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Weak Measurements for Quantum Characterization and Control Jonathan A University of New Mexico UNM Digital Repository Physics & Astronomy ETDs Electronic Theses and Dissertations Summer 7-27-2018 Weak measurements for quantum characterization and control Jonathan A. Gross University of New Mexico Follow this and additional works at: https://digitalrepository.unm.edu/phyc_etds Part of the Atomic, Molecular and Optical Physics Commons, and the Quantum Physics Commons Recommended Citation Gross, Jonathan A.. "Weak measurements for quantum characterization and control." (2018). https://digitalrepository.unm.edu/ phyc_etds/204 This Dissertation is brought to you for free and open access by the Electronic Theses and Dissertations at UNM Digital Repository. It has been accepted for inclusion in Physics & Astronomy ETDs by an authorized administrator of UNM Digital Repository. For more information, please contact [email protected]. Weak measurements for quantum characterization and control by Jonathan A. Gross B.M., Piano Performance, University of Arizona, 2011 B.S., Computer Engineering, University of Arizona, 2011 M.S., Physics, University of New Mexico, 2015 DISSERTATION Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in Physics The University of New Mexico Albuquerque, New Mexico December, 2018 c 2018, Jonathan A. Gross iii Dedication The posting of witnesses by a third and other path altogether might also be called in evidence as appearing to beggar chance, yet the judge, who had put his horse forward until he was abreast of the speculants, said that in this was expressed the very nature of the witness and that his proximity was no third thing but rather the prime, for what could be said to occur unobserved? |Cormac McCarthy, Blood Meridian: Or the Evening Redness in the West And he sayde vnto me: my grace is sufficient for the. For my stregth is made perfact thorow weaknes. Very gladly therfore will I reioyce of my weaknes that the strength of Christ maye dwell in me. Therfore have I delectacion in infirmities in rebukes in nede in persecucions in anguyshe for Christis sake. For when I am weake then am I stronge. |St. Paul, Second Letter to the Corinthians iv Acknowledgments I owe more to others than I can ever know, and I know of more that I owe to others than I can ever express. Nevertheless, it feels appropriate to make special mention of a few influences. I owe my very existence to my parents, and much of my capacity to prepare for, embark on, and complete a Ph.D. are a result of the seriousness with which they embraced the task of raising me and my brother as a God- given responsibility. My brother has also been a source of constant companionship throughout this journey of life, where transiency seems to be the norm. Numerous educators have imparted to me an enthusiasm for learning in a variety of disciplines. The beauty to which they exposed me and the care with which they regarded me did much to sustain me in pursuing a graduate education. Most recently, I have been greatly aided by the stimulating and supportive environment of the Center for Quantum Information and Control at the University of New Mexico. I could not have hoped for a more delightful or capable collection of peers and faculty to guide and support me through this learning experience. In particular, I am grateful for the direction provided by my Ph.D. supervisor, Dr. Carlton M. Caves. I suspect I will continue learning for years to come from the example he sets for how to conduct oneself as a physicist. Finally, I must acknowledge the source of the hope in which this and all my other labors are carried out: the risen Christ. v Weak measurements for quantum characterization and control by Jonathan A. Gross B.M., Piano Performance, University of Arizona, 2011 B.S., Computer Engineering, University of Arizona, 2011 M.S., Physics, University of New Mexico, 2015 Ph.D., Physics, University of New Mexico, 2018 Abstract This dissertation concerns itself with the virtues and vices of weak measurements. Weak measurements are all around us, but this does not mean that one should man- ufacture weakness on all occasions. We critically evaluate two proposals that claim weak measurements provide a novel means of performing quantum state tomography, allegedly increasing tomographic efficacy and yielding foundational insights into the nature of quantum mechanics. We find weak measurements are not an essential in- gredient for most of their advertised features. In contrast to this negative finding, we highlight an optimal tomographic scheme for which weak continuous measurements are the best known implementation, taking the opportunity to present this result in the language of quantum noise. Weak measurements are also used to describe continuous measurements and the associated