Particle Creation and Memory Effects in a Trapped Ion Quantum Simulator

Albert-Ludwigs-Universität Freiburg

Particle creation and memory eﬀects in a trapped ion quantum simulator

Matthias Wittemer

Dissertation

zur Erlangung des Doktorgrades der Fakultät für Mathematik und Physik der Albert-Ludwigs-Universität Freiburg

2019 Dekan: Prof. Dr. Wolfgang Soergel Betreuer und Erstgutachter: Prof. Dr. Tobias Schätz Zweitgutachter: apl. Prof. Dr. Bernd von Issendorﬀ

Tag der mündlichen Prüfung: 16. Dezember 2019 Prüfer: Prof. Dr. Marc Schumann apl. Prof. Dr. Thomas Filk Prof. Dr. Tobias Schätz Zusammenfassung

Die fundamentalen Gesetze der Quantenmechanik sind von unseren Alltagserfahrun- gen weit entfernt. Sie sind jedoch hervorragend geeignet um Beobachtungen in mikro- skopisch kleinen physikalischen Systemen zu erklären und vorherzusagen und werden daher bereits für verschiedene Anwendungen genutzt. Darüber hinaus wird, in Kom- bination mit relativistischen Überlegungen, grundlegenden Phänomenen der Quan- tenmechanik eine zentrale Rolle in der Entwicklung unseres Universums zugespro- chen. In der vorliegenden Arbeit verwenden wir einen Quantensimulator basierend auf gefangenen Ionen, um zwei fundamentale quantenmechanische Effekte, die seit vielen Jahrzehnten umfangreich studiert werden, zu untersuchen. In unserer ersten Untersuchung implementieren wir nicht-adiabatische Änderungen des Fallenpotenti- als der Ionen, um Fluktuationen des Quantenvakuums zu verstärken. Theoretischen Arbeiten zufolge lässt sich dieser Mechanismus als Analogie zur Teilchenerzeugung während der inflationären Phase des jungen Universums interpretieren. Solche Teil- chen gelten als Keime für die Bildung der makroskopischen Strukturen unseres heu- tigen Universums. In unserer analogen Quantensimulation wird die Erzeugung dieser Teilchen durch die Anregung eines gequetschten Zustands in der Bewegung der Io- nen nachgewiesen. Ein bemerkenswertes Merkmal dieses Mechanismus ist die damit einhergehende Erzeugung einer Quantenverschränkung. Im Falle der kosmologischen Teilchen erstreckt sich diese Verschränkung über große, kosmische Entfernungen. In unserer zweiten Untersuchung betrachten wir das Konzept offener Quantensysteme mit einem möglichst einfachen System und unter nahezu idealen Bedingungen. Jedes natürliche Quantensystem interagiert unweigerlich mit seiner Umgebung und kann daher nicht als isoliertes, sondern nur als offenes Quantensystem, welches Korrelationen mit seiner Umgebung aufbauen kann, betrachtet werden. Wir definieren ein offenes Zwei-Niveau-System im elektronischen Freiheitsgrad eines einzelnen gefangenen Ions, welches wir an eine bosonische Umgebung, bestehend aus einem Bewegungsfreiheits- grad des Ions, koppeln. Zudem implementieren wir eine einstellbare Wechselwirkung zwischen System und Umgebung, welche zur Verschränkung dieser beiden Systeme führt. Indem wir Messungen nur am offenen Quantensystem vollziehen beobachten wir nicht-Markov’sches Verhalten, welches wir mit einem rigoros definierten Maß quanti- fizieren. Dabei decken wir auf, dass die Quantifizierung solcher Quantengedächtnis- effekte durch intrinsische quantenmechanische Messunsicherheiten limitiert ist. Die Untersuchungen, die wir in der vorliegenden Arbeit präsentieren, erweitern die Mög- lichkeiten unserer Ionenfallen-Apparatur analoge Quantensimulationen fundamenta- ler Probleme der Quantenmechanik unter nahezu idealen Bedingungen durchzuführen immens. Darüber hinaus werden zukünftige Studien komplexerer Sachverhalte in rea- listischen Quantensystemen ermöglicht. Abstract

The fundamental laws of quantum mechanics are far from our everyday life experiences. However, they have been extremely successful in explaining observations in microscopic physical systems and are, already, harnessed for several applications. More- over, in combination with relativistic considerations, basic phenomena from quantum mechanics are accredited a central role in the (macroscopic) evolution of our universe. In this thesis, we employ a trapped ion quantum simulator to experimentally investigate two fundamental quantum mechanical effects that have been extensively studied for many decades. In the first investigation, we implement non-adiabatic changes of the ions’ trapping potential in order to amplify quantum vacuum fluctuations. Fol- lowing theoretical works by others, this mechanism can be interpreted as an analog to the creation of particles during cosmic inflation in the early universe. Such cosmological particles are considered as the seeds for the formation of the large-scale structures in our universe that we observe nowadays. In our analog quantum simulation, the creation of particles is evidenced by the detection of a squeezed state in the motion of the ions. A remarkable feature of this mechanism is the accompanying creation of quantum entanglement. In case of the cosmological particles, this entanglement spreads over large, cosmic distances. In the second investigation, we benchmark a theoretical concept in the framework of open quantum systems in a most basic system and under near-ideal conditions. In nature, any quantum system inevitably interacts with its environment and, thereby, needs ultimately to be considered an open system that can build up correlations and, even, entanglement with its environment. We implement a spin-1/2 system in the electronic degree of freedom of a single trapped ion, which we couple to a bosonic environment, formed by a motional degree of freedom of the ion. Further, we implement a tunable interaction between system and environment that, in turn, can lead to entanglement. By performing measurements on the open system only, we observe quantum non-Markovian behavior, which we quantify using a rigorously defined measure. Thereby, we reveal that the quantification of such quantum memory effects is fundamentally limited by fundamental quantum mechanical measurement uncertainties. The investigations presented in this thesis significantly expand the capabilities of our trapped ion platform to perform analog quantum simulations of fundamental quantum mechanical problems under near-ideal conditions, and enable future studies of increased complexity and under more realistic conditions.

Contents

1 Introduction1

2 Theoretical concepts7 2.1 Quantum mechanics basics ...... 7 2.2 Cosmology ...... 16 2.3 Open quantum systems ...... 21

3 Experimental methods 25 3.1 Experimental setup ...... 25 3.2 Spin control ...... 42 3.3 Phonon control ...... 49 3.4 Spin-phonon coupling ...... 62 3.5 Numerical calculations ...... 69

4 Experiments and results 77 4.1 Phonon pair creation ...... 77 4.2 Quantum memory eﬀects ...... 104

5 Conclusion and outlook 117

Bibliography 121

1 Introduction

To this day, quantum mechanics is our best way to describe nature at the smallest length scales. Since its development in the early 20th century, see, e.g., the works by Dirac (1930) and von Neumann (1932), the theory has made remarkable achievements in explaining and predicting phenomena that are far from our everyday experiences. Today, intriguing effects like quantum entanglement are routinely observed and harnessed for multiple applications in well isolated and microscopic physical systems. However, physicists as well as philosophers have been wondering how the fundamental laws can be translated into the macroscopic (classical) description of our world, see Bohr (1920) and Schrödinger (1926), which lead Schrödinger to his famous thought experiment (Schrödinger, 1935), addressing the bizarre consequences of such transla- tions onto macroscopic and even living objects. The fundamental laws of quantum mechanics have even more intriguing consequences when considered under relativistic conditions. Employing the framework of quantum field theory, such approaches give rise to phenomena like the Sauter- Schwinger effect (Sauter, 1931, 1932; Schwinger, 1951), Hawking radiation (Hawking, 1974, 1975), and the Unruh effect (Fulling, 1973; Davies, 1975; Unruh, 1976). All of these effects, and also other related phenomena like the static Casimir effect (Casimir, 1948) as well as its dynamical counter-part (Moore, 1970), have their origin in the intriguing description of the vacuum according to quantum field theory. While the vacuum is classically often considered as a space that is entirely devoid of matter, according to quantum field theory, the vacuum is not empty but filled with ubiqui- tous fluctuations. Taking account for Heisenberg’s uncertainty principle (Heisenberg, 1927), these fluctuations can be pictured as the random creation and annihilation of pairs of virtual particles. Certain, often considered extreme, conditions acting on the quantum fields corresponding to these particles, may then lead to the effects above. In recent theoretical investigations quantum fluctuations have been accredited an in- creasingly central role in the formation and evolution of our universe, see Brout et al. (1978), García-Bellido (1999), and Martin (2019). One key aspect of such considerations is the cosmological creation of pairs of particles in curved space-time, see Parker (1968) and Birrell and Davies (1982), and also Schrödinger (1939). In particular, during a period of cosmic expansion in the early universe, quantum fluctuations have been torn apart and amplified, and are considered to be responsible for the cosmic large-scale structures observable nowadays, cf. Liddle and Lyth (2000). Moreover, it is even considered that the creation of particles under these extreme conditions is 2 1 Introduction accompanied by the formation of quantum entanglement at large, cosmic distances (Martín-Martínez and Menicucci, 2012). Thus, Einstein’s phrasing of a “spooky action at a distance” refers in this context to an even more intriguing consequence of the fundamental laws of quantum mechanics. Further peculiarities arise when including measurements into the above described scenarios. Performing a measurement on only one partner of an entangled particle pair results in a mixed, thermal state, which is a familiar feature of, e.g., Hawk- ing radiation (Hawking, 1974, 1975). Here, the measurement traces out the other particle that is falling into the Black hole and, hence, the outcome resembles (cf. Brustein et al., 2018) a classical state, despite the underlying quantum correlations. Such situations, where measurements are performed only on a subsystem of a larger quantum system, are conveniently addressed using the framework of open quantum systems, see Breuer and Petruccione (2007). Here, one takes account of the fact that any realistic quantum system inevitably interacts with its environment and, thus, needs to be considered an open quantum system, that can build up correlations and entanglement with its environment. Although, measurements may be performed on the open quantum system only, i.e., the environment is traced out, such correlations may be identified by established methods within the open quantum systems approach, see Breuer and Petruccione (2007). Depending on the particular configuration of the system, the environment, and the particular interaction between the two, such a treatment may be capable of describing the quantum origin of our classical world (Zurek, 2003) or give rise to other intriguing phenomena such as, e.g., quantum memory effects (Breuer et al., 2016). As fascinating as the quantum effects discussed above may be, in their nature lies the fact that they can be hardly observed directly. The particle creation in the early universe, e.g., is inaccessible with electro-magnetic waves, as they only reach back to the cosmic microwave background (Penzias and Wilson, 1965; Dicke et al., 1965). Although consequences of the underlying quantum phenomena may be imprinted thereon, cf. Krauss (2014) and Ade et al. (2014, 2015), the direct observation of particle pairs created during cosmic inflation is out of reach. Similar limitations apply to investigations of open quantum systems and an experimental observation of quantum memory effects, which would correspond to quantum non-Markovian behavior. When observing a quantum system that is significantly interacting with its environment, this very interaction may lead to decoherence, in particular if the environment is of classical character, which then conceals the quantum effects one wants to investigate, i.e, the dynamics of the open system becomes Markovian, cf. Breuer and Petruccione (2007). A promising route to overcome these obstacles can be envisioned by considering analog quantum simulations. Here, the basic idea is to study the dynamics of a complex or experimentally inaccessible quantum system of interest with another, more controllable and observable quantum system with, however, analog dynamics. First 3 proposed by Richard P. Feynman (1982), quantum simulations have been proven to be a valuable tool in gaining deeper insight on complex quantum mechanical problems, whenever direct observations are unavailable and/or calculations are too demanding for classical computers, cf. Cirac and Zoller (2012). In recent years, seminal progress has been made with various experimental platforms, allowing for analog quantum simulations of most versatile problems, see Lloyd (1996), Buluta and Nori (2009), Schaetz et al. (2013), and Georgescu et al. (2014), and specialized reviews focusing on cold atoms (Bloch et al., 2012; Gross and Bloch, 2017), photons (Aspuru-Guzik and Walther, 2012), superconducting circuits (Houck et al., 2012), and trapped ions (Jo- hanning et al., 2009; Blatt and Roos, 2012; Schneider et al., 2012). Among all these different promising platforms for experimental quantum simulations, in particular trapped ions provide an excellent framework with unique performances for metrolog- ical applications as well as concepts in quantum information processing and quantum simulation, see Cirac and Zoller (1995), Wineland et al. (1998a), Cirac and Zoller (2000), Leibfried et al. (2003), and Wineland (2013). Employing state of the art laser cooling techniques, single and few trapped atomic ions can be prepared in the quantum mechanical ground state of motion with nearly unit fidelity, cf. Wineland et al. (1978), Neuhauser et al. (1978), Monroe et al. (1995), and King et al. (1998). Their motional states can be read out down to the single quantum level (Meekhof et al., 1996; Leibfried et al., 1996), and the ions’ internal (electronic) degrees of freedom can be prepared and detected with unique fidelities (Happer, 1972; Wineland et al., 1980; Nagourney et al., 1986; Sauter et al., 1986; Bergquist et al., 1986; Schaetz et al., 2005; Schmidt et al., 2005; Hume et al., 2007; Burrell et al., 2010; Harty et al., 2014), while residual couplings to noisy, decoherence-inducing surroundings can be effectively eliminated, see Langer et al. (2005), Ruster et al. (2016), and Hakelberg et al. (2018). In particular the ions’ long-ranging Coulomb interaction allows to engineer versatile Hamiltonians, which makes them a powerful platform for experimental quantum simulations (Porras et al., 2008; Cirac and Zoller, 2012). In this work, we build upon the seminal control of quantum systems provided by current state-of-the-art techniques in ion trapping. We tweak established and develop new methods in order to set up a versatile experimental platform that allows for observations of quantum mechanical phenomena on a most fundamental level. Based on these developments, we employ our trapped ion quantum simulator to investigate two of the elusive effects discussed above, see Fig. 1.1. In the first investigation, we study cosmological particle creation in the early universe, where a rapid cosmic expansion tears apart (virtual) quantum vacuum fluctuations and, thereby, creates pairs of (real) particles in an entangled state. In the second investigation, we study an open quantum system and observe quantum memory effects, that are due to the entanglement that is created by the coherent interaction between the open system and its environment. 4 1 Introduction

Space S E

Time

Figure 1.1: Sketches of the quantum simulations presented in this work. The left plot illustrates the creation of pairs of particles in an expanding space-time such as during the inflationary phase of the early universe. Here, omnipresent quantum vacuum fluctuations, pictured as virtual pairs of particle and anti-particle, are torn apart until they cannot recombine anymore and, thereby, become real particles that share entanglement (orange) at large distances. On the right we depict an open quantum system S that is interacting with an environment E. The open system is formed by a spin-1/2 (two-level) system, whereas the environment is composed of a single harmonic oscillator. The creation of entanglement (orange) between S and E gives rise to non-Markovian behavior for observables of the open quantum system. The associated quantum memory effects can be observed by measuring the flow of information I into the open quantum system.

In order to simulate cosmological particle creation, we adapt the proposals by Schützhold et al. (2007) and Fey et al. (2018). We implement non-adiabatic changes of the trapping potential of two ions in order to simulate a rapid cosmic expansion, see Fig. 1.1(left). Thereby, quantum vacuum fluctuations of the motion of the ions are amplified and, as a consequence, the ions are prepared in a squeezed state of motion. In our analog quantum simulation, the particle creation is evidenced by the detection of such squeezed states and their creation is accompanied by quantum entanglement in the ions’ motional degree of freedom. For our investigation of an open quantum system we employ a single trapped ion. We implement a spin-1/2 system in the ion’s electronic degree of freedom, which we couple to a bosonic environment, formed by a single motional mode of the ion, see Fig. 1.1(right). We implement a tunable interaction between system and environment that leads to entanglement between the two quantum systems. By performing measurements on the open system only, we monitor the flow of information and, thereby, observe quantum non-Markovian behavior. We quantify the corresponding quantum memory effects with a rigorously defined measure by Breuer et al. (2009). 5

Outline

In Chapter 2, we recapitulate the basic theoretical framework for our studies. We introduce some basic concepts of quantum mechanics before discussing the process of cosmological particle creation in the early universe and the framework of open quantum systems. Chapter 3 deals with experimental methods and numerical calculations that we employ for our experimental investigations. This includes several well-established methods as well as novel techniques that we develop for our particular endeavors. The experimental results of our studies are presented in Chap. 4. In Section 4.1 we describe our implementation of an analog quantum simulation of cosmological particle creation, which is also partially covered in Wittemer et al. (2019). Section 4.2 describes our realization of an open quantum system in our ion trap setup and the measurement of quantum non-Markovian behavior that is published in Wit- temer et al. (2018). Finally, Chapter 5 summarizes our ﬁndings and concludes with an outlook on possible extensions of our approach for future studies.

Personal contribution

I started working at the experiment in October 2015. Together with Govinda Clos, senior PhD student at that time, I implemented essential experimental methods allowing us to simulate an open quantum system in our ion trap. These results were published together with Heinz-Peter Breuer (University of Freiburg). Following this, I built an arbitrary waveform generator device and integrated it into the experimental setup. I installed a new experimental control software that was already being used at another experiment, thereby, allowing for real-time control of the ion’s trapping potential via dedicated control potentials for which I set up a boundary element method simulation of our trap. In order to also manipulate the trap’s radio-frequency potential in real-time, I set up an electrical circuit that allows for fast switching of the radio-frequency voltage amplitude on the micro second time scale. This lead to the successful quantum simulation of cosmological particle pair creation for which I set up a novel motional state analysis and numerical simulations of the implemented dynamics. The corresponding results are currently being published in collaboration with Ralf Schützhold (TU Dresden and Helmholtz-Zentrum Dresden-Rossendorf) and Christian Fey (University of Hamburg, now at MPQ in Garching). I prepared, conducted, and analyzed all of the experimental measurements presented in this thesis. Ulrich Warring, senior researcher in our group, assisted the experiments and theoretical descriptions, and supervised, together with Tobias Schätz, the overall project. 6 1 Introduction

Publications

The work presented in this thesis is also partially covered in these publications:

• M. Wittemer, G. Clos, H.-P. Breuer, U. Warring, and T. Schaetz (2018). Mea- surement of quantum memory eﬀects and its fundamental limitations. Physical Review A 97, 020102(R). doi: 10.1103/PhysRevA.97.020102 • M. Wittemer, F. Hakelberg, P. Kiefer, J.-P. Schröder, C. Fey, R. Schützhold, U. Warring, and T. Schaetz (2019). Phonon Pair Creation by Inﬂating Quantum Fluctuations in an Ion Trap. Physical Review Letters 123, 180502. doi: 10. 1103/PhysRevLett.123.180502

Publications on experiments in planar surface electrode ion traps that are, however, not covered in this thesis:

• M. Mielenz, H. Kalis, M. Wittemer, F. Hakelberg, U. Warring, R. Schmied, M. Blain, P. Maunz, D. L. Moehring, D. Leibfried, and T. Schaetz (2016). Arrays of individually controlled ions suitable for two-dimensional quantum simulations. Nature Communications 7, 11839. doi: 10.1038/ncomms11839 • H. Kalis, F. Hakelberg, M. Wittemer, M. Mielenz, U. Warring, and T. Schaetz (2016). Motional-mode analysis of trapped ions. Physical Review A 94, 023401. doi: 10.1103/PhysRevA.94.023401 • F. Hakelberg, P. Kiefer, M. Wittemer, T. Schaetz, and U. Warring (2018). Hy- brid setup for stable magnetic ﬁelds enabling robust quantum control. Scientiﬁc Reports 8, 4404. doi: 10.1038/s41598-018-22671-5 • F. Hakelberg, P. Kiefer, M. Wittemer, U. Warring, and T. Schaetz (2019). Interference in a Prototype of a Two-Dimensional Ion Trap Array Quantum Simulator. Physical Review Letters 123, 100504. doi: 10.1103/PhysRevLett. 123.100504

• P. Kiefer, F. Hakelberg, M. Wittemer, A. Bermúdez, D. Porras, U. Warring, and T. Schaetz (2019). Floquet-Engineered Vibrational Dynamics in a Two- Dimensional Array of Trapped Ions. Physical Review Letters 123, 213605. doi: 10.1103/PhysRevLett.123.213605 2 Theoretical concepts

In this Chapter, we want to provide a theoretical basis that will help to interpret our experiments. We will start by brieﬂy reviewing some general concepts of quantum mechanics. Thereafter, we will focus on cosmological particle creation in curved space- time as well as setting up a framework for the description of open quantum systems. We note that our descriptions will be kept brief and we refer to the vast variety of literature on these topics for further details.

2.1 Quantum mechanics basics

This Section is intended to set up the basic concepts for a coherent theoretical description of our experiments. Our brief explanations will closely follow J. J. Sakurai (1994), Greiner (2001), and Breuer and Petruccione (2007), thus, we refer, in particular, to these references for further reading.

2.1.1 Dynamics We want to brieﬂy recapitulate the general description of the dynamics of quantum systems. To this end, we consider a quantum system S with corresponding Hilbert space HS of dimension d, that is spanned by the basis vectors |ii. Thereby, any quantum state of S may be expressed by

d X |ψi = ci |ii , (2.1.1) i=1 Pd 2 where the ci are complex coefficients that fulfill i=1 |ci| = 1, i.e., |ψi is a normalized state vector. Further, we consider the Hamiltonian HS, the operator corresponding to the total energy of the system. The dynamics of the system is described via the well-known Schrödinger equation: d i |ψ(t)i = H |ψ(t)i (2.1.2) ~dt S Here, ~ denotes Planck’s reduced constant. The formal solution of this differential equation may be given in terms of a unitary time evolution operator U(t): |ψ(t)i = U(t) |ψ(0)i (2.1.3) 8 2 Theoretical concepts

Thereby, |ψ(t)i describes the state of the system at any time t > 0. If we consider a time-independent Hamiltonian, i.e., a closed and isolated quantum system S, cf. Breuer and Petruccione (2007), we can substitute Eq. (2.1.3) into the Eq. (2.1.2) and after integration we obtain

i − HS t U(t) = e ~ . (2.1.4)

Accordingly, for a time-dependent Hamiltonian HS = HS(t), the time evolution operator may be calculated according to

i R t 0 0 − HS (t )dt U(t) = e ~ 0 . (2.1.5)

Thus, employing the Schrödinger equation, we may calculate the time evolutions of states |ψi of a quantum system S, given its Hamiltonian HS. However, in situations where S is in a mixed state, we rather describe its dynamics using the Liouville-von Neumann equation for density operators, see Breuer and Petruccione (2007). A general density matrix ρ for the system considered here can be calculated according to ρ = |ψi hψ|, with the general state |ψi introduced above. Thus, ρ is a d × d matrix on the Hilbert space HS and, by construction, we ﬁnd tr (ρ) = 1. The Liouville-von Neumann equation is given by d i ρS(t) = − [HS, ρS(t)] . (2.1.6) dt ~ Again, we may ﬁnd the formal solution using the time evolution operator by

ρ(t) = U(t)ρ(0)U †(t). (2.1.7)

Thus, with the Liouville-von Neumann equation we can describe the quantum system’s dynamics also for mixed states. In general, mixed states can be considered as statistical ensembles of pure states. They are, in particular, relevant when considering composite quantum systems, such as in the open quantum systems approach that is discussed Sec. 2.3. We may distinguish pure from mixed quantum states employing the density operator ρ. A density matrix ρ is considered to describe a pure state if and only if ρ2 = ρ, and, thus, its 2 1 2 purity is tr ρ = 1. In contrast, a mixed state may be identiﬁed by d ≤ tr ρ < 1. Having introduced the general concept of the description of dynamics of quantum systems, we may now discuss two exemplary quantum systems that both play a central role in this work.

2.1.2 Spin-1/2 system First, we want to discuss the dynamics of the very basic spin-1/2 system. For this, we consider a quantum system with only two energy levels |↓i and |↑i, that are separated 2.1 Quantum mechanics basics 9 in energy by ~ω. In order to describe the spin-1/2 system it is convenient to introduce the Pauli operators

σx = |↑i h↓| + |↓i h↑| , (2.1.8)

σy = −i |↑i h↓| + i |↓i h↑| , (2.1.9)

σz = |↑i h↑| − |↓i h↓| . (2.1.10)

Thereby, we may write down the Hamiltonian H that describes the (free) two-level system by ω H = ~ σ . (2.1.11) 2 z Accordingly, the eigenstates |↓i and |↑i fulﬁll

ω ω H |↓i = −~ |↓i ,H |↑i = +~ |↑i , (2.1.12) 2 2 where we ﬁnd their energy separation of ~ω, see above. Further, we obtain the eigenoperators σ+ and σ− of H, that satisfy

[H, σ−] = −~ωσ−, [H, σ+] = +~ωσ+. (2.1.13)

These correspond to transitions |↑i → |↓i and |↓i → |↑i, respectively, see Fig. 2.1. In physical realizations, these are most often associated with the emission or the absorption of a photon with energy ~ω, respectively. A general state of the two-level system may be written as |ψi = c↓ |↓i + c↑ |↑i 2 2 with complex coeﬃcients c↓,↑ fulﬁlling |c↓| + |c↑| = 1. Accordingly, the corresponding density operator is then given by ρ = |ψi hψ|. Employing the Pauli operators introduced above, we may rewrite the state of the system with the Bloch vector

~rB = h~σi = tr [~σρ] , (2.1.14)

T where ~σ = (σx, σy, σz) is the vector of the Pauli operators that gives rise to the (spin) ~ ~ operator S = 2~σ, cf. J. J. Sakurai (1994). The Bloch vector provides a geometrical representation of the state of the spin-1/2 system, see Fig. 2.1. 10 2 Theoretical concepts

|↑i |↑i z x y σ σ+ −

~rB

|↓i |↓i

Figure 2.1: Illustration of a spin-1/2 system. On the left we depict the two energy eigenstates |↓i and |↑i, together with an illustration of the action of the eigenoperators σ− and σ+ of the corresponding Hamiltonian. The right side illustrates the Bloch sphere representation. We depict, as an example, the coherent action of the Pauli operator σx on the initial state |↓i. The Bloch vector (blue) rotates around the x-axis on a circular path (orange) on the surface of the Bloch sphere. Thereby, the system remains in a pure state throughout the whole interaction.

Application of the Pauli operators σi to the Bloch vector ~rB induces rotations of ~rB around the axis corresponding to σi. In Figure 2.1 we depict, as an example, the action of σx on |↓i, i.e., σx |↓i = |↑i. The Bloch vector follows a path on the surface of the Bloch sphere, corresponding to a rotation of ~rB around the x-axis. Accordingly, σy and σz, correspond to rotations around the y and z-axis, respectively. In the Bloch sphere representation, pure states correspond to points on the surface of the sphere, as they fulﬁll tr ρ2 = 1, cf. Eq. (2.1.14). In contrast, any state inside the Bloch sphere corresponds to a mixed state and we identify ~rB = 0 as the maximally mixed state, see Clos (2017). We note that under the actions of the Pauli operators, the purity of the system’s state is not changed.

For a given Bloch vector ~rB the corresponding density matrix ρ can be calculated according to

! 1 1 + hσ i hσ i − ihσ i ρ = z x y . (2.1.15) 2 hσxi − ihσyi 1 − hσzi

Thus, the expectation value of σz translates to the populations of the system’s eigenstates, whereas the expectation values of σx and σy yield the corresponding coherences. 2.1 Quantum mechanics basics 11

2.1.3 Quantum harmonic oscillator Our next example of a basic quantum system is the harmonic oscillator. We consider a particle of mass m that moves in a one-dimensional harmonic potential with frequency ω. The corresponding Hamiltonian is then given by p2 1 H = + mω2x2. (2.1.16) 2m 2 Here, p and x denote the momentum and position operators, respectively. By solving the (time-independent) Schrödinger equation H |ni = En |ni we ﬁnd the quantized energy eigenstates |ni with their corresponding energy eigenvalues given by 1 E = ω n + (2.1.17) n ~ 2 An important thing to note here is the ground state energy E0 = ~ω/2 6= 0. This zero- point energy, in agreement with Heisenberg’s uncertainty principle, corresponds to intrinsic quantum vacuum state ﬂuctuations. These are of particular interest for our experiments described in Sec. 4.1 and are further theoretically examined in Sec. 2.2.2. The eigenstates |ni are called Fock states and in Fig. 2.2 we depict the wavefunctions ψn(x) = hx|ni in the position coordinate basis for the states n = 0 to n = 8.

8 ) ω

h 6 (¯ E

4 Energy

0 4 2 0 2 4 − − Position x ( h¯/(mω)) p Figure 2.2: Eigenstates of the quantum harmonic oscillator. We depict the wavefunctions (amplitudes rescaled for clarity) of the Fock states |0i to |8i (blue) that correspond to a harmonic oscillator potential ∝ x2 (gray line). Note that all the even wavefunctions are symmetric, while all the odd states are asymmetric with respect to the x coordinate.

