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arXiv:2005.07313 · Physical Review A

Nonequilibrium quantum thermodynamics of determinantal many-body systems: Application to the Tonks-Girardeau and ideal Fermi gases

Y. Y. Atas, Arghavan Safavi-Naini, Karen V. Kheruntsyan

Published Oct 9, 2020• 9 citations
arXivHTMLPDF

Abstract

Quantum thermodynamics of a thermal Tonks-Girardeau gas driven by a periodically varying potential is investigated through the determinant representation of the many-body wave function. The approach for calculating the characteristic function of the work distribution could also be applied to a wide range of driven systems.
DOI
10.1103/physreva.102.043312

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We develop a general method for calculating the characteristic function of the work distribution of the Tonks-Girardeau (TG) gas confined to an arbitrary time-dependent trapping potential. We use the determinant representation of the many-body wave function to characterise the nonequilibrium thermodynamics of the TG gas and obtain exact and computationally tractable expressions in terms of Fredholm determinants for the mean work, the work probability distribution function, the nonadiabaticity parameter, and the Loschmidt amplitude. When applied to a harmonically trapped TG gas, our results for the mean work and the nonadiabaticity parameter reduce to those derived previously using an alternative approach. We next propose to use periodic modulation of the trap frequency in order to drive the system to highly non-equilibrium states by taking advantage of the phenomenon of parametric resonance. Under such driving protocol, the nonadiabaticity parameter may reach large values, which indicates a large amount of work being done on the system, as compared to sudden quench protocols considered previously. This scenario is realizable in ultracold atom experiments, aiding fundamental understanding of all thermodynamic properties of the system, in addition to facilitating possible optimization of high-efficiency quantum heat engine or quantum refrigeration cycles.
Describing finite-temperature nonequilibrium dynamics of interacting many-particle systems is a notoriously challenging problem in quantum many-body physics. Here we provide an exact solution to this problem for a system of strongly interacting bosons in one dimension in the Tonks-Girardeau regime of infinitely strong repulsive interactions. Using the Fredholm determinant approach and the Bose-Fermi mapping, we show how the problem can be reduced to a single-particle basis, wherein the finite-temperature effects enter the solution via an effective ``dressing'' of the single-particle wave functions by the Fermi-Dirac occupation factors. We demonstrate the utility of our approach and its computational efficiency in two nontrivial out-of-equilibrium scenarios: collective breathing-mode oscillations in a harmonic trap and collisional dynamics in the Newton's cradle setting involving real-time evolution in a periodic Bragg potential.
We derive an exact description of the nonequilibrium dynamics at finite temperature for the anyonic Tonks-Girardeau gas, extending the results of Atas et al. [Phys. Rev. A 95, 043622 (2017)] to the case of arbitrary statistics. The one-particle reduced density matrix is expressed as the Fredholm minor of an integral operator, with the kernel being the one-particle Green's function of free fermions at finite temperature and the statistics parameter determining the constant in front of the integral operator. We show that the numerical evaluation of this representation using Nystr\"om's method significantly outperforms the other approaches present in the literature when there are no analytical expressions for the overlaps of the wave functions. We illustrate the distinctive features and novel phenomena present in the dynamics of anyonic systems in two experimentally relevant scenarios: the quantum Newton's cradle setting and the breathing oscillations initiated by a sudden change of the trap frequency.
Ultracold quantum gases offer a fascinating playground for exploring and understanding the fundamentals of out-of-equilibrium behaviour of quantum matter. Experimental systems whose many-body dynamics can be described by exactly solvable models play a particularly important role. A paradigmatic example in this realm is the Tonks-Girardeau (TG) gas of impenetrable bosons. Here we investigate and predict a striking dynamical manifestation of the impenetrability--a collective many-body bounce effect. This effect, while being invisible in the evolution of the in-situ density profile of the gas, can be revealed through a nontrivial narrowing of its momentum distribution during breathing-mode oscillations. To demonstrate this, we develop an exact finite-temperature dynamical theory of a harmonically trapped TG gas, thus extending on previous zero-temperature results and opening the way to solving arbitrary dynamical protocols of the TG gas. We identify physical regimes for observing the many-body bounce effect and foresee it as a valuable probe for characterising the nonequilibrium dynamics of interacting quantum fluids.
In quantum many-body systems interactions between particles create correlations and particles can not be described independently. As a consequence the wave function of the system is a complicated object holding a large amount of information and in most cases it is not possible to obtain it exactly or analytically. Studying correlations in the context of strongly interacting onedimensional (1D) systems is interesting because in this regime we can exactly determine many particle wave function through Fermi-Bose (FB) mapping. FB mapping is in the essence of a Tonks Girardeau (TG) gas proposed in 1960, a theoretical model of 1D Bose gas with strongly repulsive contact interactions between particles. TG gas is experimentally realized in 2004 with rubidium atoms conned in tight atomic wave-guides and it is still a topic of experimental studies. Our goal is the study of correlations in TG gas through the formalism of the single particle density matrix. Single particle density matrix, among other important observable's, contains information about the momentum distribution which can be measured in experiments. We explored correlations in dark soliton-like excitations of the TG gas and found that these correlations give rise to a specific form of the momentum distribution which can be used to experimentally detect these states. In addition to the TG gas, FB mapping was used to construct exact wave functions of 1D Bose-Fermi mixtures (fermions are spin polarized). We derived the formula for the single particle density matrix of the mixture and used it to study correlations and momentum distribution of the mixture in the ground state of double well potential. Due to integrability, these systems are convenient to study stability to perturbations of many-body quantum states in different interaction regimes. We explored stability of non-equilibrium dynamics of 1D interacting Bose gases with respect to small spatial noise potential using the concept of Loschmidt echo (quantum fidelity). We used this concept to explore the pinning quantum phase transition. We found that the ground state fidelity and the Loschmidt echo can be used for diagnostics of the pinning quantum phase transition in experimentally relevant conditions.
Thermodynamics of quantum systems out-of-equilibrium is very important for the progress of quantum technologies, however, the effects of many body interactions and their interplay with temperature, different drives and dynamical regimes is still largely unknown.Here we present a systematic study of these interplays in the case of driven Hubbard chains subject to a variety of interaction (from non-interacting to strongly correlated) and dynamical regimes (from sudden quench to quasi-adiabatic), and discuss which effects all these ingredients have on the work extraction and entropy production.As treatment of many-body interacting systems is highly challenging, we introduce a simple approximation which includes, for the average quantum work, many-body interactions only via the initial state, while the dynamics is fully non-interacting.We demonstrate that this simple approximation is surprisingly good for estimating both the average quantum work and the related entropy production, even when many-body correlations are significant.