Dynamics and symmetries of a repulsively bound atom pair in an infinite optical lattice

We investigate the dynamics of two bosons trapped in an infinite one-dimensional optical lattice potential within the framework of the Bose-Hubbard model and derive an exact expression for the wave function at finite time. As initial condition we chose localized atoms that are separated by a distance of $d$ lattice sites and carry a center-of-mass quasimomentum. An initially localized pair ($d=0$) is found to be more stable as quantified by the pair probability (probability to find two atoms at the same lattice site) when the interaction and/or the center-of-mass quasimomentum is increased. For initially separated atoms ($d\ne 0$) there exists an optimal interaction strength for pair formation. Simple expressions for the wave function, the pair probability, and the optimal interaction strength for pair formation are computed in the limit of infinite time. Whereas the time-dependent wave function differs for values of the interaction strength that differ only by the sign, important observables such as the density and the pair probability do not. With a symmetry analysis this behavior is shown to extend to the $N$-particle level and to fermionic systems. Our results provide a complementary understanding of the recently observed [Winkler et al., Nature (London) 441, 853 (2006)] dynamical stability of atom pairs in a repulsively interacting lattice gas.

Figure 1

Pair probability $\left(P_{\text{pair}}\right)$ as a function of the interaction strength $\left(U\right)$ and time $\left(t\right)$ for $J=1$, $K=0$, $d=0$ (a), and $d=1$ (b). With increasing time $P_{\text{pair}}$ is converging rapidly towards an asymptotic value. For $d=0$ the pair probability increases with the interaction strength, while for $d=1$ there exists an optimal interaction strength for pair formation (except for very short times). See Sec. 2b for more details. All quantities are dimensionless.

Figure 2

Variance $\left({\sigma }_{\text{pair}}\right)$ of the pair probability as a function of the interaction strength $\left(U\right)$ and time $\left(t\right)$ for $J=1$, $K=0$, $d=0$ (a), and $d=1$ (b). Both curves have very similar characteristics, namely, a local maximum for intermediate interaction strengths and the convergence towards a constant value for large times. For $d=1$ the maximum of the pair probability and its variance coincide (see Sec. 2b for more details). Therefore, in this regime a large value of the pair probability goes hand in hand with large fluctuations. All quantities are dimensionless.

Figure 3

Density $\left[\rho \right(r,t\left)\right]$ in the relative coordinate as a function of time $\left(t\right)$ for $U=5$, $J=1$, $K=0$, and $d=0$. It can be seen that there are three wave packets propagating in time. The one propagating to the left and the one propagating to the right (with equal amplitudes due to the bosonic symmetry of the wave function) describe the separation of the two particles. The one with the largest amplitude is centered around the origin and describes the time evolution of the pair. See Sec. 2b for more details. All quantities are dimensionless.

Figure 4

Same as Fig. 3 but for $d=1$. The wave packet in the middle is less pronounced than for $d=0$. Hence, there is a smaller probability to find a pair. See Sec. 2b for more details. All quantities are dimensionless.

Figure 5

Same as Fig. 3 but for $d=5$. In the dynamics with a larger initial distance new effects such as a second splitting of the wave packet appear. See Sec. 2b for more details. All quantities are dimensionless.

Figure 6

Asymptotic pair probability $\left[P_{\text{pair}}^a\right]$ (a) and its variance $\left[{\sigma }_{\text{pair}}^a\right]$ (b) as a function of the effective interaction strength $U_Q$ for $J=1$ and $d=0$ [dashed blue (highest) curve], $d=1$ [solid violet (middle) curve], and $d=2$ [dot-dashed brown (lowest) curve]. The asymptotic pair probability has a local maximum for $d\ne 0$, the asymptotic variance for all initial distances. Since for $d\ne 0$ the asymptotic pair probability is always smaller than $\frac{1}{2}$, its maximum and the maximum of its variance coincide (see Sec. 3 for more details). All quantities are dimensionless.