DOCSLIB.ORG

Quantum Revivals and Generation of Non-Classical States in an N Spin System

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Quantum Revivals and Generation of Non-Classical States in an N Spin System Quantum revivals and generation of non-classical states in an N spin system Shane Dooley Department of Physics and Astronomy University of Leeds A thesis submitted for the degree of Doctor of Philosophy 30th September 2014 Intellectual Property and Publication Statements The candidate confirms that the work submitted is his own, except where work which has formed part of jointly authored publications has been included. The contribution of the candidate and the other authors to this work has been explic- itly indicated below. The candidate confirms that appropriate credit has been given within the thesis where reference has been made to the work of others. Chapter 5 is based on the publication: Shane Dooley, Francis McCrossan, Derek Harland, Mark J. Everitt and Timothy P. Spiller. Collapse and revival and cat states with an N-spin system. Phys. Rev. A 87 052323 (2013). Francis McCrossan contributed to the preminary investigations that led to this paper. Derek Harland and Mark Everitt contributed spin Wigner function plots. Timothy P. Spiller supervised the project. All other work is directly attributable to Shane Dooley. Section 4.4 and chapter 6 and are based on the publication: Shane Dooley and Timothy P. Spiller. Fractional revivals, multiple- Schr¨odinger-catstates, and quantum carpets in the interaction of a qubit with N qubits. Phys. Rev. A 90 012320 (2014). Timothy P. Spiller supervised the project. All other work is directly attibutable to Shane Dooley. This copy has been supplied on the understanding that it is copyright ma- terial and that no quotation from the thesis may be published without proper acknowledgement. c 2014, The University of Leeds and Shane Dooley. Acknowledgements First, I would like to thank my supervisor, Tim Spiller, for his support and patience throughout my PhD, and for always having time for me when I knocked on his door. I have been fortunate to have learned from many people over the last four years, including collaborators Mark Everitt, Derek Harland and Jaewoo Joo; fellow PhD students Terrence Farrelly, Suva De, Andreas Kurcz, Jonathan Busch, Abbas Al-Shimary, Veiko Palge, Luis Rico Gutierrez, Tom Barlow, Oleg Kim, James DeLisle and Monovie Asita; and summer students Francis McCrossan and Anthony Hayes. I would especially like to thank Tim Proctor, Adam Stokes and Paul Knott at Leeds for all the \board sessions". The last four months of my PhD were spent at NTT Basic Research Laboratories in Japan, and I am very grateful to Bill Munro for giving me that opportunity. Also at NTT, I would like to thank Yuichiro Matsuzaki and George Knee for many interesting discussions, and at NII Tokyo, Kae Nemoto, Emi Yukawa and Burkhard Scharfenberger. Finally, I would like to thank my parents, Carmel and Brendan, my brother Mark and my sister Niamh for their love and support. Abstract The generation of non-classical states of large quantum systems is an important topic of study. It is of fundamental interest because the generation of larger and larger non-classical states extends quantum theory further and further into the classical domain, and it is also of practical interest because such states are an important resource for quantum technologies. The focus of this thesis is the \spin star model" for the interaction of a single spin-1=2 particle with N other spin-1=2 particles. Although this is a simple model, we show that its dynamics include many inter- esting quantum phenomena, including fractional revival, and Jaynes- Cummings-like collapse and revival. Starting with a spin coherent state of the N spin system { an easily prepared state, in principle { we show that these dynamics can be used to generate a wide variety of non-classical states of the N spin system, including Schr¨odingercat states, GHZ states, multiple-Schr¨odingercat states and spin squeezed states. Contents 1 Introduction1 2 Background4 2.1 The quantum harmonic oscillator . .4 2.1.1 Coherent states . .6 2.1.2 Phase space pictures . .7 2.1.3 Non-classical oscillator states . 11 2.2 A system of N spin-1=2 particles . 15 2.2.1 Spin coherent states . 19 2.2.2 Spin phase space pictures . 24 2.2.3 Non-classical spin states . 26 2.3 The bosonic limit . 32 3 Motivation: Quantum Metrology 37 3.1 The standard quantum limit . 37 3.2 A quantum advantage . 44 3.3 Ultimate limits to precision . 50 3.4 Estimating a magnetic field with unknown direction . 58 4 The Spin Star Model 68 4.1 Candidates for implementation . 70 4.2 Exact solution . 72 4.3 Effective Hamiltonian: Large Detuning . 79 4.4 Effective Hamiltonian: On resonance and initial spin coherent state 81 iv CONTENTS 5 The Jaynes-Cummings Approximation 85 5.1 The Jaynes-Cummings Model . 86 5.1.1 The dispersive limit . 87 5.1.2 On resonance . 