ABSTRACT Title of dissertation: CONTROL AND VERIFICATION OF QUANTUM MECHANICAL SYSTEMS Dvir Kafri, Doctor of Philosophy, 2015 Dissertation directed by: Professor Jacob Taylor Department of Physics Quantum information science uses the distinguishing features of quantum me- chanics for novel information processing tasks, ranging from metrology to computa- tion. This manuscript explores multiple topics in this field. We discuss implemen- tations of hybrid quantum systems composed of trapped ions and superconducting circuits, protocols for detecting signatures of entanglement in small and many-body systems, and a proposal for ground state preparation in quantum Hamiltonian sim- ulators. Control and Verification of Quantum Mechanical Systems by Dvir Kafri Dissertation submitted to the Faculty of the Graduate School of the University of Maryland, College Park in partial fulfillment of the requirements for the degree of Doctor of Philosophy 2015 Advisory Committee: Christopher Monroe (Chair) Jacob M. Taylor Christopher Jarzynski Mohammad Hafezi Alexey Gorshkov c Copyright by Dvir Kafri 2015 Preface In 1935, Einstein, Podolsky, and Rosen (EPR) published a paper claiming that quantum mechanics was incomplete, that there exist ‘elements of reality’ which cannot be described by the quantum wave-function of a system [1]. The key to their argument was their definition of an element of reality: whenever a physical quantity of a system can be predicted with certainty without disturbing that system, there must necessarily be a corresponding element of reality. In quantum mechanics, an observable of a system can only be predicted with certainty if that system’s wave- function is an eigenvector of that observable. Thus two non-commuting observables (with non-complementary eigenvectors) cannot be simultaneous elements of reality. Considering wave-functions describing composites of two systems, EPR showed that by measuring one system it appeared possible to ‘steer’ the second to a wave-function corresponding to two different non-commuting observables. Therefore, by making an appropriate measurement on the first system, it seemed possible to predict the values of both non-commuting observables of the second, and therefore both observables would be elements of reality. Since this is forbidden in quantum mechanics, EPR concluded that the theory was incapable of completely describing nature 1. Building on the comments of Einstein, Podolsky, and Rosen, Erwin Schr¨odinger [2] considered a class of composite quantum states for which the value of every observ- 1Many physicists today would argue that the issue with this conclusion is not in quantum mechanics, but rather in EPR’s defintion of an ‘element of reality’. We can instead interpret EPR’s paper as showing that quantum mechanics is in direct conflict with local realism, i.e. that elements of reality must be associated with objects localized in space. ii able of one system can be inferred by measuring a corresponding observable in the other 2. An extraordinary feature of these states is that, prior to any measure- ment, one cannot make predictions with certainty about any observable of a single subsystem, and yet all of them are elements of reality. Schr¨odinger called systems displaying this bizzare property “entangled”, and stressed the importance of this phenomenon to quantum theory: “I would not call that one but rather the characteristic trait of quantum mechanics, the one that enforces its entire departure from classical lines of thought.” This unique form of correlation, and more generally the ability to form superposi- tions of composite systems, is arguably the key ingredient of quantum information science. From it stem novel applications in computation [3], communication [4], and metrology [5]. 2Today these are called maximally entangled states. In the case of two qubits (an ‘EPR pair’), they are called Bell states. iii This thesis focuses on different aspects of quantum information science, and entanglement will be relevant throughout the work. The following chapters are based on the following publications and pre-prints (in press or pending submission): Dvir Kafri, Prabin Adhikari, and Jacob M. Taylor. Dynamics of an ion coupled • to a parametric superconducting circuit. 04 2015. arXiv:1504.03993 D. Kielpinski, D. Kafri, M. J. Woolley, G. J. Milburn, and J. M. Taylor. • Quantum interface between an electrical circuit and a single atom. Phys. Rev. Lett., 108:130504, Mar 2012. Dvir Kafri and Jacob M. Taylor. A noise inequality for entanglement genera- • tion, 2013. arXiv:1311.4558 D Kafri, J M Taylor, and G J Milburn. A classical channel model for gravi- • tational decoherence. New Journal of Physics, 16(6):065020, 2014. Dvir Kafri and Jacob M. Taylor. Distinguishing quantum and classical many- • body systems. 