# Finite-temperature dynamics of a Tonks-Girardeau gas in a frequency-modulated harmonic trap

## TL;DR

This paper investigates the out-of-equilibrium finite-temperature dynamics of a harmonically trapped Tonks-Girardeau gas under periodic trap frequency modulation, providing exact solutions and stability analysis through Mathieu's equation.

## Contribution

It offers the first exact solutions for the density and momentum distributions of this system and demonstrates the equivalence of many-body and hydrodynamic stability diagrams via Mathieu's equation.

## Key findings

- Exact solutions for density and momentum distributions.
- Identification of parametric resonances in the system.
- Equivalence of many-body and hydrodynamic stability diagrams.

## Abstract

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 is exactly the same as those obtained from the exact many-body solution.