Metastability in the driven-dissipative Rabi model

We explore the long-time dynamics of the quantum Rabi model in a driven-dissipative setting and show that, as the atom-cavity coupling strength becomes larger than the cavity frequency, a new time scale emerges. This time scale, much larger than the natural relaxation time of the atom and the cavity, leads to long-lived metastable states susceptible to being observed experimentally. By applying a Floquet-Liouville approach to the time-dependent master equation, we systematically investigate the set of possible metastable states. We find that the properties of the metastable states can differ drastically from those of the steady state and relate these properties to the energy spectrum of the Rabi Hamiltonian.

Figure 1

(a) Output intensity $I_{\mathrm{out}}$ and (b) second-order correlation function $g^{\left(2\right)}\left(0\right)$ as a function of $g/{\omega }_c$ for different simulation times $\tau$. The external driving field is resonant with the transition $|{\mathrm{\Psi }}_0^{+}\rangle \to |{\mathrm{\Psi }}_1^{-}\rangle$ and its intensity is $F/\gamma =0.1$. The dissipation parameters are $\gamma =\kappa ={10}^{-2}{\omega }_c$.

Figure 2

Long-time dynamics. (a) $I_{\text{out}}$ as a function of time for different values of the coupling strength. For each point the result is obtained by averaging over one period (of the driving field). There is a clear emergence of metastability as $g$ is increased. (b) The quantity $\delta I_{\text{out}}=|I_{\text{out}}\left(\tau \right)-I_{\text{out}}\left(\infty \right)|/I_{\text{out}}\left(\infty \right)$ as a function of time. The same averaging procedure over one period is applied. Black dotted lines are exponential fits $\propto e^{{\mathrm{\Omega }}_{0,1}\tau }$, where ${\mathrm{\Omega }}_{0,1}$ is the nonzero eigenvalue of the Floquet-Liouvillian with the smallest absolute real part.

Figure 3

Separation of time scales and Liouvillian gap. Real part of the first three nonzero eigenvalues, ${\mathrm{\Omega }}_{1,0}$ (dashed blue line), ${\mathrm{\Omega }}_{2,0}$ (solid red line), and ${\mathrm{\Omega }}_{3,0}$ (yellow dotted line), as a function of $g/{\omega }_c$. The eigenvalues are labeled in such a way that $|\mathrm{Re}\left[{\mathrm{\Omega }}_{\alpha ,0}\right]|<|\mathrm{Re}{\mathrm{\Omega }}_{\alpha +1,0}|$.

Figure 4

Extremal metastable states. The output intensity and $g^{\left(2\right)}\left(0\right)$ of all possible metastable states for an arbitrary initial state lie in between the extremal metastable states values (the orange dashed line and the yellow solid line), which can be drastically different from the steady-state value (blue dotted line). The observation time is set to $\tau \gamma =1\times {10}^3$.

Figure 5

(a) Energy spectrum of the Rabi Hamiltonian (without driving). Black dotted lines indicate energy levels with an even number of excitations while red solid lines correspond to an odd number of excitations. Arrows show the available decay channels when driving the second transition $|{\mathrm{\Psi }}_0^{+}\rangle \to |{\mathrm{\Psi }}_1^{-}\rangle$. The colors match the one used in the lower panel for the transition rates. (b) Transition rates between the different dressed states, ${\chi }_{00}^{+-}$ (green squares), ${\chi }_{11}^{+-}$ (blue circles), ${\chi }_{01}^{-+}$ (inverted purple triangles), and ${\chi }_{01}^{+-}$ (yellow triangles), as a function of the coupling strength.

Figure 6

Liouvillian gap when dephasing noise is included. Real part of the first three nonzero eigenvalues, ${\mathrm{\Omega }}_{1,0}$ (solid blue line), ${\mathrm{\Omega }}_{2,0}$ (dashed red line), and ${\mathrm{\Omega }}_{3,0}$ (yellow dotted line), as a function of $g/{\omega }_c$ when dephasing noise is included. The eigenvalues are labeled in such a way that $|\mathrm{Re}\left[{\mathrm{\Omega }}_{\alpha ,0}\right]|<|\mathrm{Re}{\mathrm{\Omega }}_{\alpha +1,0}|$. The noise parameters are ${\gamma }_{\varphi }\left({\mathrm{\Delta }}_{jk}^{pp}\right)=\gamma$.