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

• What have you tried? Mar 23 at 13:17
• @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. Mar 23 at 13:19

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.

• 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. Mar 23 at 14:34
• 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? Mar 24 at 6:59

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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