We note that the wavefunctions ψn(x) are symmetric in x for even n, while they are anti-symmetric for odd n. This is of particular relevance for our experiments described in Sec. 4.1 12 2 Theoretical concepts

In an alternative picture, we may deﬁne the harmonic oscillator’s Hamiltonian using the creation and annihilation operators a† and a (J. J. Sakurai, 1994), which are deﬁned by √ a† |ni = n + 1 |n + 1i , (2.1.18) √ a |ni = n |n − 1i . (2.1.19)

Explicitly, they introduce transitions between the energy eigenstates |ni and, we ﬁnd the following representations of the position and momentum operators (J. J. Sakurai, 1994):

s x = ~ a† + a , (2.1.20) 2mω s p = i ~mω a† − a . (2.1.21) 2

This leads to the alternative expression for the system’s Hamiltonian

1 1 H = ω a†a + = ω n + , (2.1.22) ~ 2 ~ 2 where n = a†a is the number operator, whose eigenvalues correspond to the number of energy quanta in the system. We note that in theoretical descriptions of H, the † ground state energy is often omitted for simplicity, i.e., H = ~ωa a, which corresponds to a renormalization of the energy of the system. Using the Fock states |ni, a general state |ψi of the harmonic oscillator can be written as

∞ X |ψi = cn |ni , (2.1.23) n=0

P∞ 2 with complex coefficients cn, that, again fulfill n=1 |cn| = 1, cf. Sec. 2.1. Accord- ingly, the corresponding density matrix ρ = |ψi hψ| has infinite dimension, as the harmonic oscillator’s Hilbert space is infinite-dimensional. However, for finite values of the total energy of the oscillator we may introduce a cutoff Fock state and omit 2 states with negligible contributions, i.e., states with |cn| ≈ 0. This allows to explicitly calculate the density matrix ρ for a given state |ψi. Here, we want to consider three representative example states that are of particular interest for the description of our experiments. First, we discuss a thermal state of the quantum harmonic oscillator. A thermal state is an incoherent superposition of the Fock states |ni and, thus, all off-diagonal 2.1 Quantum mechanics basics 13 elements (coherences) in its density matrix are zero. The diagonal elements (populations) of the density matrix are given by

n¯n Pn(¯n) = . (2.1.24) (¯n + 1)n+1

Here, n¯ denotes the mean number of energy quanta in the statistical ensemble which the thermal state represents. Consequently, the thermal state is a mixed state and its P∞ 2 purity can be calculated according to n=0 Pn(¯n) . In Figure 2.3(left) we illustrate, as an example, the density matrix of a thermal state with n¯ = 1.0. The second state we want to consider here is a coherent state, which can be considered the most classical state of the quantum harmonic oscillator. In order to construct a coherent state, we deﬁne a displacement operator h i D(α) = exp αa† − α∗a , (2.1.25) where the complex argument α describes amplitude and direction of a displacement in phase space. Thereby, we construct a coherently displaced vacuum state by D(α) |0i or, equivalently, in its density matrix representation, by D(α) |0i h0| D†(α). The corresponding population distribution of a coherent state is a Poisson distribution −|α| k Pn(α) = e |α| /k!. In Figure 2.3(center) the density matrix of a coherent state with α = 1.5 is illustrated. Note that all of its elements are positive and real due to α = Re(α) in our example. The third and last example state discussed here is a squeezed vacuum state S(ξ) |0i h0|, which is deﬁned with the squeezing operator

1 S(ξ) = exp ξ∗a2 − ξa†2 , (2.1.26) 2 with the complex argument ξ = reiθ, from which we identify a squeezing amplitude r and a squeezing angle θ. The name squeezed state stems from their asymmetric shape in phase space. While the variance along one degree of freedom is reduced by e−2r, the variance along the other direction is accordingly widened by e2r. The characteristic feature of a squeezed state’s density matrix is that it has non-zero elements only for even states. In Figure 2.3(right) such a density matrix for ξ = 0.7 is illustrated. 14 2 Theoretical concepts

0 1.0 0 0.4 0 1.0 h | h | h | 0.8 2 2 0.3 2 0.5 h | h | h | 0.6 4 4 0.2 4 0.0 h | 0.4 h | h | 6 6 0.1 6 0.5 h | 0.2 h | h | − 8 8 8 h | 0.0 h | 0.0 h | 1.0 0 2 4 6 8 0 2 4 6 8 0 2 4 6 8 − | i | i | i | i | i | i | i | i | i | i | i | i | i | i | i Figure 2.3: Density matrix visualizations for three example states of the quantum harmonic oscillator. We depict a thermal state with n¯ = 1 (left), a coherently displaced vacuum state with α = 1.5 (center), and a squeezed vacuum state with ξ = 0.7 (right). The thermal state has zero coherences, i.e., it represents an incoherent mixture of the populated states. In contrast, the coherent state is a fully coherent superposition, indicated by the non-zero oﬀ-diagonal elements. The squeezed state shows non-zero values only for even states and coherences. Note that for the examples depicted here it is suﬃcient to depict only real values as all imaginary parts are zero due to our choice of real excitation parameters α = |α| and ξ = r.

Note that, in general, the oﬀ-diagonal elements of density matrices are complex valued. For the examples depicted in Fig. 2.3, however, all imaginary parts are zero due to our choice of real excitation parameters α and ξ.

2.1.4 Measurements in quantum mechanics In this Section we want to discuss the special role of measurements in quantum mechanics, see Dirac (1930) and von Neumann (1932). To this end we consider a single spin-1/2 that is prepared in a superposition state |ψi = c↓ |↓i + c↑ |↑i with 2 2 coefficients c↓,↑ 6= 0 fulfilling |c↓| +|c↑| = 1. To simplify the following considerations, we consider a measurement device with a corresponding measurement basis M that is aligned with the quantization axis of the spin, so that M = {|↓i , |↑i}. When a measurement on the state |ψi is performed, its wavefunction changes, i.e., it collapses to either |↓i or |↑i as described by the projection postulate, cf. von Neumann (1932). The corresponding probabilities to project the state into |↓i or |↑i are given 2 2 by |c↓| and |c↓| , respectively. We note that this mechanism applies (already) to a single measurement of the spin state |ψi. As a consequence, expectation values for the coefficients c↓,↑ can only determined by averaging over many realizations of the same experiment (or probe many spin-1/2 particles with a single measurement). In any case, the corresponding results are subjected to intrinsic quantum projection noise (QPN), cf. Itano et al. (1993). When performing r repetitions of the measurement, from which we find |↓i 2.1 Quantum mechanics basics 15

n 2 for n times, the corresponding probability p↓ = r , which is our estimator for |c↓| , lies in the following standard (Wald 1σ) conﬁdence interval of the binomial distribution:

s p (1 − p ) p ± ↓ ↓ (2.1.27) ↓ r

Here, we used the normal approximation for the binomial conﬁdence interval, which may be a valid assumption if r · min(p↓, 1 − p↓) & 5. If this cannot be ensured a Poissonian approximation can be used, cf. DasGupta et al. (2001). From Equation 2.1.27 we ﬁnd that the uncertainty related to a measurement of 2 √ |c↓| = p↓ is accompanied by an uncertainty that is proportional to 1/ r. We note that this uncertainty is an intrinsic feature of quantum mechanics and persists, no matter how accurately the state |ψi was prepared. Rather, it witnesses the probabilis- tic nature of quantum mechanics. Note that, although here discussed for a two-level system only, the projection postulate applies also to measurements on other quantum systems, and corresponding expectation values are subjected to QPN as well. Although not utilized for the experiments presented in this work, we want to mention the concept of weak measurements. Here, the idea is to interrogate only little information about the quantum system of interest and, thereby, limit the measurement induced state change of the system. We note that these measurement schemes can be derived from the (von Neumann type) measurements discussed above and we refer to Jacobs and Steck (2006) for an introduction.

2.1.5 Quantum entanglement In this Section we want to brieﬂy deﬁne the term quantum entanglement as it is a central feature of our experiments presented in Chap. 4. For a more extensive mathematical description we refer to Bruß (2002) and Eisert and Plenio (2003). We consider a bipartite quantum system with composite Hilbert space H = HA ⊗ HB, where HA and HB are the Hilbert spaces of its subsystems A and B, respectively. We call a pure state |ψi ∈ H of the system separable if and only if it can be written as a direct product

|ψi = |aiA ⊗ |biB (2.1.28) or equivalently as

By employing the individual bases |niA and |miB of the two subsystems, an arbitrary (pure) state |ψi of the total system can be written as

X |ψi = cnm |niA |miB , (2.1.30) nm with complex coeﬃcients cnm. In order to determine if such a state is separable, i.e., not entangled, we consider the reduced density matrix ρA of the full density matrix ρ = |ψi hψ|, which is obtained by tracing out subsystem B with the partial trace over the subspace HB: X ρA = trB (ρ) = hm|B ρ |miB (2.1.31) m

Thus, for a separable state as in Eq. (2.1.29), we obtain a pure state in HA:

ρA = |aiA ha|A (2.1.32)

In contrast, for an entangled state |ψi, the reduced density matrix ρA is always a 2 mixed state with purity tr(ρA) < 1. Thereby, a measure for entanglement can be constructed by employing the (von Neumann) entropy S(ρ) of a given density matrix ρ, that is deﬁned by:

S(ρ) = −tr (ρ ln ρ) (2.1.33)

We ﬁnd S(ρ) = 0 if and only if ρ is a pure state, and S(ρ) = ln d for the maximally mixed state, where d is the dimension of the Hilbert space. Consequently, in order to quantify the amount of entanglement for a given quantum state ρ of the composite Hilbert space H, we deﬁne the entanglement of entropy ES by

ES = S(ρA). (2.1.34)

We note that the above descriptions hold for pure states only, however, also mixed states may feature entanglement and there are several measures intended to quantify entanglement also in this case, see Bruß (2002).

2.2 Cosmology

In this Section we want to give a short introduction to cosmology, in order provide an appropriate context for our experiments presented in particular in Sec. 4.1. Note that, however, we will keep our explanations concise and refer to well-established textbooks on this topic for details, e.g., Mukhanov (2005), Weinberg (2008), and Liddle (2015). 2.2 Cosmology 17

2.2.1 The early universe According to the standard model of cosmology, about 13.8 billion years ago our universe was in a state of very high density and only microscopic expansion, see Gamow (1948). With the so-called big bang the universe started to expand which is considered to be still going on nowadays, as can be inferred from red-shifts in emission spectra of distant stars, see Hubble (1929). A relic of this hot start at the big bang can be found in the cosmic microwave background (CMB), see Penzias and Wilson (1965) and Dicke et al. (1965), and recent ﬁndings of the Planck mission (Ade et al., 2014). However, this standard model of cosmology left some observations unex- plained, cf. Liddle (2015):

1. Horizon problem: The CMB is highly isotropic, light seen from all parts of the sky possesses the same temperature with variations on the 10−5 level only. However, given the size and the age of the universe at the CMB, distant regions would not have the chance to interact and thermalize, i.e., should not have reached the same temperature.

2. Flatness problem: The total density of matter and energy Ωtot in our universe seems to be such that the geometry in our universe is nearly ﬂat, i.e., Ωtot ≈ 1, corresponding to an Euclidean geometry. If so, the early universe must have had a density even closer to the critical value of 1 and it is questioned why this “special” value is realized to such high accuracy, cf. Dicke (1970).

3. Monopole problem: The third problem with the hot big bang theory arises when combining it with ideas of modern particle physics. Models aiming to unify the fundamental forces predict that a signiﬁcant, i.e., measurable, amount of magnetic monopoles should have been created under the extreme conditions of the big bang. However, to this day, no magnetic monopoles have been observed.

These problems were the three primary motivations that lead eventually to the theory of cosmic inflation, first proposed in Guth (1981). Essentially, the inflationary model considers a period in the evolution of the universe during which the cosmic space- time was rapidly expanding. In fact, for a given point in space-time this expansion is considered to be even faster than the growth of the relativistic event horizon. Thereby, regions in a common event horizon are pulled apart until they are causally disconnected, i.e., they leave the common event horizon. This solves the horizon problem mentioned above, cf. Weinberg (2008). Further, the rapid cosmic expansion is considered to lead to an evolution of the density Ωtot such that it is driven towards the critical value 1, rather than away from it as would be the case without inflation. This gives an explanation for the flat universe we observe today. Finally, cosmic inflation can also be considered to solve the monopole problem. Due to the rapid 18 2 Theoretical concepts cosmic expansion, shortly after the big bang, the density of any magnetic monopoles created therein would be diluted such that none of them are observable today. Thus, we find that the model of cosmic inflation is capable of solving the problems of the standard model of cosmology. However, another puzzle, related to the horizon problem mentioned above, is again found in CMB. Although it is highly isotropic, it exhibits small fluctuations on the order of 10−5, see Bennett et al. (2003). For the same reason that different regions in the sky cannot thermalize, there is also no way to create these irregularities, cf. Weinberg (2008). This, again, can be explained with the inflationary model. Nowadays, it is believed that the rapid cosmic expansion amplified fluctuations of the quantum vacuum and, thereby, created the seeds for the large-scale structures that are imprinted on the CMB, see Liddle and Lyth (2000). We note that all these effects discussed above set certain bounds for characteristic figures of the model of inflation, i.e., the duration of the inflationary phase as well as the factor space expanded. However, as experimental observations on the subject are rare, the particular model is still under debate and open to speculations. Nevertheless, we may study the consequences that are implied by a drastic expansion of space, cf. Birrell and Davies (1982) and Fulling (1989). One intriguing aspect is the creation of particles out of the quantum vacuum, see Schrödinger (1939) and Parker (1968). This is what we will discuss in the next Section.

2.2.2 Particle creation We want to have a closer look on effects a curved space-time imprints on quantum fields, see Birrell and Davies (1982) and Fulling (1989). Here, however, we want to focus on the phenomenon of cosmic particle creation (Schrödinger, 1939; Parker, 1968). To this end, we consider a scalar field Φ, such as the inflaton or Higgs field, with mass parameter m. Accordingly, its quanta are spinless particles, however we note that analog descriptions for vector-bosons or even fermions are possible, cf. Wittemer et al. (2019). In order to study the corresponding particle creation we consider the Klein-Gordon equation ! m2c2 + Φ = 0, (2.2.1) ~2 where c is the speed of light. For simplicity we focus on 1+1 dimensions, hence, the 1 ∂2 ∂2 corresponding d’Alembert operator is given by = c2 ∂t2 − ∂x2 . In order to describe the expanding space-time, e.g., during cosmic inflation, we consider the conformal time t and the corresponding Friedmann-Lemaître-Robertson-Walker metric h i ds2 = a2(t) c2dt2 − dx2 . (2.2.2) 2.2 Cosmology 19

Here, the time-dependent scale parameter a(t) governs the cosmic expansion or contraction. For convenience, we expand the ﬁeld in the eigenfunctions fk of the Laplace operator ∆, which corresponds to a spatial Fourier transform: XZ Φ = fk(x)φk(t) (2.2.3) k

This leads to the equation of motion for the normal modes φk(t)

" 2 4 # ¨ 2 2 2 m c φk + c k + a (t) φk = 0, (2.2.4) ~2 which is the equation of motion of a harmonic oscillator with time-dependent fre- 2 quency Ωk(t) given by

2 4 2 2 2 2 m c Ωk(t) = c k + a (t) . (2.2.5) ~2 It contains the internal contribution c2k2 as well as the externally, i.e., due to cosmic expansion/contraction, implemented dynamics ∝ α2(t) in the second term. Thus, we can understand the mechanism of particle creation in the following way. If the ˙ dynamics is adiabatic, i.e., the rate of change Ωk is small compared to Ωk, the mode can oscillate freely and, thereby, remains in its ground state. However, if the external ˙ variation is too fast, i.e., the adiabadicity condition is violated Ωk 6 Ωk, the wavefunction ψk of the mode cannot adapt to the rapid changing external conditions and, as a result, deviates from the ground state and turns into an excited state. Due to the symmetry of the dynamics imprinted by a2(t), this excited state is a coherent superposition of even harmonic oscillator eigenstates only and, thus, corresponds to a squeezed state of the harmonic oscillator mode φk (Schützhold et al., 2007; Fey et al., 2018). Thus, the ﬁnal state of the mode φk is a squeezed state, which is then given by

1 |ψi = exp ξ∗a a − ξa† a† |0i , (2.2.6) k 2 k k −k k −k k with the corresponding squeezing parameter ξk, cf. Sec. 2.1.3. For small ξk we can expand the exponential in order to obtain

1 |ψi = |0i + ξ |1i |1i + O ξ2 . (2.2.7) k k 2 k k −k k

Thereby, we ﬁnd that the excitation of the mode φk is created in pairs of particles with opposite momenta ±k, complying with momentum conservation, see Schützhold et al. (2007). We note that these particles correspond to lasting excitations of the quantum 20 2 Theoretical concepts

field Φ and, thus, remain even after the external dynamics a(t) stops or slows down. Further, larger squeezing parameters can generate more than two particles, however, they are always created in pairs. In any case, the particle pairs created with the mechanism described above, share entanglement at large distances, see Martín-Martínez and Menicucci (2012). This is best illustrated by considering an alternative picture of the particle creation process. Initially, the mode φk is assumed to be in its ground state |0i. As described in Sec. 2.1.3, the energy of the ground state has a finite value of ~Ωk/2. This finite energy value takes account of the Heisenberg uncertainty principle ∆E∆t ≥ ~/2 and, accordingly, allows for quantum vacuum fluctuations. These fluctuations can be pictured as the random creation and annihilation of virtual pairs of a particle and an anti-particle. Under static conditions, i.e., for a(t) = const., these virtual particle pairs are not directly visible and have only indirect consequences, such as the Lamb shift of atomic spectral lines (Lamb and Retherford, 1947) or van-der-Waals and Casimir forces (Casimir, 1948). In addition, quantum vacuum fluctuations are considered to be responsible for the triggering of spontaneous emission, see Einstein (1917) and Dirac (1927). However, in a curved space-time, e.g., during cosmic expansion, the two partners of such a virtual particle pair can be torn apart until they do not share a common event horizon anymore. Thereby, they have lost causal contact and are unable to recombine and annihilate again, thus, they are transformed to real particles, i.e., lasting excitations of the quantum field Φ. This happens at the point where the physical wavelength λ = 2πa(t)/k of the mode φk, related to the distance of the two particles, crosses the cosmic horizon size, which is determined by the Hub- ble radius. Although now in different event horizons, unable to interact classically, the two (now real) particles are still linked via quantum entanglement, cf. Martín- Martínez and Menicucci (2012). Due to momentum conservation, the two particles have opposite momenta and, thus, measurement of the momentum on one particle immediately projects the other partner into the opposite state. As shown by Hawking (1974, 1975), the entanglement of the particles created in curve space-time (such as at the event horizon of a Black hole), manifests itself as a thermal spectrum of the emitted particles. See also Section 2.1.5 for details on the emergence of mixed states for partial measurements on entangled quantum systems. Thus, we find that curved space-time, such as the cosmic expansion during inflation in the early universe, can tear apart virtual quantum vacuum fluctuations and, thereby, create pairs of real particles that share quantum entanglement at large distances. However, we note that, of course, the actual dynamics and, thus, physical observables, e.g., the amount of entanglement created, strongly depend on the explicit evolution a(t). Since the idea of inflation came up (Guth, 1981), several different models for the potential of the inflation field V (Φ) have been constructed. Most of them are summarized in Martin et al. (2014). 2.3 Open quantum systems 21

2.3 Open quantum systems

This Section is intended to provide the basic concepts of open quantum systems, that will help in understanding the experiments presented in Sec. 4.2. Our descriptions will closely follow Breuer et al. (2016) and we refer, in particular, to Breuer and Petruccione (2007) for further details.

2.3.1 Basics The theory of open quantum systems takes account of the fact that any realistic quantum system S inevitably interacts with its environment E. Thus, the system S can be regarded as a subsystem of some larger quantum system S + E. Due to the interaction with E, we may regard S as an open quantum system. The Hilbert space of the total system is then given by

HSE = HS ⊗ HE, (2.3.1) where HS and HE denote the Hilbert spaces of S and E, respectively. Accordingly, states of the total system are described with density matrices ρSE. Considering the total system S + E to be closed, i.e., without any interactions with its surroundings, the ρSE undergo unitary dynamics that satisfy tr (ρSE) = 1 for all times. Hence, we can write down the Hamiltonian

H = HS ⊗ IE + IS ⊗ HE + HI . (2.3.2)

Here, HS and HE denote the (free) Hamiltonians of the subsystems S and E, respectively, while HI describes the interaction between both, and the IS,E denote the identity operators in HS and HE, respectively. Consequently, the dynamics of the total system is described via the von-Neumann equation: d ρ (t) = −i [H, ρ (t)] (2.3.3) dt SE SE † This equation has the formal solution ρSE(t) = U(t)ρSE(0)U (t), where the time evolution operator U is given by

− i Ht U(t) = e ~ . (2.3.4) In the framework of open quantum systems we are, however, mainly interested in the dynamics of the (open) system S. To this end, we assume an initial product state ρSE(0) = ρS(0) ⊗ ρE(0), i.e., without any initial correlations between S and E, when initialized to the states ρS(0) and ρE(0), respectively. The corresponding time evolution of the state of the open system ρS can then be written as

h † i ρS(t) = trE U(t)ρS(0) ⊗ ρE(0)U (t) (2.3.5) 22 2 Theoretical concepts

Here, trE denotes the (partial) trace over the environment. For a ﬁxed initial state of the environment ρE(0), this equation deﬁnes a linear map Φt, that maps any initial state of the open system ρS(0) to the corresponding state ρS(t) at some later time t ≥ 0:

ρS(0) 7→ ρS(t) = ΦtρS(0) (2.3.6)

The map Φt is called a quantum dynamical map and it can be shown that it is a positive map, i.e., probabilities stay positive under the action of the map. Moreover, Φt is not only positive but completely positive. Thereby, physical states of the total system S + E are mapped to physical states of S + E, even in the presence of correlations between S and E, see Breuer et al. (2016) for details.

2.3.2 Non-Markovianity: Quantum memory eﬀects

The open quantum systems approach, that we introduced in the previous Section, can be used as a framework to define non-Markovian behavior for quantum processes, see Breuer et al. (2016). Here, we recall the construction of a corresponding measure that is based on the distinguishability of quantum states of an open quantum system, originally presented by Breuer et al. (2009). However, we note that other definitions and measures of quantum non-Markovianity exist, which are based on, e.g., the divisibility of the quantum dynamical map (Rivas et al., 2010; Chruściński et al., 2011; Chruściński and Maniscalco, 2014) or the mutual information between the open system and an additional ancilla system (Luo et al., 2012). In classical probability theory, there exists a mathematical definition for stochastic processes to be Markovian in terms of conditional probability distributions, see van Kampen (1992). Here, a Markov process is defined as a stochastic process in which the probability distribution for each of a sequence of events depends only on the state attained in the previous event, not being affected by any events at earlier times. For that reason, such Markov chains are often labeled memoryless processes. In contrast, non-Markovian processes are characterized by processes where the future evolution is determined by the evolution of the previous states, i.e., these processes possess memory of their past. However, this classical definition of non-Markovian behavior cannot be translated into the quantum regime, see Vacchini et al. (2011). Due to the destructive nature of projective measurements, quantum states change randomly conditioned on the measurement outcome and, thereby, their subsequent dynamics are strongly influenced, cf. Sec. 2.1.4. The definition of quantum non-Markovianity developed by Breuer et al. (2009) considers an open quantum system S interacting with its enviroment E (see above). It employs the time evolution of the trace distance D of two quantum states of the 2.3 Open quantum systems 23

1 2 open system ρS and ρS that is given by 1 D(ρ1 , ρ2 ) = tr ρ1 − ρ2 , (2.3.7) S S 2 S S where |ρ| = pρ†ρ. The trace distance represents a metric on the corresponding state space with bounds 0 ≤ D(ρ1, ρ2) ≤ 1. We note that D(ρ1, ρ2) = 0 if and only if ρ1 = ρ2, and D(ρ1, ρ2) = 1 if and only if ρ1 and ρ2 are orthogonal. Thereby, the trace distance features a physical interpretation in terms of the distinguishability of 1 2 the quantum states ρS and ρS of the open system. It is connected to the maximal 1 2 probability Pmax to successfully distinguish ρS from ρS by (Fuchs and van de Graaf, 1999):

1 h i P = 1 + D(ρ1 , ρ2 ) (2.3.8) max 2 S S Further, we note that for any completely positive and trace preserving map Λ, the trace distance D satisﬁes

1 2 1 2 D(ΛρS, ΛρS) ≤ D(ρS, ρS) (2.3.9)

1,2 for all states ρS . Thus, considering again the time evolution of states of the open 1,2 1,2 system ρS (t) = ΦtρS (0), the quantum dynamical map Φ leads to a contradiction of the trace distance:

1 2 1 2 D ρS(t), ρS(t) ≤ D ρS(0), ρS(0) (2.3.10)

1 2 Identifying D with the distinguishability of ρS and ρS and, thereby, the amount of information encoded in the open system S we can interpret any decrease of 1 2 D ρS(t), ρS(t) as a loss of information to the environment. In contrast, any increase 1 2 of D ρS(t), ρS(t) is identified as a backflow of information from the environment into the open system. These quantum memory effects lead to the following definition of quantum non-Markovianity. A quantum process described by a dynamical map Φ 1 2 is Markovian if the trace distance D ρS(t), ρS(t) corresponding to all pairs of initial 1 2 states ρS(0) and ρS(0) is decreasing for all times t > 0. Accordingly, quantum non- 1,2 Markovian behavior is found if there is, at least, one pair of initial states ρS (0) such that the evolution of their trace distance is non-monotonic for some time t > 0. Given this definition, we may also define a measure to quantify the degree of non- Markovian behavior, see again Breuer et al. (2009). To this end, we consider the time derivative of the trace distance d σ t, ρ1,2(0) = D ρ1 (t), ρ2 (t) . (2.3.11) S dt S S 24 2 Theoretical concepts

Then, we can deﬁne a quantitative measure by Z 1,2 N (Φ) = max σ t, ρS (0) dt. (2.3.12) 1,2 σ>0 ρS (0) Explicitly, the maximum is taken over all pairs of initial states and the integration extends over all time intervals (ai, bi) in which σ is positive. Accordingly, we consider an alternative notation for the above deﬁnition:

X h 1 2 1 2 i N (Φ) = max D ρ (bi), ρ (bi) − D ρ (ai), ρ (ai) (2.3.13) 1,2 S S S S ρS (0) i Thereby, we obtain a measure for the degree of quantum non-Markovian behavior that can be evaluated for experimental measurements where the time evolution of the 1,2 quantum states ρS (t) is discretely sampled in time. In particular, we note that no prior knowledge of the dynamical map Φ is required. 3 Experimental methods

In this Chapter, we introduce the experimental tools and methods that we employ in order to perform the experiments presented in this thesis. Most of these methods are described in detail in Wineland et al. (1998a), Leibfried et al. (2003), and Wineland (2013). Therefore, we keep our descriptions brief and refer to these references for more detailed discussions.

3.1 Experimental setup

For the experiments presented in this thesis we operate an experimental setup that has been designed for the trapping of single or few charged magnesium atoms. The essential components of the experimental setup are described in detail in several dissertations (e.g. Friedenauer, 2010; Schmitz, 2009; Clos, 2017) and peer-reviewed publications (e.g. Schaetz et al., 2007). The setup consists of an ion trap, placed inside an ultra-high vacuum chamber, and several laser setups that allow to manipulate and readout the ions’ quantum states. During the course of the experiments reported in this thesis we implemented a new experimental control system (see Sec. 3.1.2), installed arbitrary waveform generators to apply control potentials in real-time (see Sec. 3.1.3), and extended the apparatus with the possibility to change the main trapping voltage also in real-time and, in particular, on short time scales (see Sec. 3.1.2). In the following Sections, we will brieﬂy describe the main parts of the apparatus, while focusing on the implemented extensions that are essential for our experiments.

3.1.1 Paul trap The heart of our experimental setup is a Paul trap (Paul and Steinwedel, 1953; Paul et al., 1958). Here, static and oscillating electric ﬁelds are employed to create a time-averaged conﬁnement for charged particles in three spatial dimensions. Our trap is a (conventional) linear Paul trap that is optimized to trap single or few ions arranged in linear chains, aligned along the trap’s symmetry axis. It was designed and assembled by Hector Schmitz (Schmitz, 2009). Figure 3.1 shows a 3D drawing of the trap’s electrodes and the support structure, as well as projections, illustrating trapping sites and coordinate axes. 26 3 Experimental methods

y x

Figure 3.1: The linear Paul trap used for our experiments. The left image shows a 3D drawing of the trap’s electrodes and the support structures. The center image illustrates the coordinate system in the radial plane (blue arrows). The RF voltage applied to the non-segmented electrodes provides a conﬁning potential in the radial plane, its minimum is marked with the blue circle. The electrode’s dimensions (in mm) and their labels are depicted in the right image. There are two distinct trapping zones with diﬀerent ion-electrode distances of 400 µm and 800 µm. The gaps between the segmented electrodes are 300 µm wide. The experiments reported in this thesis were performed with ions stored in the experimental zone between the segments labeled Z, marked with the blue cross. Ions are observed from above (y-direction) with either an EMCCD camera or a photo multiplier tube (PMT). Figure adapted from Friedenauer (2010).

The confinement along the radial directions (xy-plane) is achieved by applying oscillating voltages of several hundred volts amplitude at a frequency ΩRF/(2π) ≈ 56 MHz to the (non-segmented) radio-frequency (RF) electrodes. In order to provide a confinement along the axial direction and to fine-tune the radial trapping potential the segmented electrodes can be biased with static voltages with respect to the RF ground. There are two distinct trapping zones, a loading zone and an experimental zone, with ion-electrode distances of 800 µm and 400 µm, respectively. Typically, we trap ions in the loading zone and shuttle them to the experimental zone, where they are stored between the segments labeled Z for several hours or days. The 3D potential well provided here confines ions with secular frequencies of several MHz, that can be fine tuned with control potentials, see Sec. 3.1.3.

