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I have a question about 2 (or more) atoms coupled to an optical cavity.

The equations of motion in the interaction picture for 2 atoms in resonnance with an optical cavity starting from 1 excited atom (C_{eg,0}(t_0) = 1 )

(g is the coupling constant)

\begin{cases}

\dot{C}_{eg,0} &= - ig e^{i(\vec{k}\cdot \vec{r}_1)}C_{gg,1}\\

\dot{C}_{ge,0} &= - ig e^{i(\vec{k}\cdot \vec{r}_2)}C_{gg,1}\\

\dot{C}_{gg,1} &= -ig e^{-i(\vec{k}\cdot \vec{r}_1)}C_{eg,0} - ig e^{-i(\vec{k}\cdot \vec{r}_2)}C_{ge,0}

\end{cases}

Coming from an interaction picture Hamiltonian:

V_2 = \left[

\begin{matrix}

0 & 0 & g \\

0 &0 & g \\

g & g & 0 \\

\end{matrix}

\right]

But if i put this system in a damped cavity, something weird happens.

Instead of the system just dying down, the probability of the atoms being excited converges to 1/4.

The problem is that C_{eg0} and C_{ge0} have a phase difference of π and (C_{eg0}+C_{ge0}) \rightarrow 0 and therefore \dot{C}_{gg,1} \rightarrow 0 and the system stalls.

This is the evolution, starting from a state |eg0> (meaning first atom excited, second ground and 0 photons in the cavity) Pruduced with a standard Master equation.

As you can see, the purple line presents the state with the first atom being excited, and the yellow line the second. (the blue line is the ground state which increases as the photons leave the cavity)

For the rest, the Master equation behaves exactly as it should, a photon without atoms decays exponentially with the correct decay constant.

A closed cavity produces the correct Tavis-Cummings evolution,... everything else works just fine.

The atoms are coupled to a damped cavity so, the atoms should de-excite completely just like they do if I put them in the cavity alone. I don't see a physical reason why and how two atoms could remain somewhat excited.

Can it be that there is something wrong with the Hamiltonian? Although this Hamiltonian produces the correct evolution in all cases if there is no damping.

I must be overlooking something, but what?

Thank you for any insight.

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# Two atoms in an optical cavity.

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