Lieb-Liniger model of a dissipation-induced Tonks-Girardeau gas

We show that strong inelastic interactions between bosons in one dimension create a Tonks-Girardeau gas, much as in the case of elastic interactions. We derive a Markovian master equation that describes the loss caused by the inelastic collisions. This yields a loss rate equation and a dissipative Lieb-Liniger model for short times. We obtain an analytic expression for the pair correlation function in the limit of strong dissipation. Numerical calculations show how a diverging dissipation strength leads to a vanishing of the actual loss rate and renders an additional elastic part of the interaction irrelevant.

Figure 1

Real and imaginary parts of $k_1$ for the $N=6$ ground state with $\mathrm{Re}\left(\gamma \right)=0$. The same data are plotted in a different way as the top-left curve in Fig. 2a.

Figure 2

Parametric plot, showing the $k_j$ in the complex $k$ plane for the $N=6$ ground state. Each curve shows a trajectory traced by $k_j$ when scanning the parameter $\mathrm{Im}\left(\gamma \right)$ from 0 to $-\infty$. In all cases, the curves begin for $\mathrm{Im}\left(\gamma \right)=0$ somewhere not too far from the coordinate origin and converge for $\mathrm{Im}\left(\gamma \right)\to -\infty$ to Girardeau’s solution (shown as filled circles). (a) $\mathrm{Re}\left(\gamma \right)=0$. For $\gamma =0$ all $k_j=0$. (b) $\mathrm{Re}\left(\gamma \right)=0.25$. For $\mathrm{Im}\left(\gamma \right)=0$, the $k_j$ are approximately equally spaced on the real axis. (c) $\mathrm{Re}\left(\gamma \right)=-0.25$. For $\mathrm{Im}\left(\gamma \right)=0$, the $k_j$ are approximately equally spaced on the imaginary axis, indicating a bound state with an energy that is real and negative. However, as $\mid \mathrm{Im}\left(\gamma \right)\mid$ grows and becomes larger than $\mid \mathrm{Re}\left(\gamma \right)\mid$, the finite $\mathrm{Re}\left(\gamma \right)$ becomes irrelevant and Girardeau’s solution is reached for $\mid \gamma \mid \to \infty$.

Figure 3

Pair-correlation function and loss rate of the ground state as a function of the dissipation strength for $\mathrm{Re}\left(\gamma \right)=0$ and $N=100$. (a) The pair correlation function $g^{\left(2\right)}$ (solid line) equals $(N-1)∕N\approx 1$ at $\gamma =0$ and vanishes for $\gamma \to -i\infty$. For $\mid \gamma \mid ⪢1$ it is well approximated by Eq. (44) (dotted line). (b) The dimensionless quantity $-\mathrm{Im}\left(\gamma \right)g^{\left(2\right)}$ is proportional to the particle loss rate; see Eq. (47). For small $\mid \gamma \mid$ the loss rate increases linearly with $\mathrm{Im}\left(\gamma \right)$ because $g^{\left(2\right)}$ is approximately constant. But for large $\mid \gamma \mid$ the suppression of $g^{\left(2\right)}\propto {\mid \gamma \mid }^{-2}$ dominates and the loss rate vanishes as $1∕\mid \gamma \mid$. This is remarkable: the loss suppresses itself.