Redfield equation

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 in honor of Alfred G. Redfield, who first applied it, doing so for nuclear magnetic resonance spectroscopy.^[1] It is also known as the Redfield relaxation theory.^[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. That is, it is possible to get negative populations during the time evolution. The Redfield equation approaches the correct dynamics for sufficiently weak coupling to the environment.

The general form of the Redfield equation is

$${\displaystyle {\frac {\partial }{\partial t}}\rho (t)=-{\frac {i}{\hbar }}[H,\rho (t)]-{\frac {1}{\hbar ^{2}}}\sum _{m}[S_{m},(\Lambda _{m}\rho (t)-\rho (t)\Lambda _{m}^{\dagger })]}$$

where ${\displaystyle H}$ is the hermitian Hamiltonian, and the ${\displaystyle S_{m},\Lambda _{m}}$ are operators that describe the coupling to the environment, and ${\displaystyle [A,B]=AB-BA}$ is the commutation bracket. The explicit form is given in the derivation below.