# 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

## 1 Answer

You get a Master equation with an additional drive term that explicitly remains in the Master equation (see e.g. https://link.springer.com/book/10.1007/978-3-319-62843-1). This term is semi-classical in the sense that it involves the expectation value of the input field. For coherent input states, this is an applicable approximation.

More generally, what you are looking for is input-output theory, which studies what happens when you populate the bath of an open quantum system with interesting states (e.g. as you do in spectroscopy). There are various approaches there. Some of them involve Master, others only use Heisenberg-Langevin equations.

• So the punch line is, the BM approximation does not apply to this coherent bath, correct? By the way, I would actually be very grateful if you could point to some papers that study relevant problems (I had a look on the book you cited, it is a very long thesis, so I am very confused where I should start with) May 20, 2022 at 13:25
• The BM approximation can apply, only your Master equation looks different afterwards. This is not unusual, Master equations generally depend on the initial state of the bath (e.g. vacuum vs thermal). In the Springer Theses I can recommend chapter 2.3.5 and 2.3.6, they feature a nice presentation of the coherent state case. May 20, 2022 at 13:47
• @Wolpertinger I don't think input-output formalism really solves the problem: it separates some "interesting" modes from the rest, since but one still needs a mechanism that justifies short coherence time / randomization / truncation of equations, etc. (depending on the formalism one uses). Input-output in this context seems like kicking the can down the road. Again, I acknowledge that optics is not my primary field, but I doubt that the infinite number of modes alone is sufficient for the applicability of Born-Markov approximation. May 20, 2022 at 13:49
• @RogerVadim The applicability of the Born-Markov approximation does not depend on the bath state, it depends mostly on the system-bath Hamiltonian. May 20, 2022 at 14:12
• @Wolpertinger I think you need to expand your answer, and support it with examples and actual statements - then we will have something to talk about. For now it is essentially "reference only" answer. No offense intended. May 20, 2022 at 14:20