DOCSLIB.ORG

Microstates, Entropy and Quanta Don Koks

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Microstates, Entropy and Quanta Don Koks Microstates, Entropy and Quanta Don Koks Microstates, Entropy and Quanta An Introduction to Statistical Mechanics 123 ISBN 978-3-030-02428-4 ISBN 978-3-030-02429-1 (eBook) https://doi.org/10.1007/978-3-030-02429-1 Library of Congress Control Number: 2018960736 © Springer Nature Switzerland AG 2018, corrected publication 2020 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Cover art: Central to statistical mechanics is the idea of counting the states accessible to a system. When these states exist within some continuous space, they cannot be counted. Instead, we “tile” the space into cells, with each cell defining a state, and then we count those cells. The ball on the front cover is a schematic of this tiling of the velocity space of a free particle that moves in three spatial dimensions. For real particles, the cells are so much smaller than the size of the ball that, to all intents and purposes, the ball is a smooth sphere. The number of cells can then easily be found from the sphere’s volume. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland For my ancestors, and all those who have gone before. Preface Another book on introductory statistical mechanics? You might think that a century-old subject would have nothing left unsaid; but that is perhaps not the case. Unlike most other fields of physics, one can compare a dozen books on statistical mechanics and find a dozen different approaches to the discipline. At one extreme are authors who revel in arcane abstraction, but whose books go mostly unread. At the other extreme are very readable books that lack the mathematics to carry the reader very far beyond a set of physical assumptions. Most readers are looking for something in between; but that space is vast and subjective, with plenty of room for another book to aim for the \Goldilocks Zone" of being just right. That's why I wrote this book: I think that the field of introductory statistical mechanics still has plenty of scope for an author to try a different mix of mathematical exposition and physical reasoning. The physics part of this mix is a build of statistical mechanics from the ground up, anchored to a bedrock of physical concepts. With this approach, I hope to have revealed the necessity and importance of the subject's core ideas, such as entropy and temperature. The mathematics part of the mix has been an emphasis on a strong logical reasoning that has a clean outline, yet avoids the notational clutter and obscure discussions that are so often associated with statistical mechanics, and which can make it so hard to learn. Thus, beside the calculations of representative physical quantities, you will find here various mathematical analyses that I believe are important to physicists. Much of this mathematical foundation is given in the first chapter, such as details of integrating the gaussian function, and the correct use of infinitesimals, partial derivatives, and units of measurement. By the time you reach that chapter's end, you might be wondering whether you are really reading a book on statistical mechanics after all. And yet, you will encounter those topics time and again as you work through the rest of the book. The choice of how and where to begin describing a subject is always highly author dependent. The concepts that I introduce methodically, as needed, vii viii Preface are sometimes merely postulated with a breezy stroke of the pen in books that announce themselves as introductions. Postulatory approaches to other subjects can certainly work well; for instance, I admire Feynman's approach to electromagnetism in his Lectures on Physics, since, although he postulates Maxwell's equations at the very start, we never lose sight of the physics in his discussions. In contrast, I struggle to see any physics at all in some postulatory approaches to statistical mechanics, which can so easily ignore the difficult questions that interest physicists. I commence the subject of statistical mechanics with an archetypal obser- vation: why does a drop of ink placed in a bathtub disperse? Once dispersed, might it ever re-assemble into a drop? This question showcases the impor- tance of counting the number of ways in which a system's constituents can be arranged, and leads to statistical mechanics proper via its fundamental postulate. That discussion demands knowledge of the concept of energy, a concept that was useful and intriguing to early astronomers studying plane- tary orbits, but whose wider application was not well understood in the early days of thermodynamics, 150 years ago. With a more modern understanding of energy (or perhaps \acceptance" is a better word, since we still don't know what it is|if, indeed, asking what it is has any meaning), we are in a good position to write down the laws of thermodynamics. Then, we can explore heat engines, chemical processes and equilibria, and heat flow. The flow of heat is a stepping stone to appreciating diverse related areas, such as parti- cle diffusion and, in fact, the signal processing performed in a modern