Nonequilibrium transient dynamics of photon statistics

We investigate the transient dynamics of photon statistics through two-time correlation functions for optical field coupled to a non-Markovian environment, described by the Fano-type Hamiltonian. We exactly solve the time-evolution of an initially nonclassical state which exhibits photon antibunching. We find that the transient correlations at different times $t$ yield a smooth transition from antibunching to bunching photon statistics in the weak system-environment coupling regime. In the strong-coupling regime, the two-time correlations exhibit oscillations that persists both in the transient process and in the steady-state limit. The oscillatory behavior of photon statistics is a manifestation of strong non-Markovian memory dynamics where the system remains in nonequilibrium from its environment. We also find that the antibunching-to-bunching transition in the weak-coupling regime and the photon statistical oscillations in the strong-coupling regime are strongly influenced by the environment temperature.

Figure 1

Transient dynamics of $g^{\left(2\right)}(t,t+\tau )$ is shown for two different system-environment coupling strengths: (a) the weak coupling $\left(\eta =0.5{\eta }_c\right)$ and (b) the strong coupling $\left(\eta =1.5{\eta }_c\right)$. The other parameters are taken as ${\omega }_c=5{\omega }_0,k_{B}T=2.0\hslash {\omega }_0$, and the system is considered to be in an initial photon number state $|n_0\rangle =|5\rangle$.

Figure 2

Effect of environmental temperature on the transient dynamics of $g^{\left(2\right)}(t,t+\tau )$ for two different system-environment coupling strengths: (a) weak coupling $\left(\eta =0.5{\eta }_c\right)$ and (b) strong coupling $\left(\eta =1.5{\eta }_c\right)$. Different curves (dotted black for $T_s=1$, dashed green for $T_s=2$, dashed-dotted blue for $T_s=5$, and solid red for $T_s=100)$ represent different values of $T_s$ shown by the color legends. The other parameters are taken as ${\omega }_c=5{\omega }_0,{\omega }_{0}t=1$, and the system is considered to be in an initial photon number state $|n_0\rangle =|5\rangle$.

Figure 3

Three-dimensional plot of $g^{\left(2\right)}(t,t+\tau )$ in the steady-state limit $\left(t\to \infty \right)$ with varying temperature $T_s$ is shown for (a) weak coupling $\left(\eta =0.5{\eta }_c\right)$ and (b) strong coupling $\left(\eta =1.5{\eta }_c\right)$. We also plot $g^{\left(2\right)}(t,t+\tau )$ as a function of $\tau$ in the steady-state limit $\left(t\to \infty \right)$ with varying coupling strength $\eta$ (c) in the weak-coupling regime $\left(\eta <{\eta }_c\right)$ and (d) in the strong-coupling regime $\left(\eta >{\eta }_c\right)$. The other parameters are taken as ${\omega }_c=5{\omega }_0$, and the system is considered to be in an initial photon number state $|n_0\rangle =|5\rangle$.

Figure 4

$g^{\left(2\right)}(t,t+\tau )$ as a function of $\tau$ in the steady-state limit $\left(t\to \infty \right)$ with varying temperature $T_s$ (dotted black for $T_s=1$, dashed green for $T_s=5$, dashed-dotted blue for $T_s=10$, and solid red for $T_s=100)$ in a strong coupling $\left(\eta =1.5{\eta }_c\right)$ for the optical field initially be a coherent state $|\alpha \rangle$ with coherent amplitude $\alpha =\sqrt{2}$, i.e., the average photon number in the initial state is two. The other parameters are taken as ${\omega }_c=5{\omega }_0$.