Entanglement in the Bogoliubov vacuum

We analyze the entanglement properties of the Bogoliubov vacuum, which is obtained as a second-order approximation to the ground state of an interacting Bose-Einstein condensate. We work on one- and two-dimensional lattices and study the entanglement between two groups of lattice sites as a function of the geometry of the configuration and the strength of the interactions. As our measure of entanglement we use the logarithmic negativity, supplemented by an algorithmic check [G. Giedke et al., Phys. Rev. Lett. 87, 167904 (2001)] for bound entanglement where appropriate. The short-range entanglement is found to grow approximately linearly with the group sizes and to be favored by strong interactions. Conversely, long-range entanglement is favored by relatively weak interactions. No examples of bound entanglement are found.

Figure 1

Definition of the two subsystems. Alice and Bob are both assigned $q$ contiguous sites and the two groups have a separation of $s$ sites (gap between the extreme sites). Since we are using periodic boundary conditions—i.e., working on a ring—there are in fact two distances between the groups: $s$ and $\mid N_s-2q-s\mid$. We use the convention that $s$ is the smaller one.

Figure 2

Logarithmic negativity $E_N$ as a function of group separation $s$ for different group sizes $q$. In this plot, $N_s=321$ and $\lambda =J∕gn=20$. When both $q$ and $s$ are much smaller than $N_s$, $E_N$ is an increasing function of $q$ and a decreasing function of $s$. The small separation part $(s<5)$ can be reasonably well approximated by an exponential with decay constant $\sim 0.3$. At larger separations, the decay is faster. In fact, for small group sizes, $E_N$ and thus the distillable entanglement drop strictly to zero at quite moderate separations: $s=7$ for $q=1$ and $s=11$ for $q=2$. The inset shows a logarithmic plot for $q=3$. Here the negativity vanishes at $s=12$, but due to the finiteness of $N_s$, it reappears at $s=76$. A similar picture applies for $q=4$, while for larger $q$’s, finite-size effects always keeps $E_N$ nonzero for $N_s=321$. While the main plot does not change if $N_s$ is increased, the strict vanishing and the revival of $E_N$ is still dependent on $N_s$.

Figure 3

Dependence of entanglement upon the value of $\lambda$. The five curves show $E_N$ as a function of $s$ for $q=3$ and five different values of $\lambda$. At short distances, a higher ratio of hopping to nonlinearity leads to less entanglement, while at longer distances the opposite picture applies.

Figure 4

Assignments of sites to Alice and Bob on the 2D lattice. Note that we again use periodic boundary conditions; i.e., although the lattice is drawn as flat, we work on a torus. The “group size” denoted by $q$ gives the number of sites assigned to both Alice and Bob. The “center-of-mass separation” denoted by $s_{\mathrm{c.m.}}$ gives the distance between central sites in the two group. Finally, the “orientation” denoted by O labels different arrangements of the sites within each group.

Figure 5

Entanglement between two single sites as a function of their separation. The limit of an infinite lattice has been taken so that the sums in Eqs. (22, 23) are replaced by integrals. The ratio of the tunneling energy to the strength of the interactions is $\lambda =J∕gn=100$.

Figure 6

Results for an infinite 2D lattice with $\lambda =100$. We show the single $q=1$ curve (thick line) and results for group sizes $q=3$ (open symbols) and $q=5$ (solid symbols) where the three pairs of curves correspond to the three orientations described in Fig. 4. The fact that the curves are so similar shows that $E_N$ is approximately linearly dependent on $q$ and that $s_{\mathrm{c.m.}}$ is a sensible measure of the distance between the groups for separations where $E_N$ is still appreciably different from zero ($s_{\mathrm{c.m.}}\lesssim 11$).

Figure 7

Dependence of ${\tilde{E}}_N$ upon $\lambda$ for the orientation $\mathrm{O}=0$ the infinite 2D lattice. The dotted curves are single site results, $q=1$, the solid curves are for groups of $q=3$ sites each, and the dashed curves are for $q=5$. Results for four different separations are shown: $s_{\mathrm{c.m.}}=5$, 7, 9, and 11. The three uppermost curves are for $s_{\mathrm{c.m.}}=5$, the next three for $s_{\mathrm{c.m.}}=7$, and so on. In the displayed $\lambda$ range, all the curves show a maximum; i.e., $E_N$ is not a monotone function of $\lambda$.