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Macroscopic Quantum Phenomena in Interacting Bosonic Systems: Josephson flow in Liquid 4He and Multimode Schr¨Odingercat States

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Macroscopic Quantum Phenomena in Interacting Bosonic Systems: Josephson flow in Liquid 4He and Multimode Schr¨Odingercat States Macroscopic quantum phenomena in interacting bosonic systems: Josephson flow in liquid 4He and multimode Schr¨odingercat states by Tyler James Volkoff A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Chemistry in the Graduate Division of the University of California, Berkeley Committee in charge: Professor K. Birgitta Whaley, Chair Professor Philip L. Geissler Professor Richard E. Packard Fall 2014 Macroscopic quantum phenomena in interacting bosonic systems: Josephson flow in liquid 4He and multimode Schr¨odingercat states Copyright 2014 by Tyler James Volkoff 1 Abstract Macroscopic quantum phenomena in interacting bosonic systems: Josephson flow in liquid 4He and multimode Schr¨odingercat states by Tyler James Volkoff Doctor of Philosophy in Chemistry University of California, Berkeley Professor K. Birgitta Whaley, Chair In this dissertation, I analyze certain problems in the following areas: 1) quantum dynam- ical phenomena in macroscopic systems of interacting, degenerate bosons (Parts II, III, and V), and 2) measures of macroscopicity for a large class of two-branch superposition states in separable Hilbert space (Part IV). Part I serves as an introduction to important concepts recurring in the later Parts. In Part II, a microscopic derivation of the effective action for the relative phase of driven, aperture-coupled reservoirs of weakly-interacting condensed bosons from a (3 + 1)D microscopic model with local U(1) gauge symmetry is presented. The effec- tive theory is applied to the transition from linear to sinusoidal current vs. phase behavior observed in recent experiments on liquid 4He driven through nanoaperture arrays. Part III discusses path-integral Monte Carlo (PIMC) numerical simulations of quantum hydrody- namic properties of reservoirs of He II communicating through simple nanoaperture arrays. In addition to calculating the local superfluid density in these systems, new estimators for hydrodynamic observables and novel methods for extracting the length scale characterizing the decay of superfluidity at the system boundary from PIMC data are introduced with the aim of exploring the mechanism of superfluid weak-link formation in nanoscale containers. Part IV consists of an analysis of macroscopicity measures for a large class of Schr¨odinger cat states of N-mode photonic systems. For cat states of this class, it is shown that a well- known measure of superposition size based on the optimal distinguishability of the branches and another based on metrological usefulness of the superposition relative to its branches agree (i.e., designate the same superpositions as macroscopic) when the inner product of the branches of the superposition is sufficiently small. For certain superpositions in this class, a technique is presented for deriving a state-specific metrological macroscopicity algebra of observables. The dynamics of superposition size measures is also considered, leading to the notion of \distinguishability time," a generalization of the orthogonalization times of Mandelstam-Tamm and Margolus-Levitin. In Part V, an investigation of the nonequilibrium dynamics of a quantum dot-Josephson junction-quantum dot nanodevice is presented. The nonequilibrium action of the system is 2 calculated by making use of the Keldysh formalism and a potential use of the device for transferring macroscopic superposition states between a spin-system and an s-wave super- conductor is discussed. i Contents Acknowledgements iv Notation and list of acronyms vi Preface vii I Introduction 1 1 Quantum states of many-boson systems 2 1.1 Fock space . 2 1.1.1 Bosonic and fermionic Hilbert spaces . 2 1.1.2 Vacuum state and particle number sectors . 6 1.2 Heisenberg-Weyl coherent states . 7 1.2.1 Heisenberg-Weyl and oscillator Lie algebras . 8 1.2.2 Reduced density matrices and field operators . 9 1.2.3 Important quasiprobability distributions . 14 2 Interacting Bose gases 19 2.1 Weakly-interacting Bose gases (WIBG) . 19 2.1.1 ODLRO and Bogoliubov's approximation . 20 2.1.2 Popov approximation . 24 2.2 Liquid 4He..................................... 25 2.2.1 Landau critical velocity and Hess-Fairbank effect . 27 2.2.2 From interacting bosons to two-fluid theory . 30 II Theory of Josephson oscillations in driven liquid 4He 33 3 Josephson effect from an action principle in driven He II 34 3.1 Gauge theory of the WIBG . 35 3.1.1 Construction of gauge invariant action . 42 3.1.2 Bosonic two-body Hubbard-Stratonovich transformation . 43 ii 3.2 Josephson effect in neutral bosonic liquids . 46 3.3 Effective action for gauged WIBG . 50 3.3.1 Self-consistent Hubbard-Stratonovich field . 54 3.3.2 Gauge invariant Josephson-Anderson equation . 55 3.3.3 Fluxoid quantization, Hess-Fairbank equation . 59 3.3.4 Tunneling contribution for multiaperture arrays . 