88 5.2 The JC approximation of the spin star model . 97 5.2.1 Dynamics . 99 6 Beyond the Jaynes-Cummings Approximation { I 110 6.1 Fractional Revival . 110 6.1.1 The Kerr Hamiltonian . 111 6.1.2 The Jaynes-Cummings Hamiltonian . 112 6.1.3 The one-axis-twisting Hamiltonian . 114 6.2 The spin star model . 116 6.2.1 Fractional revivals, multiple cat states and quantum carpets 118 7 Beyond the Jaynes-Cummings Approximation { II 126 7.1 An ensemble of NV centres coupled to a flux qubit . 127 7.2 Spin squeezing . 133 8 Conclusion 146 A 150 A.1 Useful expressions . 150 A.2 The spin coherent state as an approximate eigenstate . 151 A.3 Approximating the Hamiltonian . 155 A.3.1 The Jaynes-Cummings approximation . 157 A.3.2 The one-axis twisting approximation . 158 References 172 v Chapter 1 Introduction Since the 1920's, there has been much interest in the generation of quantum states that call attention to the differences between the familiar classical world of our experience and the underlying quantum world. A famous early example is the Schr¨odinger'scat thought experiment [Schr¨odinger(1935)], where a cat is neither \alive" nor \dead" but is entangled with a decaying atom in such a way that it is in a quantum superposition of \alive" and \dead". The phrase \Schr¨odinger'scat" has since spread beyond the physics community into popular culture where { along with the Heisenberg uncertainty principle { it represents the strangeness of quantum physics. Even within physics, its technical meaning has expanded to include a quantum superposition of any two \classical" states. Steady progress in the control of quantum systems has led to generation of such states in experiments on small quantum systems. For example, a Schr¨odinger's cat state of 100 photons has been generated recently in a superconducting cavity resonator [Vlastakis et al. (2013)]. A significant challenge is to generate such \non-classical states" for larger and larger quantum systems, which would be convincing evidence that quantum physics describes the world at macroscopic scales (but that the quantum effects are usually suppressed by interaction with the environment [Schlosshauer(2007); Zurek(2003)]). Another dimension to this progress is the so-called \second quantum revo- lution": the emerging field of quantum technology [Dowling & Milburn(2003)]. The aim of quantum technology is to engineer systems based on the laws of quan- tum physics, giving an improvement in some task over what is possible within a 1 classical framework. In every quantum technology, non-classical states (like the Schr¨odingercat state) are an important resource. The most well-known example of a proposed quantum technology is a quan- tum computer: a computer that operates on the principles of quantum physics. In principle, such a computer would be much more powerful than any classical computer. However, this requires precise control of a large quantum system and careful shielding of the system from unwanted interactions with its environment. Although there has been much progress in this direction, quantum computers are still far from reaching their full potential. A more mature quantum technology is quantum communication. In particular, quantum key distribution uses the principles of quantum physics to guarantee secure communication. There are already several companies selling commercial quantum key distribution systems. Another example of quantum technology is in the field of metrology (the science of measurement). In quantum metrology, a quantum system is used as a probe to measure some other system. By preparing the probe in a \non-classical" state it turns out that it is possible to significantly improve the precision of the measurement compared to the precision that is achieved by preparing the probe in a \classical" state. In this thesis our primary motivation is the generation of non-classical states of spin systems for quantum metrology. After giving some necessary background material in chapter2, we review (in chapter3) quantum metrology, and in par- ticular, the problem of estimating an unknown magnetic field with a system of N spin-1=2 particles. Then, in chapter4 we turn to the main focus of this thesis: the spin star model. This model describes the interaction of a single central spin with N outer spins that do not interact among themselves. We suggest that there are several promising candidates systems for implementation of the spin star model and we derive two effective Hamiltonians for the model in two different parameter regimes. In chapter5 we show that there is a parameter regime where the spin star model behaves like the well-known Jaynes-Cummings model for the interaction of an harmonic oscillator and a two-level system. We find interesting dynamics, such as \collapse and revival", analogous to that in the Jaynes-Cummings model 2 and we propose that these dynamics can be used to generate Schr¨odingercat states of the N spin system.
