Quantum entanglement in the spatial-symmetry-breaking phase transition of a driven-dissipative Bose-Hubbard dimer

We theoretically explore the quantum correlation properties of a dissipative Bose-Hubbard dimer in the presence of a coherent drive. In particular, we focus on the regime where the semiclassical theory predicts a bifurcation with a spontaneous spatial symmetry breaking. The critical behavior in a well-defined thermodynamical limit of large excitation numbers is considered and analyzed within a Gaussian approach. The case of a finite boson density is also examined by numerically integrating the Lindblad master equation for the density matrix. We predict the critical behavior around the bifurcation points accompanied by large quantum correlations of the mixed steady state, in particular, exhibiting a peak in the logarithmic entanglement negativity.

Figure 1

Semiclassical mean-field results describing the spatial symmetry breaking in the driven-dissipative Bose-Hubbard dimer. The rescaled order parameter $|\langle \widehat{O}\rangle |\sqrt{U/\gamma }$, with $|\langle \widehat{O}\rangle |=|\langle {\widehat{a}}_1+{\widehat{a}}_2\rangle |$, is depicted (solid line) as a function of the dimensionless quantity $\sqrt{U}F/{\gamma }^{3/2}$. These results become exact in the thermodynamic limit with an infinite number of photons, obtained for $\sqrt{U}F$ fixed and $F\to +\infty$. There are two bifurcation points (indicated by dotted lines): in between, the symmetry is broken (non-zero-order parameter $|\langle \widehat{O}\rangle |$). Dashed lines are fits to a square root dependence in the symmetry-broken phase around the two bifurcation points. Other parameters: $J=2.5\gamma$ and $\mathrm{\Delta }=-1.5\gamma$. Inset: Sketch of the driven-dissipative Bose-Hubbard dimer and the two possible equivalent configurations: the first is characterized by a positive hopping ($J>0$) and a $\pi$ phase difference between the driving fields; the second, equivalent configuration has a negative hopping strength ($J<0$) and the same driving phase at the two sites.

Figure 2

Variance of the order parameter $V\left(\widehat{O}\right)$ as a function of $\sqrt{U}F/{\gamma }^{3/2}$. The dashed curve is the Gaussian approach, which is exact in the thermodynamic limit ($\sqrt{U}F$ fixed and $U\to 0$). Solid lines depict numerical solutions of the full master equation (3), for $U/\gamma =0.1$, 0.25, 0.5, and 1 (from top to bottom). Inset: Inverse of the variance $V{\left(\widehat{O}\right)}^{-1}$ as a function of $\sqrt{U}F/{\gamma }^{3/2}$ in the thermodynamic limit; dashed linesare fits to $V\left(\widehat{O}\right)\propto {(\sqrt{U}F-A_c)}^{-1}$ around the bifurcation points, where $A_c$ is a constant. Other parameters: $J=2.5\gamma$ and $\mathrm{\Delta }=-1.5\gamma$.

Figure 3

The logarithmic entanglement negativity $E_N$ as a function of $\sqrt{U}F/{\gamma }^{3/2}$. The dashed curve is the result for the Gaussian approach, which is valid in the thermodynamic limit ($U\to 0$ with $\sqrt{U}F$ fixed). Solid curves correspond to finite values of the nonlinearity, $U/\gamma =0.1$, 0.25, 0.5, and 1 (from top to bottom). Other system parameters are the same as in Fig. 1.

Figure 4

von Neumann entropy $S$ as a function of $\sqrt{U}F/{\gamma }^{3/2}$. The dashed curve is the Gaussian result, which is valid in the thermodynamic limit ($U\to 0$ with $\sqrt{U}F$ fixed). Solid curves correspond to finite values of the nonlinear interaction $U/\gamma =0.1$, 0.25, 0.5, and 1 (from top to bottom). Other system parameters are the same as in Fig. 1.