64 3.3.5 Discussion of effective action . 65 3.4 Current phase relations . 66 3.4.1 Current phase relation for two-aperture array . 70 3.5 Low energy current phase relations . 71 3.5.1 Weak-coupling RG approach . 71 3.5.2 Background field RG approach . 73 3.5.3 Single vs. multiple apertures . 78 3.6 Concluding remarks . 79 III Numerical simulations of constrained liquid 4He 82 4 The path integral (quantum) Monte Carlo method 83 4.1 Imaginary time picture and major approximations . 83 4.2 PIMC for weakly-coupled 4He reservoirs . 86 4.2.1 PIMC in cylindrical geometry . 87 4.3 Local hydrodynamic estimators . 89 4.3.1 Superfluid density . 89 4.3.2 Momentum and momentum correlations . 95 4.3.3 Circulation and vorticity . 101 4.3.4 Local and global healing lengths . 110 4.3.5 Evidence for weak-link formation . 115 IV Measures of superposition size in separable Hilbert space 121 5 How big is Schr¨odinger'scat? 122 5.1 Background on large quantum superposition states . 122 5.2 Superposition size via quantum binary distinguishability . 124 5.2.1 Distinguishability of branch n-RDMs . 128 5.3 Branch distinguishability examples . 131 5.3.1 General equal-amplitude two branch superpositions . 131 5.3.2 Entangled coherent states . 133 5.3.3 Gaussian product state superpositions . 138 5.3.4 Hierarchical and concatenated cat states . 141 5.3.5 (2 + 1)-D quantum critical superpositions . 144 iii 5.4 Other measurement-based measures . 147 5.4.1 Marquardt's measure . 147 5.4.2 Relative quantum Fisher information . 149 5.5 Comparison based measures . 153 5.6 Measures based on the Wigner function . 159 5.7 Entanglement of jΨi ............................... 161 6 Dynamical binary distinguishability 165 6.1 Mandelstam-Tamm uncertainty relation . 165 6.2 Margolus-Levitin orthogonalization time . 169 6.3 Independent branch evolution . 170 7 Deeper look at metrological macroscopicity 172 7.1 Spin systems . 172 7.1.1 Cat state basis for (C2)⊗N . 172 7.2 Algebras for metrological macroscopicity . 174 7.2.1 Entangled coherent states . 175 7.2.2 Multimode superposition of Fock states . 177 7.2.3 Hierarchical cat states . 177 7.2.4 Algebras for N rF macroscopicity . 179 7.2.5 General construction of jΨi with N rF (jΨi) > N . 181 V Functional integrals for QD-SNS-QD system 182 8 The QD-SNS-QD system 183 8.1 Motivation . 183 8.2 Coherent state path integrals for QD-SNS-QD . 185 8.2.1 Keldysh action for QD-SNS-SD system . 185 8.2.2 Equilibrium action for QD-SNS-QD system . 192 9 Conclusion 198 9.1 Looking ahead . 198 A Brief overview of the quantum Fisher information 201 A.1 Quantum Fisher information . 201 Bibliography 204 iv Acknowledgements This dissertation is dedicated to my family, especially to my mother and father. Through their exemplary work ethic and perseverance and love, they have provided a standard to which I am always trying to amend. Berkeley is a great place to do science mostly because of the large number of talented researchers, teachers, and visitors working there. It has been my pleasure to learn from their expertises. Of them all, I must firstly extend heartfelt thanks to my advisor, K. Birgitta Whaley (\Birgit"), for providing an excellent work environment and many opportunities for teaching and travel, and for encouraging my good ideas while amending the misguided ones and rejecting the bad ones. Her scientific influence is clearly present in the calculations and physical interpretations put forth in this dissertation. During my first semester at Berkeley, Birgit introduced me to Prof. Richard Packard because of my interest in the experiments conducted in his laboratory. Richard proceeded to instruct a personalized biweekly crash course into the rich field of superfluid dynamics; topics included vortex tangles and critical dynamics. I am thankful to him for his generous advice. Members of Richard's research group, Aditya Joshi in particular, provided valuable perspective on the experimental study of superfluids. Because of our extensive collaboration and his senior academic position, Yongkyung Kwon, of Konkuk University in Seoul, has acted as an ipso facto coadvisor. He graciously hosted me in his research group for a total of six months or so over the summers of 2011 and 2012; during the latter, he arranged a homestay for me which was an exciting immersion into Korean domestic life. His expertise in physical applications of quantum Monte Carlo techniques has been an instructive resource for me. His students, Sungjin Park and Hyeon- deok Shin, gave advice on a number of technical issues relating to path integral Monte Carlo simulations. My officemates in Hearst Memorial Mining Building and O'Brien Hall have been great sources of support and technical knowledge throughout my time in Berkeley. Conversations with Chris Herdman and longtime officemate Joel Corbo have been particularly instructive both academically and personally. I also thank Prof. Frank Wilhelm for teaching me about phenomena related to the Josephson effect in various types of weak link junctions between superconductors during one of his visits to Berkeley.
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