Load more

Recommended publications
  • Quantum Metrology with Atoms and Light
    UNIVERSITY OF WARSAW FACULTY OF PHYSICS DOCTORAL THESIS Quantum Metrology with Atoms and Light Author: Supervisor: Karol GIETKA dr hab. Jan CHWEDENCZUK´ Auxiliary Supervisor: dr Tomasz WASAK December 6, 2018 iii Streszczenie Kwantowa Metrologia z Atomami i Swiatłem´ Głównym celem tej dysertacji jest zaproponowanie metod tworzenia kwantowych stanów materii oraz ´swiatłai sprawdzenie mozliwo´sciwykorzystania˙ tych stanów do precyzyjnych pomiarów wielko´scifizycznych. Pierwsza cz˛e´s´ctego celu realizo- wana jest przy pomocy formalizmu kwantowo-mechanicznego w kontek´scieteorii ultra-zimnych gazów atomowych oraz kwantowej elektrodynamiki we wn˛ece,nato- miast druga cz˛e´s´crealizowana jest za pomoc ˛ametod teorii estymacji z informacj ˛aFi- shera w roli głównej. Poł ˛aczeniepowyzszych˙ metod jest znane ogólnie pod poj˛eciem kwantowej metrologii. W ostatnich latach wiele teoretycznego i eksperymentalnego wysiłku zostało włozonego˙ w dziedzin˛ekwantowej metrologii, poniewaz˙ dzi˛ekiniej mozliwy˙ b˛edzienie tylko rozwój technik pomiarowych daj ˛acychlepsz ˛aprecyzj˛eniz˙ te same pomiary wykonane w ramach klasycznej teorii, ale takze˙ moze˙ by´cuzyta˙ do badania fundamentalnych aspektów mechaniki kwantowej takich jak spl ˛atanie. Pierwsz ˛ametod ˛a,któr ˛arozwazamy˙ to mechanizm tworzenia tworzenia stanów spinowo-´sci´sni˛etychznany jako one-axis twisting, który moze˙ by´czastosowany na przykład w kondensacie Bosego-Einsteina uwi˛ezionegow podwójnej studni poten- cjału tworz ˛acefektywnie kondensat dwu składnikowy. Pokazujemy, ze˙ stany spinowo- ´sci´sni˛etestanowi ˛atylko mał ˛arodzin˛estanów spl ˛atanych,które mog ˛aby´cwytwo- rzone przez Hamiltonian one-axis twisting. Ta duza˙ rodzina stanów typu twisted za- wiera nawet najbardziej spl ˛atanystan znany jako kot Schroödingera. Pokazujemy równiez˙ jak wykorzysta´cte kwantowe zasoby w pomiarze nieznanego parametru, wykorzystuj ˛acnieidealne detektory atomowe oraz w przypadku kiedy oddziaływa- nie pomi˛edzyatomami nie jest dokładnie znane.
    [Show full text]
  • Degree of Quantumness in Quantum Synchronization
    Degree of Quantumness in Quantum Synchronization H. Eneriz,1, 2 D. Z. Rossatto,3, 4, ∗ F. A. Cardenas-L´ opez,´ 5 E. Solano,1, 6, 5 and M. Sanz1,† 1Department of Physical Chemistry, University of the Basque Country UPV/EHU, Apartado 644, E-48080 Bilbao, Spain 2LP2N, Laboratoire Photonique, Numerique´ et Nanosciences, Universite´ Bordeaux-IOGS-CNRS:UMR 5298, F-33400 Talence, France 3Departamento de F´ısica, Universidade Federal de Sao˜ Carlos, 13565-905 Sao˜ Carlos, SP, Brazil 4Universidade Estadual Paulista (Unesp), Campus Experimental de Itapeva, 18409-010 Itapeva, Sao˜ Paulo, Brazil 5International Center of Quantum Artificial Intelligence for Science and Technology (QuArtist) and Physics Department, Shanghai University, 200444 Shanghai, China 6IKERBASQUE, Basque Foundation for Science, Mar´ıa D´ıaz de Haro 3, E-48013 Bilbao, Spain We introduce the concept of degree of quantumness in quantum synchronization, a measure of the quantum nature of synchronization in quantum systems. Following techniques from quantum information, we propose the number of non-commuting observables that synchronize as a measure of quantumness. This figure of merit is compatible with already existing synchronization measurements, and it captures different physical properties. We illustrate it in a quantum system consisting of two weakly interacting cavity-qubit systems, which are cou- pled via the exchange of bosonic excitations between the cavities. Moreover, we study the synchronization of the expectation values of the Pauli operators and we propose a feasible superconducting circuit setup. Finally, we discuss the degree of quantumness in the synchronization between two quantum van der Pol oscillators. I. INTRODUCTION Synchronization is originally defined as a process in which two or more self-sustained oscillators evolve to swing in unison.
