Quantum regression and two-time correlators

Let a time-homogeneous master equation generate the reduced propagator. For an ordered correlation with the later operator on the left, the regression prescription starts by multiplying the earlier operator onto the left of the earlier density matrix. Propagate that product, then trace with the later observable. The initial object can be non-Hermitian, have zero trace, and need not be positive. Linearity permits its evolution even though it is not itself a physical preparation.

Reversing operator order changes that initial product and can change the correlation. This matters for emission versus absorption: lowering an excitation and then raising it probes a different occupancy than raising first and lowering later. At zero delay, always check that the expression reduces to the expectation of the ordered operator product. This provides a quick diagnostic against an incorrect initialization, especially when both one-time dipole means vanish.

For a stationary undriven qubit, take the raising operator at the later time and the lowering operator at the earlier time. In a rotating frame, the off-diagonal matrix unit decays at the transverse rate. The coefficient multiplying it is the stationary excited population, not its square. Thus the two-time dipole correlation can be nonzero even though each dipole mean is zero. The amplitude is a population factor, while the decay constant carries the linewidth information.

Fourier transforming the exponential delay dependence gives a Lorentzian around the transition frequency. Defining the spectrum with twice the real part of the one-sided transform fixes the normalization used here; other spectral conventions distribute factors differently. The half width in angular frequency is the transverse rate and the full width is twice that rate. A measured non-Lorentzian shape can indicate several transitions, slow frequency noise, driving, or memory rather than a single changed decay constant.

The underlying system and bath become correlated even if their initial preparation is a product. In the regression approximation, their relevant correlations after the earlier operator insertion are neglected or controlled in a way consistent with the Markov bath limit. A constant-rate fit to density matrices does not by itself establish this property. Even completely positive divisibility of all one-time maps is weaker than the full regression hierarchy for a general microscopic process.

At delays comparable to bath memory, or for structured reservoirs and correlated preparations, extra terms can modify two-time correlations. An exact one-time reduced equation may then fail if inserted into the standard regression formula. This is not an algebraic defect in propagating a nonpositive operator product; it is missing process information. Comparing a measured correlation with the prediction from independently determined relaxation times is therefore a substantive test of assumptions, rather than a redundant measurement of the same density matrix.