Nonreciprocal Superradiant Phase Transitions and Multicriticality in a Cavity QED System

We demonstrate the emergence of nonreciprocal superradiant phase transitions and novel multicriticality in a cavity quantum electrodynamics system, where a two-level atom interacts with two counterpropagating modes of a whispering-gallery-mode microcavity. The cavity rotates at a certain angular velocity and is directionally squeezed by a unidirectional parametric pumping ${\chi }^{(2)}$ nonlinearity. The combination of cavity rotation and directional squeezing leads to nonreciprocal first- and second-order superradiant phase transitions. These transitions do not require ultrastrong atom-field couplings and can be easily controlled by the external pump field. Through a full quantum description of the system Hamiltonian, we identify two types of multicritical points in the phase diagram, both of which exhibit controllable nonreciprocity. These results open a new door for all-optical manipulation of superradiant transitions and multicritical behaviors in light-matter systems, with potential applications in engineering various integrated nonreciprocal quantum devices.

Figure 1

Schematic illustration of the rotating dual-coupling JC model in the forward (a) and backward (b) pumps. The WGM resonator with ${\chi }^{(2)}$ nonlinearity embedded (not shown here), supports counterclockwise and clockwise cavity modes labeled as $a$ and $b$ respectively, and both modes interact with a two-level atom. The resonator rotates counterclockwise at an angular velocity $\mathrm{\Omega }$. (c),(d) Boundaries of the NP and SP for forward and backward pumps, respectively. The parameters used here are $\mathrm{\Delta }=2$, $\kappa /\mathrm{\Delta }=0.05$, $G/\kappa =1.5$, $J=0$, and ${\mathrm{\Delta }}_F/\mathrm{\Delta }=0.5$ in (c) and ${\mathrm{\Delta }}_F/\mathrm{\Delta }=-0.5$ in (d).

Figure 2

The cavity occupations $\alpha$ and $\beta$ versus the pump strength $G/\kappa$, for both the forward and backward pumps. The solid (or dotted) curve denotes the real (or imaginary) part of cavity occupations. In (a),(b), where $\lambda =1.5$, the shaded areas indicate the nonreciprocal first-order phase transition (NR 1st PT). In (c),(d), where $\lambda =1.36$, the shaded areas indicate the nonreciprocal second-order phase transition (NR 2nd PT). Here, ${\mathrm{\Delta }}_F/\mathrm{\Delta }=\pm 0.5$ for the forward and backward pumps, respectively, ${\lambda }_a={\lambda }_b=\lambda$, and other parameters are the same as in Fig. 1.

Figure 3

(a)–(c) Phase diagram of the steady-state photon number fluctuations, with $⟨c^{†}c⟩$ for the forward (b), $⟨d^{†}d⟩$ for the backward (c), and $⟨c^{†}c⟩$ (or $⟨d^{†}d⟩$) for the static microcavity (a). The blue (red) bar represents fluctuations on top of the mean-field solutions in the normal (superradiant) phase. Three regimes, denoted as I, II, and III, are identified based on the number of stable coexisting solutions. Subscripts 1 and 2 are employed to distinguish regimes with different Wigner distributions. The tricritical points are shown by blue triangles, and the multicritical points are marked by blue circles. The green curve in (b) indicates the unstable region (see details in Sec. S5 of [87]). (d) The Wigner functions in different regimes. Here, we consider the coordinates of $\lambda$ and $G/\kappa$ as $\mathrm{I}(1.0,1.1)$, ${\mathrm{II}}_1(1.42,2.6)$, ${\mathrm{III}}_1(1.8,1.98)$, ${\mathrm{II}}_2(2.04,2.24)$, and ${\mathrm{III}}_2(2.2,2.42)$, ${\mathrm{\Delta }}_q/\mathrm{\Delta }=1000$, and atomic decay $\gamma =\kappa$. (e) The critical lines at $\lambda =2$ [the black dotted line in (a)] as a function of $G$. (f),(g) Cut plots of $\ln (⟨c^{†}c⟩+1$) [black dotted line in (b)] and $\ln (⟨d^{†}d⟩+1$) [black dotted line in (c)] as a function of $\lambda$. In (e)–(g), the blue solid and red dotted lines denote cavity fluctuations in the NP and SP, respectively. We consider ${\mathrm{\Delta }}_F=0$ in (a),(e), ${\mathrm{\Delta }}_F/\mathrm{\Delta }=0.5$ in (b),(d),(f), and ${\mathrm{\Delta }}_F/\mathrm{\Delta }=-0.5$ in (c),(g).