Redfield Equation

The Redfield equation is a second-order weak-coupling master equation for the reduced density operator of an open quantum system. In common usage it means the Born–Markov equation before full secularization, so it retains some couplings between populations and coherences or between different Bohr-frequency sectors.

That makes it useful in chemistry, condensed matter, magnetic resonance, spectroscopy, exciton transport, and near-degenerate multilevel systems. It also means it is not automatically in Lindblad form and may fail positivity or complete positivity outside its regime of validity.

The safest short description is:

Redfield = weak-coupling open-system dynamics with nonsecular coherence terms retained.

Scope and Conventions

There are several closely related equations called Redfield equations. Some authors use the name for a time-nonlocal second-order equation; others use it for the time-local Born–Markov equation. This page uses the most common master-equation convention:

• Born approximation: weak system–bath correlations;
• Markov approximation: short bath memory and time-local generator;
• no full secular approximation unless stated.

The secular limit is treated separately in Secular Approximation. Completely positive Markovian generator structure is treated in Lindblad–GKSL Equation.

Hamiltonian Setup

Start with

$$H = H_S+H_E+\lambda H_I, \qquad H_I = \sum_\alpha A_\alpha\otimes B_\alpha.$$

Assume the bath reference state $\rho_E$ is stationary and centered:

$$[H_E,\rho_E]=0, \qquad \operatorname{Tr}_E(B_\alpha\rho_E)=0.$$

The interaction-picture operators are

$$A_\alpha(t) = e^{iH_St/\hbar} A_\alpha e^{-iH_St/\hbar}, \qquad B_\alpha(t) = e^{iH_Et/\hbar} B_\alpha e^{-iH_Et/\hbar}.$$

Bath correlations are

$$C_{\alpha\beta}(\tau) = \operatorname{Tr}_E \left[ B_\alpha(\tau)B_\beta(0)\rho_E \right].$$

The assumptions behind this setup are explained in System–Bath Hamiltonians, Born Approximation, and Markov Approximation.

Interaction-Picture Redfield Equation

After Born and Markov approximations, but before secularization, the interaction-picture equation can be written as

$$\begin{aligned} \frac{d}{dt}\rho_S^{I}(t) = -\frac{\lambda^2}{\hbar^2} \sum_{\alpha,\beta} \int_0^\infty d\tau\, \big( & C_{\alpha\beta}(\tau) \left[ A_\alpha(t), A_\beta(t-\tau)\rho_S^{I}(t) \right] \\ & + C_{\beta\alpha}(-\tau) \left[ \rho_S^{I}(t)A_\beta(t-\tau), A_\alpha(t) \right] \big). \end{aligned}$$

This is time local in $\rho_S^{I}(t)$, but it is not yet secular. The operators $A_\alpha(t)$ still contain several Bohr-frequency components, and cross terms between those components remain.

This equation is often the most transparent form for deriving the Redfield tensor or comparing with microscopic bath correlation functions.

Bohr-Frequency Form

Decompose the system coupling operators as

$$A_\alpha(t) = \sum_\omega e^{-i\omega t} A_\alpha(\omega),$$

where

$$[H_S,A_\alpha(\omega)] = -\hbar\omega A_\alpha(\omega).$$

Define one-sided bath transforms

$$\Gamma_{\alpha\beta}(\omega) = \int_0^\infty d\tau\, e^{i\omega\tau} C_{\alpha\beta}(\tau).$$

With common index conventions, a compact Redfield form is

$$\begin{aligned} \frac{d}{dt}\rho_S^{I}(t) = \sum_{\omega,\omega'} \sum_{\alpha,\beta} e^{i(\omega'-\omega)t} & \Gamma_{\alpha\beta}(\omega) \left( A_\beta(\omega)\rho_S^{I}(t)A_\alpha^\dagger(\omega') \right. \\ & \left. - A_\alpha^\dagger(\omega')A_\beta(\omega)\rho_S^{I}(t) \right) + \text{h.c.} \end{aligned}$$

Different references move adjoints, indices, and factors of $\hbar$ between $\Gamma_{\alpha\beta}$ and $A_\alpha(\omega)$. The important structural feature is the double sum over $\omega$ and $\omega'$ with oscillatory factors $e^{i(\omega'-\omega)t}$.