The Paul trap is located in an ultra-high vacuum chamber with a residual gas pressure of ≈ 10−10 mbar. Thereby, collisions with background gas are highly suppressed −1 to a residual mean collision rate of . 0.001 s , see Enderlein (2012). In combination with typical total trapping potential depths of ≈ 1 eV, we are able to trap ions for several hours without any laser cooling. 3.1 Experimental setup 27

3.1.2 Experimental control To operate the experimental setup we employ an experimental control system, that is schematically depicted in Fig. 3.2. It is based on a real-time pulse sequencer with timing resolution of 10 ns, that is controlled via an in-house developed software called EIOS. A detailed description of the software can be found in Kalis (2017), whereas for details on the hardware of the pulse sequencer we refer to Pham (2005).

controls

controls controls Control Potentials User EIOS AWG

RF amplitude Lowpass Filters control IF LO RF Ion control Lasers R+S RF Mixer Paul Trap Microwave

Fluorescence

Photon counts PMT

Figure 3.2: Schematic illustration of the real-time experimental control system (details see text). The user controls the setup via the software EIOS, that manages the lasers, the microwave antenna, and the arbitrary waveform generator (AWG) using the real-time pulse sequencer. In addition, EIOS counts ﬂuorescence photon events via a PMT. In the course of the experiments for this thesis we implemented the AWG, allowing for real-time control of control potentials and the RF voltage supply for the Paul trap via an RF mixer. These are highlighted in orange, whereas already present electric real-time control is marked with gray color.

During experimental sequences, ions are manipulated with electric fields, microwave, and/or laser radiation. The real-time control for microwave and laser interaction, as well as the ion fluorescence detection via a photo-multiplier tube (PMT) has been set up during previous works, cf. Friedenauer (2010), Schmitz (2009), and Clos (2017). In total, it features 4 independent direct digital synthesizer (DDS) channels, 16 transistor-transistor logic (TTL) outputs, and 4 digital analog converters (DACs). In this work, the experimental control system was extended with an arbitrary waveform generator (AWG), see Bowler et al. (2013) and Wittemer (2015) for details. It features 36 independent 16-bit DACs (Analog Devices AD9726), that are on-board amplified using high speed-amplifiers (Analog Devices AD8250) to reach an output voltage range of ±10 V. For particular applications that need high voltages, the AWG’s ouptut channels can be externally amplified by using high-voltage oper- 28 3 Experimental methods ational amplifiers (Apex PA85). The DACs are controlled by a field-programmable gate array (FPGA) (Xilinx Spartan-3E XC3S500E PQ208), that is running with a timing resolution of 10 ns, synchronized with the pulse sequencer via TTL pulses. In our setup, the AWG is capable of real-time controlling the ions’ total trapping potential via two distinct mechanisms: (1) With dedicated control potentials, applied to the Paul trap’s control electrodes, and (2) by setting the voltage amplitude on the RF electrodes. For (1) we employ 6 independent AWG channels, that are externally amplified to a voltage range of ±200 V, to apply previously calculated control potentials to the Paul trap, see Sec. 3.1.3. Using voltage adders, also based on PA85 operational amplifiers, we can, in addition, fine-tune the voltages for dedicated control electrodes with 4 additional (unamplified) AWG channels. In order to minimize electric field noise that can potentially induce incoherent ion motion (see Sec. 3.3.3), the voltages for the control electrodes are low-pass filtered outside of the vacuum chamber with typical filter cutoff frequencies of fcut ≈ 1 kHz. For (2) a single AWG output channel is connected to an RF mixer (Mini-Circuits ZMY-3), controlling the signal amplitude of the RF signal, that is generated by a Rohde&Schwarz (R+S) SMG signal generator and amplified by a narrow-band power amplifier (Dirk Fischer Elektronik). By adjusting the IF signal of the mixer we can modulate the output signal (at the RF port) and thereby modify the RF power on the Paul trap’s electrodes. This allows to manipulate the trapping potential and, thereby, the ions’ motional mode frequencies on short time scales, see Secs. 3.1.4 and 4.1. While ions in our trap are typically stored for several hours, the characteristic time scales for the dynamics of our magnesium ions is in the microsecond range. This allows for the effortless accumulation of large statistical samples with the same ion, well-suited for high-precision measurements. In general, we take experimental data with the following hierarchical structure: A single experimental data point consists of 200-500 repetitions of an experimental cycle with fixed parameters. A typical measurement sequence consists of approximately 100 data points, where only one parameter is varied. In Figure 3.3 we depict the basic framework for a single experimental cycle, that may be composed by a variety of building blocks with dedicated purposes. 3.1 Experimental setup 29

Experimental cycle

Paul trap initialization Ion state preparation Experiment Ion state detection 200 µs 3000 µs 150 µs 150 µs ≈ ≈ ≈ ≈

• Set RF voltage • Doppler cooling • Simulate open • Tomography pulse • Set control potentials • Sideband cooling quantum system • Spin-Motion mapping • Ion ordering • Motional heating • Simulate particle • Fluorescence detection • Check ion order • Optical pumping creation • MW spin ﬂip

Figure 3.3: Building blocks of a single experimental cycle, for details see text. Experiments start by initializing the Paul trap with a particular RF voltage and a dedicated set of control potentials. Subsequently, the ions’ motional and electronic quantum states are initialized. Afterwards, the analog quantum evolution is simulated. Finally, the ions’ quantum state is detected. Typically, such a cycle is repeated 200-500 times for a given set of parameters to form a single data point. Several (≈ 100) data points, where only a single experimental parameter is varied, form a typical measurement sequence.

Typically, our experiments start by initializing the Paul trap, i.e., setting the RF voltage and control potentials, see Sec. 3.1.3, implementing the ions spatial conﬁgu- ration of choice. This can also include the deterministic preparation of particular ion orders for mixed isotope ion crystals (Clos, 2017) with subsequent ordering veriﬁca- tion, see Sec. 3.3.4. Afterwards, the ions’ internal (electronic) and external (motional) degrees of freedom are initialized to the desired quantum states via established cooling, optical pumping and electron shelving tools, cf. Leibfried et al. (2003). This can involve cooling and heating to reach thermal states of desired amplitude, see Secs. 3.3.1 and 3.3.3, and a coherent preparation of the desired initial spin state using microwave (MW) or laser transitions, see Sec. 3.2.1. Following this, we implement the time evolution of the analog quantum simulation. For the experiments presented in this thesis this is either the simulation of an open quantum system, see Sec. 4.2, or the analog quantum simulation of the particle creation during a cosmic expansion, described in Sec. 4.1. The experimental cycle concludes with the detection of the ions’ quantum state(s), possibly involving full spin state tomography, see Sec. 3.2.3, and motional state detection, see Sec. 3.4.2. In the following, we describe the essential building blocks of our experiments in detail. 30 3 Experimental methods

3.1.3 Control potentials Our trap features six control electrodes at the experimental zone, that allow to su- perimpose the RF trapping potential with additional (static) control potentials. The basic procedure of the calculation of these additional control potentials is described in detail in Mielenz et al. (2016) and Wittemer (2015). In the following, the approach is briefly reviewed. Each control electrode can be biased with an individual voltage with respect to the RF ground, which leads to an electric field at the position of the ion with amplitude and direction depending on the particular control electrode. The potential Φc(Uc) corresponding to the field generated by a single control electrode biased to a voltage Uc is superimposed with the pure RF trapping potential ΦRF. Taking all six available control electrodes l into account results in the total trapping potential P6 Φtot = ΦRF + l=1 Φl(Ul). In order to be able to design control potentials of dedicated purposes, we want to calculate the individual contributions Φl(Ul). To this end we perform a boundary-element method (BEM) calculation of the electric potential landscape in our trap, cf. Nicolet (1995). For the BEM calculation the surface of each electrode is split into several triangles and the individual electric potential contribution for each triangle is calculated at a given point of interest. By superpositioning all the individual electric potentials, the total trapping potential as a function of the individual control voltages can be calculated. The BEM method we employ is based on the FastCap algorithm (Nabors and White, 1991) with the extension to solve Laplace-type problems (FastLap), that was developed at the Massachusetts Institute of Technology. Figure 3.4 exemplarily illustrates the distribution of the triangles onto our electrodes, zoomed in on the experimental zone. 3.1 Experimental setup 31

Figure 3.4: Illustration of the boundary element method. The trap electrode surfaces are split into several triangles and the ﬁeld contribution of each triangle is evaluated in all the grid points in the pink cube. The cube has a side length of 100 µm and contains 64000 equidistantly spaced grid points, that are used to interpolate the harmonic trapping potential, and calculate the electric ﬁeld vectors of each control electrode.

The accuracy of the BEM calculation depends on number and size of the triangles used, which is why we increase the triangle density in regions near the ions. In total, we distribute 44, 0000 triangles on the electrode’s surfaces. The pink cube has a side length of 100 µm and marks the region in which the individual field contributions are evaluated. It contains 403 = 64, 000 equidistantly spaced evaluation points, that characterize the relevant effect of each control potential. In this configuration the calculation with the FastLap algorithm requires ≈ 12 GB of RAM. In order to design control potentials of specific purposes we rely on the superposition of the 6 electric potentials generated by the control electrodes, i.e., we are able to specify 6 independent degrees of freedom. However, in the harmonic approximation, the potential generated by a static voltage is fully described with 8 degrees of freedom (DOF), composed by 5 potential curvatures, that define mode frequencies and orientations, and 3 electric field contributions (potential gradients). Hence, we can only partially control the harmonic potential with our electrode geometry. Moreover, the directions of the electric field vectors of the control electrodes are highly corre- lated, because all the control electrodes lie on opposing blades of the linear Paul trap. Hence, in general, dedicated control potentials may require voltage amplitudes that exceed the electric specifications of the AWG (and optional amplifiers) or the vacuum feedthrough. In order to calculate control potentials with dedicated effects and within the capabilities of our setup we employ a numerical optimization algorithm, 32 3 Experimental methods that effectively minimizes the required voltage amplitudes on the electrodes while still fulfilling the desired conditions of the created control potential. This leads to non- trivial combinations of voltages on the individual control electrodes, but provides us with versatile control potentials that, for example, apply homogeneous electric fields at the trapping minimum in all three spatial directions. This is enabled by the potential curvatures and gradients implied by the gaps between the electrodes and the trap’s finite extension along the axial direction. In the following, we briefly discuss different control potentials that are of particular interest for the experiments reported in this thesis. As a first example, we consider three electric field control potentials, Ex, Ey (radial direction), and Ez (axial direction), that allow to apply electric field amplitudes of several kV/m in three linearly independent directions. We employ these control potentials to compensate stray electric fields present in our setup, that can lead to excess micromotion, cf. Berkeland et al. (1998) and Keller et al. (2015). Typically, we coarsely compensate these stray electric fields by analyzing the correlation between fluorescence photons of the ion and the trap’s RF drive (Berkeland et al., 1998; Schmitz, 2009) and iteratively adjusting Ex, Ey, and Ez. Further, we design control potentials that manipulate the curvatures of the trapping potential and, thus, tune the ions’ motional mode frequencies and orientations. This is essential for the implementation of the desired harmonic oscillator contribution to our quantum simulations, see Secs. 4.2 and 4.1. Additionally, motional mode orientations can be critical for efficient laser cooling of the ions’ motional states, see Sec. 3.3.1. In order to calculate dedicated curvature control potentials, we constrain the optimization algorithm to vanishing homogeneous electric fields at the position of the ion, while specifically demanding significant curvatures along dedicated directions. As an example, we consider a curvature control potential Hzz, that is designed to tune the axial mode frequency in a controlled fashion. Figure 3.5 illustrates the effect of this control potential on the motional mode frequencies of a single ion. 3.1 Experimental setup 33

3 10 MHz)

π 5 2 0

1 Voltage (V) 5 −

Mode frequency (2 0 10 0 0.5 1 1.5 − 0 0.5 1 1.5

Control potential coefﬁcient κzz Control potential coefﬁcient κzz

Figure 3.5: Application of the axial curvature control potential Hzz. The left plot depicts the motional mode frequencies of a single ion as a function of the amplitude κzz of the applied control potential κzz · Hzz. The curvature design value (solid lines) for the (axial) low-frequency mode is 1 MHz for κzz = 1, from a model fit (not shown) we extract a control potential fidelity of ≈ 95 %. The right plot illustrates the individual voltages supplied to the six control electrodes used to apply Hzz. The location and size of each individual electrode leads to variable slopes, whereas the individual offsets reflect the compensation of stray electric fields using the control potentials Ex,y,z.

We depict the three individual mode frequencies ωi as a function of the amplitude- scaling coefficient κzz, i.e., for variable control potentials κzz ·Φzz. The frequencies are experimentally measured using the motion excitation method described in Sec. 3.3.2. Without any control potentials applied, the pure RF potential is only weakly confining along the (axial) z-direction, with a motional frequency of several 10 kHz only, whereas radial mode frequencies are on the order of several MHz for typical RF amplitudes. The control potential Hzz is designed to generate an axial mode frequency ω/(2π) = 1 MHz for a coefficient κzz = 1, without taking the contribution of the RF potential into account. From a model fit we extract a fidelity of ≈ 95 % with respect to the design value, which may be explained by geometric trap manufacturing imperfections. As implied by Laplace’s equation ∆Φc = 0 for static electric potentials, the increase of the axial trapping frequency due to the application of Hzz also affects the radial trapping frequencies, and, thereby, their individual orientations. The right plot of Fig. 3.5 illustrates the individual voltages used to apply Hzz, the variable slopes reflect the effect of different sizes and ion-electrode distances of the individual electrodes. In addition, the offset voltages for κzz = 0 correspond to the individual voltages used to compensate stray electric fields using the control potentials Ex,y,z. The orientations of the ions’ motional modes in the radial plane is, without any control potentials applied, given by the (slightly asymmetric) geometry of the electrodes and the RF voltage amplitude. For efficient laser cooling of the radial modes it can be necessary to rotate the modes using additional control potentials. To this end 34 3 Experimental methods

we design a control potential Hxy, that is dedicated to rotate the radial modes around the axial direction, while also keeping the axial mode frequency unaltered. This is achieved by specifically requesting non-vanishing contributions the off-diagonal elements in the control potentials curvature tensor when executing the voltage minimization algorithm, cf. Wittemer (2015). Figure 3.6 illustrates the effect of the resulting control potential on the radial motional modes of a single ion in our trap.

80 MHz) 5 π 60

40 4

Mode angle (° ) 20

Mode frequency (2 3 0 4 2 0 2 4 4 2 0 2 4 − − − − Control potential coefﬁcient κxy Control potential coefﬁcient κxy

Figure 3.6: Effect of the radial curvature control potential Hxy. The left plot depicts the motional mode frequencies in the radial plane of a single trapped ion as a function of the amplitude κxy of the applied control potential κxy · Hxy. The axial mode frequency (not shown) is not affected by Hxy within our measurement uncertainties. From the model fit (solid lines) we are able to calculate the mode orientations in the radial plane, parametrized by the angle between the mode and the x-axis, shown in the right plot. We note that already for κxy = 0, both mode angles are non-zero, which is due to the asymmetric trap geometry.

By setting off-diagonal elements in the control potentials curvature tensor, the mode frequencies and orientations of the total potential are altered. From a model fit to the measured mode frequencies we can deduce the mode angles with respect to the x- direction, cf. coordinate system in Fig. 3.1. The design of our trap ensures a non-zero mode angle already for κxy = 0 for typical RF voltage amplitudes, see Schmitz (2009). However, employing Hxy allows to actively fine-tune the radial mode orientations, which can be crucial for efficient laser cooling and the spin-phonon coupling, see Sec. 3.4.1. In addition, we note that we also design control potentials for mode rotations around the other coordinate axes, e.g., Hxz which we utilize to align the lf- mode with the trap’s symmetry axis. This is of particular importance when working with multiple ions, where we, thereby, align the axis of the ion chain with the trap axis. 3.1 Experimental setup 35

3.1.4 Radio-frequency voltage control

In addition to the control potentials applied via dedicated control electrodes we implement an AWG control channel that allows to manipulate the RF voltage U in real-time. Thereby, we can modify the RF pseudo-potential, that is mainly inﬂu- encing the trapping potential along the radial directions (xy-plane). In Figure 3.7 we depict a single ion’s motional mode frequencies, determined via the electric ﬁeld excitation method described in Sec. 3.3.2, as a function of U.

5 1

4 max

U 0.9 3 / U ) (MHz) π 2 (2 0.8 ω/ 1 Voltage

0 0.7 0.7 0.75 0.8 0.85 0.9 0.95 1 0 1 2 3 4 5 6 7 8

Trap supply voltage U/Umax Duration (µs)

Figure 3.7: RF voltage control. We depict the secular mode frequencies of a single ion in our trap, determined using the electric field excitation method described in Sec. 3.3.2, as a function of the RF voltage amplitude U. Only radial modes (orange and blue) are affected by U, while the axial mode (gray) is mainly influenced by control potentials. The switching of U can be done on short timescales (right). The blue line depicts the AWG signal that is connected to the IF port of the RF mixer. It controls the amplitude of the output signal at the RF port that is then fed through the high-power amplifier to the helical resonator of the trap, cf. Fig. 3.2. By monitoring the RF voltage with a control electrode we can infer the real-time switching of U (orange).

We show three distinct motional mode frequencies, two in the radial plane (orange and blue) and one along the axial direction (gray), that is not affected by U. Thus, this control channel can be used individually tune radial mode frequencies while keeping the axial confinement constant. In addition, by using an RF mixer to control U, the manipulation of the trapping potential can be applied on short time scales, limited by the quality factor of the helical resonator in our setup of Q ≈ 92, cf. Schmitz (2009). With our RF frequency ΩRF/(2π) = 56 MHz, this corresponds to a ring-down duration (1/e) of τ = 2Q ≈ 0.5 µs. Hence, we may apply waveforms to modify U on ΩRF these (short) time scales. We choose to switch U with a smooth ramp, intended to mitigate voltage jumps with undefined, non-reproducible behavior. To this end we consider the following waveform for a single (down-)ramp as output for our arbitrary 36 3 Experimental methods waveform generator that controls the U via the (non-linear) output of the RF mixer:

5 4 3 UAWG(t) = UAWG,high − (UAWG,high − UAWG,low) · (6t − 15t + 10t ), (3.1.1) where UAWG,high and UAWG,low are the AWG output voltages for the RF amplitudes Uhigh and Ulow, respectively, and tramp denotes the nominal ramp duration. However, we note that the effective ramp is shorter than tramp due to the choice of the smooth step waveform. In our experiments we switch between different values for U in closed cycles. After switching from UAWG,high to UAWG,low and typical holding durations of a few µs, we switch back to UAWG,high with the equivalent upramping waveform. Thereby, we ensure stable (thermal) conditions for our RF resonator setup. This is, in particular, relevant for the long-term stability of motional mode frequencies in the radial direction. Figure 3.7(right) shows an example for a real-time switching application of the RF voltage amplitude U. We depict the input waveform at the RF mixer’s IF port UAWG (blue) as well as U (orange), as inferred by the RF power on a control electrode, to which the RF signal couples capacitatively on. The depicted example waveform has a programmed, i.e., nominal, switching duration of tramp = 1 µs. However, we note a distortion of the waveform of U with respect to the input that is due to the residual, but finite bandwidth limitations of our switching capabilities. Moreover, we observe a finite shift ≈ 1 µs in U(t). In our expeirments, we account for this shift by an appropriate delay tdelay in our sequences. We note that the switching of the trapping potential on microsecond timescales is the crucial requirement for the squeezed state generation by tuning the oscillator frequency, as presented in Sec. 4.1.

3.1.5 Magnesium The atom species of choice for our experiments is magnesium, see Friedenauer (2010) and Schmitz (2009) for details. Magnesium has three stable isotopes with nuclear masses 24 u, 25 u, and 26 u, where u is the unified atomic mass unit u ≈ 1.66 × 10−27 kg. From these, only 25Mg has a nuclear spin (I = 5/2) and, thus, a hyperfine structure. In our experiments we employ the hyperfine splitting to implement a (spin-1/2) two-level system (see Sec. 2.1.2) for our quantum simulations by applying a quantization magnetic field. Figure 3.8 shows the energy levels of 25Mg+ relevant for our experiments. 3.1 Experimental setup 37

D transition Fine structure Hyperﬁne structure Zeeman splitting

F = 1 F = 2 P 3/2 F = 3 F = 4 m = -F,...,F 4, 4 P F | i

F = 2 P / 1 2 F = 3 m = -F,...,F 3, 3 F | i

1.0 PHz ≈

F = 2

|↑i 1.8 GHz S S1/2 ≈ |↓i

F = 3

mF -4 -3 -2 -1 0 1 2 3 4

Figure 3.8: Energy level scheme of 25Mg+, not to scale. The Zeeman splitting induced by a quantization magnetic field is used to encode a spin-1/2 system in the ground state manifold. To this end we define the states |↓i = S1/2 |F = 3, mF = 3i and |↑i = S1/2 |F = 2, mF = 2i as the corresponding two-level system. For our magnetic field amplitude |B| ≈ 0.585 mT the splitting between adjacent levels in the ground state manifold is in the MHz regime. For optical pumping, cooling, and state detection of the trapped ions we couple the ground state manifold to the levels P1/2 |3, 3i and P3/2 |4, 4i using ultra-violet light with λ ≈ 280 nm. 38 3 Experimental methods

We depict the D transition of 25Mg+ with the hyperfine structure including the level splitting induced by a quantization magnetic field with amplitude |B| ≈ 0.585 mT. The spin system is encoded in the hyperfine structure of the ground state manifold in the levels |↓i = S1/2 |F = 3, mF = 3i and |↓i = S1/2 |F = 2, mF = 2i. The transition frequency between |↓i and |↑i is approximately ωz/(2π) ≈ 1775 MHz, making it readily accessible with microwave radiation. For our quantization magnetic field amplitude the splitting between adjacent levels within the ground state manifold is in the MHz regime, which can be adressed with RF radiation, see Hakelberg et al. (2018). Further, laser transitions in the ultra-violet range (λ ≈ 280 nm), coupling S1/2 to P1/2 or P3/2, are employed for optical pumping and qubit state detection (see Sec. 3.2.1), as well as Doppler cooling (see Sec. 3.3.1) and two-photon stimulated-Raman (TPSR) transitions (see Sec. 3.4.1).

3.1.6 Optical setup We operate a versatile optical setup to manipulate and observe the ions in the Paul trap. This involves a photoionization laser, labeled PI, lasers for Dopper cooling and optical pumping, labeled BD and RD, and various laser beams for two-photon stimulated-Raman (TPSR) transitions, labeled B1, B3, R1, R2, and R3. For details on the optical setup we refer to Friedenauer (2010) and Clos (2017). Figure 3.9 depicts the alignment of the laser beams with respect to the Paul trap and the quantization magnetic ﬁeld axis. 3.1 Experimental setup 39

PI, BD

R2 B3

R3 B TPSR ∆~k z B1+R2 x

trap axis B3+R2

0 B1+R1 0

R1 BD, RD B σ+ B1 100 mm

Figure 3.9: Schematic of the Paul trap in the vacuum chamber illustrating the laser beams for preparation, manipulation and detection. Laser beams (blue arrows) enter the vacuum chamber through five beam ports and are aligned to either the loading zone or the experimental zone (orange dots). The application of a magnetic field (orange arrows) with air-cooled solenoids (orange bars) defines their individual polarizations. The inset table illustrates the different combinations of the TPSR beams B1, B3, R1, and R2, that allow for different configurations for the spin-phonon coupling. Ions are detected from above (y-direction) with either a PMT or an EMCCD camera (both not shown). Figure adapted from Friedenauer (2010).

Laser beams enter the vacuum chamber through five beam ports. This allows for versatile cooling schemes and different configurations for the spin-phonon coupling using TPSR transitions, see Sec. 3.4.1. The specifications for all the different laser beams are listed in Tab. 3.1. 40 3 Experimental methods

Label Source λ Pol. Transition Γnat/(2π) ∆/Γnat I/Isat (nm) (MHz) PI Dye×2 285.296 S0→P1 78.1 100 BD Fiber×4 279.635 S1/2→P3/2 41.8 BD σ+ -0.5 0.5 BDD σ+ -10 20 BDX σ+ -0.1 0.2 RD Fiber×4 280.353 S1/2→P1/2 41.3 RD σ+ -0.5 0.5 RP σ+ -0.5 1 3 5 Raman Fiber×4 279.6 S1/2→P3/2 41.8 < 5×10 < 10 B1 π < 103 B3 π < 103 R1 σ++σ− < 3×103 R2 σ+−σ− < 3×103

Table 3.1: Labels and parameters of the laser systems. We list laser sources, wavelength λ, polarization, target transition in magnesium, natural line width Γnat of the transition, relative detuning, and typical saturation parameters I/Isat for all laser beams. The table is adapted from Clos (2017).

The total laser setup, including frequency doubling and frequency stabilization techniques, is described in detail in Friedenauer et al. (2006) and Friedenauer (2010). It is optimized to allow for ion production, detection, and manipulation with high fidelities and in a broad range of parameters. In the following we briefly define labels and individual purposes of all laser beams used for our experiments. We produce magnesium ions by ionizing neutral magnesium atoms via a two-photon transition between S0 and P1 using the photo-ionization laser PI, cf. Madsen et al. (2000). By tuning its frequency to the corresponding atomic transition we are able to selectively trap particular isotopes of magnesium. This is of particular importance in order to trap hybrid magnesium ion crystals, cf. Clos et al. (2016) and Clos (2017). For Doppler cooling as well as optical pumping of the ion’s internal state we operate the Blue Doppler (BD) beams. They are σ+-polarized and tuned to the transition S1/2→P3/2. From a single frequency-quadrupled fiber laser we generate three distinct BD beams using acousto-optical modulators (AOMs). The far red-detuned (∆ = −10Γnat) beam labeled BDD (pre-)cools ions with large velocities until they can be efficiently Doppler cooled with the less detuned (∆ = −Γnat/2) beam BD to temperatures close to the Doppler limit, cf. Sec. 3.3.1. A third BD laser beam, BDX, is even less red-detuned (∆ = −Γnat/10) and serves as a detection laser for the ion’s internal state by resonance fluorescence, see Sec. 3.2.1. In order to increase the internal state preparation fidelity and for additional Doppler cooling we employ a second frequency-quadrupled fiber laser, labeled Red Doppler 3.1 Experimental setup 41

(RD). If the ion is in a state in the S1/2 manifold other than the qubit states |↓i and |↑i, repumping lasers can recover the corresponding populations. To this end the RD is tuned to the transition S1/2→P1/2 and split into two beams using AOMs. The beam RD couples to the F = 2 states of the S1/2 manifold, while the beam RP addresses the F = 3 states. Both beams are, like the BD beams, σ+-polarized in order to drive transitions with ∆mF = +1. A detailed simulation of the optical pumping effects of the RD lasers for all the states in the S1/2 manifold can be found in Schmitz (2009). Our setup includes a third frequency-quadrupled fiber laser that is used to drive the TPSR transitions. It is tuned near the S1/2→P3/2 transition. Using AOMs it is split into five distinct beams, two blue Raman beams B1 and B3, and three red Raman beams R1, R2 and R3. However, for the experiments reported here the R3 beam is not used, thus, not discussed further. In order to drive TPSR transitions a blue and a red Raman beam are turned on simultaneously. Setting their frequency difference ∆ω = ωb − ωr close to the qubit transition ωz allows to drive two-photon transitions via absorption of a photon from one beam and stimulated emission of a photon into the other beam. The momentum transfer corresponding to this transition depends on the angle between the two Raman beams and is summed up in the Table inset in Fig. 3.9. To minimize scattering of photons on the P3/2 level the absolute frequency of the Raman beams is detuned by ≈ 2π × 100 GHz. In order to still drive TPSR transitions with reasonable Rabi rates, the individual beams have UV beam powers of several mW. Further details on the TPSR transitions, with a focus to the spin-phonon coupling are given in Sec. 3.4.1. The readout of the ion’s quantum states is performed by detecting resonance fluorescence. Typically, this is done by illuminating the ions with the BDX laser and collecting the induced fluorescence with an objective. Using a motorized flip mir- ror we can choose whether to analyze the fluorescence with an EMCCD camera or a PMT, that allows for fast readout. Details on the imaging optics can be found in Friedenauer et al. (2008). In Figure 3.10 we depict exemplary EMCCD images of ions in our trap. 42 3 Experimental methods

Figure 3.10: EMCCD images (colorized) of magnesium ions in our trap. We can trap single or multiple ions in a common potential, that arrange in a linear chain (top left) or a two-dimensional structure (right), depending on the curvatures of the trapping potential. In the bottom left image, three 25Mg+ ions and one 26Mg+ (marked with the dashed circle) ion are trapped. The latter doesn’t ﬂuoresce because the detection laser is far detuned due to the isotope shift. However, it can be indirectly detected by the positions of the 25Mg+ ions. Figure adapted from Clos (2017).

3.2 Spin control

We implement a spin-1/2 system in the Zeeman sublevels of the Hyperﬁne ground state manifold of 25Mg+, see Sec. 3.1.5. The transition between the qubit states |↓i and |↑i can be realized via laser or microwave interaction. In the following sections we provide a brief overview on how to initialize, manipulate, and detect quantum states in the two-level system.

