel founder Robert Moore, has accurately predicted roughly a doubling in the transistor count per integrated circuit every 2 years since 1965. However, as the size of the transistor becomes of order the electron wave function, the technological hurtles associated with increasing the transistor count by even a few percent become seriously daunting. This physical challenge is not alone in limiting the computing power of a classical transistor based machine. There exists classes of computational problems whereby the solutions cannot be calculated either eﬃciently (i.e. in polynomial time) by a classical machine or at all. To this end, the control and readout of individual quantum systems has risen from scientiﬁc curiosity to full scale research endeavors conducted by multiple universities and private companies in an eﬀort to build a quantum computer. The work contained herein focuses on two diﬀerent quantum computing architectures utilizing superconducting circuits. The development and employment of a superconducting CPW resonator photon bus between a neutral Rydberg atom and superconducting quantum bit (qubit) is discussed. Additionally, the theory and realization of coherent qubit control via 2

Single Flux Quantum (SFQ) pulses is addressed.

1.1 Physical Information

The fundamental function of science is the extraction of information from an unknown system. Systems may be left devoid of external perturbations and monitored in eﬀort to gain information of the so called “steady state.” Perturbations may also be applied in an eﬀort to gain insight concerning dynamical behavior subject to external stimuli. Whatever the experiment may be, thermal dynamics, and more fundamentally, quantum mechanics, set bounds on the amount of information one may ultimately extract [1]. The link between information and its physicality becomes apparent if one deals in system entropies. In statistical thermodynamics, a system comprised of a multitude of particles can be described in macroscopic ensembles of microscopic states (“microstates”) [2]. In 1902, Gibbs generalized the work of Boltzmann in deriving a general form for the entropy that does not require thermodynamic isolation. Speciﬁcally, the Gibbs ensemble

N ! Ω ≡ A , (1.1) ΠNS! where NS is the number degeneracy of microstates with energy ES and NA = NS. Substi- S tuting Eq. (1.1) into the Boltzmann entropy equation yields

S ≡ kB log(Ω) = kB [log(NA!) − log(ΠNS!)] = −kB PS log(PS) , (1.2) S where PS = NS/NA is the probability of observing the system in state S. Equation Eq.(1.2) is formally known as the Gibbs entropy. In the limit that all k possible microstates exist with

equal probability PS =1/k, then Eq. (1.2) reduces to that derived by Boltzmann

S =kB log(k) (1.3) 3

The extension of Eq. (1.2) to the quantum mechanical case is straightforward. We wish to describe the amount of information available to the experimenter for a given quantum mechanical system. Most generally, we can describe any quantum state via its density matrix

ρ = pi |ψiψi| , (1.4) i where pi is the probability of measuring the system ensemble in state |ψi. Physicality forces normalization upon the ensemble such that the sum of the probabilities equates to one

pi =Tr[ρ]=1. (1.5) i Additionally, one may make the distinction of pure states. Pure quantum mechanical states have exactly one non-zero term in the density matrix.† Physically, this means that the state | | can be completely described by one wave function ψ = i ai φi with the condition that 2 |ai| =1. Mathematically, state purity may be deﬁned as

Tr[ρ2] ≤ 1 (1.6) where the equality holds for a pure state, ρ2 = ρ. Utilizing the deﬁnition of Eq. (1.4),one can immediately write down the entropy for a general quantum state, ﬁrst deﬁned by Von Neumann [1]

SVN = − pi log(pi)=−Tr [ρ log(ρ)] . (1.7) i

The Von Neumann entropy takes on values 0 ≤ SVN ≤ log(k) where the lower bound is saturated for pure states (log(1) = 0) and the upper bound is achieved for maximally mixed

states (pi = p =1/k). When the upper bound is saturated, the probability of measuring any microstate is completely random and thus any information concerning the macrostate is lost.

†Also known as the "Schmidt Number" [3] 4

If we deﬁne the amount of information to be measured in terms of the maximum amount of information to be lost, we can deﬁne a physical systems “free information”

B = Smax − SVN = log(k)+Tr[ρlog(ρ)] . (1.8)

While somewhat odd looking, Eq. (1.8) can be thought of as the informational equivalent to the Helmholtz free energy in thermodynamics. The extension of this formalism to computing and speciﬁcally the binary representation of information in what is now called “information theory” was pioneered by Shannon in 1948 after discussions with Von Neumann [4].

1.2 Quantum Bits

With the ability to harness quantum superposition, to describe qubits one must change from a discrete set of either logical 0 or logical 1 to a more continuous representation. A qubit state can be generalized as a superposition of |0 and |1 up to a global phase

|ψ =cos(θ/2) |0 + sin(θ/2)eiφ |1 . (1.9)

Borrowed from the nuclear magnetic resonance ﬁeld, the Bloch sphere [5] provides a graphical representation of a qubit state (see Fig. 1-1). A Bloch vector of unit length corresponds to a pure quantum mechanical state while lengths less than unity describe “mixed” states. A Bloch vector length of 0 is said to be maximally mixed. Single qubit logic gates are enacted via rotations about the axes. Rotations taking one logical state to another are oft referred to 5 as π-gates and are represented by the X- and Y-Pauli matrices

⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢01⎥ ⎢1⎥ ⎢0⎥ X |0 = ⎣ ⎦ ⎣ ⎦ = ⎣ ⎦ = |1 (1.10) 10 0 1 ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎢ − ⎥ ⎢ ⎥ ⎢ ⎥ ⎢0 i⎥ ⎢1⎥ ⎢0⎥ Y |0 = ⎣ ⎦ ⎣ ⎦ = ⎣ ⎦ = i |1 . (1.11) i 0 0 i

The X and Y gates are the quantum mechanical equivalent of NOT gates for quantum bits with a 90 degree phase between their axes of rotation. One may also rotate the qubit vector onto the equator of the Bloch sphere, creating a superposition of |0 and |1 via a Hadamard

gate, ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 1 ⎢11⎥ ⎢1⎥ 1 ⎢1⎥ 1 H |0 = √ ⎣ ⎦ ⎣ ⎦ = √ ⎣ ⎦ = √ (|0 + |1) . (1.12) 2 1 −1 0 2 1 2

It is important to note that, by deﬁnition, any valid qubit gate A be unitary and thus preserve the qubit vector length [6] A†A =1. (1.13)

Not only are the relative amplitudes between the qubit states crucially important but also the relative phase between them. Quantiﬁcation of a qubits performance is discussed in the next section.

1.2.1 Qubit Metrics

Perfect qubits would allow for the application of quantum gates such that arbitrary superpo- sitions of |0 and |1 would remain in the predeﬁned state until further acted upon and/or measured. However, qubits are not immune from environmental perturbations [7], thus qubit states are not inﬁnitely long lived. Qubit performance is characterized via two physical

processes giving rise to two relevant timescales. The ﬁrst, denoted as T1, encompasses the rate at which the qubit relaxes from the excited state to the ground state, irreversibly losing 6

Figure 1-1: Bloch sphere representation of a qubit. For pure states, the vector length is 1 and lives on the surface of the sphere.

energy to the surrounding environment. The rate, as described by Fermi’s golden rule, is

∗ constant in time. The second, commonly denoted to as T2 or Tφ, describes the process at which the phase between qubit states becomes no longer well deﬁned and coherence is lost. The measurement of either time involves inducing coherent rotations about the Bloch sphere followed by an adjustable idle time before ﬁnally concluding with qubit measurement.

∗ The general procedures for measuring T1 and T2 are displayed in Figures 1-2(a) and 1-2(b),

respectively. In a T1 measurement, the qubit is ﬁrst coherently driven into the |1 state via a π–rotation. It is then subsequently allowed to relax for increasing intervals of time before a measurement of the state is made. The probability of measuring the qubit in the |1 state is then plotted against the interleaved idle time between the initial gate and the

∗ measurement and ﬁt to a decaying exponential. The dephasing time T2 is measured in very much a similar manner except that the idle gate is between two π/2-rotations. In the rotating frame, when the qubit Bloch vector is placed on the equator, noise induces incoherent rotations about the z axis. These incoherent kicks of the vector randomize the phase between 7

(a) 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

1 (b) 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

Figure 1-2: Qubit lifetime measurement protocols. (a) Qubit relaxation is measured via a π-gate followed by a variable delay τ prior to measurement. (b) Qubit dephasing is measured by a pair of π/2-gates with a variable delay interceded between them. Both measurements require many repetitions.

the two computational states until such a time that all phase coherence between the qubit states is lost and the probability of measuring the |1 state decays exponentially to 0.5 where then the qubit is maximally mixed. In any physical implementation, qubit measurements project the state of the qubit onto either |0 or |1, thus measurements must be repeated many times over in order to build up necessary statistics.

1.3 Physical Qubit Realizations

Any physical quantum computing implementation must have particular attributes in order to be considered a viable pathway to a large scale quantum computing architecture [8]. Known as the “DiVincenzo criteria”, ﬁve necessary conditions must be met:

1. Physically realizable qubits that can be integrated into a larger system.

2. Qubit initialization must be robust. 8

∗ 3. Decoherence times (T1,T2 ) must be longer than the measurement times

4. A set of universal 1 and 2 qubit quantum gates.

5. The qubits must be able to be measured.

In the framework of these criteria, current eﬀorts in neutral atoms and superconducting qubits are brieﬂy reviewed.

1.3.1 Neutral Atoms

The neutral atom approach to quantum information utilizes adjacent hyperﬁne ground state energy levels to encode logical qubit states [9]. A thorough review on the state of the art is provided in [10]. The inherent atomic species homogeneity and weak magnetic dipole transition between qubit states provides a double-edged sword in the context of quantum information processing and the DiVincenzo criterion. Considering criteria 2 and 3, qubit state preparation via optical pumping is robust against atom exchange and decoherence times are intimately related with how strongly the system interacts with it’s environment. Atomic qubit state lifetimes have now reached the ∼ 10 ms range [11]. However, the same qubit homogeneity aﬀorded by using a single atomic species presents issues when trying to address single atomic qubits in multi-qubit arrays. Imperfect spatial alignment of the qubit laser probe can couple to neighboring qubits and produce errors [12]. Additionally, the weak magnetic moment between adjacent hyperﬁne states necessitates large ﬁelds be utilized to induce coherent qubit rotations at rates comparable to the relaxation and dephasing rates. Atomic qubit–qubit coupling is also severely limited by this weak moment for realizable inter–qubit spacings of a few microns. This had led to much research in dynamical coupling methods such as atom–atom collisions [13] and real / virtual photon exchange [14, 15]. Neutral atom quantum computing via Rydberg atoms aims to tackle the issues previously addressed. By exciting an electron far outside the computational subspace to a large principle quantum number n, qubit–qubit interactions are mediated via an electric dipole coupling [16], 9 with dipole-dipole interaction strength scaling as n4 [17]. This greatly increases not just the strength of the interaction for a ﬁxed distance between atoms, but also the physical range in which a strong interaction can be realized. In fact, the ﬁrst atomic two-qubit entangling gates were performed utilizing the n =50andn = 51 Rydberg states in two Rb atoms separated of order 1 cm in space [18]. Current eﬀorts in neutral Rydberg atom quantum computing have achieved 7×7 qubit arrays with single qubit gate ﬁdelities approaching 99% as measured via randomized bench- marking [19]. CNOT gates realized utilizing nearest neighbor atoms have achieved raw ﬁdelities of 77% [20]. These impressive achievements have allowed the neutral approach to satisfy the remaining DiVincenzo criteria albeit with an important caveat. Speciﬁcally, atomic state readout is not just quantum state destructive, but qubit destructive [21]. "Push out" techniques employ directionally imbalanced laser light resonant with a particular atomic state. This imbalance applies light pressure to the atom if it is in the particular state of interest. Readout is then performed by integrating the atomic ﬂuorescence signal, the level of which allows for determination of the qubit state. Once a measurement is performed, a new atomic qubit must be reloaded into the trap for further processing. A particular asset that neutral atoms do posses however is a large range of energy level transitions spanning from the microwave to optical regime. This expanse makes them attractive candidates to facilitate a microwave to optical photon transition, porting quantum information processed on one superconducting chip to another or even between diﬀerent dilution refrigerators via a photonic ﬂying qubit. While outside the scope of the work presented here, the idea is interesting and the reader is referred to recent theoretical work by Gard, B. T., et al for speciﬁcs [22].

1.3.2 Superconducting Qubits

Superconducting qubits can be aptly described as anharmonic quantum harmonic oscillators [23]. Two general properties of superconductors allow for such a description; First, the bosonic 10

F = 8 F = 7

F = 6 F = 5

F = 4

F = 3

Figure 1-3: Ground state energy level diagram for Cesium. Commonly known as “clock 2 states,” the hyperﬁne states of the 6 S1/2 level have a precisely deﬁned splitting and make for a long lived two level subspace to encode logical “0” and “1.” With the next nearest transition energy in the optical regime, neutral atom qubits are robust against qubit state leakage out of the computational manifold. 11

nature of the superconducting Cooper pair condensate allows for a single wave function description of the system dynamics. The superconducting condensate acts as a incompressible ﬂuid whereby the entirety of all the multi-particle dynamics can be be reduced to a single √ iφ complex wave function ψ = nCPe , the normalization condition of which gives the total number of Cooper pairs, nCP [24]. When subjected to harmonic boundary conditions (as in either a distributed or lumped element LC circuit), the supercurrent will “slosh” back and forth between the inductor (magnetic ﬁeld) and capacitor (electric ﬁeld) at the resonant √ frequency ω =1/ LC. The combination of the superﬂuid condensate with correct boundary conditions realizes a true physical manifestation of the quantum harmonic oscillator (QHO); The same system which every undergraduate physics student studies. However, as any undergraduate student who has done their homework should be able to tell you, the energy

spacings between all adjacent levels E|n and E|n+1 are degenerate. That is to say that

ω|n→|n+1 = ω for all QHO number states |n. To realize an addressable quantum two-level system (TLS), this degeneracy must be broken. Cooper pair tunneling, as ﬁrst theoretically described by Josephson [25] when looking at weakly coupled superconductors, gives rise to a nonlinear inductance, and provides such a degeneracy breaking mechanism. Considered in detail in section 2.1, the nonlinearity of the Josephson inductance arises naturally from the ﬁrst and second Josephson relations. When embedded in superconducting circuits, Josephson junctions break the degeneracy of QHO energy states, allowing for the realization of engineered quantum TLSs on a chip. What sets the superconducting quantum computing eﬀort apart from all other physical implementations is the ability to directly engineer parameters of the interacting quantum elements. Utilizing standard lithographical techniques, superconducting quantum circuits with a wide range of parameters can be realized for use in experiments exploring the strong and ultra-strong coupling regime of circuit quantum electrodynamics (cQED) and quantum processors. We will explore this technology in far greater depth in the following chapters. 12

1.3.3 Current Hybrid Quantum Computing Approaches

The work contained is this thesis is not the ﬁrst attempt at combining disparate quantum systems. To provide a back drop for the work contained herein, we brieﬂy review other eﬀorts.

1.3.3.1 Superconductors Coupled To Atom Clouds

Some of the foremost work done in coherently coupling superconductors to atomic clouds has been spearheaded by the group of Joseph Fortágh in collaboration with the group of Reinhold Kleiner at the University of Tüebingen. In these experiments, clouds of Rb atoms are cooled and then trapped via an on-chip superconducting "z-trap" [26, 27, 28] The z–trap is a continuous wire that is physically laid out in the shape of the letter “z.” When current is passed through the wire, the magnetic ﬁeld (and hence the trapping potential) produced has a shape similar to a cigar centered over the middle electrode. In 3 dimensions, the atoms are trapped in this cigar shaped potential. The z–trap is in close proximity to the current antinode of a superconducting CPW resonator. This setup has achieved a magnetic dipole coupling between the hyperﬁne ground state and the magnetic ﬁeld of the CPW resonator. However, in the limit of single photon,

5 single atom coupling, the Rabi frequency g/2π<1 Hz. The employment of Natom > 10 atoms is often utilized to provide a collective enhancement of the coupling. This, while also driving the resonator with a classical RF ﬁeld combats this low coupling strength with driven Rabi oscillations of strength g/2π ≈ 2 kHz now having been observed in the lab.

1.3.3.2 Superconductors Coupled to Quantum Dots

Quantum dots (QDs) aim to achieve a scalable quantum computing architecture by harnessing the power of trapping single electrons in harmonic potentials in the solid state. While typically read out with either DC transport measurements or charge sensing, there has been a growing interest in employing techniques developed in cQED to perform dispersive readout of the QDs. Recent work in both the Petta group at Princeton [29, 30, 31] and Wallraﬀ 13

(a) (b) 1.0 200

150

100 (norm.)

50

0 0 -500 -2500 250 500

Figure 1-4: (a) Cartoon of a superconducting atomic z-trap. For a DC current, the vector components of the resulting magnetic ﬁeld create a cigar shaped potential minimum centered over the middle wire. (b) Numerical simulation of the z-trap magnetic ﬁeld over the center conductor for a bias current of 1 amp. The dark blue region in the center is the ﬁeld minimum and provides the trapping potential. Larger bias currents will deepen the trap potential while also pulling the trap closer to the chip surface. group at ETH-Zurich [32, 33, 34] have demonstrated coherent coupling between QDs and superconducting CPW resonators. However, as explicitly mentioned in the Petta work, the multilayer fabrication required to realize the QD system is detrimental to the performance of the superconducting CPW resonator. Both groups report quality factors below 104. Realizing coupling limited resonator performance in the face of the challenges presented by multilayer fabrication is an active area of research. 14

Chapter 2

Josephson Physics and Superconducting Circuit Fundamentals

The career of a young theoretical physicist consists of treating the harmonic oscillator in ever-increasing levels of abstraction

— Sidney Coleman (1975)

In 1911, Kamerling Onnes, while studying properties of metals at cryogenic temperatures, came across a puzzling result. For a particular set of elements, all traces of electrical

resistance vanished below a certain critical temperature, Tc. This discovery, now known as superconductivity, has to date spawned six Nobel prizes and a wealth of scientiﬁc endeavor and physical insight spanning a large breadth of the physical sciences. A review of the discovery and some controversy surrounding it can be found in [35]. In this chapter, we will focus speciﬁcally on the applied aspects of these discoveries concentrating on the Josephson eﬀect and devices derived from it. 15

2.1 Basic Phenomena

BCS theory posits that the particles responsible for the super part of superconductivity are pairs of bound electrons, formally referred to as Cooper pairs. Bosonic in nature, these particles condense into an energy state below the Fermi energy whereby the collective lot of all the particles can be described by one encompassing wave function

iφ ψ = ψ0e , (2.1)

2 2 where the normalization condition |ψ| = |ψ0| = nCP, the number of Cooper pairs. The Josephson eﬀect extends BCS theory by looking into the tunneling nature of Cooper pairs across barriers, colloquially referred to as "weak links." Generally speaking, weak links encompass any barrier between two superconducting electrodes where the critical current, the maximum electrical current that be sustained prior to loss of superconductivity, is suppressed [36]. These barriers can be made of insulators (such as an oxide) forming Superconductor– Insulator–Superconductor (SIS) junctions or even normal non–superconducting metals forming a Superconductor–Normal Metal–Superconductor (SNS) junction. This ﬁrst fundamental discovery about such systems is that, even in the absence of an applied potential, they support a ﬁnite supercurrent across the electrode barrier. Know as a the DC Josephson eﬀect, the magnitude of the tunneling current is proportional to the sin of the phase diﬀerence between the two superconducting wave functions of either side of the barrier [37].

I(δ)=Ic sin(δ) . (2.2)

where Ic is the critical (maximum) supercurrent the junction can support and δ = φ2 − φ1 is the phase diﬀerence between electrode wave functions. The second relation derived by Josephson relates the rate of change of the phase diﬀerence to the potential diﬀerence V placed across the electrodes, 16 (a) (b)

C LJ R

Figure 2-1: (a) Cartoon depiction of a Josephson junction. The phase diﬀerence δ = φ2 − φ1 between the wave functions on either side of the junction is what allows for supercurrent to ﬂow across the boundary. (b) Resistive and Capacitively Shunted Junction circuit approximation for a Josephson junction. When current biased in the superconducting state (Ib

dδ V = . (2.3) 2e dt

Through relating Eq. (2.2) and Eq. (2.3), an inherent inductance of the Josephson element takes form

V /2e × dδ/dt Φ0 L0 LJ = = = = , (2.4) dI/dt Ic cos(δ) × dδ/dt 2πIc cos(δ) cos(δ) where Φ0 = h/2e is the superconducting magnetic ﬂux quantum. The nonlinearity of the Josephson inductance deﬁned in Eq. (2.4) will be a crucial point in the discussion of solid state superconducting qubit functionality. When no external bias current is applied, the potential energy of the junction is

Φ dδ I Φ U = IV dt = I sin(δ) 0 dt = c 0 sin(δ)dδ = E (1 − cos(δ)) (2.5) c 2π dt 2π J where the Josephson energy EJ = IcΦ0/2π. We can then directly write down the Hamiltonian 17 for the JJ (ignoring constant oﬀsets)

q2 H = − E cos(δ) , (2.6) 2C J where q is the charge on the junction capacitor plates. The addition of an explicit external

bias current, Ib, modiﬁes Eq (2.2) and consequently Eq. (2.5) in the following ways:

I(δ)=Ic sin(δ) − Ib (2.7a)

U = IV dt = Ej(1 − cos(δ) − iδ) (2.7b)

where we have introduced the reduced bias current i = Ib/Ic. This new potential modiﬁes the Hamiltonian in Eq. (2.6) and produces what is commonly referred to as the “tilted washboard potential” q2 H = − E (cos(δ) − iδ) . (2.8) 2C j

A plot of the Josephson potential as a function of reduced bias current i is shown in Figure 2-2. For i = 0, we recover the potential in Eq. (2.6). The phase particle relaxes to the minimum of the potential at phase values of δ =2nπ. As the bias current is increased, the term linear in the phase in Eq. (2.8) begins the dominate and tilts the potential. When the bias current surpasses the critical current of the junction, the phase particle freely rolls down

the potential and a voltage V =Φ0/2π × dδ/dt develops across the Josephson junction.

2.1.1 RCSJ Model

A helpful model in exploring the dynamics of Josephson junctions is the resistive and capacitvely shunted junction (RCSJ) model. In modeling the Josephson dynamics via splitting the junction into constituent elements, the RCSJ model allows for direct calculation of the dynamics through straight forward circuit analysis. Figure 2-1(b) displays a circuit schematic of a Josephson junction in the RCSJ approximation. Writing down the total 18

Figure 2-2: Potential energy landscape for a current biased Josephson junction for diﬀerent values of reduced bias current i = Ib/Ic. For no bias current (blue curve), the potential is purely (co)sinusoidal and the junction is in the supercurrent state with the phase particle living in a potential minimum at δ =2nπ and dδ/dt = 0. As the external bias current is increased to appreciable fractions of the critical current (red curve), the term linear in δ begins to dominate and tilts the cosine potential. At values of i ≥ 1, the phase particle freely rolls down the potential landscape. With the surpassing of the critical current, the junction enters the voltage state with V =Φ0/2π × dδ/dt.

current in the RCSJ model of a Josephson junction, one must account for the current in the junction, resistor, and capacitor,

V dV I = I sin(δ)+ + C . (2.9) c R dt

Making use of both the second Josephson relation and deﬁnition of the Josephson inductance, Eq. (2.9) can be recast as

I dδ d2δ = sin(δ)+τJ + τJ τRC 2 , (2.10) Ic dt dt 19

with the Josephson time constant τJ = LJ /R and the RC time constant τRC = RC. Making

the substitution dt = τJ d ﬁnally yields

I dδ d2δ = sin(δ)+ + βC 2 , (2.11) Ic d d

2 with βC = τRC/τJ =2πIcR C/Φ0. Formally known as the Stewart-McCumber parameter

[38, 39], βC is the ratio of the RC time constant to the Josephson time constant. Junction

dynamics in two limits of βC is worth further discussion. Over damped Regime

When the RC time constant is much shorter than the Josephson time constant, βC < 1

and the junction is said to be over damped. For values of βC 1, the last term in Eq. (2.9) can be ignored and the equation solved in both the steady state (i<1 ,dδ/dt=0)and dynamical phase (i>1) limits.

In the steady (superconducting) state, the total current I

I δ = arcsin . (2.12) Ic

When the bias current surpasses the critical current value, a voltage develops across the junction and the phase evolves in time. After rearrangement of terms in Eq. (2.11), we arrive at the integral equation

⎡ ⎛ ⎞ ⎤ 2 − (I/Ic) − 1 I/Ic sin(δ) ⎣ 2 ⎝ ⎠ ⎦ δ = dt = 2 arctan 1 − (Ic/I) tan t + Ic/I (2.13) τJ τJ

2 The junction phase can be seen to be evolving at a frequency ω = (I/Ic) − 1/τJ . The time averaged voltage then across the junction is then

ω V (t) =Φ = IR 1 − (I /I)2 (2.14) 0 2π c 20

for I>Ic. Equation Eq. (2.14) is single valued and junctions in this regime do not exhibit hysteresis. Underdamped Regime

In the under damped regime, βC ≥ 1 and the second order derivative in time of δ in Eq. (2.11) can not be ignored. For currents below the critical current, the results are the same as in Eq. (2.12) again with no voltage developing. Upon return from the ﬁnite voltage regime, under damped junctions do not immediately reenter the superconducting state with

the condition Ib

2 μm Nb-AlOx-Nb trilayer Josephson junction is shown in Figure 2-3(c). Above the critical current of approximately 40 μA, the junction switches to the voltage state. Above the

superconducting gap voltage Vg =2Δ/e, where Δ is the superconducting gap energy and e is the fundamental electronic charge, the junction is completely normal and exhibits a linear IV dependence that, when extrapolated to zero bias current, passes directly through zero voltage as Ohm’s law demands.

2.1.2 Superconducting Quantum Interference Devices

When a loop of superconducting metal is interrupted by a JJ, interference of the supercon- ducting wave function phase φ becomes directly measurable. Any superconducting loop comprised of one or more JJ is referred to as a superconducting quantum interference device (SQUID). Because all elements of a SQUID are superconducting, the single–valuedness of the BCS wave function imposes a constraint on the phase of junctions within loops

2πΦj δi + =2πn , (2.15) i j Φ0 21

(a) (b) 2 2

1.5 1.5 c 1 c 1 I/I I/I

Ir 0.5 0.5

0 0 0123456 0123456 V/I R V/I R c c (c) 160 120 80 40 0 -40 -80 Bias Current (uA) -120 -160

-5 -4 -3 -2 -10 1 2 3 4 5 Voltage (mV)

Figure 2-3: (a) Representative nonhysteretic IV curve of a overdamped junction. (b) Representative hysteretic IV curve of an underdamped junction. The retrapping current Ir for an unshunted junction should be very near 0 A. (c) IV curve of a 4 μm2 area Nb-Al trilayer junction measured at T = 4.2 K. The critical current of this particular junction was approximately 40 μA. The retrapping current is ∼ 1μA. The dashed line is drawn to show the linear dependence on bias current for voltages above the superconducting gap voltage. 22

th th where δi is the phase of the i junction and Φj the ﬂux of the j inductor. In the case of a DC–SQUID, a loop interrupted by two parallel JJs, the total current through the loop is simply sum of the current in both branches

IT = Ic1 sin(δ1)+Ic2 sin(δ2) , (2.16)

| | | | Taking Ic1 = Ic2 = Ic and expanding Eq. (2.15), the relation between junction phases is

Φ ΦA δ1 − δ2 +2π = δ1 − δ2 +2π + βL =2πn . (2.17) Φ0 Φ0 with the total ﬂux Φ = ΦA +ΦLoop the sum of an external applied ﬂux ΦA and the loop ﬂux

ΦLoop = LIcirc and βL =2πLIcirc/Φ0. The SQUID circulating current is then

1 I = I [sin(δ ) − sin(δ )] . (2.18) circ 2 c 1 2

Combining Eq. (2.16) and Eq. (2.17) for n = 0 yields

ΦA DC–SQUID IT =2Ic cos π sin(δ)=Ic sin(δ) (2.19) Φ0 wherewehavemadethesubstitutionδ ≡ (δ1 + δ2)/2. Equation (2.19) allows for the description of a symmetric DC-SQUID as simply a single JJ with an external ﬂux dependent

DC–SQUID critical current Ic =2Ic cos (πΦA/Φ0). Further discussions of SQUIDs utilized in superconducting classical and quantum computing can be found in Chapter 5 and Appendix A.