Full secularization keeps only $\omega=\omega'$ blocks, producing the standard weak-coupling Lindblad form when the rate matrices are positive semidefinite. When the population block closes, its diagonal entries reduce to Pauli Rate Equations. Redfield keeps nonsecular blocks.

Redfield Tensor

In an energy eigenbasis, the same equation is often written as a linear equation for density-matrix elements:

$$\frac{d}{dt}\rho_{mn} = -i\omega_{mn}\rho_{mn} + \sum_{k,\ell} R_{mn,k\ell}\rho_{k\ell}.$$

Here

$$\omega_{mn} = \frac{E_m-E_n}{\hbar},$$

and $R_{mn,k\ell}$ is the Redfield tensor. It is a fourth-rank object that couples populations and coherences according to the system–bath coupling and bath spectra.

This representation is common in molecular relaxation theory and spectroscopy because it directly describes relaxation rates, dephasing rates, coherence transfer, and population transfer in the energy basis.

What Redfield Keeps

Compared with a fully secular Lindblad equation, a Redfield equation can keep:

• coherence transfer between near-degenerate transitions;
• population-coherence coupling;
• quantum beats between close transition frequencies;
• interference between decay pathways;
• corrections important on short and intermediate times;
• line-shape information that a coarse secular model can erase.

This is why Redfield dynamics remains useful even though it lacks the universal safety of a Lindblad generator.

Positivity and Complete Positivity

A Redfield equation is constructed to preserve trace and Hermiticity when the coefficients are assembled consistently. It is not guaranteed to preserve positivity for every density operator, and it is not generally a completely positive semigroup.

The issue is structural. Complete positivity of a time-independent Markovian semigroup is guaranteed by Lindblad–GKSL form. Nonsecular Redfield equations usually do not have that form.

This does not make every Redfield calculation useless. It means the equation has a domain of validity. Small positivity violations can appear when the equation is pushed beyond the perturbative regime, beyond the intended time window, or onto initial states outside the prepared domain. Large or immediate negativity is a warning that the assumptions or implementation should be revisited.

Practical checks include:

• verify trace and Hermiticity preservation algebraically;
• test positivity on eigenstates, superpositions, and states near the boundary of the density-operator set;
• compare with a secular, partial-secular, or coarse-grained equation;
• check steady states and detailed balance;
• avoid interpreting results after populations become negative;
• report whether the equation is used as a microscopic approximation or a phenomenological model.

For the channel-level condition, see Completely Positive Maps.

When Redfield Is Useful

Redfield dynamics is often useful when weak coupling is plausible but full secularization is too coarse.

Typical settings include:

• nearly degenerate transitions in multilevel atoms or molecules;
• exciton transport with coherence transfer;
• nuclear magnetic resonance relaxation;
• molecular spectroscopy and line broadening;
• spin relaxation in condensed-matter environments;
• short- and intermediate-time dynamics before asymptotic secular behavior dominates.

The most favorable regime is weak coupling with a bath correlation time short enough for Markov closure, but with transition splittings not all large compared with dissipative rates.

Relation to Other Equations

Redfield sits between the Born–Markov memory-shortened equation and the fully secular Lindblad equation.

The chain is:

$$\text{microscopic Hamiltonian} \to \text{Born equation} \to \text{Born–Markov equation} \to \text{Redfield equation} \to \text{secular Lindblad equation}.$$

The last arrow is optional. It improves complete-positivity structure but can remove physically important near-degenerate coherence effects.

Partial secular and coarse-grained equations interpolate between Redfield and fully secular Lindblad descriptions. They try to keep slow cross terms while recovering better positivity properties.

