Quantum correlations in the one-dimensional driven dissipative $XY$ model

We study the nonequilibrium steady state (NESS) of a driven dissipative one-dimensional system near a critical point, and explore how the quantum correlations compare to the known critical behavior in the ground state. The model we study corresponds to a cavity array driven parametrically at a two photon resonance, equivalent in a rotating frame to a transverse field anisotropic $XY$ model [C.-E. Bardyn and A. Imamoğlu, Phys. Rev. Lett. 109, 253606 (2012)]. Depending on the sign of transverse field, the steady state of the open system can be either related to the ground state or to the maximum energy state. In both cases, many properties of the entanglement are similar to the ground state, although no critical behavior occurs. As one varies from the Ising limit to the isotropic $XY$ limit, entanglement range grows. The isotropic limit of the NESS is, however, singular, with simultaneously diverging range and vanishing magnitude of entanglement. This singular limiting behavior is quite distinct from the ground state behavior; it can, however, be understood analytically within spin-wave theory.

Figure 1

Cartoon illustrating coupled cavity array with hopping $J$, two-cavity pumping $\Omega$, and loss rate $\kappa$.

Figure 2

Negativity $\mathcal{N}$ vs transverse field $g$, comparing MPO numerical solution for ${\chi }_{\max }=20$ (blue, solid) to exact diagonalization (red, dashed) for a four-site Ising model. Parameters (in units of $J$): $\Delta =1$, $\tilde{\kappa }=0.5$.

Figure 3

Panel (a) showing spin-spin correlations $\langle {\sigma }_j^x{\sigma }_{j+l}^x\rangle$ as a function of transverse field. The different lines correspond to different separations. Panel (b) showing decay of spin-spin correlations $\langle {\sigma }_j^x{\sigma }_{j+l}^x\rangle$ as a function of separation $l$ between the spin sites. Both panels plotted for the Ising limit ($\Delta =1$). It is clearly seen that the NESS exhibit FM and AFM ordering for negative and positive values of transverse field ($g=\pm 1$). Inset [panel (c)] shows short-range incommensurate order for lower values of transverse field ($g=\pm 0.1$). The axes in the inset are the same as in the main plot. Other parameters (in units of $J$): $\tilde{\kappa }=0.5$ and MPO calculation performed for $N=40$ site chain, with ${\chi }_{\text{max}}=20$.

Figure 4

Mean-field phase diagram for the nonequilibrium steady state of Eq. (4) as a function of dimensionless parameters $g,\Delta ,\tilde{\kappa }$.

Figure 5

Spin-spin correlations $\langle {\sigma }_j^x{\sigma }_{j+1}^x\rangle$ as a function of transverse field strength $g$. Parameters (in units of $J$): $\Delta =1,\tilde{\kappa }=0.5$ and MPO calculation performed for the $N=40$ site chain, with ${\chi }_{\text{max}}=20$.

Figure 6

Evolution of quantum correlations with transverse field $g$ in the Ising limit. Panel (a) shows negativity $\mathcal{N}$, and panel (b) geometric quantum discord $\mathcal{D}$. In addition the integrated susceptibility $S_{\text{int}}^{xx}={\sum }_j\langle {\sigma }_i^x{\sigma }_j^x\rangle$ is shown in (c); at the equilibrium critical point this would show a power-law divergence with system size. Note that only one line is shown in panel (a) because entanglement vanishes beyond nearest neighbors at $\Delta =1$. Parameters as in Fig. 3.

Figure 7

Evolution of quantum correlations with transverse field $g$ near the isotropic limit $\Delta =0.05$. Panels (a) and (b) show negativity $\mathcal{N}$ and geometric quantum discord $\mathcal{D}$ as in Fig. 6. Panel (c) shows spin-spin correlation $\langle {\sigma }_j^x{\sigma }_{j+l}^x\rangle$ as in Fig. 3. Panel (d) shows spatial dependence of correlations for the anisotropic $X$$Y$ model. Parameters (in units of $J$): $\tilde{\kappa }=0.5$ and MPO calculation performed for the $N=40$ site chain, with ${\chi }_{\text{max}}=20$.

Figure 8

Evolution of quantum correlations with anisotropy $\Delta$. Panel (a) shows negativity $\mathcal{N}$ and (b) geometric quantum discord $\mathcal{D}$. Parameters (in units of $J$): $g=-1,\tilde{\kappa }=0.5$ and MPO calculation performed for the $N=40$ site chain, with ${\chi }_{\text{max}}=20$.

Figure 9

Evolution of quantum correlations with $\tilde{\kappa }$. Panel (a) shows negativity $\mathcal{N}$, and panel (b) shows geometric quantum discord $\mathcal{D}$, for both Ising limit and small anisotropy limit. Parameters (in units of $J$): $g=-1$ and MPO calculation performed for the $N=40$ site chain, with ${\chi }_{\text{max}}=20$.

Figure 10

(a) Correlation length ${\xi }_c$ at $\Delta =0.005$ vs decay rate $\tilde{\kappa }$. (b) Nearest-neighbor correlations $|\langle {\sigma }_j^{-}{\sigma }_{j+1}^{-}\rangle |$ vs anisotropy $\Delta$ for several values of $\tilde{\kappa }$. In both panels MPO numerics (points) are compared to spin-wave theory (lines). Parameters (in units of $J$): $g=-1$, MPO calculation is performed for the $N=40$ site chain with ${\chi }_{\text{max}}=20$.