Momentum relaxation of a mobile impurity in a one-dimensional quantum gas

We investigate the time evolution of the momentum of an impurity atom injected into a degenerate Tonks-Girardeau gas. We establish that given an initial momentum $p_0$ the impurity relaxes to a steady state with a nonvanishing momentum $p_{\infty }.$ The nature of the steady state is found to depend drastically on whether the masses of the impurity and the host are equal. This is due to multiple coherent scattering processes leading to a resonant interaction between the impurity and the host in the case of equal masses. The dependence of $p_{\infty }$ on $p_0$ remains nontrivial even in the limit of vanishing interaction between the impurity and host particles. In this limit $p_{\infty }\left(p_0\right)$ is found explicitly.

Figure 1

Upper panels: Kinematically allowed regions for a single pairwise scattering in the cases of the light (left) and heavy (right) impurity. The final momentum of the impurity, $k$, is shown vs the initial momentum of the impurity, $q.$ Lower panels: The impurity momentum at infinite time, $p_{\infty },$ as a function of the initial momentum, $p_0$, for the light (left) and heavy (right) impurity. Solid blue line shows an iterative solution (two iterations) of Eq. (5). Shaded area (green online) represents the maximum error: The exact solution of Eq. (5) lies inside this area. Notice a much better convergence of iterations in the case of light impurity.

Figure 2

Overlaps $|\langle {\psi }_{\lambda }|\mathrm{in}\rangle {|}^2$ versus energy $E$ (relative to the in-state energy $E_{\mathrm{in}}$) for $N=135$, $L=405$, $\gamma =3$, and $p_0=1.2k_F$. Several families of states are clearly visible. The diagram in the upper left corner shows schematically the structure of sets $\lambda$ for the top three families, including the dominant family $\mathfrak{s}$ (maroon). The dashed curve corresponds to the asymptotic expression obtained in the limit ${\gamma }^{2}lnN\to 0,$ ${\gamma }^{2}N\to \infty .$ Inset demonstrates the finite-size dependence of $p_{\infty }$ at $p_0=1.6k_F$ and $\gamma =3$.