    [Show full text]
  • Multi-Time Correlations in the Positive-P, Q, and Doubled Phase-Space Representations
    Multi-time correlations in the positive-P, Q, and doubled phase-space representations Piotr Deuar Institute of Physics, Polish Academy of Sciences, Aleja Lotników 32/46, 02-668 Warsaw, Poland A number of physically intuitive results for the in quantum information science (see [58] and [129] for calculation of multi-time correlations in phase- recent reviews) and for the simulation of large-scale space representations of quantum mechanics are quantum dynamics. Negativity of Wigner, P, and other obtained. They relate time-dependent stochastic phase-space quasi-distributions is a major criterion for samples to multi-time observables, and rely on quantumness and closely related to contextuality and the presence of derivative-free operator identi- nonlocality in quantum mechanics [58, 65]. The inabil- ties. In particular, expressions for time-ordered ity to interpret the Glauber-Sudarshan P [67, 132] and normal-ordered observables in the positive-P Wigner distributions in terms of a classical probabil- distribution are derived which replace Heisen- ity density is the fundamental benchmark for quan- berg operators with the bare time-dependent tum light [88, 128]. Quasi-probability representations stochastic variables, confirming extension of ear- arise naturally when looking for hidden variable de- lier such results for the Glauber-Sudarshan P. scriptions and an ontological model of quantum theory Analogous expressions are found for the anti- [16, 58, 93, 127]. Wigner and P-distribution negativ- normal-ordered case of the doubled phase-space ity are considered a resource for quantum computation Q representation, along with conversion rules [8, 58] and closely related to magic state distillation among doubled phase-space s-ordered represen- and quantum advantage [138]; the Glauber-Sudarshan tations.
    [Show full text]
  • Density Operator: the Coherent State and Squeezed State Masatsugu Sei Suzuki Department of Physics, SUNY at Binghamton (Date: January 21, 2017)
    Phase space representation of density operator: the coherent state and squeezed state Masatsugu Sei Suzuki Department of Physics, SUNY at Binghamton (Date: January 21, 2017) Here we discuss the phase space representation of the density operator, including Wigner function for the coherent state and squeezed state in quantum optics. The density operator of a given system includes classical as well as quantum mechanical properties. __________________________________________________________________ 1. P function representation for the Density operator We start with a density operator defined by ˆ d 2P() where P() is called the Glauber-Sudarshan P function (representation) and is an coherent state. We note that Tr[ˆ] Tr d 2P() d 2P()Tr[ ] d 2P() d 2P() 1 or Tr[ˆ] Tr d 2P() 1 d 2P() d 2 1 2 d 2P() d 2 1 2 2 d 2P() d 2 exp( * * ) 1 2 2 d 2P()exp( ) d 2 exp( 2Re[ * ])) 1 2 d 2P() exp( )exp( 2 ) d 2P() 1 where 1 d 2 1ˆ (formula) 2 exp{ 2 2 2Re[ * ]} Note that and are complex numbers, x iy , a ib Since Re[ *] Re[(a ib)(x iy)] ax by we get the integral as d 2 exp( 2 2Re[ *])) dxexp(x2 2ax)dx dyexp(y2 2by) exp(a2 b2 ) exp( 2 ) We also note that the density operator can be rewritten as 1 ˆ d 2 d 2 ˆ 2 1 d 2 d 2 ˆ 2 using the formula 1 d 2 1ˆ ((Example)) The average number of photons can be written as nˆ aˆ aˆ Tr[ˆaˆ aˆ] Tr[ d 2P() aˆ aˆ] d 2P() aˆ aˆ d 2P() 2 So P() is normalized as a classical probability distribution.