unconditional and conditional master equations. Though this description vi has been presented before, we provide a new perspective by using tools and termi- nology from the discipline of quantum information and quantum computing. We do so by showing how the continuous-time evolution of these master equations arises from discretizing in time the interaction between a system and a probe field and by formulating quantum-circuit diagrams for the discretized evolution. We then refor- mulate this interaction by replacing the probe field with a bath of qubits, one for each discretized time segment, reproducing all of the standard quantum-optical mas- ter equations. This provides an economical formulation of the white-noise theory of continuous measurements, highlighting its fundamental underlying assumptions and limitations. White-noise theory is incapable of describing photon-counting measurements in the presence of thermal and squeezed noise. We accommodate such scenarios by considering an environment that includes traveling wave packets that are squeezed, deriving a hierarchy of equations similar to those used to describe traveling wave packets with fixed photon number. Squeezing introduces qualitatively different ef- fects, however, complicating numerical solution of these hierarchies. We provide preliminary numerical analysis of the formalism and showcase its utility by calculat- ing the resonance fluorescence of a two-level atom in squeezed vacuum with squeezing bandwidth narrower than the atomic linewidth, a regime previous techniques cannot explore. vii Contents List of Figures xii List of Tables xxvii 1 Introduction1 2 Weak measurements7 2.1 What is a weak measurement?......................7 2.2 Measurement statistics and state update................ 11 2.3 Continuous outcome spaces....................... 15 2.4 Iterated weak measurements....................... 16 2.5 Dynamics................................. 21 3 Continuous measurements with probe fields 23 3.1 Quantum noise and input-output formalism.............. 31 4 Weak measurements for characterization 35 viii Contents 4.1 Evaluation criteria............................ 36 4.1.1 Novelty.............................. 36 4.1.2 Efficacy.............................. 38 4.1.3 Foundational significance..................... 40 4.2 Direct state tomography......................... 40 4.3 Weak-measurement tomography..................... 53 4.4 Spin-coherent-state measurement.................... 59 5 Weak measurements for control 66 5.1 Notational Conventions.......................... 67 5.2 Quantum trajectories for vacuum field and qubit probes........ 70 5.2.1 Z basis measurement: Photon counting or direct detection.. 71 5.2.2 X basis measurement: Homodyne detection.......... 76 5.2.3 Generalized measurement of X and Y: Heterodyne detection. 80 5.2.4 Summary of qubit-probe measurement schemes........ 88 5.3 Quantum trajectories for Gaussian probe-qubit states......... 89 5.3.1 Gaussian problem for probe fields................ 90 5.3.2 Gaussian problem for probe qubits............... 92 5.3.3 Coherent states and mean-field stochastic master equation.. 94 5.3.4 Thermal states.......................... 97 5.3.5 Pure and thermal squeezed states................ 103 ix Contents 6 Systems illuminated by squeezed wave-packet modes 116 6.1 Formalism................................. 117 6.2 Squeezed state temporal decomposition................. 118 6.3 State update................................ 126 6.3.1 Photon counting......................... 127 6.3.2 Compact notation......................... 134 6.3.3 Homodyne............................. 136 7 Numerical applications 141 7.1 Visualizing evolution........................... 141 7.1.1 Markovian evolution....................... 141 7.1.2 Non-Markovian evolution..................... 144 7.2 Resonance fluorescence in narrow-bandwidth squeezed vacuum.... 146 8 Conclusion 151 A Optimal estimators 155 A.1 Fidelity.................................. 155 A.2 Trace distance............................... 156 B Temporal ordering for quantum noise 158 C Pure and thermal squeezed baths 162 x Contents D Mixed squeezed states 164 D.1 Araki-Woods............................... 166 D.2 Two-qubit setup analogous to two-mode squeezing........... 168 D.3 Two-qubit squeezed-thermal state.................... 169 D.4 Qutrit................................... 170 E Numerical performance 172 E.1 Simulations................................ 175 F Quantum regression theorem 182 xi List of Figures 2.1 Equivalence between a direct (left) and ancilla-coupled (right) mea- surement of σz. Note that for the CNOT gate in the ancilla-coupled measurement, the application of the NOT gate to the probe is con- trolled on g = 1 , as shown algebraically in Eq. (2.1). The single j i j i wires
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