radar receiver. But no system is ever truly isolated; and the question of how to analyse a system in contact with the wider world brings us to the Boltzmann dis- tribution, with examples in paramagnetism, atomic energy levels, molecular and crystal heat capacities, and data-transmission theory. The Boltzmann distribution also sheds light on the motion of gas particles. I use that theory to explore an atmosphere, as well as the molecular details of viscosity and thermal conductivity. Quantum ideas then emerge, via Einstein's and Debye's theories of heat capacity. The notion of fermions and bosons forms a springboard to the study of electronic heat capacity, electrical conduction, thermal noise in electric circuits, the spectra of light produced by hot bodies, some cosmology, the greenhouse effect, and the modern technologies of light-emitting diodes and the laser. I have sprinkled the text with occasional short digressions, discussing top- ics such as the factorial function in number theory, the energy{momentum tensor in relativity, a little bit of signal processing, and decrying the short- comings of modern analytical astronomy. Hopefully, these asides will only enrich your interest without being a distraction. Unlike some books on statistical mechanics, I have chosen to discuss a lot of material before introducing the Boltzmann distribution. Thus, in those pre- Boltzmann chapters, I invoke the equipartition theorem to approximate the Preface ix particles in an ideal gas of temperature T as each having, say, translational energy 3=2 kT . Later, when studying the Boltzmann distribution, we learn that only their average translational energy is 3=2 kT . Some authors will avoid this initial simplification by introducing the Boltzmann distribution very early on. But I think that using the simple approximation initially, and leaving Boltzmann for later, is useful pedagogically. A subject as old as statistical mechanics is bound to carry baggage picked up along the way, created as a normal part of its development, when physicists and chemists were searching for the best path through the new forest they had discovered. The choice of what might best be discarded is a little subjective. I have tended to minimise the use of phrases and topics that appear to be generally confusing, unattractive, or not useful. For example, I cannot imagine that the conventional vernacular that de- scribes various flavours of ensemble, along with free energies, partition func- tions, and Maxwell relations, does anything to attract new adherents to sta- tistical mechanics. Pseudo wisdom that you can find in books on the subject, such as \The trick to solving this problem is to use the grand canonical en- semble", is apt to give the impression that statistical mechanics is all about finding the right trick using the right ensemble to get the right answer. The language of ensembles is not especially deep, and after explaining what it means and how it's used, I tend to avoid it, because the \correct ensemble to use" should be clear from the context being studied; it is not some arbitrary choice that we make. Free energies have a range of uses in thermodynamics (and I certainly use them in this book), but they are probably more relevant to the history of the subject, when early physicists and chemists worked hard to ascertain the nature of what was then a cutting-edge new quantity called energy. Nowadays, we view free energies as useful combinations of more fun- damental parameters such as energy, temperature, and entropy.
Load more

Recommended publications
  • 8.1. Spin & Statistics
    8.1. Spin & Statistics Ref: A.Khare,”Fractional Statistics & Quantum Theory”, 1997, Chap.2. Anyons = Particles obeying fractional statistics. Particle statistics is determined by the phase factor eiα picked up by the wave function under the interchange of the positions of any pair of (identical) particles in the system. Before the discovery of the anyons, this particle interchange (or exchange) was treated as the permutation of particle labels. Let P be the operator for this interchange. P2 =I → e2iα =1 ∴ eiα = ±1 i.e., α=0,π Thus, there’re only 2 kinds of statistics, α=0(π) for Bosons (Fermions) obeying Bose-Einstein (Fermi-Dirac) statistics. Pauli’s spin-statistics theorem then relates particle spin with statistics, namely, bosons (fermions) are particles with integer (half-integer) spin. To account for the anyons, particle exchange is re-defined as an observable adiabatic (constant energy) process of physically interchanging particles. ( This is in line with the quantum philosophy that only observables are physically relevant. ) As will be shown later, the new definition does not affect statistics in 3-D space. However, for particles in 2-D space, α can be any (real) value; hence anyons. The converse of the spin-statistic theorem then implies arbitrary spin for 2-D particles. Quantization of S in 3-D See M.Alonso, H.Valk, “Quantum Mechanics: Principles & Applications”, 1973, §6.2. In 3-D, the (spin) angular momentum S has 3 non-commuting components satisfying S i,S j =iℏε i j k Sk 2 & S ,S i=0 2 This means a state can be the simultaneous eigenstate of S & at most one Si.
