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 } {\displaystyle \{B_{i}\}} such that the (Markovian) master equation implies a closed linear system of first-order differential equations for their expectation values, d d t ⟨ B i ( t ) ⟩ = ∑ j G i j ⟨ B j ( t ) ⟩ , {\displaystyle {\frac {d}{dt}}\langle B_{i}(t)\rangle =\sum _{j}G_{ij}\,\langle B_{j}(t)\rangle ,} with a (time-independent) coefficient matrix G i j {\displaystyle 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 τ ≥ 0 {\displaystyle \tau \geq 0} ), d d τ ⟨ B i ( t + τ ) B ℓ ( t ) ⟩ = ∑ j G i j ⟨ B j ( t + τ ) B ℓ ( t ) ⟩ , {\displaystyle {\frac {d}{d\tau }}\langle B_{i}(t+\tau )\,B_{\ell }(t)\rangle =\sum _{j}G_{ij}\,\langle B_{j}(t+\tau )\,B_{\ell }(t)\rangle ,} for each fixed index ℓ {\displaystyle \ell } (and similarly for other operator orderings, with the appropriate convention). Equivalently, writing the reduced dynamics as a dynamical map Φ τ {\displaystyle \Phi _{\tau }} (for example Φ τ = e L τ {\displaystyle \Phi _{\tau }=e^{{\mathcal {L}}\tau }} for a time-homogeneous generator L {\displaystyle {\mathcal {L}}} ), one may express two-time correlations in terms of an auxiliary operator evolved by the same map: ⟨ A ( t + τ ) B ( t ) ⟩ = T r [ A Φ τ ( B ρ ( t ) ) ] , τ ≥ 0 , {\displaystyle \langle A(t+\tau )\,B(t)\rangle =\mathrm {Tr} \!\left[A\,\Phi _{\tau }\!{\bigl (}B\,\rho (t){\bigr )}\right],\qquad \tau \geq 0,} (with the product B ρ ( t ) {\displaystyle B\,\rho (t)} replaced by ρ ( t ) B {\displaystyle \rho (t)\,B} if the chosen convention requires it). Higher-order multi-time correlations follow by repeated application of Φ {\displaystyle \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.
References
