The Born–Markov Approximation

The goal: a closed equation for the system alone

Exact dynamics of $S+B$ are unitary but intractable — the bath has astronomically many degrees of freedom. We want a closed equation of motion for $\rho_S(t)$ alone. There is no free lunch: getting one requires approximations. The standard route, valid for weak coupling to a large memoryless bath, is the Born–Markov approximation. It rests on three assumptions, applied in the interaction picture where the coupling is treated as the small perturbation.

Assumption 1 — Born (weak coupling)

The system–bath coupling $H_{SB}$ is weak enough to keep only second order in it (a perturbative expansion in the coupling strength). Equivalently, the bath is so large that the system barely perturbs it, so the total state stays approximately a product at all times,

\rho_{SB}(t) \approx \rho_S(t)\otimes \rho_B,

with $\rho_B$ a fixed (typically thermal) bath state. System–bath correlations build up but are small and are neglected when they feed back into the system's evolution. This is the Born approximation.

Assumption 2 — Markov (no memory)

The bath correlation functions

C_{\alpha\beta}(\tau) = \mathrm{Tr}_B\!\left[\, B_\alpha(\tau)\,B_\beta(0)\,\rho_B \,\right]

decay to zero over a short bath correlation time $\tau_B$ . If $\tau_B$ is tiny compared with the timescale $\tau_S$ on which the system relaxes, the bath effectively forgets the system instantly. Two consequences follow:

The system's rate of change at time $t$ depends only on $\rho_S(t)$ , not on its history — the evolution becomes time-local (the Redfield equation): $\dot\rho_S(t) = \mathcal{L}\,\rho_S(t)$ .
In the integral over the bath's memory, the upper limit can be extended to infinity, since the integrand has already decayed.

The separation of scales is the heart of Markovianity:

\tau_B \ll \tau_S .

Assumption 3 — Secular (rotating-wave) approximation

A further averaging removes terms that oscillate rapidly at the system's Bohr frequency differences $\omega - \omega'$ during the slow relaxation. Dropping these fast-rotating terms (the secular or rotating-wave approximation) is what converts the time-local Redfield equation into a generator of the special Lindblad form, which alone guarantees that $\rho_S$ stays a valid density matrix (completely positive, trace preserving) for all times.

When it is valid — and when it is not

The Born–Markov–secular chain is excellent for systems weakly coupled to broad, structureless baths at not-too-low temperature: spontaneous emission of an atom into the vacuum, a qubit damped by a flat electromagnetic continuum. It fails when the bath has sharp spectral features (a cavity mode, a structured phonon spectrum), when coupling is strong, or at very low temperature, where $\tau_B$ grows and the bath's memory matters. Those regimes are non-Markovian, treated at the end of this module. For now we accept the approximations and, in the next lesson, write down the master equation they produce.