    [Show full text]
  • Many Particle Orbits – Statistics and Second Quantization
    H. Kleinert, PATH INTEGRALS March 24, 2013 (/home/kleinert/kleinert/books/pathis/pthic7.tex) Mirum, quod divina natura dedit agros It’s wonderful that divine nature has given us fields Varro (116 BC–27BC) 7 Many Particle Orbits – Statistics and Second Quantization Realistic physical systems usually contain groups of identical particles such as spe- cific atoms or electrons. Focusing on a single group, we shall label their orbits by x(ν)(t) with ν =1, 2, 3, . , N. Their Hamiltonian is invariant under the group of all N! permutations of the orbital indices ν. Their Schr¨odinger wave functions can then be classified according to the irreducible representations of the permutation group. Not all possible representations occur in nature. In more than two space dimen- sions, there exists a superselection rule, whose origin is yet to be explained, which eliminates all complicated representations and allows only for the two simplest ones to be realized: those with complete symmetry and those with complete antisymme- try. Particles which appear always with symmetric wave functions are called bosons. They all carry an integer-valued spin. Particles with antisymmetric wave functions are called fermions1 and carry a spin whose value is half-integer. The symmetric and antisymmetric wave functions give rise to the characteristic statistical behavior of fermions and bosons. Electrons, for example, being spin-1/2 particles, appear only in antisymmetric wave functions. The antisymmetry is the origin of the famous Pauli exclusion principle, allowing only a single particle of a definite spin orientation in a quantum state, which is the principal reason for the existence of the periodic system of elements, and thus of matter in general.
    [Show full text]
  • The Conventionality of Parastatistics
    The Conventionality of Parastatistics David John Baker Hans Halvorson Noel Swanson∗ March 6, 2014 Abstract Nature seems to be such that we can describe it accurately with quantum theories of bosons and fermions alone, without resort to parastatistics. This has been seen as a deep mystery: paraparticles make perfect physical sense, so why don't we see them in nature? We consider one potential answer: every paraparticle theory is physically equivalent to some theory of bosons or fermions, making the absence of paraparticles in our theories a matter of convention rather than a mysterious empirical discovery. We argue that this equivalence thesis holds in all physically admissible quantum field theories falling under the domain of the rigorous Doplicher-Haag-Roberts approach to superselection rules. Inadmissible parastatistical theories are ruled out by a locality- inspired principle we call Charge Recombination. Contents 1 Introduction 2 2 Paraparticles in Quantum Theory 6 ∗This work is fully collaborative. Authors are listed in alphabetical order. 1 3 Theoretical Equivalence 11 3.1 Field systems in AQFT . 13 3.2 Equivalence of field systems . 17 4 A Brief History of the Equivalence Thesis 20 4.1 The Green Decomposition . 20 4.2 Klein Transformations . 21 4.3 The Argument of Dr¨uhl,Haag, and Roberts . 24 4.4 The Doplicher-Roberts Reconstruction Theorem . 26 5 Sharpening the Thesis 29 6 Discussion 36 6.1 Interpretations of QM . 44 6.2 Structuralism and Haecceities . 46 6.3 Paraquark Theories . 48 1 Introduction Our most fundamental theories of matter provide a highly accurate description of subatomic particles and their behavior.
