In quantum mechanics, the Redfield equation is a Markovian master equation that describes the time evolution of the reduced density matrix ρ of a strongly coupled quantum system that is weakly coupled to an environment.
The equation is named after Alfred G. Redfield, who first applied it, doing so for nuclear magnetic resonance spectroscopy.
[2] There is a close connection to the Lindblad master equation.
If a so-called secular approximation is performed, where only certain resonant interactions with the environment are retained, every Redfield equation transforms into a master equation of Lindblad type.
Redfield equations are trace-preserving and correctly produce a thermalized state for asymptotic propagation.
However, in contrast to Lindblad equations, Redfield equations do not guarantee a positive time evolution of the density matrix.
The Redfield equation approaches the correct dynamics for sufficiently weak coupling to the environment.
Consider a quantum system coupled to an environment with a total Hamiltonian of
The starting point of Redfield theory is the Nakajima–Zwanzig equation with
projecting on the equilibrium density operator of the environment and
[3] An equivalent derivation starts with second-order perturbation theory in the interaction
[4] In both cases, the resulting equation of motion for the density operator in the interaction picture (with
is some initial time, where the total state of the system and bath is assumed to be factorized, and we have introduced the bath correlation function
in terms of the density operator of the environment in thermal equilibrium,
that gives the typical time scale on which the correlation functions decay.
holds, then the integrand becomes approximately zero before the interaction-picture density operator changes significantly.
In this case, the so-called Markov approximation
In the Schrödinger picture, the equation then reads
Secular (Latin: saeculum, lit.
The time evolution of the Redfield relaxation tensor is neglected as the Redfield equation describes weak coupling to the environment.
Therefore, it is assumed that the relaxation tensor changes slowly in time, and it can be assumed constant for the duration of the interaction described by the interaction Hamiltonian.
In general, the time evolution of the reduced density matrix can be written for the element
Given that the actual coupling to the environment is weak (but non-negligible), the Redfield tensor is a small perturbation of the system Hamiltonian and the solution can be written as
is not constant but slowly changing amplitude reflecting the weak coupling to the environment.
This is also a form of the interaction picture, hence the index "I".
We can integrate this equation on condition that the interaction picture of the reduced density matrix
, therefore the contribution of one element of the reduced density matrix to another element is proportional to time (and therefore dominates for long times
is not approaching zero, the contribution of one element of the reduced density matrix to another oscillates with an amplitude proportional to
It is therefore appropriate to neglect any contribution from non-diagonal elements (
The only elements left in the Redfield tensor to evaluate after the Secular approximation are therefore: