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arXiv:1912.09648

Theory of thermalization in an isolated Bose-Einstein condensate

Che-Hsiu Hsueh, Chi-Ho Cheng, Tzyy‐Leng Horng, W. C. Wu

Published Dec 20, 2019• 0 citations
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Abstract

Thermalization in an isolated oscillating Bose-Einstein condensate in a disordered trap is investigated. We show Shannon entropy in $x$ or $p$ representation is the eligible one to describe the thermalization. Besides, we show that multiple scattering with the disorder generates more and more incoherent thermal particles and condensed and thermal particles act as mutual heat bath that results in the thermalization of the whole system. We also demonstrate that Loschmidt's paradox can be resolved in the present system.

DOI
10.48550/arxiv.1912.09648

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Thermalization in an isolated oscillating Bose-Einstein condensate in a disordered trap is investigated. We show Shannon entropy in xxx or ppp representation is the eligible one to describe the thermalization. Besides, we show that multiple scattering with the disorder generates more and more incoherent thermal particles and condensed and thermal particles act as mutual heat bath that results in the thermalization of the whole system. We also demonstrate that Loschmidt's paradox can be resolved in the present system.
Previously we numerically showed that thermalization can occur in an oscillating Bose-Einstein condensate (BEC) with a disordered harmonic trap when healing length ξξξ of the condensate is shorter than the correlation length σDσ_{D}σD​ of the disorder [see, for example, the experiment reported in Phys. Rev. A \textbf{82}, 033603 (2010)]. In this work, we investigate the weakly interacting or Anderson localization regime ξ>σDξ>σ_{D}ξ>σD​ and show that the oscillating BEC can also exhibit a relaxation process from nonequilibrium to equilibrium. In such an isolated quantum system, energy and particle number are conserved and the irreversible evolution towards thermodynamic equilibrium is induced by the disorder. The thermodynamic equilibrium is evidenced by the maximized entropy S[nk]S\left[n_{k}\right]S[nk​] in which the waveaction spectrum nkn_{k}nk​ follows the Rayleigh-Jeans distribution. Besides, unlike a monotonic irreversible process of thermalization to equilibrium, the Fermi-Pasta-Ulam-Tsingou recurrence arises in this system, manifested by the oscillation of the non-equilibrium entropy.
We numerically simulate an oscillating Bose-Einstein condensate in a disordered trap [Phys. Rev. A 82, 033603 (2010)] and the results are in good agreement with the experiment. It shows that the disorder acts as a medium, which results in a relaxation from nonequilibrium to equilibrium, i.e., thermalization. An algebraic localization is realized when the system approaches the equilibrium, and if the system falls into the regime when the healing length of the condensate exceeds the correlation length of the disorder, exponential Anderson localization is to be observed.
Abstract If and how an isolated quantum system thermalizes despite its unitary time evolution is a long‐standing, open problem of many‐body physics. The eigenstate thermalization hypothesis (ETH) postulates that thermalization happens at the level of individual eigenstates of a system's Hamiltonian. However, the ETH requires stringent conditions to be validated, and it does not address how the thermal state is reached dynamically from an initial non‐equilibrium state. We consider a Bose‐Einstein condensate (BEC) trapped in a double‐well potential with an initial population imbalance. We find that the system thermalizes although the initial conditions violate the ETH requirements. We identify three dynamical regimes. After an initial regime of undamped Josephson oscillations, the subsystem of incoherent excitations or quasiparticles (QP) becomes strongly coupled to the BEC subsystem by means of a dynamically generated, parametric resonance. When the energy stored in the QP system reaches its maximum, the number of QPs becomes effectively constant, and the system enters a quasi‐hydrodynamic regime where the two subsystems are weakly coupled. In this final regime the BEC acts as a grand‐canonical heat reservoir for the QP system (and vice versa), resulting in thermalization. We term this mechanism dynamical bath generation (DBG).
A Bose gas in a double-well potential, exhibiting a true Bose-Einstein condensate (BEC) amplitude and initially performing Josephson oscillations, is a prototype of an isolated, nonequilibrium many-body system. We investigate the quasiparticle (QP) creation and thermalization dynamics of this system by solving the time-dependent Keldysh-Bogoliubov equations. We find avalanchelike QP creation due to a parametric resonance between BEC and QP oscillations, followed by slow, exponential relaxation to a thermal state at an elevated temperature, controlled by the initial excitation energy of the oscillating BEC above its ground state. The crossover between the two regimes occurs because of an effective decoupling of the QP and BEC oscillations. This dynamics is analogous to elementary particle creation in models of the early universe. The thermalization in our setup occurs because the BEC acts as a grand canonical reservoir for the quasiparticle system.
We question the validity of the grand canonical ensemble for the description of Bose-Einstein condensation of small ideal Bose gas samples in isolated harmonic traps. While the ground state fraction and the specific heat capacity can be well approximated with the help of the conventional grand canonical arguments, the calculation of the fluctuation of the number of particles contained in the condensate requires a microcanonical approach. Resorting to the theory of restricted partitions of integer numbers, we present analytical and numerical results for such fluctuations in one- and three-dimensional traps, and show that their magnitude is essentially independent of the total particle number.