Sudden Expansion of a One-Dimensional Bose Gas from Power-Law Traps

We analyze free expansion of a trapped one-dimensional Bose gas after a sudden release from the confining trap potential. By using the stationary phase and local density approximations, we show that the long-time asymptotic density profile and the momentum distribution of the gas are determined by the initial distribution of Bethe rapidities (quasimomenta) and hence can be obtained from the solutions to the Lieb-Liniger equations in the thermodynamic limit. For expansion from a harmonic trap, and in the limits of very weak and very strong interactions, we recover the self-similar scaling solutions known from the hydrodynamic approach. For all other power-law traps and arbitrary interaction strengths, the expansion is not self-similar and shows strong dependence of the density profile evolution on the trap anharmonicity. We also characterize dynamical fermionization of the expanding cloud in terms of correlation functions describing phase and density fluctuations.

Figure 1

Illustration of the evolving local quasimomentum distribution $f(k,x;t)=f(k,x-\hslash kt/m;0)$, where $v_{\max }=\hslash k_{\max }/m$.

Figure 2

Examples of the initial density profile $\rho (x)$ (black solid lines) and the quasimomentum distribution $g(k)$ (blue dash-dotted lines), which determines the shape of the asymptotic density profile ${\rho }_{\infty }(x,t)$ and the momentum distribution $n_{\infty }(k,t)$, Eqs. (5) and (7), with $Q$ being the maximum (quasi)momentum. (a) Expansion from a harmonic trap, with the solid and dash-dotted lines corresponding to the numerical results for ${\gamma }_0=1$. The analytic results (coinciding with the hydrodynamic self-similar solutions) in the weakly (${\gamma }_0\ll 1$) and strongly (${\gamma }_0\to \infty$) interacting regimes are shown, respectively, by the dashed-magenta and dotted-red lines. The numerical results in these regimes, obtained for ${\gamma }_0=2.5\times {10}^{-4}$ and ${\gamma }_0=200$, are indistinguishable from the respective analytic curves and are omitted from the graphs for clarity. (b) Main curves are for a highly anharmonic trap with $\nu =14$ [27] and ${\gamma }_0\ll 1$; the semicircle (dashed red line) corresponds to the limiting behavior of $g(k)$ for a box potential ($\nu \to \infty$). (c) Strongly interacting regime, for the same $\nu =14$.