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Assessing the Nonequilibrium Thermodynamics in a Quenched Quantum Many-Body System via Single Projective Measurements

L. Fusco1, S. Pigeon1, T. J. G. Apollaro2,1, A. Xuereb1,3, L. Mazzola1, M. Campisi4, A. Ferraro1, M. Paternostro1, and G. De Chiara1

  • 1Centre for Theoretical Atomic, Molecular and Optical Physics, School of Mathematics and Physics, Queen’s University, Belfast BT7 1NN, United Kingdom
  • 2Dipartimento di Fisica, Università della Calabria, 87036 Arcavacata di Rende (CS), Italy
  • 3Department of Physics, University of Malta, Msida MSD 2080, Malta
  • 4NEST, Scuola Normale Superiore & Istituto di Nanoscienze-CNR, I-56126 Pisa, Italy

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Phys. Rev. X 4, 031029 – Published 19 August, 2014

DOI: https://doi.org/10.1103/PhysRevX.4.031029

Abstract

We analyze the nature of the statistics of the work done on or by a quantum many-body system brought out of equilibrium. We show that, for the sudden quench and for an initial state that commutes with the initial Hamiltonian, it is possible to retrieve the whole nonequilibrium thermodynamics via single projective measurements of observables. We highlight, in a physically clear way, the qualitative implications for the statistics of work coming from considering processes described by operators that either commute or do not commute with the unperturbed Hamiltonian of a given system. We consider a quantum many-body system and derive an expression that allows us to give a physical interpretation, for a thermal initial state, to all of the cumulants of the work in the case of quenched operators commuting with the unperturbed Hamiltonian. In the commuting case, the observables that we need to measure have an intuitive physical meaning. Conversely, in the noncommuting case, we show that, although it is possible to operate fully within the single-measurement framework irrespectively of the size of the quench, some difficulties are faced in providing a clear-cut physical interpretation to the cumulants. This circumstance makes the study of the physics of the system nontrivial and highlights the nonintuitive phenomenology of the emergence of thermodynamics from the fully quantum microscopic description. We illustrate our ideas with the example of the Ising model in a transverse field showing the interesting behavior of the high-order statistical moments of the work distribution for a generic thermal state and linking them to the critical nature of the model itself.

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Popular Summary

Nonequilibrium thermodynamics deals with the properties of physical, chemical, and biological systems that are not in thermodynamic equilibrium and the transformations that involve them. Fundamental fluctuation theorems have been formulated to relate fluctuations of physical quantities for nonequilibrium systems (such as work) to their equilibrium features. These relations can be exported to the quantum domain, where they play key roles in understanding the thermodynamic consequences of physically interesting processes occurring at the microscopic level. However, the general lack of direct observability in the quantum domain of thermodynamically important quantities hinders the collection and interpretation of data. We study in detail the conditions that a quantum mechanical process should satisfy in order to guarantee the direct accessibility, and thus the measurability, of work.

We apply our framework to the statistical features of the work done on or by a quantum many-body system and seek to understand the onset of quantum criticality by analyzing out-of-equilibrium dynamics. Our goal is to link nonequilibrium thermodynamics and macroscopically observable properties of a system. To this end, we employ sudden thermal quenching, which is decidedly nonadiabatic. Interestingly, the general noncommutative nature of Hamiltonians describing quantum processes turns out to be key for the direct observation of work in such systems. Our findings suggest that it is possible to observe signatures of quantum many-body criticality in the statistical moments of work.

We expect that our findings will be valuable for developing an experimentally certifiable framework of finite-time thermodynamics in quantum systems, which will be crucial for applications in nanotechnology and quantum technology.