3.2.1 Initialization and detection

Typically, we initialize an ion’s spin state by preparing |↓i via optical pumping with the σ+-polarized laser beams RD and BD. Every absorption of a σ+-polarized photon induces a transition with ∆mF = +1, successively populating |↓i. Once the state |↓i is reached, the ion undergoes a cycling transition between |↓i = S1/2 |3, 3i and P3/2 |4, 4i with the BD lasers. For the RD lasers |↓i is a dark state, as there is no mF = 4 level in the P1/2 manifold. From calibration measurements (see below) we extract preparation infidelites for |↓i below the percent level, which we attribute to imperfect laser polarizations. In order to prepare hyperfine states other than |↓i, we employ electron shelving techniques, cf. Leibfried et al. (2003). Starting from |↓i, the state |↑i can be prepared with the application of a resonant microwave (or TPSR) pulse of appropriate length to form a π-pulse. Analogously, other hyperfine states of the F = 2 manifold may be reached with microwave pulses of different frequencies, while other states of the F = 3 manifold can be prepared via combinations of microwave π-pulses, or, alternatively, 3.2 Spin control 43 via RF pulses, that induce direct transitions within the F = 3 manifold, see Hakelberg et al. (2018). At the end of each experimental cycle (cf. Sec. 3.1.2) we detect the spin state using the cycling transition of the BD lasers. The BDX laser is tuned close to resonance to the transition (see Sec. 3.1.6), inducing fluorescence only for |↓i, while the scattering of photons is highly suppressed if the ion is in |↑i. Taking into account the finite aperture of the imaging optics as well as their individual efficiencies, we calculate an overall detection efficiency of ≈ 5.6 ‰ (Friedenauer, 2010). For typical intensities of the BDX laser (I ≈ 0.2 · Isat) this corresponds to a theoretical photon count rate with the PMT on the order of ≈ 1 × 105 Hz for |↓i and negligible count rates for |↑i. In order to extract the populations P↓,↑ of the states of the two-level system from the fluorescence measurement we analyze the histograms of the photon counts. Figure 3.11 illustrates the basic procedure of the analysis.

1 R 130 kHz R 2 kHz 0.8 F ≈ F ≈ λ = 0.01 0.6 ↑ w = 0.999(1) w = 0.017(4) ↓ ↓ 0.4

Probability λ = 5.43 0.2 ↓ 0 0 2 4 6 8 10 12 0 2 4 6 8 10 12 0 2 4 6 8 10 12 Photon counts in 40 µs

Figure 3.11: Analysis of the photon count histograms to extract the populations of |↓i and |↑i, P↓ and P↑. The data presented here corresponds to the microwave flop depicted in Fig. 3.12. Initially, we fit two Poissonians to the combined histogram of all the data points (left plot) to extract the individual fluorescence levels λ↓,↑ of the states |↓i and |↑i. Subsequently, for each individual data point we fit two Poissonians with fixed positions λ↓,↑, but free weights w↓ and w↑. These correspond to the populations P↓ and P↑. The data in the center plot correspond to the spin state after the initialization routine, whereas the right plot corresponds to the state after an additional microwave π-flip to prepare |↑i.

Typically, we repeat each experiment 200-500 times and count fluorescence photons, see Sec. 3.1.2. These counting events follow Poissonian statistics, hence, the corresponding histograms are superpositions of two Poissonian distributions, one for each level. From a model fit to the combined histogram of all data points for a given measurement sequence we determine the individual photon count rates λ↓,↑ for |↓i and |↑i, see Fig. 3.11 (left). For each data point of the measurement sequence we fix 44 3 Experimental methods

the λ↓,↑, the positions of two Poissonian distributions, and fit only their statistical weights w↓,↑. These weights correspond to the populations of the two-level system P↓ and P↑, including their statistical uncertainties. Thereby we are able to translate fluorescence photon count rates RF to qubit state populations. From the data shown in Fig. 3.11 we can extract a combined preparation and detection fidelity of 99.9(1) % for |↓i and 98.3(4) % for |↑i. We note that, here, the detection error corresponds to measurements along the quantization axis only, cf. Sec. 2.1.4. For the full spin state tomography, i.e., complementary measurements along different axes in the Bloch sphere, see Sec. 3.2.3.

3.2.2 Manipulation

The frequency splitting of the qubit states is ωz/(2π) ≈ 1775 MHz. Thus, the corresponding energy gap can be readily addressed with microwave radiation. To this end, we use a quarter-wave antenna to couple microwave radiation into the RF resonator. Typically, we use ≈ 2 W of microwave radiation, which allows for coherent manipulation of the ion’s spin state with a Rabi rate ΩMW/(2π) ≈ 80 kHz. In Figure 3.12 (left) we show a corresponding Rabi ﬂop, starting in |↓i.

0.8 ↓ P 0.6

0.4

Population 0.2

0 0 20 40 60 80 100 0 20 40 60 80 100 Microwave interaction duration (µs) TPSR interaction duration (µs)

Figure 3.12: Coherent spin state manipulation of a single 25Mg+ ion using microwave radiation (left) and a TPSR transition with co-propagating laser beams (right). We initialize |↓i, drive the interaction for variable duration and detect the population P↓. Data points depict experimental results (error bars partially smaller than symbol size) and solid lines indicate model ﬁts. Decoherence effects are increased for the laser coupling due to beam pointing and intensity ﬂuctuations.

The qubit coupled to resonant microwave radiation corresponds to an interaction ~ΩMW ~ΩMW Hamiltonian Hint = 2 (σ+ + σ−) = 2 σx, see Sec. 2.1.2. For typical experimental time scales, in particular for π and π/2 pulses, we can neglect any decoherence eﬀects and consider the corresponding evolution to be unitary. This is of particular interest to eﬃciently prepare |↑i, after initializing |↓i via optical pumping, cf. Sec. 3.2.1 3.2 Spin control 45

The coherent coupling of the qubit states, can also be realized with TPSR transitions. Figure 3.12 (right) illustrates the driving of the two-level system with a motional-insensitive (∆~k = 0) Raman beam combination B1+R1. In principle, this ~Ω0 leads to the equivalent interaction term Hint = 2 σx, where Ω0 is the Rabi rate, given by the intensities of the Raman beams. Here, however, decoherence effects can be observed, which we attribute to laser beam pointing and intensity fluctuations. We note that the spin state can be coherently manipulated using motional-sensitive TPSR transitions as well. In this case, the corresponding P↓, depend on the ion’s motional state, see Sec. 3.4.1. For both, laser as well as microwave interaction, certain phase control can be implemented. We use direct digital synthesizers (DDSs) to set frequency and phase offsets of the corresponding interaction fields. For the microwave, the DDS phase directly corresponds to a phase offset of the microwave field. In case of the laser interaction, the DDS phase is converted to a phase of the acoustic wave inside of the AOM before it is translated to a phase offset of the light field. In general, with our experiments we are sensitive to relative phase shifts of the interaction fields, which we may employ to perform the full spin state tomography, that is described in the next Section.

3.2.3 Spin tomography In this Section, we describe how the projective measurements along the quantization axis (see Sec. 3.2.1) and the coherent driving of the spin system that is explained in the previous Section are combined to perform a full spin state tomography. The method is based on unitary rotations of the Bloch vector ~rB, that corresponds to a spin state |si. From the resonance fluorescence measurement we extract the probability P↓(|si), that is equivalent to the expectation value of the spin observable along the quantization axis hσzi = 2P↓(|si) − 1. In order to extract the projections onto the axes orthogonal to the quantization axis, hσxi and hσyi, we perform unitary rotations of ~rB, before the resonance fluorescence measurement is applied. The rotation pulses can be described in terms of the Pauli matrices σx,y,z, see Sec. 2.1.2. In order to measure hσxi we apply a rotation of θ = π/2 around the σy-axis, given π −i σy by the rotation matrix Ry = e 4 . This rotates the σx-component of the Bloch vector onto the measurement axis, such that for the subsequent fluorescence analysis we find hσxi = 2P↓ (Ry |si) − 1. Analogously, for hσyi, a rotation around the σx-axis π −i σx with Rx = e 4 is applied and hσyi = 2P↓ (Rx |si) − 1. In our experiments, the axis of rotation for a tomography pulse is chosen via the (relative) phase ∆φ of the interaction field, as illustrated in Fig. 3.13. 46 3 Experimental methods

Rotation Fluorescence Interaction Laser off pulse Laser off detection

φ = 0 P σx ↓ → h i ∆φ = π/2

P σy ↓ → h i ∆φ = 0

P σz ↓ → h i

Figure 3.13: Procedure for the spin state tomography. When the interaction field is turned on, the relative phase φ = 0 is defined. After the desired interaction duration, the laser (or microwave) field is turned off, however, the corresponding DDS oscillator keeps oscillating (dashed line). When the unitary rotation pulse (solid lines) is applied a phase offset with respect to the initial phase is chosen in order to select a dedicated measurement axis. In the subsequent fluorescence detection the spin state population P↓ is extracted, from which the expectation values hσx,y,zi are calculated.

When the interaction with the light (or microwave) field is turned on, a relative phase φ = 0 is defined. For subsequent pulses (of the same origin) we can define a relative phase shift via the phase of the DDS signal that generates the interaction field. This phase allows to perform the unitary rotation pulse for the tomography around different axes, and, thereby, enables the full spin state tomography. Hence, the tomography scheme relies on coherent, unitary rotation pulses. For our experiments, unitarity of rotation pulses is best fulfilled with microwave pulses, cf. Fig. 3.12. Here, Rabi rates and fidelities are insensitive to the ion’s motional state and decoherence effects are insignificant. In Figure 3.14 (top) we depict a microwave flop, starting in |↓i and performing the spin state tomography (with microwave pulses). In order to perform the spin state tomography after driving TPSR transitions, we cannot rely on the microwave. This is due to the fact that there is no fixed phase relation between the Raman laser beams and the microwave, thus, we cannot drive coherent microwave pulses with respect to the TPSR interaction, that defines the relative phase offset. For that reason, when driving TPSR interactions, we perform the rotation pulses for the tomography likewise with TPSR transitions. Corresponding examples are depicted in Fig. 3.14 (center and bottom). 3.2 Spin control 47

0.5 z i z , y ,

x 0 σ h

0.5 y −

1 − 0 5 10 15 20 25 x Microwave interaction duration (µs) 1

0.5 z i z , y ,

x 0 σ h

0.5 y −

1 − 0 20 40 60 x TPSR interaction duration (µs) 1

0.5 z i z , y ,

x 0 σ h

0.5 y −

1 − 0 2 4 6 8 10 x TPSR interaction duration (µs)

Figure 3.14: Examples for the spin state tomography. We depict the individual spin state components hσx,y,zi (gray, orange, and blue, respectively) as a function of the interaction duration with the microwave (top) and the blue sideband transition (center and bottom). In addition, we show the corresponding Bloch vector evolutions. While for the top and center plots we prepared intially |↓i, the dynamics in the bottom graphs correspond to an initial spin state on the equator of the Bloch sphere. Data points depict experimental results, solid lines indicate theoretical predictions. 48 3 Experimental methods

We note that, in principle, a fixed phase relation between microwave and TPSR transitions can be realized. This would require a beat node, including an active phase stabilization mechanism. However, in the current state of the setup this is not implemented and subject to future improvements. The same requirement applies for different Raman beam combinations. Fixed phase relations are, currently, only established for the same beam combinations, coherence between different combinations would likewise require phase locking mechanisms. Hence, for our experiments presented in Sec. 4.2, the tomography rotation pulses are performed with a carrier transition using motional-sensitive Raman beams. Com- paring our measurements with numerical predictions, we estimate a corresponding fidelity for the spin state tomography on the order of 99 %, cf. Wittemer et al. (2018).

3.2.4 Spin coherence In this Section we want to briefly discuss the coherence properties of the spin that we encode in the ion’s electronic degree of freedom. To this end, we perform Ramsey interferometry (Ramsey, 1950), that is well-suited to measure frequency stabilities. The method is based on the pulsed application of two microwave π/2-pulses, resonant to the transition |↓i ↔ |↑i. Starting in |↓i, the first pulse creates the superposition state 1 √ |↓i + eiφ |↑i , (3.2.1) 2 where φ is a phase. After a waiting duration long compared to the microwave pulse length we apply a second pulse with variable phase. If no coherence is lost, the second pulse can, depending on its phase, transfer the ion’s state to |↓i or |↑i, which corresponds to the full Ramsey contrast. However, due to decoherence effects, the phase sensitivity might be decreased, resulting in a reduced Ramsey contrast. We extract the Ramsey contrasts from the phase scans for variable waiting durations in order to quantify the decay of coherence. In Figure 3.15 we depict corresponding experimental results. 3.3 Phonon control 49

0.8

0.6

0.4

0.2

Norm. Ramsey contrast 0 0 50 100 150 200 0 5 10 15 20 25 Waiting duration (µs) Waiting duration (ms)

Figure 3.15: Measurement of the ion’s spin state coherence using Ramsey interferometry. We depict the normalized Ramsey contrast as a function of the waiting duration between two π/2-pulses for the qubit states (left) and for the transition |3, 0i ↔ |2, 0i (right), that is less sensitive to magnetic field fluctuations. Data points depict experimental results (error bars smaller than symbol size), solid lines indicate model fits.

For the qubit transition (left) we observe a Gaussian loss of coherence and extract a (1/e2-)coherence time of τ = 94(6) µs. In contrast, for the transition |3, 0i ↔ |2, 0i we observe a significantly increased coherence duration (note the different x-axis scale), as well as a partial revival around 20 ms, that corresponds to the 50 Hz oscillation period of the power grid. Therefore, we consider magnetic field fluctuations as the dominating mechanism for decoherence of the ion’s internal degrees of freedom. We note that several techniques in order to overcome this limitation exist. In our group, we already realized the generation of quantization magnetic fields using perma- nent magnets in a different setup, as well as using field-insensitive clock transitions, see Hakelberg et al. (2018). By this we achieved a coherence time of greater than 6 s for electronic superpositions states of 25Mg+. In addition, the magnetic field noise level can be further reduced via shielding against external stray magnetic fields as demonstrated in Ruster et al. (2016).

3.3 Phonon control

The ions in our trap are confined in a 3D potential well, that is a superposition of an RF generated pseudo-potential and additional control potentials applied via static electric fields. Due to the large dimensions of the trap’s electrode geometry the ions experience a highly harmonic potential on typical ion oscillation amplitudes. Thus, for the remainder of this work we consider the total trapping potential to be purely harmonic. For a detailed discussion of the motion of ions in trapping potentials with significant anharmonic contributions we refer to Home et al. (2011). 50 3 Experimental methods

First, we consider a single ion of charge e and mass m that is placed in to a 3D harmonic potential Φ(x, y, z). Its potential energy can then be written as a function ∂2 ∂2 ∂2 of the potential’s curvatures Hxx = ∂x2 Φ, Hyy = ∂y2 Φ, and Hzz = ∂z2 Φ, along three orthogonal directions x, y and z, whose origins are set to the potential minimum. The ion’s potential energy is then given by

1 V = e H x2 + H y2 + H z2 . (3.3.1) 2 xx yy zz From this expression we can infer the normal mode frequencies for the ion’s motion in the potential: e e e ω2 = H , ω2 = H , ω2 = H (3.3.2) xx m xx yy m yy zz m zz When displaced from the potential minimum the ion undergoes harmonic motion, oscillating at these (secular) frequencies. We note that, in general, the directions of the motional mode vectors are not aligned with our trap’s (externally defined) coordinate system. In particular off-diagonal elements in control (or stray) potential curvature tensors induce rotations of the normal modes, see Sec. 3.1.3. Hence, it is convenient to label motional modes according to the value of their eigenfrequencies rather than their individual orientations. In the following, we identify three individual motional modes for a single ion, a low-frequency (lf) mode ωlf, which corresponds to the axial mode ωzz, and a mid-frequency (mf) mode ωmf and a high-frequency (hf) mode ωhf in the radial plane. Typically, the lf-mode is aligned along the trap axis, so that its eigenfrequency ωlf is mainly tuned by control potentials and independent from the RF voltage amplitude, while ωmf and ωhf are influenced by the total potential. Next, we want to discuss the situation when N ions are trapped in the potential landscape discussed above. In general, this leads to the following equation of motion for the kth ion with position coordinate ~rk:

 ω2 0 0  mf N ~r − ~r ~r¨ +  0 ω2 0  · ~r = γ X k l (3.3.3) k  hf  k |~r − ~r |3 2 l6=k k l 0 0 ωlf

2 Here the constant γ = e /(4π0m) encodes the Coulomb coupling strength between the ions. Note that we chose a coordinate system such that ωmf is aligned along the x-direction and ωlf points along the (axial) z-direction, cf. Eq. (3.3.2) and Sec. 3.1.3. In the following we consider small displacements δ around the equilibrium positions 0 0T ~rk = 0, 0, zk of the ions. We note that analytical solutions for the equilibrium positions exist for up to N = 3, while for higher N numerical calculations can be employed, see James (1998), Steane (1997), and Kielpinski et al. (2000). Considering 3.3 Phonon control 51 only small δ allows to Taylor expand the potential energy. Thereby, we can linearize the Coulomb interaction term and rewrite the equations of motion, cf. Fey (2014). Along the axial direction we obtain ! d2 N + ω2 δz + 2 X M δz = 0, (3.3.4) dt2 lf k kl l l and along the radial x-direction ! d2 N + ω2 δx − X M δx = 0. (3.3.5) dt2 mf k kl l l Here, the matrix M is given by  PN γ for l = k  j6=k 0 0 3 |zk−zj | Mkl = (3.3.6) − γ for l 6= k  0 0 3 |zk−zl | 2 Diagonalization of M yields N eigenvalues ωM that allow us to write down the normal mode equations of motion. Along the axial direction we obtain ! d2 + ω2 + 2ω2 δz = 0, (3.3.7) dt2 lf M M while for the radial x-direction we find ! d2 + ω2 − ω2 δx = 0. (3.3.8) dt2 mf M M Thus, the Coulomb interaction of the ions, expressed via the matrix M, gives rise to additional out-of-phase modes and in total, considering all three spatial dimensions, we find 3N normal modes. We now want to discuss the case of N = 2 in more detail, as this is the configuration for our experiments described in Sec. 4.1. In this case we can calculate the 0 0 equilibrium separation z0 = z1 − z2 along the z-direction analytically. By requesting ω2 z0 = γ we obtain lf 1,2 0 0 2 |z1 −z2 | !1/3 2e 1/3 2e2 z0 = = 2 . (3.3.9) 4π0Hzz 4π0mωlf Following this, we find the explicit form of the Coulomb interaction matrix M: ! γ 1 −1 M = 3 (3.3.10) z0 −1 1 52 3 Experimental methods

2 Diagonalizing this matrix yields the two eigenvalues ωM = {0, ωlf}. Thus, we ﬁnd the following explicit expressions for the motional mode frequencies of our two-ion chain using Eq. (3.3.7):

ωz,i = ωlf, (3.3.11) √ ωz,o = 3 · ωlf (3.3.12) Here, we distinguishing an in-phase or center-of-mass (COM) oscillation frequency ωz,i from an out-of-phase oscillation with frequency ωz,o, that is often referred to as stretch mode, see Fig. 3.16. Analogously, employing Eq. (3.3.8), along the x-direction we ﬁnd two mode frequencies

ωx,i = ωmf, (3.3.13) q 2 2 ωx,o = ωmf − ωlf, (3.3.14) where, again, ωx,i is the COM mode and ωx,o is the out-of-phase oscillation, often referred to as rocking mode. In Figure 3.16 the motional modes of the two-ion chain are illustrated.

1 ion 2 ions lf-mode lf COM lf stretch

mf-mode mf COM mf rocking

Figure 3.16: Illustration of the motional modes for one and two ions. A single ion (left) may be used to identify a low-frequency (lf) mode and a mid-frequency (mf) mode. When two ions are placed into the same trapping potential (right), their mutual Coulomb repulsion, gives rise to additional motional modes. We distinguish between in-phase or center-of-mass (COM) modes and out-of-phase modes. Along the axial direction the out-of-phase mode is labeled stretch mode, while along the radial direction it is referred to as rocking mode.

We note that in our discussion above, we considered only one radial degree of freedom. However, the derivation for the motional modes associated to ωhf is completely analog to ωmf. 3.3 Phonon control 53

In addition, it is important to note that in our descriptions above, we considered two ions with equal mass m. However, for two ions of different masses m1 and m2, as in our experiments presented in Sec. 4.1, the motional mode frequencies are shifted. A general calculation of the shifted mode frequencies as a function of the mass ratio µ = m2/m1 can be found in Kielpinski et al. (2000), Morigi and Walther (2001), and Wübbena et al. (2012). Here, however, we are (only) interested in the mode frequencies for a two-ion chain, consisting of a 25Mg+ and a 26Mg+, i.e., for a mass ratio of µ = 26/25, as this is the configuration discussed in Sec. 4.1. The results of the analytical calculations according to Wübbena et al. (2012), for our experimentally realized mode configuration, are given in Tab. 3.2.

ions I 1 ion 2 ions H mode H 1 × 25 25+25 25+26 ∆ axial in-phase ωz,i/(2π) 1.30 MHz 1.30 MHz 1.28 MHz −1.5 % axial out-of-phase ωz,o/(2π) - 2.23 MHz 2.23 MHz −0.2 % radial in-phase ωx,i/(2π) 2.88 MHz 2.88 MHz 2.83 MHz −1.7 % radial out-of-phase ωx,o/(2π) - 2.57 MHz 2.50 MHz −2.7 %

Table 3.2: Mode frequencies for a mixed-isotope two-ion crystal 25Mg+ + 26Mg+. We show numerical values, calculated according to Wübbena et al. (2012), for the motional mode conﬁguration that is implemented in our experiments discussed in Sec. 4.1. The trapping potential is characterized with a single 25Mg+, yielding the single- ion mode frequencies ωlf and ωmf. For two ions in the same potential the mode frequencies depend on the mass ratio µ of the two ions. We calculate mode frequencies for µ = 1 and µ = 26/25 and the corresponding relative shift ∆.

The numerical values for the motional mode frequencies for our ion combination 25Mg+ + 26Mg+ with µ = 26/25 indicate frequency shifts on the percent level (only). However, we note that the mass dependence of mode frequencies may become more significant for other ion combinations, in particular, for crystals of different atom species, cf. Tan et al. (2015). In general, the motional modes of single and multiple trapped ions allow for most versatile applications, apart from their popular applications as data buses for quantum information processing, cf. Kielpinski et al. (2000). Here we want to mention only a few very recent applications: While in the experiments described Sec. 4.2 a single motional mode is used to form a controlled environment for a spin-1/2 system, in Clos et al. (2016) ion crystals with up to 5 ions have been used to engineer large environments with dense mode spectra. Recently, it has been shown that the mode frequencies associated to different structural orders for a 34 ion crystal of 24Mg+ can be experimentally resolved and excited (Brox et al., 2017). Besides, motional modes of ions, trapped in two-dimensional arrays, have been employed to couple the oscillations of individually trapped ions by their Coulomb repulsion, either by a resonant 54 3 Experimental methods interaction (Hakelberg et al., 2019) or via a Floquet-engineering approach (Kiefer et al., 2019). In addition, recently it has been shown that motional modes of trapped ions prepared in squeezed states can be employed as a platform for implementing qubits for quantum information processing, see Flühmann et al. (2019). In the following Sections, we will try to provide a brief but complete overview of the methods to manipulate motional states of trapped ions that are relevant for the experiments discussed in Chap. 4. We will focus on how classical and quantum states can be initialized using state of the art techniques for trapped (atomic) ions, many of our techniques are also extensively discussed in Wineland et al. (1998a) and Leibfried et al. (2003).

3.3.1 Ground state preparation In this Section we briefly recall the different laser cooling schemes, allowing to prepare motional states of low excitation. Here we focus on the techniques used in this work, i.e., Doppler cooling and resolved sideband cooling using TPSR transitions, and we refer to Metcalf and van der Straten (1999) and Foot (2005) for a more detailed discussion as well as other laser cooling techniques. Doppler cooling of ions in our trap is achieved with cooling lasers from a single direction, see Fig. 3.9. This, however, requires a finite overlap of all the motional mode vectors with the wavevector of the cooling beam. To this end the mode configuration might be appropriately adapted using dedicated control potentials, see Sec. 3.1.3. When Doppler cooling via an internal atomic transition with natural linewidth Γ using a laser beam that is detuned by −Γ/2, the corresponding minimal temperature reached is given by (Leibfried et al., 2003): √ ~Γ 1 + s Tmin(Γ, s, ξ) = (1 + ξ) (3.3.15) 4kB

Here, s = I/Isat denotes the saturation parameter that takes account for power broad- ening, and ξ incorporates the directionality of the emitted photons. Typically, we cool with intensities half the saturation intensity (s = 1/2) and for dipole-radiation we obtain ξ = 2/5, see Leibfried et al. (2003). Equation (3.3.15) denotes the Doppler limit for a free particle. However, when laser-cooling a trapped ion we are, in particular, interested in the mean motional state excitation n¯ for a motional mode with mode frequency ω aligned along the mode vector ~kω. To this end we consider a Doppler ~ cooling beam with wavevector kBD and obtain the Doppler limit along the direction of the motional mode:

|~k | · |~k | T = T (Γ, s, ξ) · BD ω (3.3.16) mode min ~ ~ |kBD · kω| 3.3 Phonon control 55

The corresponding mean motional state excitation is then given by 1 n¯(T , ω) = . (3.3.17) mode ~ω e kB Tmode − 1 For our optical setup, see Fig. 3.9, and typical mode configurations we obtain n¯ ≈ 10 after Doppler cooling. In order to further reduce the motional state excitation we can, subsequently to the Doppler cooling, perform resolved sideband cooling using TPSR transitions. For a detailed description we refer, again, to Leibfried et al. (2003). Resolved sideband coolig relies on the repeated application of red sideband pulses that transfer the motional state from |ni to |n − 1i, while flipping the spin state from |↓i to |↑i, see Sec. 3.4.1. In between different sideband pulses the spin state is (re-)initialized to |↓i using the repumping lasers RD and RP, that, on average, do not change the motional state. We note that the incoherent spontaneous emission in the repumping step is responsible for the decoherence that is required in order to reduce the entropy of the motional state and, thereby, effectively yield lower thermal states. In order to cool multiple motional modes such cooling cycles acting on different motional modes can be concatenated, see Clos (2017). Although this technique is not fundamentally limited by a minimal temperature, in practice sideband cooling has to compete with intrinsic motional heating mechanisms, see Wineland et al. (1998b) and Turchette et al. (2000). Heating rates depend, e.g., the motional mode, electrode geometries, electric filtering, electrode surface textures, and the trap’s temperature. In our setup, heating rates are on the order of tens of quanta per second on axial modes (n¯˙ ≈ 10 s−1) and thousands of quanta per second (n¯˙ ≈ 103 s−1) on radial modes. In contrast, we can implement resolved sideband cooling that reduces motional excitations with rates n¯˙ ≈ −1 × 104 s−1 in both directions. As this is significantly faster than the typical heating rates, we can efficiently prepare n¯ ≈ 0 in all three spatial dimensions, i.e., preparing the 3D motional ground state, cf. Kalis et al. (2016).

3.3.2 Classical ion motion Due to their charge, trapped ions can be readily manipulated using (classical) electric fields. In this Section we briefly recall a motional excitation method that allows to create coherent states, known as the tickle method, for details see Wineland et al. (1998a), Leibfried et al. (2003), and Kalis et al. (2016). The method is based on the application of an oscillating electric field that leads to an oscillating force on the trapped ion, thereby forming a driven harmonic oscillator. When the frequency of the electric field is close to a motional mode frequency the ion gains excitation due to the drive and, thereby, coherent states of large amplitude can be prepared. These can be considered the most classical states of the quantum harmonic oscillator. Typically, for coherent states motional excitations with n¯ & 100 can be detected via a reduction 56 3 Experimental methods in the ion’s fluorescence. In Figure 3.17 we depict the fluorescence signal as a function of the frequency of the electric driving field near a single ion’s axial mode frequency. This method allows for precise measurements of motional mode frequencies, where the measurement resolution is, in our case, limited only by the stabilities of the motional mode frequencies themselves, cf. Sec. 3.3.5.

1 15

0.8 10 )

0.6 3 (10 ¯ 0.4 n 5 0.2 Norm. ﬂuorescence 0 0 1.9140 1.9145 1.9150 1.9155 1.9160 0 5 10 15 20 Excitation frequency (MHz) Excitation duration (ms)

Figure 3.17: Motional mode excitation with classical driving fields. The left plot depicts the normalized fluoresence of a single trapped ion as a function of the frequency of an electric driving field, exciting motion along the axial direction. Here the excitation duration was set to 3 ms, from the model fit (solid line) we can extract the motional mode frequency ωlf/(2π) = 1.914 86(5) MHz. On the right we show the motional state excitation as a function of the excitation duration. Here, however, we set the driving frequency detuned by 450 Hz to the motional mode frequency in order to coherently excite and de-excite ion motion.

When mode frequency and orientation with respect to the detection laser are known, i.e., extracted from independent calibration measurements, a reduction in fluorescence can be assigned to an excitation of a coherent state |αi with n¯ ∝ |α|2 quanta, cf. Kalis et al. (2016) and Hakelberg et al. (2019). The right plot in Fig. 3.17 illustrates the motional excitation as a function of the electric field excitation duration when the driving frequency is slightly detuned from the motional mode frequency. This leads to a coherent excitation and de-excitation illustrating the creation of coherent states, rather than thermal states, and allows to estimate the motional coherence properties, see Sec. 3.3.5. We note that the detection of a motional excitation via the fluorescence method works on the level of several hundreds of quanta only. For a motional state reconstruction on the level of single quanta, see Sec. 3.4.2.

