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 need not preserve positivity or complete positivity for every initial state. The size of this defect and its effect on a requested prediction must be assessed even at weak coupling.

Helpful background. The Born approximation controls perturbative order, the Markov approximation replaces short memory, and secular averaging separates rotating frequency blocks.

The nonsecular weak-coupling generator

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: a second-order expansion with negligible higher-order feedback;
• 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

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

Take $\lambda$ dimensionless, $A_\alpha$ dimensionless and Hermitian, and $B_\alpha$ Hermitian with energy units. Assume an initially factorized state and stationary bath reference state with centered coupling operators:

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

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

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

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 those definitions, the compact frequency form is

$$\frac{d}{dt}\rho_S^{I}(t) = \frac{\lambda^2}{\hbar^2} \sum_{\omega,\omega'} \sum_{\alpha,\beta} e^{i(\omega'-\omega)t} \Gamma_{\alpha\beta}(\omega) \left( A_\beta(\omega)\rho_S^{I}(t)A_\alpha^\dagger(\omega') - A_\alpha^\dagger(\omega')A_\beta(\omega)\rho_S^{I}(t) \right) + \text{h.c.}$$

The “h.c.” means the Hermitian conjugate of the whole summed contribution for Hermitian $\rho_S^I$, including its coefficient and phase. The same $\lambda^2/\hbar^2$ prefactor applies to both contributions. Here $\Gamma_{\alpha\beta}$ has units energy squared times time, so omitting that prefactor would give the wrong rate units. The double sum retains terms with $\omega\ne\omega'$.

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 a nonsecular equation can improve a particular weak-coupling prediction when full secularization removes the coherence being measured.

Trace preservation does not guarantee positive states

The commutator structure preserves trace and, with consistently paired terms, Hermiticity. Positivity is a separate condition. Nonsecular combinations can contain a dissipative coefficient matrix with negative eigenvalues, even though each individual transition rate is positive.

For an explicit illustration, let $A_j=|g\rangle\langle e_j|$ for two decay pathways. A nonsecular coefficient matrix formed from unequal positive rates can have the arithmetic-mean form

$$C=\gamma\begin{pmatrix}1&5/2\\5/2&4\end{pmatrix}, \qquad \gamma>0,$$

whose determinant is $-9\gamma^2/4$. Consider the dissipative terms $\sum_{ij}C_{ij}(A_j\rho A_i^\dagger-\{A_i^\dagger A_j,\rho\}/2)$ and the normalized initial state

$$|\psi\rangle=\frac{2|e_1\rangle-|e_2\rangle}{\sqrt5}.$$

Its ground population is initially zero, but

$$\left.\frac{d p_g}{dt}\right|_0 =\frac15(2,-1)C\binom{2}{-1} =-\frac{2\gamma}{5}.$$

It becomes negative immediately. An added Hamiltonian commutator cannot change this derivative for a state initially supported entirely in the excited subspace. This example shows why positivity cannot be inferred from positive diagonal rates or a trace-preserving implementation.

A small negative eigenvalue may fall within a known approximation error, but it is never a physical negative probability. Establish the error scale by comparison with a more accurate model or controlled limit. Clipping negative eigenvalues afterward changes the evolution and does not repair the microscopic derivation. A secular or carefully constructed coarse-grained GKSL equation can preserve positivity, but may have different errors in coherences near degeneracy. See Complete Positivity for the stronger reference-system requirement.

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 secular and coarse-grained equations

In the convention used here, the Born–Markov equation before frequency averaging is the Redfield equation; there is no extra approximation between those two names. Full secular averaging removes distinct-frequency cross terms and gives the usual weak-coupling GKSL form. Near degeneracy, partial-secular and coarse-grained constructions can retain slow cross terms, but their coefficient matrix must be checked rather than assumed positive.

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.

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 $5\Gamma$ pair is the better candidate, but a factor of five is not a strong asymptotic separation: its relative phase advances only five radians, less than one full cycle, in one relaxation time. Accuracy still needs checking. The $0.2\Gamma$ pair rotates much more slowly and should generally be treated within a near-degenerate block or with another justified nonsecular method.

References

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