Redfield equation

In quantum mechanics, the Redfield equation is a Markovian master equation that describes the time evolution of the density matrix $ρ$ of a quantum system that is weakly coupled to an environment.

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

{\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 $H$ is the Hermitian Hamiltonian, and the $S_{m},\Lambda _{m}$ are operators that describe the coupling to the environment. Their explicit form is given in the derivation below.

Derivation

Let us consider a quantum system coupled to an environment with a total Hamiltonian of $H_{{\text{tot}}}=H+H_{{\text{int}}}+H_{{\text{env}}}$ . Furthermore, we assume that the interaction Hamiltonian can be written as $H_{{\text{int}}}=\sum _{n}S_{n}E_{n}$ , where the $S_{n}$ act only on the system degrees of freedom, the $E_{n}$ only on the environment degrees of freedom.

The starting point of Redfield theory is the Nakajima–Zwanzig equation with ${\mathcal {P}}$ projecting on the equilibrium density operator of the environment and ${\mathcal {Q}}$ treated up to second order.^[1] An equivalent derivation starts with second-order perturbation theory in the interaction $H_{{\text{int}}}$ .^[2] In both cases, the resulting equation of motion for the density operator in the interaction picture (with $H_{{0,S}}=H+H_{{\text{env}}}$ ) is

{\frac {\partial }{\partial t}}\rho _{I}(t)=-{\frac {1}{\hbar ^{2}}}\sum _{{m,n}}\int _{{t_{0}}}^{t}dt'{\biggl (}C_{{mn}}(t-t'){\Bigl [}S_{{m,I}}(t),S_{{n,I}}(t')\rho _{I}(t'){\Bigr ]}-C_{{mn}}^{\ast }(t-t'){\Bigl [}S_{{m,I}}(t),\rho _{I}(t')S_{{n,I}}(t'){\Bigr ]}{\biggr )}

Here, $t_{0}$ 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 $C_{{mn}}(t)={\text{tr}}(E_{{m,I}}(t)E_{{n}}\rho _{{\text{env,eq}}})$ in terms of the density operator of the environment in thermal equilibrium, $\rho _{{\text{env,eq}}}$ .

This equation is non-local in time: To get the derivative of the reduced density operator at time t, we need its values at all past times. As such, it cannot be easily solved. To construct an approximate solution, note that there are two time scales: a typical relaxation time $\tau _{r}$ that gives the time scale on which the environment affects the system time evolution, and the coherence time of the environment, $\tau _{c}$ that gives the typical time scale on which the correlation functions decay. If the relation

\tau _{c}\ll \tau _{r}

holds, then the integrand becomes approximately zero before the interaction-picture density operator changes significantly. In this case, the so-called Markov approximation $\rho _{I}(t')\approx \rho _{I}(t)$ holds. If we also move $t_{0}\to -\infty$ and change the integration variable $t'\to \tau =t-t'$ , we end up with the Redfield master equation

{\frac {\partial }{\partial t}}\rho _{I}(t)=-{\frac {1}{\hbar ^{2}}}\sum _{{m,n}}\int _{0}^{\infty }d\tau {\biggl (}C_{{mn}}(\tau ){\Bigl [}S_{{m,I}}(t),S_{{n,I}}(t-\tau )\rho _{I}(t){\Bigr ]}-C_{{mn}}^{\ast }(\tau ){\Bigl [}S_{{m,I}}(t),\rho _{I}(t)S_{{n,I}}(t-\tau ){\Bigr ]}{\biggr )}

We can simplify this equation considerably if we use the shortcut $\Lambda _{m}=\sum _{n}\int _{0}^{\infty }d\tau C_{mn}(\tau )S_{n,I}(-\tau )$ . In the Schrödinger picture, the equation then reads

{\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 }]

References

↑ Volkhard May, Oliver Kuehn: Charge and Energy Transfer Dynamics in Molecular Systems. Wiley-VCH, 2000 ISBN 3-527-29608-5
↑ Heinz-Peter Breuer, Francesco Petruccione: Theory of Open Quantum Systems. Oxford, 2002 ISBN 978-0-19-852063-4

brmesolve Bloch-Redfield master equation solver from QuTiP.
Redfield, A. G. (1965). "The Theory of Relaxation Processes". Adv. Magn. Res. 1: 1. doi:10.1016/B978-1-4832-3114-3.50007-6.

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