    [Show full text]
  • Arxiv:0805.0285V1
    Generalizations of Quantum Statistics O.W. Greenberg1 Center for Fundamental Physics Department of Physics University of Maryland College Park, MD 20742-4111, USA, Abstract We review generalizations of quantum statistics, including parabose, parafermi, and quon statistics, but not including anyon statistics, which is special to two dimensions. The general principles of quantum theory allow statistics more general than bosons or fermions. (Bose statistics and Fermi statistics are discussed in separate articles.) The restriction to Bosons or Fermions requires the symmetrization postu- late, “the states of a system containing N identical particles are necessarily either all symmetric or all antisymmetric under permutations of the N particles,” or, equiv- alently, “all states of identical particles are in one-dimensional representations of the symmetric group [1].” A.M.L. Messiah and O.W. Greenberg discussed quantum mechanics without the symmetrization postulate [2]. The spin-statistics connec- arXiv:0805.0285v1 [quant-ph] 2 May 2008 tion, that integer spin particles are bosons and odd-half-integer spin particles are fermions [3], is an independent statement. Identical particles in 2 space dimensions are a special case, “anyons.” (Anyons are discussed in a separate article.) Braid group statistics, a nonabelian analog of anyons, are also special to 2 space dimensions All observables must be symmetric in the dynamical variables associated with identical particles. Observables can not change the permutation symmetry type of the wave function; i.e. there is a superselection rule separating states in inequivalent representations of the symmetric group and when identical particles can occur in states that violate the spin-statistics connection their transitions must occur in the same representation of the symmetric group.
    [Show full text]
  • The Basis of the Second Law of Thermodynamics in Quantum Field Theory
    The Basis of the Second Law of Thermodynamics in Quantum Field Theory D.W. Snoke1 , G. Liu Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, PA 15260, USA S. Girvin Department of Physics, Yale University, New Haven, CT 06520, USA Abstract The result that closed systems evolve toward equilibrium is derived entirely on the ba- sis of quantum field theory for a model system, without invoking any of the common extra- mathematical notions of particle trajectories, collapse of the wave function, measurement, or intrinsically stochastic processes. The equivalent of the Stosszahlansatz of classical statistical mechanics is found, and has important differences from the classical version. A novel result of the calculation is that interacting many-body systems in the infinite volume limit evolve toward diagonal states (states with loss of all phase information) on the the time scale of the interaction time. The connection with the onset of off-diagonal phase coherence in Bose condensates is discussed. 1. Introduction The controversy over the formulation of the Second Law of thermodynamics in terms of statistical mechanics of particles is well known [1]. Boltzmann presented his H-theorem as a way of connecting the deterministic, time-reversible dynamics of particles to the already well known, time-irreversible Second Law. This theorem, in Boltzmann’s formulation using classical mechanics, says that a many-particle system evolves irreversibly toward equilibrium via collision processes of the particles which can be calculated using known collision kinetics. Many scientists in his day found his arguments unpersuasive, however, partly because they involved notions of coarse graining which were ill-defined, and partly because the outcome of irreversibility seemed to contradict well-established ideas of time-reversibility such as Poincare cycles.
