Quantum phase-space analysis of population equilibration in multiwell ultracold atomic systems

We examine the medium time quantum dynamics and population equilibration of two-, three-, and four-well Bose-Hubbard models using stochastic integration in the truncated Wigner phase-space representation. We find that all three systems will enter at least a temporary state of equilibrium, with the details depending on both the classical initial conditions and the initial quantum statistics. We find that classical integrability is not necessarily a good guide as to whether equilibration will occur. We construct an effective single-particle reduced density matrix for each of the systems, using the expectation values of operator moments, and use this to calculate an effective entropy. Knowing the expected maximum values of this entropy for each system, we are able to quantify the different approaches to equilibrium.

Figure 1

Schematic of our three-mode Bose-Hubbard system. The ${\widehat{a}}_i$ are the bosonic annihilation operators for each mode, while $J$ represents the coupling rate between the modes. In this article, we always set $J=1$, which sets the units of time.

Figure 2

Schematic of our four-mode Bose-Hubbard system. The ${\widehat{a}}_i$ and ${\widehat{b}}_i$ are the bosonic annihilation operators for each mode, while $J$ represents the coupling rate between the modes.

Figure 3

The time development of the Lyupanov exponents for the three- and four-well systems. For (a) the reference initial condition was with all 300 atoms equally distributed between wells one and two, with none in well three, and the perturbed initial condition had 149 in wells one and two, with 2 atoms in well three. $\chi N_{\text{tot}}=0.5J$, with $J=1$. The solid line is $L_1$, the dash-dotted is $L_2$ (dynamics indistinguishable from $L_1$) and the dotted line is $L_3$. For (b), the reference initial condition was with 200 atoms in each of wells $a_1$ and $b_2$, with the other two unoccupied. The perturbed initial condition had 199 in wells $a_1$ and $b_2$, with 1 atom in each of the other two. $\chi N_{\text{tot}}=0.5J$, with $J=1$. The four exponents are indistinguishable. All quantities plotted in this and subsequent plots are dimensionless.

Figure 4

The mean-field population dynamics and pseudoentropy for the two-well system, with $N_1\left(0\right)=200$, $N_2\left(0\right)=0$, $\chi N_{\text{tot}}=0.5J$, and $J=1$, calculated using the truncated Wigner representation. The solid (blue) lines are the average over $1.57\times {10}^5$ trajectories for an initial Fock state, while the dash-dotted lines (red) are averaged over $2.33\times {10}^5$ trajectories for an initial coherent state. (a) gives the expectation values of the population in well one and (b) gives the calculated pseudentropies.

Figure 5

The first mean-field partial revival and the probabilities for each number in well one at $Jt=900$, calculated using the exact method for the same parameters and initial conditions as in Fig. 4, but with initial Fock states.

Figure 6

The mean-field population dynamics of well one and the pseudoentropy for the same parameters and initial conditions as the reference state in Fig. 3a in the three-well model. The solid (blue) lines are the average over $1.04\times {10}^9$ trajectories for initial coherent states, while the dash-dotted lines (red) are averaged over $8\times {10}^4$ trajectories for initial Fock states. The total atom number was 300, with half of these initially in well one and half in well two. (a) gives the expectation values of the population in well one, and (b) gives the calculated pseudoentropies.

Figure 7

The population dynamics of well $a_1$of the four-well system and the pseudoentropy for the same parameters and initial conditions as the reference state of Fig. 3b. The solid (blue) lines are the average over $1.36\times {10}^5$ trajectories for initial Fock states, while the dash-dotted lines (red) are averaged over $4.05\times {10}^5$ trajectories for initial coherent states. The total atom number was 400, with half of these initially in well $a_1$ and half in well $b_2$. (a) gives the expectation values of the population in well $a_1$, and (b) gives the calculated pseudoentropies.