    [Show full text]
  • A Brief Course of Quantum Theory
    A Brief Course of Quantum Theory DONG-SHENG WANG October 20, 2019 Contents 1 Qubits9 1.1 Bit, pbit................................9 1.2 Qubit................................. 10 1.3 Unitary evolution........................... 11 1.3.1 Rabi oscillation........................ 12 1.3.2 Periodic Hamiltonian..................... 12 1.3.3 Parameter-dependent Hamiltonian.............. 13 1.3.4 Quantum control....................... 16 1.4 Non-unitary evolution......................... 18 1.4.1 Quantum channels...................... 18 1.4.2 Thermodynamics....................... 21 1.5 Observable and Measurement..................... 22 1.5.1 Projective measurement................... 23 1.5.2 Non-commuting set of observable.............. 24 1.5.3 Tomography, Estimation, Discrimination.......... 25 1.6 The power of qubit.......................... 27 1.6.1 Black-box encoding..................... 27 1.6.2 Quantum key distribution................... 28 1.6.3 Measure classical values................... 29 1.6.4 Leggett–Garg inequalities.................. 30 1.6.5 Contextuality......................... 31 1.7 Zoo of qubits............................. 32 2 Basic Formalism 35 2.1 Hilbert space............................. 35 2.1.1 States............................. 36 2.1.2 Norm............................. 40 2.1.3 Dynamics........................... 41 2.1.4 Measurement......................... 43 2.2 Quantum notions........................... 46 2.2.1 Information and entropy................... 46 2.2.2 Entanglement......................... 48 i
    [Show full text]
  • Phase Space Entanglement Spectrum 2
    Phase Space Entanglement Spectrum Vatsal Dwivedi1, Victor Chua2 1Institute for Theoretical Physics, University of Cologne, Z¨ulpicher Straße 77a, 50937 Cologne, Germany 2Department of Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland E-mail: [email protected] Abstract. We generalize the position- and momentum-space entanglement cutsto a family of cuts corresponding to regions in the classical phase space. We explicitly compute the entanglement spectra of free fermionic many-body wavefunctions for a family of phase space entanglement cuts that continuously interpolates between position- and momentum-space cuts. For inversion symmetric wavefunctions, the phase space entanglement spectrum possess a chiral symmetry, to which a topological index can be associated. Keywords: Entanglement, Classical phase space, Noninteracting fermions arXiv:1802.09531v1 [math-ph] 26 Feb 2018 Phase Space Entanglement Spectrum 2 1. Introduction Quantum entanglement is arguably the most intriguing aspect of our current understanding of nature. Besides its implications for the ontology of quantum mechanics, it has also stimulated and enriched a diverse array of fields within theoretical physics[1] such as condensed matter[2, 3, 4], quantum information[5, 6], quantum field theory and string theory[7]. Insights from the study of quantum entanglement have led to the development of many analytical tools for characterization and simulation of extended many-body quantum systems, notable examples being topological entanglement entropy[8] and the density matrix renormalization group (DMRG)[9]. In broad terms, quantum entanglement is the appearance of non-local correlations between local measurements on different parts of a system. Mathematically, this is encoded in the non-purity of the reduced density matrix, which can be quantified using the entanglement spectrum and associated entropies.
    [Show full text]
  • Verification of Nonclassicality in Phase Space
    Verification of nonclassicality in phase space Dissertation zur Erlangung des akademischen Grades doctor rerum naturalium (Dr. rer. nat.) der Mathematisch-Naturwissenschaftlichen Fakult¨at der Universit¨atRostock vorgelegt von Thomas Kiesel, geb. am 07.02.1983 in Wismar aus Rostock Rostock, 29. Juni 2011 Gutachter: Prof. Dr. Werner Vogel, Universit¨atRostock Prof. Dr. John Klauder, University of Florida Prof. Dr. Adam Miranowicz, University Pozna´n Tag der Verteidigung: 12. Oktober 2011 Abstract Nonclassicality has been introduced to examine the question whether experimental observations can be explained within a classical physical theory. In quantum op- tics, it is defined as the failure of the so-called Glauber-Sudarshan quasiprobability to be interpreted as a classical probability density. Although this definition has been accepted for several decades, there has been a