3.3.3 Quantum state initialization In this Section, we want to discuss methods to implement diﬀerent types of quantum states in the motion of our trapped ions. First, we consider two diﬀerent methods to prepare thermal states with variable mean phonon excitations n¯, as required for the 3.3 Phonon control 57 experiments presented in Sec. 4.2. As discussed in Sec. 3.3.1, after Doppler cooling, motional states with n¯ ≈ 10 are obtained, while a subsequent sideband cooling phase results in n¯ ≈ 0. In order to create thermal states with intermediate excitations we implement variable heating phases after preparing the ground state. To ensure a mixed, thermal character of the thereby creates states, we require an incoherent excitation mechanism. One method to realize this is to add white noise on control electrodes. Here, the incoherently oscillating voltages lead to incoherent motional excitations that prepare the ion in a thermal motional state. Another possibility is to heat the motional state by scattering photons, e.g., from the BDX laser. The spontaneous emission process involved here causes incoherent momentum kicks that give the ion’s motional state its thermal character. In Figure 3.18 we show examples for both these methods, where initially a single ion’s axial motion has been prepared in its ground state. ¯ n

2.5 2 1.5 1 0.5

Thermal state excitation 0 0 2 4 6 8 10 0 1 2 3 4 5 Electric ﬁeld heating duration (µs) BDX heating duration (µs)

Figure 3.18: Controlled motional state heating. We excite thermal states with variable mean excitations n¯ by using either incoherent voltage noise on a control electrode (left) or incoherent momentum kicks by photons from the BDX laser (right). Both mechanisms can be operated in regimes with signiﬁcantly increased excitation rates with respect to the background heating rate of n¯˙ = 10 s−1 (orange lines).

We note that, in principle, the intrinsic heating rate apparent in our setup could be employed as well to excite thermal states out of the ground state. However, the corresponding rate (gray lines in Fig. 3.18) happens on time scales long compared to our (motional and internal) coherence times, which signiﬁcantly limits the applicability of this method to our experiments. With the controlled heating mechanisms the excitation rate can be tuned by the voltage noise amplitude or the laser power, respectively, to realize more convenient heating durations. Next, we want to consider the excitation of squeezed and coherent states out of the motional ground state. The corresponding methods are described in detail in Meekhof et al. (1996), Wineland et al. (1998a), and Leibfried et al. (2003). As discussed in 58 3 Experimental methods

Sec. 3.3.2, coherent states can be created with oscillating electric ﬁelds that excite coherent motion. As shown in Fig. 3.19(left) this method can also applied to low (n¯ ≈ 0 − 10) excitation amplitudes. Here the excitation method is essentially the same as in Sec. 3.3.2, however the amplitude of the oscillating voltage was reduced in order to be able to ﬁne tune the motional state excitation amplitude. The detection of the motional states created has to be performed on the level of single quanta, the corresponding method is described in Sec. 3.4.2. | r α

| 2.5 2 1 1.5 1 0.5 0.5 Excitation amplitude Excitation amplitude 0 0 0 5 10 15 0 20 40 60 80 100 Electric ﬁeld excitation duration (µs) Laser squeezing duration (µs)

Figure 3.19: Phonon state initialization with coherent and squeezed states. We depict the coherent state excitation amplitude |α| as a function of the electric ﬁeld excitation duration (left). The right plot depicts the squeezing amplitude r as a function of the duration a parametric excitation at twice the motional mode frequency is applied via an optical dipole force. Motional states are recon- structed via the spin-phonon mapping with TPSR transitions, see Sec. 3.4.2.

Squeezed states of motion on a mode with frequency ω can be created by a parametric drive at 2ω, see Heinzen and Wineland (1990), Wineland et al. (1998a), and Leibfried et al. (2003). This can be realized, for example, by modulating the trapping potential itself as in Burd et al. (2019) or by applying a modulated optical dipole force, cf. Meekhof et al. (1996). In Figure 3.19 we illustrate the creation of squeezed states via a parametrically modulated optical dipole force using the beams R2 and R3, generating a traveling standing wave along the trap’s axial direction. In our trap, these methods work for squeezing amplitudes up to r ≈ 2, for higher excitations we observe increasing distortions of the Fock state distributions, which may be due to finite motional mode instabilities and the fact that squeezed states with large r effectively sample the (fluctuating) trapping potential. We note that, alternatively, squeezed states can be created by non-adiabatic changes in the motional mode frequency as discussed in, e.g., Janszky and Yushin (1986), Graham (1987), and Heinzen and Wineland (1990), and experimentally realized recently in our group, see Wittemer et al. (2019) and Sec. 4.1. 3.3 Phonon control 59

3.3.4 Spatial ordering When performing experiments with mixed-isotope (or mixed-species) ion chains main- taining a particular ion order may be desirable in order to ensure stable conditions for subsequent experimental runs. In this Section, we will discuss a spatial ordering technique, based on techniques described in Hume (2010) and Clos (2017), and extended with a qubit-state independent post-selection scheme in this work. Typical ion-ion separations within linear chains in our trap are of a few micro meters. This can be compared to typical laser beam waists of 40 µm and emphasizes the need for spatial ordering, in particular when laser beams are tuned to drive coherent transitions. Here, we will discuss an ordering technique in the case for a two-ion crystal composed of a 25Mg+ and a 26Mg+ ion, which is aligned along the axial direction. Typically, we tune our laser frequencies to the transitions of 25Mg+ in order to coherently manipulate and readout its spin state. Consequently, we align the laser beams to the 25Mg+ and wish to maintain this (axial) ion order throughout the consecutive repetitions of an experimental cycle, cf. Sec. 3.1.2. The method of doing so is based on a drastic change of the ratio ωmf/ωlf of the (weaker) radial confinement ωmf and the axial confinement ωlf. Two ions of the same mass align along the axial direction for ωmf/ωlf > 1. However, for different masses this critical value changes as the heavier ion is less tightly confined in the radial direction. In our case of 25Mg+ and 26Mg+, i.e., a mass ratio of µ = 26/25, the critical value is given by ωmf/ωlf & 0.98, cf. Hume (2010). When ωmf/ωlf is switched below the critical value the axially oriented ion chain transitions into a radially oriented chain, i.e., both ions are at the same position along the z-axis and, thereby, the axial ion order is reset. In order to realize a particular axial order, we introduce an isotope-dependent asymmetry by applying a light force that pushes the 25Mg+ along the axial direction using the BDX beam, while the light pressure is highly suppressed for the off-resonant 26 + Mg . Switching ωmf/ωlf back above the critical value for the axial chain can result in increased probabilities for a particular axial ion order. In our experiments, we optimize the time-dependence of ωmf/ωlf and the light pressure amplitude via the beam’s intensity and, thereby, accomplish ordering efficiencies ≈ 99 %. Nevertheless, we want to be able to remove the remaining “wrong” orders from our data analysis as these may have been manipulated with significantly different Rabi rates. In principle, different ion orders can be distinguished by different fluorescence rates, given that the detection laser and the sensitive window for the PMT have been optimized for the “right” order. However, from the fluorescence measurement after a given experimental sequence, this might not be directly evidenced, because decreased fluorescence can also be due to a change in the spin state of the 25Mg+ ion. To this end, we implement an additional post-selection fluorescence channel that is evaluated at the beginning of each of the N repetitions of a given measurement sequence, see Sec. 3.1.2. The corresponding post-selection scheme is illustrated in Fig. 3.20. 60 3 Experimental methods

1x N repetitions 1

Check Ordering Experiment 0.5 order 0 1 0.2 0.5 0.1 Norm. Fluorescence Probability 0 0 0 4 8 12 16 0 4 8 12 16 0 5 10 15 20 25 30 Photon counts in 40 µs Microwave interaction duration (µs)

Figure 3.20: Ion order control. When working with mixed-isotope crystals we establish a dedicated ion order before every N repetitions of an experimental cycle (left). In addition, every experimental cycle starts with an order check. In our post- measurement analysis (right), we identify different ion orders by decreased fluorescence rates during the order check (gray data points, top right). When analyzing data for the actual experimental (here a MW flop between |↓i and |↑i) these “bad” data points are left out (gray data points, bottom right). Thereby, we are able to identify “wrong” ion orders, independently from the spin state evolution that can potentially comprise decreased fluorescence rates. Note that for this example the ordering step was not included in order to visualize both fluorescence levels.

We discuss the post-selection scheme with an example for a microwave flop between |↓i and |↑i. Here, for particular durations the spin state reaches |↑i, which is a dark state for the detection laser BDX and, hence, decreased fluorescence rates are detected. However, we must distinguish between decreased fluorescence due to the spin state evolution and due to a “wrong” ion order. To this end, every N repetitions of an experimental cycle are preceded by an ordering sequence. In addition, at the beginning of each cycle we include an order checking step that consist of a Doppler cooling and optical pumping period to prepare |↓i and subsequent resonance fluorescence detection, cf. Sec. 3.2.1. We identify different ion orders by decreased fluorescence rates during this step, which is independent of the subsequent spin state evolution. Data points following a “bad” ion order are then left out for the post-measurement analysis (gray data points in Fig. 3.20). We note that for simple single-tone Rabi oscillations (as in the example discussed above) this might not seem necessary. However, when the spin state undergoes more complex time evolutions we require an independent and unbiased control channel in order to remove “bad” data points for our measurement analysis. 3.3 Phonon control 61

3.3.5 Phonon coherence

Similar to the ion’s spin state, also the motional (phonon) states have only finite coherence times. The limitations are mainly due to finite mode frequency stabilities but we also note that motional state heating (see Sec. 3.3.3) and anharmonic contributions of the trapping potential (Hakelberg et al., 2019) can disrupt the ions’ motional coherence. In order to quantify the motional mode coherence of a single trapped ion, we perform a Ramsey-type experiment (Ramsey, 1950), similar to the technique discussed for the spin state coherence measurement, see Sec. 3.2.4. We excite a coherent state with an oscillating electric field resonant to the motional mode frequency, see Sec. 3.3.2. After a waiting duration, long compared to the excitation duration, another oscillating electric field pulse with variable phase is applied. In Figure 3.21 we depict the (normalized) contrast of the phase scan as a function of the waiting duration between the pulses for the axial (left) and a radial (right) mode.

0.8

0.6

0.4

0.2

Norm. Ramsey contrast 0 0 10 20 30 40 0 100 200 300 400 500 Waiting duration (ms) Waiting duration (µs)

Figure 3.21: Motional mode coherence. Using Ramsey-type experiments with excitations of coherent states of several hundreds of motional quanta we estimate the motional coherence times of τ = 116(27) ms along the axial direction (left), and τ = 246(14) µs along the (radial) mf-mode (right). Data points depict experimental results, solid lines indicate model ﬁts.

From fits of exponential decay models to the experimental data we extract mode a coherence time (1/e) of 116(27) ms for the axial (ωlf/(2π) ≈ 1.32 MHz) mode and (only) 246(14) µs for the radial (ωmf/(2π) ≈ 2.88 MHz) mode. The significant difference between both modes might be due to the different contributions of the RF created pseudopotential to the corresponding confinement. While the axial mode frequency is controlled via (static) control potentials, the radial modes depend on the RF voltage and, thereby, are disturbed by fluctuations of the RF voltage amplitude. We note that the coherence times stated here correspond to a free evolution of the motional state. When continuously manipulating the motional state, e.g., when driving TPSR transitions on motional sidebands, the associated dynamical decoupling 62 3 Experimental methods leads to longer coherence times. This in in particular relevant for the phonon state analysis on the single quantum level that is described in Sec. 3.4.2.

3.4 Spin-phonon coupling

In this Section, we will brieﬂy discuss our method to coherently couple spin and motional degrees of freedom of single and multiple trapped ions. The method discussed here is based on two-photon stimulated Raman transitions, however, we note that recently spin-motion coupling for trapped ions has been also realized using the near ﬁeld of microwave currents, see, e.g., Ospelkaus et al. (2011).

3.4.1 Two-photon stimulated Raman transitions

We employ two-photon stimulated-Raman (TPSR) transitions using a virtual level 25 + detuned by ∆R/(2π) ≈ 100 GHz from the P3/2 manifold of Mg to coherently manipulate the spin state, cf. Sec. 3.1.5. The technique in general is well described in Wineland et al. (1998a) and Leibfried et al. (2003), whereas particular details on the implementation in our setup are given in Friedenauer (2010) and Clos (2017). In the following, we will briefly recall the basic mechanism, focused onto the parameter regimes of the results discussed in Chap. 4. In order to drive TPSR transitions, we require two photons; one from a blue Raman beam (B1 or B3) and one from a red Raman beam (R1, R2, or R3), see Sec. 3.1.6. Thereby, the electronic state of the ion can undergo a two-photon transition by ab- sorbing a photon from one beam and emitting one into the other beam. To this end, the frequency difference of the red and the blue Raman beam used has to be set close to the spin state’s qubit transition frequency ωz. Such a Raman transition is accompanied by a momentum transfer, that depends on the effective wavevector ∆~k of the two-photon field, given by the difference of the wavevectors of the individual Raman ~ ~ ~ beams, i.e, ∆k = kb − kr. In our setup we have implemented several beams to be able to realize effective wavevectors along both the x and the z-direction or with ∆~k ≈ 0, see Fig. 3.9. When driving TPSR transitions with ∆~k||x or ∆~k||z, the transition is sensitive to the ions’ motion along the radial or axial direction, respectively. In contrast, when ∆~k = 0 the transition rates are insensitive to the ions’ motional state. The spin-phonon coupling strength can be expressed with the so-called Lamb-Dicke parameter ηLD. For a motional mode with frequency ωx that is oriented along a normalized mode vector ~kx, the spin-phonon coupling strength established via TPSR transitions via an effective wavevector ∆~k is given by

~ ~ ~ ηLD = x0|∆k| cos ∠ kx, ∆k , (3.4.1) 3.4 Spin-phonon coupling 63

p where x0 = ~/(2mωx) is the extent of the ground state wavefunction, m is the mass ~ ~ of the ion (or the ion chain), and ∠ kx, ∆k is the angle between the mode vector and the eﬀective wavevector, cf. Mielenz et al. (2016). With a given ηLD one can express the spin-phonon coupling Hamiltonian as (Porras et al., 2008):

Ω † † H = ~ σ eiηLD(a +a) + σ e−iηLD(a +a) (3.4.2) I 2 + −

This interaction term couples the spin flip operators σ± = (σx ± iσy) /2 to the harmonic oscillator creation and annihilation operators a and a† and, thereby, allows for transitions between different Fock states |ni, cf. Sec. 2.1.3. The corresponding coupling strength to individual motional modes can be tuned by choice of a particular ∆~k. Here we want to consider two different cases that are relevant for our experimental results. In Section 4.2 we couple the spin state of a single ion to its motion along the axial mode ωlf using the Raman beams B1 and R2 that ~ ~ ~ create an effective wavevector ∆k along the z-direction. This configuration (∆k||klf) allows to dominantly couple to the axial mode and, consequently, transition rates and population transfer efficiencies are insensitive to any motion along the radial directions. For the experiments discussed in Sec. 4.1 we employ the spin-phonon coupling to analyze the motional state along the radial direction. To this end we use the Raman beams B3 and R2, creating an effective wavevector ∆~k||x and allowing for TPSR transitions that are insensitive to the axial ion motion. Figure 3.22 shows ~ ~ scans of the Raman laser frequency difference near ωz both, for ∆k||klf (with a single ion), and for ∆~k||x (with two ions). 64 3 Experimental methods

↓ 0.8 P 0.6 0.4

Population 0.2 0 1

↓ 0.8 P 0.6 0.4

Population 0.2 0 5 4 3 2 1 0 1 2 3 4 5 − − − − − Relative TPSR detuning (MHz)

Figure 3.22: Frequency scans with motion-sensitive TPSR transitions. We measure the spin state population P↓ as a function of the two-photon detuning from the carrier resonance (|↓, ni ↔ |↑, ni, black lines) for one ion with ∆~k||z (top), and for two ions with ∆~k||x (bottom). With the choice of the orientation of ∆~k, the motional sensitivity along particular directions can be controlled. This leads to the emergence of red and blue sidebands for both, COM modes (solid) as well as out-of-phase modes (dashed lines). Mode frequencies are extracted from independent calibration measurements by using the method described in Sec. 3.3.2.

From the TPSR frequency scans we can infer the different coupling strengths to specific motional modes by the choice of ∆~k. Thereby, one can use TPSR couplings in order to measure the motional mode configuration, i.e., frequency, orientation, and motional state distribution, see Kalis et al. (2016). We note that in our experiments we fine-tune the alignments of our ∆~k in order to minimize any crosstalk to “orthogonal” modes by adjusting the individual Raman beam alignments. By choice of the frequency difference of the two Raman beams used we can either drive carrier transitions that act only on the spin state |↓, ni ↔ |↑, ni or sideband transitions that reduce or enhance the motional quantum state when the spin is flipped, e.g., a first blue sideband transition |↓, ni ↔ |↑, n + 1i or a first red sideband transition |↓, ni ↔ |↑, n − 1i. Typically, we use red sidebands up to the second order (∆n = −2) for efficient sideband cooling, cf. Sec. 3.3.1. In any case, it is important to note that the Rabi rates for individual transitions depend on ∆n and the particular 3.4 Spin-phonon coupling 65 phonon state n that is manipulated. This is essential for our phonon state analysis that is discussed in the next Section.

3.4.2 Mode-resolved motional state analysis One application of TPSR transitions that is of particular importance for the results presented in Sec. 4.1 is the motional state analysis. The experimental method used here is already well described in Hakelberg (2015) and has been implemented for the experiments presented in Mielenz et al. (2016) and Kalis et al. (2016). For the experiments presented in this work, we extend the protocol with a parametrized Gaussian state analysis, in order to estimate the full motional density matrix rather than just the Fock state populations (diagonal elements). In the following we will brieﬂy discuss the method. The basic idea of the mode-resolved motional state analysis is to map the motional state of a trapped ion to its internal (spin) state, which can be readout with high ﬁdelity, cf. Wineland et al. (1998a) and Leibfried et al. (2003). When driving motion- sensitive (ηLD > 0) TPSR transitions the Rabi rate that corresponds to a transition |↓, ni → |↑, n0i, i.e., ∆n = n0 − n, is given by s −η2 /2 |∆n| n!

0 α where n< (n>) is the lesser (greater) of n and n and the Ln(X) denote the generalized Laguerre polynomials, cf. Wineland et al. (1998a). This dependence of the Rabi rate on the individual Fock state number n is used to distinguish individual Fock states. In our experiments we employ this by driving TPSR transitions for variable durations t and record the spin state population P↓. When coupling to a single mode on a carrier or sideband resonance the corresponding spin-state evolution is given by (Leibfried et al., 2003):

∞ ! 1 X P (t) = 1 + P cos Ω 0 t (3.4.4) ↓ 2 n n,n n=0 Here, we neglect any finite detuning to the TPSR resonance, decoherence effects and couplings to other motional modes. For a full description of P↓(t) including these we refer to Hakelberg (2015). From a parametrized model fit we extract individual contributions of frequencies that correspond to particular Fock states and, thereby, obtain their individual populations Pn. In principle this works for carrier transitions as well as for sideband transitions. However, in particular for couplings where ∆~k has overlap with multiple motional modes, carrier transitions are affected by all those modes, whereas (resolved) sideband transitions can be employed to readout motional modes individually. Moreover, in order to increase the accuracy of the motional state 66 3 Experimental methods detection, we can perform simultaneous fits to time evolutions of TPSR couplings on different sideband or carrier transitions. Figure 3.23 depicts examples of such phonon analyses for different motional quantum states of a single ion on both, axial and radial motional modes. 3.4 Spin-phonon coupling 67

1 1

0.8 0.8 n ↓ P P 0.6 0.6

0.4 0.4 Population Population 0.2 0.2

0 0 0 20 40 60 80 100 0 1 2 3 4 5 6 7 8 1 1

0.8 0.8 n ↓ P P 0.6 0.6

0.4 0.4 Population Population 0.2 0.2

0 0 0 20 40 60 80 100 0 1 2 3 4 5 6 7 8 1 1

0.8 0.8 n ↓ P P 0.6 0.6

0.4 0.4 Population Population 0.2 0.2

0 0 0 20 40 60 80 100 120 0 1 2 3 4 5 6 7 8 1 1

0.8 0.8 n ↓ P P 0.6 0.6

0.4 0.4 Population Population 0.2 0.2

0 0 0 20 40 60 80 100 120 0 1 2 3 4 5 6 7 8 TPSR interaction duration (µs) Phonon number state n

Figure 3.23: Examples for the phonon state analysis. We record P↓ for variable TPSR coupling durations (left) and extract individual phonon state populations Pn (right, data points with error bars partially smaller than symbol size) from a combined model fit (solid lines). From a fit of a Gaussian state (right, bars) to the Pn the corresponding Wigner function can be depicted (inset). For the axial mode (top row) the phonon analysis can be done with a carrier (orange) and a sideband (blue) transition. Here, a thermal state of n¯ = 2.37(4) is extracted. In the radial direction, we choose to perform the analysis of a single mode with corresponding sideband flops only in order to be less sensitive on the other radial mode(s). For the mf-mode we depict a ground state analysis with n¯ = 0.07(1) (2nd row), a coherent state with |α| = 1.69(2) (3rd row), and a squeezed and displaced state with |α| = 0.19(4) and r = 1.08(3) (bottom row). 68 3 Experimental methods

We depict examples of a motional state analysis of a thermal state on the axial mode (ωlf/(2π) ≈ 1.9 MHz) as well as ground state, coherent state and squeezed and displaced state on the radial mf-mode (ωlf/(2π) ≈ 2.8 MHz). In the axial direction, we can record the carrier and the first blue sideband to analyze the motional state along ωlf. In contrast, along the radial directions a mode-resolved state analysis is more conveniently performed with sideband flops only because ∆~k has a finite overlap on both the mf-mode and the hf-mode. In particular, when analyzing motional states where we are interested in the particular shape of the Fock state population distribution (rather than parameters of a parametrized distribution) we record red and blue sidebands. From model fits to recorded TSPR flops we obtain the individual Fock state populations Pn, i.e., the diagonal elements of the corresponding motional density matrix. We note that this method does not give any information about off-diagonal elements of the motional density matrix, i.e., no phase information is acquired. How- ever, it can be extended by performing additional measurements to obtain the full density matrix and Wigner function, as described in Leibfried et al. (1996). As an alternative, we may also perform model fits of parametrized Gaussian states to the measured Pn in order to estimate the underlying Wigner function. To this end, we consider the most general Gaussian state, which is a squeezed and coherently displaced thermal state. Such a state ρSDT can be generated by applying a squeezing 1 ∗ 2 †2 iθ operator S(ξ) = exp 2 ξ a − ξa , with ξ = re , and a displacement operator † ∗ D(α) = exp αa − α a to a thermal density matrix ρth (¯nth):

† † ρSDT (ξ, α, n¯th) = S(ξ)D(α)ρth (¯nth) D (α)S (ξ) (3.4.5)

The final state ρSDT is then a function of the squeezing parameter ξ, the coherent displacement parameter α, and the mean phonon number of a thermal state n¯th (which is real because it contains no phase information). However, in order to perform fits of the diagonal elements of ρSDT to experimentally measured Pn we can reduce the number of free fit parameters to three excitation amplitudes r, |α|, and n¯th, and a single relative phase ϕ between squeezing and displacement. Fits of such kind generate the bar charts in Fig. 3.23(right) and the corresponding Wigner functions. Again, we note that this method does not provide a measurement of the motional Wigner function, but the obtained Wigner function is the most probable one if the state analyzed is a Gaussian state, which should be fulfilled for all our experiments. However, we note that it cannot be applied to other states such as, e.g., single Fock states, which can also be constructed in trapped ion systems, see Meekhof et al. (1996) and McCormick et al. (2019). 3.5 Numerical calculations 69

3.5 Numerical calculations

In this Section, we want to discuss diﬀerent numerical methods that may be employed in order to describe our experimental results. Our experiments aiming to create squeezed states by tuning the frequency of a harmonic oscillator in Sec. 4.1, may be described with a numerical simulation of a time-dependent harmonic oscillator. For the experiments presented in Sec. 4.2 we require a method to calculate time evolutions of the spin-phonon coupling, cf. Sec. 3.4.1. Both techniques will be brieﬂy presented in the following.

3.5.1 Time-dependent harmonic oscillator In our experiments presented in Sec. 4.1, we essentially implement a quantum harmonic oscillator with time-dependent frequency. In the corresponding literature, the description of such quantum systems is often simpliﬁed by either the adiabatic approximation or, if adiabadicity cannot be assured, by the assumption of instantaneous (sudden) frequency jumps, cf. Janszky and Yushin (1986) and Graham (1987). How- ever, in our experimental realization, we are operating in the intermediate regime between the adiabatic limit and instantaneous frequency jumps. To this end, we require a method to numerically calculate harmonic oscillator excitations for non-trivial, non- adiabatic time evolutions of its frequency. We implement such a numerical method in QuTiP (Johansson et al., 2013), based on the analytic description of the Hamiltonian of the harmonic oscillator with time-dependent frequency ω(t) by Silveri et al. (2017): 1 H(t) = ω(t) a† (t)a (t) + ~ ω ω 2 1 i d ln (ω(t)) h i = ω(t) a†a + − ~ a2 − a†2 (3.5.1) ~ 2 4 dt

† Here, aω(t) and aω(t) denote the time-dependent creation and annihilation operators, that are transformed to the time-independent basis operators a† and a, defined for the initial mode frequency ω0 = ω(0). We note that, although this is an analytic expression, for our numerical calculations we still have to introduce a cutoff in Fock space in order to obtain finite-dimensional density matrices. In our experiments, residual technical imperfections lead to an additional time- dependent force term

† HF (t) = x0F0(t)(a + a ), (3.5.2) p where F0(t) denotes an eﬀective force amplitude and x0 = ~/ (2mω0) is the width of the ground state wave function of the oscillator with mass m. We note that in 2 our experiments, we ﬁnd F0(t) ∝ 1/ω , and, thus, in our numerical calculations we parametrize it accordingly. 70 3 Experimental methods

While a variation ω(t) can lead to squeezing excitations, the time-dependent force term may induce coherent displacements of the oscillator. We note that both excitations depend on the explicit time evolutions ω(t) and F (t), i.e, they are functions R t 0 0 of the evolution of the oscillator’s phase ϕ(t) = 0 ω(t )dt . This is, in particular, relevant in the parameter regime where changes of ω cannot be assumed to occur instantaneously. In Figure 3.24 we depict an example for a numerical calculation of the time-dependent quantum harmonic oscillator, where ω/(2π) is evolving similar to our experiments described in Sec. 4.1, i.e., switched from 2.5 MHz to 0.5 MHz within 1 µs and back up after a waiting duration of 2 µs. 3.5 Numerical calculations 71

0 2 F ,

ω 1 0

0.5 i p h ,

i 0 x h 0.5 − 0.6

| 0.4 α | 0.2 0 2

) 1.5 p ∆

( 1 , 2 )

x 0.5 ∆ (

0.6 0.4 r 0.2 0 0 1 2 3 4 5 Duration (µs)

Figure 3.24: Numerical simulation of the time-dependent quantum harmonic oscillator. We employ QuTiP to simulate the evolution of a harmonic oscillator whose frequency ω (orange) is switched from between 2.5 MHz and 0.5 MHz, where a single ramp takes 1 µs, and that is subjected to a time-dependent force term (blue) F (t) ∝ 1/ω2 (arb. units). The calculation provides us with expectation values for position hxi (blue) and momentum hpi (orange) operators, from which we can calculate the amplitude of a coherent displacement |α| In addition, we obtain the variances (∆x)2 (blue) and (∆p)2 (orange), that allow us to calculate a squeezing amplitude r. 72 3 Experimental methods

We depict the input of the simulation, i.e., the evolutions ω(t) and F0(t), as well as the corresponding result, intended to obtain expectation values for a coherent displacement amplitude |α| and the amplitude of a squeezing excitation r. The QuTiP code provides us with the expectation values hxi and hpi for the position operator p † p † x = ~/ (2mω0) a + a and the momentum operator p = i ~mω0/2 a − a as well as their variances (∆x)2 = hx2i − hxi2 and (∆p)2 = hp2i − hpi2. From the expectation values we can calculate the amplitude of a coherent displacement by q |α| = hxi2 + hpi2, (3.5.3) whereas their variances allow to determine a squeezing amplitude by

1 h i r = arccosh (∆x)2 + (∆p)2 . (3.5.4) 2 As shown in Fig. 3.24, both excitations, |α| and r, evolve non-trivially due to the evolution ω(t). However, in our experiments we will be most interested in the final excitation amplitudes |α| and r after a given evolution of ω. Accordingly, in Figure 3.25(left) we depict final |α| and r when ω/(2π) is switched from 0.5 MHz to 2.5 MHz with a single ramp of variable duration tramp. In addition, in Figure 3.25(right) we illustrate final |α| and r as a function of tramp for a single pulse of ω, i.e., switching ω/(2π) from 2.5 MHz to 0.5 MHz and switching it back after a fixed holding duration thold = 0.5 µs, cf. Fig. 3.24. 3.5 Numerical calculations 73

0.8 1.5 0.6

r 1 0.4 0.2 0.5 0 0

0.3 0.4

| 0.2 α | 0.2 0.1 0 0 0 1 2 3 0 0.5 1 1.5 2

Ramp duration tramp (µs) Ramp duration tramp (µs)

Figure 3.25: Numerical simulation of the time-dependent harmonic oscillator for variable ramp durations. We depict the squeezing amplitude r and the amplitude of a coherent displacement |α| when the harmonic oscillator frequency ω/(2π is switched from 0.5 MHz to 2.5 MHz in a smooth step (left) and for a pulse consisting of two ramps with interleaved holding duration 2 µs (right). For the single ramp, r approaches its maximum value corresponding to an instantaneous switching of rmax ≈ 0.8 for short tramp, whereas we observe r → 0 for adiabatic evolutions. In contrast, |α| shows an oscillating behavior as a function of tramp that is due to interferences between diﬀerent parts of the ramp ω(t). Interferences between the two ramps of a single pulse lead to oscillating behaviors for both, r and |α|.

