Spontaneous symmetry breaking in a quadratically driven nonlinear photonic lattice

We investigate the occurrence of a phase transition, characterized by the spontaneous breaking of a discrete symmetry, in a driven-dissipative Bose-Hubbard lattice in the presence of two-photon coherent driving. The driving term does not lift the original $\text{U}\left(1\right)$ symmetry completely and a discrete ${\mathbb{Z}}_2$ symmetry is left. When driving the bottom of the Bose-Hubbard band, a mean-field analysis of the steady state reveals a second-order transition from a symmetric phase to a quasicoherent state with a finite expectation value of the Bose field. For larger driving frequency, the phase diagram shows a third region, where both phases are stable and the transition becomes of first order.

Figure 1

White line: Contour separating the regions where $max\left[\mathrm{Im}\left({\omega }_k\right)\right]{|}_k<0$ (left) and $max\left[\mathrm{Im}\left({\omega }_k\right)\right]{|}_k>0$ (right) on the $(J,G)$ plane. Color plot: The order parameter $|\langle \widehat{a}\rangle |$ computed self-consistently, at steady state, from the analytical solution of Ref. [25]. Parameters: $\mathrm{\Delta }=-J,U=\eta =1$.

Figure 2

(a) The quantities $|\langle \widehat{a}\rangle {|}^2$ and $n=\langle {\widehat{a}}^{†}\widehat{a}\rangle$ plotted as a function of $J$ for $G=3$. The horizontal dashed line denotes the symmetric solution $n_s$ corresponding to $|\langle \widehat{a}\rangle |=0$, and the vertical dashed line marks the fitted critical value $J_c=0.3305$. (b) Double-logarithmic plot of $|\langle \widehat{a}\rangle |$ as a function of $J-J_c$ for three values of $G$. The dashed lines denote the critical behavior $|\langle \widehat{a}\rangle |\propto |J-J_c{|}^{1/2}$ obtained by fitting the data close to the critical point $J=J_c$. (c) and (d) Color plot of the Wigner function $W\left(z\right)$ computed for $G=3$ and for values of $J$ corresponding, respectively, to the symmetric ($J=0.25$) and broken-symmetry ($J=0.5$) phases. (e) and (f) Trajectories on the $(\mathrm{Re}\left(\langle \widehat{a}\rangle \right),\mathrm{Im}\left(\langle \widehat{a}\rangle \right))$ plane as computed for $G=3$ and, respectively, $J=0.25$ (a) and $J=0.5$ (b). Different trajectories correspond to an initial coherent state $|{\alpha }_0\rangle$ for different values of ${\alpha }_0$, and the arrows indicate the direction of time. Circles denote the initial coherent states, while the squares mark the fixed points reached at steady state. Parameters: $\mathrm{\Delta }=-J,U=\eta =1$.

Figure 3

(a) White line: Contour separating the regions where $max\left[\mathrm{Im}\left({\omega }_k\right)\right]{|}_k<0$ and $max\left[\mathrm{Im}\left({\omega }_k\right)\right]{|}_k>0$. Color plot: The order parameter $|\langle \widehat{a}\rangle |$ computed self-consistently at steady state. The white circle marks the onset of the bistable region of the phase diagram. (b) Double-logarithmic plot of $|\langle \widehat{a}\rangle |$ as a function of $J-J_c$ for two values of $G$. The dashed line denotes the critical behavior $|\langle \widehat{a}\rangle |\propto |J-J_c{|}^{1/2}$ for $G=8$, obtained by fitting the data close to the critical point $J=J_c$. (c) and (d) Plots of $|\langle \widehat{a}\rangle \left(t\right)|$ and $P\left(t\right)=\mathrm{Tr}\left[{\widehat{\rho }}^2\left(t\right)\right]$, respectively, as a function of time, for $J=1$ and $G=3.7$. Different curves correspond to different initial coherent states $|{\alpha }_0\rangle$, with ${\alpha }_0=2.0,1.0,0.5,0.25,0.1,0.05$. (e) Dispersion of $\mathrm{Im}\left({\omega }_k\right)$ for the least stable excitation of the symmetric steady state, as computed for three different values of $J$, and $G=4$. Parameters: $\mathrm{\Delta }=0,U=\eta =1$.