2.2 Quantization Of Electrical Circuits

The ability to perform quantum algorithms with superconducting circuits means that the constituent components comprising the processor must be behaving quantum mechanically. The previous section discussed basic Josephson physics beginning with the two Josephson 23

relations. The RCSJ model and Stewart-McCumber parameter are very important is discussing and understanding junction behavior even when the system temperature is an appreciable

percentage of the superconducting critical temperature, Tc. However, when the relevant energy scales of the circuit Hamiltonian are much larger than the bath temperature, new phenomena arise that can only be explained through quantum mechanics. The energy stored in an unshunted, unbiased Josephson junction was derived in Eq. (2.6) and is reprinted here for convenience,

q2 H = − E cos(δ) . (2.20) 2C J

The relevant temperature scale for a junction with area A =0.1 μm2 , critical current

2 2 density J =10A/cm, capacitance per unit area C =10fF/μm are EJ /kb ≈ 230 mK

2 and Ec/kb ≈ 900 mK, where the junction charging energy Ec = e /2C and C = C×A.

For temperatures T EJ /kb ,Ec/kb, any quantum mechanical behavior is washed out by incoherent processes due to the thermal bath. However, at temperatures achievable at the cold stage of dilution refrigerators, the quantum mechanical behavior becomes manifestly apparent. The Hamiltonian for the Josephson element in Eq. (2.20) does not lend itself to direct calculation. Approximation methods do however provide the necessary clarity to understand how to make an artiﬁcial TLS and thus a qubit from a Josephson junction. We begin by expanding the potential to second order in the phase and dropping constant terms which yields q2 E H = + J δ2 . (2.21) 2C 2

While this expression may look straightforward, the choice of variables is a bit awkward. Expressing δ is terms of the circuit node ﬂux φ is a more natural choice. The relationship between the junction phase and node ﬂux is found by integrating Eq. (2.3) yielding 24

2π 2π δ = Vdt= φ. (2.22) Φ0 Φ0

Substituting this back in to Eq. (2.21) yields

q2 E 2π 2 H = + J φ2 (2.23) 2C 2 Φ0

To this level of approximation, the Josephson junction is nothing more than a harmonic oscillator with conjugate variables q and φ. In very much the same way as with the traditional QHO [40], we can promote the conjugate variables to operators and provide them with the appropriate commutation relation [φ, q]=i having units of action. Equating this Hamiltonian to that of the traditional QHO with q ↔ p, φ ↔ x,and C ↔ m, we can ﬁnd the equally spaced energy levels in this approximation.

1 E 2π 2 Cω2 = j (2.24) 2 2 Φ 0 E 2π2e 2 ω2 = j (2.25) C h 2 2 2 e 2 ω = EJ (2.26) C

ω = 8EcEJ (2.27)

In the language of 2nd quantization, the full Hamiltonian for this QHO is then

H = ω(a†a +1/2) (2.28) with the raising (a†) and lowering (a) operators satisfying [a, a†]=i and

Cω i a = φ + q (2.29a) 2 2 Cω i a† = φ − q . (2.29b) 2 2 25

While this illustrates how to go from the language of circuit parameters to that of standard quantum mechanics, the results are lackluster. To this order of expansion, a Josephson junction is a QHO with degenerate energy level spacings ω. This would make a very poor qubit indeed as addressing any two neighboring energy states with a resonant tone would invariably lead to leakage out of the computational manifold. Thankfully however the cos term is an inﬁnite sum and we can expand to another order in φ. The quartic term in the expansion is

E 2e 4 E 2e 4 2 V = − J φ4 = − J (a + a†)4 . (2.30) 4! 4! 2Cω

The expansion of the last term is tedious as the order of the operators must be kept consistent. However, only 6 terms in the expansion conserve excitation number n = a†a and yield non-zero ﬁrst order corrections. When added together, these 6 terms provide a correction to the energy of the form

2 En = ω(n +1/2) − Ec(6n +6n +3)/3 , (2.31) where now the energies between adjacent levels are

E10 = E1 − E0 = ω − 4Ec (2.32a)

E21 = E2 − E1 = ω − 8Ec (2.32b)

α = E21 − E10 = −4Ec . (2.32c)

We can see now that the degeneracy has been broken and an anharmonicity α exists of order

the charging energy Ec. With this anharmonicity now in place, we can truncate the Hilbert space of the Josephson junction to the ﬁrst two energy levels and adjust the Hamiltonian to

ω H = ω(n +1/2) → σ (2.33) 2 z 26

where σz is the Pauli operator. The exact same treatment can be applied to a true LC oscillator circuit where the same QHO Hamiltonian can be arrived at albeit without the higher order corrections. The quantizing of a superconducting LC oscillator along with the justiﬁcation of the Josephson junction as a quasi TLS motivates the next chapter (and arguably the past 20 years worth of cQED and superconducting quantum computing) and it’s discussions of Jaynes-Cummings physics on a superconducting circuit chip.

2.3 Superconducting Qubit Zoo

Before moving on to describing just how Jaynes-Cummings physics and QED style interactions are realized on a chip, it is worth covering a few physical implementations of Josephson based superconducting qubits and reviewing the diﬀerences between them.

2.3.1 Phase qubit

Initial demonstrations of quantum like behavior were ﬁrst demonstrated in current biased Josephson junctions [41]. Seminal work from the lab of John Clark at UC-Berkeley showed that the Josephson phase particle, when subjected to the tilted washboard potential of a current biased Josephson junction, exhibited quantized energy levels [42, 43]. It was these levels that were used to make some of the very ﬁrst superconducting qubits. Large capacitively

shunted Josephson junctions were current biased such that the ω10 transition was in the microwave regime. Microwaves were applied directly through the DC bias line to excite superpositions of the |0 and |1 state. Readout was performed via driving the |1→|2 transition. If the qubit was in the |1 state, it’s probability of tunneling through the barrier would increase via the transition to the |2 state and a measurable voltage would develop. If the qubit was in the ground state, the drive would be oﬀ resonant and the qubit would not be excited. This is what is meant by "phase" in the qubit description; Is the qubit in the 27 superconducting or resistive state? This style of qubit suﬀers from a multitude of issues. The current bias used to tilt the potential is inherently noisy and limits the qubit phase performance as the energy spectrum is directly dependent upon the bias stability. Additionally, the large capacitors used to shunt the qubits have traditionally been made from amorphous dielectric materials that exhibit densely populated energy gap spectrums that can absorb the qubit excitation [44]. Furthermore, the readout is qubit state destructive with only the tunneling event (or lack there of) being the indicator of the qubit state.

2.3.2 Flux Qubit

Flux qubits consist of a superconducting loop interrupted by one or more Josephson junctions [45, 46, 47]. As a consequence of the single valued-ness of the superconducting wave function at any point in space, the ﬂux through a superconducting loop is quantized in units of the

superconducting magnetic ﬂux quantum Φ0 = h/2e [48]. These loops are then ﬂux biased via an external current inductively coupled to the loop to a point where the induced ﬂux is Φ0/2. This fractional ﬂux quantum frustrates the circulating current in the loop into superpositions of clockwise and counter-clockwise propagation around the ring. The counter propagating currents are mapped onto the qubit subspace and readout is performed via an external SQUID magnetometer. Historically, ﬂux qubits have been plagued by fabrication issues leading to junction assymetry [49] and ultimately poor performance. Recent work has focused on capacitively shunted (colloquially referred to as C-Shunt) ﬂux qubits [50, 51] that have begun to show promise in achieving better relaxation and coherence times.

2.3.3 Transmon

The transmon qubit [52], ﬁrst developed at Yale in the lab of Rob Schoelkopf, can be described exactly in the same fashion as the unbiased Josephson junction. The main diﬀerence is that the junction (or two arranged as a DC SQUID for frequency tunability, see Appendix A) 28 is capacitively shunted to ground through a large (∼ 100 fF) planar capacitor. This shunt capacitance all but eliminates the qubits sensitivity to charge noise by eﬀectively dividing out the kinetic q dependant term in the Hamiltonian. The trade oﬀ, however, is that the large shunt capacitance also lowers the charging energy Ec and consequently the anharmonicity generally enjoyed by its phase qubit counterpart. This small anharmonicity α ∼ 100 MHz is an aﬀordable price to pay in gaining the charge insensitivity that plagued earlier charge qubit designs and has allowed for the building and control of multi-qubit processors with individual

∗ qubit T1 and T2 times approaching 100 μs[53]. The work detailed in the latter half of this thesis employs a “rectmon” transmon qubit [54, 55] with a 75 fF shunt capacitance and a center frequency of 4.8 GHz. 29

Chapter 3

Circuit Quantum Electrodynamics

Circuit Quantum Electrodynamics (cQED) allows for the ground breaking, Nobel prize wining work pioneered by Serge Haroche [56] and David Wineland [57, 58] in cavity quantum electrodynamics to be faithfully emulated and extended utilizing superconducting circuit elements. Playing the role of the three dimensional cavity is the superconducting coplanar waveguide (CPW) resonator. It’s atomic counterpart is played by the transmon qubit. The electric dipole interaction between the two is mediated via a coupling capacitor. The ability to directly engineer the on-chip elements has allowed for the exploration of a plethora of regimes from strong to more recently ultra–strong coupling [59, 60]. In the latter, the excitation exchange g rate between the cavity and qubit is of order the resonant frequency of the cavity and/or qubit. Moreover, it has allowed for the exploration and development of the dispersive

coupling regime in which the detuning Δ ≡ ωr − ωq g between the resonator and qubit is larger than the coupling strength. It is this regime that quantum processors are operated in. We will brieﬂy review the physics of the coupled cavity–qubit system. Afterwards, we will elucidate how to bring up a single cavity–qubit system in the lab, focusing on tangible quantities. 30

3.1 Cavity–Qubit Hamiltonian

The coupled cavity-qubit system circuit diagram is displayed in Figure 3-1. The resonator and qubit terms are annotated with the subscripts “r” and “q”, respectively with the coupling

capacitor denoted by Cc. We can write the Lagrangian down for this circuit by inspection utilizing the node ﬂuxes

2 L 1 ˙2 − φr 1 ˙2 2πφq 1 ˙ − ˙ 2 = Crφr + Cqφq + EJ cos + Cc φr φq . (3.1) 2 2Lr 2 Φ0 2

The conjugate momenta, in this case the charges qr and qq, are found utilizing Hamilton’s equations and are L d ˙ ˙ − ˙ qr = ˙ = Crφr + Cc φr φq , (3.2a) dφr L d ˙ − ˙ − ˙ qq = ˙ = Cqφq Cc φr φq . (3.2b) dφq

This is conveniently expressed in matrix notation as

˙ q = Cφ (3.3) with

⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎢ ⎥ ⎢ ˙ ⎥ ⎢ − ⎥ ⎢qr⎥ ˙ ⎢φr⎥ ⎢Cr + Cc Cc ⎥ q = ⎣ ⎦ , φ = ⎣ ⎦ ,C= ⎣ ⎦ . (3.4) ˙ qq φq −Cc Cq + Cc

The Hamiltonian now is ˙ H = φ T q −L. (3.5)

To eliminate the ﬂux derivative term, we utilize the relation

˙ ˙ − − φ T q = φ T CC 1q = q T C 1q. (3.6)

Substituting Eq. (3.6) into Eq. (3.5) and expanding we arrive at 31

Figure 3-1: Lumped element schematic of a resonator coupled to a qubit via coupling capacitor Cc.

2 2 2 qr Cq + Cc φr qq Cr + Cc 2πφq Cc H = + + − Ej cos + qrqq , (3.7) 2 det |C| 2Lr 2 det |C| Φ0 det |C| where the det notation mean the determinant of the argument. The ﬁrst two terms in Eq. (3.7) are the adjusted resonator terms which in the language of second quantization can be expressed as

2 2 qr Cq + Cc φr † Hr = + = ωr(a a +1/2) (3.8) 2 det |C| 2Lr

The third and fourth terms are the re-expressed qubit Hamiltonian which can be truncated into the TLS formalism

2 qq Cr + Cc 2πφq ωq Hq = − Ej cos = σz (3.9) 2 det |C| Φ0 2

The ﬁnal term is the coupling between the resonator and qubit. We can express this is the same formalism by expanding the charges into the appropriate raising and lowering operators for the resonator and qubit. Utilizing the raising and lowering operator relations derived 32

earlier in Eq. (2.29a) and Eq. (2.29b), qq and qr equate to

| | ωq det C − + qq = −i (σ − σ ) (3.10a) 2(Cr + Cc) ωr det |C| † qr = −i (a − a ) . (3.10b) 2(Cq + Cc)

Substituting these two into the coupling Hamiltonian

C H = c q q (3.11) c det |C| q r

C ω det |C| ω det |C| = − c q r (a − a†)(σ− − σ+) det |C| 2(Cr + Cc) 2(Cq + Cc)

√ Cc ωqωr = − (aσ− − aσ+ − a†σ− + a†σ+) . 2 (Cr + Cc)(Cq + Cc)

The outer two terms do not conserve excitation number and can be ignored. Once done, the

ﬁnal form of Hc is

√ C ω ω c q r + † − Hc = (aσ + a σ ) , (3.12) (Cq + Cc)(Cr + Cc) where the coupling between the qubit and resonator is

√ Cc ωqωr g ≡ . (3.13) (Cq + Cc)(Cr + Cc)

Combining the results of Eq. (3.9),Eq. (3.8),Eq. (3.12), and Eq. (3.13), the full Hamiltonian takes the form

ω H = ω (a†a +1/2) + q σ + g(aσ+ + a†σ−) , (3.14) r 2 z 33 which is exactly the Jaynes-Cummings Hamiltonian for an atom interacting with a cavity ﬁeld. We will brieﬂy touch on two diﬀerent limits of this Hamiltonian. Speciﬁcally, we will look

at both the resonant (ωq = ωr) dispersive (|ωq − ωr| g) coupling regimes.

3.1.1 Resonant Jaynes–Cummings Hamiltonian

When the cavity and qubit frequencies are on resonance, they swap an excitation back forth at rate g. The interaction term couples neighboring resonator-qubit states, where now the eigenstates of the systems are a linear superposition of |0,n and |1,n− 1. Re-expressing the Janes-Cummings Hamiltonian in terms of state kets provides valuable intuition for the resonant interaction,

HJC = ωr(n |n n| +1/2) ω + q (|1 1|−|0 0|) (3.15) 2 √ + g n (|0,n 1,n−1| + |1,n−1 0,n|) .

The Schrödinger equation for the wave function describing the superposition of an excitation

−1/2 in either the qubit or the resonator |ψ± =(2) (|0,n±|1 ,n−1)nowreads

ω √ H |ψ = ω (n +1/2) − q |0,n / 2 JC ± r 2 ω √ ± ω (n − 1/2) + q |1,n−1 / 2 (3.16) r 2 √ √ + g n (|1,n−1±|0,n) / 2 ,

√ where the diﬀerence in energy between adjacent hybridized levels is ΔE± =2g n. What the eigenenergies and wave functions say is that when strongly coupled and on 34

5.403 1.0

5.402 0.8

5.401 0.6 5.400 0.4 5.439

0.2 5.438 Normalized Transmission Resonator Frequency (GHz) 5.437 0.0 5.37 5.38 5.395.40 5.41 5.42 5.43 Qubit Frequency (GHz)

Figure 3-2: Resultant qubit-cavity spectrum from master equation simulation of the Jaynes- Cummings Hamiltonian using QuTip. As the qubit is tuned into resonance with the cavity, the system hybridizes and the cavity mode splits with splitting 2g = 6 MHz.

resonance, the coupled components lose their individual nature and the system as a whole can no longer be described as a mere sum of the components. Figure 3-2 displays the results of a master equation simulation of the Jaynes–Cummings Hamiltonian performed utilizing QuTip. In the simulation, the qubit frequency (X–axis) is gradually tuned into resonance with the readout resonator frequency at ωr =5.4 GHz. The coupling strength g for the simulation was set to 3 MHz.

3.1.2 Dispersive Jaynes-Cummings Hamiltonian

In the opposite limit, when ωq − ωr ≡ Δ g, the cavity-qubit system is said to be “dispersively” coupled and no direct excitation exchange between the two can occur. In order to gain insight to the dynamics of this regime, we treat the coupling term as a perturbation to the independent eigenenergies / wave functions of the uncoupled cavity and qubit. 35

The perturbation term couples states of diﬀering photon number and qubit state so, to ﬁrst order in perturbation theory, the eﬀect is zero. To second order, we ﬁnd

2 |1,n− 1| H |0,n| E|0,n = E|0,n + (3.17) E|0,n − E|1,n−1

2 = E|0,n − ng /Δ 2 |0,n| H |1,n− 1| E|1,n−1 = E|1,n−1 + (3.18) E|1,n−1 − E|0,n

2 = E|1,n−1 + ng /Δ , with the unperturbed energies

E|0,n = 0,n| H0 |0,n (3.19)

= ωr(n +1/2) − ωq/2

= (ωrn − Δ/2) ,

E|1,n−1 = 1,n− 1| H0 |1,n− 1 (3.20)

= ωr(n − 1/2) + ωq/2

= (ωrn +Δ/2) .

What this tells us is that the interaction can be represented in the both the photon number and spin basis allowing a rewriting of the perturbing Hamiltonian as

† 2 H = a aσz g /Δ . (3.21)

The full Hamiltonian now has the form 36

H = H0 + H (3.22)

† † 2 = ωr(a a +1/2) + ωqσz/2+ a aσz g /Δ .

Eq. (3.22) can be looked at in two distinct ways. In the ﬁrst, the interaction term imparts a cavity photon number state dependent shift on the qubit frequency where now the qubit Hamiltonian is rewritten as

H = ω σ /2+ a†aσ g2/Δ → (ω +2a†ag2/Δ)σ (3.23) q q z z 2 q z

This Hamiltonian allows for a direct calibration of the superconducting cavity photon number by performing qubit spectroscopy. This is a very powerful tool as it allows one not just to distinguish the photon number, but also the photon number state distribution. This allows one to distinguish between diﬀerent distributions be they Bose-Einstein, thermal, or Fock. The other view point of the interaction term is its eﬀect on the resonator where the center frequency is now inﬂuenced by the state of the qubit.

† † 2 2 † Hr = ωra a + a aσz g /Δ → (ωr + σz g /Δ)a a (3.24)

Where we have dropped constant terms. Looking at this form of the interaction, we see that the resonator frequency is qubit state dependent. This allows for what is referred to as a “weak” measurement; The interaction Hamiltonian commutes with the unperturbed Hamiltonian and through interrogating the state of the weakly coupled resonator, we can determine the qubit state without directly probing it. This form of the Hamiltonian is the basis for all qubit quantum non-demolition (QND) readout. 37

3.2 Qubit-Cavity Bringup

There have been a great many review articles and theses covering in great mathematical detail the nuances of Eq. (3.23) and Eq. (3.24). Instead of covering and/or reviewing the material again, the following sections outline in detail how one measures a superconducting transmon qubit in the lab. It is the hope of the author that this overview will provide a general guideline in experimental procedure.

3.2.1 Coupling The Qubit–Cavity System to the Outside World

Having analyzed the circuit in Figure 3-1, we now have to extract the resonator frequency encoded qubit information from the system so that we can determine and manipulate the qubit state. We have two choices as to how to hook up the readout resonator to the outside world. We can do it by either capacitively or inductively coupling the readout resonator to a microwave feed line. Figures 3-3(a) and 3-3(b) illustrate the inductive and capacitive coupling schemes for an LCR tank resonator circuit (we will ignore the qubit–resonator coupling for now and just concentrate on the resonator–feed line coupling). While shown as discrete elements, readout resonators are almost always distributed elements such as CPW resonators where the resonant frequency is set by geometrical standing wave boundary conditions. We can treat these CPW resonators as open / shorted transmission lines and apply microwave transmission line theory [61] to derive the scattering parameters. We follow the formalism given in [62]. For small internal loss, the input impedance of a shorted transmission line resonator with

characteristic impedance Z0 is

Z Z = 0 (3.25) αl + iπΔω/2ω0 where α is the attenuation constant and Δω = ω − ω0. Relating the internal quality factor

Qi = π/4αl and substituting, Eq. (3.25) takes the form 38

(a) (b) Z0 Z0 Z0 Z0 M Cc

RC L RC L

Figure 3-3: Readout resonator can be coupled to a microwave feed line either capacitively or inductively. (a) Inductively coupled LCR tank circuit with mutual coupling inductance M. (b) Capacitively coupled LCR tank circuit with coupling capacitance Cc.

4Z Q /π Z = 0 i , (3.26) 1+i2Qidx where dx =Δω/ω0. Coupling the resonator to a feed line renormalizes the impedance by the

external coupling quality factor Qe

Z0Qe Z = (1 + i2Qidx) . (3.27) 2Qi

Figure 3-3(a) shows a resonant tank circuit coupled to a feed line via a mutual inductance M. The external quality factor for a inductively coupled resonator is

2ωL QM = (3.28) e R∗ where L is the equivalent inductance of the LRC circuit given by

1 4Z L = = 0 , (3.29) ω2C πω

and R∗ is the Norton equivalent circuit shunt resistance with form

2 2 2 2 ∗ ω M /Z0 ≈ ω M R = 2 2 2 . (3.30) (1 + ω M /Z0 ) Z0 39

Substituting Eq. (3.29) and Eq. (3.30) back into Eq. (3.28), we arrive at

8Z2 QM = 0 (3.31) e π(ωM)2

Alternatively, the resonator may be coupled capacitively as in Figure 3-3(b). The external quality factor for a capacitively coupled resonator is given by

ωR∗C QC = (3.32) e 2 where C is the equivalent capacitance of the LRC circuit

π C = . (3.33) 4ωZ0

R∗ again is the Norton equivalent shunt impedance which in the capacitive coupling case equals

2 2 2 ∗ 1+ω Cc Z0 ≈ 1 R = 2 2 2 2 . (3.34) ω Cc Z0 ω Cc Z0

Substituting Equations (3.33) and (3.34) into Eq. (3.32), we arrive at

C π Qe = 2 2 . (3.35) 8Z0 (ωCc)

The forward scattering matrix element S21 for a shunt impedance is

Z S−1 =1+ 0 . (3.36) 21 2Z

By substituting in Eq. (3.27) for Z and reorganizing, we ﬁnally arrive at

min Qe + i2QiQedx S21 + i2QT dx S21 = = , (3.37) (Qe + Qi)+2iQiQedx 1+i2QT dx

min where S21 = Qe/(Qe +Qi) is the minimum of the matrix element when on resonance (dx =0) 40

and QT is the total quality factor given by

1 1 1 = + . (3.38) QT Qi Qe

For qubit experiments, the external quality factor sets the time scale for qubit cavity photon loading and unloading which limits the cavity photon ring up time and ultimately determines the amount of time needed for averaging during cavity readout. It is important to

keep the cavity photon leak out rate κe ≡ ω/Qe greater than the cavity photon dissipation rate

ki ≡ ω/Qi such that the measurement photons leak into the detector instead of dissipating in

5 the cavity. This reasoning demands large Qi ∼ 10 such that reasonable couplings can be set.

3.2.1.1 Cavity Photon Number

The previous analysis of the coupling dynamics of a microwave resonator also allows for a direct calculation of the cavity photon number as a function of the internal and coupling

quality factor. The power leaking in to the resonator Pi from an applied RF tone of power

Pa is

κe PA Pi = PA = . (3.39) ω Qc

Photons both leak out of and are dissipated in the cavity which yields an output rate Po

κe + κi Pcav Po = Pcav = . (3.40) ω QT

At equilibrium, the power leaking in and out of the cavity will be equal which yields the relation between the intracavity power and the applied power

QT Pcav = PA . (3.41) Qe 41

In the absence of an applied drive tone, the resonator itself can be treated as a power source that delivers power to the surrounding environment at rate

n¯ω2 Pcav =¯nωκT = , (3.42) QT where n¯ is the average photon number inside the cavity. Equating Eq. (3.41) and Eq. (3.42), the average photon number inside the cavity can be directly related to the applied drive power

2 × QT n¯ = PA 2 . (3.43) Qeω

3.2.2 Qubit Signs of Life

A particularly nice feature of the transmon qubit is that devoid of any microwave excitation or control, the mere presence of it’s coupling to the readout resonator imparts a dispersive frequency shift on the resonator. When renormalizing the frequencies of both the resonator and the qubit for the weak interaction with the transmon, levels beyond the [|0 , |1] subspace must be accounted for due to the qubit’s weak anharmonicity. The renormalized resonator frequency, when coupled to a transmon qubit, takes the form

χ ω = ω − 12 , (3.44) r r 2 where g2 g2 χij = ≡ . (3.45) ωij − ωr Δij

This renormalized resonator frequency tells the experimentalist two important pieces of information: The ﬁrst is whether or not the qubit is functioning. The second, depending on

the sign of χ12, tells the experimentalist whether the qubit frequency is above or below the bare, high power resonator frequency and limits the range required for search in order to 42

6.165 -20

6.164 -25 6.163

6.162 -30 6.161

6.16 -35 (dB) 21

6.159 S -40 6.158 Frequency (GHz)

6.157 -45

6.156

6.155 -50 -120 -115 -110 -105 -100 -95 -90 Power (dBm)

Figure 3-4: Qubit readout resonator spectroscopy vs. readout power. In the high power regime (P > −99 dBm), the resonator frequency is simply the bare resonator frequency ωr. As the power of the spectroscopy tone is lowered, the resonator frequency becomes unstable and diﬃcult to measure in the average mode of the vector network analyzer. Finally, below a certain applied power (in this case -108 dBm), the full dispersive shift of the qubit on the resonator can be clearly seen. The upward shift in resonator center frequency denotes that the qubit frequency is below the cavity frequency, yielding a negative χ12. determine the qubit frequency. Figure 3-4 displays a two-dimensional vector network analyzer (VNA) plot where the frequency and power of the microwave readout tone are being swept. For explicit on-chip microwave powers greater than -98 dBm (not accounting for microwave losses due to cabling, reﬂection, etc.), the resonator is at it’s bare resonant frequency that is power independent. Between -98 and -108 dBm, the system is in an intermediate regime where the applied power to the resonator is loading the cavity with a photon occupancy of order the critical photon

2 2 number ncrit =Δ/4g . Below -108 dBm of explicit applied power, the cavity frequency shifts, becoming fully dressed by the interaction with the qubit. The upward shift in the 43

-5 6.176

6.174

6.172 -10

6.17

6.168

-15 (dB)

6.166 21 S 6.164

-20 Frequency (GHz) 6.162

6.16

6.158 -25 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 Qubit Flux Bias Current (mA)

Figure 3-5: Qubit readout resonator spectroscopy vs. qubit ﬂux bias current. The sinusoidal dependence of the qubit and thus resonator frequency is clearly seen. The spectrum does not diverge at Φ = Φ0/2 however do to the 2-to-1 junction size ratio for the qubit SQUID. The mutual inductance MFB between the bias line and the SQUID loop was modeled to be 2.2 pH, thus the current to achieve a full ﬂux quantum of evolution is IFB =Φ0/MFB ≈ 1 mA. center frequency of the resonator as measured in Figure 3-4 indicates that the qubit |1→|2 transition (but more importantly, the |0→|1 transition) is below the cavity. Additionally, the shift magnitude (∼ 2 MHz) allows one to estimate the qubit |1→|2 transition frequency to be approximately 2.5 GHz below the cavity for a designed coupling g = 100 MHz. If the transmon capacitor is shunted with a dc-SQUID as opposed to a single junction, one may adjust the qubit transition frequency via the application of an external ﬂux to the SQUID loop. To understand this is more detail, we need only to look at the Josephson energy

EJ of the SQUID loop 44

HSQUID = −EJ1 cos(δ1) − EJ2 cos(δ2) . (3.46)

The condition of ﬂux quantization enforces that the equation for the phase diﬀerence between the two junctions take the form

η ≡ δ2 − δ1 =2nπ +2πΦ/Φ0 , (3.47)

where Φ is an externally applied ﬂux. For identical junctions, EJ1 = EJ2 = EJ and

ESQUID = −2EJ cos(πΦ/Φ0) . (3.48)

In the case of an asymmetric SQUID comprised of junctions with diﬀering critical currents, we can deﬁne a percentage diﬀerence

E − E d = J2 J1 , (3.49) EJ1 + EJ2 where then the Hamiltonian in Eq. (3.46) takes the form

¯ 2 2 H = EJ cos(πΦ/Φ0) 1+d tan (πΦ/Φ0) , (3.50)

¯ where EJ = EJ1 + EJ2. Figure 3-5 shows the readout resonator spectrum as a function of qubit ﬂux bias current. The non-divergent sinusoidal dependency of the frequency means that the qubit has a roughly 2-to-1 junction size ratio with a d =1/3. At the upper most frequency, the χ12 shift is now 12 MHz, yielding an approximate upper qubit frequency of 5.25 GHz again assuming a coupling g ≈ 100 MHz. 45

3.2.3 Finding the Qubit Frequency

The previous section detailed how to initially conﬁrm that the qubit is at a minimum coupling to the readout resonator in the expected way and, if designed for it, properly ﬂux tunes. The next step in bringing up the cavity-qubit system is determining the qubit frequency. The previous sections gave upper and lower bounds for where in frequency space the fundamental qubit transition should lie. That band was between 3.6 and 5.2 GHz given an estimated coupling of 100 MHz. Experimentally, the way to determine a qubit’s frequency at a particular ﬂux bias point (should the qubit be tunable) is to apply a microwave drive at the resonator center frequency and monitor the level of the output signal as a function of qubit microwave drive frequency. The qubit drive can be applied directly to the qubit via an independent control line or through

the readout resonator. The qubit drive tone is applied for times T Tπ such that the qubit state saturates at some superposition of |0 and |1 ensuring a measurable readout resonator frequency shift. Spectroscopy data of a transmon qubit is shown in Figure 3-6 where both the qubit drive tone frequency and signal level were swept. The important feature to note is how the qubit frequency line width becomes smaller as a function of qubit drive attenuation. This eﬀect is due to stimulated emission. Qubit drive photons can drive not just the |0→|1 transition but also drive the |1→|0 transition which in spectroscopy looks like a parallel qubit loss channel yielding a larger qubit transition line width. As the drive power is reduced, less stimulated emission events occur and the extraneous loss channel diminishes reducing the qubit line width.

