Redfield Equations

The Redfield equations are obtained from the second-order QME by making two key approximations: the bath remains in thermal equilibrium, and the system density matrix varies slowly compared to the bath correlation time. The derivation follows Section 2.2 of [Her23].

Assumptions

1. Weak system–bath coupling. The excitonic basis (eigenstates of $\hat{H}_S$) is the natural choice, since these are nearly stationary:

\[\frac{\partial}{\partial t}\rho(t) = -\frac{i}{\hbar}[H_S, \rho(t)], \qquad \rho_{\alpha\alpha}(t) \approx \text{const}, \qquad \rho_{\alpha\beta}(t) = \rho_{\alpha\beta}(0)e^{-i\omega_{\alpha\beta}t}.\]

1. Bath in equilibrium. The RBP is approximated as constant: $\hat{w}_{cd}^{(I)}(t) \approx \hat{w}_{eq}$, so the bath state factorises from the system at all times.

2. Markov approximation. The density matrix inside the memory integral is evaluated at the current time rather than at the integration variable: $\hat{\rho}^{(I)}(t - \tau) \approx \hat{\rho}^{(I)}(t)$.

Derivation

Starting from the QME for the RDM (see Quantum Master Equation) with a bath in equilibrium and switching to the Schroedinger picture for the RDM:

\[\frac{\partial}{\partial t}\hat{\rho}^{(I)}(t) = -\frac{i}{\hbar}[\hat{H}_S(t), \hat{\rho}(t)] -\frac{1}{\hbar^2}\int_0^{t}\mathrm{d}\tau\, U_S(t)\operatorname{tr}_B\bigl\{ [\hat{H}_I^{(I)}(t), [\hat{H}_I^{(I)}(t-\tau), \hat{\rho}^{(I)}(t) \oplus \hat{w}_{eq}\,|c\rangle\langle d|]] \bigr\}U_S^\dagger(t).\]

Applying the Markov approximation and expanding the double commutator using the interaction Hamiltonian $\hat{H}_I = \sum_n \Delta\hat{V}_n \hat{K}_n$ (where $\hat{K}_n = |n\rangle\langle n|$), the bath traces reduce to correlation functions:

\[\operatorname{tr}_B\{\Delta V_m(-\tau)\Delta V_n \hat{w}_{eq}\} = C_n^*(\tau)\,\delta_{nm}.\]

Final form

Defining the $\Lambda$ operator:

\[\Lambda_n(t) = \int_0^{t}\mathrm{d}\tau\, C_n(\tau) U_S(\tau) \hat{K}_n U_S^\dagger(\tau),\]

the Redfield equation in the Schroedinger picture reads

\[\frac{\partial}{\partial t}\rho(t) = -\frac{i}{\hbar}[H_S, \rho(t)] + \frac{1}{\hbar^2}\sum_n \Bigl[ K_n \rho(t) \Lambda_n^\dagger(t) + \Lambda_n(t) \rho(t) K_n - K_n \Lambda_n(t) \rho(t) - \rho(t) \Lambda_n^\dagger(t) K_n \Bigr].\]

In the excitonic basis ($|\alpha\rangle, |\beta\rangle, \ldots$ being eigenstates of $H_S$), the matrix elements of $\Lambda$ are

\[\langle\alpha|\Lambda_n(t)|\beta\rangle = \int_0^{t}\mathrm{d}\tau\, C_n(\tau) e^{-i\omega_{\alpha\beta}\tau} \langle\alpha|\hat{K}_n|\beta\rangle,\]

and the Redfield equations become

\[\frac{\partial}{\partial t}\rho(t) = -i\omega_{\alpha\beta}\rho_{\alpha\beta} + \frac{1}{\hbar^2}\sum_n \sum_{\gamma\delta}\bigl[ K_{\alpha\gamma}^n \rho_{\gamma\delta}(t)(\Lambda_{\delta\beta}^n)^*(t) + \Lambda_{\alpha\gamma}^n(t)\rho_{\gamma\delta}(t)K_{\delta\beta}^n - K_{\alpha\gamma}^n \Lambda_{\gamma\delta}^n(t)\rho_{\delta\beta}(t) - \rho_{\alpha\gamma}(t)\Lambda_{\gamma\delta}^{n*}(t)K_{\delta\beta}^n \bigr],\]

where $K_{\alpha\beta}^n = \langle\alpha|\hat{K}_n|\beta\rangle$.

Relationship to other methods

The Redfield equations are the simplest approximate QME and serve as a baseline:

• Constant ansatz ($\hat{w}^{(I)}_{ab}(t) = \hat{w}_{eq}$) gives the same bath approximation but keeps $\hat{\rho}^{(I)}(\tau)$ inside the integral (no Markov approximation). This is the QME_sI_ansatz solver with ansatz=:const_sch.
• Iterative QME systematically improves beyond Redfield by refining the bath state.
• Modified Redfield treats diagonal fluctuations non-perturbatively.

Implementation

In the package, QME_sI_Redfield solves the Redfield equations in the interaction picture, with the memory integral evaluated numerically at each time step. The correlation function $C_n(\tau)$ depends on $t - \tau$ and is constructed from the vibrational modes of each molecule.