lack of simple and complete criteria to check whether a given quantum state of light is nonclassical. In this thesis, the experimental use of some nonclassicality criteria is examined. Then, so-called nonclassicality quasiprobabilities are introduced, which allow a complete investigation of nonclassicality of an arbitrary quantum state in a simple manner. Their theoretical background is carefully elaborated, and their use is demonstrated with experimental data. Finally, some issues of the relation between nonclassicality and the special effect of entanglement are discussed, again with the application of experimental measurements. Zusammenfassung Der Begriff der Nichtklassizit¨atwurde
    [Show full text]
  • Get Full Text
    Multi-time correlations in the positive-P, Q, and doubled phase-space representations Piotr Deuar Institute of Physics, Polish Academy of Sciences, Aleja Lotników 32/46, 02-668 Warsaw, Poland A number of physically intuitive results for the in quantum information science (see [58] and [129] for calculation of multi-time correlations in phase- recent reviews) and for the simulation of large-scale space representations of quantum mechanics are quantum dynamics. Negativity of Wigner, P, and other obtained. They relate time-dependent stochastic phase-space quasi-distributions is a major criterion for samples to multi-time observables, and rely on quantumness and closely related to contextuality and the presence of derivative-free operator identi- nonlocality in quantum mechanics [58, 65]. The inabil- ties. In particular, expressions for time-ordered ity to interpret the Glauber-Sudarshan P [67, 132] and normal-ordered observables in the positive-P Wigner distributions in terms of a classical probabil- distribution are derived which replace Heisen- ity density is the fundamental benchmark for quan- berg operators with the bare time-dependent tum light [88, 128]. Quasi-probability representations stochastic variables, confirming extension of ear- arise naturally when looking for hidden variable de- lier such results for the Glauber-Sudarshan P. scriptions and an ontological model of quantum theory Analogous expressions are found for the anti- [16, 58, 93, 127]. Wigner and P-distribution negativ- normal-ordered case of the doubled phase-space ity are considered a resource for quantum computation Q representation, along with conversion rules [8, 58] and closely related to magic state distillation among doubled phase-space s-ordered represen- and quantum advantage [138]; the Glauber-Sudarshan tations.
    [Show full text]
  • Arxiv:1310.1298V2 [Quant-Ph] 10 Feb 2014 [4, 5, 10]
    Laser Theory for Optomechanics: Limit Cycles in the Quantum Regime Niels Lörch,1, 2 Jiang Qian,3 Aashish Clerk,4 Florian Marquardt,5, 6 and Klemens Hammerer1, 2 1Institut für Gravitationsphysik, Leibniz Universität Hannover and Max-Planck-Institut für Gravitationsphysik (Albert-Einstein-Institut), Callinstraße 38, 30167 Hannover, Germany 2Institut für Theoretische Physik, Leibniz Universität Hannover, Appelstraße 2, 30167 Hannover, Germany 3Arnold Sommerfeld Center for Theoretical Physics, Center for NanoScience and Department of Physics, Ludwig-Maximilians-Universität München, Theresienstrasse 37, 80333, München, Germany 4Department of Physics, McGill University, Montreal, Quebec, Canada H3A 2T8 5Friedrich-Alexander-Universität Erlangen-Nürnberg, Staudtstr. 7, D-91058 Erlangen, Germany 6Max Planck Institute for the Science of Light, Günther-Scharowsky-Straße 1/Bau 24, D-91058 Erlangen, Germany (Dated: February 11, 2014) Optomechanical systems can exhibit self-sustained limit cycles where the quantum state of the mechanical resonator possesses nonclassical characteristics such as a strongly negative Wigner density, as was shown re- cently in a numerical study by Qian et al. [Physical Review Letters, 109, 253601 (2012)]. Here we derive a Fokker-Planck equation describing mechanical limit cycles in the quantum regime which correctly reproduces the numerically observed nonclassical features. The derivation starts from the standard optomechanical master equation, and is based on techniques borrowed from the laser theory due to Haake and Lewenstein. We com- pare our analytical model with numerical solutions of the master equation based on Monte-Carlo simulations, and find very good agreement over a wide and so far unexplored regime of system parameters. As one main conclusion, we predict negative Wigner functions to be observable even for surprisingly classical parameters, i.e.