Examples

Two-level atom far from degeneracy

For a two-level atom with transition frequency $\omega_0$ much larger than the decay rate $\Gamma$, nonsecular cross terms rotate at frequencies of order $2\omega_0$. Redfield and secular Lindblad predictions usually agree well after short transients.

Nearly degenerate excited states

In a V-type system with two excited states decaying to a common ground state, cross terms between the two optical transitions can be slow if the transition frequencies are close. Redfield can retain interference between decay pathways that full secularization would discard.

Exciton transport

In molecular aggregates, energy gaps and environmental relaxation rates can be comparable. Redfield-type equations are often used to model coherence transfer and population relaxation, but positivity and steady-state behavior must be checked against the physical regime.

Common Mistakes

Calling Redfield a Lindblad equation

Time-local does not mean Lindblad. A generator can be local in time and still fail complete positivity.

Treating positivity violations as irrelevant

Small violations outside the intended perturbative window may be diagnostic rather than fatal. Negativity in the regime used for predictions is not harmless.

Assuming nonsecular always means more accurate

Keeping more terms can preserve important coherences, but it can also keep terms outside the controlled approximation. Accuracy is judged by time scales, observables, and consistency checks.

Forgetting initial conditions

Standard Redfield derivations usually assume a factorized or weakly correlated initial state. Equilibrated system–bath states require additional care.

Ignoring convention differences

Redfield tensors differ across books by signs, adjoints, Fourier-transform conventions, and whether Lamb shifts are included in $R_{mn,k\ell}$ or in a separate Hamiltonian.

Exercises

Identify the secular step

In the Bohr-frequency Redfield equation, which terms are removed by full secularization?

Solution

The terms with $\omega\ne\omega'$ carry factors

$$e^{i(\omega'-\omega)t}.$$

Full secularization drops these rapidly rotating cross terms when $|\omega-\omega'|\gg\Gamma$ and keeps the $\omega=\omega'$ blocks. Exact degeneracies with the same Bohr frequency remain inside the same block.

Trace preservation

Why does a commutator expression such as

$$\operatorname{Tr} \left[ A, B\rho \right]$$

vanish?

Solution

The trace of a commutator vanishes by cyclicity:

$$\operatorname{Tr}([A,B\rho]) = \operatorname{Tr}(AB\rho) - \operatorname{Tr}(B\rho A) = 0.$$

This is why the commutator structure of the Redfield equation is compatible with trace preservation, even though positivity is a separate issue.

Redfield tensor size

For an $N$-level system, how many components does the object $R_{mn,k\ell}$ have before symmetries or sparsity are used?

Solution

Each of $m,n,k,\ell$ can take $N$ values, so the tensor has $N^4$ components before constraints are applied. In practice, Hermiticity, trace preservation, symmetries, selection rules, and sparse couplings reduce the independent data.

Choosing between Redfield and secular Lindblad

Two transitions have splitting $|\omega-\omega'|=5\Gamma$. A different pair has splitting $0.2\Gamma$. Which pair is safer to secularize?

Solution

The pair with splitting $5\Gamma$ is safer to secularize because its cross terms rotate several times on the dissipative time scale. The pair with splitting $0.2\Gamma$ is near-degenerate on that scale, so Redfield, partial secularization, or another nonsecular method may be needed.

References

• A. G. Redfield, “On the theory of relaxation processes,” IBM Journal of Research and Development 1, 19–31 (1957).
• A. G. Redfield, “The theory of relaxation processes,” Advances in Magnetic and Optical Resonance 1, 1–32 (1965).
• H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems, Oxford University Press, 2002.
• Á. Rivas and S. F. Huelga, Open Quantum Systems: An Introduction, Springer, 2012.
• C. W. Gardiner and P. Zoller, Quantum Noise, 3rd ed., Springer, 2004.
• V. May and O. Kühn, Charge and Energy Transfer Dynamics in Molecular Systems, 3rd ed., Wiley-VCH, 2011.