Microscopic derivation of the Jaynes-Cummings model with cavity losses

In this paper we provide a microscopic derivation of the master equation for the Jaynes-Cummings model with cavity losses. We single out both the differences with the phenomenological master equation used in the literature and the approximations under which the phenomenological model correctly describes the dynamics of the atom-cavity system. Some examples wherein the phenomenological and the microscopic master equations give rise to different predictions are discussed in detail.

Figure 1

Populations $P_{0,g}\left(t\right)$ and $P_{0,g}^{\mathrm{ph}}\left(t\right)$ vs $\tau =2\Omega t$, when the system starts from the state $\mid 0,e⟩$, with $\gamma ∕2\Omega =0.1$. The solid line refers to the predictions given by the phenomenological master equation, while the dashed line refers to the predictions given by the microscopic one. In the inset it is shown the long time behavior of the same quantities, for $0⩽\tau ⩽100$.

Figure 2

Populations $P_g\left(t\right)$ and $P_g^{\mathrm{ph}}\left(t\right)$ vs $\tau =2\Omega t$ for the initial state $\mid 0,e⟩$, with $\gamma ∕2\Omega =0.1$. The solid line refers to the predictions given by the phenomenological master equation, while the dashed line refers to the predictions given by the microscopic one. In the large panel, where one can see the long time behavior, the two lines are indistinguishable, while in the inset we have emphasized the dynamics for $21⩽\tau ⩽26$, to put into evidence the presence of the frequency shift.

Figure 3

Populations $P_g\left(t\right)$ and $P_g^{\mathrm{ph}}\left(t\right)$ vs $\tau =2\Omega t$, when the system starts from the state $\mid E_{1,+}⟩$, with $\gamma ∕2\Omega =0.1$. The solid line refers to the predictions given by the phenomenological master equation, while the dashed line refers to the predictions given by the microscopic one. In the inset one can see the long time behavior of the same quantities for $0⩽\tau ⩽100$.