Dynamic control and probing of many-body decoherence in double-well Bose-Einstein condensates

We present an approach to dynamic decoherence control of finite-temperature Bose-Einstein condensates in a double-well potential. Due to the many-body interactions the standard “echo” control method becomes less effective. The approach described here takes advantage of the interaction-induced change of the spectrum, to obtain the optimal rate of $\pi$ flips of the relative phase between maximally distinguishable collective states. This method is particularly useful for probing and diagnosing the decoherence dynamics.

Figure 1

(Color online) Decay of coherence as a function of time, for a noninteracting system coupled to a thermal reservoir with the Hamiltonian $\widehat{H}=-2E_J/N{\widehat{L}}_x+\Delta E{\widehat{L}}_z$, with and without modulation. The energy scale is given by the average of the tunneling energy $\overline{E_J}$. The reservoir correlation time is $t_c=0.05\left[1/\overline{E_J}\right]$ and amplitude $A=0.1\overline{E_J}$, and the modulation amplitude of the tunneling energy is $10\overline{E_J}$. Solutions of the Schrödinger equation are compared to exponential decay with a modified rate due to the modulation (see text).

Figure 2

(Color online) (a) Energy ladder in the ${\widehat{L}}_x$ basis, and the effect of interactions in first-order perturbation theory. This is compared to the exact diagonalization shown in the gray-shaded area (for $Ec=4E_J/N^2$). (b) Effective mean-field potential as a function of the relative phase $\varphi$ [7]. The green particle presents the system in the stable 0-phase state, while the red particle presents the system in the unstable $\pi$-phase state.

Figure 3

(Color online) Decay control using pulsed sign changes of $E_J$, for different periods of the modulation, for $E_C=8E_J/N^2$. The total atom number is $N=100$. For extremely fast modulations (dashed blue line), the tunneling averages to zero, and the dynamics exhibit simple phase diffusion, as for $E_J=0$. The optimal modulation frequency is $4{\omega }_p/2\pi$ (see text).

Figure 4

(Color online) Bloch-sphere trajectories for the $\pi$-oscillation dynamics ($\pi$-phase initial state) with (green diamonds) and without (red circles) pulse control at the optimal period of $1/4f_p$, for $E_C=8E_J/N^2$. The total atom number is $N=100$, and the simulation lasts until $t\simeq 45\left[1/f_p\right]$. The $\pi$ flip is realized through a finite-duration $\pi$ rotation around the ${\widehat{L}}_z$ axis (as a result the bloch vector reaches $S_y=\pm 1$ during these rotations). (a) Mean-field dynamics. (b) Exact many-body dynamics. The difference between many-body and mean-field dynamics stems from the vicinity of the initial state to the classical bifurcation point. (c) $Q$ representation in the atomic coherent-state basis of the system without modulation at time $t=4\left[1/4f_p\right]$. (d) Same as (c) with optimal modulation. All parameters are as in Fig. 3.

Figure 5

(Color online) Decay control using pulse flips, in the presence of thermal reservoirs with different correlation times, at a temperature of $\sim 100\text{nK}$, which is approximately 1/3 of the chemical potential. The total atom number is $N=100$. Decay is reduced for both a Markovian reservoir (a), and a non-Markovian reservoir ($t_c\sim 15\text{ms}$ $\sim 1/4f_p$) (b).

Figure 6

(Color online) The coherence of the system as a function of time and as a function of the modulation period, without thermal noise, for $E_C=8E_J/N^2$, $N=100$. Such results constitute a Fourier decomposition of the many-body dynamics of the system. As such, the coherence displays a broad peak centered at the characteristic system dynamics time of ${\left(4{\omega }_p/2\pi \right)}^{-1}\simeq 0.9\left[1/E_J\right]$. Inset: the dephasing rate as a function of the pulse period, calculated from the main figure using exponential fits to the dynamics.