Article Text

References (69)

  1. P. Talkner, E. Lutz, and P. Hänggi, Fluctuation Theorems: Work Is Not an Observable, Phys. Rev. E 75, 050102R (2007).
  2. J. P. Pekola, P. Solinas, A. Shnirman, and D. V. Averin, Calorimetric Measurement of Quantum Work, arXiv:1212.5808; V. Vedral, An Information-Theoretic Equality Implying the Jarzynski Relation, J. Phys. A 45, 272001 (2012); K. Micadei, R. M. Serra, and L. C. Celeri, Thermodynamic Cost of Acquiring Information, Phys. Rev. E 88, 062123 (2013); D. Kafri and S. Deffner, Holevo’s Bound from a General Quantum Fluctuation Theorem, Phys. Rev. A 86, 044302 (2012).
  3. O. Abah, J. Rossnagel, G. Jacob, S. Deffner, F. Schmidt-Kaler, K. Singer, and E. Lutz, Single-Ion Heat Engine at Maximum Power, Phys. Rev. Lett. 109, 203006 (2012).
  4. V. A. Ngo and S. Haas, Demonstration of Jarzynski’s Equality in Open Quantum Systems Using a Stepwise Pulling Protocol, Phys. Rev. E 86, 031127 (2012); T. Albash, D. A. Lidar, M. Marvian, and P. Zanardi, Fluctuation Theorems for Quantum Processes, 88, 032146 (2013).
  5. M. Campisi, P. Talkner, and P. Hänggi, Fluctuation Theorem for Arbitrary Open Quantum Systems, Phys. Rev. Lett. 102, 210401 (2009).
  6. R. Dorner, S. R. Clark, L. Heaney, R. Fazio, J. Goold, and V. Vedral, Extracting Quantum Work Statistics and Fluctuation Theorems by Single-Qubit Interferometry, Phys. Rev. Lett. 110, 230601 (2013); L. Mazzola, G. De Chiara, and M. Paternostro, Measuring the Characteristic Function of the Work Distribution, 110, 230602 (2013); Detecting the Work Statistics through Ramsey-like Interferometry, Int. J. Quantum. Inform. 12, 1461007 (2014).
  7. J. Goold and K. Modi, Measuring the Heat Exchange of a Quantum Process, arXiv:1401.4088.
  8. J. Goold, M. Paternostro, and K. Modi, A Nonequilibrium Quantum Landauer Principle, arXiv:1402.4499.
  9. A. Sindona, J. Goold, N. Lo Gullo, and F. Plastina, Statistics of the Work Distribution for a Quenched Fermi Gas, New J. Phys. 16, 045013 (2014).
  10. A. Sindona, J. Goold, N. Lo Gullo, S. Lorenzo, and F. Plastina, Orthogonality Catastrophe and Decoherence in a Trapped-Fermion Environment, Phys. Rev. Lett. 111, 165303 (2013).
  11. S. Deffner and E. Lutz, Nonequilibrium Work Distribution of a Quantum Harmonic Oscillator, Phys. Rev. E 77, 021128 (2008).
  12. P. Talkner, P. Sekhar Burada, and P. Hänggi, Statistics of Work Performed on a Forced Quantum Oscillator, Phys. Rev. E 78, 011115 (2008).
  13. S. Deffner, O. Abah, and E. Lutz, Quantum Work Statistics of Linear and Nonlinear Parametric Oscillators, Chem. Phys. 375, 200 (2010).
  14. H. T. Quan and C. Jarzynski, Validity of Nonequilibrium Work Relations for the Rapidly Expanding Quantum Piston, Phys. Rev. E 85, 031102 (2012).
  15. P. Talkner, M. Morillo, J. Yi, and P. Hänggi, Statistics of Work and Fluctuation Theorems for Microcanonical Initial States, New J. Phys. 15, 095001 (2013).
  16. M. Esposito, U. Harbola, and S. Mukamel, Nonequilibrium Fluctuations, Fluctuation Theorems, and Counting Statistics in Quantum Systems, Rev. Mod. Phys. 81, 1665 (2009).
  17. M. Campisi, P. Hänggi, and P. Talkner, Colloquium: Quantum Fluctuation Relations: Foundations and Applications, Rev. Mod. Phys. 83, 771 (2011); Colloquium: Quantum Fluctuation Relations: Foundations and Applications 83, 1653 (2011).
  18. U. Seifert, Stochastic Thermodynamics, Fluctuation Theorems and Molecular Machines, Rep. Prog. Phys. 75, 126001 (2012).
  19. H. Tasaki, Jarzynski Relations for Quantum Systems and Some Applications, arXiv:cond-mat/0009244v2; J. Kurchan, A Quantum Fluctuation Theorem, arXiv:cond-mat/0007360v2; S. Mukamel, Quantum Extension of the Jarzynski Relation: Analogy with Stochastic Dephasing, Phys. Rev. Lett. 90, 170604 (2003).