    [Show full text]
  • Arxiv:1106.1166V2 [Quant-Ph] 16 Apr 2013 Fermions, Bosons Or Particles with Fractional Statistics
    Observing fermionic statistics with photons in arbitrary processes Jonathan C. F. Matthews,1 Konstantinos Poulios,1 Jasmin D. A. Meinecke,1 Alberto Politi,1, 2 Alberto Peruzzo,1 Nur Ismail,3 Kerstin W¨orhoff,3 Mark G. Thompson,1 and Jeremy L. O'Brien1, ∗ 1Centre for Quantum Photonics, H. H. Wills Physics Laboratory & Department of Electrical and Electronic Engineering, University of Bristol, Merchant Venturers Building, Woodland Road, Bristol, BS8 1UB, UKy 2Presently at Center for Spintronics and Quantum Computation, University of California Santa Barbara, USAy 3Integrated Optical Microsystems Group, MESA+ Institute for Nanotechnology, University of Twente, Enschede, The Netherlands Quantum mechanics defines two classes of particles|bosons and fermions|whose exchange statis- tics fundamentally dictate the dynamics of quantum systems. Here we develop a scheme that uses multi-partite entanglement to directly observe the correlated detection statistics of any number of fermions in any physical process. This approach relies on sending each of the entangled particles through identical copies of the process and by controlling a single phase parameter in the entangled state, the correlated detection statistics can be continuously tuned between bosonic and fermionic statistics. We implement this scheme via two entangled photons shared across the polarisation modes of a single photonic chip to directly mimic the fermion, boson and intermediate behaviour of two-particles undergoing a continuous time quantum walk with an average similarity of 93.6±0.2% with ideal models. The ability to simulate fermions with photons is likely to have applications for verifying computationally hard boson scattering in linear optics and for observing particle correla- tions (including fractional statistics) in analogue simulation using any physical platform that can prepare the entangled state prescribed here.
    [Show full text]
  • Field Theory of Anyons and the Fractional Quantum Hall Effect 30- 50
    NO9900079 Field Theory of Anyons and the Fractional Quantum Hall Effect Susanne F. Viefers Thesis submitted for the Degree of Doctor Scientiarum Department of Physics University of Oslo November 1997 30- 50 Abstract This thesis is devoted to a theoretical study of anyons, i.e. particles with fractional statistics moving in two space dimensions, and the quantum Hall effect. The latter constitutes the only known experimental realization of anyons in that the quasiparticle excitations in the fractional quantum Hall system are believed to obey fractional statistics. First, the properties of ideal quantum gases in two dimensions and in particular the equation of state of the free anyon gas are discussed. Then, a field theory formulation of anyons in a strong magnetic field is presented and later extended to a system with several species of anyons. The relation of this model to fractional exclusion statistics, i.e. intermediate statistics introduced by a generalization of the Pauli principle, and to the low-energy excitations at the edge of the quantum Hall system is discussed. Finally, the Chern-Simons-Landau-Ginzburg theory of the fractional quantum Hall effect is studied, mainly focusing on edge effects; both the ground state and the low- energy edge excitations are examined in the simple one-component model and in an extended model which includes spin effets. Contents I General background 5 1 Introduction 7 2 Identical particles and quantum statistics 10 3 Introduction to anyons 13 3.1 Many-particle description 14 3.2 Field theory description
    [Show full text]
  • Statistical Mechanics
    Statistical Mechanics Henri J.F. Jansen Department of Physics Oregon State University October 12, 2008 II Contents 1 Foundation of statistical mechanics. 1 1.1 Introduction. 1 1.2 Program of statistical mechanics. 4 1.3 States of a system. 5 1.4 Averages. 10 1.5 Thermal equilibrium. 14 1.6 Entropy and temperature. 16 1.7 Laws of thermodynamics. 19 1.8 Problems for chapter 1 . 20 2 The canonical ensemble 23 2.1 Introduction. 23 2.2 Energy and entropy and temperature. 26 2.3 Work and pressure. 28 2.4 Helmholtz free energy. 31 2.5 Changes in variables. 32 2.6 Properties of the Helmholtz free energy. 33 2.7 Energy fluctuations. 35 2.8 A simple example. 37 2.9 Problems for chapter 2 . 39 3 Variable number of particles 43 3.1 Chemical potential. 43 3.2 Examples of the use of the chemical potential. 46 3.3 Di®erential relations and grand potential. 48 3.4 Grand partition function. 50 3.5 Overview of calculation methods. 55 3.6 A simple example. 57 3.7 Ideal gas in ¯rst approximation. 58 3.8 Problems for chapter 3 . 64 4 Statistics of independent particles. 67 4.1 Introduction. 67 4.2 Boltzmann gas again. 74 III IV CONTENTS 4.3 Gas of poly-atomic molecules. 77 4.4 Degenerate gas. 79 4.5 Fermi gas. 80 4.6 Boson gas. 85 4.7 Problems for chapter 4 . 87 5 Fermions and Bosons 89 5.1 Fermions in a box. 89 5.2 Bosons in a box.
