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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? $$H_2=\omega_{c1} a^{\dagger}a+\omega_{c2} b^{\dagger}b+\omega_a\sigma_z+\Omega_1(a^{\dagger}\sigma_{-}+a\sigma_{+})+\Omega_2(b^{\dagger}\sigma_{-}+b\sigma_{+}).$$

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  • $\begingroup$ What have you tried? $\endgroup$ Mar 23 at 13:17
  • $\begingroup$ @JasonFunderberker I tried to solve it in the same way as people use when solving single-harmonics oscillator. But that approach doesn't seem to work. $\endgroup$
    – Tan Tixuan
    Mar 23 at 13:19

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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.

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  • $\begingroup$ Right, this is what I had in mind. But I was afraid the fact that the coupling matrix grows size with the excitation is not a good thing, because it severely limits our numerical capability. $\endgroup$
    – Tan Tixuan
    Mar 23 at 14:34
  • $\begingroup$ Hi, thanks for the answer again. Are you aware of any approximation to solve the dynamics reduced density matrix of the two-level system when the initial environment state is a coherent state? $\endgroup$
    – Tan Tixuan
    Mar 24 at 6:59
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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)

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  • $\begingroup$ While this link may answer the question, it is better to include the essential parts of the answer here and provide the link for reference. Link-only answers can become invalid if the linked page changes. - From Review $\endgroup$
    – Miyase
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