  20. G. E. Crooks, Entropy Production Fluctuation Theorem and the Nonequilibrium Work Relation for Free Energy Differences, Phys. Rev. E 60, 2721 (1999).
  21. C. Jarzynski, Nonequilibrium Equality for Free Energy Differences, Phys. Rev. Lett. 78, 2690 (1997).
  22. J. Liphardt, S. Dumont, S. B. Smith, I. Tinoco Jr., and C. Bustamante, Equilibrium Information from Nonequilibrium Measurements in an Experimental Test of the Jarzynski Equality, Science 296, 1832 (2002).
  23. D. Collin, F. Ritort, C. Jarzynski, S. B. Smith, I. Tinoco Jr, and C. Bustamante, Verification of the Crooks Fluctuation Theorem and Recovery of RNA Folding Free Energies, Nature (London) 437, 231 (2005).
  24. F. Douarche, S. Ciliberto, A. Petrosyan, and I. Rabbiosi, An Experimental Test of the Jarzynski Equality in a Mechanical Experiment, Europhys. Lett. 70, 593 (2005).
  25. S. Toyabe, T. Sagawa, M. Ueda, E. Muneyuki, and M. Sano, Experimental Demonstrations of Information-to-Energy Conversion and Validation of the Generalized Jarzynski Equality, Nat. Phys. 6, 988 (2010).
  26. O.-P. Saira, Y. Yoon, T. Tanttu, M. Möttönen, D. V. Averin, and J. P. Pekola, Test of the Jarzynski and Crooks Fluctuation Relations in an Electronic System, Phys. Rev. Lett. 109, 180601 (2012).
  27. T. B. Batalhão, A. M. Souza, L. Mazzola, R. Auccaise, I. S. Oliveira, J. Goold, G. De Chiara, M. Paternostro, and R. M. Serra, Experimental Reconstruction of Work Distribution and Verification of Fluctuation Relations at the Full Quantum Level, arXiv:1308.3241.
  28. G. Huber, F. Schmidt-Kaler, S. Deffner, and E. Lutz, Employing Trapped Cold Ions to Verify the Quantum Jarzynski Equality, Phys. Rev. Lett. 101, 070403 (2008).
  29. M. Heyl and S. Kehrein, Crooks Relation in Optical Spectra: Universality in Work Distributions for Weak Local Quenches, Phys. Rev. Lett. 108, 190601 (2012).
  30. J. P. Pekola, P. Solinas, A. Shnirman, and D. V. Averin, Calorimetric Measurement of Work in a Quantum System, New J. Phys. 15, 115006 (2013).
  31. R. Dorner, J. Goold, C. Cormick, M. Paternostro, and V. Vedral, Emergent Thermodynamics in a Quenched Quantum Many-Body System, Phys. Rev. Lett. 109, 160601 (2012).
  32. M. Campisi, R. Blattmann, S. Kohler, D. Zueco, and P. Hänggi, Employing Circuit QED to Measure Nonequilibrium Work Fluctuations, New J. Phys. 15, 105028 (2013).
  33. A. Silva, Statistics of the Work Done on a Quantum Critical System by Quenching a Control Parameter, Phys. Rev. Lett. 101, 120603 (2008).
  34. D. G. Joshi and M. Campisi, Quantum Hertz Entropy Increase in a Quenched Spin Chain, Eur. Phys. J. B 86, 157 (2013).
  35. P. Smacchia and A. Silva, Work Distribution and Edge Singularities for Generic Time-Dependent Protocols in Extended Systems, Phys. Rev. E 88, 042109 (2013).
  36. E. Mascarenhas, H. Bragança, R. Dorner, M. França Santos, V. Vedral, K. Modi, and J. Goold, Work and Quantum Phase Transitions: Quantum Latency, Phys. Rev. E 89, 062103 (2014).
  37. A. Carlisle, L. Mazzola, M. Campisi, J. Goold, F. L. Semião, A. Ferraro, F. Plastina, V. Vedral, G. De Chiara, and M. Paternostro, Out of Equilibrium Thermodynamics of Quantum Harmonic Chains, arXiv:1403.0629.
  38. T. J. G. Apollaro, G. Francica, M. Paternostro, and M. Campisi, Work Statistics, Irreversible Heat and Correlations Build-up in Joining Two Spin Chains, arXiv:1406.0648.
  39. P. Talkner, P. Hänggi, and M. Morillo, Microcanonical Quantum Fluctuation Theorems, Phys. Rev. E 77, 051131 (2008).

