# One-dimensional Bose gas driven by a slow time-dependent harmonic trap

## TL;DR

This paper investigates the dynamics of a one-dimensional hard-core boson cloud under slow, time-dependent harmonic trap release, providing analytical solutions and exploring effects like log-periodic modulation during expansion.

## Contribution

It offers exact and approximate analytical solutions for the cloud's evolution under various trap ramp protocols, including a novel case with a time-inverse amplitude variation.

## Key findings

- Derived a typical length scale related to the ramp
- Compared perturbative and exact solutions for linear ramp
- Discovered log-periodic modulation in cloud expansion

## Abstract

We consider the unitary time evolution of a one-dimensional cloud of hard-core bosons loaded on a harmonic trap potential which is slowly released in time with a general ramp $g(t)$. After the identification of a typical length scale $\ell(t)$, related to the time ramp, we focus our attention on the dynamics of the density profile within a first order time-dependent perturbation scheme. In the special case of a linear ramp, we compare the first order predictions to the exact solution obtained through Ermakov-Lewis dynamical invariants. We also obtain an exact analytical solution for a cloud released from a harmonic trap with an amplitude that varies as the inverse of time. In such situation, the typical size of the cloud grows with a power law governed by an exponent that depends continuously on the initial trap frequency. At high enough initial trap amplitude, the exponent acquires an imaginary part that leads to the emergence of a log-periodic modulation of the cloud expansion.