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Nonequilibrium quantum thermodynamics of the Tonks-Girardeau gas

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

Published May 15, 2020• 0 citations

Abstract

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.

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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.

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.

We report on the derivation of determinant representations for the Green's functions and spectral function of the trapped Tonks-Girardeau gas on the lattice and in the continuum. Our results are valid for any type of statistics of the constituent particles, at zero and finite temperature and arbitrary confining potentials, including nonequilibrium scenarios induced by sudden changes of the external potential. In addition, they are also extremely efficient and easy to implement numerically with the main computational effort being represented by the calculation of partial overlaps of the dynamically evolved single-particle wave functions. In the lattice case we show that the spectral function of a system with a strong harmonic potential presents only two singular lines compared with three singular lines in the case of a homogeneous system.

We study the out-of-equilibrium dynamics of a finite-temperature harmonically trapped Tonks-Girardeau gas induced by periodic modulation of the trap frequency. We give explicit exact solutions for the real-space density and momentum distributions of this interacting many-body system and characterize the stability diagram of the dynamics by mapping the many-body solution to the solution and stability diagram of Mathieu's differential equation. The mapping allows one to deduce the exact structure of parametric resonances in the parameter space characterized by the driving amplitude and frequency of the modulation. Furthermore, we analyze the same problem within the finite-temperature hydrodynamic approach and show that the respective solutions to the hydrodynamic equations can be mapped to the same Mathieu equation. Accordingly, the stability diagram and the structure of resonances following from the hydrodynamic approach are exactly the same as those obtained from the exact many-body solution.

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.