# Master equation with a coherent bath

When we consider an oscillator $$a$$ acting with a bath of oscillators $$b_i$$ with the interaction Hamiltonian reads $$H_{int}=\sum_{i}g_ia b_i^{\dagger}+g_i^*a^{\dagger}b_i,$$ with the free Hamiltonian: $$H_{free}=\omega_0a^{\dagger}a+\sum_{i}\omega_ib^{\dagger}b$$

It is a standard text book exercise (See, for example, Chapter 7 of this book) to derive the Born-Markov master equation for the density matrix of oscillator $$a$$ when the bath oscillator $$b_i$$ are all in thermal ensemble, i.e. $$\rho_{b_i}\propto exp(-\beta b^{\dagger}b)$$

However, I also tried a different scenario, when the bath are in a pure coherent ensemble instead of thermal ensemble, i.e.

$$\rho_{b_i}\propto |\alpha_i\rangle\langle \alpha_i|$$ $$\rho_{bath}=\otimes_i |\alpha_i\rangle\langle \alpha_i|$$

Where $$|\alpha_i\rangle$$ is the standard quantum harmonic oscillator coherent state. However, it seems that no similar Born-Markov master equation can be derived in this case due to some ill-defined integral in the Born-Markov master equation. Do anyone know any reference for this problem?

Edit: It seems that this problem of applying the Born-Markov approximation resides in the fact that there is some coherence time scale which is very large for the bath. I would actually be very grateful if some one can make this statement more explicit(e.g. by stating what are the things to calculate so that we can evaluate whether BM is a good approximation.)

• Can you really call it bath in this case? More importantly: does the Born-Markov approximations really apply? - They are grounded in thermodynamic reasoning, which implies residual interactions (not explicitly included in the Hamiltonian) leading to relaxation to thermal equilibrium. May 18, 2022 at 10:58
• @RogerVadim Could you be more specific? I thought that the Born-Markov approximation can be applied as long as the bath is large enough and it is irrespective of the form of the density matrix of the bath. So I can apply it regardless whether it is a thermal bath or a coherent bath. But my understanding may be wrong. May 18, 2022 at 11:06
• I am not a great expert here, but my understanding is that Born-Markov implies a short coherence time of the bath... without thermodynamic assumptions the coherence time is very long or even infinite - you can calculate the correlation function yourself. May 18, 2022 at 11:41
• @RogerVadim I don't think these objections are valid. What the OP is asking about is essentially input-output theory (see my answer). May 20, 2022 at 13:14