The Bloch-Redfield master equation is one of the foundamental time-local EOM in open quantum system, which considers only the Born-Markov approximation.
Consider the interaction Hamiltonian between the system and environment, which is given by
$$\hat{H}_\text{int} = \sum_\alpha \hat{A}^\alpha \otimes \hat{B}^\alpha$$
where $\hat{A}^\alpha$ and $\hat{B}^\alpha$ are the operators acting on the system and environment, respectively. The Bloch-Redfield master equation can be expressed in the eigenbasis (labeled as $a,b,c,\cdots$) of the bare system Hamitonian, namely
$$\begin{aligned}
\dot{\rho}_{ab} = -i\omega_{ab} \rho_{ab} -\frac{\hbar^{-2}}{2} \sum_{\alpha,\beta}\biggl[
&\sum_n \hat{A}^\alpha_{an} \hat{A}^\beta_{nc} \rho_{cd} \delta_{db} S^{\alpha\beta}(\omega_{cn})
- \hat{A}^\beta_{ac} \rho_{cd} \hat{A}^\alpha_{db} S^{\alpha\beta}(\omega_{ca})\\\
&+\sum_n \delta_{ac} \rho_{cd} \hat{A}^\alpha_{dn} \hat{A}^\beta_{nb} S^{\alpha\beta}(\omega_{dn})
- \hat{A}^\beta_{ac} \rho_{cd} \hat{A}^\alpha_{db} S^{\alpha\beta}(\omega_{db})
\biggr],
\end{aligned}$$
where $S^{\alpha\beta}(\omega) = \int_{-\infty}^{\infty} d\tau e^{i\omega\tau} \langle \hat{B}^\alpha(\tau) \hat{B}^\beta(0)\rangle$ is the environmental spectrum. The entire right-hand side can be realized as a superoperator matrix called Bloch-Redfield tensor.
Thus, to align with brmesolve in Python QuTiP, the followings should be implemented:
- function
brterm: the contribution of one coupling operator (specific $\alpha$) to the Bloch-Redfield tensor.
- function
bloch_redfield_tensor: Constructs the entire Bloch-Redfield tensor as a single SuperOperator.
- function
brmesolve: The solver which solves the time evolution based on Bloch-Redfield tensor.
- the docstrings for the above functions
- Corresponding runtests for the above functions
Note that the function names, argument names, argument orders, and user experience should all align with Python QuTiP.