  40. In what follows, we indicate physical quantities related to thermal states with a subscript G (Gibbs). In Sec. V, instead we are interested only in the thermal state, so we will drop the subscript.

  41. A. B. Migdal, Qualitative Methods in Quantum Theory (Addison-Wesley, Reading, MA, 1989).
  42. P. Jurcevic, B. P. Lanyon, P. Hauke, C. Hempel, P. Zoller, R. Blatt, and C. F. Roos, Quasiparticle Engineering and Entanglement Propagation in a Quantum Many-Body System, Nature (London) 511, 202 (2014); M. Cheneau, P. Barmettler, D. Poletti, M. Endres, P. Schau, T. Fukuhara, C. Gross, I. Bloch, C. Kollath, and S. Kuhr, Light-Cone-Like Spreading of Correlations in a Quantum Many-Body System, 481, 484 (2012); S. Trotzky, Y.-A. Chen, A. Flesch, I. P. McCulloch, U. Schollwöck, J. Eisert, and I. Bloch, Probing the Relaxation Towards Equilibrium in an Isolated Strongly Correlated One-Dimensional Bose Gas, Nat. Phys. 8, 325 (2012); S. Braun, M. Friesdorf, S. S. Hodgman, M. Schreiber, J. P. Ronzheimer, A. Riera, M. del Rey, I. Bloch, J. Eisert, and U. Schneider, Emergence of Coherence and the Dynamics of Quantum Phase Transitions, arXiv:1403.7199.
  43. A. Engel and R. Nolte, Jarzynski Equation for a Simple Quantum System: Comparing Two Definitions of Work, Europhys. Lett. 79, 10003 (2007).
  44. K. Binder, Finite Size Scaling Analysis of Ising Model Block Distribution Functions, Z. Phys. B 43, 119 (1981).
  45. K. Binder, Critical Properties from Monte Carlo Coarse Graining and Renormalization, Phys. Rev. Lett. 47, 693 (1981).
  46. F. Schwabl, Statistical Mechanics (Springer-Verlag, Berlin, 2005)
  47. D. C. Mattis, The Theory of Magnetism: Statics and Dynamics (Springer-Verlag, Berlin, 1988); W. Nolting and A. Ramakanth, Quantum Theory of Magentism (Springer-Verlag, Berlin, 2010).
  48. L. E. Reichl, A Modern Course in Statistical Physics, 2nd ed. (Wiley-VCH, Berlin, 1997).
  49. In the general case, the magnetic susceptibility is given by the formula that can be found in R. Kubo, Statistical-Mechanical Theory of Irreversible Processes. I. General Theory and Simple Applications to Magnetic and Conduction Problems, J. Phys. Soc. Jpn. 12, 570 (1957).
  50. F. Hellmann, M. Mondragon, A. Perez, and C. Rovelli, Multiple-Event Probability in General-Relativistic Quantum Mechanics, Phys. Rev. D 75, 084033 (2007).
  51. G. Watanabe, B. Prasanna Venkatesh, and P. Talkner, Generalized Energy Measurements and Modified Transient Quantum Fluctuation Theorems, Phys. Rev. E 89, 052116 (2014).
  52. A. E. Allahverdyan, Non-equilibrium Quantum Fluctuations of Work, arXiv:1404.4190.
  53. M. Campisi, Quantum Fluctuation Relations for Ensembles of Wave Functions, New J. Phys. 15, 115008 (2013).
  54. S. Deffner, Quantum Entropy Production in Phase Space, Europhys. Lett. 103, 30001 (2013).
  55. B. Leggio, A. Napoli, A. Messina, and H.-P. Breuer, Entropy Production and Information Fluctuations Along Quantum Trajectories, Phys. Rev. A 88, 042111 (2013).
  56. J. M. Horowitz, Quantum-Trajectory Approach to the Stochastic Thermodynamics of a Forced Harmonic Oscillator, Phys. Rev. E 85, 031110 (2012).
  57. F. W. J. Hekking and J. P. Pekola, Quantum Jump Approach for Work and Dissipation in a Two-Level System, Phys. Rev. Lett. 111, 093602 (2013).
  58. A. E. Allahverdyan and Th. M. Nieuwenhuizen, Fluctuations of Work from Quantum Subensembles: The Case Against Quantum Work-Fluctuation Theorems, Phys. Rev. E 71, 066102 (2005).
  59. G. N. Bochkov and Y. E. Kuzovlev, Nonlinear Fluctuation-Dissipation Relations and Stochastic Models in Nonequilibrium Thermodynamics: I. Generalized Fluctuation-Dissipation Theorem, Physica (Amsterdam) 106A, 443 (1981).
  60. F. Schlögl, Zur statistischen Theorie der Entropieproduktion in nicht abgeschlossenen Systemen, Z. Phys. 191, 81 (1966).
  61. S. Vaikuntanathan and C. Jarzynski, Dissipation and Lag in Irreversible Processes, Europhys. Lett. 87, 60005 (2009).
  62. S. Deffner and E. Lutz, Generalized Clausius Inequality for Nonequilibrium Quantum Processes, Phys. Rev. Lett. 105, 170402 (2010).
  63. S. Sotiriadis, A. Gambassi, and A. Silva, Statistics of the Work Done by Splitting a One-Dimensional Quasicondensate, Phys. Rev. E 87, 052129 (2013).

