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Deterministic quantum master equation for non-Markovian signal processing

Guilherme De Sousa, Diogo O. Soares-Pinto

Published Jan 1, 2026• 0 citations
Publisher
DOI
10.2139/ssrn.6530288

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In this work, we derive a deterministic master equation to model a general, possibly non-Markovian, feedback. The master equation describes a system with a general evolution and measurement operation, with feedback being applied in terms of signal processing. The feedback signal has an arbitrary structure with dimensionality that indicates the degree of non-Markovianity of the information processing. We present examples to illustrate how such a master equation can be used to model systems with memory feedback and non-trivial frequency dependence.
It is shown that non-Markovian master equations for an open system which are local in time can be unravelled through a piecewise deterministic quantum jump process in its Hilbert space. We derive a stochastic Schrödinger equation that reveals how non-Markovian effects are manifested in statistical correlations between different realizations of the process. Moreover, we demonstrate that possible violations of the positivity of approximate master equations are closely connected to singularities of the stochastic Schrödinger equation, which could lead to important insights into the structural characterization of positive non-Markovian equations of motion.
In this paper, we derive a general master equation for continuous feedback based on arbitrary linear signals. This result is an extension of the Quantum Fokker-Planck Master Equation derived by Annby-Andersson et al (Phys. Rev. Lett. 129, 050401) to the case where the experiment has a general filtering structure. The filtering operation can include delayed information, memory effects, and non-Markovian signal processing. Using this general master equation, we derive analytical results for the ground state cooling of a quantum harmonic oscillator. We compared our results with those derived by De Sousa et al (Phys. Rev. E 111, 014152). Our framework aims to capture more realistic situations, to allow experiments to be better modeled, and to study non-Markovian effects in the spectral properties of the measurement signal.