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Quantum Fokker-Planck Master Equation with general signal filtering

Guilherme de Sousa

TL;DR

The paper extends the Quantum Fokker-Planck Master Equation (QFPME) to incorporate arbitrary linear filtering in the feedback loop, enabling memory and non-Markovian effects in continuous quantum measurement. It derives a generalized master equation in an enlarged filter space for the joint density $\hat{\rho}({\bf G},t)$ and demonstrates how cascaded low-pass, band-pass, and general kernel filters alter the system dynamics, with explicit forms for each case. The framework is then applied to cooling a quantum harmonic oscillator under various filtering schemes, yielding analytical expressions for the asymptotic energy and showing that appropriately engineered filters can improve cooling in certain parameter regimes while exposing potential stability limits. This ensemble-level approach provides a practical tool for modeling realistic signal processing in quantum feedback experiments, with potential applications to optomechanics, entanglement stabilization, and squeezing, and clarifies when Markovian embeddings suffice or when memory effects must be retained.

Abstract

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.

Quantum Fokker-Planck Master Equation with general signal filtering

TL;DR

The paper extends the Quantum Fokker-Planck Master Equation (QFPME) to incorporate arbitrary linear filtering in the feedback loop, enabling memory and non-Markovian effects in continuous quantum measurement. It derives a generalized master equation in an enlarged filter space for the joint density and demonstrates how cascaded low-pass, band-pass, and general kernel filters alter the system dynamics, with explicit forms for each case. The framework is then applied to cooling a quantum harmonic oscillator under various filtering schemes, yielding analytical expressions for the asymptotic energy and showing that appropriately engineered filters can improve cooling in certain parameter regimes while exposing potential stability limits. This ensemble-level approach provides a practical tool for modeling realistic signal processing in quantum feedback experiments, with potential applications to optomechanics, entanglement stabilization, and squeezing, and clarifies when Markovian embeddings suffice or when memory effects must be retained.

Abstract

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.

Paper Structure

This paper contains 14 sections, 61 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. A QFPME with general filtering
  3. Modeling various filters
  4. Multiple low-pass filters
  5. Band-pass filtering
  6. General kernel filtering
  7. Cooling a harmonic oscillator
  8. One layer of filtering
  9. Two layers of filtering
  10. Three layers of filtering
  11. Band-pass filter
  12. Discussion and outlook
  13. Deriving the general QFPME
  14. Cooling a quantum harmonic oscillator

Figures (2)

  • Figure 1: Illustration of the signal processing leading to the measurement outcome. The underlying quantum dynamics, possibly with feedback, yield a noisy measurement outcome given by $z(t)$. The noisy signal is then processed (filtered) by different channels, each contributing to a different component of the overall signal. The information processing is modeled using a general linear system.
  • Figure 2: Phase diagram indicating which protocol is able to cool down the particle to the lowest energy for a given value of $(\gamma,\Omega)$. Each region indicates which protocol yields the lowest energy using Eqs. (\ref{['eq:H1-infty']},\ref{['eq:H2-infty']},\ref{['eq:H3-infty']},\ref{['eq:band_pass_energy']}). The vertical dashed lines indicate the transition regions where different low-pass filters have the best cooling in the limit $\Omega \rightarrow \infty$ (see Eq. \ref{['eq:g_l_decision_region']}).