Cavity elimination in cavity-QED: a self-consistent input-output approach

Abstract

Simplifying composite open quantum systems through model reduction is central to enable their analytical and numerical understanding. In this work, we introduce a self-consistent approach to eliminate the cavity degrees of freedom of cavity quantum electrodynamics (CQED) devices in the non-adiabatic regime, where the cavity memory time is comparable with the timescales of the atom dynamics. To do so, we consider a CQED system consisting of a two-level atom coupled to a single-mode cavity, both subsystems interacting with the environment through an arbitrary number of ports, within the input-output formalism. A self-consistency equation is derived for the reduced atom dynamics. This allows retrieving an exact expression for the effective Purcell-enhanced emission rate and, under reasonable approximations, a set of self-consistent dynamical equations and input-output relations for the effective two level atom. The resulting reduced model captures non-Markovian features, characterized through an effective Lindblad equation exhibiting two decoherence rates, a positive and a negative one. In the continuous-wave excitation regime, we benchmark our approach by computing effective steady states and output flux expressions beyond the low-power excitation regime, for which a semi-classical treatment is usually applied. We also compute two-time correlations and spectral densities, showing an excellent agreement with full cavity quantum electrodynamics simulations, except in the strong-coupling, high-excitation regime. Our results provide a practical framework for reducing the size of CQED models, which could be generalized to more complex atom and cavity configurations.

Figures (8)

• Figure 1: Scheme of an atom coupled to a single mode cavity, both the atom and the cavity being coupled through several ports to the external environment. The atom is here coupled to the environment through four ports, each one with damping rates $\gamma_l$ with $l\in\{1,2,3,4\}$ and input-output operators $\hat{c}^{(l)}_{\textrm{in}},\hat{c}^{(l)}_{\textrm{out}}$. Similarly, the cavity is here coupled to the environment through two ports, with damping rates $\kappa_j$ with $j\in\{1,2\}$ and input-output operators $\hat{b}^{(j)}_{\rm in},\hat{b}^{(j)}_{\rm out}$. The number of ports can be arbitrarily increased to model more realistic geometries.
• Figure 2: Reflected (first column), transmitted (second column) and emitted (third column) fluxes in units of the input flux $\phi_{\rm in}$. The full CQED simulations (dotted red) are compared with the analytical $\textrm{\cancel{C}QED}$ calculations (blue) and the numerical $\textrm{\cancel{C}QED}$ simulations (dotted green). Each row corresponds to a different incoming photon flux $\phi_{\rm in}$ with (from the top to the bottom row): $N_{\rm in}$=5; 1; 0.1 and $10^{-4}$ respectively. The first set of parameters in Table 1 has been used, corresponding to a high cooperativity $C=40$ and an absence of atom-cavity detuning.
• Figure 3: Reflected (first column), transmitted (second column) and emitted (third column) fluxes in units of the input flux $\phi_{\rm in}$. The full CQED simulations (dotted red) are compared with the analytical $\textrm{\cancel{C}QED}$ calculations (blue) and the numerical $\textrm{\cancel{C}QED}$ simulations (dotted green). Each row corresponds to a different incoming photon flux $\phi_{\rm in}$ with (from the top to the bottom row): $N_{\rm in}$=5; 1; 0.1 and $10^{-4}$ respectively. The second set of parameters in Table 1 has been used, corresponding to a moderate cooperativity $C=1$ and to an atom-cavity detuning $\omega_{\rm a}-\omega_{\rm c}=0.5\kappa$.
• Figure 4: Spectral density of the scattered incoherent light, summed over all CC ports (first column) and AC ports (second column). The full CQED simulations (dotted red) are compared with the analytical $\textrm{\cancel{C}QED}$ calculations (blue) and the numerical $\textrm{\cancel{C}QED}$ simulations (dotted green). Each row corresponds to a different incoming photon flux $\phi_{\rm in}$ with (from the top to the bottom row): $N_{\rm in}$=5; 1; 0.1 and $10^{-4}$ respectively. (a): First set of parameters in Table. \ref{['table-of-parameters']}, with $C=40$ and $\omega_{\rm a}=\omega_{\rm c}$. (b): second set of parameters in Table. \ref{['table-of-parameters']}, with $C=1$ and $\omega_{\rm a}-\omega_{\rm c}=0.5\kappa$.
• Figure 5: Second-order auto-correlation functions $g^{(2)}_{\rm CC,1}$ (reflected photons, left columns) and $g^{(2)}_{\textrm{AC},l}$ (emitted photons, right columns). The full CQED simulations (dotted red) are compared with the analytical $\textrm{\cancel{C}QED}$ calculations (blue) and the numerical $\textrm{\cancel{C}QED}$ simulations (dotted green). Each row corresponds to a different incoming photon flux $\phi_{\rm in}$ with (from the top to the bottom row): $N_{\rm in}$=5; 1; 0.1 and $10^{-4}$ respectively. (a) First set of parameters in Table. \ref{['table-of-parameters']}, with $C=40$ and $\omega_{\rm a}=\omega_{\rm c}$. (b) Second set of parameters in Table. \ref{['table-of-parameters']}, with $C=1$ and $\omega_{\rm a}-\omega_{\rm c}=0.5\kappa$.