    [Show full text]
  • Landauer's Principle in a Quantum Szilard Engine Without Maxwell's
    entropy Article Landauer’s Principle in a Quantum Szilard Engine without Maxwell’s Demon Alhun Aydin 1,2 , Altug Sisman 3 and Ronnie Kosloff 1,* 1 Fritz Haber Research Center, Institute of Chemistry, The Hebrew University of Jerusalem, 91904 Jerusalem, Israel; [email protected] 2 Nano Energy Research Group, Energy Institute, Istanbul Technical University, 34469 Istanbul, Turkey 3 Department of Physics and Astronomy, Uppsala University, 75120 Uppsala, Sweden; [email protected] * Correspondence: [email protected] Received: 19 February 2020; Accepted: 1 March 2020; Published: 3 March 2020 Abstract: Quantum Szilard engine constitutes an adequate interplay of thermodynamics, information theory and quantum mechanics. Szilard engines are in general operated by a Maxwell’s Demon where Landauer’s principle resolves the apparent paradoxes. Here we propose a Szilard engine setup without featuring an explicit Maxwell’s demon. In a demonless Szilard engine, the acquisition of which-side information is not required, but the erasure and related heat dissipation still take place implicitly. We explore a quantum Szilard engine considering quantum size effects. We see that insertion of the partition does not localize the particle to one side, instead creating a superposition state of the particle being in both sides. To be able to extract work from the system, particle has to be localized at one side. The localization occurs as a result of quantum measurement on the particle, which shows the importance of the measurement process regardless of whether one uses the acquired information or not. In accordance with Landauer’s principle, localization by quantum measurement corresponds to a logically irreversible operation and for this reason it must be accompanied by the corresponding heat dissipation.
    [Show full text]
  • Particle Statistics Affects Quantum Decay and Fano Interference
    Particle Statistics Affects Quantum Decay and Fano Interference † Andrea Crespi,1 Linda Sansoni,2,* Giuseppe Della Valle,3,1 Alessio Ciamei,2 Roberta Ramponi,1,3 Fabio Sciarrino,2, ‡ Paolo Mataloni,2 Stefano Longhi,3,1, and Roberto Osellame1,3,§ 1Istituto di Fotonica e Nanotecnologie, Consiglio Nazionale delle Ricerche, Piazza Leonardo da Vinci 32, I-20133 Milano, Italy 2Dipartimento di Fisica, Sapienza Università di Roma, Piazzale Aldo Moro 5, I-00185 Roma, Italy 3Dipartimento di Fisica, Politecnico di Milano, Piazza Leonardo da Vinci 32, I-20133 Milano, Italy (Received 6 October 2014; published 2 March 2015) Quantum mechanical decay, Fano interference, and bound states with energy in the continuum are ubiquitous phenomena in different areas of physics. Here we experimentally demonstrate that particle statistics strongly affects quantum mechanical decay in a multiparticle system. By considering propagation of two-photon states in engineered photonic lattices, we simulate quantum decay of two noninteracting particles in a multilevel Fano-Anderson model. Remarkably, when the system sustains a bound state in the continuum, fractional decay is observed for bosonic particles, but not for fermionic ones. Complete decay in the fermionic case arises because of the Pauli exclusion principle, which forbids the bound state to be occupied by the two fermions. Our experiment indicates that particle statistics can tune many-body quantum decay from fractional to complete. PACS numbers: 42.82.Et, 03.65.Xp, 03.67.Ac, 42.50.−p Decay of excited states has been a topic of great interest (e.g. electrons) nature. However, in such previous experi- since the early times of quantum mechanics [1].A ments the peculiar role of particle statistics in the decay metastable state decaying into a continuum usually shows process was not disclosed.