3.2.4 Orthogonal Qubit Axis Control

With the qubit |0→|1 transition frequency characterized via spectroscopy, coherently controlling the qubit state vector is the next task. This is done via the application of a resonant microwave drive tone that, in the lab frame, couples to the qubit via the interaction 46

5.25 5.5

5.2 5 5.15

5.1 4.5

5.05

4 5

4.95 3.5 Qubit Frequency (GHz)

4.9

3 4.85 S21 Cavity Transmission (arb. units)

4.8 0246810121416 Qubit Attenuation (dB)

Figure 3-6: Qubit Spectroscopy Data. At high drive powers, the line width is large due to an additional stimulated emission loss channel. As the drive power is reduced, the resonance line width reduces to a constant value the width of which can provide an estimate of the qubit relaxation lifetime T1.

HD = Γ(t)cos(ωqbt + γ)σx (3.51)

To achieve both amplitude and phase modulation, the qubit drive tone is usually pro- duced via single sideband (SSB) modulation achieved through the utilization of an IQ (In-phase/Quadrature) mixer. A standard IQ mixer has 4 ports: The local oscillator “LO” port, an in-phase “I” and quadrature “Q” port, and ﬁnally an output “RF” port. A constant

microwave tone applied to the LO port is multiplied by the signals AI (t)andAQ(t) applied at the I and Q ports, respectively. The resulting signal at the RF port is

ARF (t)=AI (t)cos(ωLOt)+AQ(t) sin(ωLOt) . (3.52) 47

For SSB modulation where ωq = ωLO − ωSSB, the signals applied at the I and Q ports should be

AI (t)=Γ(t)cos(ωSSB + γ) , (3.53a)

AQ(t)=−Γ(t) sin(ωSSB + γ) . (3.53b)

The output tone at the RF port now takes the form

ARF (t)=Γ(t)cos(ωqt + γ) . (3.54)

The Hamiltonian for the qubit plus the drive tone in the lab frame now takes the form

H = Γ(t)cos(ωqt + γ)σx + ωq/2 σz . (3.55)

The natural question to ask when looking at this equation is how one achieves independent

qubit X/Y axis control given only a σx operator. The answer lies in considering how the qubit “sees” the application of the drive Hamiltonian in it’s own frame. We can transform the Hamiltonian given in Eq. (3.55) for an arbitrary drive frequency

ωd into the interaction picture via the following transformation

iH0t/ −iH0t/ VI = e Ve (3.56) iωqtσz/2 −iωqtσz/2 = e Γcos(ωdt + γ)σxe .

After putting terms into matrix form and multiplying,

⎡ ⎤ ⎢ iωqbt⎥ ⎢ 0 e ⎥ VI = Γcos(ωdt + γ) ⎣ ⎦ . (3.57) e−iωqbt 0

We now expand the cos term into it’s exponential form and multiply it throughout resulting in 48

⎡ ⎤ ⎢ (i{ωd+ωq}t+γ) (i{ωq−ωd}t−γ)⎥ Γ ⎢ 0 e + e ⎥ VI = ⎣ ⎦ (3.58) 2 e(−i{ωd+ωq}t−γ) + e(i{ωd−ωq}t+γ) 0

In the interaction picture, wavefunctions are constant with respect to time while the ob- servables are not. The solution to the Schrödinger equation in the interaction picture given Eq. (3.58) is trivial, yielding 4 terms

−iΓ ψ(t)= ψ V dt , (3.59) 2 0 I where

⎧ ⎪ ⎪ 11 22 ⎪ V dt = V dt = C ≡ 0 ⎪ I I ⎪ ⎪ (i{ωd+ωq}t+γ) (i{ωq−ωd}t−γ) ⎨ 12 e e VI dt = + − . VI dt = ⎪ i(ωd + ωq) i(ωq ωd) (3.60) ⎪ ⎪ ⎪ (−i{ωd+ωq}t−γ) (i{ωd−ωq}t+γ) ⎪ e e ⎪ 21 ⎩⎪ VI dt = + −i(ωd + ωq) i(ωd − ωq)

For drive frequencies near resonance, the ﬁrst term in the oﬀ-diagonal elements for ψ(t) become vanishingly small. We can then neglect these terms, which in combination of driving

on resonance (ωd = ωq) gives the interaction terms in Eq. (3.58) the form

⎡ ⎤ ⎢ −γ⎥ Γ ⎢ 0 e ⎥ VI = ⎣ ⎦ (3.61) 2 eγ 0 ⎡ ⎤ ⎢ − ⎥ Γ ⎢ 0cos(γ) i sin(γ)⎥ = ⎣ ⎦ (3.62) 2 cos(γ)+i sin(γ)0 Γ = [cos(γ)σ + sin(γ)σ ] . (3.63) 2 x y 49

It can be seen now directly that the phase γ of the sideband tone applied to the I and Q ports of the IQ mixer determines which axis the qubit is rotated about. While in this analysis we have explicitly dropped the time dependence of Γ, it is important to note that qubit drive amplitude modulation is a key component in achieving precise rotation angles about the Bloch sphere. For the work in this thesis, Gaussian proﬁles were utilized.

3.3 Qubit and Qubit Gate Benchmarking

Figure 3-7 displays a sample dilution refrigerator (DR) wiring schematic for a single qubit / resonator pair. Independent RF sources are required for the qubit (red) and resonator (blue). In this schematic, the qubit and readout tones are produced via SSB modulation and tied together onto a single microwave line with a 3 dB microwave splitter / coupler. The tying together of signals is not necessary if the qubit has an independent addressing line (not shown). These signals are then attenuated through multiple microwave attenuators at diﬀerent temperature stages of the DR before ﬁnally being applied to the device under test (DUT). The DUT is generally encased in a superconducting sample box that helps to reduce the magnetic ﬁeld in the nearby vicinity due to Meissner shielding. In addition, extra layers of radiation and magnetic shielding are applied around the sample. This can be done by painting the inside of a mu-metal or Cryoperm can with ECCOSORB. After either driving the qubit or sampling the resonator, the tones are then passed out of the sample and through a series of microwave isolators and ampliﬁers. The isolators serve to reduce the noise and hence back action emanating backwards through the chain from the input of the microwave ampliﬁers. Once properly ampliﬁed, the signal coming out of the fridge is mixed back down (green) to the sideband frequency with the I and Q ports then being digitized for analysis. 50

Timing / Instructions / Data Acquisition

QB DAC RO DAC 20 dB Attenuator ADC

3 dB Splitter

LO LO LO IQ IQ IQ 20 GHz LPF RF RF RF

Circulator / Isolator

HEMT 300 K

60 K

HEMT 3 K

mK

Radiation Shielding JPA

Magnetic Shielding DUT

Figure 3-7: Sample dilution refrigerator wiring schematic for a single qubit / resonator pair experiment. 51

3.3.1 Characterizing and Calibrating the Qubit States

Figure 3-8(a) shows a sample IQ “blob” distribution for the application of an idle (blue) gate and a π-gate (orange) to a transmon qubit.The π-gate time was determined with a Rabi oscillation experiment where the qubit is driven on resonance for variable amounts of time resulting in an oscillatory output signal. Utilizing a high power readout protocol [63], large separation in IQ space between the two states can be achieved. To convert this raw signal into proper qubit states, the centroids (yellow crosses) of the two distributions are found and a bisecting (black, dashed) line drawn between them. The data are then rotated and centered onto the origin and binned according their distance along the bisector line. The result of this manipulation is a histogram with two Gaussian distributions as can be seen in Figure 3-8(b). Two distinct measures of the qubit readout chain may be obtained from the process. The ﬁrst, commonly referred to as the separation ﬁdelity, is a measure of how well resolved

(a) (b) 60 1 60 Idle Gate Idle Gate Pi Gate 0.9 55 Pi Gate 50 0.8 Single Shot = 85.8% Fidelity 50 0.7 40 0.6 45 Separation 30 Fidelity = 100% 0.5

40 Counts 0.4 Occupation 20 0.3 35 Quadrature (mV) 0.2 10 30 0.1

0 0 -5 0 5 10 15 20 25 -25 -20 -15 -10 -5 0 5 10 15 20 25 In-Phase (mV) Separation (mV)

Figure 3-8: (a) Representative IQ data set for applied idle gate and π-gate. The large separation between the blobs allows for accurate qubit state discrimination. (b) Centered, rotated, and binned data IQ blob data. Two distinct measures of ﬁdelity may be determined from the processing of IQ data. The ﬁrst is the separation ﬁdelity which is a measure of how well separated the data are. The second is the single shot readout ﬁdelity which is a measure of the max distance between the integrals (dashed orange / dashed blue) of the two signals. The integral of the signal is also utilized to normalize the data such that qubit states (and the superposition of them) may be put in normalized units. 52

the |0 and |1 qubit states are. To determine the separation ﬁdelity, each resulting qubit gate histogram is ﬁt to a Gaussian (solid orange and blue curves). These gaussian ﬁts are then integrated over the separation range resulting in error functions (erf). The resulting maximum diﬀerence between the erf(|1) and erf(|0) is the separation ﬁdelity. Figure 3-8(b) displays a qubit measurement that yields a 100% separation ﬁdelity. The other measure, known as the single shot ﬁdelity, is deﬁned as

F =1− P (1|0) − P (0|1) (3.64) where P (x|y) deﬁnes the probability of measuring state x when state y has been prepared. Experimentally, this is determined via the diﬀerence between the integrated π-gate signal (dashed orange line, Figure 3-8(b)) and the integrated idle gate signal (dashed blue line, Figure 3-8(b)). The single shot ﬁdelity can be thought of in terms of extractable information from the system. For instance, in the extreme limit that the ﬁdelity is 0, the probability of measuring either qubit state |1 or qubit state |0 are equal after having prepared state |0. The state then is maximally mixed with a saturated entropy and no information can be faithfully measured. Finally, the calibration of the measurement chain to obtain results in qubit state occu- pation is achieved by normalizing the integrated qubit state histograms (right hand y-axis, Figure 3-8(b)). Using this calibration we can see that even when the qubit is in the ground state, there is measurable population of the |1 state. This ground state preparation error permeates along for the qubit rotations as a non-negligible amount of |0 state population can be seen for an applied π-gate. The single shot ﬁdelity previously discussed is not only limited by imperfect gates, but also by imperfect state preparation. 53

3.3.2 Randomized Benchmarking of Qubit Gates

Now that we have discussed how to implement arbitrary rotations about the Bloch sphere and how to properly calibrate and measure the state of the qubit, the natural question to ask is how well can one perform precise rotations / qubit gates and how does one characterize said performance? For a single qubit, there are 24 unique rotations that preserve the octahedral symmetry. These rotations belong to a class known as the Cliﬀord group. Each Cliﬀord maps one point on the Bloch sphere to another. An easy way to think about these rotations is by considering mapping the positive Z axis of the Bloch sphere to any one of the other 5 vertices of which, when holding the Z axis then stationary, there are 4 unique rotations. The number of unique Cliﬀords C for a single qubits is then

C = number of axes × rotations per axis = 6 × 4=24. (3.65)

Table 3.1 lists the 24 unique Cliﬀords for a single qubit. Randomized benchmarking (RB) aims to characterize the ﬁdelity of qubit gates utilizing the power of randomization. The RB protocol to characterize average qubit gate ﬁdelity is to: (I) Apply a random sequence

of m Cliﬀord operations to the qubit, (II) calculate and apply the unique Cliﬀord CI that inverts the Cliﬀord sequence and brings the qubit back to |0, and (III) measure the ground state population averaged over k diﬀerent trials of the m Cliﬀords. The randomness of the protocol depolarizes the environmental noise and is resilient against state preparation and measurement (SPAM) errors. The resulting sequence ﬁdelity is ﬁt to

F = Apm + B. (3.66)

The average error per Cliﬀord, r, is related to the ﬁt parameter p via

(1 − p)(d − 1) r = (3.67) d 54

Single Qubit Cliﬀords Pauli I X Y Y, X 2π/3 ±X/2, ±Y/2 ±Y/2, ±X/2 π/2 ±X/2 ±Y/2 -X/2, Y/2, X/2 -X/2, -Y/2, X/2 Hadamard X, ±Y/2 Y, ±X/2 X/2, Y/2, X/2 -X/2, Y/2, -X/2 Table 3.1: List of single qubit Cliﬀords. with d =2n where n is the number of qubits. SPAM errors are encoded in the coeﬃcient A and linear oﬀset B. While this protocol provides insight in the average ﬁdelity per gate, it provides no insight as to which gates, if any, may be bad apples and limiting the overall sequence ﬁdelity. A subtle but important change to the RB protocol is the insertion of a target gate into the m random Cliﬀord sequence that we wish to measure the ﬁdelity of. The interleaving of this target gate (hence the name interleaved randomized benchmarking), allows for direct comparison against of sequence of m Cliﬀords. The interleaved gate sequence decay pIG allows for the calculation of error per interleaved gate

(1 − p /p)(d − 1) r = IG (3.68) IG d

Figure 3-9(a) displays the interleaved RB experiment protocol. A reference experiment is

run with only randomized Cliﬀord sequences of length m followed by an inversion gate CI . The experiment is then re-run only with an interleaved gate, denoted here by G . The data

are then compared to extract the gate ﬁdelity rIG. Sample data for taken in the lab at 40 55

(a) Reference ()

Experiment ()

(b)

0.85 Interleaved Gate: I Interleaved Gate: X/2 0.8

0.75

0.7

0.65

Sequence Fidelity 0.6

0.55

5 10152025303540455055 Number of Cliffords,

Figure 3-9: (a) Interleaved randomized benchmarking experiment sequence. A randomized Cliﬀord gate array is applied m times followed by subsequent inversion to the qubit |0 state and measurement. The sequence is then repeated except for the interleaving of a desired gate, G, in the Cliﬀord sequence. (b) Randomized benchmarking data taken for an idle (reference) gate and an X/2 gate applied to a transmon qubit in the lab at 40 mK. The curves are independently ﬁt with Eq. (3.66) and the extracted p’s are utilized in the calculation of the error per gate utilizing Eq. (3.68). 56 mK is displayed in Figure 3-9(b). The applied gate was an X/2 gate with extracted ﬁdelity

F =1− rIG =99.7(3) %. 57

Chapter 4

Hybrid Atom–CPW cQED Experiment

Any one physical implementation of a qubit is bound to have speciﬁc strengths and weaknesses associated with it. The scaling up of any one physical qubit system to realize a quantum architecture comprised of the millions of qubits deemed necessary to achieve a fault tolerant processor will have to deal with the scaling of these particular weaknesses. Hybrid quantum architectures aim to to combine physically disparate quantum systems in eﬀort to circumvent any one physical implementations weaknesses while exploiting their individual strengths. The following sections detail eﬀorts undertaken towards the demonstration of coherent coupling between a superconducting CPW resonator and a single Rydberg Cs atom; The realization of which would be a major milestone towards designing and building a larger integrated atom-superconductor quantum processor.

4.1 By The Numbers

The commonality shared between cavQED and cQED quantum computing architectures is that of the superconducting microwave resonator. Serving the same purpose in both physical systems, the microwave resonator allows one to weakly probe the qubit system, be it atomic 58

(a) (b)

Figure 4-1: (a) Diagram of a single Rydberg Cs atom coupled to a planar superconducting CPW resonator. For maximal coupling, the atom is placed at the voltage antinode. (b) Energy level diagram for an atom–CPW photon Cz conditional phase gate. The Cs atom is excited out of the hyperﬁne ground state manifold (which is utilized as the 2 state qubit manifold) to a large principle quantum number Rydberg state |r. The CPW mode is resonant with the |r→|r transition which evolves the state picking up a factor of 2π in the phase depending on whether or not a photon resides in the cavity. The atom is then de-excited back down into the qubit state manifold.

or artiﬁcial. The ﬁrst step in the integration of the two disparate technologies is to show that the large three–dimensional cavities commonly used in atomic experiments can be replaced with the planar distributed LC oscillators used in most cQED experiments. We explore the possibility of such a replacement by looking at the relevant numbers and timescales to

perform a conditional phase gate Cz between a single Rydberg atom and a superconducting CPW resonator [64]. A schematic of the proposed coupling scheme is shown in Figure 4-1(a). An atom is placed at the voltage antinode of the resonator so as to maximize the electric dipole coupling. Figure 4-1(b) illustrates the pulse sequence required to realize a conditional phase gate between the atom and CPW cavity photon. The qubit states are encoded in the hyperﬁne levels of the ground state. The |0→|1 transition frequency for Cs is ω01 =2π × 9.19263 GHz 59

(see Figure 1-3). A π–pulse excites the atom out of the computational manifold out to a large principle quantum number n Rydberg state |r via the application of a classical laser pulse of

duration τ|r = π/Ω. The transition frequency between |r and a nearby Rydberg state |r is then tuned into resonance with the cavity mode of the CPW resonator. If a photon is present in the resonator, the Rydberg state is coherently Rabi driven between |r and |r picking up a 2π evolution of the phase in time τrr = π/g,whereg is the vacuum Rabi frequency between the atom and the CPW resonator. The resultant Rydberg state is then mapped back down to the computational |1 state via the application of another classical laser pulse

of time τ|r. The total time for the completion of the pulse sequence is

τTotal =2τ|r + τrr = π(2/Ω+1/g) . (4.1)

The resulting gate from this series of operations is the conditional phase gate

Cz = |00 00|−|01 01| + |10 10| + |11 11| , (4.2) where the ﬁrst state designation is the Fock state of the resonator and the second the state of the atomic qubit. When combined with single qubit rotations, the controlled phase allows for the entanglement of the atom–cavity system providing a path for direct microwave to optical photon conversion. Teasing out the coupling scheme further, we look to the preparation of a Bell state utilizing

the Cz gate. The Bell state we wish to prepare is

1 |Ψ+ = √ (|01 + |10) , (4.3) 2 where again the ﬁrst designation in the ket notation refers to the photon number inside the resonator and the second the state of the atomic qubit. The gate sequence to prepare the Bell state is 60

ˆ ˆ ˆ ˆ B+ = Ha Cz Ha (4.4)

ˆ where Ha is the Hadamard gate applied to the atom. The initial states required to realize the √ | | | | | Bell state are ψ a = 0 and ψ r =1/ 2(0 + 1 ) . Preparation errors for the resonator

superposition state are accounted for via a simulated qubit - cavity coupling of gqc =2π × 100

MHz and a qubit T1 =2μs. However, perfect qubit state initialization and preparation are assumed. The Bell state preparation ﬁdelity F for a range of simulated couplings g and internal cavity quality factors Q was calculated utilizing

√ √ F =Tr ρ+ρ ρ+ , (4.5)

+ + where ρ+ = |Ψ Ψ | is the ideal density matrix for the Bell state. The Cz gate implementation described above is considered explicitly for the electronic transition between neighboring

Rydberg states 90s1/2 and 90p3/2 with dipole moment

drr = 90s, m =1/2| d |90p, m =3/2 = 2/9 × 8360 ea0 , (4.6)

where e is the fundamental charge and a0 is the Bohr radius. The radiative decay rates

−1 −1 incorporated into the simulation were γr = 820 μs and γr =2ms for the |90p, m =1/2 and |90p, m =3/2 states, respectively. The time evolution of the density matrix ρ was calculated via the Lindblad master equation

i ˆ 1 † − † − † ρ˙ = ρ, H + 2ciρci ci ciρ ρci ci . (4.7) 2 i

The superoperators ci considered for the simulation are listed in Table 4.1. Operators c1 and

c2 describe the relaxation of the Rydberg states |r and |r , respectively. Operators c3 and

c4 describe the loss and acquisition of cavity photons as a function of the cavity line width −1 −ωr/kbT κr = ωr/QT and the thermal cavity photon numbern ¯ = e − 1 . 61

Superoperator Physicality Mathematical Form √ c Rydberg state |r relaxation γ σ− 1 √ r r − c Rydberg state |r relaxation γ σ 2 r r c Cavity photon number lowering κ(¯n +1)a 3 √ † c4 Cavity photon number raising κna¯

Table 4.1: The superoperators utilized in calculating the Bell state density matrix ρ. c1,2 describe the relaxation of Rydberg states |r / |r at rates γr and γr , respectively. c3,4 describe the coupling of photons in and out of the cavity as a function of the thermal photon numbern ¯.

Figure 4-2 shows the resulting contour plot of the Bell state preparation ﬁdelity as a function of the atom - cavity coupling strength g and the quality factor Q at T = 0 K. In the

limit that T =0K,n¯ =0andthusc4 → 0, the physics is dominated solely by the cavity photon loss rate κ γ and the atom–cavity coupling g. We see that for reasonable coupling rates in the single MHz regime and a quality factor no less that Q ≥ 105, the state can be prepared with a ﬁdelity F≥99%. We will return to the eﬀects of accounting for thermal photons in a following section.

4.2 Experimental Apparatus

Ideally, experiments to test the theory outlined in the preceding section would take place at millikelvin temperatures on the cold stage of a dilution refrigerator (DR). However, the engineering challenges associated with integrating the milliwatt power lasers required for single atom trapping and manipulation with the modest cooling powers (∼ 10 μW) at the mK stage of modern dilution refrigerators was deemed as challenge too great to tackle for a ﬁrst iteration experiment. Instead, we chose to build a custom UHV LHe cryostat as a jumping oﬀ point to demonstrate that the two disparate quantum systems be integrated at the most basic of levels. The design implemented tandem suspended LN2 / LHe reservoirs from a 16.5" conﬂat ﬂange (see Figure 4-3). The LN2 vessel (red) was connected to the conﬂat ﬂange via G10 62

Figure 4-2: Contour plot showing the Bell state preparation ﬁdelity as simulated for diﬀerent coupling strengths g and cavity quality factors Q. Regularly achieved single photon regime 5 internal quality factors of Qi ≈ 10 when combined with moderate coupling strengths in the singles of MHz, provides preparation ﬁdelities greater than 99 %.

support struts (green) 2.065"×1.750"×0.060" in size. Made from 0.055" rolled stainless steel, the outer diameter of the LN2 vessel was 10" with a total length of 10". The LHe vessel (blue) was also made from 0.055" rolled stainless steel with an overall length of 20". The liquid capacity of the LN2 and LHe tanks was 10 and 27 liters, respectively. Each reservoir had a 3003 series Al alloy wiring feed through plate bolted to the bottom of it in order to facilitate the necessary wiring and electronics for the resonator measurement while also providing real estate for thermal heat sinking while also serving as a radiation baﬄe. Each reservoir base was made from an explosively joined 304 stainless steel-OFHC copper bi-metal plate. Surrounding the LHe tank was a 1100 series Al alloy radiation shield (orange). This shield is thermally sunk to the base plate of the LN2 reservoir. Bolted to the bottom of the LHe reservoir was an oxygen free high-purity Copper (OFHC) sample cold ﬁnger (brown) that extends down below the outer vacuum shield into the science chamber. The entire 63

cold space was enclosed via two custom made 17" long by 14" OD stainless steel full nipples connectorized with 16.5" conﬂat ﬂanges on either side. Below these, a 16.5" → 6" reducing full nipple conﬂat ﬂange was employed such that the UHV sample chamber with optical access could be bolted on. Microwave wiring was introduced into the cryostat via UHV rated SMA feed throughs (Kurt J. Lesker P/N IFDCG012011) welded into a 2.75" conﬂat ﬂange. The interior microwave coax had a copper-nickel (CuNi) outer conductor with a silver–plated CuNi inner conductor. Each signal line was thermally sunk to each successive temperature stage so as to mitigate the heat load seen by the base of the LHe reservoir. Copper foil straps for heat sinking were soldered on to wires and bolted to the top of the liquid nitrogen reservoir, the 77 K stage plate, the top of the LHe reservoir and ﬁnally to the 4 K stage plate. Two wires were utilized in the S21 through microwave measurement of the resonator sample. Sixty dB of attenuation was placed on the 4 K stage plate and another 10 dB at the sample mount allowed for measurements in both the high and low power regime. The output signal was ampliﬁed at 4 K via a Caltech Microwave HEMT ampliﬁer (BW = 4-12 GHz, T <5Knoisetemperature, P/N CITCRYO4-12A) followed by a room temperature NARDA HEMT ampliﬁer. Design and material choices were made in eﬀort to simultaneously satisfy two (and often contradictory) criteria; Minimize the heat load to the LHe vessel while preserving a clean ultra-high vacuum (UHV) environment. While an exhaustive list of design choices used in eﬀort to satisfy these conditions would approach tome levels in length, it is worthwhile to discuss a few particulars and their relevancy towards a working UHV cryostat.

Conﬂat UHV Flanges

While perhaps an obvious choice when compared to viton compression seals, it is worth a short discussion of why Conﬂat all-metal seals are important for this design. Conﬂat seals provide a thermally bakeable vacuum shield. Temperatures greater than 200 C are generally employed to fully outgas adsorbed water oﬀ of the stainless steel surface while the copper gaskets 64

Figure 4-3: Machine drawing of the custom build UHV cryostat (left) and a picture of the installed cryogen vessels (right). Custom pant leg seals allow for cryogen transfer into the reservoirs. Not shown are the top plate wiring feed throughs or the optical access sample chamber. 65

Mat. A(cm2) LUR (empty, Torr L/sec) LUR (Inserted, Torr L/sec) LUR (Torr L/(sec cm2)) G10 90 1.0 × 10−5 1.3 × 10−5 3.3 × 10−8 Rogers 4000 series 13 2.8 × 10−8 1.2 × 10−7 7.1 × 10−9 Table 4.2: Measured leak-up rates (LUR) for G10 support struts and Rogers 4000 series microwave circuit board.

provide a seal capable of withstanding the 12-14 orders of magnitude pressure diﬀerence between the inside of the vacuum space and the surrounding atmosphere environment. While the maximum bakeable temperature of this system was limited by indium seals utilized in the packaging of the Caltech HEMT microwave ampliﬁer to below 110 C, the conﬂat ﬂanges still provided the suﬃcient and necessary vacuum interface to achieve the required pressures for the experiment.

