Bose-Einstein condensates in accelerated double-periodic optical lattices: Coupling and crossing of resonances

We study the properties of coupled linear and nonlinear resonances. The fundamental phenomena and the level crossing scenarios are introduced for a nonlinear two-level system with one decaying state, describing the dynamics of a Bose-Einstein condensate in a mean-field approximation (Gross-Pitaevskii or nonlinear Schrödinger equation). An important application of the discussed concepts is the dynamics of a condensate in tilted optical lattices. In particular the properties of resonance eigenstates in double-periodic lattices are discussed, in the linear case as well as within mean-field theory. The decay is strongly altered, if an additional period-doubled lattice is introduced. Our analytic study is supported by numerical computations of nonlinear resonance states, and future applications of our findings for experiments with ultracold atoms are discussed.

Figure 1

Real (left) and imaginary (right) part of the eigenvalues (3) as a function of $ϵ$ for $\gamma =1$. A type-I crossing is found for $v=1.01>\gamma$ (upper figures), a type-II crossing is found for $v=0.99<\gamma$ (lower figures).

Figure 2

Real (left) and imaginary (right) part of the eigenvalues of the four-level Hamiltonian (4) as a function of $ϵ$ for $\gamma =1$, $v=0.98$, $w=0.04$, and $\delta =0.2$.

Figure 3

Imaginary part of the eigenvalues of the four-level Hamiltonian (4) as a function of $1∕F$ and $\delta =0.02$ (left) and $\delta =0.05$ (right). See text for details.

Figure 4

Real (left) and imaginary (right) part of the eigenvalues (9) as a function of $ϵ$ for $\gamma =1.00$, $c=0.3$, and $v=1.05>\gamma$ (upper figures), $v=1.00=\gamma$ (middle figures), $v=0.95<\gamma$ (lower figures). The linear levels $(c=0)$ are plotted as dotted lines for comparison.

Figure 5

Real (left) and imaginary (right) part of the eigenvalues (9) as a function of $ϵ$ for $\gamma =1.00$, $c=0.9$, and $v=1.01>\gamma$ (upper figures), $v=1.00=\gamma$ (middle figures), $v=0.99<\gamma$ (lower figures).

Figure 6

Eigenenergies (upper panel) and decay rates (lower panel) of the two most stable Wannier-Stark ladders for the potential $V\left(x\right)=\cos x$.

Figure 7

(Color online) Resonant tunneling in a double-periodic Wannier-Stark system. Decay rates of the four most stable resonance for $V_0=2$, $\varphi =0$, and $\delta =0$ (dashed black line), $\delta =0.05$ (thin red lines), and $\delta =0.1$ (thick blue lines).

Figure 8

(Color online) Explanation of resonantly enhanced tunneling (RET). See text for details.

Figure 9

(Anti)Crossing of the real and the imaginary part of two eigenenergies of the Wannier-Stark resonances in a double-periodic Wannier-Stark system for $V_0=2$, $\varphi =0$, and $\delta =0.075$ (upper panels), $\delta =0.08$ (lower panels).

Figure 10

(Color online) Upper panel: Decay rates of the two most stable resonances as a function of the relative phase $\varphi$ and the inverse field strength $1∕F$ for $V_0=2$ and $\delta =0.05$. Lower panel: For fixed Stark field $F=0.12$.

Figure 11

Effects of fluctuations of the relative phase $\varphi$. (a) Decay rate of the two most stable resonances for $V_0=2$, $\delta =0$, and $\varphi =0$. (b) Position $F_{1,2}$ of the RET peaks vs the relative phase $\varphi$. (c) Uncertainty $\mathrm{\Delta }F$ of the positions of the RET peaks vs fluctuation $\mathrm{\Delta }\varphi$ of the phase. (d) Relative fluctuation $\mathrm{\Delta }\Gamma ∕{\Gamma }_0$ of the decay rate for $F=1∕8$ vs $\mathrm{\Delta }\varphi$.

Figure 12

(Color online) Decay rates of nonlinear Wannier-Stark resonances in double periodic optical lattices with $V_0=2$, $\delta =0.05$, $\varphi =0$, and $g=0$ (dashed black line), $g=-0.1$ (dotted green line), $g=-0.2$ (dash-dotted red line), and $g=-0.3$ (solid blue line), respectively.

Figure 13

Shift of the RET peaks. Shown is the position $F_{\mathrm{res}}$ of the RET peaks (lower panel) and the resonant decay rate ${\Gamma }_{\mathrm{res}}$ (upper panel) in dependence of the nonlinearity $g$.

Figure 14

(Color online) Asymmetry of the nonlinear RET peaks. Shown is the decay rate $\Gamma \left(F\right)$ of the two peaks (left and right) around the respective peak positions $F_{\mathrm{res}}$ for $g=0$ (black dashed line) and $g=-0.3$ (solid blue line).

Figure 15

Pulsed output from a tilted optical lattice for $V_0=0.8$, $F=1∕18$ and $\delta =0$, $g=0$. Shown is the atomic density $\mid \psi (x,t){\mid }^2$ in a grey-scale plot.

Figure 16

Pulsed output from a tilted optical lattice for $g=0$. Shown is the atomic density for $t=14T_B$ for $\delta =0$ (top), $\delta =0.2$ and $\varphi =0$ (middle), and $\delta =0.2$ and $\varphi =\pi ∕2$ (bottom).

Figure 17

Fidelity of the output switch in dependence of the standard deviation $\mathrm{\Delta }\varphi$ of the phase fluctuations.