Dissipation-induced coherence and stochastic resonance of an open two-mode Bose-Einstein condensate

We discuss the dynamics of a Bose-Einstein condensate in a double-well trap subject to phase noise and particle loss. The phase coherence of a weakly interacting condensate, experimentally measured via the contrast in an interference experiment, as well as the response to an external driving becomes maximal for a finite value of the dissipation rate matching the intrinsic time scales of the system. This can be understood as a stochastic resonance of the many-particle system. Even stronger effects are observed when dissipation acts in concurrence with strong interparticle interactions, restoring the purity of the condensate almost completely and increasing the phase coherence significantly. Our theoretical results are backed by Monte Carlo simulations, which show a good qualitative agreement and provide a microscopic explanation for the observed stochastic resonance effect.

Figure 1

(Color online) The open double-well trap considered in the present paper.

Figure 2

(Color online) Decay rate $\kappa$ of the quasi-steady-state (14) as a function of the tunneling rate $J$ and the dissipation rate $1/T_1$ for ${\gamma }_p=5{\text{s}}^{-1}$ and $U=ϵ=0$.

Figure 3

(Color online) Contrast $\alpha$ of the quasi-steady-state (14) as a function of the tunneling rate $J$ and the dissipation rate $1/T_1$ (a) for ${\gamma }_p=5{\text{s}}^{-1}$ and $U=ϵ=0$ and (b) for a fixed value of the tunneling rate $J=2{\text{s}}^{-1}$ and (c) a fixed value of the dissipation rate $1/T_1=2{\text{s}}^{-1}$. The dashed-dotted red lines represent the approximations [Eq. (19)] for small and large values of $J$.

Figure 4

Relaxation to the quasi-steady-state for ${\gamma }_p=5{\text{s}}^{-1}$, $T_1^{-1}=1{\text{s}}^{-1}$, $ϵ=10{\text{s}}^{-1}$, and $U=0$. (a) Relaxation of the contrast $\alpha \left(t\right)$ for $J=4{\text{s}}^{-1}$. (b) Decay of the particle number $n\left(t\right)$ for $J=4{\text{s}}^{-1}$. (c) Development of the SR maximum of the contrast $\alpha \left(J\right)$.

Figure 5

Steady state values of the contrast $\alpha$ as a function of the tunneling rate (a) for $U=0$ and different values of the energy bias $ϵ$ and (b) as a function of the effective interaction strength $g=Un$ for $ϵ=0$. The remaining parameters are ${\gamma }_p=5{\text{s}}^{-1}$ and $T_1^{-1}=2{\text{s}}^{-1}$.

Figure 6

(Color online) (a) Histogram of the probabilities to measure the relative phase $\varphi$ and the relative population imbalance $s_z$ in a single experimental run after $t=1.5\text{s}$ obtained from a MCWF simulation of the many-body dynamics. The initial state was chosen as a pure BEC (i.e., a product state) with $s_z=n/2$ and $n\left(0\right)=100$ particles and the remaining parameters are ${\gamma }_p=5{\text{s}}^{-1}$, $T_1=0.5\text{s}$, $ϵ=10{\text{s}}^{-1}$, and $U=0.1{\text{s}}^{-1}$. (b) Average contrast $\alpha =\sqrt{s_x^2+s_y^2}/n$ (solid black line) after $t=1.5\text{s}$ compared to $\sqrt{s_x^2+s_y^2}/\left|\mathbf{s}\right|$ and $\left|\mathbf{s}\right|/n$ (dashed red lines).

Figure 7

Dynamics of the relative population imbalance $s_z\left(t\right)/n\left(t\right)$ in a weakly driven double-well trap for three different values of the tunneling rate: (a) $J_0=0.5{\text{s}}^{-1}$, (b) $J_0=1.5{\text{s}}^{-1}$, and (c) $J_0=5{\text{s}}^{-1}$. The amplitude of the forced oscillations is maximal for intermediate values of $J_0$ as shown in part (b). The remaining parameters are $T_1^{-1}=2{\text{s}}^{-1}$, $U=0$, $ϵ=0$, ${\gamma }_p=5{\text{s}}^{-1}$, and $J_1/J_0=10\%$. Please note the different scalings.

Figure 8

(Color online) (a) Response [amplitude of the oscillations of $s_z\left(t\right)/n\left(t\right)$] of a weakly driven double-well trap vs $T_1^{-1}$ and $J_0$ calculated within linear-response theory. (b) For a fixed value of the tunneling rate $J_0=2.5{\text{s}}^{-1}$. (c) For a fixed value of the dissipation rate $T_1^{-1}=2{\text{s}}^{-1}$. The remaining parameters are $U=0$, $ϵ=0$, ${\gamma }_p=5{\text{s}}^{-1}$, and $J_1/J_0=10\%$.

Figure 9

(Color online) Dynamics of the coherence (a) $s_x\left(t\right)/n\left(t\right)$ and the relative population imbalance (b) $s_z\left(t\right)/n\left(t\right)$ for a double-well trap with a driven energy bias $ϵ$ for $J_0=2{\text{s}}^{-1}$ and $T_1^{-1}=4{\text{s}}^{-1}$. (c) Response [amplitude of the oscillations of $s_x\left(t\right)/n\left(t\right)$] vs $T_1^{-1}$ and $J_0$ calculated within linear-response theory. The remaining parameters are $U=0$, ${ϵ}_1=1{\text{s}}^{-1}$, and ${\gamma }_p=5{\text{s}}^{-1}$.

Figure 10

(a) Time evolution of the purity $p$ and the contrast $\alpha$ for $J=U=10{\text{s}}^{-1}$, $ϵ=0$, and $T_1=0.5\text{s}$. (b) Time evolution without interactions $\left(U=0\right)$ and (c) without dissipation $\left(1/T_1=1/T_2=0\right)$ for comparison. The occasional revivals are artifacts of the small particle number. The initial state is a pure BEC with $s_z=n/2$ and $n\left(0\right)=100$ particles. The results of a MCWF simulation averaged over 100 runs are plotted as a thin solid line in (a) and (c), while the mean-field results are plotted as a thick line in (a) and (b). Note that the mean-field approximation is exact in case (b), whereas it breaks down in case (c) and is thus not shown (cf. [32, 35]).

Figure 11

(Color online) Mean-field dynamics (a) without interactions and dissipation, (b) with interactions $Un=40{\text{s}}^{-1}$, and (c) with interactions and dissipation ${\gamma }_a=10{\text{s}}^{-1}$. The remaining parameters are $J=10{\text{s}}^{-1}$ and $ϵ=0$. To increase the visibility we have plotted the rescaled Bloch vector $\mathbf{s}/n$ and we have artificially fixed the particle number so that $n=\text{const}$.

Figure 12

Time evolution of the purity $p$ and the contrast $\alpha$ for $J=U=10{\text{s}}^{-1}$, $ϵ=0$, and (a) $1/T_1=0.5{\text{s}}^{-1}$, (b) $1/T_1=1.5{\text{s}}^{-1}$, and (c) $1/T_1=2.5{\text{s}}^{-1}$. The initial state is a pure BEC with $s_z=n/2$ and $n\left(0\right)=100$ particles. The results of a MCWF simulation averaged over 100 runs are plotted as a thin solid line, while the mean-field results are plotted as a thick line.

Figure 13

(a) Purity $p$ and (b) contrast $\alpha$ after $t=2\text{s}$ as a function of the dissipation rate $1/T_1$ for different values of the interaction strength $Un$ calculated within the mean-field approximation. The remaining parameters are chosen as in Fig. 10a.