Bloch oscillations of Bose-Einstein condensates: Breakdown and revival

We investigate the dynamics of Bose-Einstein condensates in a tilted one-dimensional periodic lattice within the mean-field (Gross-Pitaevskii) description. Unlike in the linear case the Bloch oscillations decay because of nonlinear dephasing. Pronounced revival phenomena are observed. These are analyzed in detail in terms of a simple integrable model constructed by an expansion in Wannier-Stark resonance states. We also briefly discuss the pulsed output of such systems for stronger static fields.

Figure 1

The squared modulus ${\mid \psi (x,t)\mid }^2$ of the wave function for $g=0$ shows the familiar Bloch oscillations.

Figure 2

Expectation values of the position ${⟨x⟩}_t$ and width $\Delta x_t$ for the wave function shown in Fig. 1.

Figure 3

Same as Fig. 1, but for a nonlinear interaction $g=+5$ (left) and $-5$ (right).

Figure 4

Expectation values of the position ${⟨x⟩}_t$ and width $\Delta x_t$ for the wave function shown in Fig. 3 for a repulsive nonlinearity $g=+5$.

Figure 5

Same as Fig. 4, but for an attractive nonlinearity $g=-5$.

Figure 6

Expectation value ${⟨x⟩}_t$ of the position and width $\Delta x_t$ for a repulsive nonlinearity $g=+10$. The inset shows a magnification of the time interval between $8T_B$ and $16T_B$.

Figure 7

Width $\Delta x_t$ of the wave packet shown in Fig. 6 for a repulsive nonlinearity $g=+10$.

Figure 8

Bloch oscillations: Expectation value of position ${⟨x⟩}_t$ of the wave packet for a moderate nonlinearity $g=1$. The propagation was done with the split-operator method (dashed line) respectively with approximation (15) (solid line).

Figure 9

As Fig. 8, but for a strong nonlinearity $g=10$.

Figure 10

Bloch oscillations for a strong nonlinearity $g=10$. The propagation was done with the split-operator method (dashed line) and with approximation (18) (solid line).

Figure 11

Bloch oscillations in momentum space. The wave function $\mid \psi (k,t)\mid$ shown for $t=0$ (solid line) moves uniformly under the envelope of the Wannier-Stark function $\mid {\Psi }_0\left(k\right)\mid$ (dashed line).

Figure 12

Time evolution of the function $\tilde{C}(k,t)$ in Eq. (22) for $g=10$. The function $\tilde{C}$ is scaled as $\tilde{C}(0,0)=1$.

Figure 13

Function $\mid \tilde{C}(k,t)\mid$ [Eq. (27)] for $gt=90T_B$ and $\gamma =0.15$. The integral was evaluated numerically (solid line) and using the stationary phase method (dashed line).

Figure 14

Pulsed output for different nonlinearities $g=0$, $-5$, 5, and 10 (from top to bottom). The wave function ${\mid \psi (k,t)\mid }^2$ is displayed for $t=9.1T_B$.