Redfield equation

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

The starting point of Redfield theory is the Nakajima–Zwanzig equation with ${\displaystyle {\mathcal {P}}}$ projecting on the equilibrium density operator of the environment and ${\displaystyle {\mathcal {Q}}}$ treated up to second order.[2] An equivalent derivation starts with second-order perturbation theory in the interaction ${\displaystyle H_{\text{int}}}$.[3] In both cases, the resulting equation of motion for the density operator in the interaction picture (with ${\displaystyle H_{0,S}=H+H_{\text{env}}}$) is

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

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

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 ${\displaystyle \tau _{r}}$ that gives the time scale on which the environment affects the system time evolution, and the coherence time of the environment, ${\displaystyle \tau _{c}}$ that gives the typical time scale on which the correlation functions decay. If the relation

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

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

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

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

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

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