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

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 $g\left(t\right)$. After the identification of a typical length scale $\ell \left(t\right)$, 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 situations, the typical size of the cloud grows with a power law governed by an exponent that depends continuously on the initial trap frequency. At a 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. (paper)