G10 Support Struts

G10 support struts were utilized in the tandem hanging of the LN2 and LHe reservoirs. G10 provides structural integrity and rigidity while also possessing a low thermal conductivity of 0.1 W/mK constant over a temperature range of 4.2 K to 293 K. The major concern with G10 in this implementation, however, was its outgassing properties under UHV conditions. The vacuum leak-up rate (LUR) for the G10 struts at room temperature was measured by monitoring the vacuum leak-up rate for a turbo pumped stainless steel conﬂat full nipple both empty and with the G10 strut present in the vacuum chamber. The results are tabulated in table 4.2. The room temperature LUR for one G10 strut was measured to be 1.3 × 10−5 Torr L/sec. At room temperature, a LUR of this magnitude would be a major hurdle to achieving the necessary base pressures required for single atom experiments, but at cryogenic temperatures, the beneﬁts of the low thermal conductivity far outweighs whatever slight LUR they may maintain when at 77 K or 4.2 K. 66

Al 1100 / 3003 77 Kelvin shield

Two diﬀerent alloys of aluminum were utilized in the construction of the 77 K shield. Aluminum alloy 3003 (Al3003) was utilized for the fabrication of any and all ﬂanges where drilling and / or tapping of holes was required. Al3003 has a good machine-ability while also possessing a

low emissivity ( 3003 ≈ 0.03) when compared to it’s more commonly found 6061 counterpart

( 6061 ≈ 0.1). The tubing for the shield was fabricated out of near elemental 1100 series

aluminum (Al1100) purely based on it’s near elemental emissivity value 0.01 < 1100 < 0.03.

Wiring and Wiring Materials

The choice of materials related to microwave wiring was based on microwave signal performance alone, relying on cryogenic operation to usurp any deleterious outgassing eﬀects from the intrinsic materials. PTFE Teﬂon insulated 50 Ω 860/200 μm outer/inner conductor coaxial wire (Coax Co. P/N SC-086/50-SCN-CN) was utilized to route microwave signals from 300 K to 4.2 K. The outer (inner) conductor was made of (silver–plated) cupronickel with a manufacturer speciﬁed thermal conductivity at 4.2 K of 7.0×10−3 W/m K. Normally one would choose the cupronickel inner and outer conductor option for thermal conductivity concerns (the thermal conductivity of the all cupronickel wire is speciﬁed as 9.84×10−4 W/m K) when wiring from room temperature but in a trade oﬀ to maximize microwave performance, a single length of the silver-plated option was chosen to go between 293 K and the 4.2 K stage plate with copper strap heat sinks placed at the top of the LN2 vessel, the 77 K stage plate, and the top of the LHe vessel. Each wire prior to installation was cleaned in subsequents baths of Alconox, acetone, and isopropyl alcohol (IPA). Rogers 4000 series microwave circuit board was utilized to transition from the SMA cabling to the sample mount and provide a wire bonding surface in which to route the microwave signals to the resonator. Again chosen speciﬁcally based upon the materials microwave performance, the outgassing properties required testing. The results of the outgassing experiments are tabulated in Table 4.2. At room temperature, the measured 67

material LUR was 7.1×10−9 Torr L/(sec-cm2), which, for a board size of 1 cm2, would only require a pumping speed of 1 L/sec to maintain pressures below 10−8 Torr. When cooled to LHe temperatures, this rate becomes even more negligible. Choices related to DC wiring (HEMT power, electric ﬁeld compensation pins, etc) were based on minimizing outgassing. All DC wires were Kapton clad solid–core copper wire. The gauge of the wire was chosen based upon operating current / voltage considerations. Kapton / polyimide outgassing rates for similar cables have been measured for the LIGO experiment with reported rates as low as 1.0 × 10−8 Torr L/s/cm2 at 286 C.

4.2.1 Cryostat Performance

The base temperature of the sample mount when fully assembled reached Tbase =5.04 K. The limiting factors in achieving a lower temperature at the sample mount were direct exposure to 300 K radiation necessitated by the experimental design and the poor base temperature of the LHe reservoir base plate of T =4.6 K. While no explicit tests were done to investigate the mechanism for such a poor base plate temperature, it is currently believed that the explosively joined bi–metal reservoir plate is to blame. The base pressure of the cryostat with no active pumping beyond that provided by the vessel cryopumping was measured to be below 10−9 Torr with an ion gauge and further conﬁrmed with a measured Cs MOT lifetime of 1.5 seconds, corresponding to a background pressure of 6×10−10 Torr. The vibrational motion of the cryostat was also measured utilizing a pair of power balanced 852 nm laser beams. The experimental setup is displayed in Figure 4-4(a). A collimated 852 nm laser beam was split into two power balanced arms utilizing a 50/50 beam splitter. One arm was used as a reference beam so as to calibrate out any time dependent drifts in the beam intensity. The other arm was passed into the cryostat and aligned so as 1/2 half of the beam illuminated a razors edge that was bolted onto the cryostat cold ﬁnger. The motion calibrated diﬀerence spectrum of the measurement is displayed in Figure 4-4(b). Motion 68

calibration was performed by walking the measurement beam oﬀ the razors edge in predeﬁned increments and monitoring the beam diﬀerence level. Integrating the motion spectrum out

to the knee frequency fknee = 1 KHz and taking the square root yielded an RMS motion of 1.3 μm.

4.3 Superconducting CPW Resonator Design

With the new temperature constraints imposed by the LHe cryostat, it is worth revisiting the ﬁdelity calculations for the Bell state preparation but now taking into account a ﬁnite temperature. The average photon number in the resonator as a function of temperature is given by the Bose-Einstein distribution

−1 n¯ = e−ωr/kB T − 1 . (4.8)

These photons incoherently drive the |r→|r Rydberg transition in addition to limiting the cavity superposition state preparation. Figure 4-5 displays the calculated ﬁdelity for the Bell state preparation as a function of the cavity temperature. Above 50 mK, incoherent driving from the now non–negligible thermal photon occupation begins to play an appreciable roll in the dynamics and the ﬁdelity begins to decrease quickly. With this in mind, the new ﬁgure of merit chosen to deem the implementation of the

superconductor-atom interface a success is the number of Rabi oscillations nRabi or coherent excitation swaps between the disparate systems before total photon loss. The number of coherent excitation swaps between a resonator and a qubit is deﬁned as

2g n = (4.9) Rabi κ + γ where κ and γ and the loss rates of the resonator and qubit, respectively. The dipole coupling strength between the two systems is g and Q is the total quality factor. The photon loss 69

(a) Cryostat / Sample Measurement Beam

852 nm Reference Beam 50/50 Beam Splitter

(b) /Hz 2 um

Figure 4-4: (a) Diagram depicting the experimental setup to measure the vibrational motion of the cryostat cold ﬁnger. A collimated beam of 852 nm laser light was split utilizing a 50/50 beam splitter the arms power balanced referenced to each other. One arm was used as a reference beam so as to be able to calibrate against drifts in the overall beam power. The other was passed into the cryostat, passing by a razor’s edge that was bolted onto the cold ﬁnger. (b) Vibrational motion data of the cryostat cold ﬁnger taken at 4.2 K. Integrating the spectrum gives an RMS motion of 1.3 μm. 70

100

95

90

85

80 Fidelity (%)

75

70

65 0.0 0.1 0.2 0.3 0.4 0.5 Temperature (mK)

Figure 4-5: Simulation of the Bell state preparation ﬁdelity as a function of temperature. Above approximately 50 mK, the ﬁdelity begins to roll oﬀ as the thermal photon number in the cavity becomes appreciable and incoherent driving begins to play a dominant role in the dynamics.

−1 rates γR for Rydberg atoms is of order ∼ 1ms , which, when compared to the photon loss

−1 rate in state of the art superconducting planar resonators κ = ωr/Q ≈ 50 ms for a center

5 frequency ωr =2π × 5 GHz and Q =10, can be ignored. Eq. (4.9) now reduces to

2g 2gQ nRabi = → . (4.10) κ ωr

For a ﬁxed frequency ωr, we have only the product gQ to maximize in order to achieve the

strong coupling limit deﬁned as nRabi > 1. At the base temperature of the cryostat, the primary microwave loss mechanism limiting the Q factor is thermal quasiparticles. We will review the basics of CPW resonator geometry followed by a discussion of thermal quasiparticles and the Mattis-Bardeen conductivity. We 71 will then combine these two to form a full description of thermal quasiparticle dominated loss in superconducting CPW resonators and how engineering the quality factor can be achieved through judicious choice of resonator geometry. Engineering the coupling strength g will also be addressed. It will be shown that for standard CPW geometries, the electric ﬁeld and thus the coupling between the atom and resonator is very small at realistic atomic laser trap distances from the CPW surface. We will show how to properly engineer the CPW electric ﬁeld to extend far beyond the chip surface, greatly enhancing the spatial extent of the ﬁeld while introducing no additional loss.

4.3.1 Engineering the Quality Factor

4.3.1.1 CPW Basics

The CPW geometry can be thought of as a two dimensional coaxial wire. It is comprised of a center trace surrounded on each side by a ground plane. The relevant physical dimensions are the width of the center trace, W , and the gap between the center trace and the ground plane,

S. Conformal mapping techniques [65] allow for closed form equations of the capacitance Cg

and inductance Lg per unit length the CPW cross sectional geometry, given by

K(k0) Cg =4 0 eﬀ (4.11) K(k0) μ0 K(k0) Lg = , (4.12) 4 K(k0)

where 0,μ0 are the permittivity and permeability of free space, respectively. The eﬀective

dielectric constant eﬀ, to ﬁrst order, is the average between the relative dielectric substrate permittivity and the permittivity of free space

+1 = r . (4.13) eﬀ 2 72

The K terms are the complete elliptic integrals of the ﬁrst kind. The arguments k0 and k0 are dependent on the CPW geometry in the following manner

W k0 = (4.14) W +2S − 2 k0 = 1 k0 . (4.15)

An important warning to heed is that K (k0)andK(k0) are often interchanged in the literature concerning CPWs. The impedance of the line is deﬁned in the normal way as

Z = Lg/Cg . (4.16)

Terminating a CPW at either end with a capacitive open or inductive short will impose standing wave boundary conditions. If both ends of the CPW are either shorted or open, the CPW will support resonant frequencies at half wave integer multiples of the fundamental. Should the terminations of the CPW be mixed and matched with one end being capacitive with the other inductive, the CPW will support resonant frequencies at quarter wave integer multiples.

4.3.1.2 Mattis-Bardeen Conductivity

The notion of describing the ﬂow of electrons in a solid as a two component ﬂuid arises naturally even when considering the most basic of models. The Drude model oﬀers a classical interpretation of electrical conductivity through the application of Newton’s second law. The force exerted on a particle of charge q from an electric ﬁeld E is

dv m = qE − mv/τ (4.17) dt 73 where v is the velocity and τ the relaxation time. Utilizing the relationship between current density and the velocity of charge carriers,

J = nqv (4.18) where n is the charge carrier volumetric density, Eq. (4.17) becomes

m dJ m = qE − J. (4.19) nq dt nqτ

iωt Substituting Ohm’s law for an alternating electric ﬁeld E ≡ E0e , the force equation takes the form

nq2τ σiωτE eiωt + σE eiωt = E eiωt . (4.20) 0 0 m 0

Dropping common terms, one arrives at a complex conductivity

σ σ σ = 0 + iωτ 0 (4.21) 1+ω2τ 2 1+ω2τ 2

2 where σ0 = nq τ/m. Only in the limits of very large frequency or very long relaxation time does the imaginary part of Eq. (4.21) begin to contribute to the overall conductivity. This simple model of complex (normal) conductivity fails in the limit of low temperature / long relaxation time and high frequencies as the ﬁeld does not exponentially fall to 0 inside the metal as predicted by the classical skin depth. This phenomena, known as the anomalous skin eﬀect, was ﬁrst derived by Chambers [66] and formally extended for superconductors by D.C. Mattis and John Bardeen [67]. In their seminal paper (culminating in the majority of Mattis’ PhD thesis), Mattis and Bardeen provide a full quantum mechanical treatment of conductivity for both normal and superconducting metals. For the superconducting case, the real and complex contributions

to the conductivity σ = σ1 − iσ2 are given by 74

0 10 1.00

í 10 0.75

í 10 0.50

í 10 0.25

í 10 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure 4-6: Normalized Mattis-Bardeen conductivities σ1 and σ2 verse reduced temperature T/Tc. As the temperature approaches 0, the real part of the conductivity (blue curve) responsible for DC resistance goes to 0. In the opposite limit (T → Tc), the real part of the conductivity approaches the normal state conductivity at Tc. Conversely, the imaginary part of the conductivity approaches 0 as the critical temperature is reached.

σ 2 ∞ (E2 +Δ2 + ωE) 1 − = [f(E) f(E + ω)] dE (4.22) σn ω Δ (Δ2 − E2) (E + ω)2 − Δ2

1 −Δ (E2 +Δ2 + ωE) + [1 − 2f(E + ω)] dE ω Δ−ω (Δ2 − E2) (E + ω)2 − Δ2

Δ − 2 2 σ2 1 [1 2f(E + ω )](E +Δ + ωE) = dE (4.23) σn ω Δ−ω (Δ2 − E2) (E + ω)2 − Δ2

where σn is the normal state conductivity at the critical temperature Tc, and Δ is the BCS superconducting energy gap. The second term in Eq. (4.22) is zero unless ω 2Δ. Figure 4-6 displays the normalized real and imaginary parts of the Mattis-Bardeen

conductivity verse the reduced temperature T/Tc.AsT → Tc, the real part of the conductivity

approaches the normal state conductivity when evaluated at Tc. Conversely, the imaginary part of the conductivity approaches 0. In this opposite limit, the real part goes to 0 while 75 the imaginary part saturates. The Mattis-Bardeen complex conductance gives rise to a complex impedance of the form

1 1 σ1 σ2 ≡ Zs = = 2 2 + i 2 2 Rs + iωLk (4.24) σd d σ1 + σ2 σ1 + σ2 where d is the thickness of the metal and we have equated the real and imaginary parts to an

eﬀective surface resistance Rs and what is commonly referred to as a kinetic inductance Lk. The ratio of the imaginary to real parts of the impedance deﬁne the quality factor Q of the

system, which, when combined with the geometric contribution Lg from a CPW yields the relation

ω(L + L ) ωL L σ L Q = k g = k 1+ g = 2 1+ g . (4.25) Rs Rs Lk σ1 Lk

The ramiﬁcation of Eq. (4.25) is that at temperatures of an appreciable fraction of the superconducting critical temperature, the quality factor of a CPW resonator isn’t only just temperature dependent but also geometry dependent. The kinetic contribution to the total inductance is deﬁned by looking at the energy stored in the supercurrent,

1 1 I2L = μ λ2 j2dS , (4.26) 2 k 2 0 where μ0 is the permeability of free space, λ is the superconducting penetration depth, j is the supercurrent density and the integral is taken over the cross sectional area dS of the CPW. Rearranging Eq. (4.26), one can deﬁne the kinetic inductance in terms of the supercurrent density j,

2 2 j dS Lk = μ0λ (4.27) ( jdS)2

The problem of calculating the kinetic inductance has now been reduced to calculating the 76

cross sectional supercurrent density. The cross sectional supercurrent density is in general highly non-uniform necessitating numerical evaluation. However, in the thin ﬁlm limit where λ2/d W , a closed form equation for the kinetic inductance can be derived and is given by Clem [68] as

μ λ L = 0 q(d/λ)g(k, ) (4.28) k W where q(d/λ) is a thin-ﬁlm thickness correction given by

q(x) = (sinh(x)+x)/8 sinh2(x/2) (4.29) and g(k, ) is an of order unity unit-less geometric correction given by

1 2(1 − k) g(k, )= ln , (4.30) 2(1 − k)K2(k) (1 + k) where 1. Figure 4-7 displays the closed form and numerically evaluated kinetic inductance per unit length for diﬀerent CPW geometries. The penetration depth λ and metal thickness d simulated were λ = d = 100 nm. Very good agreement can be seen between the numerical analysis and the closed form provided by Clem for most values of the center trace width and the gap. For wider center traces, cutoﬀ procedures utilized in [68] to achieve closed form solutions begin to annex non-negligible contributions to the supercurrent. With the kinetic and geometric inductances now fully in hand, the thermal quasiparticle limited internal quality factor can be directly modeled. Figure 4-8 displays the numerically calculated quality factor Q as given in Eq. (4.25) for niobium with a superconducting gap Δ = 1 mV at a temperature of 4.2 K again with λ/d =1.

In the limit of S W , The geometric inductance scales as Lg → μ0S/W leading to the

ratio Lg/Lk ∝ S/λ. Conversely, when S ≈ W , the geometric contribution Lg ∼ μ0 while the

kinetic contribution reduces to Lk = μ0λ/W . Combined, this gives the ratio Lg/Lk ∝ W/λ. 77

(a)

x 10ï 4 (b) W = 5, Numerical W = 5, Clem 3.5 W = 10, Numerical W = 10, Clem 3 W = 15, Numerical W = 15, Clem 2.5 W = 20, Numerical W = 20, Clem

2 Lk (Henry/Meter) 1.5

1

0.5 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 Gap Width, S (um)

Figure 4-7: (a) Normalized supercurrent density averaged over the thickness of the traces for CPW geometry W =6μm, S =3μm and thickness d = 100 nm. (b) Closed form and numerical evaluation of the kinetic inductance per meter for CPWs of diﬀerent geometries. For S/W ≈ 1, the kinetic inductance is primarily a function of W . Conversely, for small S, the kinetic inductance has a non-negligible gap dependence. 78

Factors of 5–6 improvement in the internal quality factor are predicted with modest increases with both the CPW center trace width and CPW gap.

4.3.2 Measuring the Quality Factor

To test the geometric dependence of the quality factor, Nb CPW resonators of diﬀering geometries were fabricated at the Wisconsin Center for Applied Microelectronics (WCAM) on the University of Wisconsin - Madison campus. Three inch (75 mm) diameter, 500 μm thick sapphire wafers were loaded into a magnetron sputter system. Prior to sputtering, the 3–inch Nb target was cleaned with a 400 W argon plasma for 2 minutes. For deposition, the

power was increased to 500 W and the three inch wafers were rotated at ωsample ≈ 2π × 1 Hz. Sputtering for 2 minutes yielded a ﬁlm of 90 nm, which was veriﬁed with proﬁlometry. Photolithography began by spinning SPR-955 0.9CM photoresist on the wafers at 4500 RPM for 60s followed by a pre-exposure bake at 100 C for 90 seconds. Wafers were then loaded into a Karl Suss MA6 contact aligner. Hard contact mode was utilized with an exposure wavelength of 365 nm and exposure time of 3.6 seconds. Wafers were then removed and post-exposure baked (PEB) on a hotplate at 110 C for 90 seconds. Development was done via immersion in Microchem CD-26 TMAH developer for 60s followed immediately by a DI

H2O rinse for another 60s. The resulting pattern was then checked via optical microscopy. Subsequent reactive ion etching (RIE) of the pattern was carried out in either a Unaxis Plasmatherm 790 RIE tool utilizing a sulfer hexaﬂouride (SF6) etch or a Unaxis Plasmatherm 770 RIE tool utilizing a boron trichloride etch depending on the tool availability. The sapphire substrate utilized for these experiments has a negligible etch rate for either etch recipe. Figure 4-9(a) shows the resulting sample. Each chip contained six quarter-wave CPW

Tool Gases Flow (SCCM) RF Power RIE/ICP (W) Rate (nm/min) 790 SF6 / O2 15 / 20 150 / NA 100 770 BCl3 / Cl2 / Ar 18/9/18 50 / 300 45 Table 4.3: Parameters used for etching Nb ﬁlms in either a PT 790 or 770. 79

7000

W = 5 m W = 10 m 6000 W = 15 m i W = 20 m

5000

4000

3000 Internal Quality Factor Q 2000

1000 0 5 10 15 20 25 30 CPW Gap, S ( m)

Figure 4-8: Numerical evaluation of Eq. (4.25) for a Nb ﬁlm of thickness d = 100 nm, penetration depth λ = 100 nm, and superconducting gap Δ/e = 1 mV simulated at T =4.2 K. Just as with the kinetic inductance, in the limit of small gap S, the quality factor is predicted to be relatively width independent and scale predominantly with the gap. As the CPW gap is widened, this dependence lessons and is replaced with a strong dependence of the width W .

resonators multiplexed in frequency between 4.5 and 5.0 GHz. Each resonator had a capacitive

4 coupling to the common feedline of 5 fF resulting in a coupling quality factor Qc ≈ 10 . Samples were aluminum wire bonded inside an aluminum box. The sample box was then wired to a LHe dip probe and surrounded with a high permeability metal shield. This was then immersed in LHe and the forward scattering matrix element was measured utilizing a vector network analyzer (VNA). The resulting spectrum for one of the samples is displayed in

Figure 4-9(b). Each transmission dip was ﬁt to Eq. (3.37) with ﬁtting parameters Qi, Qe,and

ω0. The extracted internal quality factors Qi as a function of sample geometry are plotted in Figure 4-10. The solid lines are the predicted Mattis–Bardeen quality factors from Eq. (4.25) 80

(a)

(b) -4

-6

-8

(dB) -10 21

S -12

-14

-16 4.5 4.6 4.7 4.8 4.9 Frequency (GHz)

Figure 4-9: (a) Optical micrograph of the multiplexed quarter-wave resonator chip. Bright regions are Nb while dark are silicon. Each chip was 6×6mm2 containing 6 resonators multiplexed in frequency from 4.5 - 5.0 GHz. (b) Representative S21 transmission spectrum. Each sample was aluminum wire bonded inside an aluminum sample box. The box was then attached to the SMA leads of a custom LHe dip probe and encased in a mu-metal shield. Samples were submerged in LHe and the transmission spectra taken for each dip. 81

for Nb with a measured critical temperature Tc =8.8 K, zero-temperature penetration depth

λ0 = 87 nm, and superconducting gap Δ/e = 1 mV. There is a non-negligible dependence on the resonant frequency for the Mattis–Bardeen conductivities, thus in order to obtain better agreement between the theory and the data, the extracted center frequencies from the

measured resonators were used in the calculation of σ1 and σ2. This is why discrete steps are seen in the solid prediction lines as frequencies were recycled every 6 measured resonators. As history as shown, anything bearing the name “Bardeen” tends to be correct and great agreement can be seen between the extracted resonator quality factor and that predicted by Eq. (4.25). The geometry dependence predicted and thusly measured means that the loss can be directly engineered simply by changing the geometrical layout of the CPW resonator with almost an order of magnitude increase in the internal quality factor from a [W, S]=[5, 1] μm to a [W, S]=[50, 30] μm pairing.

4.3.3 Engineering the atom-CPW coupling strength

The second but equally important term to maximize in the quest to reach strong coupling between the CPW resonator and a single Rydberg atom is the vacuum coupling strength or Rabi frequency g, deﬁned as

E · d g = (4.31) where E is the photonic electric ﬁeld vector and d is the dipole moment of the atom. The electric ﬁeld of a CPW resonator is highly concentrated in the gap between the center electrode and corresponding ground planes. Figure 4-11(a) shows the numerically calculated zero-point electric ﬁeld proﬁle for a CPW resonator with center trace W =20μm and a gap width S =10μm. The largest magnitude ﬁeld is in the CPW gap between the center trace and ground planes and falls oﬀ exponentially near the surface of the chip. Figure 4-11(b) plots the absolute magnitude of the electric ﬁeld and the coupling strength g (assuming the 82

16 W = 5 μm

) W = 10 μm 3 14 W = 20 μm 12 W = 30 μm W = 50 μm 10

8

6

4

2

0 Internal Quality Factor (x10 0 5 10 15 20 25 30 CPW Gap, S (μm)

Figure 4-10: Extracted internal quality factor Qi for diﬀerent CPW geometries. Error bars for the ﬁts are smaller than the symbol size. The solid lines are the predicted Mattis-Bardeen thermal quasiparticle limited quality factors for a superconducting gap Δ/e =1mVanda zero temperature penetration depth λ0 = 87 nm. The discrete steps in the predictions result from utilizing the extracted resonant frequencies for the calculation of the Mattis–Bardeen conductivities.

ﬁeld and dipole moment are aligned) as a function of height above the chip surface. The dipole moment vector utilized in the calculation was again for the electronic transition

between neighboring Rydberg states 90s1/2 and 90p3/2 with magnitude

drr = 90s, m =1/2| d |90p, m =3/2 = 2/9 × 8360 ea0 (4.32)

where e is the fundamental charge and a0 is the Bohr radius. For physically realizable atomic trapping heights (z 25 μm) above the chip, the coupling strength is below 1 MHz resulting 83

4 in a nRabi < 1 even for a quality factor in excess of 10 . In eﬀort to combat the low coupling rate and provide a reasonable distance at which the atom could be trapped from the chip surface while also minimizing the amount of stray laser light scattered on the chip surface, a thick–ﬁlm copper electroplating process was developed such that the voltage antinode of the resonator could be physically extended far into the z-axis. Figure 4-12(a) displays a CAD drawing of the “science” end of the resonator. To facilitate strong coupling to the atom, 2×∼50 μm tall copper pillars would be plated at the capacitively shunted end of a quarter-wave CPW resonator. The superconducting Nb is colored blue while the sapphire substrate is colored yellow. The electroplated copper pillars are colored pink and the trapping laser is colored green. The red sphere denotes the optimal trap placement for the atom.

