Purity oscillations in coupled Bose-Einstein condensates

Distinct oscillations of the purity of the single-particle density matrix for many-body open quantum systems have been shown to exist [D. Dast, D. Haag, H. Cartarius, and G. Wunner, Phys. Rev. A 93, 033617 (2016)]. They are found in $\mathcal{PT}$-symmetric Bose-Einstein condensates, in which the coherence of the condensate drops and is almost completely restored periodically. For this effect the presence of a gain and loss of particles turned out to be essential. We demonstrate that it can also be found in closed quantum systems whose subsystems experience a gain and loss of particles. This is shown with two different lattice setups for cold atoms, viz., a ring of six lattice sites with periodic boundary conditions and a linear chain of four lattice wells. In both cases pronounced purity oscillations are found, and it is shown that they can be made experimentally accessible via the average contrast in interference experiments. This shows that it is possible to identify this characteristic effect in closed quantum systems which have been proposed to realize $\mathcal{PT}$-symmetric quantum mechanics via separation into subsystems.

Figure 1

Schematic representation of the setup discussed in Sec. 2: Two coupled three-site subsystems forming a six-site Hermitian system.

Figure 2

Time evolution of the filling levels for $N=70$, $g=J_1=1$, and $J_2=2$ for two initially fully separated BECs. Relatively stable oscillations are established after some onset period. Due to the symmetry of the system and conservation of the total particle number, there is a phase difference of $\pi$ between the oscillations of the central site (solid line) and those of the outer sites (dashed line).

Figure 3

Time dependence of the purity $P$ (solid lines) of one subsystem and the purity $P_{\mathrm{tot}}$ (dashed lines) of the total system for different values (a) $J_2=0.5$, (b) $J_2=1$, (c) $J_2=2$, and (d) $J_2=3$ of the coupling strength between the two systems. The other parameters are $N=70$ and $g=J_1=1$. The purity $P$ undergoes oscillations that can be very distinct and of a large amplitude for higher values of $J_2$. The total purity stays almost constant at $P_{\mathrm{tot}}=0.4$ and decreases only slightly over time, which can be explained solely by statistical reasons. The amplitude and average of the oscillations of $P$ decrease over time as well, especially for the highest value of the tunneling strength, $J_2=3$ (d).

Figure 4

Average contrast ${\nu }_{12}$ after time $t$ in an interference experiment of sites 1 and 2 compared with the purity $P_{12}$, the squared particle imbalance $I_{12}$ of the two sites, and the overall purity $P$ of the subsystem for the same parameters as in Fig. 2. The average contrast undergoes distinct oscillations close in shape and size to those of the purities $P_{12}$ and $P$. The moderate particle imbalance is not able to disturb this behavior, partly since its minima almost coincide with the maximum of the purity.

Figure 5

Schematic representation of the chain setup, in which four lattice sites are coupled. The left pair of wells forms subsystem 1 and the right pair establishes subsystem 2.

Figure 6

Particle numbers in the subsystems of the chain of four wells. The initial state consists of 20 particles in subsystem 1 and only one particle in subsystem 2. For a coupling constant $J=2$ and no particle-particle interactions ($g=0$), the dynamics of the system shows very uniform oscillations.

Figure 7

Purity oscillations in the two subsystems of the chain of four wells for the same parameters as in Fig. 6. The oscillations in both halves of the system alternate with a phase shift of $\pi$.

Figure 8

Average contrast in the two subsystems of the chain of four wells for the same parameters as in Fig. 6. The contrast shows a dynamic behavior similar to the purity oscillations shown in Fig. 7.