Tunneling dynamics of a few bosons in a double well

We study few-boson tunneling in a one-dimensional double well. As we pass from weak interactions to the fermionization limit, the Rabi oscillations first give way to highly delayed pair tunneling (for medium coupling), whereas for very strong correlations multiband Rabi oscillations emerge. All this is explained on the basis of the exact few-body spectrum and without recourse to the conventional two-mode approximation. Two-body correlations are found essential to the understanding of the different tunnel mechanisms. The investigation is complemented by discussing the effect of skewing the double well, which offers the possibility to access specific tunnel resonances.

Figure 1

(Color online) Two-atom dynamics. (a) Population of the right-hand well over time, $p_{\text{R}}\left(t\right)$, for different interaction strengths $g=0$ (—), $g=0.2$ (---), $g=4.7$ $(\cdots )$, and $g=25$ $\left(\text{−}\cdot \text{−}\right)$. (b) Snapshots of the one-body density $\rho \left(x\right)$ for different times $t$ in the strongly correlated case $g=25$. (All quantities in harmonic-oscillator units throughout, see text.)

Figure 2

(Color online) (a) Single-particle spectrum $\left\{{ϵ}_a\right\}$ of a double well with barrier height $h=8$. (b) Two-particle spectrum as a function of the interaction strength $g$. Inset: Doublet formation with increasing $g$.

Figure 3

(Color online) Top: Probability $p_2\left(t\right)$ of finding two atoms in the same well for $g=0,0.2,25$. Bottom: Snapshots of two-body correlation function ${\rho }_2\left(x_1,x_2\right)$ at equilibrium points, $\delta \left(t_{\ast }\right)=0$, for $g=0\left(t_{\ast }=44\right)$, $g=0.2\left(t_{\ast }=128\right)$, and $g=25\left(t_{\ast }=53\right)$—from left to right.

Figure 4

(Color online) Time evolution $p_{\text{R}}\left(t\right)$ of (a) $N=3$, (b) $N=4$ atoms initially in one well. Shown are the coupling strengths $g=0$ (—), $g=0.2$ (---), $g=4.7$ $(\cdots )$, and $g=25$ $\left(\text{−}\cdot \text{−}\right)$. Insets: Long-time behavior for $g=0.2$ (the longer period) and $g=4.7$. (Observe the different time scales in both insets.)

Figure 5

(Color online) Partial-imbalance effects in the fermionization limit $\left(g=25\right)$. (a) Small-imbalance oscillations (scenario 1) for $N=3,4$ atoms. Plotted is the population of the right-hand well, $p_{\text{R}}\left(t\right)$. Bottom: Density evolution $\rho \left(x;t\right)$ for $N-1=2$ (b) and $N-1=3$ atoms (c) initially in the right-hand well if exactly one atom is present on the left (scenario 2).

Figure 6

(Color online) Maximum single-atom probability ${\overline{p}}_1$ as a function of the tilt parameter $d$. Solid line, uncorrelated tunneling, $g=0$; dashed line, correlated tunneling, $g=0.2$. Inset: Near the fermionization limit, $g=25$. Note that the resonances are not symmetric in $d$ owing to the unsymmetric initial state $\Psi \left(0\right)$.

Figure 7

Single-particle spectrum $\left\{{ϵ}_a\left(d\right)\right\}$ of a double well as a function of the asymmetry $d$.

Figure 8

(Color online) Two-body spectrum $\left\{E_m\left(d\right)\right\}$ in a tilted double well. (a) $g=0,0.2$ and (b) $g=25$.