Experimental observation of a generalized Gibbs ensemble
Detecting multiple temperatures
Most people have an intuitive understanding of temperature. In the context of statistical mechanics, the higher the temperature, the more a system is removed from its lowest energy state. Things become more complicated in a nonequilibrium system governed by quantum mechanics and constrained by several conserved quantities. Langen et al. showed that as many as 10 temperature-like parameters are necessary to describe the steady state of a one-dimensional gas of Rb atoms that was split into two in a particular way (see the Perspective by Spielman).
Abstract
The description of the non-equilibrium dynamics of isolated quantum many-body systems within the framework of statistical mechanics is a fundamental open question. Conventional thermodynamical ensembles fail to describe the large class of systems that exhibit nontrivial conserved quantities, and generalized ensembles have been predicted to maximize entropy in these systems. We show experimentally that a degenerate one-dimensional Bose gas relaxes to a state that can be described by such a generalized ensemble. This is verified through a detailed study of correlation functions up to 10th order. The applicability of the generalized ensemble description for isolated quantum many-body systems points to a natural emergence of classical statistical properties from the microscopic unitary quantum evolution.
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References and Notes
1
C. E. Shannon, W. Weaver, The Mathematical Theory of Communication (Univ. of Illinois Press, Urbana, IL, 1949).
2
Jaynes E. T., Information theory and statistical mechanics. Phys. Rev. 106, 620–630 (1957).
3
Jaynes E. T., Information theory and statistical mechanics. II. Phys. Rev. 108, 171–190 (1957).
4
K. Huang, Statistical Mechanics (Wiley, New York, 1987).
5
Polkovnikov A., Sengupta K., Silva A., Vengalattore M., Colloquium: Nonequilibrium dynamics of closed interacting quantum systems. Rev. Mod. Phys. 83, 863–883 (2011).
6
Gring M., Kuhnert M., Langen T., Kitagawa T., Rauer B., Schreitl M., Mazets I., Smith D. A., Demler E., Schmiedmayer J., Relaxation and prethermalization in an isolated quantum system. Science 337, 1318–1322 (2012).
7
Ronzheimer J. P., Schreiber M., Braun S., Hodgman S. S., Langer S., McCulloch I. P., Heidrich-Meisner F., Bloch I., Schneider U., Expansion dynamics of interacting bosons in homogeneous lattices in one and two dimensions. Phys. Rev. Lett. 110, 205301 (2013).
8
Caux J.-S., Essler F. H., Time evolution of local observables after quenching to an integrable model. Phys. Rev. Lett. 110, 257203 (2013).
9
Calabrese P., Essler F. H. L., Fagotti M., Quantum quench in the transverse-field Ising chain. Phys. Rev. Lett. 106, 227203 (2011).
10
Kinoshita T., Wenger T., Weiss D. S., A quantum Newton’s cradle. Nature 440, 900–903 (2006).
11
Rigol M., Dunjko V., Yurovsky V., Olshanii M., Relaxation in a completely integrable many-body quantum system: An ab initio study of the dynamics of the highly excited states of 1D lattice hard-core bosons. Phys. Rev. Lett. 98, 050405 (2007).
12
Rigol M., Breakdown of thermalization in finite one-dimensional systems. Phys. Rev. Lett. 103, 100403 (2009).
13
Cazalilla M. A., Effect of suddenly turning on interactions in the Luttinger model. Phys. Rev. Lett. 97, 156403 (2006).
14
Caux J.-S., Konik R. M., Constructing the generalized Gibbs ensemble after a quantum quench. Phys. Rev. Lett. 109, 175301 (2012).
15
Rigol M., Dunjko V., Olshanii M., Thermalization and its mechanism for generic isolated quantum systems. Nature 452, 854–858 (2008).
16
Kollar M., Wolf F. A., Eckstein M., Generalized Gibbs ensemble prediction of prethermalization plateaus and their relation to nonthermal steady states in integrable systems. Phys. Rev. B 84, 054304 (2011).
17
Vosk R., Altman E., Many-body localization in one dimension as a dynamical renormalization group fixed point. Phys. Rev. Lett. 110, 067204 (2013).
18
Lieb E. H., Liniger W., Exact analysis of an interacting Bose gas. I. The general solution and the ground state. Phys. Rev. 130, 1605–1616 (1963).
19
V. Korepin, N. Bogoliubov, A. Izergin, Quantum Inverse Scattering Method and Correlation Functions (Cambridge Univ. Press, Cambridge, 1993).
20
Kuhnert M., Geiger R., Langen T., Gring M., Rauer B., Kitagawa T., Demler E., Adu Smith D., Schmiedmayer J., Multimode dynamics and emergence of a characteristic length scale in a one-dimensional quantum system. Phys. Rev. Lett. 110, 090405 (2013).
21
Langen T., Geiger R., Kuhnert M., Rauer B., Schmiedmayer J., Local emergence of thermal correlations in an isolated quantum many-body system. Nat. Phys. 9, 640–643 (2013).
22
Petrov D. S., Shlyapnikov G. V., Walraven J. T. M., Regimes of quantum degeneracy in trapped 1D gases. Phys. Rev. Lett. 85, 3745–3749 (2000).