In case of the single ramp, the squeezing amplitude r approaches the value of an ω(0) instantaneous switching, given by rmax = log ω(t) ≈ 0.8 for tramp → 0, whereas we find r → 0 if tramp approaches adiabatic values. In contrast the coherent displacement amplitude |α| shows an oscillating behavior as a function of tramp that is a result of interference effects of excitations arising during different parts of the ramp ω(t) and that are affected by the evolution of the oscillator’s phase ϕ(t). For large values of tramp the oscillator equilibrates at a finite displacement |α| which is due to our 2 implementation with F0(t) ∝ 1/ω(t) .

When ω is switched in a pulsed manner, both excitation amplitudes, |α| and r, indicate interference of the excitations created by the two ramps. For an appropriately chosen thold, these interference eﬀects may be employed in order to enhance the ﬁnal squeezing amplitude. This method was, already, proposed by Janszky and Adam (1992), and is realized in our experiments presented in Sec. 4.1, where we utilize the interference of the excitations of two pulses, i.e., four ramps of ω. 74 3 Experimental methods

3.5.2 Spin-phonon coupling In order to calculate the time evolution of the spin-phonon coupling presented in Sec. 3.4.1, we can employ numerical simulations of the total Hamiltonian as presented in Porras et al. (2008). This is in particular required when Rabi rates Ω are not small compared to motional frequencies ω, i.e., when we leave the resolved sideband regime as in Clos et al. (2016). Although in this work we solely operate our spin-phonon couplings in the weak coupling regime, there are still off-resonant couplings to transitions other than the one considered in analytic approximations like in Eq. (3.4.4). As this can significantly affect our experimental findings presented in Sec. 4.2, we rather compare them to numerical calculations of the implemented dynamics. To this end, we employ exact numerical diagonalization methods to calculate time evolutions under the action of the spin-phonon Hamiltonian given in Eq. (4.2.2), and repeated here for convenience:

ω Ω h † i H = ~ z σ + ω a†a + ~ 0 σ eiη(a +a) + H.c. (3.5.5) 2 z ~ E 2 + The calculation is based on a Matlab code by Diego Porras, see Porras et al. (2008). It calculates the full eigensystem (all eigenvalues and eigenvectors) of the total Hamil- tonian, i.e., for spin as well as phonon degrees of freedom. In order to make the corresponding matrix descriptions finite-dimensional, we introduce a cutoff ncut in the phononic Fock space. This cutoff has to be chosen such that it discards only negligible amounts of state populations not only for the initial state but also for the whole state space that is explored during the dynamics. Details on the choice of ncut as well as more involved discussions of the numerical calculations and the breakdown of the analytic approximation can be found in Clos et al. (2016) and Clos (2017). In Figure 3.26 we depict an example calculation of the spin-phonon Hamiltonian when driving a blue sideband for an initial state |↓, n = 0i. This corresponds to the parameter regime for the experiments described in Sec. 4.2. 3.5 Numerical calculations 75 i z

σ 1 h , i y

σ 0 h , i x

σ 1

h − 1 | B r

~ 0.5 |

0 1 n

P 0.5

0 1 i

n 0.5 h

0 0 10 20 30 40 50 60 70 80 90 100 TPSR interaction duration (µs)

Figure 3.26: Numerical simulation of the spin-phonon Hamiltonian. We calculate the time evolution for the initial state |↓, n = 0i that is coupled on the ﬁrst blue sideband on the axial lf-mode, i.e., ωz = ωlf = 2π × 1.9 MHz, with Rabi rate Ω0 = 0.1 MHz. This corresponds to only a weak oﬀ-resonant driving of other transitions. We depict the time evolutions of the expectation values of the spin observables hσx,y,zi (blue, gray, orange), the corresponding length of the q 2 2 2 Bloch vector |~rB| = σx + σy + σz , and for the phononic degree of freedom,

the populations Pn of the Fock states n = 0, 1, 2 (orange, blue, gray) as well as the mean phonon occupation number hni. For this calculation we introduced a Fock state cutoﬀ at ncut = 20.

The exact numerical diagonalization calculates, up to a cutoff ncut in Fock space, the total density matrix of the coupled system. Thereby, we are able to illustrate time evolutions for expectation values of spin observables hσx,y,zi, the associated q 2 2 2 length of the Bloch vector |~rB| = σx + σy + σz , and the populations Pn of the first three Fock states (n = 0, 1, 2) as well as the mean phonon excitation hni in the phononic degree of freedom. We note that for the blue sideband transition |↓, ni ↔ |↑, n + 1i the Bloch vector length |~rB| is not constant, which is the basis of our implementation to observe quantum memory effects, see Sec. 4.2. A visualization of the corresponding dynamics of the Bloch vector, obtained from experimental data, is shown in Fig. 3.14 in Sec. 3.2.3. While in our experiments the phononic degree of feedom is, usually, traced out by our measurements, in the numerical calculations 76 3 Experimental methods we obtain information on the Fock state space as well and can identify durations where spin-phonon entanglement is created. In the example in Fig. 3.26 this occurs at t ≈ 8 µs, where a Schrödinger cat like state |ψi = √1 (|↓, n = 0i + |↑, n = 1i) is 2 created. With regard to our findings in Sec. 4.2, we note that the spin observables hσxi and hσyi show small but observable oscillations that are due to off-resonant couplings to other transitions, e.g., the carrier transition |↓, ni ↔ |↑, ni. These justify the necessity to employ numerical calculations of the implemented dynamics in the first place. 4 Experiments and results

This Chapter discusses the experimental results of our analog quantum simulations. In Section 4.1 we reenact cosmological particle creation by creating squeezed states in the motion of our trapped ions with non-adiabatic changes of their trapping potential. The corresponding experimental results are partially covered in Wittemer et al. (2019). In Section 4.2 we employ a single trapped ion in order to investigate an open quantum system. We observe quantum memory eﬀects and quantify the corresponding non-Markovian behavior. These results were published in Wittemer et al. (2018).

4.1 Phonon pair creation

As described in Sec. 2.2.2, the process of particle pair creation in our early universe can be described via a quantum harmonic oscillator with time-dependent frequency Ωk(t). For convenience, we repeat the equation of motion for a single mode φk of a ﬁeld Φ in an expanding space-time, for the full derivation please see Sec. 2.2.2:

" 2 4 # ¨ 2 2 2 m c ¨ 2 φk + c k + a (t) φk = φk + Ωk(t)φk = 0 (4.1.1) ~2

Here, c is the speed of light, m is the mass, and the time-dependent scale parameter a(t) governs the cosmic expansion. The cosmological process of particle pair creation is then understood in the follow- ˙ 2 ing way. For a non-adiabatic evolution of Ωk, i.e., Ωk 6 Ωk, the mode’s quantum state is excited due to evolution Ωk(t). If we describe the state of the mode via its wavefunction ψk the mechanism can be pictured in the following way, see Fig. 4.1 for 0 an illustration. For a fast, i.e., non-adiabatic change of Ωk → Ωk, the wavefunction cannot adapt to the (external) change and, as a consequence, represents an excited 0 state in the new potential with frequency Ωk. If Ωk is switched symmetrically, i.e., without any displacements of the potential minimum or similar, only symmetric harmonic oscillator wavefunctions contribute to the excited state, see Sec. 2.1.3 and, in particular, Fig. 2.2. Therefore, the excitation is realized as a superposition of symmetric (even) wavefunctions which is a squeezed state, see Fig. 4.1(right). In the particle picture, the tearing apart of quantum vacuum ﬂuctuations, i.e., virtual pairs of particle and anti-particle, results in real particle pairs with opposite momenta ±~k. 78 4 Experiments and results

In the squeezed state excitation, this is evidenced by increased populations of even states only. We note that when considering populations of the particle number states only, we disregard the corresponding phases, i.e., the oﬀ-diagonal elements of the density operator. As a consequence, the signs of the momenta ±~k are not resolved. In addition, we note that populations of higher particle number states (n = 4, 6, ...), that are present in a squeezed state excitation, correspond to higher numbers of particle pairs. Figure 4.1 illustrates the particle pair creation mechanism via the squeezing operation of a quantum harmonic oscillator discussed above. 4.1 Phonon pair creation 79 ) ) x x ( ( k k ψ ψ

Position x Position x

+~k n 0.8 P 0.6

Space 0.4 ~k k Population 0.2 ~ ± 2~k − ± 0 t0 t1 0 1 2 3 4 5 Time Number of particles n

Figure 4.1: Cosmological particle pair creation in curved space-time. (Top row) The ground state wavefunction ψk(x) (black solid line) of a harmonic oscillator mode φk has 2 ﬁnite energy and a spatial spread given by the variance (∆x) ∝ 1/Ωk. When Ωk changes (dashed parabola), ψk(x) evolves accordingly. If the change of Ωk is fast, ψk(x) cannot follow adiabatically and, thus, represents an excited, squeezed state (solid line) within the new potential (right plot). The excitation is characterized by a decreased spread (∆x)2 compared to the new ground state (dashed line). (Bottom left) Schematic of curved space-time during cosmic expansion. Quantum ﬂuctuations are depicted as a pair of virtual particles at some time t0. Due to a rapid cosmic expansion the two virtual particles are torn apart until their distance, related to the physical wavelength of the corresponding mode (shaded area), becomes too large for them to recombine and annihilate (at t1). Thereafter, the two particles have become real and move into opposite directions with momenta ±~k. However, they remain linked via quantum entanglement. (Bottom right) The Fock state populations of the mode φk indicate the excitation of a squeezed state, that corresponds to the pairwise creation of particles with momenta ±~k. This state contains excitations for even numbers of particles only, while odd states remain unpopulated. Figure adapted from Wittemer et al. (2019). 80 4 Experiments and results

With our focus on a single mode of a scalar field, we effectively reduce the experimentally inaccessible dynamics during cosmic expansion to a well-controllable quantum system. This is the basis for analog quantum simulations and, correspondingly, we may set up a quantum simulator and interpret its dynamics in terms of cosmological particle pair creation, see Fig. 4.1. With regard to trapped ions, there are two theoretical proposals for an experimental implementation, see Schützhold et al. (2007) and Fey et al. (2018). The common basis for both approaches is the realization of non-adiabatic frequency changes of a quantum harmonic oscillator that results in a squeezed state, which is then analog to the creation of cosmological particle pairs. Hence, we may define the corresponding task for the experimenter: Create squeezed states in a quantum harmonic oscillator by tuning the oscillator’s eigenfrequency. In the following Sections we will briefly recapitulate our way to the first successful experimental implementation of this scheme.

4.1.1 Precursor experiments In this Section, we will briefly give an overview of some pioneering attempts to implement the phonon pair creation in our group. We note that also other groups have proposed (Alonso et al., 2013) and reportedly failed (Leupold, 2015) to implement squeezed states by fast changes of the motional frequencies of trapped ions. Thus, we want to illustrate some of the difficulties that are inherent for an implementation of such non-adiabatic potential changes. The original protocol by Schützhold et al. (2007) considered a single ion, trapped in a potential with a time-dependent axial motional mode frequency, but fixed radial confinement. A first implementation of this protocol in our ion trap was already (attempted) by Schmitz (2009). However, the limited real-time control over the trapping potential resulted in the creation of large coherent states rather than squeezed state excitations, similar to the experiments presented in Alonso et al. (2016). With our improved real-time control, using arbitrary waveform generators (AWGs), see Sec. 3.1.2, and dedicated control potentials, see Sec. 3.1.3, we may give the proposal another shot. The corresponding protocol, as well as the experimental results are depicted in Fig. 4.2. 4.1 Phonon pair creation 81

| 6

Initialize Cool Pulse Hzz TPSR Detect α |

4 0

10 2 −

20 Excitation amplitude Voltage change (V) − 0 0 5 10 15 20 0.28 0.3 0.32 0.34 0.36 0.38

Waveform duration (µs) Pulse depth ∆ωlf/(2π) (MHz)

Figure 4.2: Experimental results for the non-adiabatic switching of the axial mode with a single trapped ion. We employ voltage overshooting techniques in order to allow for fast switching of the control potential Hzz that tunes ωlf. On the left we depict the experimental protocol and the corresponding AWG waveform (gray, one electrode only) to overshoot the low-pass ﬁlters of the control electrodes. The measured output waveform (orange) is in good agreement with the desired output (blue). On the right we show the resulting motional excitation as a function of the pulse depth ∆ωlf. We detect coherent states of variable amplitudes |α| preventing the detection of signiﬁcant squeezing amplitudes r as these are, estimated from numerical simulations, below 1 × 10−4 in the probed parameter regime.

We employ overshooting techniques (Bowler et al., 2013) to allow for fast waveforms of the control potential Hzz and find good agreement between desired and measured output. However, we find only coherently displaced motional states our trapped ions, in a parameter regime where squeezing is, estimated from numerical calculations, negligible. Thus, with this attempt we face the same difficulties as in the previous implementation by Schmitz (2009). We attribute this to a shift of the trapping potential minimum when Hzz is pulsed, that we are not mitigate sufficiently, despite the new level of control that we obtain by upgrading the experimental apparatus with the real-time control. This may be due to finite irregularities in the electrical characteristics of our trap’s electrodes. Thus, in a next step we realize the proposal by Fey et al. (2018), that considers two trapped ions in, essentially, the same time-dependent trapping potential as the proposal by Schützhold et al. (2007). Focussing on the radial degrees of freedom of the two ions, significant squeezing is proposed for the rocking mode. We employ the fast switching capabilities of the control potential Hzz (see Fig. 4.2) in order to realize the proposed sequence, the corresponding results are depicted in Fig. 4.3. 82 4 Experiments and results

1 1 0.8 0.8 0.6 0.4 0.6 0.2 0 0.4 0.8 0.6 0.2 0.4

Norm. Fluorescence 0.2 0 0 0 1 2 3 4 5 0 5 10 15 20

Holding duration thold (µs) TPSR interaction duration (µs)

Figure 4.3: Experimental results for the non-adiabatic switching of the axial mode with two ions. We realize the protocol depicted in Fig. 4.2(left) with two ions, where all motional modes prepared in their ground state. On the left we depict the resulting modulation of the normalized ﬂuorescence of the ions as a function of the holding duration thold between the two ramps of Hzz. This excitation of the stretch mode corresponds to a coherent state of large amplitude (n¯ 100). Du to the non-linear Coulomb interaction of the ions, this excitation is coupled onto the rocking mode and, as a consequence, the motional state analysis thereon (right, top and bottom for thold = 0.6 µs and thold = 1.0 µs, respectively) does not allow for the detection of a squeezed state.

Here, we detect coherent excitations of large amplitudes (n¯ 100) on the stretch mode, due to the modulation of the distance of the two ions via the applied waveform of Hzz. However, due to the non-linear coupling of the ions via their Coulomb interaction, this excitation is also transferred onto the rocking mode, similar to a cross-Kerr type coupling, cf. Ding et al. (2017a,b) and see also Palmero et al. (2015). As a consequence, the motional state analysis, see Sec. 3.4.2, on the rocking mode does not allow for the detection of any created squeezing. As an alternative to the modulation of the ion’s trapping potential via dedicated control potentials applied onto the control electrodes of our trap, see Sec. 3.1.2, we may also manipulate the RF induced pseudo-potential in real-time. This is enabled by the dynamic control of the RF voltage U, see Sec. 3.1.4, that allows for ramping durations tramp ≈ 1 µs. In a ﬁrst implementation, we realize the sequence depicted in Fig. 4.4 with a single ion. 4.1 Phonon pair creation 83

r 1.5 , Initialize Cool Pulse U TPSR Detect | α |

100 1 (%) U

50 0.5 RF voltage 0 Excitation amplitude 0 0 5 10 15 2.5 2.6 2.7 2.8

Waveform duration (µs) Pulse depth ∆ωmf/(2π) (MHz)

Figure 4.4: Experimental results for the non-adiabatic switching of the radial trapping potential with a single ion. We employ our real-time control of the RF voltage U to switch the radial mode frequency ωmf to a lower value with an adiabatic ramp and back up with a non-adiabatic ramp. On the left we depict the experimental protocol and an oscilloscope measurement of the switching of U that indicates an eﬀective ramp duration tramp ≈ 1 µs for the up-ramp. On the right we show the resulting motional excitation on the radial mf-mode as a function of the pulse depth ∆ωmf of the protocol. We detect both, coherent and squeezing excitation with variable amplitudes |α| (blue) and r (orange), that can be compared to the results of numerical simulations (solid lines).

In order to study the underlying dynamics induced by our real-time control of U in a controlled way, we employ an asymmetric waveform. We slowly ramp down U, thereby, lowering ωmf adiabatically, before shooting U back up with an instantaneous switching of the control channel on the RF mixer, see Sec. 3.1.4. Due to the finite bandwidth of the helical resonator of our trap, the corresponding up-ramp of U takes tramp ≈ 1 µs, which is significantly non-adiabatic for any ωmf/(2π) < 1 MHz. We realize the protocol for variable switching amplitudes in U and perform the motional state analysis described in Sec. 3.4.2. Thereby, we detect non-zero squeezing amplitudes r that are in agreement with numerical simulations of the implemented dynamics. However, due to finite stray electric fields present in our setup, the squeezing excitation is accompanied by coherent displacement of even larger amplitudes |α|. As a consequence, the corresponding Fock state distributions do not exhibit a squeezed state structure with populations in even states only, cf. Sec. 2.1.3. Nevertheless, the experimental results of this protocol represent a first signal for a squeezed state excitation. Thus, the implementation of the underlying mechanism may be fine-tuned in order to optimize the measurement signal. In the following Sections we describe how the above sequence is adapted in order to achieve an unbiased observation of a squeezed state. 84 4 Experiments and results

4.1.2 Quantum simulator setup For our (successful) implementation of the phonon pair creation in our trap we consider two ions in a time-dependent radial confinement ωrad(t) along the x-direction, but with fixed axial confinement ωax along the z-direction. For small radial displacements δx from the equilibrium positions, we distinguish the center-of-mass mode q 2 2 ω1(t) = ωrad(t) from the rocking mode ω2(t) = ωrad(t) − ωax, see Sec. 3.3 for details. Explicitly, considering a displacement δq2 along the rocking mode the equation of motion is given by

h 2 2 i δq¨2 + ωrad(t) − ωax δq2 = 0. (4.1.2)

Thereby, we can identify the following analogies between our trapped ion setup and the cosmic particle pair creation. Equation (4.1.2) is analog to Eq. (4.1.1), where the motional mode δq2 corresponds to the Fourier mode φk of the field Φ. The 2 Coulomb interaction of the two ions leads to the term −ωax which corresponds to the internal (propagating) dynamics c2k2, whereas the (external) trapping potential 2 2 4 2 2 ωrad simulates the mass term m c /~ . Its time-dependence ωrad(t) represents the external influence and is analog to the cosmic expansion a(t). Further, we identify an 2 expanding universe with an increasing radial confinement ωrad(t). In order to experimentally realize particle pair creation in the motional modes of our trapped ions, we identify the following essential criteria:

1. Fast switching of ωrad in order to induce non-adiabatic dynamics for ω2(t). 2. Motional ground state preparation in order for quantum vacuum ﬂuctuations to dominate the dynamics.

3. Motional state readout on the single phonon level.

For the first point we employ the real-time control over the trap’s RF voltage U, described in Sec. 3.1.4. Here, the time scale for the switching of U and, consequently ωrad, is limited by the trap’s helical resonator to (minimal) switching durations tramp ≈ 1 µs. The measurement in Fig. 3.7 verifies the real-time switching capabilities of U via an oscilloscope measurement, that we can only assume to yield the desired dynamical evolution of ωrad. However, we want to verify these results using our trapped ions and measure the real-time switching of the motional mode frequencies ω1 and ω2 directly. To this end, we employ a spin-motional echo sequence, based on the method described in Leibfried et al. (2002). The corresponding sequence, together with exemplary data for ω1, is shown in Fig. 4.5(left). We initialize U to Uhigh, a pre-determined value that allows for efficient sideband cooling with radial mode freqencies of several MHz, cf. Sec. 3.3.1. After initializing the motional ground state with n¯1,2 ≈ 0 and preparing |↓i, we implement the following spin echo protocol. 4.1 Phonon pair creation 85

With a π/2 pulse on the blue sideband transition√ |↓, ni ↔ |↑, n + 1i on the corresponding mode, ω1 or ω2, a superposition 1/ 2 (|↓, 0i + |↑, 1i) is created. During the first arm of the echo sequence, we ramp down U, hold at Ulow for a duration thold, and ramp back up to Uhigh. We account for a finite delay between UAWG(t) and U(t) with an appropriate waiting duration tdelay after ramping up U(t). The corresponding waveform, for exemplary values Uhigh and Ulow, incorporating the non-linear atten- uation by the RF mixer is shown in Fig. 3.7 in Sec. 3.1.4. In the second arm, after a π-pulse (echo) on the blue sideband, U is kept high (no pulse), and the sequence concludes with another π/2-pulse on the blue sideband, and subsequent spin-state detection. Overall, this sequence represents an interferometer with effective arm length tse = 2 tramp + thold + tdelay, that is sensitive to differential phase accumulations of both arms. Hence, we can employ this sequence to detect mode frequency differences. Repeating measurements for variable thold and, accordingly variable tse, we record√ final P↓. Due to the spin-phonon entanglement created by the state 1/ 2 (|↓, 0i + |↑, 1i), the spin signal P↓(thold) oscillates sinusoidally, with the oscillation rate corresponding to the difference of the motional mode frequency during thold in both arms. We show an example evolution of P↓(thold) in Fig. 4.5(left), indicating a frequency difference ∆ω1/(2π) = 1.53(3) MHz, that is in agreement with the (static) calibration presented in Sec. 3.1.4. We benchmark the experimental parameter range for the real-time switching of motional mode frequencies by repeating these measurements for both, ω1 and ω2 for variable Ulow and depict corresponding results in Fig. 4.5(right). 86 4 Experiments and results

3.5 tse tse π/2 π π/2 3 2.5 1 2

0.8 ) (MHz) π 1.5 (2

↓ 0.6

P 1 0.4 ω/ 0.2 0.5 0 0 0 1 2 3 4 5 0.7 0.75 0.8 0.85 0.9 0.95 1

Holding duration thold (µs) Trap supply voltage U/Umax

Figure 4.5: Verifying real-time mode frequency switching via spin-phonon entanglement. We depict the experimental sequence for the spin-phonon echo sequence and exemplary data for the COM mode ω1 on the left. From the oscillation rate of P↓(thold) we extract mode frequency diﬀerences during thold in both arms of the interferometer. Since the oscillation rate is constant, for long as for short thold, we can infer that the frequency shift is complete after each ramp of length tramp = 1 µs. The right plot illustrates the agreement of the mode frequencies ω1 (blue) and ω2 (orange) extracted from this dynamical method (data points) with the static calibration measurement (solid lines), determined via the oscillating electric ﬁeld excitation method, see Sec. 3.3.2. Figure adapted from Wittemer et al. (2019).

We set the offset for the (relative) real-time switching verification protocol described above by measuring the (absolute) mode frequency at Uhigh with the oscillating electric field excitation method described in Sec. 3.3.2. Thereby we are able to compare our dynamic calibration data with the static values and find good agreement for a broad range of U. Hence, we may conclude that the dynamic RF voltage control is capable of switching (radial) motional modes in our trap with ramping durations tramp = 1 µs, although we note that the effective ramping duration, due to the choice of a smooth step, is effectively shorter than 1 µs.

Nevertheless, if we conservatively consider tramp = 1 µs, we may define an upper bound motional mode frequency 2π × 1 MHz for which corresponding dynamics are non-adiabatic. With regard to our intended protocol to employ a non-adiabatic switching for the creation of a squeezed state in the ions’ motion, we find an advantage of focusing on the rocking mode ω2. Due to the Coulomb interaction of the two ions, the mode frequency variation as a function of U is boosted and, as a consequence, we find ω2 → 0 already for ωrad ≈ ωax. For this reason, the non-adiabatic parameter regime is reached for conditions that still allow for stable trapping, significantly reducing the risk of losing ions when switching U. In addition, we note that, for the same reason, the slope ω2(U) is increased with respect to ω1(U) for same switching 4.1 Phonon pair creation 87 amplitudes in U. Thus, for the following considerations we will focus on the rocking mode ω2 and choose Uhigh and Ulow for a non-adiabatic evolution accordingly. In a first, step we coarsely tune the motional mode configuration, aligning the two ions on the trap’s axial symmetry axis. In particular, we optimize both, electric fields as well as curvatures with dedicated control potentials, see Sec. 3.1.3. Further, we choose ωax/(2π) ≈ 1.3 MHz and Uhigh so that ω1/(2π) ≈ 2.8 MHz and, consequently, q 2 2 ω2 ≈ ω1 − ωax ≈ 2π × 2.5 MHz. For the high-frequency mode along the radial direction we find ωhf(Uhigh/(2π)) ≈ 4.4 MHz, which is significantly separated from the other modes so that we may disregard it in the following. By this choice of ωax and Uhigh we ensure well resolved sidebands for all modes (COM and out-of-phase) along the radial directions to allow for efficient ground state cooling via the sideband cooling technique described in Sec. 3.3.1. The corresponding mode spectrum is shown in Fig. 3.22(bottom). We note that for the experiments described in the following, we trap two magnesium ions with different isotopes, 25Mg+ and 26Mg+. Thereby, motional mode frequencies are shifted with respect to a homogeneous chain. However, the shifts are only on the few percent level, see Sec. 3.3. By using only one 25Mg+, thereby, implementing only one spin-1/2 system, we can conveniently prepare and detect the motional state of the ion chain by the mode-resolved motional state analysis described in Sec. 3.4.2. To this end, we couple the spin state of the 25Mg+ to the motion of both ions with TPSR transition laser beams. In order to ensure stable Rabi rates, we maintain and verify a dedicated ion order with the ordering procedure described in Sec. 3.3.4. Accordingly, we align the laser beams for both, TPSR transitions and spin state detection, to the 25Mg+. With the ground state cooling optimized, we can check the motional state on the rocking mode ω2. As presented in Sec. 3.4.2, we couple the spin system in the elec- 25 + tronic state of Mg to the motion along ω2 for variable durations and extract individual Fock state populations Pn from the different oscillation rates of the spin- state population P↓. For the rocking mode we drive both, red and blue sidebands in subsequent sequences and extract the Pn from a combined model fit. In Figure 4.6 we depict experimental results for the motional state of the rocking mode after sideband cooling. 88 4 Experiments and results

1 1

0.8 n 0.8 P 0.6 0.6 ↓ P 0.4 0.4

0.2 Population 0.2 0 0 0 20 40 60 80 100 120 0 1 2 3 4 5 6 7 8 TPSR interaction duration (µs) Phonon number state n

Figure 4.6: Analyzing the motional ground state on the rocking mode. We record P↓ for variable TPSR coupling durations (left) and extract the individual phonon state populations Pn (right, data points with error bars partially smaller than symbol size) from a combined model ﬁt (solid lines). From a ﬁt of a Gaussian state (right, bars) to the Pn, the corresponding Wigner function can be depicted (inset). Here, a thermal state with n¯ = 0.03(6) is detected, accordingly the Wigner function outlines a circle with an area ≈ ~, in accordance with Heisenberg’s uncertainty principle. Figure adapted from Wittemer et al. (2019).

From combined model fits to the measured spin signal we extract individual Fock state populations Pn. In particular, we find a population P0 > 90 % for the ground state n = 0. We estimate the residual thermal excitation n¯ by performing a model fit of a Gaussian state ρth(¯nth) to the measured Pn and find n¯th = 0.03(6), see Sec. 3.4.2 for details. In a subsequent step we can calculate the Wigner function corresponding to ρth(¯nth), the result is depicted in the inset in Fig. 4.6. The Wigner function extends over ≈ ±10 nm (for the 1/e2 amplitude) along the position axis, corresponding to the p extent of the ground state wavefunction ~/(2mω2), cf. Wineland et al. (1998a) and Leibfried et al. (2003). Thus, we have completed the checklist of prerequesites to follow our protocol of creating (and measuring) squeezed motional states by switching the motional mode frequency of our trapped ions. We are able to prepare the motional ground state and detect the state on the single phonon level. Moreover, we have verified that we are able to ramp the radial confinement within tramp = 1 µs. Consequently, we choose Ulow such that the weak confinement features a rocking mode frequency ω2/(2π) ≈ 0.5 MHz. This is well below 1 MHz to ensure non-adiabatic dynamics, but well-above the point where the axially oriented chain mutates into a radially oriented chain at ωax ≈ ωrad, i.e., ω2 ≈ 0. For convenience, in Tab. 4.1 we summarize the motional frequencies ω1, ω2, and ωax that are relevant for our envisioned protocol, at both, Ulow and Uhigh. 4.1 Phonon pair creation 89

Ulow Uhigh ωax/(2π) 1.31(1) MHz 1.31(1) MHz ω1/(2π) 1.40(1) MHz 2.83(1) MHz ω2/(2π) 0.50(1) MHz 2.50(1) MHz

Table 4.1: Relevant motional modes for the squeezed state generation. The radial mode frequencies at Uhigh are measured via the electric field excitation method, while the corresponding values at Ulow are determined by the measured frequency differences using the spin-phonon echo sequence. In our protocol, we switch between both configurations with a smooth step within tramp = 1 µs. We note that the remaining two motional modes along the radial directions are > 2π × 3 MHz for both configurations, so that we may omit them in our analysis.