    [Show full text]
  • Coherent Representation of Fields and Deformation Quantization
    Coherent representation of fields and deformation quantization Jasel Berra-Montiela,b,∗, Alberto Molgadoa,b aFacultad de Ciencias, Universidad Aut´onoma de San Luis Potos´ı Av. Salvador Nava S/N Zona Universitaria, San Luis Potos´ı, SLP, 78290 Mexico bDual CP Institute of High Energy Physics, Mexico Abstract Motivated by some well known results in the phase space description of quan- tum optics and quantum information theory, we aim to describe the formal- ism of quantum field theory by explicitly considering the holomorphic rep- resentation for a scalar field within the deformation quantization program. Notably, the symbol of a symmetric ordered operator in terms of holomorphic variables may be straightforwardly obtained by the quantum field analogue of the Husimi distribution associated with a normal ordered operator. This relation also allows to establish a c-equivalence between the Moyal and the normal star-products. In addition, by writing the density operator in terms of coherent states we are able to directly introduce a series representation of the Wigner functional distribution which may be convenient in order to calculate probability distributions of quantum field observables without per- forming formal phase space integrals at all. Keywords: Deformation Quantization, Star-product, Holomorphic representation 2000 MSC: 81S30, 53D55, 81R30 arXiv:2005.14333v1 [quant-ph] 28 May 2020 1. Introduction Deformation quantization, also referred to as phase space quantum me- chanics by many authors, consists in a general quantization procedure based ∗Corresponding author Email addresses: [email protected] (Jasel Berra-Montiel), [email protected] (Alberto Molgado) Preprint submitted to Elsevier June 1, 2020 on the idea that a quantum system is obtained by deforming both the ge- ometrical and the algebraic structures of classical phase space [1], [2].
    [Show full text]
  • The Wehrl Entropy Has Gaussian Optimizers 3
    Letters in Mathematical Physics manuscript No. (will be inserted by the editor) The Wehrl entropy has Gaussian optimizers Giacomo De Palma Abstract We determine the minimum Wehrl entropy among the quantum states with a given von Neumann entropy, and prove that it is achieved by thermal Gaussian states. This result determines the relation between the von Neumann and the Wehrl entropies. The key idea is proving that the quantum- classical channel that associates to a quantum state its Husimi Q representa- tion is asymptotically equivalent to the Gaussian quantum-limited amplifier with infinite amplification parameter. This equivalence also permits to deter- mine the p q norms of the aforementioned quantum-classical channel in the two particular→ cases of one mode and p = q, and prove that they are achieved by thermal Gaussian states. The same equivalence permits to prove that the Husimi Q representation of a one-mode passive state (i.e. a state di- agonal in the Fock basis with eigenvalues decreasing as the energy increases) majorizes the Husimi Q representation of any other one-mode state with the same spectrum, i.e. it maximizes any convex functional. Keywords Wehrl entropy von Neumann entropy Husimi Q representation quantum Gaussian states ·Schatten norms · · · 1 Introduction The Husimi Q representation [20] is a probability distribution in phase space that describes a quantum state of a Gaussian quantum system, such as an harmonic oscillator or a mode of electromagnetic radiation [2,24]. It coincides I acknowledge financial support from the European Research Council (ERC Grant Agree- ment no 337603), the Danish Council for Independent Research (Sapere Aude) and VILLUM FONDEN via the QMATH Centre of Excellence (Grant No.
    [Show full text]
  • An Overview of Definitions and Applications in Quantum Mechanics
    Entropy in Physics: An Overview of Definitions and Applications in Quantum Mechanics Fabian Immanuel IJpelaar January 22, 2021 Abstract In modern physics, entropy is a well-known and important quantity. At its core, the entropy is a function of the probabilities of a distribution that is meant to describe the uncertainty in outcome of a random process represented by that distribution. However, it has been used in many different fields and, as a consequence, has many interpretations. Moreover, a lot of different functions have been grouped under the name of entropy, all with their own uses and interpretations. In this work, we discuss the definitions, origins, and interpretations of many of these functions as well as how they fit together. We will also explicitly cover some of the applications that the entropies have found within physics, in particular in quantum physics. These applications include thermodynamics, measurement uncertainty in quantum mechanics, measures of mixedness in the density matrix formalism and in phase space formalisms, measures of entanglement, and parton distributions in QCD. Master's Thesis Physics Van Swinderen Institute Supervisor: Dani¨elBoer Second Examiner: Elisabetta Pallante 1 Contents Introduction 5 1 The Entropies of Statistical Mechanics and Information Theory 9 1.1 Statistical Mechanics and the Classical Entropy . 10 1.1.1 The Liouville Equation . 10 1.1.2 The Microcanonical Ensemble . 11 1.1.3 The Microcanonical Entropy . 11 1.1.4 Extensivity of the microcanonical Entropy and Temperature . 12 1.1.5 The Canonical Ensemble . 13 1.1.6 The Canonical Entropy . 13 1.1.7 Closing Remarks . 14 1.2 The Shannon Entropy .
    [Show full text]