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arXiv:2006.09424 · Physical Review A

Nonequilibrium dynamics of the anyonic Tonks-Girardeau gas at finite temperature

Ovidiu I. Pâţu

Published Oct 2, 2020• 12 citations
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Abstract

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.

DOI
10.1103/physreva.102.043303

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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.
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 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 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.
Using a scaling transformation we exactly determine the dynamics of an harmonically confined Tonks-Girardeau gas under arbitrary time variations of the trap frequency. We show how during a one-dimensional expansion a ``dynamical fermionization'' occurs as the momentum distribution rapidly approaches an ideal Fermi gas distribution, and that under a sudden change of the trap frequency the gas undergoes undamped breathing oscillations displaying alternating bosonic and fermionic character in momentum space. The absence of damping in the oscillations is a peculiarity of the truly Tonks regime.
We summarize in the present work exact results obtained for Tonks-Girardeau gases on one-dimensional optical lattices both for the ground state and nonequilibrium dynamics. On the theoretical side, impenetrable bosons offer the opportunity to study strongly interacting systems in one-dimensional lattices exactly, by means of the Jordan-Wigner transformation, and hence contribute to the topic of strong correlations at the center of interest in both condensed matter physics and quantum gases. This motivation is further enhanced by recent experimental realizations of such systems with ultracold atoms. After having shown their universal properties in equilibrium, we concentrate on their nonequilibrium dynamics. It will be shown that, starting from a pure Fock state, quasi-long-range correlations develop dynamically and lead to the formation of quasicondensates with a momentum determined by the underlying lattice. We expect this effect to be relevant for atom lasers with full control of the wavelength. Then, we will show that the free evolution of an initially confined Tonks-Girardeau gas leads to a momentum distribution that approaches at long times that of the equivalent fermionic system, giving rise to a bosonic gas with a Fermi edge, and hence a fermionization that can only be obtained out of equilibrium. Remarkably, although the momentum distribution function of the Tonks-Girardeau gas becomes equal to the one of the fermions, no loss in coherence is observed in the system, as reflected by a large occupation of eigenstates of the one-particle density matrix.
Abstract We provide a thorough characterisation of the zero-temperature one-particle density matrix of trapped interacting anyonic gases in one dimension, exploiting recent advances in the field theory description of spatially inhomogeneous quantum systems. We first revisit homogeneous anyonic gases with point-wise interactions. In the harmonic Luttinger liquid expansion of the one-particle density matrix for finite interaction strength, the non-universal field amplitudes were not yet known. We extract them from the Bethe Ansatz formula for the field form factors, providing an exact asymptotic expansion of this correlation function, thus extending the available results in the Tonks–Girardeau limit. Next, we analyse trapped gases with non-trivial density profiles. By applying recent analytic and numerical techniques for inhomogeneous Luttinger liquids, we provide exact expansions for the one-particle density matrix. We present our results for different confining potentials, highlighting the main differences with respect to bosonic gases.