With the mode conﬁgurations set by Ulow and Uhigh (and additional curvature control potentials) we may perform a numerical simulation of the dynamics. In particular, we are interest for the squeezing amplitude r that is excited by a single ramp of duration tramp = 1 µs with the mode frequencies listed in Tab. 4.1. To this end, we employ the simulation of the time-dependent quantum harmonic oscillator that is described in Sec. 3.5.1, the results are depicted in Fig. 4.7. 90 4 Experiments and results

3 6 2 a 5 2 4

) (MHz) 3 π

(2 1 2 / 2

ω 1

0 Scale parameter 0.75 0.8 0.85 0.9 0.95 Trapping voltage U (Umax)

3 7 2 5 2 ) (MHz) a π 1 3 (2 / 2 0 1 ω 5 10 2 2

/ω 3 2 5 (MHz) ˙ ω 2 ˙ 1 a 0 0.2 0.4 sq r

0.1 ¯ 0.2 n 0 0 0 0.2 0.4 0.6 0.8 1 Duration (µs)

Figure 4.7: Experimental parameter regime and numerical simulation for the squeezed state generation. (Top) Frequency of the rocking mode ω2 as a function of U, data points are measured with the oscillating electric ﬁeld excitation method (error bars smaller than symbol size), the solid line shows a model ﬁt to the data. The chosen frequency range for the dynamical protocol (shaded area) is compared to the analog scale parameter a2 (right axis). The lower graphs depict a numerical simulation of a single ramp, switching ω2/(2π) within tramp = 1 µs from 0.5 MHz to 2.5 MHz. We depict ω2(t) (solid line) as well as the evolution of the scale pa- 2 rameter a2(t) (dashed line, right axis) and their derivatives, ω˙ 2/ω2 (solid line) and a˙ 2 (dashed line, right axis). The bottom graph illustrates the accumulation of a squeezing amplitude r and the corresponding instantaneous average particle number n¯sq, that is induced by the non-adiabatic evolution ω2(t). Figure adapted from Wittemer et al. (2019). 4.1 Phonon pair creation 91

The calibration curve ω2(U) can be compared to the scale parameter a2(t) = ω2(t)/ω2(0). For a ramp from ω2(Ulow) to ω2(Uhigh) we find a2(tramp) = 5, equivalent to an expansion of space by 1.6 e-foldings in our analogy of cosmic expansion. We take the non-linear relation ω2(U) into account and calculate the time evolutions ω2(t) and 2 a2(t) for tramp = 1 µs as well as their time derivatives ω˙ 2/ω2 and a˙ 2(t). From these we can evaluate the non-adiabadicity of the dynamics. In particular at the beginning of 2 the ramp where the slope in ω2(U) is large, we find ω˙ 2/ω2 ≥ 5, significantly increased 2 with respect to the adiabatic limit ω˙ 2/ω2 = 1. From our numerical simulations of the dynamics we obtain the time evolution of the density matrix ρ2 that describes the motional state of the rocking mode. From ρ2 we can extract the instantaneous 2 squeezing amplitude r, as well as the average particle number n¯sq = sinh (r), see Sec. 3.5.1 for details. The simulation yields an oscillating accumulation of r and n¯sq that visualizes the impact of our finite ramp duration. During the ramp the mode’s R t 0 0 phase ϕ2(t) = 0 ω2(t )dt evolves and thus, infinitesimal excitations from different parts of the ramp can interfere according to ϕ2. The simulated final r ≈ 0.25 for a single ramp ω2(t) corresponds to an average particle number n¯sq ≈ 0.07, larger than the residual thermal excitation of the initial state n¯th = 0.03(6). Thus, in principle, already for a single ramp particle pair creation could be detected. However, we note that for our experiments we choose to apply such non-adiabatic evolutions in closed cycles, i.e., we form pulses of ω2 by combining ramps switching from ω2(Uhigh) to ω2(Ulow) and vice versa. Explicitly, after switching to Ulow within tramp = 1 µs and an appropriate holding duration thold = 2 µs we ramp U back up. Thereby, we can perform state initialization and detection with equal mode configurations at Uhigh, where sidebands (in particular for ω2) are well resolved. In addition, we note that by the choice of thold we may induce constructive or destructive interference of the individual excitations (squeezing or displacement) of both ramps, cf. Sec. 3.5.1.

4.1.3 Pair creation with a single pulse As a ﬁrst experimental realization we discuss the pair creation with a single pulse of ω2. The protocol (not to scale) is depicted in Fig. 4.8(top). We initialize U to Uhigh and perform sideband cooling (SBC) of all radial motional modes before preparing either |↓i or |↑i. Subsequently, we pulse U, inducing two non-adiabatic evolutions for ω2. Afterwards we map the motional state of the rocking mode to the ions’ spin state using TPSR transitions on the blue or red sideband, see Sec. 3.4.2. We conclude the sequence by measuring the spin state using resonance ﬂuorescence detection, see Sec. 3.2.1. 92 4 Experiments and results

Init. spin Pulse U TPSR coupling Detect P Init. U SBC ↓ 1 1

0.8 n 0.8 P 0.6 0.6 ↓ P 0.4 0.4

0.2 Population 0.2 0 0 0 20 40 60 80 100 120 0 1 2 3 4 5 6 7 8 TPSR interaction duration (µs) Phonon number state n

Figure 4.8: Phonon pair creation with a single pulse. The protocol starts with initialization of U and subsequent resolved sideband cooling (SBC) of all radial modes and preparation of the spin state. Afterwards, U is pulsed, each ramp has a nominal duration of tramp = 1 µs. We detect the motional state of the rocking mode by mapping it to the spin state and measuring P↓. The bottom row depicts the corresponding motional state analysis. We record red and blue sidebands (left) in subsequent sequences and extract individual Fock state populations Pn (data points, right) from our model ﬁts. The resulting motional state shows excitation due to the pulse. We decompose the excitation into contributions by squeezing r = 0.54(8) and coherent displacement |α| = 0.88(6) with a model ﬁt of a Gaussian state (bars) from which we can visualize the corresponding Wigner function (inset). Figure adapted from Wittemer et al. (2019).

In our experiments, we record red and blue sideband flops and obtain individual Fock state populations Pn from our combined model fit. The experimental results, depicted in Fig. 4.8, indicate significant motional excitation of the rocking mode due to the pulse. However, the Fock state population distribution is of mixed character rather than clearly indicating a squeezed state. Therefore, we decompose the excitation into a thermal share n¯th, a squeezing amplitude r, and a coherent displacement amplitude |α|, by fitting a Gaussian state ρSDT(r, |α|, n¯th) to the measured Pn. The result is r = 0.54(8) and |α| = 0.88(6), where we kept the thermal excitation at −1 n¯th = 0.03, because heating effects for the rocking mode (n¯˙ ≤ 20 s ) remain negligible on the timescales of our experimental sequences (≈ 100 µs). The underlying coherent displacement can be attributed to residual stray potentials in our trap. We note that the rocking mode is insensitive to homogeneous electric fields that displace both ions equally. However, electric field gradients, that lead to a differential force on both ions, can lead to a coherent state on the rocking mode when ωrad is switched. In order to visualize the impact of a residual (differential) force F2 under the evolution ω2(t), we employ the numerical simulation of the time-dependent quantum harmonic oscillator, see Sec. 3.5.1. We include a (differential) electric stray field amplitude 4.1 Phonon pair creation 93 of 13 mV/m and simulate the quantum dynamics of the rocking mode during the evolution of a single pulse of U. The result is depicted in Fig. 4.9. 94 4 Experiments and results

3 (a.u.) 2 F 2

1 MHz), π

(2 0 2 ω 0.6

| 0.4 α | 0.2 0 0.6 0.4 r 0.2 0 0.6 0.4 ¯ n 0.2 0 0 1 2 3 4 5 Duration (µs)

Figure 4.9: Numerical simulation of the quantum dynamics of the rocking mode ω2 during a single pulse, including a finite differential electric stray field amplitude of 13 mV/m. We depict the evolution of the oscillator frequency ω2 (orange) and the associated effective force F2 (gray). From our numerical simulations we obtain the coherent displacement amplitude |α| and the squeezing amplitude r that show qualitatively different behaviors. In addition, we depict in the bottom graph the instantaneous mean phonon number n¯ for the actual state (solid blue) and the individual phonon numbers corresponding to the coherent displacement 2 2 n¯coh = |α| (dashed gray) and the pure squeezing excitation n¯sq = sinh (r) (dashed orange). We note that the sum n¯coh +n ¯sq does not reproduce n¯, which indicates the impact of the oscillator’s phase evolution ϕ2(t) with respect to the phases of the individual excitations. 4.1 Phonon pair creation 95

We show the evolution of the rocking mode frequency ω2 and the corresponding evolution of the effective force F2. From our simulation, we obtain the time evolution of the full density matrix and, thus, are able to calculate the amplitude of a coherent displacement |α| and the squeezing amplitude r. Our numerical results indicate qualitatively different behaviors for |α| and r. The coherent state amplitude |α| oscillates predominantly between the two ramps, where ω2 and, thus, F2 is constant. This indicates the finite inertia of the oscillator ω2. In contrast, we find constant r when ω2 is constant and increasing behavior occurs predominantly when variations ω˙ 2 are fast compared to ω2, i.e., at the end of the downwards ramp and at the beginning of the upwards ramp of ω2. Thereby, the criticality of non-adiabatic dynamics is highlighted by our numerical results. We also note that the evolution of both, |α| and r, is different during both ramps, indicating the influence of the phase evolution ϕ2(t). In addition to the excitation parameters, we depict the total instantaneous mean number of phonons n¯. Here we, in particular, note that n¯ is not equal to the sum of the instantaneous phonon number corresponding to each individual excitations, 2 2 n¯coh = |α| and n¯sq = sinh (r). Instead, n¯ depends on the individual state and its distinct phase evolution that leads to a differential phase between both excitations, which gives rise to constructive or destructive interference. In conclusion, we find motional excitation due to the switching of U, and thereby ω2, in our experiments as well as in our numerical simulation. With the latter we can reproduce the ambiguous Fock state population distribution, that explains our results as a superposition of a finite coherent displacement and a significant squeezing excitation. Thereby, we may consider these results as the successful creation of a squeezed state by tuning the frequency of a quantum harmonic oscillator. However, the ambiguous Fock state distribution does not allow for a direct verification of the squeezed state, rather we must rely on the fitting of an appropriately parametrized Gaussian state. Therefore, residual doubts of the squeezing excitation may be allowed. These, however, will be dispelled in the next Section.

4.1.4 Signal puriﬁcation with two pulses Under the assumption that a single pulse of U creates a superposition of a squeezed and coherently displaced state, we can think of an echo sequence that reverses the coherent displacement. To this end, we apply two equal pulses of U, each of them displacing and squeezing the oscillator, that are separated by a variable free evolution duration tfree. The corresponding phase space dynamics are illustrated in Fig. 4.10. 96 4 Experiments and results

Initialize Free evolution: tfree Readout )

α tfree Im( Re(α)

Figure 4.10: Purifying squeezing with an echo sequence. By applying two equal pulses of U, separated by a free evolution duration tfree, we are able to reverse the coherent displacement that is induced by a single pulse. We depict the protocol as well as corresponding Wigner functions, illustrating the dynamics in phase space. The coherent excitation is reversed, if the duration between the two pulses is tπ = π/ω2 (or odd multiples). For these durations, the state has traveled from +α to −α, i.e., acquired a phase of π for the coherent excitation. The squeezing excitation, however, oscillates with 2ω2 and, thus, has acquired a phase of 2π, so that the second pulse can even increase the squeezing amplitude. Figure adapted from Wittemer et al. (2019).

We consider the following procedure to create a purely squeezed state on the rocking mode. Similar to the one pulse sequence, we initialize the motional state close to the ground state. When U is pulsed, squeezing is created together with a coherent displacement α. During the free evolution duration tfree, when ω2/(2π) is back at 2.5 MHz, the state can oscillate freely, i.e., not influenced by (external) variations ω2(t). If a second pulse of U is applied after a duration of free evolution of tπ = π/ω2 (or odd multiples) the coherent excitation can be reversed. After tπ, the state has traveled on a circular path in phase space from +α to −α so that the second pulse reverses the coherent displacement to α ≈ 0. In contrast, the squeezing excitation 2 †2 accumulates its phase with 2ω2 due to the terms a and a in the definition of the squeezing operator. Hence, after tπ the squeezing has acquired a phase of 2π and the second pulse can constructively interfere with the first pulse and even increase the final squeezing amplitude. However, we note that due to a finite phase difference between the two different excitations (squeezing and displacement) tπ is not necessarily the duration after which squeezing is maximized with the second pulse. In order to illustrate the dynamics corresponding to the two-pulse sequence, we employ again the numerical simulation of the time-dependent harmonic oscillator from Sec. 3.5.1. We depict numerical results for the evolutions of |α| and r for three representative tfree = {0, 1, 1.5}· tπ in Fig. 4.11. 4.1 Phonon pair creation 97

2 ) (MHz) π

(2 1 / 2

ω 0

1 | α | 0.5

r 0.5

0 0 1 2 3 4 5 6 7 8 Duration (µs)

Figure 4.11: Numerical simulation of the two-pulse sequence. We depict the evolution ω2(t) and corresponding excitation amplitudes |α| and r for three representative tfree = {0, 1, 1.5}· tπ (blue, orange, gray). In particular, we find that final excitation amplitudes |α| and r depend on tfree due to their different oscillation rates ω2 and 2ω2, respectively. This enables the purification of the squeezing excitation from the underlying coherent displacement when tfree = tπ.

We find that, for tfree = 0, excitations arising from the second pulse constructively interfere with those from the first pulse because no phase difference has accumulated. As a consequence, both final amplitudes, r and |α|, are increased. In contrast, when tfree = tπ the coherent displacement has accumulated a phase of π, thus, the second pulse reverses the excitation to α ≈ 0. For the squeezing tπ corresponds to an accumulated phase of 2π, i.e., the second pulse can, again, increase the squeezing amplitude r. Accordingly, for tfree = 1.5tπ, which corresponds to an effective phase difference of π for r, the final squeezing amplitude is zero, as the individual squeezing excitations from the two pulses cancel out.

We note that the exact duration tπ that is required to realize the anticipated echo effect (such that α ≈ 0) depends on the particular waveform ω2(t). For our realization of a smoothstep waveform, see Sec. 3.1.4, there are durations with ω˙ 2(t) ≈ 0 at the end of the ramp that effectively contribute to the duration of free evolution and, thereby, lead to an offset for tπ. In our numerical simulations in Fig. 4.11 this offset is compensated by appropriately chosen tfree. For our experimental realization, however, we optimize the echo effect by realizing the two-pulse sequence for variable tfree and 98 4 Experiments and results

ﬁnd tπ from the evolutions |α|(tfree) and r(tfree). The corresponding experimental results are shown in Fig. 4.12.

| 1 α |

, 0.8 r 0.6 0.4

Excitation 0.2 0 30.1 30.2 30.3 30.4 30.5

Free evolution duration tfree (µs)

Figure 4.12: Experimentally optimizing the echo effect of the two-pulse sequence. We realize the two-pulse sequence for variable tfree and extract |α| (blue) and r (orange). For comparison, we depict results of numerical calculations (solid lines) with an estimated effective stray field amplitude 13 mV/m. The individual evolutions |α|(tfree) and r(tfree) illustrate the different (dominant) oscillation rates ω2 and 2ω2, respectively. As a consequence, for tfree = 30.2 µs we find decreased |α| and increased r with respect to the results for a single pulse. Figure adapted from Wittemer et al. (2019).

In our experiments, we implement the two-pulse sequence with variable tfree ∈ [30.15, 30.45]. Thereby, we take account of residual slow drifts of U due to finite bandwidth effects, similar to the included tdelay for a single pulse, see Sec. 3.1.4. We note that this does not significantly introduce additional decoherence effects as the motional coherence time of the rocking mode is & 100 µs, cf. Sec. 3.3.5. Comparing our experimental data |α|(tfree) and r(tfree) with numerical simulations, including an estimated effective stray field amplitude 13 mV/m, we recognize the different (dominant) oscillation rates ω2 and 2ω2, respectively. However, in particular for |α|(tfree) we find additional frequency components in the signal that we attribute to our finite ramp duration tramp = 1 µs. When tramp is finite the oscillator’s phase ϕ2 evolves during the ramp and, thus, the final excitation is the result of several coherent displacements with different phases, i.e., displacements along different directions in phase space. As these individual excitations have different phases they rephase after different durations, i.e., they effectively have individual tπ. We note that this effect may even be enhanced due to the superimposed squeezed state evolution. Nevertheless, analyzing our experimental results, we recognize the anticipated echo effect. For tfree ≈ 30.2 µs we find decreased |α| and increased r with respect to the single pulse sequence. Hence, in Figure 4.13 we depict the motional state analysis corresponding to the two-pulse sequence with tfree = 30.2 µs. 4.1 Phonon pair creation 99

1 1

0.8 n 0.8 P 0.6 0.6 ↓ P 0.4 0.4

0.2 Population 0.2 0 0 0 20 40 60 80 100 120 0 1 2 3 4 5 6 7 8 TPSR interaction duration (µs) Phonon number state n

Figure 4.13: Motional state analysis for the optimized two-pulse sequence with tfree = 30.2 µs. We record red and blue sidebands (left) in subsequent sequences and extract individual Fock state populations Pn (data points, right) from a combined model ﬁt. From the Pn the squeezed state excitation is directly evidenced by increased populations for even Fock states while odd states remain nearly unpopulated. Note that, this corresponds to the unambiguous evidence of the squeezing excitation as the Pn are determined without any constraints regarding the motional state. Nevertheless, we employ the Gaussian state analysis in order to quantify the corresponding excitation amplitudes r as well as residual |α|. The ﬁt (bars) yields r = 0.83(8) and |α| = 0.29(15) and the corresponding Wigner function is depicted in the inset. Figure adapted from Wittemer et al. (2019).

From our combined model fit to the recorded red and blue sideband flops on the rocking mode, we extract individual Fock state populations Pn, see Sec. 3.4.2. Here, the squeezed state is directly evidenced by the Pn. Populations are increased for even Fock states only, while odd states remain nearly unpopulated. This is the clear signature of a squeezed state and, thus, we may conclude that we created squeezing by tuning the frequency of a harmonic oscillator. In our analog to cosmological particle creation, this corresponds to the creation of pairs of particles, here phonons, that are created out of quantum fluctuations from the (vacuum) ground state. For example, we detect single phonon pairs (n = 2) in P2 ≈ 20 % and double pairs (n = 4) in P4 ≈ 5 % of our realizations. In order to quantify the corresponding squeezing amplitude r and, thereby, the mean number of created particles n¯sq, we employ again the Gaussian state analysis. From the model fit of a squeezed displaced thermal state we obtain r = 0.83(8) and |α| = 0.29(15). Accordingly, we can calculate the Wigner function that illustrates a suppression of the phase space variance along the position coordinate (∆x)2 by 7.2 dB. 100 4 Experiments and results

4.1.5 Entanglement The characteristic squeezing excitation on the rocking mode of a two-ion chain is accompanied by entanglement in the motional degree of freedom, analog to the entanglement of the cosmological particles, see Sec. 2.2.2. Here, we will briefly outline the basic framework of the created entanglement, while referring to Retzker et al. (2005), Serafini et al. (2009), and Fey et al. (2018) for details. For our description, we consider the two ions, labeled A and B, and focus on the spatial degree of freedom that is aligned along the motion of the rocking mode ω2. We define individual, single ion oscillator states |niA, B with corresponding (uncoupled) eigenfrequencies ωA = ωB ≈ ω1. Due to the finite distance between the two ions (≈ 5.5 µm, given by ωax/(2π) ≈ 1.3 MHz, see Sec. 3.3) and their interaction via the Coulomb repulsion, the two harmonic oscillators |niA and |niB are coupled with a rate ωcoupl/(2π) ≈ 300 kHz, cf. Hakelberg et al. (2019). We note that for long timescales t 1/ωcoupl, the motion of the two ions is typically decomposed into the common motional modes ω1 (COM) and ω2 (rocking) with corresponding number states |ni1,2. For convenience, we assume an initial vacuum state, cf. Retzker et al. (2005), that can be written as

−β p p p with e = (κ − 1/2)/(κ + 1/2) and κ = 1/4 ω1/ω2 + ω2/ω1 ≈ 0.5008 for our experimental parameters. In the basis of the single ion oscillator states |niA, B, this (already) corresponds to an entangled, squeezed state. In our experiments, we amplify the spatial entanglement by the coherent generation of squeezing in the out-of-phase mode ω2, while squeezing of the in-phase mode ω1 is, estimated from the numerical simulations of the implemented dynamics, negligible. To simplify our description, we assume pure squeezing S(ξ) with ξ = reiθ, i.e., a motional state without any superimposed coherent displacement. Thereby, we estimate our ﬁnal state in the basis of the |ni1,2: ξ |ﬁni = |0i |0i + √ |0i |2i + O(r2) (4.1.5) 1 2 2 1 2

Transferring this into the basis |niA, B of the individual ion motion yields ξ ξ |ﬁni = |0i |0i − |1i |1i − √ (|0i |2i + |2i |0i ) + O(r2). (4.1.6) A B 2 A B 8 A B A B Here, the entanglement is identiﬁed, in particular, by the second term that corresponds to the intrinsic anti-correlation for the motion of the rocking mode. In a 4.1 Phonon pair creation 101

• • • simplified picture, it resembles a Schrödinger cat state ∝ r (| •i + |• i), where | •i • and |• i indicate the ions’ non-classical anti-correlation. In order to quantify the entanglement that is associated to the state |fini in the basis of the |niA, B, we consider Gaussian states and employ the criterion by Serafini et al. (2004) to calculate the entanglement of formation EF . Here we consider initial thermal states, described by average phonon numbers n¯1,2. Following Serafini et al. (2004), we find

! ! (1/2 + χ)2 (1/2 + χ)2 (1/2 − χ)2 (1/2 − χ)2 E (χ) = ln − ln , (4.1.7) F 2χ 2χ 2χ 2χ where χ = min(λ1, λ2) is the smaller of the so-called symplectic eigenvalues: √ √ √ −r 1 + 2¯n2 1 + 2¯n1 ω1 λ1 = e √ (4.1.8) 2 ω2 √ √ √ r 1 + 2¯n2 1 + 2¯n1 ω2 λ2 = e √ (4.1.9) 2 ω1

The symplectic eigenvalues and, therefore, EF are functions of the realized squeezing −5 amplitude r. For our initial (thermal) state we obtain EF (r = 0) ≈ 10 , whereas for the squeezed state depicted in Fig. 4.13, we ﬁnd a signiﬁcantly increased entanglement of formation EF (r = 0.83) ≈ 0.41. We note that for the above calculations we assume that the motional states involved remain Gaussian throughout our experimental sequences. Thereby, we neglect excitations other than squeezing, heating or coherent displacements. For our experimental realization, however, this does not constitute a limitation.

4.1.6 Discussion In the experiments described above, we create squeezed states in the motion of two trapped ions by non-adiabatically changing the corresponding motional mode frequency. While the theoretical framework for such a mechanism has been setup several years ago (Janszky and Yushin, 1986; Graham, 1987), an experimental implementation remained elusive due to the fragile quantum dynamics involved. In our realization we switch the rocking mode frequency with a single pulse and obtain a significant squeezing amplitude r = 0.54(8), which is, however, superimposed with a coherent displacement |α| = 0.88(6). This coherent displacement is due to finite differential stray electric fields acting on both ions, which may, in principle, be further compensated in future implementations using more accurate control potentials. Following an alternative approach, we suppress the coherent displacement by implementing a purifying echo sequence. After optimization this yields final values r = 0.83(8) and 102 4 Experiments and results

|α| = 0.29(15). Here, the residual coherent displacement may result from finite imperfections in the timing optimization as well as bandwidth limitations in the switching mechanism of the RF voltage, see Sec. 3.1.4. In order to overcome the latter, we may implement transfer function precompensation techniques such as the voltage overshooting by Bowler et al. (2013) that we employed for the experiments depicted in Fig. 4.2. As the total (relevant) transfer function in our case might be challenging to determine, such precompensation attempts may be assisted by machine learning. In addition, we note that for the echo sequence finite mode frequency instabilities have a pronounced detrimental impact. This however, may be suppressed for future studies by using stabilization techniques for the RF voltage, cf. Johnson et al. (2016). Moreover, ultimately it must be considered that squeezing and coherent displacement operators do not commute. As a consequence, the amplitude in U for the second pulse must optimized for an optimal redisplacement of the Wigner function to the phase space origin. As derived in (Schützhold et al., 2007; Fey et al., 2018), our squeezing sequences can be interpreted as an experimental analog to cosmological particle creation in the early universe (Wittemer et al., 2019). We note that the particles created in our case are phonons, and in our experimental results we find evidence for their bosonic nature. Our data indicate excitations for multiple particle pairs (n = 4, 6, 8), which is forbidden for fermionic particles due to the Pauli exclusion principle. In analogy to the cosmological particle pairs, the phonon pairs (single or multiple) can be identified ~ ~ ~ with particles with opposite momenta +~|k2| and −~|k2|, where k2 is the mode vector of the rocking mode ω2. This gives rise to entanglement, which is in our analog encoded in the motional states of the two ions. The intrinsic anti-correlation of the individual ion motion of the rocking mode is amplified by the squeezing excitation to yield significant values for the entanglement of formation. In future studies we can separate the two ions, i.e., transferring them into individual trapping potentials, which would allow to effectively turn off their classical connection. However, their quantum entanglement can be preserved, yielding individual particles in an entangled motional state, that resembles a Schrödinger cat state. Similar to Hawking radiation, a measurement on one ion only would then result in a mixed, thermal state, cf. Sec. 2.1.5. Additionally, using TPSR transitions the motional entanglement can be transferred onto the ions’ internal degrees of freedom for further processing. With our experimental analog of cosmological particle pair creation, we establish an experimental platform to test different cosmological models. By implementing different shapes and durations of the ramps we may simulate different dynamics of the related cosmic scale parameter. In addition, adding durations of parametric driving may allow to increase the number of particles created. Considering couplings to additional environments and noise fields may allow to study the causal connections of squeezing, pair creation, and entanglement in realistically extended analogs. 4.1 Phonon pair creation 103

Further, our method represents a novel (experimental) tool to create squeezed states of motion. This can aid in gaining sensitivity for quantum metrology applications, cf. Burd et al. (2019) and see also Aasi et al. (2013). In addition, we note that squeezing has also been proposed to substantially enhance eﬀective spin-spin interactions for trapped ions, see Ge et al. (2019). Thereby, experimental quantum simulations of interest may be implemented, cf. Cirac and Zoller (2012). Recently, squeezed motional states of trapped ions have also been used to implement qubits for quantum information processing (Flühmann et al., 2019). Finally, our results regarding squeezing and its puriﬁcation from coherent excitations can be considered when implementing (multi-ion) entangling gates on multiplex trap architectures. Here, rapid changes of the potential landscapes are required for scaling towards a universal quantum computer (Cirac and Zoller, 2000; Kielpinski et al., 2002). 104 4 Experiments and results

4.2 Quantum memory eﬀects

As discussed in Sec. 2.3.2, the coherent interaction between an open quantum system S and its environment E can lead to an exchange of information between the two subsystems of the total quantum system S + E. When observing S only, i.e., when tracing out the environment, a loss of information may be associated with either a coherent transfer of information to E or with decoherence effects, e.g., due to unwanted couplings to other, additional surroundings. In contrast, any backflow of information to S, is due to the coherent interaction between S and E, and might be employed to characterize E or possible correlations (such as entanglement) between S and E. In order to quantify the flow of information that is exchanged between S, E, and possible correlations of the two subsystems, we utilize the measure for the degree of non-Markovian behavior by Breuer et al. (2009), cf. Sec. 2.3.2. However, in order to define an experimentally more feasible measure, we adapt the definition of N : t Xmax N = [D(t) − D(t − ∆t)]>0 (4.2.1) t=∆t Here, we consider (only) two initially orthogonal states of the open quantum system, rather than maximizing over all possible initial state pairs, cf. Sec. 2.3.2. We track the evolution of the distinguishability of these two states via the trace distance D(t). Explicitly, the sum in Eq. (4.2.1) extends over all positive changes of D(t), i.e., any growth of D is identified as a backflow of information to S, and accounted for by an increase of N . In addition, we take account of a finite temporal measurement resolution rather than assuming an analytic expression for D(t). In particular, we consider measurements of D(t) up to a maximum interaction duration tmax, and we assume that measurements are performed within the interval [0, tmax], equally separated by ∆t. We note that the corresponding sampling rate γ = 1/∆t and 1/tmax define the highest and lowest frequency, respectively, with which a growth of D can be detected. Consequently, this sets a spectral bound for our experimental detection window with which we can observe quantum memory effects.

4.2.1 Quantum simulator setup In order to study the dynamics of an experimental open quantum system on a fundamental level, we choose a basic configuration for our trapped ion quantum simulator. We use a single trapped ion to implement both, an open quantum system S, and an environment E, that it is interacting with. We define the ion’s internal degree of freedom, its pseudo-spin representing a spin-1/2 system, as the open quantum system. Accordingly, we write the Hamiltonian of the open system as HS = ~ωzσz/2, where σz is the Pauli matrix with eigenstates |↓i and |↑i and effective energy splitting ~ωz, 4.2 Quantum memory effects 105 cf. Secs. 2.1.2 and 3.2. The environment is formed by the ion’s motional degree of freedom along the axial mode with frequency ωE, which acts as a quantum harmonic oscillator, cf. Secs. 2.1.3 and 3.3. Consequently, we define the environment’s Hamil- † † tonian by HE = ~ωEa a, with the annihilation and creation operators a and a . In order to implement an interaction between S and E, we employ the spin-phonon coupling discussed in Sec. 3.4. For the experiments presented here, we tune the coupling close to the blue sideband in order to realize the following Hamiltonian:

H = HS + HE + HI

ω Ω h † i = ~ z σ + ω a†a + ~ 0 σ eiη(a +a) + H.c. (4.2.2) 2 z ~ E 2 +

Here, we express the interaction term HI by the (carrier) Rabi rate Ω0, spin-flip operators σ± = (σx ± iσy)/2, Pauli matrices σx,y, and the Lamb-Dicke parameter η. In order to quantify quantum memory effects in this context, we investigate the evolution of initial product states ρ(0) = ρS(0) ⊗ ρE(0). For our experiments, we 1 2 choose two representative, orthogonal, initial spin states: ρS(0) = |↑i h↑| and ρS(0) = |↓i h↓|. By preparing the ion’s axial motion in thermal states of only few residual n¯, we ensure that energies of S and E remain comparable. Thereby, we are able to monitor the flow of information between S and E, allowing for distinct features of quantum memory effects. In Figure 4.14 we depict a numerically calculated exemplary time 1,2 evolution ρS (t) and changes in Fock-state populations Pn that indicate a transfer of information from S to E.

