Interaction-induced Bloch oscillation in a harmonically trapped and fermionized quantum gas in one dimension

Motivated by a recent experiment by Meinert et al. [arXiv:1608.08200], we study the dynamics of an impurity moving in the background of a harmonically trapped one-dimensional Bose gas in the hard-core limit. We show that due to the hidden “lattice” structure of background bosons, the impurity effectively feels a quasiperiodic potential via impurity-boson interactions that can drive the Bloch oscillation under an external force, even in the absence of real lattice potentials. Meanwhile, the inhomogeneous density of trapped bosons imposes an additional harmonic potential to the impurity, resulting in a similar oscillation dynamics but with a different period and amplitude. We show that the sign and strength of the impurity-boson coupling can significantly affect the above two potentials in determining the impurity dynamics.

Figure 1

Schematic of the lattice potential $V\left(x\right)=g_{ib}{\sum }_i{\rho }_i\left(x\right)$ for an impurity moving in the background of $N$ hard-core bosons. ${\rho }_i\left(x\right)(i=1,2,...,N)$ is the density distribution of the $i\mathrm{th}$ ordered boson in the trap, and $g_{ib}$ is the impurity-boson coupling strength. In the plot we take $N=10$, and $g_{ib}<0.a_{ho}$ is the typical length of the harmonic confinement. The dashed line shows the deviation of $V\left(x\right)$ from the ideal lattice potential due to the density inhomogeneity of the bosons in a harmonic trap.

Figure 2

(a) Mean displacement $\langle x\rangle$ and (b) mean momentum $\langle k\rangle$ of the impurity as time evolves. Here ${\tilde{g}}_{ib}=-2,\tilde{F}=0.05$. The exact results [solid lines, using potential $V$ Eq. (5)] are compared with those by replacing $V$ with $V_L+V^{\prime }$ (dashed lines). Here the units of length, momentum, and time, respectively, are $a_{ho},1/a_{ho}$, and $1/{\omega }_{ho}$.

Figure 3

Momentum distribution $n\left(k\right)$ of the impurity on the parameter plane of momentum $k$ and time $t$. Here ${\tilde{g}}_{ib}=-2$; the forces are $\tilde{F}=0.05$ in (a) and 0.3 in (b). The units are the same as used in Fig. 2.

Figure 4

(a1) and (a2) The period and amplitude of $\langle x\rangle$ as functions of attractive $g_{ib}$. (b1) and (b2) The period and amplitude of $\langle k\rangle$ as functions of attractive $g_{ib}$. The results from different models are compared. Here $\tilde{F}=0.05$, and $g_{ib}$ is in the unit of $\left({\omega }_{ho}{a}_{ho}\right)$. Other units are the same as in the caption of Fig. 2.

Figure 5

The same as Fig. 3 but for repulsive $g_{ib}$. In order to compensate the concave potential $V^{\prime }\left(x\right)$, we activate an additional (convex) harmonic potential $V^{\prime \prime }\left(x\right)$ for the impurity with frequency ${\omega }^{\prime \prime }={\omega }^{\prime }+0.2{\omega }_{ho}$.