(a) (b)

Figure 4-11: (a) False color contour plot of the zero-point (ncav = 0) electric ﬁeld proﬁle for a CPW resonator with center trace width W =20μm and a gap width S =10μm. (b) Absolute magntiude zero-point electric ﬁeld (left y-axis) and coupling strength g (right y-axis) as a function of height z above the chip surface. The solid blue line is taken to be directly over the center trace while the dashed red line is calculated in the gap between the center trace and ground plane. The coupling rate g was calculated for the dipole moment between Rydberg states 90s1/2 and 90p3/2. Even for the large dipole moment between these neighboring electronic Cs Rydberg states, for realizable trapping distances (z 25 μm) above the chip surface, the coupling strength g 1 MHz. 84

Figures 4-12(b-c) show COMSOL simulations for the electric ﬁeld between the plated structures for a zero point voltage

2 1/2 V = ωr/2CCPW =2μV (4.33)

placed on the center trace of the CPW where ωr =2π × 5.4 GHz is the center frequency of the resonator and CCPW =0.44 pF. Figure 4-12(b) is a side-on view looking down the x axis. The plated structures are outlined in black. The false color maps the electric ﬁeld strength while the white arrows denote the ﬁeld vector orientation. For the majority of the height of the copper electrodes, the electric ﬁeld is uniform in both magnitude (0.06 V/m) and orientation (yˆ). Figure 4-12(c) is an overhead view of the modeled ﬁeld looking in the −zˆ direction. The electric ﬁeld again is uniform in magnitude and orientation this time throughout the 30 μm width of the copper pillars. Given this ﬁeld magnitude, the coupling is calculated to be g/2π ≈ 3 MHz, a factor of ∼ 10 higher than without the posts. The fabrication of the copper pillar structures is outlined pictorially in Figure 4-13. A base layer of 180 nm thick Nb was sputtered on to a 3-in, 500 μm thick single-side polished sapphire wafer. Positive photoresist SPR-955 0.9cm was manually dispensed over the surface of the wafer spin-coated at 4000 RPM followed by a pre-exposure bake on a contact hotplate at 100 C for 90s. The photoresist was then lithographically patterned utilizing a 5:1 projection i-line (365 nm) stepper. The wafer was then post exposure baked on a contact hotplate at 110 C for 90s. Development was performed via submersion and agitation in Microchem CD-26 TMAH developer for 60s. The wafer was then submerged and agitated in a bath of deionized H20 for another 60s. The pattern was visually inspected under a microscope prior to etching. Etching of the Nb base layer was performed in a Unaxis 790 reactive ion etcher utilizing the recipe outline in Table 4.3. The plasma etch time was 2 min. The purposeful over etching time was set to ensure a complete etch. The low reactivity of the sapphire substrates allows this to be possible without trenching. Once etched, the sample was left submerged in room temperature acetone for 12+ hours to ensure proper removal of the photoresist and residual 85

(a)

(b) (c)

Figure 4-12: (a) 3D CAD drawing of proposed electroplated structures at the voltage antinode of a CPW resonator. (b) Side proﬁle detailing zero-point electric ﬁeld strength and proﬁle as simulated in COMSOL. The False color maps the ﬁeld strength in V/m while the arrows denote the ﬁeld vector orientation. (c) Overhead proﬁle detailing zero-point electric ﬁeld strength and proﬁles as simulated in COMSOL. Notice that in both simulations the ﬁeld between the plates is homogeneous in both magnitude and vector orientation. 86 organics. Initial experiments in directly trying to electroplate Cu onto blanket wafers of sputtered Nb showed little success. The Cu ﬁlms had very large grain size and required large plating potentials to achieve the manufacturer recommended plating current densities of 10 A / cm2. It was decided that the poor conductivity of Nb at room temperature in addition to it’s native amorphous oxide was to blame. To combat this, an interstitial layer comprised of electron beam evaporated titanium and palladium was added. Poor adhesion of Pd necessitates the Ti adhesion layer. Negative photoresist AZ-5214 was spin coated at 4000 RPM followed by a pre-exposure bake at 95 C for 60s. The wafer was then loaded on to a Karl Suss MA6 contact aligner and put into hard contact with the mask. The streets between the dies and the desired areas intended for electroplating were left unexposed while the rest of the wafer is exposed to 365 nm , 10 mW / cm2 light for 30 s. The wafer was then image reversal baked at 110 C for 60s followed by a UV ﬂodd exposure for 60s. The development protocol is the same as for SPR-955. Once fully developed, the wafer was placed inside a custom 4-pocket hearth electron beam evaporator. The patterned wafer was argon ion milled for 17 seconds at a backing pressure of 4×10−5 Torr prior to deposition. After, a 3 nm Nb↔Pd adhesion layer of titanium was evaporated at 0.5 nm/s as measured with a crystal thickness monitor adjacent to the sample stage. Thirty nanometers of Pd was then evaporated at 0.5 nm/s so as to provide a base metal for the electro-deposition of the Cu structures that does not natively oxidize. Once ﬁnished, the wafer was removed from the evaporator and left to sit in a bath of room temperature acetone for 12+ hours so as to lift oﬀ the evaporated metal on top of the exposed negative resist. Mild sonication could be used to accelerate the lift oﬀ process. Once complete, the lift oﬀ left the streets between sample dies along with the capacitor ends of the resonators covered with the Ti/Pd bilayer. The electro-deposition of the Cu capacitor plates began with the patterning of a photoresist hard mask. Intervia BPN-65A negative photoresist was spun ﬁrst at 400 RPM for 1 minute with a ramp up to 800 RPM for 2+ minutes. While not explicitly measured, according to 87

Cu SO Sol’n Al2O3 Substrate AZ-5214 2 4

Niobium Ti/Pd EP’d Cu

SPR-955 Intervia BPN

(a) (b)

(c) (d)

(e) (f)

Cu+ Cu+ Cu+ Cu+ Cu+ Cu+ Cu+

Cu+

------

Figure 4-13: Process for electroplating Cu on thin ﬁlm Nb. (a) The process began with a 180 nm thick sputtered layer of Nb (pink) on a 500 μm thick sapphire substrate (silver). (b) The base layer Nb was optically patterned with SPR-955 positive photoresist (dark blue) followed by a SF6 plasma etch. (c) Deposition of the Ti/Pd (yellow) interstitial plating adhesion layer was done via a negative resist (AZ-5214, purple) and electron beam evaporation followed by a lift oﬀ via submersion in acetone. Note that the undercut in the photoresist was important to ensure that contact is broken between metal deposited in the developed areas and that on top of the photoresist. (d) Intervia BPN–65A photoresist (green) was spun on exposed and developed to expose Ti/Pd surface. (e) The wafer was submerged in Enthone copper electroplating solution (blue) with a negative voltage potential applied across the face of the wafer that attracts the positively charged Cu ions. (f) Once full plated, the wafer was removed from the solution, rinsed, dried, and diced. It was only then that the thick photoresist is removed revealing the lone standing copper structures (brown). 88

the spin-speed curve provided by the manufacturer, this should provide a resist thickness of 60 um. The resist was then pre-exposure baked (no post-exposure bake is required) at 60 C for 3 minutes followed by 120 C for 9 minutes. The second bake time can be adjusted if the resist is still soft after completion of initial 9 minute time. The resist was then allowed to cool to room temperature. Once cool, the resist was exposed in a Karl Suss MA6 contact aligner with 365 nm, 10 mW/cm2 UV light for 60s. Development was performed via puddle develop in Microchem CD-26 developer for a minimum of 3×1 min intervals, with fresh developer added every 1 minute. If, upon inspection after this ﬁrst 3 minutes, the resist was not fully developed, the process was continued until complete. Post development, many of the developed areas had organic residue left over. This was subsequently removed via an

oxygen plasma ash descum of 50 mT O2 × 250 W × 5 min. The now fully patterned wafer was placed into a single-wafer electroplating holder from Advanced Micromachining Tools (AMMT.com). DC electrical contact between the holder and the wafer was made via gold plated copper springs pressed into the front side of the wafer. These ﬁnger springs were screwed into a gold plated copper ring housed in the interior of the Teﬂon holder. The interior ring further made electrical contact with copper wires that ran up the interior of a Teﬂon rod, eventually terminating in a female 3.5 mm socket. Prior to submersion in the copper sulfate plating bath, the bath had to ﬁrst be heated to 25 C and stirred at 200 RPM. It was important to use a properly leached polypropylene tub as the acidic plating solution would become contaminated from impurities in the plastic if not done. The plating solution used was commercially available and produced by Enthone (www.enthone.com). Once heated and stirred, the prepared sample and a plating cathode were both submerged facing each other into the plating bath approximately 10 inches apart. The cathode used for this process was a 3"×6" rectangular Ti mesh. Potential was applied to the Ti cathode and the wafer anode via a Dynatronix Pulse Series (dynatronix.com) plating supply. The average forward plating current was set to provide a current density of 10 A/cm2 across the exposed wafer surface with the forward–to–reverse plating current and 89

plating time ratios set to 2:1 and 20:1, respectively. The total charge transfer was monitored during the plating process. A total charge transfer of 1500 A·s resulted in structures of ∼ 50 μm in height. Once complete, the sample was removed from the sample holder and both were rinsed thoroughly in DI water. Failure to do so would allow the plating solution to evaporate depositing diﬃcult to remove copper sulfate crystals on the hardware / wafer. To ensure survival of the freshly plated structures, the wafer was diced prior to the Intervia BPN photoresist removal. Finally, removal of the Intervia photoresist was done via overnight submersion in 75 C Intervia BPR photostripper. Figure 4-14(a) displays the ﬁnal product of the fabrication. The CPW geometry chosen for the sample was W =50μmandS =25μm, yielding an impedance Z = 50 Ω. Samples were 16 mm long by 4 mm wide tapering down to a with of 150 μm at the science end (red box) so as to minimize the amount of stray trapping laser light deposited on the chip. The Rayleigh length for a focused laser beam is

πω2 x = 0 (4.34) R λ where ω0 is the radius of the beam at the focus and λ is the laser light wavelength. For a

trap laser light of λ = 780 nm and a focus ω0 =3μm, the Rayleigh length is xR =36μm. The beam radius as a function of distance from the focus is given by

x 2 w(x)=w0 1+ (4.35) xR

For a maximum allowable beam radius w(x) ≡ wmax =25μm, half the max height for the copper electroplated structures, the max width of the chip at the trapping sight was

2 wmax xmax = xR − 1 = 215 μm . (4.36) w0

The tapering of the chip proﬁle was achieved via a dicing saw with the overall width at the position of the copper structures brought down to 150 μm. 90

(a) (b)

100 um

(c) (d)

50 um (e) (f)

Figure 4-14: (a) Micrograph of the fabricated superconducting quarter wave CPW resonator. The chip was 16 mm long by 4 mm wide tapering down to a width of 150 μm at the science end. The resonator was inductively coupled to a feed line and capacitively shunted to ground. Image courtesy of Edward M. Leonard. (b) Proﬁle view of the electroplated structures. (c) SEM image of electroplated structures. (d) Head on cartoon proﬁle√ illustrating the necessity of a tapered chip (gray) proﬁle. The beam (red) waist w scales by 2 every Rayleigh length xR away from the focus. (e) Microwave transmission across an electroplated resonator taken 4K × 4 at T = 4.2 K. The internal quality factor was ﬁtted to be Qi =3.0 10 . (f) Microwave transmission data taken across another electroplated resonator at T= 100 mK. At powers 100mk × 5 corresponding to single photon occupation, the ﬁtted quality factor was Qi =1.5 10 . 91

Figures 4-14(e-f) show microwave transmission data taken for two diﬀerent samples; One sample was measured at T = 4.2 K (e) and the other T = 100 mK (f). The T = 4.2 K

4K × 4 quality factor was power independent with an extracted value of Qi =3.0 10 . This value matches with the predicted Mattis–Bardeen quasiparticle limited CPW quality factor for a ﬁlm of thickness d = 180 nm. This is important point because it could then be said that the judicious placement of the copper towers at the voltage antinode of the resonator induce no additional microwave loss. The extracted quality factor at T = 100 mK was power dependent,

100mK × 5 with a single–photon occupation quality factor Qi =1.5 10 . This value is comparable with state of the art planar Nb CPW resonators at millikelvin temperatures. When measured at the base temperature of the custom UHV cryostat, the internal quality

4 factor Qi =1.0 × 10 , in agreement with predictions from Mattis-Bardeen for Nb at T =5

K. This value for Qi, when combined with the electroplated structure enhanced anticipated Rabi frequency of g/2π = 3 MHz, yields a predicted number of coherent excitation swaps

nRabi ≈ 12 1, placing the interaction securely in the strong coupling regime [69].

4.3.4 Stray Electric Field Compensation

Residual gas adsorbates that condense on cryogenic surfaces have been shown to produce non-negligible time dependent stray DC electric ﬁelds [70]. When combined with the large DC polarizability of Rydberg atoms, the dynamics of atoms near cryogenic surfaces becomes unpredictable. In eﬀort to counteract the chaotic and deleterious eﬀects of these adsorbates, DC electric ﬁeld compensation pins were incorporated with the sample mount. Figure 4-15(a-b) show a proﬁle and head-on view of the sample, respectively. The sample mount is bolted on to the cold ﬁnger with 2×8-32 socket head cap screws with no thermal grease applied between them. The four DC electric ﬁeld compensation pins were arranged around the atom trapping region (green circle, Figure 4-15(b)). COMSOL simulations were performed to model the ﬁeld proﬁles generated with 1 volt applied independently to each of the 4 electrodes with the sample mount held at ground. The resultant electric ﬁeld vectors 92

(a) (b) Sample Mount Sample 4K 1 2 Shield Bracket RF

Cold Finger 4 3

DC E-Field compensation pins

Figure 4-15: (a) Proﬁle photograph of 1st generation sample and sample mount bolted on to the cold ﬁnger. (b) Head-on view of sample and sample mount. Four DC electrodes were placed symmetrically about the single atom trapping site (green circle). Each electrode was capable of being independently controlled via a custom high voltage control module.

Pin Voltages (V) Electric Field (V/m)

Pin 1 Pin 2 Pin 3 Pin 4 Ex Ey Ez 1 0 0 0 -6.2 -4.2 -5.3 1 1 0 0 -16.6 -7.0 -5.7 1 0 1 0 -16.6 7.0 -5.7 1 0 0 1 -6.2 4.2 -5.3 Table 4.4: COMSOL simulation results of electric ﬁeld vector at atomic trapping region when placing a potential of 1 V in four diﬀerent conﬁgurations.

for the four diﬀerent conﬁgurations are compiled in Table 4.4 It is important to note that the problem is over constrained with 4 voltages to vary but only three spatial dimensions; Thus, any two of the pins can be tied together and held at the same potential while maintaining full 3 dimensional ﬁeld control. 93

Figure 4-16: First image of Cs MOT (bright point, mid-frame) trapped directly beneath the superconducting CPW chip. Image courtesy of Joshua Isaacs thesis.

A custom bi-polar high voltage controller was built in order to automate the control of the pin voltages and thus the electric ﬁeld environment. Output voltages between ±1kV were controlled via a the output of a 20 bit (1 ppm) DAC fed with a control voltage of 10.5 V.

4.4 Current Status

As of this writing, the sample has been installed in the cryostat and the cryostat has been installed on the optical table. To date, a MOT has been formed just beneath the chip (see Figure 4-16) and eﬀorts are currently underway toward achieving single atom trap loading, cooling, and transport. 94

Chapter 5

Single Flux Quantum Qubit Control

The marriage of superconducting classical and quantum processor technology, on it’s face, is a very natural path forward toward scaling up to qubit numbers of order the 1 million or so required for implementation of fault tolerant surface codes. Both are mature tech- nologies sharing many similar traits in terms of materials, fabrication, and operation. In this chapter we review modern classical superconducting computing based on single ﬂux quantum technologies. Afterwards, we describe experiments and data demonstrating the ﬁrst successful implementation of SFQ based control of a transmon qubit. We describe in detail the fabrication of the SFQ–qubit sample. We illustrate how to obtain orthogonal qubit axis control which allows for orthogonal SFQ based qubit gates. We show data on the interleaved randomized benchmarking (RB) performance of these gates. Furthermore, we investigate limitations to SFQ gate performance by looking at non-equilibrium quasiparticles (QP) generated by the SFQ circuit. Finally, we will describe future experiments where the classical SFQ control circuit is decoupled from the transmon qubit by employing a multi–chip module (MCM)/ﬂip-chip architecture. 95

5.1 Superconducting Supercomputing with Single

Flux Quanta

The basic tenant of classical computing with Josephson based superconductor technology is the regulation, storage, and transmission of SFQ pulses. For junctions that are critically

to over damped (βc ≥ 1), the junction phase δ can evolve in steps of 2π between successive minimums of the junction potential without free cascade down the washboard before removal of the bias current. When combined with the second Josephson relation, these successive steps in the phase produce a voltage pulse in time with quantized area

A = Vdt= dδ =Φ . (5.1) 2e 0

The energetics of a SFQ JJ phase slip are displayed in Figure 5-1. A necessary condition for the generation of SFQ phase slips is that βC ≤ 1 which generally involves the use of external shunt resistors. In further circuit diagrams, the use of shunt resistors is implied for all junctions but not explicitly shown.

In SQUID loops with βL 1, stable circulating current ﬂux states diﬀer by ±Φ0 which then are used to encode and store the logical “0” and “1”. Figure 5-2(a) displays the circuit diagram for a 1–bit D latch SFQ memory circuit [71]. The dc–SQUID is comprised of junctions J1 and J2 and inductor LS. At the beginning of the clock cycle, the current in

the SQUID is propagating counter–clockwise corresponding the “0” state. Static bias Ib is

applied very close to the critical current of J1. When an SFQ pulse is applied to the input, it

drives the current of J1 above the critical current and changes the ﬂux state of the SQUID

to “0”, lowering now the total current through J1 and biasing J2. A CLK pulse entering the

circuit now drives J2 to switch producing a SFQ pulse at the output and resetting the ﬂux state of the SQUID. If, however, no pulse arrives at the input during the clock cycle, the CLK

pulse is instead dropped across junction J3 and no output is produced. For proper operation

of the circuit, the circulating current Icirc =Φ0/LS produced by the input SFQ pulse must 96

Figure 5-1: Phase particle evolution for a SFQ phase slip. A static bias i = Ib/Ic is applied tilting the washboard potential. An current pulse is then applied to force i>1 allowing the phase particle to traverse 2π after which it is returned to its static value. The voltage pulse developed across the junction is quantized in area.

not exceed the critical current of J2. Additionally, the critical current of J3 should be less

than J2 such that when no pulse is applied to the input, the CLK pulse is dropped across it. The transmission of information from processor node to processor node is performed by the ballistic propagation of SFQ voltage pulses. However, the picosecond timescale of the pulses implies enormous frequency content giving rise to pulse dispersion during propagation. To combat this dispersion, SFQ repeater circuits can be used to transfer pulses over long distances. Figure 5-2(b) displays the circuit diagram of a 2–stage Josephson transmission line

(JTL). Independent biases (or a common bias rail) are used to bring junctions J1 and J2 close

to their respective critical currents. An incoming SFQ pulse drops across junction J1 driving

it normal and producing another SFQ pulse at L2. This SFQ pulse is then subsequently

dropped across junction J2 producing yet again another SFQ pulse at the output. The choice

of series inductors and hence βL for each single junction RF–SQUID is important as setting

the max loop ﬂux IcL<Φ0/2 forces forward propagation of the SFQ pulse instead of storage as was the case in the D latch circuit. With picosecond pulses ﬂying back and forth between circuits, propagation non–reciprocity is crucial for downward streaming of information. Figure 5-2(c) displays a two junction buﬀer

stage. For proper operation, the critical current of J1 should be larger than J2. When input 97

(a) (b)

0 / 1

(c) (d)

Figure 5-2: Common SFQ circuits. (a) 1–bit SFQ D latch. (b) Two junction JTL. (c) Unidirectional SFQ buﬀer stage. (d) SFQ 1×2 pulse splitter.

SFQ pulse arrive at L1 it subsequently drives J1 normal producing a SFQ pulse which is subsequently dropped again across J2 reproducing the pulse at the output. However, if a

pulse arrives from the output, it drives J2 normal ﬁrst dropping the voltage again at the output not allowing it to pass to the input. The ability to multiplex multiple circuit inputs with a single SFQ pulse is necessary to fan out information and increase computing complexity. Figure 5-2(d) displays an 1 × 2SFQ √ fanout. In parallel, junctions J2 and J3,eachwithequalcriticalcurrentsI2 =I3 =I=I1 2, √ look like a single junction with total critical Ic= 2I1. When an SFQ pulse drives the input, the resulting SFQ pulse drives both branches producing a complimentary SFQ pulse at each output without loss of amplitude. Many diﬀerent inceptions of SFQ computing have come along in the past 30+ years. We brieﬂy review the major implementations here. 98

5.1.1 Rapid Single Flux Quantum

First developed in 1985 by Likharev, Mukhanov, and Semenov [72], Rapid Single Flux Quantum (RSFQ) technology was the ﬁrst fully theoretically investigated superconducting computing paradigm that showed promise to compete with CMOS. Junctions in RSFQ are both externally shunted and biased via on–chip thin ﬁlm resistors. While initial tests and operating speeds showed promise, the static power consumption from the bias rails has limited the scaling of the processor size to a few kb.

5.1.2 Energy Eﬃcient SFQ

A variant of RSFQ, energy eﬃcient SFQ (ee–SFQ, ERSFQ) [73] aimes to tackle the quiescent power consumption issues previously discussed by replacing the bias resistors with supercon- ducting inductors and current limiting JJs. During standby (no junction switching), zero power is dissipated in the circuit. Additionally, during bring–up, local phase imbalance across the junctions in the circuit will self–correct as the junctions will continuously switch until an approximately global phase balance is reached.

5.1.3 Reciprocal Quantum Logic

Developed at Northrop Gumman in 2011, reciprocal quantum logic (RQL) [74] isn’t an extension of RSFQ but instead an entirely new form of superconducting classical computing. Instead of static biases, circuit elements are inductively coupled to and powered by a distributed common AC rail which also serves as a global clock, serving to reduce and/or eliminate the accumulation of timing jitter between circuit stages. Additionally, logical 0 and 1 are encoded in pairs of opposite polarity SFQ pulses, providing a natural ﬂux set / reset of the destination Josephson circuit. For the work contained in this chapter, traditional RSFQ circuitry is employed. 99

5.2 SFQ Control of a Transmon Qubit

A detailed theoretical analysis of SFQ based transmon qubit control can be found in [75]. Here we outline the calculations in that paper with an eye towards circuit design and experimental control. A circuit diagram of the SFQ control idea is given in Figure 5-3(a). A generalized SFQ producing circuit is capacitively coupled to a transmon qubit. The Hamiltonian for this circuit is

2 ˆ − 2 Q CcV (t) Φˆ H = + , (5.2) 2CT 2Lq where CT = Cq + Cc is the total capacitance of the circuit. Expanding the ﬁrst term in Eq. (5.2) yields

Qˆ2 Φˆ 2 C H = + − c V (t)Qˆ (5.3) 2CT 2Lq CT where we have dropped constant, non-operator dependent terms. The ﬁrst two terms in Eq. (5.3) are just the qubit self energy. Utilizing the methodologies outlined in Chapter 3, the qubit terms can be re-expressed again in the TLS formalism

ˆ2 ˆ 2 Q Φ ωq Hq = + → σz . (5.4) 2CT 2Lq 2

The last term in Eq. (5.3) is the interaction of the qubit with a capacitively coupled voltage pulse V (t). To better illustrate the eﬀect of this coupling, we expand the charge operator into it’s truncated qubit state raising and lowering operators

ω C Qˆ = −i q T (σ− − σ+) . (5.5) 2

Substituting this back into the coupling Hamiltonian yields

ωq − + HSFQ = iCcV (t) (σ − σ ) . (5.6) 2CT 100

The additive combination of the truncated raising and lowering operators equate exactly to the Pauli Y matrix

− + σ − σ = −iσy . (5.7)

Finally substituting this last relation back into Eq. (5.6) yields

ωq HSFQ = CcV (t) σy . (5.8) 2CT

We can calculate the eﬀect of this Hamiltonian on a general state |ψ via the Schrödinger equation

d |ψ −i = HSFQ |ψ (5.9a) dt d |ψ H = i SFQ dt (5.9b) |ψ |ψ(t) ω ω ln = C q σ V (t)dt = iC Φ δ(t − Δt) q σ (5.9c) |ψ c 2C y c 0 2C y 0 T T ωq |ψ(t) =exp iCcΦ0δ(t − Δt) σy |ψ0 , (5.9d) 2CT

where we have used the relation V (t)dt =Φ0δ(t − Δt) for a train of delta-like SFQ pulses

arriving at intervals of Δt =2nπ/ωq. By relating Eq. (5.9d) to the quantum mechanical rotation operator D,

D(δθ) ≡ exp [iδθσy/2] , (5.10) 101

(a) (b) (c)

SFQ Source

Figure 5-3: (a) Circuit diagram depiction of a transmon qubit capacitively coupled to a generalized SFQ source (b) Voltage timing diagram of the output of the SFQ driver. The area underneath each pulse is exactly one ﬂux quantum. The pulse-to-pulse spacing to coherently drive a qubit is at integer multiples of the qubit Bloch vector rotation period. (c) Bloch sphere illustration of the qubit state vector trajectory when irradiated by trains of SFQ pulses. The vector is displaced an angle dθ for each pulse. Between subsequent pulses, the state vector precesses about the z axis at the qubit frequency ωq a number n times before the next pulse arrives where n is an integer. we can estimate a displacement angle δθ per SFQ pulse of

ω δθ/2=C Φ q (5.11a) c 0 2C T 2ωq δθ = CcΦ0 . (5.11b) CT

The ramiﬁcations of Eq. (5.11b) is that the accuracy of an arbitrary qubit rotation when performed via SFQ pulses is now determined by a fundamental constant of Nature and a lithographically adjustable circuit parameter. The absolute error due to the discrete rotation angle is

δθ2 1 −F =1− cos(δθ/2) . (5.12) SFQ 4

for small δθ 1. Figure 5-3(b) depicts a time trace of an SFQ pulse train used to coherently drive a transmon qubit. The area underneath each pulse is exactly 1 superconducting magnetic ﬂux 102

Coupling Capacitor (aF) (a) (b) 400 200 100 50 1 100 n SFQ = 10 0.9 n = 25 SFQ n = 50 0.8 SFQ n = 100 -1 SFQ 10 0.7 n = 1000 SFQ 0.6

-2

0.5 P 10

0.4

0.3

Normalized Amplitude 10-3 0.2

0.1 -4 0 10 4 4.5 5 5.5 6 0 50 100 150 200 250 300 350 400 Frequency (GHz) nSFQ

Figure 5-4: (a) Normalized frequency spectrum of SFQ pulse trains of diﬀerent number of SFQ pulses. The SFQ pulse–to–pulse spacing was 200 ps corresponding to a drive frequency of 5 GHz. For a small number of pulses, signiﬁcant spectral weight can be seen at frequencies above and below the central peak. Frequency content below the central peak can drive higher order qubit transitions. (b) Maximum qubit |2 state leakage probability as a function of the number of SFQ pulses required to achieve a qubit π–rotation. For a π–rotation requiring π ≈ −3 nSFQ 200 SFQ pulses, the leakage error is below 10 . The corresponding coupling capacitance Cc required to yield a set number of SFQ pulses is plotted on the top x–axis.

quantum Φ0 with the inter-pulse timing set to an integer multiple of the qubit period. Figure 5-3(c) depicts the corresponding trajectory of the qubit Bloch vector (red) when irradiated by such a pulse train. The Bloch vector undergoes discrete rotations of magnitude δθ about the yˆ axis for every pulse. Between pulses, the Bloch vector precesses freely about the zˆ (|0) axis at frequency ωq. A dominant source of error to consider is qubit state leakage into the |2 state. A singular SFQ pulse has an enormous bandwidth in frequency space, scaling as ∼ 1/σ,whereσ is the SFQ pulse width. The reduced anharmonicity of a transmon qubit combined with this large bandwidth will couple higher order qubit states out of the computational manifold resulting in a non-negligible occupation of the qubit |2 state. Figure 5-4(a) shows the Fourier spectrum of a sequence of SFQ pulses in time for diﬀering number of SFQ pulses. The pulse–to–pulse spacing was set to be 200 ps resulting in a central frequency component centered at 5 GHz.

For pulse sequence numbers below nSFQ ≈ 100, signiﬁcant spectral weight resides in the 103

side lobes of the spectrum. These side lobes can drive qubit rotations out of the computational manifold resulting in poor control and large error. In order to model the leakage, we extend the Hamiltonian to include a third level which now takes the form

⎡ ⎤ ⎢ ⎥ ⎢00 0 ⎥ ⎢ ⎥ |2 ⎢ ⎥ H = ω ⎢ ⎥ (5.13) Free q ⎢01 0 ⎥ ⎣ ⎦ 001+ωq/ωq − | →| where ωq =(E|2 E|1)/ is the frequency for the qubit 1 2 transition. The Hamiltonian describing leakage out to the qubit |2 state is given by

⎡ ⎤ ⎢ − ⎥ ⎢0 10⎥ ⎢ √ ⎥ |2 2ωq ⎢ ⎥ H = C Φ ⎢ − ⎥ . (5.14) SFQ c 0 ⎢10 2⎥ Cq ⎣ √ ⎦ 0 20

We model the system for a qubit fundamental transition frequency ωq =2π × 5 GHz,

anharmonicity ωq/ωq =0.95, and qubit capacitance Cq = 100 fF. Figure 5-4(b) displays the results from the full master equation simulation of the qutrit system as modeled in QuTiP [76]. The maximum |2 state population is plotted against the required number of SFQ pulses

to achieve a π–rotation (bottom x–axis) and the corresponding coupling capacitance Cc (top π | x–axis). For SFQ π–pulse lengths nSFQ 200, the maximum 2 state leakage is found to be below 10−3. The corresponding coupling capacitances required to achieve this low–level of leakage are well below 1 fF.

5.3 Bridging the Quantum–Classical Divide

In order to demonstrate the coherent control of a transmon qubit with SFQ pulses, an experiment had to be designed that not only was extensible enough to provide deﬁnitive proof of principle but also fabricable with the tools available in the lab and the clean room. 104

In the following sections, we describe the design, modeling and fabrication of superconducting Nb experiments that combined both classical SFQ circuitry with a transmon qubit–CPW resonator quantum circuit.