Open system S Environment E |↑i ⊗ n: 0 1 2 0.8 n

ρ1 (t) P S D(0) 0.6 D(t1) 0.4 ρ2 (t) S 0.2 Population 0 0 t1 |↓i Interaction duration t Figure 4.14: Exemplary time evolution of the quantum dynamics of the open system S (left) and the environment E (right). We depict numerical calculations for the 1,2 dynamics of the two initial spin states ρS in the Bloch sphere representation as well as the time evolutions for the populations of the ﬁrst three Fock states Pn. Due to the coupling between S and E, information is exchanged, leading to a decrease of the trace distance D between the two spin states. Note that, for this illustrating example, we chose Ω0/(2π) = 500 kHz, in order to implement 1,2 non-trivial time evolutions ρS (t). Figure adapted from Wittemer et al. (2018). 106 4 Experiments and results

When setting the spin-phonon coupling close to a motional sideband transition, 1 2 initially pure states of S, such as ρS and ρS, are driven into the Bloch sphere, effectively reducing the length of their Bloch vectors, when the environmental degree of freedom is traced out. This corresponds to a flow of information from S to E or to correlations/entanglement between the two systems, that can be identified with the 1 2 time evolution of the distinguishability of the two states ρS and ρS, measured by their trace distance D(t). A reduction of D(t) corresponds to a flow of information from S to E (or correlations), while a backflow of information to S may be identified by an increase of D(t). We note that we consider the dynamics of the total quantum system S +E to be unitary, i.e., any decoherence effects by residual couplings to surroundings remain insignificant. In order to quantify the amount of information in the open system S, the environment must be traced out, corresponding to a measurement of the partial density matrix ρS, see Sec. 2.3. The choice of system and environment within the total system S + E is, in principle, arbitrary. For our experiments, however, it is the most convenient choice as we can routinely perform a full state tomography of the ion’s spin-state, as described in Sec. 3.2.3, while a measurement of the full density matrix of the ion’s motional state, as described in Leibfried et al. (1996), requires a significant experimental overhead. In our experiments, we implement the following parameters to ensure elementary quantum dynamics. We consider the electronic degree of freedom of a single 25Mg+ as a spin-1/2 system S with the eigenstates |↓i and |↑i with a frequency splitting of 2π × 1775 MHz, cf. Sec. 3.2. In order to engineer a controlled environment E for the open system, we implement a quantum harmonic oscillator, formed by the ion’s motion along the axial direction with a mode frequency of ωE/(2π) = 1.920(3) MHz. We implement the coherent interaction between S and E by employing TPSR transitions with ∆~k along the axial direction, realizing a spin-phonon coupling with Lamb-Dicke parameter η = 0.32. In order to ensure elementary quantum dynamics, the Hamil- tonian in Eq. (4.2.2) is realized with Ω0/(2π) ≈ 100 kHz so that Ω0 ωE. Thereby, we mitigate any off-resonant driving of transitions other than the one selected by our choice of ωz. By choosing ωz near ωE, we experimentally realize a single spin-1/2, that is coupled (only) to a single quantum harmonic oscillator, enabling a study of an open quantum system on a very fundamental level. We note that we choose the coupling with the axial mode due to its frequency stability with a measured coherence time of 116(27) ms, that exceeds typical coupling durations (≈ 100 µs) by orders of magnitudes, see Sec. 3.3.5. Together with the coherence of the spin, that is stabilized via dynamical decoupling (Viola et al., 1999) due to the TPSR driving, we may neglect any decoherence effects for our combination of S and E, and consider the corresponding dynamics of the total system to be unitary, cf. Clos et al. (2016). We note that, consequently, a flow of information and, thereby, quantum memory effects can only be detected for subsystems (S or E), 4.2 Quantum memory effects 107 where measurements correspond to partial traces that trace out the rest of the total system S + E, cf. Sec. 2.3. In order to assess systematic effects of our measurements (see below), we employ numerical simulations of the implemented quantum dynamics. To this end, we perform numerical calculations of the time evolution of the total system S + E under the coupling governed by the spin-phonon Hamiltonian in Eq. (4.2.2), as described in Sec. 3.5.2. However, in order to take account of quantum projection noise (QPN), induced by a finite number of measurement repetitions r, see Sec. 2.1.4 and Itano et al. (1993), as well as a finite temporal sampling rate γ, we distinguish between numerical simulations with two different sets of parameters. First, we conduct numerical simulations, intended to simulate “ideal” measurements, where we set γ = 100γ0, where γ0 is the sampling rate of our experimental realization (see below), and without including any QPN into our simulations, equivalent to r → ∞. To this end we, simulate the time evolution of the Hamiltonian in Eq. (4.2.2), yielding the “true” values for expectation values of the spin-observables 1,2 hσlitrue (l = x, y, z) for both initial spin states ρS . From these, we extract the time evolution of the trace distance D(t) and, according to Eq. (4.2.1), the non- Markovianity which we refer to as the “true” values Ntrue. Second, we conduct numerical simulations where we set γ and r according to our experimental values γ0 and r0. In order to include the effect of QPN into our simulations, we consider the following steps:

1. Simulate true values for the spin expectation values hσlitrue (l = x, y, z) for both 1,2 initial states ρS . 2. Calculate the QPN-induced uncertainty, given by the standard deviation of the binomial distribution: s 1 hσ i + 1 hσ i + 1 δhσ i = 2 l true 1 − l true (4.2.3) l true r 2 2

3. Generate random numbers hσli with a Gaussian probability distribution around hσlitrue with width (1 standard deviation) δhσlitrue.

4. Calculate the trace distance D from the hσli. 5. Extract N from the time evolution of these D(t) according to Eq. (4.2.1) to yield a single realization of the non-Markovianity with incorporated QPN.

In order to estimate the mean time evolution D(t) under the influence of QPN, we repeat steps 1-4 for k times and average the resulting D. Accordingly, in order to obtain N with incorporated QPN impact, we repeat steps 1-5 for k0 times and average the resulting N . For our simulations, we choose k = k0 = 50 and, thereby, fluctuations 108 4 Experiments and results between different runs with equal parameters remain insignificant. Thereby, we obtain realistic numerical estimates for our measurements that we refer to as Dsim and Nsim.

4.2.2 Time evolution

Here, we want to discuss an example of an experimentally measured time evolution of 1 2 the trace distance D of the initial states ρS and ρS. We initialize the environment in a thermal state with n¯ = 1.0(0) by using the electric ﬁeld heating method presented in Sec. 3.3.3, and implement the spin-phonon coupling according to the Hamiltonian in Eq. (4.2.2). To this end, we set ωz/ωE = 1.000(2), i.e., we drive the ﬁrst blue sideband for variable durations t. The experimental results, together with numerical simulations, are depicted in Fig. 4.15.

1 ) t (

D 0.8 0.6 0.4 0.2 Trace distance 0 2.5 N 2 1.5 1 0.5

Non-Markovianity 0 0 2 4 6 8 Interaction duration t/τ

Figure 4.15: Time-resolved measurement of the trace distance D(t) and the measure for 1 2 non-Markovianity N . We record D for the two initial spin states ρS and ρS, with the environment initialized to n¯ = 1.0(1), and the interaction tuned to ωz/ωE = 1.000(2). Every measurement of D is composed of 3 spin measurements (tomography) for both initial states. We compare our data to “ideal” numerical simulations Dtrue and Ntrue (orange lines), as well as to “realistic” numerical simulations Dsim and Nsim (blue lines). Any increase in D is identified as a backflow of information to S and accounted for by an increase of the measure N . Due to intrinsic measurement noise, a significant bias B (shaded area) with respect to Ntrue accumulates for both, our experimental data and Nsim. Figure adapted from Wittemer et al. (2018). 4.2 Quantum memory effects 109

We perform time-resolved measurements of D(t) by implementing the Hamiltonian −1 in Eq. (4.2.2) for variable durations of t, with a sampling rate of γ0 ≈ 15τ within the observation interval [0, 9τ]. Here, both, γ0 and tmax, are set by the characteristic time scale τ = 2π/Ω0 of the Hamiltonian. Each measurement of D is obtained by performing a full spin state tomography for both initial states, and each datapoint represents the average of r0 = 500 repetitions. 1 2 Initially, the spin states ρS and ρS are orthogonal, i.e., lie on opposite points on the surface of the Bloch sphere, corresponding to the maximal distinguishability with D(0) = 1. Due to the coupling on the blue sideband (ωz = ωE), information is transferred from S to E and/or correlations and, thereby, D(t) is decreasing. However, for particular durations, we observe increasing D(t) that we identify with a backflow of information to the open quantum system S. Accordingly, any increase of D(t) is accounted for by an increase of N , due to our definition of the measure for non- Markovianity in Eq. (4.2.1). We note that our experimental results for D(t) agree with the simulated evolution for both, the ideal simulation D(t)true as well as the realistic simulation D(t)sim, that incorporates QPN and the finite sampling rate γ0. However, we observe the accumulation of a significant bias B in the evolution of N (t) with respect to Ntrue. As the bias is apparent in our experimental data as well as in the realistic simulation Nsim, we attribute it to the intrinsic QPN, induced by our finite choice of r0. In the following, we will analyze the impact of QPN in more detail, revealing its fundamental impact on the measure N .

4.2.3 Impact of quantum projection noise As illustrated in Fig. 4.15, our experimental data allows to resolve the impact of intrinsic QPN onto the measure N . We note that, in principle, QPN can suppressed √ by 1/ r. However, its origin is fundamentally due to the nature of projective measurements in quantum mechanics, see Sec. 2.1.4. Hence, our experimental platform offers the unique possibility to study the fundamental impact of intrinsic measurement uncertainties onto the quantification of quantum non-Markovian behavior. In order to investigate the impact of QPN and finite γ on N , we consider results for fixed tmax = 9τ, for the time evolution depicted in Fig. 4.15. To this end, we depict functions N (γ, r) in Fig. 4.16. 110 4 Experiments and results

r0 γ0 N

101

− 0

0.8

0.5 0.3 0

− 10 − 0.3 Non-Markovianity 1 1 10− 101 102 103 100 101 102 0 0.3 1

− Repetitions− r Sampling rate γ/τ −

0.8 0.5

0.3 5 1 −

Bias (γ, r)/ true B N 0.3 3 10 0 5 20 r 1

0.5 0.3 − 50 2 0.3 10 − 0.8

− 5 Repetitions 1 20

0 101 0.3 0.3 50 0.8 0.5 − − − 1 5 100 101 20 102 1 Sampling rate γ/τ −

Figure 4.16: Variable impact of QPN and finite sampling rates γ on the measure for non- Markovianity N . In the top row, we depict functions N (γ = γ0, r) (left) and N (γ, r = r0) (right) for our experimental data (data points, error bars omitted for clarity), Nsim (blue lines) and Ntrue (orange lines). The bias B(γ, r) = N (γ, r)−Ntrue (shaded areas) is a function of both, γ and r, and can lead to significant under- or overestimation of Ntrue. The non-trivial relations are further illustrated by the relative bias B(γ, r)/Ntrue in the bottom plot. The orange line depicts the zero bias line that corresponds to an equilibrium between overestimation of N due to QPN and underestimation of N due to finite sampling rates γ. The blue dot corresponds to the experimental parameter pair (γ0, r0), the dotted lines correspond to the cuts depicted in the top row. Figure adapted from Wittemer et al. (2018). 4.2 Quantum memory effects 111

For fixed γ = γ0, we vary r by postselecting random subensembles of the r0 experimental realizations, generate resampled evolutions D(t), and evaluate N (γ0, r). Averaging over 50 random realizations of the postselection procedure results in functions N (γ0, r) for our experimental data and Nsim that illustrate the suppression of the overestimation of Ntrue for larger r. However, we note that for r r0 we find significant underestimation of Ntrue, i.e., B < 0, which evidences the impact of finite γ on N . Therefore, we vary the mean sampling rate γ by random postselection of data points D(t) for fixed r = r0, thereby, generating functions N (γ, r0). Here, the underestimation of Ntrue can be enhanced for sampling rates that are too slow to resolve fast dynamical features in D(t). In contrast, sampling rates that approach and exceed reasonable values lead to an overestimation of Ntrue due to the amplified impact of QPN. The bottom plot in Fig. 4.16 summarizes the complex interplay between γ and r, generating variable B. The zero bias line, where overestimation due to QPN and underestimation due to finite γ cancel out, is a non-trivial function of γ and r, and characteristic for the particular time evolution D(t).

4.2.4 Local quantum probing In this Section, we want to investigate the consequences of our findings onto envisioned applications of quantum non-Markovianity measures. Here, we will focus on the idea of employing quantum memory effects for local quantum probing of the open quantum system’s environment, see Breuer et al. (2016). We note that a decrease of the amount of information stored in the open quantum system S may be associated with either a coherent exchange with its environment E or with (incoherent) decoherence effects that correspond to information that is lost from the total system S + E. In contrast, any backflow of information to S, is due to the coherent interaction between S and E and associated correlations, and, hence, might be employed to characterize E. To this end, quantifying the backflow of information to an open quantum system might allow for local quantum probing of properties of the system’s environment. Here, we can employ our experimental platform in order to benchmark these envisioned applications (Laine et al., 2010; Breuer et al., 2016; Gessner and Breuer, 2019) in the light of our findings, i.e., with regard to the fundamental limitations of quantum non-Markovianity measures due to the nature of projective measurements in quantum mechanics. To this end, we keep our system in parameter regimes where numerical simulations are still available, allowing us to quantify the bias B for different quantum probing applications. However, we note that our experimental platform can be tuned beyond the regime of numerical tractability as demonstrated in Clos et al. (2016). We illustrate the basic idea of local quantum probing for our combination of S and E in Fig. 4.17. 112 4 Experiments and results

1 ) t ( D

0 1 ) t ( D

0 0 1 2 3 4 5 6 7 8 9 Interaction duration t/τ

Figure 4.17: Examples for different time evolutions D(t), allowing for local quantum probing. Data points depict experimental results (error bars partially smaller than symbol size), solid lines indicate numerical calculations with no free fit parameters. In the top plot we tune the coupling to ωz/ωE = 1.000(2) and initialize the environment to a thermal state with n¯ = 0.09(2). Thereby, we ensure basic quantum dynamics, dominated by a sinusoidal oscillation with frequency Ω = ηΩ0. When the interaction is detuned to ωz/ωE = 0.900(2) (center plot), oscillations of D are accelerated but attenuated, corresponding to the detuned driving of the dominant transition. The bottom plot shows D(t) for resonant driving ωz/ωE = 1.000(2), but with the environment initialized to a thermal state with n¯ = 0.80(2). Here, the dynamics is composed of several frequency components corresponding to the different Fock states of the harmonic oscillator that are involved in the dynamics. Figure adapted from Wittemer et al. (2018). 4.2 Quantum memory effects 113

If the environment is initialized close to the ground state with n¯ = 0.09(2), the coupling on the blue sideband with ωz/ωE = 1.000(2), realizing the Hamiltonian in Eq. (4.2.2), the dominant frequency component of the dynamcis D(t) is given by the sideband coupling rate Ω = ηΩ0. As the environment is in a pure quantum state, we observe a coherent exchange of population between the harmonic oscillator Fock states n = 0 and n = 1, while population transfers into other states are suppressed. When tuning the coupling off-resonant to ωE, effective coupling rates increase accord- q 0 2 2 ing to Ω ∝ Ω + (ωz − ωE) . However, due to the off-resonant driving, population transfer is suppressed by a factor Ω2/Ω02. Hence, by quantifying the amount of information transferred to S, the coherent coupling of S to its environment E can, in principle, be probed. We note that, in particular for the driving with finite detuning and the correspondingly increased Rabi rates, residual impact of decoherence can be observed in the experimental data D(t), see Fig. 4.17(center). This is due to fluctuations of the laser beams implementing the TPSR coupling, as coherence times of the uncoupled systems S and E significantly exceed the coupling durations implemented here, cf. Secs. 3.2.4 and 3.3.5. By comparing measured and simulated D(t), we ex- −1 tract a decoherence rate Γdec ≈ 0.06τ . In the following, this effect is neglected and we consider our total system S + E completely isolated from external baths, cf. Clos et al. (2016). When initializing the environment to a thermal states with finite n¯ = 0.80(2), the evolution D(t) gains additional frequency components corresponding to additional Fock states that participate in the dynamics. While overall amplitudes are reduced with respect to n¯ = 0.09(2), the dynamics is partially accelerated as higher Fock states are driven with higher Rabi rates by the spin-phonon Hamiltonian, cf. Sec. 3.4. Thereby, the time evolution D(t) is sensitive to different quantum states of the environment E and, hence, we may employ N to probe characteristics of E. In our experiments, we employ both parameters, finite detunings and finite thermal excitations of E, to benchmark the applicability of N for local quantum probing, cf. Laine et al. (2010), Breuer et al. (2016), and Gessner and Breuer (2019). In order to allow for a time-resolved estimation, we evaluate N for three distinct tmax = {2, 5, 9}τ. First, we consider N for fixed environmental states with n¯ = 0.09(2), but for variable ωz near ωE in order to probe variable couplings between S and E. The results are depicted in Fig. 4.18(left). We observe resonances near ωz ≈ ωE with different shapes depending on tmax. In particular, for small tmax, recorded N (ωz) exhibit a double-peak√ structure. This reflects the increase in the effective coupling rate Ω0 ∝ Ω2 + δ2 for a detuning δ from the resonance. In contrast, for larger tmax, line shapes are dominated by the resonant coupling of the blue sideband at 0 ωz = ωE as oscillation amplitudes in evolutions D(t) scale width Ω/Ω , i.e., they are maximal for ωz = ωE, see above. By comparing our data to both, Nsim and Ntrue, we can estimate the contribution of the bias B. Here, we find on average a 114 4 Experiments and results

relative contribution of B/Ntrue ≈ 18 % with only small variations within the probed parameter regime. Thereby, we are able to experimentally resolve features predicted by Ntrue, despite the impact of QPN and finite γ. Second, we benchmark the probing of different environmental states with N . To this end, we fix ωz = ωE, and vary the initial thermal excitation n¯ of the environment, by using the electric field heating method described in Sec. 3.3.3, subsequently to preparing the ground state via resolved sideband cooling, see Sec. 3.3.1. We observe increasing N (¯n) for short tmax, that are a consequence of the Fock state dependence of the Rabi rates Ωn,n0 , cf. Eq. (3.4.3) in Sec. 3.4. In contrast, for large tmax, true values suggest decreasing N (¯n). For larger n¯, the spin interacts with a larger number of Fock states, yielding a less trivial time evolution D(t). With multiple Rabi rates contributing to the dynamics, the time evolution is partially faster, but corresponding oscillation amplitudes are decreased, cf. Fig. 4.17. As a consequence, the predicted features in N (¯n) cannot be resolved in our experiments. We find that the bias B is a non-trivial function of n¯, due to the fact that the complex time evolutions for larger n¯ cannot be resolved with constant significance for fixed values of γ and r.

3 N 2.5 2 1.5 1 0.5 Non-Markovianity 0 0.9 0.95 1 1.05 1.1 0 0.2 0.4 0.6 0.8 1 1.2 1.4

Interaction frequency ωz /ωE Initial average occupation number n¯

Figure 4.18: Benchmarking local quantum probing applications of N . We depict experimental results of N for distinct tmax = {2, 5, 9}τ (triangles, squares, circles) and compare them to Ntrue (orange lines) and Nsim (blue lines). The left plot illustrates the probing of the coupling between S and E via the dependence of N on the coupling frequency ωz. Here, the environment is initialized to n¯ = 0.09(2). We observe distinct resonances near ωE that differ in shape depending on tmax, giving insight on the different time-scales for the interaction between S and E for finite detunings. Contributions of QPN and finite γ (shaded areas) are near constant in the probed parameter regime. In the right plot we benchmark the probing of different environmental states with fixed ωz/ωE = 1.000(2). Here, different slopes in N (¯n) are observed, indicating in- creasingly complex quantum dynamics for larger n¯. The impact of QPN and finite γ varies significantly, preventing the experimental detection of features predicted by true values Ntrue. Figure adapted from Wittemer et al. (2018). 4.2 Quantum memory effects 115

In general, we observe good agreement between our experimental N and “realistic” numerical simulations Nsim. However, in particular for long tmax, large detunings from ωz = ωE lead to increased experimental values N with respect to Nsim. We attribute the neglected but ﬁnite decoherence rates in our setup for these. As coupling rates increase and oscillation amplitudes decrease (see above) the relative impact of a decohering envelope in evolutions D(t) is enhanced, leading to an ampliﬁed impact of QPN and, thereby, overestimation of Ntrue.

4.2.5 Discussion

With the experiments described above, we present a new experimental platform, well- suited to observe dynamics of open quantum systems. We employ a single trapped ion to implement a single spin-1/2 in its electronic degree of freedom, representing an open quantum system. Further, we define the ion’s motional degree of freedom as the bosonic environment of the open quantum system. We coherently couple both systems by implementing TPSR transitions, tuned close to a blue sideband transition in order to allow for elementary quantum dynamics. With the high level of isolation of the total system (spin and motion) from its surroundings that we achieve, we mitigate decoherence effects so that we may neglect them in our analysis, cf. Secs. 3.2.4 and 3.3.5. We implement full spin state tomography, allowing to perform time-resolved measurements of the trace distance of two initially orthogonal spin states, that quantify the amount of information encoded in the open quantum system. Thereby, we are able to observe quantum memory effects, for the first time in a matter-based quantum system, see Wittemer et al. (2018). Our apparatus allows for high-resolution measurements that allow to resolve the effect of intrinsic quantum projection noise on our experimental data. The thereby emerging bias between real measurement data and theoretical predictions that we reveal represents a fundamental limitation of quantum non-Markovianity measures. We employ our experimental platform to quantify and characterize the bias in detail with respect to envisioned applications of non-Markovianity measures, cf. Breuer et al. (2016). To this end, we employ our open system as a local quantum probe to detect different couplings to and different quantum states of its environment. While we note that our system can be scaled up and tuned to more complex environments and couplings, we choose to study the effects reported here on a very fundamental level. Thereby, our approach can act as an experimental reference platform studying the relations between non-Markovian dynamics, fundamental fluctuations in quantum mechanics, and time scales in, and beyond, numerically tractable regimes. Hence, it can aid understanding of physical systems in which parameters are less controlled, and other noise sources contribute substantially to an excess bias. We note that our findings are by no means limited to our platform, as we expect other experimental 116 4 Experiments and results platforms and even numerical approaches, such as Monte Carlo simulations, to be affected by our findings as well. In general, our findings imply questions concerning the general applicability of measures for quantum non-Markovianity. The effect of QPN on other measures, which are based on, e.g., the divisibility of the dynamical map (Rivas et al., 2010; Chruściński et al., 2011; Chruściński and Maniscalco, 2014) or the mutual information between the open quantum system and an additional ancilla system (Luo et al., 2012), needs to be studied, since we expect them to be significantly influenced by QPN as well. Accordingly, the envisioned applications of current non-Markovianity measures are fundamentally limited. Thus, it is required to extend the current definitions of quantum non-Markovianity measures by taking QPN into consideration. In addition, we note that time scales and the related flow of information depend on the particular experimental realization, which should be included into extended definitions as well in order to enable a comparison of different platforms. In any case, an upper limit for the sampling rate might be given, e.g., by the so-called quantum speed limit (Deffner and Lutz, 2013), which, in turn, would limit the impact of QPN. 5 Conclusion and outlook

In this thesis, we set up a quantum simulator based on trapped atomic ions to study fundamental predictions of quantum mechanics. To this end, we develop new and implement existing state-of-the-art methods that allow us to initialize, manipulate, and measure the quantum states of our ions in real-time and with unprecedented precision. We employ the thereby obtained control to investigate two fundamental phenomena from quantum mechanics. In our first investigation, we study the mechanism of cosmological particle pair creation, see Sec. 4.1.6 for a detailed discussion. We implement non-adiabatic changes of the trapping potential of two ions and, thereby, create phonons out of the motional (vacuum) ground state. Following the works by Schützhold et al. (2007) and Fey et al. (2018) this mechanism may be interpreted as an analog to the creation of pairs of particles in curved space-time such as during the inflationary period in the early universe. In our implementation, the particle pair creation is evidenced by the detection of squeezed states in the motion of the ions. The corresponding mechanism, i.e., squeezing by non-adiabatically switching the frequency of a harmonic oscillator, has been theoretically proposed several years ago, see Janszky and Yushin (1986), Graham (1987), and Heinzen and Wineland (1990). However, experimental implementations in our and other groups had remained unsuccessful due to the parasitic excitation of large coherent displacements, cf. Schmitz (2009) and Leupold (2015). With our high-fidelity apparatus and an optimized purifying echo sequence we successfully realize the unambiguous detection of squeezed states via such a protocol for the first time. Still, in our realizations a finite residual coherent displacement is observed that we attribute to stray electric fields and imperfections of our real-time control. In future implementations these may be overcome by high-precision control potentials as well as more involved techniques for our real-time control, see Sec. 4.1.6. For our second investigation, we study a most basic open quantum system using a single trapped ion, see the detailed discussion in Sec. 4.2.5. We define a spin-1/2 system in the ion’s electronic degree of freedom and couple to it to a bosonic environment, formed by a single motional (harmonic oscillator) mode of the ion. This bipartite quantum system can be assumed to be isolated from any additional environments and, thereby, allows to study quantum phenomena for open quantum systems under near-ideal conditions. The interaction between system and environment that we implement leads to entanglement and, as a consequence, we detect non-Markovian behavior for the time evolution of the open quantum system. We 118 5 Conclusion and outlook quantify these quantum memory effects with the measure developed by Breuer et al. (2009). Here, our high-resolution measurements reveal the fundamental impact of intrinsic quantum mechanical measurement uncertainties onto the quantification of quantum non-Markovian behavior, which was not taken account of in any previous theoretical consideration. In the light of these findings, we experimentally evaluate the feasibility of envisioned applications of quantum non-Markovian behavior for local quantum probing, cf. Laine et al. (2010), Breuer et al. (2016), and Gessner and Breuer (2019). Thereby, we highlight that the fundamental impact of projective measurements in quantum mechanics needs to be integrated into definitions and measures for quantum non-Markovian behavior, in order to make them reliable for experimental observations. The experimental platform that we introduce in this thesis allows for observations of quantum mechanical phenomena on the most fundamental levels. We provide a versatile testbench not only for fundamental predictions of pure quantum mechanical effects but also for intriguing phenomena that live on the elusive intersection between quantum field theory and general relativity. In future experiments, we can continue these studies for various related analogies involving, but not being limited to, the Sauter-Schwinger effect or Hawking radiation, cf. Schützhold (2008) and Wit- temer et al. (2019). Moreover, by incorporating a period of parametric excitation into our experimental sequences, we may increase the number of created particles and, thereby, simulate the cosmic reheating of the universe after inflation (Kofman et al., 1994, 1997). Further, we may alter shapes and durations of the non-adiabatic potential changes in order to test different much-debated theoretical models of cosmic inflation, see Martin et al. (2014). Combining such endeavors with the framework of open quantum systems will allow to simulate realistic extensions for such analogs. To this end, our approach may be extended with the coupling to additional (classical or quantum) environments, formed by ancilla ions or controlled couplings to optical and/or electric fields. Thereby, we can realize more complex configurations and less ideal conditions for our open quantum systems, cf. Clos (2017), that may be continuously tuned, even into regimes beyond numerical tractability, see Clos et al. (2016). The unique control over the coupling between open quantum systems and precisely controlled environments our setup offers, might even allow to observe fundamental quantum effects on the transition from the quantum regime to the classical limit, cf. Zurek (2003). Moreover, with the work presented in this thesis, we introduce a new experimental technique to create squeezed states of motion. With our method of purifying squeezing from coherent excitations with echo sequences in phase space, our protocol may be employed for applications in quantum information processing (Flühmann et al., 2019), metrology (Burd et al., 2019), and for enhancing effective spin-spin interactions for trapped ions (Ge et al., 2019), thereby allowing to perform experimental quantum simulations of interest, see Cirac and Zoller (2012). Furthermore, as shown 119 in Sec. 4.1.5, the squeezing excitation that we create is accompanied by the formation of quantum entanglement. Here, we note existing proposals of harnessing this effect for, e.g., quantum information processing, see Eisert and Plenio (2003) and Alonso et al. (2013). With our work, we present the first successful experimental implementation of this effect. A unique feature of the entanglement created by our approach is its applicability to scalable ion trap architectures, see, e.g., Mielenz et al. (2016) and Hakelberg et al. (2019). Here, the simultaneous entanglement of many individual quantum entities may be implemented with the techniques presented in this thesis, cf. Eisert et al. (2004). Thus, our approach may push quantum information processing into novel regimes, and, thereby, propel ambitious endeavors (Cirac and Zoller, 1995, 2000; Kielpinski et al., 2002; Monroe and Kim, 2013) of realizing a universal quantum computer based on trapped ions.

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