Temporal correlations of a driven 2LS

The dynamics of a driven two-level system (2LS) is described through a Hamiltonian (in the frame rotating with the frequency of the laser, and setting $\hbar=1$): \begin{equation} \label{eq:ThuSep21152420CEST2017} H = (\omega_\sigma-\omega_\mathrm{L}) \sigma^\dagger \sigma + \Omega (\sigma + \sigma^\dagger)\,. \end{equation} Here $\sigma = |g\rangle \langle e|$ is the operator that lowers the state of the 2LS, $\omega_\sigma$ is the frequency of the 2LS and $\omega_\mathrm{L}$ is the frequency of the laser that drives the 2LS with intensity $\Omega$. The description is done by supplementing the Hamiltonian dynamics with the dissipation, introducing the master equation in the Lindablad form: \begin{equation} \label{eq:ThuSep21152659CEST2017} \partial_t \rho = i[\rho,H] + \frac{\gamma_\sigma}{2} \mathcal{L}_\sigma\rho + \frac{P_\sigma}{2} \mathcal{L}_{\sigma^\dagger}\rho\,, \end{equation} where $\gamma_\sigma$ and $P_\sigma$ are the rates of decay and incoherent driving of the 2LS, respectively; and $\mathcal{L}_c = 2c\rho c^\dagger - c^\dagger c\rho - \rho c^\dagger c$.

Incoherent driving

For this case, we solve the master equation in (\ref{eq:ThuSep21152659CEST2017}) setting $\Omega=0$ and $\omega_\mathrm{L}=0$, and we find that the second order correlation of the 2LS is given by \begin{equation} \label{eq:ThuSep21153253CEST2017} g^{(2)}(\tau) = 1- e^{-(\gamma_\sigma+P_\sigma)\tau}\,. \end{equation}

When the light emitted is filtered in frequency with a detector of width $\Gamma$ at resonance with the 2LS, the correlation becomes \begin{equation} \label{eq:FriSep14155921CEST2018} g^{(2)}_\Gamma (\tau) = 1 + \big[ 2 e^{-(\Gamma+\Gamma_\sigma)\tau/2}\, \Gamma \Gamma_\sigma (5\Gamma -\Gamma_\sigma) - e^{-\Gamma_\sigma \tau} \,\Gamma^2(3\Gamma + \Gamma_\sigma) + e^{-\Gamma \tau} \,\Gamma_\sigma (\Gamma_\sigma^2 - 3\Gamma \Gamma_\sigma -2\Gamma^2)\big] \big/\big [ (\Gamma-\Gamma_\sigma)^2 (3\Gamma+\Gamma_\sigma)\big]\,, \end{equation} where $\Gamma_\sigma = \gamma_\sigma+P_\sigma$. Furthermore, when the light is collected over all the frequencies, the correlations become \begin{equation} \label{eq:FriSep14160942CEST2018} g^{(2)}_\Gamma (\tau) = 1 -\frac{\Gamma}{\Gamma^2 - \Gamma_\sigma^2} \left(\Gamma e^{-\Gamma_\sigma \tau}-\Gamma_\sigma e^{-\Gamma \tau} \right )\,. \end{equation}

Coherent driving

For this case, we solve the master equation in (\ref{eq:ThuSep21152659CEST2017}) setting $P_\sigma=0$ and $\omega_\mathrm{L}=\omega_\sigma$, and we find that the second order correlation of the 2LS is given by \begin{equation} \label{eq:ThuSep21153341CEST2017} g^{(2)}(\tau) = 1- e^{-3\gamma_\sigma\tau/4}\Big [\cos(R\tau/4) + \frac{3\gamma_\sigma}{R} \sin(R\tau/4) \Big]\,, \end{equation} with $R=\sqrt{64\Omega^2-\gamma_\sigma^2}$. In the limit of vanishing driving, i.e., $\Omega\rightarrow 0$, the correlation becomes \begin{equation} \label{eq:ThuSep21153813CEST2017} g^{(2)}(\tau) = \big( 1- e^{-\gamma_\sigma\tau/2}\big)^2\,, \end{equation} which approaches 1 slower than the analogue version of Eq. (\ref{eq:ThuSep21153253CEST2017}) (in which $P_\sigma\rightarrow 0$).

Likewise, when the light emitted is filtered with a finite width $\Gamma$ but collecting the light over all the frequencies, the correlations become \begin{equation} \label{eq:FriSep14161659CEST2018} g^{(2)}_\Gamma (\tau) = 1 - \frac{\Gamma}{4\Gamma^4 - 5\Gamma^2 \gamma_\sigma^2 +\gamma_\sigma^4} \big [ 3 \gamma_\sigma^3 e^{-\Gamma \tau}+ 8\Gamma (\Gamma^2-\gamma_\sigma^2)e^{-\gamma_\sigma \tau/2} - \Gamma(4\Gamma^2-\gamma_\sigma^2) e^{-\gamma_\sigma \tau} \big]\,. \end{equation}

\begin{multine} \label{eq:FriSep14161659CEST2018} g^{(2)}_\Gamma (\tau) = 1 - \frac{\Gamma}{4\Gamma^4 - 5\Gamma^2 \gamma_\sigma^2 +\gamma_\sigma^4} \big [ 3 \gamma_\sigma^3 e^{-\Gamma \tau}+\\ 8\Gamma (\Gamma^2-\gamma_\sigma^2)e^{-\gamma_\sigma \tau/2} - \Gamma(4\Gamma^2-\gamma_\sigma^2) e^{-\gamma_\sigma \tau} \big]\,. \end{multilane}