23
J. Reichel, V. Vuletic, Eds., Atom Chips (Wiley-VCH, Berlin, 2011).
24
Schumm T., Hofferberth S., Andersson L. M., Wildermuth S., Groth S., Bar-Joseph I., Schmiedmayer J., Krüger P., Matter-wave interferometry in a double well on an atom chip. Nat. Phys. 1, 57–62 (2005).
25
See supplementary materials on Science Online.
26
Betz T., Manz S., Bücker R., Berrada T., Koller Ch., Kazakov G., Mazets I. E., Stimming H. P., Perrin A., Schumm T., Schmiedmayer J., Two-point phase correlations of a one-dimensional bosonic Josephson junction. Phys. Rev. Lett. 106, 020407 (2011).
27
Berges J., Borsányi S., Wetterich C., Prethermalization. Phys. Rev. Lett. 93, 142002 (2004).
28
Gasenzer T., Berges J., Schmidt M. G., Seco M., Nonperturbative dynamical many-body theory of a Bose-Einstein condensate. Phys. Rev. A 72, 063604 (2005).
29
Berges J., Gasenzer T., Quantum versus classical statistical dynamics of an ultracold Bose gas. Phys. Rev. A 76, 033604 (2007).
30
Eckstein M., Kollar M., Werner P., Thermalization after an interaction quench in the Hubbard model. Phys. Rev. Lett. 103, 056403 (2009).
31
Kitagawa T., Imambekov A., Schmiedmayer J., Demler E., The dynamics and prethermalization of one-dimensional quantum systems probed through the full distributions of quantum noisez. New J. Phys. 13, 073018 (2011).
32
Bistritzer R., Altman E., Intrinsic dephasing in one-dimensional ultracold atom interferometers. Proc. Natl. Acad. Sci. U.S.A. 104, 9955–9959 (2007).
33
Kitagawa T., Pielawa S., Imambekov A., Schmiedmayer J., Gritsev V., Demler E., Ramsey interference in one-dimensional systems: The full distribution function of fringe contrast as a probe of many-body dynamics. Phys. Rev. Lett. 104, 255302 (2010).
34
P. Barmettler, C. Kollath, http://arxiv.org/abs/1312.5757 (2013).
35
I. G. Hughes, T. Hase, Measurements and Their Uncertainties (Oxford Univ. Press, Oxford, 2010).
36
Hübener R., Mari A., Eisert J., Wick’s theorem for matrix product states. Phys. Rev. Lett. 110, 040401 (2013).
37
Berrada T., van Frank S., Bücker R., Schumm T., Schaff J. F., Schmiedmayer J., Integrated Mach-Zehnder interferometer for Bose-Einstein condensates. Nat. Commun. 4, 2077 (2013).
38
Lesanovsky I., Schumm T., Hofferberth S., Andersson L., Krüger P., Schmiedmayer J., Adiabatic radio-frequency potentials for the coherent manipulation of matter waves. Phys. Rev. A 73, 033619 (2006).
39
Adu Smith D., Gring M., Langen T., Kuhnert M., Rauer B., Geiger R., Kitagawa T., Mazets I., Demler E., Schmiedmayer J., Prethermalization revealed by the relaxation dynamics of full distribution functions. New J. Phys. 15, 075011 (2013).
40
Whitlock N. K., Bouchoule I., Relative phase fluctuations of two coupled one-dimensional condensates. Phys. Rev. A 68, 053609 (2003).
41
Geiger R., Langen T., Mazets I., Schmiedmayer J., Local relaxation and light-cone-like propagation of correlations in a trapped one-dimensional Bose gas. New J. Phys. 16, 053034 (2014).
42
Meyer H.-D., Manthe U., Cederbaum L. S., The multi-configurational time-dependent Hartree approach. Chem. Phys. Lett. 165, 73–78 (1990).
43
Alon O. E., Streltsov A. I., Cederbaum L. S., Multiconfigurational time-dependent Hartree method for bosons: Many-body dynamics of bosonic systems. Phys. Rev. A 77, 033613 (2008).
44
Grond J., Schmiedmayer J., Hohenester U., Optimizing number squeezing when splitting a mesoscopic condensate. Phys. Rev. A 79, 021603 (2009).
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Published In
Science
Volume 348 | Issue 6231
10 April 2015
10 April 2015
Copyright
Copyright © 2015, American Association for the Advancement of Science.
Submission history
Received: 5 June 2014
Accepted: 2 March 2015
Published in print: 10 April 2015
Acknowledgments
We acknowledge discussions with E. Demler, E. Dalla Torre, K. Agarwal, J. Berges, M. Karl, V. Kasper, I. Bouchoule, M. Cheneau, and P. Grisins. Supported by the European Union (SIQS and ERC advanced grant QuantumRelax), the Austrian Science Fund (FWF) through the Doctoral Programme CoQuS (W1210) (B.R. and T.S.), the Lise Meitner Programme M1423 (R.G.) and project P22590-N16 (I.E.M.), Deutsche Forschungsgemeinschaft grant GA677/7,8 (T.G.), the University of Heidelberg Center for Quantum Dynamics, Helmholtz Association grant HA216/EMMI, and NSF grant PHY11-25915. T.L., T.G., and J.S. thank the Kavli Institute for Theoretical Physics, Santa Barbara, for its hospitality.
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