Beyond mean-field dynamics in open Bose-Hubbard chains

We investigate the effects of phase noise and particle loss on the dynamics of a Bose-Einstein condensate in an optical lattice. Starting from the many-body master equation, we discuss the applicability of generalized mean-field approximations in the presence of dissipation as well as methods to simulate quantum effects beyond mean field by including higher-order correlation functions. It is shown that localized particle dissipation leads to surprising dynamics, as it can suppress decay and restore the coherence of a Bose-Einstein condensate. These effects can be applied to engineer coherent structures such as stable discrete breathers and dark solitons.

Figure 1

Numerical test of the BBR method for a leaky double-well trap with loss in the second well. (a) and (b) Dynamics of the population imbalance $\langle {\mathrm{n̂}}_2-{\mathrm{n̂}}_1\rangle$ for two different values of the loss rate, comparing the BBR approximation (solid blue line) to numerically exact results (thick red line). (c) Condensate fraction ${\lambda }_0/n_{\mathrm{tot}}$ as a function of time and the loss rate ${\gamma }_2$. (d) Trace distance (12) between the exact rescaled SPDM $\sigma (t)/n(t)$ and the respective BBR approximation. In all cases the initial state is assumed to be a pure BEC with equal population and a phase difference of $\pi$ between the two modes. The remaining parameters are $J=10\hspace{0.5em}{\mathrm{s}}^{-1}$, $\kappa =0$, $U=0.5\hspace{0.5em}{\mathrm{s}}^{-1}$, and $n(0)=200$ atoms.

Figure 2

Dynamics of a BEC in a triple-well trap with boundary dissipation: (a) atomic density $\langle {\mathrm{n̂}}_j(t)\rangle$, (b) total particle number, and (c) the condensate fraction ${\lambda }_0/n_{\mathrm{tot}}$ for $J=5\hspace{0.5em}{\mathrm{s}}^{-1}$, $\gamma =20\hspace{0.5em}{\mathrm{s}}^{-1}$, $\kappa =0$, $U=30\hspace{0.5em}{\mathrm{s}}^{-1}$, and $n(0)=60$ atoms. (d) Total particle number after a fixed propagation time $t_{\mathrm{final}}=0.2\hspace{0.5em}\mathrm{s}$ as a function of the interaction strength $U$. Mean-field ($-·-$) and BBR (—) results are compared to numerically exact simulations with the quantum jump method averaging over 200 trajectories (thick red line).

Figure 3

Decay of a discrete breather state for $J=5\hspace{0.5em}{\mathrm{s}}^{-1}$, $\gamma =20\hspace{0.5em}{\mathrm{s}}^{-1}$, $\kappa =0$, and $U=15\hspace{0.5em}{\mathrm{s}}^{-1}$. Numerical results calculated with the BBR method (—) are compared to the analytic estimates (13) and (14), respectively ($---$).

Figure 4

Formation of vacancies by localized loss at the central lattice site. [(a),(b)] Evolution of the atomic density $\langle {\mathrm{n̂}}_j(t)\rangle$ (color scale as in Fig. 2). [(c),(d)] Final value of the total particle number after a fixed propagation time $t_{\mathrm{final}}$ as a function of the loss rate $\gamma$, without (solid lines) and with strong phase noise (dashed lines, $\kappa =50\hspace{0.5em}{\mathrm{s}}^{-1}$). The remaining parameters are $J=5\hspace{0.5em}{\mathrm{s}}^{-1}$, $U=0.2\hspace{0.5em}{\mathrm{s}}^{-1}$, and $n(0)=1000$ particles [(a)–(c)] and $U=2\hspace{0.5em}{\mathrm{s}}^{-1}$ and $n(0)=50$ particles (d). The dynamics has been simulated with the BBR (thin blue lines) and the quantum jump method (thick red lines).

Figure 5

Coherence of a vacancy generated by loss from the central site: [(a),(b)] atomic density, [(c),(d)] scaled eigenvalues ${\lambda }_m/n_{\mathrm{tot}}$ of the SPDM, and [(e),(f)] phase coherence $g^{(1)}$ between the two BEC fragments. Parameters are the same as in Fig. 4 with $\gamma =100\hspace{0.5em}{\mathrm{s}}^{-1}$ and $\kappa =0$. Results of a quantum jump simulation are plotted as thick red lines and BBR results as thin blue lines.

Figure 6

Generation of dark solitons using loss imprinting at a rate $\gamma =100\hspace{0.5em}{\mathrm{s}}^{-1}$ at the central site [(a),(c)] and phase imprinting in the lower half of the lattice [(b),(c)], both for times $t<0.1\hspace{0.5em}\mathrm{s}$ only. Shown are the atomic density (left, color scale as in Fig. 2) and the scaled eigenvalues of the SPDM (right) calculated with the BBR method. Parameters are $J=5\hspace{0.5em}{\mathrm{s}}^{-1}$, $U=0.1\hspace{0.5em}{\mathrm{s}}^{-1}$, $\kappa =0$, and $n(0)=1000$ particles.