04 2015. arXiv:1504.01187 Dvir Kafri and Jacob M. Taylor. Algorithmic cooling of a quantum simulator, • 07 2012. arXiv:1207.7111 iv Table of Contents List of Figures vii List of Abbreviations viii 1 Introduction 1 1.1 Decoherence and controlled interactions. ..... 3 1.2 Verificationofquantumsystems . 4 1.3 Statepreparation ............................. 6 1.4 Thesisoutline ............................... 7 2 Dynamics of an Ion Coupled to a Parametric Superconducting Circuit 10 2.1 Introduction................................ 10 2.2 PhysicalSystemandHamiltonian . 13 2.3 Linearization of the Parametric Oscillator . ..... 18 2.4 Time-Dependent Quantum Harmonic Oscillator . .. 22 2.5 InteractionPicture ............................ 25 2.6 Couplingstrength............................. 28 2.7 Conclusions ................................ 33 2.8 Appendix–LinearizationProcedure . 34 3 Quantum interface between an electrical circuit and a singleatom 37 3.1 Introduction................................ 37 3.2 Realisationofion-CQEDcoupling . 40 3.3 LC/spinprotocols............................. 44 3.4 Resistancetomotionalheating. 46 3.5 Outlook .................................. 48 4 Auniversaltestforentanglingdynamics 49 4.1 Introduction................................ 49 4.2 Mainresult ................................ 51 4.3 Circuit model for long-range interactions . ..... 56 4.4 Experimentalapplication. 62 4.5 Conclusion................................. 67 v 4.6 Appendix ................................. 68 4.6.1 Calculation of the generator and physical considerations . 68 4.6.2 ProofoftheTheorem. 70 5 Aclassicalchannelmodelforgravitationaldecoherence 78 5.1 Introduction................................ 78 5.2 Combining quantum and gravitational physics . .... 81 5.3 Gravity as a classical measurement channel . .... 85 5.4 An experimental test of gravitational decoherence . ...... 89 5.5 Conclusion................................. 93 6 DistinguishingQuantumandClassicalMany-BodySystems 95 6.1 Introduction................................ 95 6.2 Toymodel................................. 97 6.3 Generalprotocol .............................103 6.4 Onequbitprotocol ............................105 6.5 Outlook ..................................108 7 GroundStatePreparationbyQuantumSimulatedCooling 109 7.1 Introduction................................109 7.2 Grover’sAlgorithmbySimulatedCooling. 112 7.3 QuantumSimulatedCooling . .117 7.4 Perturbative tools, Simulation errors, and Timing . .......120 7.5 CoolingaQuantumCircuit . .126 7.6 Extension .................................131 7.7 ConcludingRemarks . .137 7.8 AppendixA:MathematicalTools . .138 7.9 AppendixB:DeterministicQSCAnalysis. 151 7.9.1 ReducedAlgorithm: . .155 7.9.2 ProofsofLemmas7.9.2-7.9.5. .157 7.10 AppendixC:ExtensionAnalysis. .173 Bibliography 185 vi List of Figures 2.1 Simplifieddiagramofexperimentalsetup. ... 15 2.2 External flux drive used for sinusoidal modulation of inductance. 21 2.3 Stability diagram of Mathieu’s equation in the resonant regime. 31 2.4 The periodic component of the characteristic function . ........ 36 3.1 Equivalent-circuit model of scheme for ion-circuit coupling. 39 3.2 Schematic of device coupling ions and circuit . ..... 41 3.3 Quantum buses enabled by LC/motion coupling. ... 46 4.1 Circuit model underlying dynamical equation. ..... 59 4.2 Opticalcavityexperiment. 65 5.1 Harmonic oscillator model of gravitational coupling. ........ 82 6.1 Entanglingprotocols. 98 6.2 2DContinuoustimequantumwalk. .102 7.1 Sensitivity of the Grover cooling scheme to errors in bath detuning. 116 7.2 Spectrum of HˆS + HˆB during the cooling of energy level j. ......121 vii List of Abbreviations EPR Einstein, Podolsky and Rosen RWA Rotating Wave Approximation BAW Bulk-Acousting-Wave (microelectromechanical modulator) LOCC Local Operations with Classical Communication MBS Many-body System QSC Quantum Simulated Cooling TCP Trace-preserving, Completely Positive viii Chapter 1: Introduction While quantum information science has seen impressive advances in the last two decades, it has also generated unique challenges. Experiments in atom and ion traps have been approaching the coherence and size limits required to simu- late exotic states of matter [6–8]. Success in these efforts will provide insight into strongly correlated many-body systems in condensed matter [8–10], as well as quan- tum field theories that cannot be studied with classical numerical methods [11,12]. Yet it is exactly the fact that they simulate such quantum systems that makes confirming the output of these simulators so difficult [13–15]. Along similar lines, progress in quantum optics has led to some of the first industrial applications of quantum information, including provably