Born, Markov, and secular approximations

Write the joint Hamiltonian as system plus bath plus a weak interaction, and work in the interaction picture. Iterating the exact equation once creates a double commutator involving the joint density operator at an earlier time. The Born closure replaces this operator inside the second-order integral by a product of the corresponding reduced state and a fixed stationary bath state. The bath is assumed large enough that its properties are not appreciably changed by the system.

This replacement does not assert that the actual joint state remains unentangled. Small correlations generate the leading dissipative effect; the closure specifies which orders are retained after tracing. A nonzero average bath force can first be absorbed into the system Hamiltonian. Initial system–bath correlations can introduce additional terms, and strong coupling changes equilibrium as well as dynamics. One must assess the approximation on the relaxation times of interest, because small coupling does not justify truncating a short-time Taylor series at arbitrarily late times.

Let the bath correlation decay on a time $\tau_B$, while the reduced interaction-picture state changes on a much longer relaxation time $\tau_R$. Replacing the delayed reduced state inside the integral by its present value gives a time-local approximation. Extending the integration limit to infinity is justified only after the bath correlation has decayed and before appreciable reduced relaxation. A large bath alone does not ensure fast correlation decay: narrow spectra and band edges can retain long memory.

The secular step additionally averages terms oscillating at differences of system Bohr frequencies. A coarse-graining time must resolve each discarded frequency difference while remaining short relative to relaxation. Exactly equal frequencies belong in the same block rather than being discarded. Near-degenerate transitions can violate this separation even when the bath is fast and the coupling weak. Partial secularization or another controlled treatment may then be needed; deleting all coherence couplings by habit can remove relevant dynamics.

Born and Markov steps often lead to a Redfield equation. Its trace and Hermiticity can be correct while some predicted eigenvalues become negative. Complete positivity is a stronger structural requirement. With stationary positive bath spectra, a properly resolved secular construction yields a positive coefficient matrix and hence a GKSL generator. That mathematical guarantee does not establish that the chosen generator accurately approximates the microscopic system: a perfectly valid channel can still model the wrong physics.

Useful diagnostics include comparing bath memory with observed decay, checking small transition separations against decay linewidths, and testing predictions against exact solvable limits. A constant-rate equation should not be trusted for times shorter than the coarse-graining window. Conversely, numerical positivity violations in a nominal channel solver may be integration errors, rather than evidence of physical memory. Separate the validity of the derivation, the mathematical properties of the equation, and the accuracy of its numerical solution before interpreting a discrepancy.