Lieb-Liniger gas in a constant-force potential

We use Gaudin’s Fermi-Bose mapping operator to calculate exact solutions for the Lieb-Liniger model in a linear (constant-force) potential (the constructed exact stationary solutions are referred to as the Lieb-Liniger-Airy wave functions). The ground-state properties of the gas in the wedgelike trapping potential are calculated in the strongly interacting regime by using Girardeau’s Fermi-Bose mapping and the pseudopotential approach in the $1/c$ approximation ($c$ denotes the strength of the interaction). We point out that quantum dynamics of Lieb-Liniger wave packets in the linear potential can be calculated by employing an $N$-dimensional Fourier transform as in the case of free expansion.

Figure 1

The single-particle density ${\rho }_{B,c}(x)$ (solid black line) of $N=10$ Lieb-Liniger bosons in a wedgelike potential ($c=40$, $\alpha =1$). Dashed blue line shows the density in the Tonks-Girardeau limit.

Figure 2

Evolution of $N=3$ Lieb-Liniger bosons in the linear potential $\alpha x$ ($\alpha =3$) from the ground state of a box with infinitely high walls. Single-particle density in time for various interaction strengths $c$: (a) $c=0.25$, (b) $c=3$, and (c) $c=10$. Red dotted lines are for $t=0$, solid black lines are for $t=1$, and blue dashed lines are for $t=2$.

Figure 3

Evolution of the momentum distribution. The colors and lines for different $c$ and $t$ are identical as in Fig. 2.