    [Show full text]
  • Observing Fermionic Statistics with Photons in Arbitrary Processes SUBJECT AREAS: Jonathan C
    Observing fermionic statistics with photons in arbitrary processes SUBJECT AREAS: Jonathan C. F. Matthews1, Konstantinos Poulios1, Jasmin D. A. Meinecke1, Alberto Politi1*, QUANTUM MECHANICS Alberto Peruzzo1, Nur Ismail2, Kerstin Wo¨rhoff2, Mark G. Thompson1 & Jeremy L. O’Brien1 QUANTUM OPTICS QUANTUM INFORMATION 1Centre for Quantum Photonics, H. H. Wills Physics Laboratory & Department of Electrical and Electronic Engineering, University of THEORETICAL PHYSICS Bristol, Merchant Venturers Building, Woodland Road, Bristol, BS8 1UB, UK, 2Integrated Optical Microsystems Group, MESA1 Institute for Nanotechnology, University of Twente, Enschede, The Netherlands. Received Quantum mechanics defines two classes of particles-bosons and fermions-whose exchange statistics 22 February 2013 fundamentally dictate quantum dynamics. Here we develop a scheme that uses entanglement to directly observe the correlated detection statistics of any number of fermions in any physical process. This approach Accepted relies on sending each of the entangled particles through identical copies of the process and by controlling a 28 February 2013 single phase parameter in the entangled state, the correlated detection statistics can be continuously tuned Published between bosonic and fermionic statistics. We implement this scheme via two entangled photons shared across 27 March 2013 the polarisation modes of a single photonic chip to directly mimic the fermion, boson and intermediate behaviour of two-particles undergoing a continuous time quantum walk. The ability to simulate fermions with photons is likely to have applications for verifying boson scattering and for observing particle correlations in analogue simulation using any physical platform that can prepare the entangled state prescribed here. Correspondence and requests for materials should be addressed to particularly striking departure from classical physics in quantum mechanics is the existence of two J.L.O.
    [Show full text]
  • Particle Statistics and Lossy Dynamics of Ultracold Atoms in Optical Lattices
    PHYSICAL REVIEW A 97, 053614 (2018) Particle statistics and lossy dynamics of ultracold atoms in optical lattices J. Yago Malo,1 E. P. L. van Nieuwenburg,2,3 M. H. Fischer,2 and A. J. Daley1 1Department of Physics and SUPA, University of Strathclyde, Glasgow G4 0NG, Scotland, United Kingdom 2Institute for Theoretical Physics, ETH Zurich, 8093 Zurich, Switzerland 3Institute for Quantum Information and Matter, Caltech, Pasadena, California 91125, USA (Received 12 September 2017; published 21 May 2018) Experimental control over ultracold quantum gases has made it possible to investigate low-dimensional systems of both bosonic and fermionic atoms. In closed one-dimensional systems there are many similarities in the dynamics of local quantities for spinless fermions and strongly interacting “hard-core” bosons, which on a lattice can be formalized via a Jordan-Wigner transformation. In this study, we analyze the similarities and differences for spinless fermions and hard-core bosons on a lattice in the presence of particle loss. The removal of a single fermion causes differences in local quantities compared with the bosonic case because of the different particle exchange symmetry in the two cases. We identify deterministic and probabilistic signatures of these dynamics in terms of local particle density, which could be measured in ongoing experiments with quantum gas microscopes. DOI: 10.1103/PhysRevA.97.053614 I. INTRODUCTION the eigenstates and for out-of-equilibrium dynamics induced by changing local trap quantities—will be equal as well. In the past few years, there has been rapid progress in However, single-particle loss can generate differences in local the characterization and control of dissipative dynamics for quantities due to the different exchange symmetries in the ultracold atoms in optical lattices.
    [Show full text]