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 such that the (Markovian) master equation implies a closed linear system of first-order differential equations for their expectation values, with a (time-independent) coefficient matrix . 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 ), for each fixed index (and similarly for other operator orderings, with the appropriate convention).
Equivalently, writing the reduced dynamics as a dynamical map (for example for a time-homogeneous generator ), one may express two-time correlations in terms of an auxiliary operator evolved by the same map: (with the product replaced by if the chosen convention requires it). Higher-order multi-time correlations follow by repeated application of 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.
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