Momentum distribution of a freely expanding Lieb-Liniger gas

We numerically study free expansion of a few Lieb-Liniger bosons, which are initially in the ground state of an infinitely deep hard-wall trap. Numerical calculation is carried out by employing a standard Fourier transform, as follows from the Fermi-Bose transformation for a time-dependent Lieb-Liniger gas. We study the evolution of the momentum distribution, the real-space single-particle density, and the occupancies of natural orbitals. Our numerical calculation allows us to explore the behavior of these observables in the transient regime of the expansion, where they are nontrivially affected by the particle interactions. We derive analytically (by using the stationary phase approximation) the formula which connects the asymptotic shape of the momentum distribution and the initial state. For sufficiently large times, the momentum distribution coincides (up to a simple scaling transformation) with the shape of the real-space single-particle density (the expansion is asymptotically ballistic). Our analytical and numerical results are in good agreement.

Figure 1

(Color online) Contour plots of ${\left|{\psi }_{B,c}\left(0,x_2,x_3,t\right)\right|}^2$ for $c=1$ at (a) $t=0$ and (b) $t=3$. As the time $t$ increases, the probability density at the hyperplanes where particles are in contact decreases.

Figure 2

(Color online) The lowest natural orbital ${\lambda }_1\left(t\right)$ as a function of time for three values of $c$. Red diamonds (dashed line) is for $c=0.25$, black circles (solid line) for $c=1$, and blue squares (dotted line) for $c=5$; the lines connecting the markers are guides for the eyes. The corresponding horizontal lines without markers denote the asymptotic occupancies calculated from the asymptotic wave functions [13].

Figure 3

(Color online) Evolution of the $x$-space density in time for various interaction strengths $c$: (a) $c=0.25$ at $t=0$ (red dotted line), $t=2$ (solid black line), and $t=4$ (blue dashed line); (b) $c=1$ at $t=0$ (red dotted line), $t=2$ (solid black line), and $t=4$ (blue dashed line); (c) $c=10$ at $t=0$ (red dotted line), $t=1$ (solid black line), $t=3$ (blue dashed line). The asymptotic $x$-space density ${\rho }_{\infty }\left(\xi \right)$ (circles) is plotted as a function of $x=\xi t$ corresponding to the largest time in each subplot.

Figure 4

(Color online) Evolution of the momentum distribution in time for various interaction strengths $c$. The lines and colors for different values of $c$ and $t$ are identical as in Fig. 3. Solid black and blue dashed lines are almost indistinguishable.

Figure 5

(Color online) Asymptotic expansion velocity ${\xi }_{\infty }$ (squares, dashed line) and the square root of the total energy $\sqrt{E}$ (circles, solid line) for various interaction strengths $c$; lines serve as guides for the eyes (see text for details).