# Hamiltonian in a Master Equation

I am going through this paper on the complete positive map with memory. The bath operators $$\Gamma_k (t)$$ are told to satisfy the correlation $$\langle \Gamma_j(t) \Gamma_k(t^\prime) \rangle = a_k^2 e^{-|t-t^\prime|/\tau_k} \delta_{jk}$$. However, the type of reservoir (whether it is thermal or something else) is not mentioned. How could one figure out the Hamiltonian of the reservoir used in this case? The only Hamiltonian given is eq-8 which is the interaction Hamiltonian ($$H_I = \hbar \sum\limits_{i=1}^3 \Gamma_i(t) \sigma_i$$, they use the symbol $$H$$ for $$H_I$$).

My question: How to figure out the system and environment Hamiltonians given the correlation function and the interaction Hamiltonian?

• Presumably, interaction Hamiltonian seems to be written in interaction picture with respect to free reservoir Hamiltonian. Commented Mar 25, 2019 at 20:17

There is no interaction picture. There is a time-dependent Hamiltonian given, as you say, by

$$H(t) = \hbar \sum_j \Gamma_j(t) \sigma_j .$$

The classical fields $$\Gamma_j(t)$$ are generated with some other process, which is not shown. These fields are in fact Gaussian random fields with a certain covariance matrix. In a sense we have a quantum system interacting with a classical one. This is indeed a particular case of open system (simply the reservoir is classical). In any case if the time scale associated with the classical (external) bath is very short, you can "integrate out" the classical, external bath and write a master equation for the quantum system, their Eq. (10). Although they don't say that explicitly, this requires a certain approximation to be valid.

• Thanks, @lcv. The knowledge of the bath is essential since the form of the correlations is determined by the nature of the bath/environment. Commented Mar 26, 2019 at 9:35
• The bath is a classical Gaussian, quadratic form in $\Gamma$. It is simply the inverse of the covariance matrix
– lcv
Commented Mar 26, 2019 at 9:46
• Could you kindly provide some details in the answer? Commented Mar 26, 2019 at 9:56
• Yes absolutely. I can be more explicit tomorrow (it's a bit late one the West coast)
– lcv
Commented Mar 26, 2019 at 9:57
• Thank you. It would help if you could also comment on the form of the correlation <\Gamma(t_j) \Gamma_k(t')> the way it appears in the paper. Commented Mar 26, 2019 at 10:00