Generalized classes of continuous symmetries in two-mode Dicke models

As recently realized experimentally [Nature (London) 543, 87 (2017)], one can engineer models with continuous symmetries by coupling two cavity modes to trapped atoms via a Raman pumping geometry. Considering specifically cases where internal states of the atoms couple to the cavity, we show an extended range of parameters for which continuous symmetry breaking can occur, and we classify the distinct steady states and time-dependent states that arise for different points in this extended parameter regime.

Figure 1

(a) Atoms in a cavity interact with two independent modes. These could be realized through differing polarizations or different spatial profiles. (b) Level scheme showing effective coupling of two low-lying levels through Raman transitions. The strength of each transition can be controlled independently by varying the pump Rabi frequency and detunings, as well as the overlap of the pump beams with each mode.

Figure 2

Phase diagrams for $\gamma =0$ showing normal state (hatched blue), inverted state (hatched red), and two-mode superradiance (pink). (a) $\omega \equiv {\omega }_a={\omega }_b$ vs $g$. We see the two normal states are stable for low $g$, while for higher $g$ the superradiant state becomes stable as expected. (b) ${\omega }_a$ vs ${\omega }_b$ for fixed $g=3$ kHz. The superradiant phase exists only close to the line ${\omega }_a={\omega }_b$, but (c) for ${\omega }_{\mathrm{diff}}\equiv {\omega }_a-{\omega }_b$ small but nonzero there are rotating superradiant solutions.

Figure 3

Phase diagram in the ${\omega }_a\text{−}{\omega }_b$ plane at $\gamma =1$. The plane is separated into $\mathrm{SR}\alpha$ (green), $\mathrm{SR}\beta$ (orange), oscillatory limit cycle (white) phases, and symmetry-broken two-mode superradiance (pink). These modes are equivalent along two lines of U(1) symmetry, ${\omega }_a={\omega }_b$ and ${\kappa }^2=4{\omega }_a{\omega }_b$, marked by the pink lines. Dots show the ${\omega }_a\text{−}{\omega }_b$ plane positions of the limit cycles in Fig. 5. The black line shows the position of the spectrum in Fig. 4. The inset shows a magnified view of the region close to the origin.

Figure 4

Imaginary parts of linear-stability eigenvalues, illustrating the Hopf bifurcation that occurs near to the ${\omega }_a={\omega }_b$ line. Note that, as shown in panel (b), the Hopf bifurcation is slightly displaced from ${\omega }_a={\omega }_b$, but the boundary where a mode switches from stable $\left[\text{Im}\right(\lambda )<0]$ to unstable $\left[\text{Im}\right(\lambda )>0]$ is precisely at ${\omega }_a={\omega }_b$, so for $\gamma =1$, the U(1) symmetry breaking only occurs precisely on this line. Plotted for ${\omega }_b=-{\omega }_a+40\mathrm{MHz}$, $\gamma =1$, and $g=3$ kHz (a) along the line illustrated in Fig. 3 and (b) close to the point ${\omega }_a={\omega }_b$.

Figure 5

(a) Long-time limit cycles of $\mathit{S}=(S^x,S^y,S^z)$ plotted on the Bloch sphere crossing the region in Fig. 3 where there are no stationary steady states. The trajectory evolves between orbits of the $\mathrm{SR}\alpha$ fixed point and that of the $\mathrm{SR}\beta$ phase. (b) Time evolution of the intensity of the light fields at the point $({\omega }_a,{\omega }_b)=(-5.5,5.5)$ MHz showing persistent oscillations after 4 ms of time evolution.

Figure 6

Evolution of the phase diagram in ${\omega }_a\text{−}{\omega }_b$ plane, at $g=3\mathrm{kHz}$, as one varies $\gamma$ from $\gamma =0$, as shown in Fig. 2 to $\gamma =1$, as shown in Fig. 3.

Figure 7

${\omega }_a$ vs $\gamma$ phase diagram for $g=3\mathrm{kHz}$ and ${\omega }_b=5\mathrm{MHz}$.

Figure 8

Level scheme, as in Fig. 1, showing bare parameters.