Physics:Quantum regression theorem

Quantum regression theorem (QRT) is a result in quantum statistical mechanics and quantum optics that provides a rule for computing multi-time correlation functions from the same reduced dynamics that describes one-time expectation values of an open quantum system.

Statement

A common formulation (used in open-systems and quantum-optics texts) is the following. Suppose there exists a set of system operators $\{B_i\}$ such that the (Markovian) master equation implies a closed linear system of first-order differential equations for their expectation values, $$\frac{d}{dt}⟨B_i(t)⟩=\sum _j{G}_{ij}⟨B_j(t)⟩,$$ with a (time-independent) coefficient matrix $G_{ij}$. Then the quantum regression theorem states that the corresponding two-time correlation functions satisfy the same system of equations (as a function of the time difference $\tau \ge 0$), $$\frac{d}{d\tau }⟨B_i(t+\tau )B_{\ell }(t)⟩=\sum _j{G}_{ij}⟨B_j(t+\tau )B_{\ell }(t)⟩,$$ for each fixed index $\ell$ (and similarly for other operator orderings, with the appropriate convention).

Equivalently, writing the reduced dynamics as a dynamical map ${\mathrm{\Phi }}_{\tau }$ (for example ${\mathrm{\Phi }}_{\tau }=e^{{ℒ}\tau }$ for a time-homogeneous generator ${ℒ}$), one may express two-time correlations in terms of an auxiliary operator evolved by the same map: $$⟨A(t+\tau )B(t)⟩=\mathrm{T}\mathrm{r}\left[A{\mathrm{\Phi }}_{\tau }\right(B\rho (t)\left)\right],\tau \ge 0,$$ (with the product $B\rho (t)$ replaced by $\rho (t)B$ if the chosen convention requires it). Higher-order multi-time correlations follow by repeated application of $\mathrm{\Phi }$ between successive operator insertions.

Use and limitations

The QRT is widely used to compute spectra and noise properties (for example, fluorescence and resonance fluorescence spectra) in Markovian open-system models. Its validity is commonly tied to the approximations used to derive a Markovian master equation (such as negligible memory effects and suitable initial system–environment factorization). When these assumptions fail, especially for strongly non-Markovian dynamics, the QRT can become inaccurate and may require modifications.