Quantum dynamics of the driven and dissipative Rabi model

The Rabi model considers a two-level system (or spin 1/2) coupled to a quantized harmonic oscillator and describes the simplest interaction between matter and light. The recent experimental progress in solid-state circuit quantum electrodynamics has engendered theoretical efforts to quantitatively describe the mathematical and physical aspects of the light-matter interaction beyond the rotating-wave approximation. We develop a stochastic Schrödinger equation approach which enables us to access the strong-coupling limit of the Rabi model and study the effects of dissipation and ac drive in an exact manner. We include the effect of Ohmic noise on the non-Markovian spin dynamics, resulting in Kondo-type correlations, as well as cavity losses. We compute the time evolution of spin variables in various conditions. As a consideration for future work, we discuss the possibility of reaching a steady state with one polariton in realistic experimental conditions.

Figure 1

States of the double spin path. Through the Feynman-Vernon formulation, the spin paths and elements of the spin (reduced) density matrix are depicted through the classical variables $\eta$ and $\xi$.

Figure 2

Spin path $\eta \left(t\right)={\sum }_{j=0}^{2n}{{\rm Y}}_j\theta (t-t_j)$ in red; $\xi \left(t\right)={\sum }_{j=1}^{2n}{\Xi }_j\theta (t-t_j)$ in dashed blue. The spin path starts and ends in the state $A$.

Figure 3

Spin path $\eta \left(t\right)={\sum }_{j=0}^{2n-1}{{\rm Y}}_j\theta (t-t_j)$ in red; $\xi \left(t\right)={\sum }_{j=1}^{2n-1}{\Xi }_j\theta (t-t_j)$ in dashed blue. Here the spin path ends in the blip state $B=|+-\rangle$. The initial state is still $A=|++\rangle$.

Figure 4

Spin path $\eta \left(t\right)={\sum }_{j=1}^{2n}{{\rm Y}}_j\theta (t-t_j)$ in red; $\xi \left(t\right)={\sum }_{j=0}^{2n}{\Xi }_j\theta (t-t_j)$ in dashed blue. Here the spin path starts and ends in the blip state $B=|+-\rangle$.

Figure 5

(a) JC ladder of polaritons. (b) Absolute value of the Bloch-Siegert shift $|\overline{\delta }|$ versus $g/{\omega }_0$, with perturbation theory in red [106], and results from our method in blue (dots). (c) Example of dynamics of $\langle {\sigma }^z\rangle$ with the initial condition $|{+}_z\rangle$, which is a linear superposition of $|1-\rangle$ and $|1+\rangle$, for quite strong couplings. Parameters are set to $g/{\omega }_0=0.7$ and $\Delta /{\omega }_0=0.2$. The Rabi solution from our method is shown in dashed blue; within the RWA; the JC solution would rather read $\langle {\sigma }^z\left(t\right)\rangle =1-2g^2{sin}^2(\sqrt{g^2+{\delta }^2}t/2)/(g^2+{\delta }^2)$ and is shown in red.

Figure 6

Dynamics of $\langle {\sigma }^x\rangle$ with the initial condition $|{+}_x\rangle$, for $\Delta /{\omega }_0=0.1$ and $g/{\omega }_0=0.3$. The analytical solution within the adiabatic approximation is $\langle {\sigma }^x\left(t\right)\rangle =cos\Omega t$. Inset: $\Omega /\Delta =exp(-g^2/2{\omega }_0^2)$ versus $g/{\omega }_0$; the red curve is from the exact result in the adiabatic limit while the dots represent our numerical results.

Figure 7

Dynamics of $\langle {\sigma }^z\rangle$ with the initial condition $|{+}_z\rangle$ which is a linear superposition of the first two polaritons. We consider $g/{\omega }_0=0.01$, $\Delta /{\omega }_0=0.97$, ${\omega }_c/{\omega }_0=100$, and several values of $\alpha$ until $\alpha \approx 0.1$. We observe a relaxation towards a nontrivial final state by increasing $\alpha$ and the value of $\langle {\sigma }^z\rangle$ is in accordance with Bethe ansatz calculations [110, 111, 112, 120, 121].

Figure 8

Dynamics of $\langle {\sigma }^z\rangle$ with the initial condition $|{+}_z\rangle$. The blue curve shows the damped evolution in the presence of photon leakage out of the cavity whereas the dashed red curve represents the undamped case. Parameters are $g/{\omega }_0=5.0\times {10}^{-2}$, $\Delta /{\omega }_0=0.8$, $\Gamma /{\omega }_0=1.0\times {10}^{-5}$, and $\alpha =0$.

Figure 9

Dynamics of $\langle {\sigma }^z\rangle$ with the system initially in its ground state; parameters are $g/{\omega }_0=0.02$, $\Delta /{\omega }_0=0.9$, and $V_0/{\omega }_0=0.1$. A driving ac field is applied until the system has reached the first lower polariton; $t_s$ refers to the time at which we switch off the drive. The blue curve is the ideal dissipationless case $(\alpha =0)$ and diagonalization in the dressed-states basis gives the same result with the resolution of the figure. The dashed red curve is for $\alpha ={10}^{-5}$ and the dotted green curve for ${10}^{-4}$; ${\omega }_c=100{\omega }_0$ (see also the inset).

Figure 10

Dynamics of the mean number of polaritons $\langle N\left(t\right)\rangle$ in the weak-coupling $g$ limit, and without dissipation. Parameters are $g/{\omega }_0=0.02$, $\Delta /{\omega }_0=0.9$, and $V_0/{\omega }_0=0.1$. From the ground state, the system is brought into a nontrivial polaritonic final state by driving the cavity. The black dashed line refers to the moment when the ac coherent drive is switched off. Inset: Standard deviation with the same parameters.

Figure 11

Probabilities of level occupancies for the driven and dissipative case. The probability of finding the system in the ground state is in blue. The probabilities corresponding to the “$-$” subladder are in green, while the probabilities corresponding to the “+” subladder are in red. The system initially is in its ground state; parameters are $g/{\omega }_0=0.02$, $\Delta /{\omega }_0=0.9$, $V_0/{\omega }_0=0.1$, and $\kappa /{\omega }_0=0.04$. A driving ac field is applied; $t_s$ refers to the time at which we switch off the drive. We remark that the green “plateau” with one polariton $|1-\rangle$ is far from 1 and drops off rapidly. The polaritonic state is not stable.