5.3.1 SFQ Driver Design

One of the most basic SFQ circuits is the dc/SFQ converter. Despite its misleading name, the circuit actually converts a continuous RF tone into a time series array of SFQ pulses. The circuit diagram of a three Josephson junction dc/SFQ converter is displayed in Figure

5-5(a). A junction JJT and shunt inductor LS form a RF SQUID. These two components

represent the “set” portion of circuit. The reset portion is comprised of two junctions JJR1

and JJR2 in parallel. The operation of the circuit is depicted in Figures 5-5(b) and 5-5(c). During the ﬁrst π phase evolution of the trigger tone (blue, top trace), the DC bias current (purple) and trigger current add together in the trigger junction (green, middle trace) arm of the RF SQUID. The two currents together are enough the exceed the critical current of the trigger junction, causing it to go normal, creating a ﬂuxon-antiluxon voltage pulse pair. The ﬂuxon propagates to the output of the circuit while the antiﬂuxon sets the circulating current ﬂux state of the RF SQUID (orange arrows, Figure 5-5(b)). This circulating current ﬂows opposite in direction to the RF and DC bias currents bringing the total current through the trigger junction back down to below it’s critical current value and allowing the Josephson phase to only progress 2π down the tilted washboard potential (see Figure 5-1). During the second π phase evolution of the trigger tone, the RF and DC current add together in the reset arm of the converter triggering a SFQ pulse (red trace) that resets the state of the RF SQUID. Every 2π phase evolution of the trigger current, the dc/SFQ converter arms, sending out one SFQ pulse to the output and is subsequently reset. For proper ﬂux set and reset of the RF SQUID, it is important that the RF SQUID can support such a propagating current

and thus puts limits on βL, deﬁned as 105

(a)

IDC JJR2 I RF JJ R1 Out

L S JJT

(b) IRF t

VT Out t

VR t

(c) IRF t

VT Out t

VR t

Figure 5-5: (a) Circuit diagram of a dc/SFQ converter. The circuit is comprised of 3 Josephson junctions. One junction (JJT) serves as the SFQ trigger while the other 2 (JJR1/R2) serve to reset the circuit. (b) During the ﬁrst π phase evolution of the RF tone, the DC bias current (purple) and the trigger current (blue) add together exceeding the critical current of the trigger junction (green). The junction brieﬂy goes into the normal state and expels 2 ﬂuxon pulses. One ﬂuxon pulse (green) propagates to the output of the circuit while the other sets the ﬂux state (orange arrows) of the RF SQUID formed by JJT and LS. The circulating current in the SQUID quickly reduces the total current through the trigger junction allowing for only 1 SFQ pulse to be generated. (c) During the second π phase evolution of the trigger tone, the DC and RF currents add in the reset arm (red) of the dc/SFQ converter, triggering a reset of the ﬂux state of the RF SQUID. 106

2πIc,TLs βL = . (5.15) Φ0

For proper operation, 2 <βL < 6[77]. To better illustrate these limits, we solve for the Josephson phase in the trigger junction via circuit analysis. We begin by combining the reset

junctions JJR1 and JJR2 into a single junction JJR. The sum of the bias currents running

through each of the 2 legs of the SQUID is equal the the applied bias current IB,

IDC = IJJT + IJJR = Ic,T sin(δT)+Ic, R sin(δR) . (5.16)

The current shunted through the inductor is the sum of the trigger current and the reset current

IL = IRF + Ic,R sin(δR) . (5.17)

Finally, the sum of the phases of the circuit must equate to an integer multiple of 2π yielding the equation

2π ILL+δR − δT =2nπ . (5.18) Φ0

Combining equations Eq. (5.16),Eq.(5.17), and Eq. (5.18), we can derive a transcendental

equation of the trigger junction phase δT,

2π IDC − Ic,T sin(δT) L(IRF + IDC) − βL sin(δT) + arcsin = δT +2nπ (5.19) Φ0 Ic,R

Figures 5-6(a) and 5-6(b) show the simpliﬁed circuit diagram and plots of the LHS (color)

and RHS (dashed lines) of Eq. (5.19) as a function of δT, respectively. The LHS values have been artiﬁcially displaced in the y–axis so as to display them one on top of one another. As

βL is increased, more solutions, and thus more stable RF SQUID ﬂux states, can be found 107

(a) (b) 30

20 IDC

10 IRF JJR 0 LHS

-10 L S JJT -20

-30 -4-3-2-101234

Figure 5-6: (a) Simpliﬁed dc/SFQ converter circuit diagram. (b) Plot of the LHS of Eq. (5.19) as a function of the trigger junction phase. Dashed lines plot the RHS side of the transcendental equation for varying values of n.AsβL is increased, more solutions (simultaneous overlap of colored and dashed lines) are found which physically equates to more stable ﬂux states in the RF SQUID.

for Eq. (5.19).ForβL values below 1, the SQUID cannot support a single ﬂux quantum equivalent circulating current. Due to it’s relative simplicity in terms of operation (1 DC bias current + 1 RF trigger current) in addition to the low number of junctions, the dc/SFQ circuit was chosen as a ﬁrst pass circuit to demonstrate SFQ based qubit control. The step ﬁrst in integrating the converter with a qubit was to design it in such a way that a minimum amount of input RF power was required to trigger SFQ pulses. The onset power of SFQ pulses can be measured via the appearance of Shapiro steps [78] in the junction IV curve. When irradiated with RF tones, at particular biases a junction will switch into the voltage state at a rate commensurate with the driving frequency ω. So long as the junction is properly shunted so as to not be

hysteretic (βC 1), the junction will switch into and reset from the voltage state without a reduction of the static bias down to the retrapping current. Equating the energy required to break cooper pairs with the photon energy of the RF driving tone, a very simple relation develops relating the magnitude of voltage steps seen in the Josephson IV curve to the 108

Inductor Value (pH)

L1 3.35

L2 1.27

L3 0.69

L4 1.29

L5 0.21

L6 0.18

L7 1.14

L8 0.08

L9 1.74

L10 0.13

L11 2.11 Table 5.1: dc/SFQ wiring inductances used in WRSpice circuit model. frequency of the RF tone,

2eV = ω (5.20a) ω V = . (5.20b) 2e

Whiteley Research Spice (WRSpice) [79] was used to model and engineer the driver at the circuit element level for low (−60 dBm) RF input power operation. The modeled circuit is shown in Figure 5-7(a). The design is based oﬀ of work in [80, 81] and the RSFQ cell library of Stony Brook University [82]. The circuit is exactly the same as discussed previously with the only change being the addition of a single junction Josephson transmission line (JTL) to the output. Values for the modeled wiring inductances can be found in Table 5.1. The resulting simulated IV curve for the driver with a modeled critical current density of

2 Jc =1kA/cm, shunt inductor LS = 5 pH, and no applied RF power is displayed in Figure 5-7(b). The total critical current of the driver is the sum of the individual critical currents 109

(a) (b) 200 IDC 175 150 L8 L4 125 P L L L RF L L 7 9 11 100 L1 2 5 75

L3 50 DC Bias Current (uA) L L 25 S L6 10 0 0 102030405060 Driver Voltage (uV) (c) (d) 80 80 200 175 70 70 175 60 150 150 60 50 125 125 50 100 100 40 40 75 30 75 30

50 (uV) Driver Voltage 50 (uV) Driver Voltage DC Bias Current (uA) 20 DC Bias Current (uA) 20 25 10 25 10 0 0 -70 -60 -50 -40 -30 -70 -60 -50 -40 -30 RF Drive Power (dBm) RF Drive Power (dBm)

80 (e) 175 70 150 60 125 50 100 40 75 30

50 (uV) Driver Voltage

DC Bias Current (uA) 20 25 10

0 10 20 30 40

LS (pH) (f) (g) 70 175 80 175 60 70 150 150 60 50 125 125 50 40 100 100 40 30 75 75 30 20 Driver Voltage (uV) Driver Voltage 50 (uV) Driver Voltage 50 DC Bias Current (uA) 20 DC Bias Current (uA) 25 10 25 10 0 0 10 20 30 40 10 20 30 40

LS (pH) LS (pH)

Figure 5-7: (a) dc/SFQ circuit modeled in WRSpice. Values for the inductances are listed in table 5.1. (b) Simulated IV curve for the dc/SFQ circuit in (a) for no applied RF with 2 Jc =1kA/cm and LS = 5 pH. (c-d) False color dc/SFQ IV curves as a function of applied 5 GHz RF power for shunt inductors LS = {10, 20} pH. The onset of a clearly deﬁned Shapiro step isn’t until the power is above -40 dBm. (e) Constant power (-60 dBm) IV curve as a function of shunt inductance LS. Shunt inductance values corresponding to a 2.5 <βL < 5 provide the largest operating region in DC bias current. (f-g) IV curves as a function of LS 2 for Jc = {0.9, 1.1} kA/cm . Changing the critical current density by as much as ±10% still yields an operable driver for proper choice of LS. 110

of the constituent junctions totaling 165 μA. Figure 5-7(c) plots the driver voltage (false color) as a function of DC bias current (y–axis) and applied RF power (x–axis) for a driving

frequency of 5 GHz and shunt inductor LS = 10 pH. The onset of the n = 1 ﬁrst Shapiro step voltage V = ×(2π ×5 GHz)/2e ≈ 11.2 μV Shapiro step can be seen for biases IDC 100 μA

and powers PRF −45 dBm. The same simulation is displayed in Figure 5-7(c) only for a shunt inductor LS = 20 pH. The two–fold increase in βL allows for more stable trigger junction phase states and a resultant lowering in the required RF power to trigger single 2π phase slips in the trigger junction. Figure 5-7(e) displays again the simulation results

for the dc/SFQ driver but now sweeping the LS shunt inductor value from 1–50 pH for a

ﬁxed driving frequency of 5 GHz and power PRF = -60 dBm. For shunt values LS< 20 pH with corresponding βL < 2.5, only reductions in the overall critical current of the driver are seen with no clear Shapiro step arising in the IV curves. For values of shunt inductance

20 < LS < 40 with corresponding 2.5 <βL < 5.0, a clear n=1 Shapiro step develops for bias

currents 110 μA < IDC < 170 μA. For higher shunt inductances, more than one Shapiro step arises, limiting the operating DC bias current range.

This βL “sweet spot” for the low power onset of Shapiro Steps echoes and conﬁrms the

results of Eq. (5.19) / Figure 5-6(b). Namely that for too low of a shunt inductance (βL), only 1-2 solutions of the transcendental equation exist requiring large amounts of RF power

to drive transitions between stable states. As βL is increased, the number of stable phase points increase allowing for the driving of 2π phase slips at lower RF power. It is worth mentioning that while the simulations and design mind–set were for engineering drivers that minimized the required applied RF power, they did not explicitly try to minimize the on–chip dc/SFQ power dissipation. The energy dissipated for one 2π junction phase slip is

E = Pdt = IcVdt= Ic Vdt= IcΦ0 . (5.21)

For a junction being triggered at a clock rate f, the power dissipated then is just 111

PSFQ = E × f = IcΦ0f. (5.22)

For Ic = 100 μA and clock rate / drive frequency f =5GHz,PSFQ ≈ 1nW. A feature of the dc/SFQ circuit and RSFQ circuits in general is the robustness to processing / fabrication induced variation. Figures 5-7(e–f) display the resultant IV curves again for the driver depicted in Figure 5-7(a) but for ±10% variation in the critical current density with (e)

2 showing the results for a critical current density Jc =0.9kA/cm and (f) showing results for

2 the a critical current density Jc =1.1kA/cm. When Jc is lowered (increased), the required shunt inductance to obtain to widest operating margin in DC bias current simply shifts up (down) by roughly the same percentage. This η ≈ 20% operating margin for a proper design allows for devices to survive and operate appropriately even when exposed to the chaotic, fractal-like nature of microfabrication processes and tools generally seen at university level general user clean room facilities.

5.3.2 SFQ–Qubit Circuit Layout

Figure 5-8(a) displays a circuit diagram of the coupled SFQ–Qubit system. The output of

the dc/SFQ driver (red) is coupled to the qubit (blue) via a capacitor Cc. The qubit is a ﬂux–

tunable transmon with mutual inductance MΦ to an external bias line (yellow). Readout of

the qubit is performed via capacitively coupling Cg to a quarter wave CPW resonator. Qubit state dependent dispersive readout of the resonator is performed in microwave transmission

measurements via an inductive coupling Mc to a common feed line (black). Figure 5-8(b) shows an overhead micrograph of the completed circuit. The colored outlines correspond to the colored circuit elements in 5-8(a). The SFQ–qubit coupling capacitor was in–plane with the qubit capacitor pad (blue). The qubit design was modeled from the Syracuse University style “rectmon” transmon. The qubit capacitor pad was 40×400 μm in footprint with a COMSOL / HFSS modeled self capacitance Cq ≈ 76 fF. The qubit capacitor pad was 112

(a) (b)

(c)

Figure 5-8: (a) Circuit diagram layout of coupled dc/SFQ driver to a transmon qubit. The output of the driver (red) is capacitively coupled to the qubit (blue) via a coupling capacitor Cc. The qubit capacitor pad Cq is shunted to ground via an asymmetric 2:1 DC SQUID that is externally ﬂux biased (yellow) via a mutual coupling MΦ. The qubit is capacitively coupled to a quarter wave readout resonator (blue) that in turn is inductively coupled to a microwave feed line (black). (b) Micrograph of fabricated integrated dc/SFQ driver and qubit. Colored boxes outline the corresponding colored circuit elements shown in (a). (c) Zoomed in micrograph of fabricated dc/SFQ converter. The Josephson junctions are 1–to–1 placed with respect to the corresponding circuit diagram. 113

shunted with two Josephson junctions forming a dc SQUID with area/energy asymmetry α = 2 (see Appendix A) providing a value for d =1/3. Large modularity in physical design was aﬀorded by the use of optical projection lithogra-

phy. Lithography masks were made such that a multitude of SFQ–qubit couplings Cc could be tested ranging from sub–attoFarad to the single femtoFarad range. In addition, the entire driver placement on the chip was modular with three physical distances of 0.5, 1.5, and 3.5 mm between the output of the driver and the qubit able to be realized. Furthermore,

readout resonator frequency ωr, feed line coupling Mc, and qubit coupling Cg could also be modulated.

5.3.3 Fabrication

For the past 12+ years, traditional wisdom [83] has been that incorporating amorphous dielectrics into the design of superconducting quantum circuits is a fool’s errand and that doing so only serves to reduce the performance of the individual quantum circuit elements by increasing the microwave loss. However, all the circuit diagrams, Hamiltonians, and numerical simulations in the world mean nothing if the above proposed experiment can’t be physically realized on a chip. With the minimum dc/SFQ stack requiring 3 layers (ground plane / JJ electrode + wiring dielectric + JJ counter electrode), traditional wisdom had to be discarded. The following describes in detail the fabrication required to integrate the necessary mutlti–layer stack of SFQ fabrication with the traditional monolayer fabrication employed in making superconducting quantum circuits. An overview of the 8 additive layer (6 metal + 2 dielectric) process is displayed in Table 5.2.

Layer M0 The fabrication process began with the submersion of a bare intrinsic Si100 three–inch wafer in room temperature hydroﬂuoric (HF) acid for 60 seconds to strip the

native SiOx oxide. The wafer was then loaded into a DC magnetron sputtering tool with base

−9 pressure Pb < 5 × 10 T. Prior to deposition, the Nb target was plasma cleaned at argon

pressure PAr=4.25 mT at 400 W × 2 min. The wafer was rotated at ∼1 Hz during deposition 114

Layer Thickness Surface Deposition Materials Patterning Etching Purpose ID (nm) Pre-treatment Method Ground Plane Projection M0 Nb 180 HF DC Sputter ICP Resonator i-line Qubit Projection V1 SiO2 130 – PECVD RIE Ground Vias i-line Projection M1 Nb 90 ion–mill DC Sputter RIE SFQ JJ Base Electrode i-line Projection V2 SiO2 180 – PECVD RIE JJ Vias i-line DC Sputter + Projection SFQ Bias Wiring M2 Nb/Al-AlOx-Al/Nb 100 ion–mill RIE in-situ oxidation i-line JJ Barrier / Counter Electrode Projection R Ti/Pd 3 + 20 ion–mill E-beam Evap – SFQ Shunt Resistors i-line Projection QP Ti/Cu/Pd 93 ion–mill E-Beam Evap – Quasiparticle traps i-line E-Beam QB Al-AlOx-Al 100 ion–mill E-Beam Evap – Qubit JJs Litho Table 5.2: Layer–by–layer overview of the dc/SFQ–Qubit chip fabrication. Overall 8 additive layers (6 metal + 2 dielectric) were required.

in which Nb was sputtered at 2×500 W×2 min with 10 minutes between sputters for a total ﬁlm thickness of 180 nm as measured with proﬁlometry. For patterning, SPR-955 0.9 cm positive photoresist (PR) was spun at 4500 RPM for 60 seconds followed by a pre-exposure bake at 100 C × 90 seconds. Pattern exposure was done with a 365 nm i-line 5:1 projection stepper. The wafer was then post-exposure baked at 110 C × 60s. Development of the pattern was done via submersion and agitation in Microposit CD26 TMAH developer for 60s followed immediately by submersion in DI water for another 60s. The wafer was then actively dried with dry nitrogen gas. Etching of the Nb was performed in a PlasmaTherm PT770 with

aCl2/BCl3 inductively coupled plasma (ICP). The etch was inspected visually prior to PR stripping. The PR was stripped overnight in room temperature acetone followed subsequently by baths in IPA and deionized (DI) water. The depth of the etch was characterized with surface proﬁlometry with an average trench depth of 210 nm corresponding to an over etch of 30 nm.

Layer V1 TheﬁrstoftwoSiOx wiring dielectrics, layer V1 was deposited using a 115

PlasmaTherm PT-70 plasma enhanced chemical vapor deposition (PECVD) tool. Prior to wafer insertion, the tool platen was heated to 250 C. The chamber was then cleaned in an

O2 plasma for 2×250s periods followed by a recipe pre-seed of 500s. Only afterwards was

the wafer loaded into the tool. SiOx was deposited for 250s with a target thickness of 130 nm. Lithographic patterning was performed in the same manner as described above for layer M0 with one additional step. A PR reﬂow bake at 125 C×180 s was performed post development to induce ∠45◦ sloped sidewalls into the developed PR edges. The dielectric etch was performed in a PlasmaTherm 790 reactive ion etcher (RIE) with a carbon platen. Prior to loading the sample, the empty chamber was cleaned with an oxygen plasma for 7

minutes. The wafer was then etched with a CHF3 based RIE where the balance between physical and chemical etching transfered the sloped sidewalls of the resist into the dielectric. The sloped dielectric sidewall was crucial in eliminating step coverage issues in subsequent layers. Post etch, the PR was stripped again in subsequent baths of acetone, IPA and DI water. Proﬁlometry yielded an average layer thickness of 132 nm. Layer M1 Filling the role of the JJ bottom electrode, layer M1 began with a Ar–ion mil

−4 at PAr =2×10 T for 45 s to promote good metal–to–metal contact between M1 and M0 by milling the native oxide on M0. Deposition of the Nb was carried out the same way as described for M0 with an important caveat. Speciﬁcally, the sputter pressure was adjusted to provide a slightly compressive (200 MPa) intrinsic Nb ﬁlm stress. Achieving a compressive ﬁlm stress for the JJ electrode layer was crucial as JJs made with tensile stress ﬁlms have been shown to have large leakage currents [84]. The sputtering pressure used varied as a function of the target life increasing from approximately 4 mT for a newly installed target and ending at 6.5 mT after 500 minutes of target sputter time. Patterning of the layer was done in the same way as M0 with an additional PR reﬂow step as described above. Etching

of the layer was performed with an SF6 RIE plasma. Layer V2 Layer V2 deﬁned the SFQ JJ area. It was deposited, patterned and etched in the same way as V1 only with a deposition time now of 350 seconds for a target thickness of 116

180 nm. Layer M2 This Nb layer deﬁned the SFQ JJ counter electrode. Involving the co– deposition of both Nb and Al for the trilayer junctions, particular care was taken to follow the exact steps detailed in the following:

(i) The wafer was loaded in the sputter system load lock which was then pumped out via a roughing and turbo pump for t ≥ 1h.

(ii) Commensurate with the load lock pump–out, the main sputter chamber was seeded

with 100 mT of O2 gas. This seeding process conditions the chamber chemistry provided better fabrication run–to–run process uniformity.

(iii) After the pre-seed, the wafer was transferred from the load lock to the main chamber. The pressure was allowed to recover to P≤ 5 × 10−8 T.

(iv) The wafer was Ar–ion milled for 25 seconds to prepare the exposed Nb surface of layer M1 by removing the native oxide.

(v) The Al sputter target was cleaned at 200 W×2 m. Al deposition was then performed at 40 W×2 m for a target thickness of 8 nm.

(vi) O2 gas was ﬂowed into the chamber achieving a background pressure of 1 mT. All pumps were then shut oﬀ from the main chamber and the pressure was allowed to rise to 100 mT. The wafer was allowed to oxidize for 10 min with a target critical current

2 density Jc =1kA/cm .

(vii) After oxidation, the O2 gas was pumped out via the load lock roughing pump, turbo pump and main chamber cryopump.

(viii) Al was again sputtered following the procedure described in (v) only this time for 90 s for a target thickness of 6 nm, 117

(ix) A ﬁnal cap of Nb was deposited for 2 min following the cleaning/sputtering steps outlined earlier. The target thickness was 90 nm.

This counter electrode layer was coated with SPR955 photoresist and patterned with the stepper. Etching of the Nb was performed in the Unaxis 790 RIE tool utilizing the already

mentioned SF6 RIE plasma. Once completed and prior to venting, a second 50 W Ar plasma was run for 80 seconds prior to venting so as to remove residual SF6 plasma residue from the wafer. The now exposed Al was then etched in a room temperature bath of MF-24A TMAH PR developer. Test JJ patterns incorporating JJs of areas A =2,4,8,and16μm2 were

then 4–wire probed to determine their room-temperature normal state resistance Rn.The Ambegaokar–Baratoﬀ relation [85]

π 2Δ (J × A)R = (5.23) c n 4 e with A the junction area, Δ the SC gap energy and e the fundamental charge, allowed for

determination of the critical current density Jc.

Layer V1/V2 Removal Up until this point, much of the PECVD SiOx dielectric was still covering an area over layer M0 where the qubit and readout resonators needed to be patterned. The patterning and etching of these layers to reveal this area of M0 was carried in similar fashion to either of the individual dielectric etching procedures. Qubit and Resonator Patterning With V1 and V2 layers now removed, the qubit and resonator were patterned and etched utilizing the same methods described for layer M0. Layer R AZ–5214 PR was spun on the wafer and baked at 95 C for 60 s. Pattern exposure was carried out in the stepper. An image reversal (IR) bake was then done at 110 C for 60 seconds followed by an 60 s, 365 nm ﬂood IR exposure in a Karl Suss MA6 contact aligner. Development of the pattern was performed in room temperature CD-26 for

45 seconds followed by a 30 second DI water rinse and dry N2 gas. When properly processed, the resist should has a ∠45◦ undercut. The wafer was then loaded into a custom electron 118

beam evaporator. After loading into and pumping out the load lock, the sample was opened to the main chamber and the pressure allowed to recover to below 5 × 10−7 T. An Ar–ion mill was performed for 17 seconds at 2 × 10−4 T backing pressure. A Ti adhesion layer was then evaporated onto the wafer at a rate of 0.5 Å/s for a total of 3 nm. Pd was then evaporated at a rate of 0.5 Å/s for a total of 30 nm. Once the evaporations were completed, the wafer was placed in a Teﬂon tripod holder submerged in room temperature acetone overnight to complete the liftoﬀ process. Once all the extraneous Ti/Pd was lifted oﬀ, test resistor features of 1.1 and 2.2 squares were 4–wire probed. The residual resistance ratio (RRR) the Ti/Pd resistors was measured to be approximately 2 between 293 K and 4 K. Layer QP Fabricating Cu quasiparticle (QP) traps began with spinning, patterning, exposing, and developing AZ–5214 PR in the exact same manner as was done for layer R. Once complete, the wafer was loaded into the evaporator and allowed to pump down again to below 5 × 10−7 T. Following the same ion mill procedure described for the resistor layer, Ti was evaporated at a rate of 0.5 Å/s for a total of 3 nm. Copper was then evaporated at a rate of 1 Å/s for a total of 80 nm. Another layer of Ti was then evaporated again at 0.5 Å/s for another 3 nm. Finally, the resistor stack was completed with an evaporation of Pd at a rate of 0.5 Å/s for a total of 10 nm. The total trap area ﬁll factor was ∼ 35%. While only the Cu plays a role in the trapping of QPs, the exposed Cu surface was found to react negatively (color change) with the subsequent processing required for the qubit JJ fabrication. An additional protective layer stack of Ti/Pd on top of the copper was found to counteract this adverse reaction. Layer QB The ﬁnal step in the process was the patterning and double angle evaporation of the QB JJs. This step in the process was carried out by our Syracuse collaborators. The

process began with a 150 W×30 second O2 plasma descum of the wafer prior to spinning on the MMA / PMMA e–beam resist stack. The MMA was spun on ﬁrst at 2500 RPM for 60 seconds for a target coating of 600 nm. The wafer was then baked at 170 C for 10 minutes. The PMMA resist was then spun on at 3000 RPM for 60 seconds targeting a thickness 119

of 70 nm. The wafer was then baked again at 170 C for another 10 minutes. The Dolan bridges [86] were written with a typical e–beam dosage of ∼ 1100 μC/cm2 . The exposed resist was then developed in a 3:1 ratio IPA–to–methylisobutylketone (MIKB) mixture for 60 seconds with agitation. The Dolan bridge on average was 130 nm wide. The e–beam written junction trace widths were 180 and 360 nm with a designed junction overlap of 150 nm yielding a SQUID asymmetry ratio α = 2 (see Appendix A, Eq. (A.14)). The wafer was then loaded into a custom e–beam evaporator. An Ar–ion mill was performed for 8 seconds prior to Al deposition. The two Al evaporations were carried out at ±∠11◦. The base and counter electrodes were evaporated to thickness of 55 nm and 35 nm, respectively with an

intermediary P= 9 T, 5% O2 in Ar oxidation carried out for 9 minutes. The subsequent liftoﬀ was performed in room temperature dichloromethane. A photo of an undiced fully fabricated sample is displayed in Figure 5-9(a). Each chip was 8×8mm2 and incorporated two independent experiments. Figure 5-9(b) displays an overhead false–color optical image of the driver with the 4 junctions comprising the dc/SFQ driver plus the JTL colored yellow. Layers M1 and M2 are colored blue and red, respectively. Finally, a cross–sectional false–colored SEM image is shown in Figure 5-9(c). A focused ion beam was used to mill into the sample so that the image could be taken. Nb layers are colored

red while SiOx layers are colored blue. The substrate has been colored yellow. The trilayer junction formed by layers M1 and M2 is indicated by the black ×.

5.4 Measurement

In the following, we describe the process of how fully processed samples were prescreened at multiple temperatures / environments such that only viable candidates were measured at millikelvin temperatures of the DR. This multi–temperature prescreen process became known colloquially as “The great ﬁlter.”†

†A reference to Robert Hanson and Nick Bostrom’s work concerning the Fermi Paradox 120

(a) (b)

2 mm

M2 V2 M1 V1 M0

(c)

Figure 5-9: (a) Glamor shot of an undiced full processed chip. Two experiments were able to be fabricated per chip. (b) overhead false–color optical image of the dc/SFQ driver. M1, M2 and the junctions are colored blue, red, and yellow, respectively. (c) False color cross-sectional SEM image of the dc/SFQ driver layer stack. Nb is colored red while SiOx is colored in blue. The chip substrate in colored yellow. The junction formed between M1 and M2 is indicated by the black ×. Images (a) and (c) courtesy of Edward M. Leonard, Jr. Image (b) courtesy of JJ Nelson. 121

5.4.1 LHe SFQ Circuit Pre-screening

The all Nb process for the SFQ drivers allowed for samples to be prescreened at 4.2 K in LHe as a ﬁrst step in ﬁltering out poorly performing chips. Samples were placed on and aluminum wire bonded to a custom dip probe mount containing bias and ﬁltering electronics allowing for a three–wire measurement of the driver along with four–wire measurements of unshunted test junctions. Once bonded, the mount+sample were placed at the end of a custom dip probe and surrounded by a high permeability magnetic shield. This was then submerged in a 100L LHe storage dewar which was then placed inside a custom built Faraday cage to reduce environmental RF noise. The measurement setup is depicted in Figure 5-10(a). The three–wire driver bias lines each incorporated low–pass T-ﬁlters. The RF drive line had 20 dB of attenuation placed on the sample mount. IV curves were taken by applying a 1 Hz bias tone through the I port of the three–wire setup and monitoring the voltage. Baseline curves with no applied RF power were taken along with data at diﬀerent drive frequencies and applied power. Figure 5-10(b) shows baseline data along with data for a 5 GHz drive and a 1.25 GHz drive at an explicit applied on–chip power of –40 dBm (no calibration was done for microwave reﬂections or line losses). The baseline critical current of 130 μA corresponded

2 to a junction critical current density of Jc =0.73 kA/cm . Shapiro steps of 10.3 μVand 2.6 μV can clearly be seen for the respective drive frequencies at bias currents exceeding

|I| > 100μA. Only samples exhibiting Shapiro steps at explicit applied powers of PRF < –35 dBm were moved on to more extensive testing the lower temperatures aﬀorded by an ADR.