  64. For a classical instantaneous quench H0(z)→H1(z)=H0(z)+ΔE(z), the classical work generating function reads as follows [17]: Gcl(u)=∫dzexp[iu(H1(z)−H0(z))]exp[−βH0(z)]/Z=∫dzexp[iuΔE(z)]exp[−βH0(z)/Z], which is the generating function of the statistics of the classical observable ΔE(z).

  65. A. Imparato and L. Peliti, Work-Probability Distribution in Systems Driven out of Equilibrium, Phys. Rev. E 72, 046114 (2005).
  66. A. Cuccoli, A. Taiti, R. Vaia, and P. Verrucchi, Extracting Signatures of Quantum Criticality in the Finite-Temperature Behavior of Many-Body Systems, Phys. Rev. B 76, 064405 (2007).
  67. J. Thompson, M. Gu, K. Modi, and V. Vedral, Quantum Computing with Black-Box Subroutines, arXiv:1310.2927v5.
  68. S. Sachdev, Quantum Phase Transitions (Cambridge University Press, Cambridge, England, 1998).
  69. H. T. Quan, Z. Song, X. F. Liu, P. Zanardi, and C. P. Sun, Decay of Loschmidt Echo Enhanced by Quantum Criticality, Phys. Rev. Lett. 96, 140604 (2006).

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Phys. Rev. X 4, 031029– Published 19 August, 2014
  • Received 28 April 2014
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