Effective three-body interactions for bosons in a double-well confinement

When describing the low-energy physics of bosons in a double-well potential with a high barrier between the wells and sufficiently weak atom-atom interactions, one can, to a good approximation, ignore the high-energy states and thereby obtain an effective two-mode model. Here we show that the regime in which the two-mode model is valid can be extended by adding an on-site three-body interaction term and a three-body interaction-induced tunneling term to the two-mode Hamiltonian. These terms effectively account for virtual transitions to the higher-energy states. We determine appropriate strengths of the three-body terms by an optimization of the minimal value of the wave-function overlap within a certain time window. Considering different initial states with three or four atoms, we find that the resulting model accurately captures the dynamics of the system for parameters where the two-mode model without the three-body terms is poor. We also investigate the dependence of the strengths of the three-body terms on the barrier height and the atom-atom interaction strength. The optimal three-body interaction strengths depend on the initial state of the system.

Figure 1

The fidelity ${\mathcal{F}}_{\min }$ as a function of the coefficients $W$ and $Q$, for the case $\lambda =1,g=0.7E_R/k$. A single, clear peak is visible, with the maximum fidelity corresponding to the point $W_0\approx -0.0147,Q_0\approx 0.0068$.

Figure 2

The time evolution of the population in the right well $N_R\left(t\right)$ as governed by the full Hamiltonian $\widehat{\mathcal{H}}$ (black, solid line), the standard two-mode Hamiltonian ${\widehat{\mathcal{H}}}_{\mathtt{2}}\mathtt{mode}$ (red, dashed), and the extended two-mode Hamiltonian with three-body interactions ${\widehat{\mathcal{H}}}_{\mathtt{eff}}$ (blue, dotted). The optimal three-body parameters $W_0,Q_0$ are determined by maximising ${\mathcal{F}}_{\min }$. Note, that appropriate three-body terms lead to very good predictions of the exact dynamics. Coefficients $(W_0,Q_0)$ and interaction strength $g$ are given in units of $E_R$ and $E_R/k$, respectively.

Figure 3

The time evolution of the fidelity $\mathcal{F}\left(t\right)$ for the standard two-mode Hamiltonian ${\widehat{\mathcal{H}}}_{\mathtt{2}}\mathtt{mode}$ (red, dashed line) and the extended two-mode Hamiltonian with three-body interactions ${\widehat{\mathcal{H}}}_{\mathtt{eff}}$ (blue, solid line), for various interaction strengths in a shallow well ($\lambda =1.0$). The fidelity in the extended ${\widehat{\mathcal{H}}}_{\mathtt{eff}}$ model stays near 1 even for long times, unlike the fidelity of the standard ${\widehat{\mathcal{H}}}_{\mathtt{2}}\mathtt{mode}$ model. Coefficients $(W_0,Q_0)$ and interaction strength $g$ are given in units of $E_R$ and $E_R/k$, respectively.

Figure 4

The values of $W_0/U$ (top) and $Q_0/U$ (bottom) as a function of interaction $g$ and barrier height $\lambda$ for a system initially prepared in the state $|\mathtt{0},\mathtt{3}\rangle$. In the considered range of interactions a linear regression of $W_0/U$ indicates that $W_0$ is quadratic in $g$. Inset: The slope of a linear fit to $W_0/U$ for different barrier heights $\lambda$, for repulsive (black lower line) and attractive (red upper line) interactions. Note that the slope decreases with increasing $\lambda$.

Figure 5

Left panel: The fidelity ${\mathcal{F}}_{\min }$ as a function of $g$ when the dynamics is governed by the standard two-mode Hamiltonian ${\widehat{\mathcal{H}}}_{\mathtt{2}}\mathtt{mode}$ [red dashed (lower) line], the extended two-mode Hamiltonian ${\widehat{\mathcal{H}}}_{\mathtt{eff}}$ (blue solid line), and the extended two-mode Hamiltonian ${\widehat{\mathcal{H}}}_{\mathtt{eff}}$ without interaction-induced tunneling ($Q=0$) [green dashed (upper) line]. For $Q=0$, a value of $W_0$ was chosen that maximizes ${\mathcal{F}}_{\min }$ within the $Q=0$ constraint. Results are shown for various barrier heights $\lambda$. For shallow wells the fidelity of the standard two-mode model ${\widehat{\mathcal{H}}}_{\mathtt{2}}\mathtt{mode}$ drops rapidly close to 0. Three-body on-site interaction improves significantly the accuracy of the model. For deep wells the tunneling correction controlled by $Q_0$ does not play a significant role. Right panel: Time evolution of the right-well population $N_R\left(t\right)$ as governed by the full Hamiltonian $\widehat{\mathcal{H}}$ (black solid line), the Hamiltonian ${\widehat{\mathcal{H}}}_{\mathtt{eff}}$ with (blue dotted line), or without (green dashed line) the single-particle tunneling correction term. The inaccuracy that results from omitting the tunneling correction is apparent for shallow wells.

Figure 6

The values of $W_0$ and $Q_0$ obtained for different initial states $|\mathtt{0},\mathtt{3}\rangle$ (blue line, squares), $|\mathtt{1},\mathtt{2}\rangle$ (red line, crosses), and $|\mathtt{0},\mathtt{4}\rangle$ (green line, circles) and different values of repulsive (top panel) and attractive (bottom panel) interactions ($g$ is in the range $-1.5$ to 1.5 with a step size of 0.1, in units of $E_R/k$). All results are obtained for a specific height of the barrier, $\lambda =1$.