Irregular Dynamics in a One-Dimensional Bose System

We study many-body quantum dynamics of $\delta$-interacting bosons confined in a one-dimensional ring. Main attention is paid to the transition from the mean-field to the Tonks-Girardeau regime using an approach developed in the theory of interacting particles. We analyze, both analytically and numerically, how the Shannon entropy of the wave function and the momentum distribution depend on time for weak and strong interactions. We show that the transition from regular (quasiperiodic) to irregular (“chaotic”) dynamics coincides with the onset of the Tonks-Girardeau regime. In the latter regime, the momentum distribution of the system reveals a statistical relaxation to a steady state distribution. The transition can be observed experimentally by studying the interference fringes obtained after releasing the trap and letting the boson system expand ballistically.

Figure 1

Entropy as a function of the rescaled time $gt$ for fixed $m=6$ and different values of $n/g$. Data are given for $N=6$. The saturation of $S(t)$ for small values of $n/g$ manifests the onset of the Tonks-Girardeau regime. The dashed line corresponds to the theoretical prediction (2).

Figure 2

Rescaled entropy as a function of $n/g$ for $N=6$ particles and different $m$ as indicated in the inset.

Figure 3

Particle density in the momentum representation for four values of $n/g$, and for different rescaled times $gt=0$, 0.2, 0.4, 0.6, and 0.8 (lines are thicker for later times).

Figure 4

Visibility as a function of time, for four values of rescaled interaction $n/g$. Here is $N=6$ and $m=6$.