Dynamics of dissipative Bose-Einstein condensation

We resolve the real-time dynamics of a purely dissipative $s=1/2$ quantum spin or, equivalently, hard-core boson model on a hypercubic $d$-dimensional lattice. The considered quantum dissipative process drives the system to a totally symmetric macroscopic superposition in each of the $S^3$ sectors. Different characteristic time scales are identified for the dynamics and we determine their finite-size scaling. We introduce the concept of cumulative entanglement distribution to quantify multiparticle entanglement and show that the considered protocol serves as an efficient method to prepare a macroscopically entangled Bose-Einstein condensate.

Figure 1

The jump operators $L_{xy}$ act on all $d{L}^d$ pairs of nearest-neighbor sites on the regular periodic lattice with coordination number $n_c=2d$. This is illustrated above for the example of the two-dimensional lattice: (a) For each local operator, as, e.g., the local spin $s_x^a$, we attach $n_c$ jump operators (red) that contribute according to Eq. (4). (b) For two-point functions, as, e.g., $C_{xy}$, we need to consider two cases: If $x$ and $y$ are nearest neighbors, we attach one jump operator that connects both sites $x$ and $y$ (blue) [cf. Eq. (6)], while $n_c-1$ jump operators are assigned separately to $x$ and $y$ (red) [cf. Eq. (7)]. On the other hand, if $x$ and $y$ are nonadjacent, we attach $n_c$ jump operators to both $x$ and $y$.

Figure 2

Dissipative gap $\mathrm{\Delta }$ in units of $\gamma$ as a function of system size (total particle number) $N$ on a double-logarithmic scale for dimensions $d=1$ (squares), $d=2$ (dots), and $d=3$ (triangles).

Figure 3

Real-time evolution of a system in $d=3$ dimensions consisting of $N={36}^3$ particles initialized in the infinite-temperature thermal ensemble. Selected Fourier modes ${\mathcal{C}}_p\left(t\right)$, with $p_{\mu }=2\pi n{\delta }_{\mu ,1}/L, n=1,...,L/2$, are shown, illustrating the growth and subsequent decay of correlations. The dashed lines indicate the asymptotic values ${\mathcal{C}}_p^{\infty }$; different points mark the two characteristic time scales $t_1\sim |p{|}^{-2}$ (dots) and $t_2\sim |p{|}^{-1/3}$ (squares).

Figure 4

Real-time evolution in $d=2$ dimensions for a system consisting of $N={216}^2$ particles prepared in an incoherent initial ensemble with total spin $S^3=0$. We show the time-dependent cumulative entanglement distribution $\mathcal{E}\left(t\right)$ (blue) and compare it to the behavior of the Fourier modes ${\mathcal{C}}_p\left(t\right)$, with $p_{\mu }=2\pi n_{\mu }/L, n_{\mu }=1,...,L/2$ (labeled). The dashed line indicates the asymptotic value for the entanglement distribution ${\mathcal{E}}^{\infty }=N$.