Multi-mode Jaynes-Cummings Model The Jaynes Cummings Model describes that a qubit coupled to a harmonics oscillator. The Hamiltonian of this model can be written as
$$H_1=\omega_c a^{\dagger}a+\omega_a\sigma_z+\Omega(a^{\dagger}\sigma_{-}+a\sigma_{+}).$$
This Hamiltonian is exactly solvable, as can be found in the Wikipedia. My question is whether this model, when extended to multi-harmonic oscillator case, is exactly solvable?
i.e. whether the following Hamiltonian is exactly solvable?
$$H_2=\omega_{c1} a^{\dagger}a+\omega_{c2} b^{\dagger}b+\omega_a\sigma_z+\Omega_1(a^{\dagger}\sigma_{-}+a\sigma_{+})+\Omega_2(b^{\dagger}\sigma_{-}+b\sigma_{+}).$$
 A: I will give some hints on how the single mode case is solved, such that the solution of the multi-mode case can be found analogously.
The central realization for solving the single mode version is to note that the Jaynes-Cummings Hamiltonian is excitation number conserving. This means that states of a given excitation number only couple to states of the same excitation number.
This allows us to design an ansatz for the eigenstates of the Hamiltonian. For a given excitation number $n$, there are exactly two states that couple: $|n\rangle|g\rangle$ and $|n-1\rangle|e\rangle$. Using these states as a basis, finding the eigenstates then turns into the problem of finding the eigenvectors and -values of a 2x2 coupling matrix.
For the multi-mode case, we can straightforwardly apply this scheme. Only now, there are more states...
If we have $n$ excitations, there can either be $n$ or $n-1$ excitations in the bosonics sectors (the atom has maximum 1 excitation). However, there are $\sum_{l=0}^{n} l=\frac{n(n+1)}{2}$ ways to distribute $n$ excitations over 2 bosonic modes. Even the number of states of the resulting eigenvalue problem is therefore dependent on the excitation number and the coupling matrix has size $n(n+1)$.
As a result, this problem $-$ unlike the JC model $-$ is likely only solvable analytically for a restricted number of excitations.
A: people linked that kind of systems to the so-called para-particles (https://arxiv.org/abs/1803.00654), and even made a nice experimental demonstration using trapped ions (https://arxiv.org/abs/2108.05471)
