Impurity transport through a strongly interacting bosonic quantum gas

Using near-exact numerical simulations, we study the propagation of an impurity through a one-dimensional Bose lattice gas for varying bosonic interaction strengths and filling factors at zero temperature. The impurity is coupled to the Bose gas and confined to a separate tilted lattice. The precise nature of the transport of the impurity is specific to the excitation spectrum of the Bose gas, which allows one to measure properties of the Bose gas nondestructively, in principle, by observing the impurity; here we focus on the spatial and momentum distributions of the impurity as well as its reduced density matrix. For instance, we show it is possible to determine whether the Bose gas is commensurately filled as well as the bandwidth and gap in its excitation spectrum. Moreover, we show that the impurity acts as a witness to the crossover of its environment from the weakly to the strongly interacting regime, i.e., from a superfluid to a Mott insulator or Tonks-Girardeau lattice gas, and the effects on the impurity in both of these strongly interacting regimes are clearly distinguishable. Finally, we find that the spatial coherence of the impurity is related to its propagation through the Bose gas.

Figure 1

Schematic diagram for the (a) impurity, (b) Bose gas, and (c) interaction Hamiltonians.

Figure 2

Motion of a single impurity in a tilted lattice with and without a scattering process. (a) The impurity decoupled from any environment will coherently Bloch oscillate at frequency $\Delta /\hslash$, but its expected position will not drift [5, 29]. Here we plot the evolution in time $t$ for just over one and a half Bloch oscillations. The impurity is initially localized in a Wannier state at the middle site of the 101-site lattice. We set the tilt to be $\Delta /J_a=0.1$ and $\langle {\widehat{n}}_{a,i}\rangle$ is the impurity density at lattice site $i$. (b) Introducing a scattering mechanism under the relaxation-time approximation results in the Esaki-Tsu dependence of drift on tilt in Eq. (3). We plot this dependence here.

Figure 3

Transport of an impurity through a bosonic bath. An example of the evolution of the impurity and boson densities for the case $\Delta /J_b=0.3$, $U_b/J_b=2$, $M=101$, and $N=50$. The color map in the $xy$ plane shows the expected impurity density at each lattice site $\langle {\widehat{n}}_{a,i}\rangle$, and similarly for the surface plot and boson density $\langle {\widehat{n}}_{b,i}\rangle$.

Figure 4

Determining the bandwidth and gap from the dependence of displacement on tilt. (a) The dependence of the displacement of the impurity $\langle {\widehat{x}}_a\rangle$ at $t_{\mathrm{sim}}$ as a function of tilt $\Delta$ for three regimes of the Bose gas: the superfluid regime $U_b/J_b=0.1$ with $N=50$, $M=101$ (SF: displacement scaled down by a factor of $0.1$); the Tonks-Girardeau limit $U_b/J_b\to \infty$ with $N=150,M=151$ (TG); and the Mott insulating regime $U_b/J_b=10$ with $N=M=101$ (MI: displacement scaled up by a factor of 100). (b) The same quantity is plotted for several bosonic systems in the Tonks-Girardeau limit, demonstrating that the qualitative shape and the positions of the bandwidth suppressions are independent of the filling (so long as it is incommensurate) of the Bose gas.

Figure 5

Suppression of average drift in the strongly interacting regimes. Here we show the peak displacement (maximized over all tilts) as a function of interaction strength $U_b/J_b$ for commensurate and incommensurate fillings of a 101-site bosonic lattice.

Figure 6

Quasimomentum distribution of the impurity after traveling through the Bose gas. Here $\Delta /J_b=0.063,M=101$ and ${\widehat{n}}_{a,p}$ is the number operator for an impurity quasimomentum of $2\pi p/aM$. (a) The momentum distribution of the impurity at $t_{\mathrm{sim}}$ for a range of interaction strengths $U_b/J_b$ and incommensurate filling $N=50$. (b) The same as (a), but for commensurate filling $N=101$.

Figure 7

Decoherence of an impurity traveling through an incommensurately filled system $N=50,M=101$. (a) Absolute values of the elements of the reduced density matrix for the impurity $|\langle {\widehat{a}}_i^{†}{\widehat{a}}_j\rangle |$ with $U_b/J_b=2$ and $\Delta /J_b=0.1$, calculated at $t_{\mathrm{sim}}$. (b) The same as (a), but for $U_b/J_b=0.1$. (c) Also at $t_{\mathrm{sim}}$ we have plotted the purity of the density matrix $\mathrm{Tr}\left({\widehat{\rho }}_a^2\right)$ as a function of tilt and interaction strength.