Pulse propagation in the interacting one-dimensional Bose liquid

We study wave propagation in interacting Bose liquid, where the short-range part of the interaction between atoms is of a hard-core type, and its long-range part scales with a distance as a power law. The cases of Coulomb, dipole-dipole and van der Waals interaction are considered. We employ a hydrodynamic approach, based on the exact solution of the Lieb–Liniger model, and study the evolution of a density pulse instantly released from a potential trap. We analyze semiclassical Euler and continuity equations and construct the corresponding Riemann invariants. We supplement our analysis with numerical calculations and discuss experimental applications for ultracold atom experiments.

Figure 1

Setup. Initial density disturbance is created by the application of a potential $U\left(x\right)$ which is then switched off at $t=0$.

Figure 2

Schematic illustration for the characteristics in a unilaterally propagating pulse.

Figure 3

Density profile after the shock for the case of a Coulomb interaction, $\gamma \gg 1$ (top) and $\gamma \ll 1$ (bottom).

Figure 4

Density profile after the shock for the case of a dipole interaction, $\gamma \gg 1$ (top) and $\gamma \ll 1$ (bottom).

Figure 5

Density profile after the shock for the case of a van der Waals interaction, $\gamma \gg 1$ (top) and $\gamma \ll 1$ (bottom).

Figure 6

A schematic setup of a one-dimensional interferometer. The matter wave is emitted with amplitudes ${\mathrm{\Psi }}_1$ and ${\mathrm{\Psi }}_2$ from the opposite endpoints to the screen. A potential trap that induces a pulse is connected to the right arm of the interferometer.

Figure 7

Profile of the velocity field for the dipole interaction, $\gamma \gg 1$ (top) and $\gamma \ll 1$ (bottom).

Figure 8

Profile of the $\theta$ field for the dipole interaction, $\gamma \gg 1$.

Figure 9

Interference pattern for dipole interaction, $\gamma \gg 1$ (top) and $\gamma \ll 1$ (bottom).