Driven-Dissipative Time Crystalline Phases in a Two-Mode Bosonic System with Kerr Nonlinearity

For the driven-dissipative system of two coupled bosonic modes in a nonlinear cavity resonator, we demonstrate a sequence of phase transitions from a trivial steady state to two distinct dissipative time crystalline phases. These effects are already anticipated at the level of the semiclassical analysis of the Lindblad equation using the theory of bifurcations and are further supported by the full quantum treatment. The system is predicted to exhibit different dynamical phases characterized by an oscillating nonequilibrium steady state with nontrivial periodicity, which is a hallmark of time crystals. We expect that these phases can be directly probed in various cavity QED experiments.

Figure 1

Schematic representation of our setup: A cavity with two bosonic modes $a$ (orange cloud) and $b$ (blue cloud) coupled with a strength $g$ to each other, see Eq. (1). A Kerr interaction with a strength $U$ is generated by a nonlinear element (red square) for the $b$ mode. The cavity is driven coherently by a single photon drive $E_1$. The cavities allow for the decay of the modes with the single-photon rates ${\gamma }_{a,b}$ and two-photon rate ${\chi }_b$ (orange and blue arrows).

Figure 2

(a) Solid (dashed) red curve depicts the semiclassical stable (unstable) steady state solution for the $b$-mode particle number, $\sqrt{n_b}=|z|$, as a function of the driving amplitude ${\mathcal{E}}_1$. The points $H_1$ and $H_2$ are sub- and supercritical Hopf bifurcations, respectively. Stable (unstable) limit cycles emerging from $H_2$ ($H_1$) are depicted using the solid (dashed) blue lines, indicating the absolute values in between which the oscillations occur. Stable and unstable limit cycles annihilate at the limit point cycle (LPC). $S{N}_{1,2}$ are saddle-node points of the optical bistability. ${\mathrm{PD}}_1$ (${\mathrm{PD}}_2$) corresponds to a period-doubling bifurcation when passing this point from upper (lower) values of ${\mathcal{E}}_1$. Between the points ${\mathrm{PD}}_1$ and ${\mathrm{PD}}_2$ is the region of limit cycles with a double loop structure as shown in (c). The magenta and green arrows indicate the forward and backward sweeping paths as outlined in the main text. (b),(c) Examples of normal and period doubled limit cycles (solid blue line) and unstable limit cycle (dashed blue line) in the ($\mathrm{Re}[z]$, $\mathrm{Im}[z]$) plane. The red dot represents a stationary state lying on the lower red curve from (a). (d),(e) The Fourier spectrum of $\mathrm{Re}[z]$ of the dynamics is represented by stable limit cycles shown in (b) and (c). The system parameters are ${\gamma }_a={\chi }_b=1$, $g=U={\mathrm{\Delta }}_a=10$ and ${\mathrm{\Delta }}_b=-20$ (computed in units of ${\gamma }_b$).

Figure 3

(a) The dissipative gap as a function of the driving amplitude ${\tilde{\mathcal{E}}}_1/{\gamma }_b$. Qualitatively different phases are colored in by hand, based on the underlying gap closure behaviors. (b) Eigenvalues of the first few decaying modes ${\lambda }_i$ for ${\tilde{\mathcal{E}}}_1/{\gamma }_b=5$ and different values of $N$. Here, the gap does not close as we increase $N$. (c) The spectrum for ${\tilde{\mathcal{E}}}_1/{\gamma }_b=13$. The arrows enclosing symbols depict how the eigenvalues converge towards the imaginary axis [$\mathrm{Re}({\lambda }_i)=0$] as a function of $N$. (d) Spectrum for ${\tilde{\mathcal{E}}}_1/{\gamma }_b=16$. An additional eigenvalue enclosed in rectangles features gap closing behavior with increasing $N$, having an imaginary part that is about half as large as the original eigenvalues enclosed within the arrows. This eigenvalue resembles the period doubling found in the semiclassical case. (e) Periods of quantum oscillations, $T=2\pi /\mathrm{Im}({\lambda }_i)$, as a function of the driving amplitude. All system parameters are chosen the same as in Fig. 2. The results from the quantum trajectories are obtained by averaging 3000 trajectories for a duration of $t{\gamma }_b=15$ with ${\gamma }_{b}dt=0.005$ and using the software provided in [61].