5.4.2 100 mK Quantum Circuit Pre-screening

For samples containing dc/SFQ drivers that passed the initial 4.2 K screening, the next step in discerning suitability for the millikelvin stage of the DR was a repackaging into a custom superconducting Al sample box and brief stint on the 100 mK stage of an ADR. The wire bonded sample and sample box are shown in Figure 5-11(a). An ADR wiring diagram for sample characterization is shown in Figure 5-11(b). A 6:1 coaxial relay was utilized 122

#" $ $

!

Figure 5-10: (a) Circuit diagram showing 4.2 K measurement setup for characterizing the dc/SFQ driver. A three–wire measurement was done with all passive bias electronics at 4.2 K. (b) IV curves for diﬀerent RF drive frequencies. The baseline IV curve shows a total driver critical current of Ic ≈ 130 μA. Shapiro steps of 10.3 μV and 2.6 μV in size can be seen for 5 GHz and 1.25 GHz drives, respectively. The power listed is the explicit power applied at the chip not accounting for microwave line losses or reﬂections.

to switch between testing the SFQ drivers (RF1 / RF2) and passing in microwaves to the quantum circuit (RO). Each line passed through a custom high frequency Eccosorb ﬁlter at the millikelvin stage. The readout output was passed through another Eccosorb ﬁlter at the millikelvin stage followed subsequently by a microwave isolator, K&L low pass microwave ﬁlter and a HEMT microwave ampliﬁer all at 3 K. Qubit ﬂux bias lines were biased through 1 kΩ resistors sunk on the 3 K stage and passed through low-pass (-60 dB at 3 GHz) Eccosorb ﬁlters on the cold stage. Three–wire measurements for driver characterization were comprised of 2 kΩ and 1 kΩ resistors for the voltage and current bias respectively tied together either on the 3 K plate or at the millikelvin stage depending on whether the cool down was shared with experiments or not. At this temperature only qubit signs on life (see section 3.2.2) could be investigated. Simple measurements reading out qubit cavities in transmission using a 2–port VNA allowed for rapid conclusions to the questions (1) “is the qubit alive?” and (2) “does it ﬂux tune appropriately?” Characteristics curves displaying a positive response to 123

(a) IV1

RF1 FB1

RO RO

IV2 FB2

RF2

(b) K&L HEMT K&L

40 2 k1 k 2 k 1 k 1 k 1 k

6:1 Relay 3 K HF ESRB HF ESRB HF ESRB HF ESRB LF ESRB LF ESRB mK LF ESRB LF ESRB

IV1 RF1 FB1

RO RO

IV2 FB2 RF2 MuMetal Shield

Figure 5-11: (a) Bonded sample in sample mount. Each chip contained 2 independent experiments (labels 1 & 2). Readout resonators were inductively coupled to a common feed line (RO). (b) ADR Wiring diagram for T = 100 mK sample prescreening. 6:1 channel relays were used to increase wiring ﬂexibility. Each quantum and classical circuit pair could be tested independently. 124

both these questions are displayed in Figures 3-4 and 3-5, respectively. IV curves of the SFQ drivers at T = 100 mK often displayed a slight amount of hysteresis near the critical current. This was due to either the critical current exceeding such a value

that βC > 1 or, as was not found until imaged under a SEM, the shunt resistors failing after thermal cycling. Figure 5-12 displays IV curves taken for a driver at T = 100 mK for no applied RF power (blue) and for a 5 GHz drive at -40 dBm (orange). With no applied RF, the

driver was slightly hysteretic despite βC < 1 even for a slightly larger than designed 180 μA critical current. However, as can be seen by the orange curve, the deleterious hysteresis could often be suppressed with the application of microwave power to the driver. A clear Shapiro step can be seen at a bias current of approximately 110 μA. This is not in the hysteretic regime of the driver and thus constituted a perfectly suitable operating point. Upon discovering the thermal cycling issue with drivers becoming hysteretic at 100 mK despite being not at 4.2 K and only a modest increases in the critical current and thusly

βC , prescreened drivers exhibiting this behavior were imaged by SEM. Figure 5-13 shows a depth proﬁle SEM image of a dc/SFQ driver shunt resistor. Again the Nb metalization layers are in pink, SiOx wiring dielectric is in blue and the shunt resistor is in green. The process parameters for achieving a sloped side wall etch in Nb detailed in the previous sections drifted resulting in a near vertical side wall proﬁle of M2. With a thickness of 100 nm, the vertical side wall all but broke many shunt resistor connections between the junction and ground. Multiple thermal cycles of the samples exasperated the issue as resistors that did make galvanic connection during deposition would break from the thermal stress.

5.4.3 Initial SFQ–Qubit Bringup

Only a handful of samples across multitudes of wafers survived the multi–temperature prescreen procedure described in the previous sections. The few precious samples that did survive were further tested on the millikelvin stage of a DR. Figure 5-14 displays the initial wiring diagram aﬀorded by the space available in the DR. Microwaves utilized to control 125

300

200 No RF 5 GHz, -40 dBm

100

0

-100 Voltage -200

-300 -300 -200 -100 0 100 200 300 Current

Figure 5-12: dc/SFQ IV curve for both no applied microwave power (blue) and a -40 dBm, 5 GHz drive (orange) measured at 100 mK. For no applied microwave power, the driver was slightly hysteretic. The application of microwaves, however, lowered the critical current value and thusly βC to such a value that the hysteresis was suppressed and a DC bias current operating point of I = 110 μA could be found.

R M2

M1

M0

Figure 5-13: Depth proﬁle of SFQ driver layer stack showing the Nb metalization layers (pink), the SiOx wiring dielectric (blue) and the JJ shunt resistor (green). It is plain to see from this image that the reﬂow bake plus etch of layer M2 did not result in a sloped side wall proﬁle of the Nb. This caused many shunt resistors to break upon deposition or during thermal cycling resulting in hysteretic dc/SFQ drivers. 126

and readout the qubit were SSB mixed in IQ mixers with the output of 1 GS/s DACS. These signals were then combined with a 3 dB splitter at room temperature and fed into the DR. Once at the cryogenic plates, the signal passed through stages of attenuation (50 dB in total) in addition to Eccosorb and K&L 12 GHz low–pass ﬁlters before ﬁnally reaching the input of the sample. At the output side, 3 microwave isolators sunk to the cold stage separated the chip from the 4 K HEMT ampliﬁer to prevent deleterious eﬀects stemming from back-action ampliﬁer noise. The HEMT ampliﬁed signal was further ampliﬁed by two Narda 2–8 GHz microwave ampliﬁers before ﬁnally being heterodyne mixed back down to the side band frequency and digitized with a 1 GS/s Alazar analog to digital converter (ADC). The qubit ﬂux bias line began with a Stanford Research Systems (SRS) SIM928 voltage source passed through a 10 kΩ resistor sunk on the 4 K plate. The DC current was then subsequently ﬁltered with a 250 MHz low–pass and low–frequency Eccosorb ﬁlter. The ﬁnal step taken in signal preparation was to insert a superconducting Nb wire into the chain to help thermalize any hot electrons. The dc/SFQ driver was biased with a SRS SIM928 voltage source passed through a 10 kΩ/1 kΩ 3–wire measurement combined at the 4 K stage. The resulting single wire was then low–pass ﬁltered, connected to a superconducting Nb wire and subsequently fed into the dc/SFQ bias line. Initial tests of the quantum/classical interface involved triggering the dc/SFQ driver direct from a coherent room temperature microwave source where the signal could be turned on and oﬀ with a fast (∼ 1 ns) microwave switch. The signal was attenuated by 40 dB and low–pass ﬁltered before being applied to the RF input of the dc/SFQ driver. The results reported from here on are for sample 070717C-C4. Prior to SFQ experimentation, baseline measurements of the qubit lifetime and dephasing rate were performed with standard microwave manipulation. The T1 and T2* curves are shown in Figure 5-15(a) and Figure 5-15(b) respectively, for the qubit tuned to its upper sweet

spot frequency of ωq =4.85 GHz. The ﬁtted relaxation curve yielded a qubit T1 =23.6±0.6 μs. | ∗ ± The ﬁtted Ramsey decay of the 1 population yielded a T2 =24.4 0.8 μs. While these times were impressive for such a heavily processed sample, they were not indicative of the 127

Variable 12 GHz Low 20 dB I 20 dB 1 GS/s Attenuator Pass Filter Attenuator I Arbitrary 12 GHz 1 GS/s AWG 20 GHz Low Waveform Microwave 10 dB ADC Analog-to-Digital HF Ecc Pass Eccosorb 10 dB Generator Generator Attenuator Converter Filter Q Q I ECL IQ Voltage LO R microwave - + Mixer Source Q switch 3 dB 300 MHz Low High Electron Nb coaxial LF Ecc Pass Eccosorb Cryogenic HEMT Mobility Combiner Nb stub Filter Isolator Transistor

Qubit Flux Bias 4 K mK - + LF Ecc Nb 10 kOhm 250 MHz Qubit Control

I I AWG LO R Al box Q Q fabricated chip quantum circuit 20 dB 20 dB HF Ecc 10 dB Cavity Readout 12 GHz I I AWG LO R Q classical Q circuit 12 GHz

I I ADC LO R HEMT Q Q

DC/SFQ Bias

- + LF Ecc Nb 1 kOhm 2 GHz

10 kOhm

ECL microwave DC/SFQ Trigger switch 20 dB 20 dB

12 GHz

Figure 5-14: Wiring diagram for initial testing. The control of the SFQ tone was done with a simple ECL triggered microwave switch. Standard single side band mixing was utilized to on-demand create and shape the readout and qubit drive tones. These tones were coupled together at room temperature via a 3 dB splitter and fed into the DR. 128

average qubit performance measured across many diﬀerent samples. The average measured ¯ ¯∗ ≈ relaxation and dephasing times across all samples were T1 = T2 5 μs. After standard qubit manipulation and characterization with microwaves had been estab- lished, basic functionality of the SFQ–qubit driving scheme was tested. SFQ driven Rabi oscillations were measured by applying qubit transition frequency subharmonic drives to the dc/SFQ circuit and dispersively measuring the qubit state. Subharmonic drives were required due to microwave bleed through of the dc/SFQ driver, preventing the turning oﬀ coherent driving of the qubit even for no DC bias current applied to the dc/SFQ circuit. The false color image in Figure 5-16(a) displays the qubit occupation (false color) as a function of SFQ drive time (y–axis) and SFQ bias current (x–axis) for a drive frequency

ωD = ωq/3. Above approximately 90 μA of bias current, fast oscillations of the qubit population can be seen for times T< 500 ns. The corresponding dc/SFQ drive IV curve is displayed below the false color image. A clear Shapiro step can be see at the same bias current indicating the onset of consistent in–phase phase slips of the driver circuit. Figure 5-16(b)

(a) (b)

1 1

1 0.5 1 0.75 P P

0 0.5 20 60 100 20 60 100

Figure 5-15: (a) Measured (black diamonds) exponential decay of SFQ–qubit P|1 lifetime. The extracted T1 (red line) was 23.6±0.6 μs. (b) Measured (black diamonds) dephasing time T2∗. The extracted coherence time (red line) for the qubit was 24.4±0.8 μs. Average ¯ ¯∗ ≈ times for all measured qubits were T1 = T2 5 μs. 129

displays another scan focused on the driver bias turn–on region. The turn–on of coherent driving of the qubit is seen to correspond 1–to–1 with the onset of a Shapiro step in the driver IV curve. An important note is that the oscillations are stable with respect to the bias current over a range of approximately 15 μA allowing for bias noise resistant performance. With the data taken in Figures 5-16(a-b) indicating appropriate dc/SFQ bias and mi- crowave drive parameters, more standard SFQ based qubit metrology could be performed. Figure 5-16(c) displays the classic chevron pattern of Rabi oscillations as a function a SFQ

drive frequency again centered at ωq/3. Ramsey fringes as a function of SFQ drive frequency are displayed in Figure 5-16(d). It is worth noting that the chevron pattern in Figure 5-16(c) and the Ramsey fringes in Figure 5-16 denote diﬀerent qubit transition frequencies with

the center of the chevron pattern suggesting ωq/3=1.650 GHz and the Ramsey fringes

ωq/3=1.653 GHz, the latter of which is the true third subharmonic of the qubit frequency. While more thoroughly addressed in section 5.4.6, we brieﬂy mention here that continuous driving of the dc/SFQ circuit as is done in the Rabi chevron experiment consistently gen- erates quasiparticles that add a complex dissipative loss channel [87] to the qubit shifting it’s fundamental frequency. The asymmetry in the 2D Ramsey experiment in Figure 5-16 displays the qubit frequency shifting back to its natural resonance after an interleaving I gate between SFQ driven X/2 gates of length T = 200 ns. While phase control of the SFQ drive tone was not aﬀorded by the initial wiring displayed in Figure 5-14, the high bandwidth of the microwave switch allowed for exploration of deep subharmonic qubit dynamics. Figure 5-17(a) displays the results of a 36 hour scan over which the drive frequency to the dc/SFQ circuit was swept from 0.48–1.25 GHz and the qubit state monitored for 500 ns. Coherent oscillations of the qubit state can be seen for a multitude of

frequencies ωd = ωq/n for subharmonic number n ≥ 4. With coherent subharmonic dynamics seen down to below 1 GHz, direct digital synthesis (DDS) was utilized to look at the qubit dynamics at deeper subharmonic frequencies. Figure 5-17(b) displays the coherent Rabi

oscillation of the qubit when irradiated with pulses from the driver driven at ωq/41 ≈ 125 130

(a) (b) 4.0 500 3.5 0.8 3.0 400 0.6 2.5 300 2.0 1.5 200 0.4 Time ( s) Time Time (ns) Time 1.0 100 0.2 0.5 0 0 -2.324 -2.328 4 (mV) ( V) -2.332 2 -2.336 V Driver -2.340 V 0 50 60 70 80 90 100 110 120 90 100 110 120

(c) (d)

500 500 0.8 0.8 400 400 0.6 0.6 300 300

200 0.4 200 0.4 Time (ns) Time Time (ns) Time 100 100 0.2 0.2 0 0 1.64 1.645 1.65 1.655 1.66 1.648 1.65 1.652 1.654 1.656

Figure 5-16: (a) Qubit occupation as a function of time and dc/SFQ driver bias current for a SFQ drive frequency of ωd ≈ ωq/3. The corresponding uncorrected driver IV curve is displayed underneath the false color plot. For bias currents Ib > 90μA, coherent oscillations of the qubit state can be seen. (b) Zoom-in on the driver bias region between 90 and 125 μA. Again the turn on of fast Rabi oscillations can be seen to coincide with the onset of a Shapiro step in the IV curve of the dc/SFQ driver displayed below the false color plot. The magnitude of the step was measured to be 3.46 μV in direct relation to the drive frequency. (c) Rabi chevron oscillations as a function of the dc/SFQ drive frequency centered at ωq/3. (d) Ramsey oscillations as a function of the dc/SFQ drive frequency centered again at ωq/3. 131

MHz. Discrete steps in the occupation can be seen every 1/125×10−6 s = 8 ns corresponding the application of a single SFQ pulse being applied to the qubit once a drive period. At the time of this writing, to our knowledge, this is the ﬁrst demonstration of truly digital control of a qubit state where, as previously mentioned, the precision of control is now set lithographically through the coupling capacitor between the digital circuit and the quantum circuit; lower the capacitance, increase the precision (with the cost paid for in gate time). Subharmonic control also opens the door to rapid fundamental transition frequency determination. In contrast to the procedures outlined in section 3.2.3 where a saturating microwave tone is swept in frequency looking for a change in the readout tone level, a swept SFQ tone positioned in the deep subharmonic frequency band would produce a forest of peaks (or dips, depending on the read out resonator scheme); These peaks provide a spectroscopic signature of the qubit fundamental transition. To look at this more concretely, we deﬁne the consecutive qubit subharmonics as

ω ω = q (5.24a) n n ω ω = q . (5.24b) n+1 n +1

Dividing the above equations by one another

ω n +1 n = . (5.25) ωn+1 n

Straightforward algebra ﬁnally yields

ω n = n+1 . (5.26) ωn − ωn+1

The ramiﬁcation of Eq. (5.26) is that the measurement of any 2 consecutive qubit subharmonic

frequencies provides the subharmonic number relating back to ωq. Applying this to the data 132

500 P (a) 0.9 1 450 0.8 400

350 0.7

300 0.6 250

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Figure 5-17: (a) Qubit oscillations as a function of SFQ drive frequency. Subharmonic Rabi oscillations as low as ωq/10 = 498 MHz can be seen. (b) Quarter period Rabi oscillation for an SFQ drive frequency of ωq/41 ≈ 125 MHz. The qubit occupation measurement sampling frequency was set to 1 GHz. (inset) Close–up view of quantized steps in the Rabi oscillation. For a 125 MHz SFQ drive, 8 points were recorded per step in the occupation corresponding to the application of 1 SFQ pulse every T =1/125 × 10−6 = 8 ns. 133

displayed in Figure 5-17(a), we see coherent Rabi oscillations at roughly both 1.24 GHz and 0.98 GHz. Substituting these numbers into Eq. (5.26) yields a value for n of approximately 3.8. Rounding this up to the next whole integer and multiplying by 1.24 GHz yields a

fundamental qubit frequency of ωqb ≈ 4.96 GHz, directly in line with the qubit spectroscopy data for this sample displayed in Figure 3-6.

5.4.4 Establishing Orthogonal Axis Control

While the simple turning on and oﬀ of the microwave drive to the dc/SFQ converter allowed for coherent oscillations of the qubit and quantized steps in the Rabi oscillation to be realized, it did not aﬀord phase control over the SFQ drive tone. To realize phase control of the SFQ drive tone, the microwave switch was removed in favor for a standard single side band mixing protocol as outlined in blue in Figure 5-18. Revisiting traditional orthogonal axis control with shaped microwaves, we reprint the equation detailing to on resonance interaction between a single side band mixed tone and a qubit in the rotating frame

Γ(t) V = [cos(γ)σ + sin(γ)σ ] , (5.27) I 2 x y where Γ(t) is the time varying amplitude of the mixed side band tone and γ it’s phase. Extending this treatment to driving with SFQ pulses as opposed to shaped microwaves is straightforward if one considers the dynamics in the time domain. Figure 5-19 displays circuit diagrams a SFQ driven X gate and a SFQ driven Y gate. By adjusting the phase of the mixed side band tone, a variable timing oﬀset develops between SFQ driven X and Y rotations. To measure and conﬁrm SFQ gate orthogonality in the lab, an SFQ pulse sequence was

∗ applied to the qubit very similar to a T2 measurement only instead of the intermittent period between subsequent X/2 gates being ﬁlled with an idle gate, a variable length and phase SFQ driven Y gate was applied. Figure 5-20(a) displays the SFQ pulse timing diagram of

the orthogonality experiment. A XSFQ/2 pulse is applied to the qubit to set the control axis. 134

Variable 12 GHz Low 20 dB I 20 dB 1 GS/s Attenuator Pass Filter Attenuator I Arbitrary 12 GHz 1 GS/s AWG 10 dB Waveform Microwave 10 dB ADC Analog-to-Digital Generator Generator Attenuator Converter 20 GHz Low Q I HF Ecc Pass Eccosorb Q IQ Voltage LO R Filter - + Mixer Source Q 3 dB 300 MHz Low High Electron Nb coaxial LF Ecc Pass Eccosorb Cryogenic HEMT Mobility Combiner Nb stub Filter Isolator Transistor

Qubit Flux Bias 4 K mK - + LF Ecc Nb 10 kOhm 250 MHz Qubit Control

I I AWG LO R Al box Q Q fabricated chip quantum circuit 20 dB 20 dB HF Ecc 10 dB Cavity Readout 12 GHz I I AWG LO R Q classical Q circuit 12 GHz

I I ADC LO R HEMT Q Q

DC/SFQ Bias

- + LF Ecc Nb 1 kOhm 2 GHz

10 kOhm

DC/SFQ Trigger

I I AWG LO R 20 dB 20 dB Q variable based 12 GHz Q on desired tone

Figure 5-18: Wiring diagram for advanced SFQ–qubit testing including qubit state rotation orthogonality and randomized benchmarking of SFQ gates. The diﬀerence between this diagram and that displayed in Figure 5-14 is highlighted in blue. 135

Afterwards, a YSFQ gate of variable time t and phase γ is applied followed by another XSFQ/2 gate. Figure 5-20(b) displays overdetermined tomograms for SFQ subharmonic drives at the 3textrd and 41st subharmonic. In both cases when t = 0, the pulse sequence equates to a single

XSFQ gate. For the n = 3 case, there are 3 axes / values of γ for which orthogonality can be achieved. This over-determination of axis orthogonality is simple to see when plotting the phase evolution of two sine waves with frequencies 3 times that of each other as done on the left hand side of Figure 5-20(c). Over the ﬁrst π–phase evolution of the each wave, the phases of the two waves align 2 times (black dots) indicating parallelism which results in the continuous coherent driving of the qubit (blue lobes in LHS tomogram, 5-20(b)). The right hand side of Figures 5-20(b) and 5-20(c) show the same dynamics only at a subharmonic drive of n = 41. Now, instead of 3 orthogonal axes, there are correspondingly 41 orthogonal values of 0 ≤ γ<πin which YSFQ may be set to. The signal overlaps (black dots) shown on the RHS of Figure 5-20(c) indicate the phase matching of the qubit and SFQ drive tones.

I I AWG LO R dc/SFQ Q Q

AWG dc/SFQ

Q

Figure 5-19: Single side band mixing allows for direct phase control over the output tone from the IQ mixers. These out of phase tones (red / blue) induce timing shifts between pulses for what is deﬁned as a SFQ driven X gate and a SFQ driven Y gate. Once a particular phase / clock is set for the SFQ X gate, setting the SFQ Y gate is simply a matter of changing the phase γ of the side band tone. 136

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0 Amplitude (arb. units) Amplitude (arb. units)

Phase (rad) Phase (rad)

Figure 5-20: (a) Gate sequence to test SFQ based qubit gate orthogonality. Two SFQ based XSFQ/2 rotations were separated by a variable length and phase YSFQ gate. (b) Overdetermined tomograms for driving the qubit at ωd = ωq/3andωd = ωq/41. Values of γ = m × π/3(m × π/41) were in phase with the surrounding XSFQ gates and thus continued driving the qubit state along those axes. (c) diagrammatic representation of the phase evolution of the qubit verses that of the drive frequency. The LHS plot shows 2 phase overlaps (black dots) of the qubit evolution (red) and the drive evolution (blue) between 0 ≤ γ<π. The RHS plot shows the 40 overlaps between the qubit evolution and the drive tone again for 0 ≤ γ<π. 137

5.4.5 Randomized Benchmarking of SFQ–Qubit Gates

The combination of coherent qubit state driving with orthogonal axis control allowed for the establishment of SFQ based single qubit gates. Prior to characterization with IRB (see section 3.3.2), it was thought prudent to secure the fact that all 24 single qubit Cliﬀords could be realized. To test this, each Cliﬀord gate was applied and subsequently inversed. The |0 state population was then measured and taken as a measure of the ﬁdelity of applying only the one gate. Mathematically, the experiment took the form

CC−1 ≤ 1 , (5.28) where SPAM errors were not explicitly corrected for. Resulting ﬁdelities for the application of and subsequent inverse of the 24 single qubit Cliﬀords are displayed in Figure 5-21 for the n =3,n =4,andn = 39 subharmonic. It is important to note that while traditionally the diﬀerence between a π and a π/2 rotation when utilizing microwaves is in the amplitude of the

pulse, for SFQ based gates the diﬀerence is the length of the pulse sequence, i.e. τX =2×τX/2. The ﬁdelities averaged over all 24 Cliﬀords were 74±8%, 65 ± 11%, and 71 ± 8% for the n =3,n =4,andn = 39 subharmonic, respectively. While promising in it’s own right that the entire single qubit Cliﬀord gate series could be realized with SFQ pulses, more investigation was required into why the overall ﬁdelities for the application and inverse of single gates were so low. To do this, T1 measurements of the qubit were performed utilizing standard microwave rotations but with an oﬀ–resonant SFQ drive applied while the measurement was in progress. Figure 5-22 displays the relaxation of the qubit from the |1 state for both no SFQ signal applied during the experiment and for a detuned, 2 GHz SFQ drive. For SFQ drive lengths of 250 ns, the occupation of the qubit reduces quickly to a value of ∼ 0.7 before falling oﬀ exponentially towards a background level of ∼ 0.2. This initial fall oﬀ of the occupation for the background SFQ signal explains not just the average ﬁdelity achieved for the Cliﬀords displayed in Figure 5-21 but also the gate length dependency of 138

1

0.9

0.8

0.7

0.6

0.5

0.4 Sequence Fidelity 0.3

0.2

0.1

0 I,I X/2,-X/2 -X/2,X/2 X,X Y/2,-Y/2 -Y/2,Y/2 Y,Y -X/2,Y/2,X/2,-X/2,-Y/2,X/2-X/2,-Y/2,X/2,-X/2,Y/2,X/2Y,X,Y,X Y,X/2,Y,X/2Y,-X/2,Y,-X/2X,Y/2,X,Y/2X,-Y/2,X,-Y/2X/2,Y/2,X/2,X/2,Y/2,X/2-X/2,Y/2,-X/2,-X/2,Y/2,-X/2Y/2,X/2,-X/2,-Y/2Y/2,-X/2,X/2,-Y/2-Y/2,X/2,-X/2,Y/2-Y/2,-X/2,X/2,Y/2X/2,Y/2,-Y/2,-X/2X/2,-Y/2,Y/2,-X/2-X/2,Y/2,-Y/2,X/2-X/2,-Y/2,Y/2,X/2

Gate Sequence

Figure 5-21: The raw ﬁdelities of the 24 single qubit Cliﬀords as applied by driving the SFQ driver at the n =3,n =4,andn = 39 subharmonic. The average gate ﬁdelity for the n =3 subharmonic was 74±8%. A trend between sequence ﬁdelity and overall gate length in time can be seen with gates requiring full rotations (and thus driven for longer) having lower overall ﬁdelity.

that ﬁdelity. The inset in Figure 5-22 displays a high density time series plot of the qubit T1 probability for both no background SFQ signal and with it. With the background SFQ signal applied, the max obtainable probability falls oﬀ linearly with time for the ﬁrst 250 ns. For n = 3, the average length of all the gates (including their inversions) shown in Figure 5-21 is 76 ns. The corresponding max obtainable ﬁdelity for a oﬀ–resonant SFQ pulse at ∼ 75 ns is

80%. The average obtained ﬁdelity for the ωd = ωq/3 driven Cliﬀords was 75%. A natural explanation for the sudden loss of occupation is that the driver is pushing the qubit to higher states. However, this is not seen as only two qubit state IQ distributions are ever resolved during the T1 experiments. Instead, what is seen is a spurious increase in the |0 state population over the ﬁrst 250 ns of the SFQ drive, indicative of quasiparticle generation and subsequent heating. This put an overall limit on the max obtainable sequence 139

0.9 0.95

0.9

0.85

0.8 0.8

0.75

0.7 0.7 0.65 0 50 100 150 200 250 300 350 Time (ns) 0.6

0.5

0.4

No BG SFQ 0.3 W/ BG SFQ

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Time ( s)

Figure 5-22: Qubit T1 experiment with either the SFQ driver oﬀ (blue curve) or driven during the entire sequence at ωSFQ =2π × 2 GHz. For the driver on, P|1 falls oﬀ immediately to below 0.7 with the background level rising to above 0.2. (inset) First 350 ns of P|1 relaxation experiment. A fast linear decrease in the occupation can be seen for the ﬁrst 250 ns.

ﬁdelity for the interleaved RB experiments to be discussed in the following sections. As discussed in section 3.3.2, interleaved RB provides a protocol to actively test the ﬁdelity of single gate operations by looking at the overall change in the sequence ﬁdelity as a function of the number of Cliﬀord gates executed m. A subtle but important change to implementing interleaved RB for SFQ based gates was the elimination of the identity or idle gate I. The dynamic qubit frequency apparent in the asymmetry of the 2–dimensional Ramsey plot of Figure 5-16(d) but not in the Rabi chevron plot of Figure 5-16(c) meant that in order to stay on resonance with the qubit throughout the m Cliﬀord gate long sequence, no idle time could be aﬀorded allowing the qubit to relax back to it’s unperturbed fundamental transition frequency. Figure 5-23 displays interleaved RB data for both the n =3andn = 41 subharmonics. Fidelities F≥90% were achieved for all applied single qubit gates at both subharmonics (see Table 5.3). As discussed above, the best obtainable sequence ﬁdelity for low Cliﬀord number was limited by quasiparticle heating with a maximum value F≈70%. The combination 140

(a)

0.7 0.7 Interleaved Gate: None Interleaved Gate: None Interleaved Gate: X Interleaved Gate: Y 0.65 0.65 Interleaved Gate: X/2 Interleaved Gate: Y/2

0.6 Interleaved Gate: -X/2 0.6 Interleaved Gate: -Y/2

0.55 0.55

0.5 0.5 Sequence Fidelity Sequence Fidelity 0.45 0.45

0.4 0.4

0.35 0.35

24681012141618 2 4 6 8 10 12 14 16 18 Number of Cliffords Number of Cliffords (b)

0.65 Interleaved Gate: None Interleaved Gate: None 0.65 Interleaved Gate: X Interleaved Gate: Y 0.6 Interleaved Gate: X/2 Interleaved Gate: Y/2 Interleaved Gate: -X/2 0.6 Interleaved Gate: -Y/2

0.55 0.55

0.5 0.5 Sequence Fidelity Sequence Fidelity

0.45 0.45

0.4 0.4 12345678 12345678 Number of Cliffords Number of Cliffords

Figure 5-23: (a) Interleaved randomized benchmarking of SFQ driven qubit gates at ωSFQ = ωq/3. X–rotations are displayed in the left–hand column and Y–rotations in the right–hand column. Gate ﬁdelities greater than 90% were achieved for all 6 SFQ gates. (b) Interleaved randomized benchmarking of SFQ driven qubit gates at ωSFQ = ωq/41. Again, gate ﬁdelities above 90% are achieved. The ∼ 70% max ﬁdelity for low Cliﬀord number is due to heating issues stemming from running the SFQ driver discussed in the text.

of long gate times (τgate > 150 ns) and limited on–board DAC memory allowed for only 8 interleaved Cliﬀords to be applied at the n = 41 subharmonic necessitating less precision and larger error in the ﬁts.

5.4.6 SFQ Generated Quasiparticle Studies

As previously mentioned, the max gate sequence ﬁdelity and invariably the gate ﬁdelities themselves were limited by quasiparticle generation due to running of the dc/SFQ driver. But, how can we say this with such assuredness? To deﬁnitively prove that it was in fact 141

Gate Fn=3 Fn=41 X 94.0 ± 1.0% 91 ± 2% X/2 95.5 ± 0.8% 93 ± 3% -X/2 95.7 ± 0.5% 95 ± 3% Y 93.9 ± 0.5% 94 ± 3% Y/2 96.9 ± 0.6% 92 ± 3% -Y/2 94.6 ± 0.6% 95 ± 2% Table 5.3: SFQ gate ﬁdelities determined with interleave randomized benchmarking for the n =3andn = 41 subharmonic.

QPs limiting the operation ﬁdelity, further experiments had to be carried out. Quasiparticles relax through two main physical channels; They either recombine or are trapped [88]. These

two physical mechanisms lead to two diﬀerent rates at which the average QP density nqp in a system decays

dn qp = −an2 − bn . (5.29) dt qp qp

The quadratic and linear terms model recombination and trapping, respectively. Dropping the recombination term yields a simple exponential decay of the quasiparticle density

−bt nqp(t)=nqp(0)e . (5.30)

The qubit energy relaxation rate Γ is directly proportional to the number of QPs in the vicinity of the qubit with the form

Γ(t)=cnqp(t)+Γ0 . (5.31)

The rate of decay of the qubit excited state is then

dP|1 =Γ(t)P| , (5.32) dt 1 which, when integrated, yields a super–exponential form [89] for P|1 142

nqp(exp[−t/Tqp]−1) −t/T1 P|1 = e e , (5.33)

where nqp is the average number of QPs, Tqp is the relaxation induced by 1 QP, and T1 the

relaxation time associated with all other processes. With this direct dependency of P|1 on

the number of QPs, an experiment could be devised to extract the nqp as a direct result of running the dc/SFQ driver. Figure 5-24(a) displays the measurement protocol for extracting the average number of QPs generated during a variable time, oﬀ–resonant SFQ drive applied prior to a microwave

π–rotation of the qubit. The protocol is simply a standard microwave T1 experiment run

after a SFQ poisoning pulse. Figure 5-24(b) displays the P|1 decay probability (false color) as a function SFQ drive time. The eﬀect of running the SFQ driver for only a few microseconds is seen immediately in the increase in decay. These curves were subsequently ﬁt with Eq.

(5.33). The extracted nqp for the varying SFQ drive time is displayed in Figure 5-24(c). The extracted background QP number was 0.10 for no applied poisoning pulse saturating up to 2.1 for poisoning times τSFQ > 45 μs. The inset of Figure 5-24 displays the ﬁrst 5 μs where the time axis has been scaled by ωSFQ = 133 MHz so that nqp could be displayed

as a function of the number of SFQ phase slips. The background nqp =0.1 obscures the dynamics for the ﬁrst 200 phase slips after which a linear rise in the extracted QP number is seen with slope (1.6 ± 0.2) × 10−3 QPs coupling to the qubit per dc/SFQ driver phase slip. With some light now shed on the generation of QPs from the running of the dc/SFQ driver, measuring their relaxation properties was subsequently performed. Figure 5-25(a)

displays the QP relaxation / T1 recovery experimental protocol. A poisoning pulse of ﬁxed

length was applied followed by variable recovery time after which a standard T1 measurement

protocol was performed. Figure 5-25(b) displays the P|1 decay (false color) as a function of recovery time. The recovery saturates at approximately 70 μs. Figure 5-25(c) displays the

ﬁtted nqp as a function of recovery time. The QP decay constant b from Eq. (5.30) was ﬁtted to a value of b−1 =17.6 ± 0.3 μs. The inset shown in Figure 5-25(c) shows the same 143

(a) SFQ

(b) (c) 2.5 100 0.9 0.8 2 80 0.7 1.5 60 0.6 1.6(2)e-3 0.6 0.5 1 40 0.4 0.4 0.2 20 0.3 0.5 0 0 200 400 600 0.2 0 Phase Slips

Readout Delay (us) 0 0 1020304050 0 1020304050 SFQ Drive Time (us) SFQ Drive Time (us)

Figure 5-24: (a) Experimental protocol to extract nqp from running the dc/SFQ driver. An oﬀ–resonant SFQ drive at frequency ωSFQ = ωqb/41 + δ is performed for a variable length of time τSFQ immediately followed by a microwave π–pulse. A variable length of idle time τRO is then between the π–pulse and measurement. (b) Data on a series of T1 experiments as a function of oﬀ–resonant SFQ drive time. The P|1 probability decay (false color) was measured out to 110 μsforSFQdrivesoutto50μs. (c) Extracted nqp from the ﬁts to the data in (b). The number of QP saturates at approximately 45 μsatnqp≈2.1. (Inset) Zoom in view of the ﬁrst 5 μs scaled by the SFQ drive frequency. The extracted linear ﬁt to the data, once out of the background, yielded a coupling of 1.6 × 10−3 QPs/phase slip to the qubit.

data as in (c) but plotted on a semi–log scale. The linearity of the data when plotted this way conﬁrms that the QP decay is due to recombination solely and the earlier dropping of the quadratic term in Eq. (5.29) was justiﬁable. Just as in the Mattis and Bardeen work concerning thermal QPs, non–equilibrium QPs also give rise to a complex impedance [87]; The real part responsible for the additional loss discussed and measured above with the complex contribution giving rise to a frequency shift of the qubit [90, 91]. To study the frequency eﬀects of the SFQ generated non-equilibrium

QPs, a ω = ωq − 5MHz detuned Ramsey fringe experiment was run preceded by an SFQ poisoning pulse of variable length. The pulse timing sequence / experimental protocol is outlined is Figure 5-26(a). A detuning of 5 MHz was chosen so that many Ramsey fringes could be measured within the pulse sequence timing / memory limits of the DAC/ADC 144

(a) SFQ (b) (c)

0.9 3 100 0 10 0.8 80 0.7 2

60 0.6 -1 10 20 40 60 80 100 0.5 40 0.4 1

20 0.3

0.2

Readout Delay (us) 0 0 0 20 40 60 80 100 20 40 60 80 100 120 Recovery Time (us) Recovery Time (us)

Figure 5-25: (a) Experimental protocol to examine the relaxation of QPs generated from the dc/SFQ driver. The driver was run for a nqp saturating length of 50 μs producing a background QP level nqp∼2.5 followed by a variable recovery period before a standard T1 measurement was performed. (b) Data on a series of T1 experiments as a function of recovery time after the QP saturating SFQ drive. (c) Extracted nqp from the ﬁts to the data in (b) as a function of recovery time. Eq. (5.30) was ﬁt to the data with a decay constant b−1 =17.6 ± 0.3 μs. (Inset) Decay of the average QP number plotted on semi–log axes. The linearity conﬁrms that the dominant QP relaxation mechanism was trapping.

hardware allowing for low–error ﬁts. The measured Ramsey fringes as a function of SFQ drive time are displayed in Figure 5-26(b). As the SFQ drive time increased, the max visibility of the fringes decreased and a slight broadening of the oscillations could be seen in addition to slight frequency shifts. Figure 5-26(c) displays the ﬁtted Ramsey fringe frequencies as a function of SFQ drive time. After an initial turn–on period of approximately 50 μs, the qubit frequency and hence the detuning shifts down linearly with the SFQ drive time. In the limit where qubit relaxation is solely dominated by non–equilibrium QPs, the latter term in Eq. (5.31) becomes negligible and the relaxation rate reduces to

Γ(t) → cnqp(t) . (5.34) 145

(a) SFQ (b) (c) 4.5 0.9 4.995 4.0 4.990 0.8 3.5 4.985 0.7 4.980 3.0 0.6 4.975 2.5 4.970 0.5 2.0 4.965 1.5 0.4 4.960 1.0 0.3 4.955 Delay Time (us) 0.5 4.950 0.2 Ramsey Detuning (MHz) 4.945 0 0 100 200 300 400 500 600 700 0 100 200 300 400 500 600 700 SFQ Drive Time (ns) SFQ Drive Time (ns)

Figure 5-26: (a) Experimental protocol to examine induced qubit frequency shifts from non–equilibrium QPs. Prior to a detuned Ramsey fringe experiment, a variable length SFQ poisoning pulse was run. (b) 5 MHz qubit Ramsey fringes as a function of oﬀ–resonant SFQ drive time. A decrease in the max visibility along with a slight broadening and expansion of the fringes are commensurate with longer SFQ drives. (c) Fitted detuning frequency as a function of SFQ drive time. After approximately a 50 μs delay, a linear shift in the qubit frequency can be seen as a function of SFQ drive time.

The prefactor, determined from Fermi’s golden rule and the complex QP impedance, is given by

ω 2Δ c = q (5.35) π ωq

Additionally, the frequency shift of the qubit δωq is calculated to be (see [91, 92]) directly proportional to the QP density as well with form

⎡ ⎤ nqpωq ⎣ 1 2Δ ⎦ δωq = − +1 . (5.36) 2 π ωq

Combined, Eq. (5.34) and Eq. (5.36) provide a fundamental, materials constant only relation 146

⎡ ⎤ δω 1 ω q = − ⎣1+π q ⎦ ≡ m (5.37) Γ 2 2Δ

Figures 5-27(a–b) display two diﬀerent experimental protocols for testing the relations

in Eq. (5.37). In 5-27(a), an oﬀ–resonant (ωSFQ = ωq/3 − δ =1.6 GHz) SFQ drive is applied for a variable amount of time followed immediately by either a qubit dephasing or relaxation experiment. For 5-27(b), the length of the oﬀ–resonant SFQ drive is held constant followed by a variable recovery period before performing either the dephasing or relaxation experiment. The results of the variable poisoning and recovery time experiments are displayed in Figures 5-27(c) and 5-27(d), respectively. For the variable poisoning length, a linear relationship between the relaxation rate Γ and number of SFQ phase slips was measured with a commensurate linear decrease in the qubit frequency where the explicit detuning ∗ ∼ for the T2 measurement was 5 MHz. Data in Figure 5-27(d) displays an exponential decrease in the QP induced relaxation as a function of recovery time. This is to be expected

as the contribution from the leading term in Eq. (5.33) is exponential in nqp which was measured to decrease exponentially as a function of time (see Figure 5-25). The return of the Ramsey frequency to the purposefully detuned 7.5 MHz can be seen to coincide as well with the decrease in nqp. Figures 5-27(e) and 5-27(f) plot the extracted Ramsey detuning against the extracted relaxation rates for the poisoning and recovery experiments,

respectively. For both experiments, the predicted linear relationship between dωqb and Γ can be seen albeit with diﬀerent slopes. The ﬁtted slope for the poisoning experiment was

mPoison = −5.4 while the ﬁtted slope for the recovery experiment was mRecovery = −1.313 . When substituted back into Eq. (5.37), neither of these values produce a reasonable estimate to the Al gap energy of Δ ∼ 160 μeV. The variable length poisoning experiment predicts a

gap energy of ΔPoison =1μeV while the recovery experiment predicts a closer yet still low

ΔRecovery =37μeV. Exactly why the two rates diﬀer from each other is currently not yet known. The data however is not an outlier as experiments done at Yale also looking at QP poisoning also saw a 147

(a) (b) recovery poison poison (c) (d) 5.00 110 7.53 500

7.51 450 4.99 100

7.49 400 Relaxation Rate (kHz) Relaxation Rate (kHz)

4.98 90 7.47 350

4.97 80 7.45 300

7.43 250 4.96 70 7.41 200

60 Ramsey Detuning (MHz) Ramsey Detuning (MHz) 4.95 7.39 150

7.37 100 4.94 50 7.35 50 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 0 50 100 150 Number of Phase Slips Recovery Time (us) (e) 0 (f) 0 150 4.5 -100 -50 ) 4.0 Number of Phase Slips -1 ) -1 -200 3.5 Recovery Time (us) -100 100 3.0 -300 -150 2.5 -400

-200 2.0 -500 50 1.5 -250 -600 1.0 Qubit Shift (rad. ms Qubit Shift (rad. ms -300 -700 0.5

-350 0 -800 0 0 5 10 15 20 25 30 35 40 45 50 55 0 50 100 150 200 250 300 350 400 450 Relaxation Rate (kHz) Relaxation Rate (kHz)

Figure 5-27: (a) QP Poisoning experiment protocol to study the relationship between qubit state relaxation Γ and frequency shift δωq. (b) Complimentary experiment to again study the relationship between Γ and δωq by measuring the relaxation properties of each after a ﬁxed length QP poisoning pulse. (c) QP poisoning / shifting of the decay rate Γ and qubit frequency. A linear increase (decrease) was measured in the qubit relaxation (detuning). (d) QP relaxation data. The qubit relaxation rate was seen to decrease exponentially in time in line with the exponential decay in average QP number measured earlier. (e) Measured QP induced qubit frequency shift δωq Vs. Γ for the poison experiment. The ﬁtted slope was mPoison = −5.4 ± 0.09. (f) Measured QP induced qubit frequency shift δωq vs. Γ for the recovery experiment. The ﬁtted slope was mRecovery = −1.313 ± 0.006. 148

linear trend between δωq and Γ albeit with a slope corresponding to too large of a gap energy.

It is worth noting that plotting the absolute qubit frequency change δfq = δωq/2π against the change in Γ for the poisoning experiment predicts a value of m = −0.891 corresponding to an Al gap energy Δ = 165 μeV; This value is much more in line with direct tunnel–junction gap measurements [93].

5.5 Next Generation Design and Outlook

With the demonstration of coherent qubit control utilizing SFQ pulses now having been demonstrated and the limits to the ﬁdelity of that control being thoroughly investigated, the hybrid SFQ–qubit experiment has been warmed up and put out to pasture in it’s current incantation. We describe here brieﬂy plans currently in action towards realizing the next generation of SFQ–qubit control.

5.5.1 Flip Chip Design

As previously mentioned, one of if not the main limitation to the ﬁdelity of the initial SFQ control scheme was the on–chip generation of QPs from the Cooper pair breaking inherent to the operation of the dc/SFQ converter circuit. In order to combat this limitation, the next generation design will acoustically decouple the dc/SFQ driver from the quantum circuit via a multi–chip module (MCM) ﬂip–chip architecture. Recent work [94] has shown that QP poisoning over large (several cm) distances can be mediated by long–range phonons produced by QP recombination at one physical locale breaking Cooper pairs at another remote location. The introduction of an acoustic mismatch / decoupling between the dc/SFQ driver and the quantum circuit will serve to mediate if not wholly eliminate QP poisoning of the qubit. Figure 5-28(a) displays the circuit layout design of the proposed and currently in fabrication ﬂip–chip experiment. An 8×8mm2 control chip (green) will house all of the microwave readout, qubit control, and SFQ electronics. Enough lines are aﬀorded to have two redundant 149

experiments per MCM. Indium bump bonds (blue squares) provide galvanic connection between the two ground planes of the control and “quantum” chip (orange). Beyond connecting the ground planes of the two chips, no galvanic signal transmission is currently planned. Figure 5-28(b) displays a close–up of the microwave feed line on the control chip and the shorted inductive coupler of one of the quarter-wave CPW resonators on the quantum chip. Mutual coupling M across the ﬂip–chip gap can be tuned by simply adjusting the length of the coupler. Figure 5-28(c) displays a zoomed–in view of the resonator–qubit coupling plus the output of the dc/SFQ driver and ﬂux bias line. Again, capacitive or inductive coupling across the air gap between the chips will facilitate signal transfer. Figure 5-28(d) shows a

close–up view of the Hypres designed dc/SFQ driver. A single bias current rail Ib supplies the necessary current to both the dc/SFQ driver and the output JTLs. The input microwave

current RFSFQ drives phase slips in the dc/SFQ converter which are then multiplexed into two separate outputs by the pulse splitter (PS) circuit. The output SFQ pulses are relegated to ∼ either qubit by switch currents IS1 and IS2 which can be turned on by a fast ( 1 ns) current supply. The SFQ pulses are then sent through a multi–stage JTL feeding a transmission line to the qubit. An interesting mode operation aﬀorded by driving the qubit with SFQ pulses is over

clocking of the microwave drive tone to harmonics of the qubit frequency ωSFQ =4× ωq. Without any form of gating, in this mode of operation pulses from a dc/SFQ driver would arrive when the qubit vector was aligned with the ±xˆ and ±yˆ axes of the Bloch sphere. This by itself would produce a zero–sum rotation as two of the discrete rotations would be in phase with the other two out. However, replacing the simple dc/SFQ driver instead with

a bit–shift register slave to a clock frequency ωCLK =4× ωq where subsequent SFQ bit pairs were alternating high or low, qubit gate times could be halved as compared to their on–resonance counterparts. 150

(a)

(b)

5 mm (c) 8 mm

(d)

(b) (c)

(d)

Figure 5-28: (a) Layout schematic showing the bump bonded ﬂip chip architecture for the SFQ–qubit project. An 8×8mm2 carrier chip (green) houses the microwave and SFQ control hardware for two redundant experiments. Flip–chip bonded to it via In bump bonds (blue) will be a 5×5mm2 “quantum” chip (orange) housing both the qubits and readout resonators. (b) Close–up of the proposed inductive coupling scheme between the common feed line and the resonators. No galvanic connections to facilitate signal ﬂow is planned. Instead, all coupling between the two chips with either be capacitive or inductive. (c) Close–up of the readout resonator–qubit pair plus the output coupling capacitor of the dc/SFQ circuit (bottom, green) and the ﬂux bias line (right, green). (d) Zoom in of the dc/SFQ driver circuit. 151

Appendix A

Josephson Energy of a dc SQUID

In section 2.2, the Hamiltonian for a single unbiased Josepshon junction was derived and in section 2.3.3 it was stated how the transmon qubit was nothing more than a capacitively shunted junction, making the trade oﬀ between charge noise sensitivity an anharmonicity. If, however, the transmon qubit capacitor is shunted by two junctions forming a dc SQUID loop, the energy of the SQUID loop must be taken into account. Looking at just the potential terms for the junctions, the Hamiltonian for a general dc SQUID is

H = −E1 cos(δ1) − E2 cos(δ2) , (A.1) where the subscripts refer to either junction 1 or 2 of the dc SQUID. As a consequence of ﬂux quantization, the phase diﬀerence between the two junctions in constrained by the magnetic ﬂux quantum

2πΦ δ1 − δ2 =2nπ + , (A.2) Φ0 where Φ is an externally applied ﬂux. We deﬁne the total Josephson energy EΣ as the sum of the individual junction energies 152

EΣ = E1 + E2 (A.3)

Substituting (A.2) and (A.3) into the second term of (A.1) and collecting common terms gives

H = −EΣ cos(δ2) − E1(cos(δ1) − cos(δ2)) (A.4) δ + δ δ − δ = −E cos(δ ) − E −2sin 1 2 sin 1 2 (A.5) Σ 2 1 2 2 πΦ = −E cos(δ )+2E sin(δ)sin nπ + (A.6) Σ 2 1 Φ 0 πΦ πΦ = −E cos(δ )+2E sin(δ) sin(nπ)cos +cos(nπ)sin (A.7) Σ 2 1 Φ Φ 0 0 πΦ = −EΣ cos(δ2)+2E1 sin(δ)sin , (A.8) Φ0

where we have deﬁned δ ≡ (δ1 + δ2)/2. This relation, when combined with (A.2) allows for expansion of the ﬁrst term in (A.1)

πΦ πΦ H = −E cos nπ − − δ +2E sin(δ)sin (A.9) Σ Φ 1 Φ 0 0 nΦ πΦ πΦ = −E cos(δ)cos − E sin(δ)sin +2E sin(δ)sin (A.10) Σ Φ Σ Φ 1 Φ 0 0 0 nΦ E − E πΦ = −E cos(δ)cos + 1 2 sin(δ)sin 0 (A.11) Σ Φ E + E Φ 0 1 2 nΦ α − 1 πΦ = −E cos(δ)cos + sin(δ)sin 0 (A.12) Σ Φ α +1 Φ 0 nΦ πΦ = −E cos(δ)cos + d sin(δ)sin 0 (A.13) Σ Φ Φ 0 nΦ πΦ0 = −EΣ cos cos(δ)+d sin(δ)tan , (A.14) Φ0 Φ

where we have deﬁned the ratio between the two Josephson energies α ≡ E1/E2 for E1 >E2 and d =(α − 1)/(α + 1). The ﬁnal step is to note that the bracketed terms in (A.14) are 153

a generalized trigonometric identity relating sin and cos, which, when exploited, yields the ﬁnal form for the energy a general ﬂux biased dc SQUID to be

πΦ 2 2 πΦ H = −EΣ cos 1+d tan . (A.15) Φ0 Φ0

The square root term prevents the energy from diverging as the external ﬂux takes on values

approaching Φ0. In the limit of symmetric junctions E1 = E2 = EJ , d =0,EΣ =2EJ and (A.15) takes on the more familiar form

πΦ H = −2EJ cos . (A.16) Φ0 154

Appendix B

Preserving High Quality Nb

In order to preserve and protect the 180 nm Nb ground plane (layer M0) from 7 additional layers of processing as described in Chapter 5.3.3, sacriﬁcial protection layers were employed to shield the base layer from the subsequent processing. Initially, an 8 nm cap of Al was employed because it could be sputtered sequentially with the Nb base layer without breaking vacuum and Al has a low reactivity with the diﬀerent plasma etch processes used in the processing of subsequent layers. Until the fabrication of the joint SFQ–qubit circuit, the practice of employing Al as a sacriﬁcial cap layer had been used to great success in the fabrication of other multi–layer circuits such as the SLUG microwave ampliﬁer. However, during initial attempts to fabricate the SFQ–qubit circuit, the ﬁnal base layer etch (“Qubit and Resonator Patterning”, Ch.5.3.3) left behind a metallic scum in the pockets where Nb should have been fully etched away. Figure B-1 displays a dark ﬁeld false colored optical microscopy image of the readout resonator–qubit coupling area. The resonator, qubit, and ﬂux bias line are colored yellow, red, and purple, respectively with the surrounding ground plane in blue. The etched pockets between all these features should have been fully devoid of metal leaving only the surface of the Si substrate behind. However, as is evident in Figure B-1, a silvery heterogeneous residue was left behind after the ﬁnal Nb-Al etch. 155

(a)

(b)x 104 (c) 12 14000 O1s Si2p Nb4p Si2s 12000 10 C1s Al2s Al2p Nb3d 10000 8 Nb3p 8000 6 6000 Counts / s Counts / s 4 4000

2 2000

0 0 1200 1000 800 600 400 200 0 200 150 100 50 0 Binding Energy (eV) Binding Energy (eV)

Figure B-1: (a) Dark ﬁeld false colored optical microscopy image of the resonator (yellow) and qubit (red) coupling area. Ground plane has been color blue. The black areas are where the base Nb+Al protection layer should have been fully removed. A thin metallic scum had been left on the wafer after processing that inhibited the quality factor of the readout resonators. (b) X–ray surface spectrogram of a fully processed SFQ–qubit sample utilizing an Al protection layer. (c) Zoom in on the 0–200 eV binding energy region in (b) (red box). Al 2p peaks were observed on the surface of the sample despite a full Al etch prior to patterning M0. 156

Circuits with this residue had high–power resonator internal quality factors below 1 × 103 preventing eﬃcient readout of the coupled qubits. In order to determine what the residual scum was comprised oﬀ, circuits were placed inside an x–ray photoelectron spectroscopy (XPS) tool and the surface chemistry was probed with an 11 keV Al source x–ray beam. The resulting x–ray surface spectrogram is displayed in Figures B-1(b–c). Figure B-1(b) displays a wide energy scan of the M0 surface. The oxygen and carbon signatures are from handling and atmosphere exposure. Between 0–200 eV binding energy (red box, Figure B-1(b), B-1(c)), residual traces of Al are seen to remain on the sample despite a chemical wet etch of the protection layer. The Al was either being etched and subsequently redeposited or resisting the etch and subsequently acting as a non–reactive barrier for the ﬁnal M0 plasma etch. This problem was solved by removing the Al protection layer altogether and simply not etching the area of layer V1 covering the resonator–qubit pocket until the junction processing was complete. Figure B-2(a) shows another dark ﬁeld false colored optical microscopy image oﬀ a sample processed without the Al protection layer. The corresponding x–ray spectrogram of a fully completed sample is displayed in Figure B-2(b). Resonators processed with this stack showed low–power, single photon level internal quality factors above 105. 157

(a)

(b)

Figure B-2: (a) Dark ﬁeld false colored optical microscopy image of the resonator (yellow) and qubit (red) coupling area processed without an Al protection layer. Ground plane has been color blue. No more silvery metallic scum can be seen on the surface of the substrate or etched metal layers. (b) X–ray surface spectrogram of a fully processed SFQ–qubit sample utilizing without an Al protection layer. 158

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