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On the non-Markovian quantum stochastic network dynamics

Haijin Ding, Guofeng Zhang

TL;DR

This work presents a kernel-based quantum stochastic framework for non-Markovian dynamics in waveguide-QED networks, where delays from photon propagation create memory effects encoded in integral kernels between atom operators and input noises. By deriving non-Markovian QSDEs and associated master equations, the authors enable coherent feedback, state filtering, and delayed control in both semi-infinite and infinite waveguide geometries, including multi-point coupling. The approach reveals how non-Markovian commutators and Itô rules shape system dynamics and measurement outputs, and it provides Markovian approximations to connect with standard QSDEs. Numerical simulations illustrate filtering behavior in few-atom networks, highlighting how delays and drive fields influence entanglement, excitation transfer, and observer performance, with potential applications in robust quantum networking and quantum control.

Abstract

In this paper, we investigate non-Markovian quantum dynamics from the perspective of quantum noises in a network of atoms mediated by a waveguide. In such networks, quantum coherent feedback control becomes achievable when coherent fields (or quantum noises) in the format of photons with continuous modes propagate through the waveguide. Different from traditional Markovian quantum systems, the non-Markovian quantum network can be regarded as a quantum system interacting with multiple input quantum noise channels with different time delays, and the \rm{Itō} relationships among different quantum noise channels can be represented with a non-Markovian integral process with integral kernels determined by the distances among atoms and their coupling strengths to the waveguide. Then the non-Markovian dynamics of the quantum network can be modeled with the quantum stochastic differential equation (QSDE) containing integral kernels determined by the \rm{Itō} relationships among quantum noises. Utilizing this stochastic approach related to quantum noises, the filtering of quantum states can be modulated by parameters such as atom-waveguide coupling strengths, quantum control fields, and measurement results collected at the output end of the waveguide.

On the non-Markovian quantum stochastic network dynamics

TL;DR

This work presents a kernel-based quantum stochastic framework for non-Markovian dynamics in waveguide-QED networks, where delays from photon propagation create memory effects encoded in integral kernels between atom operators and input noises. By deriving non-Markovian QSDEs and associated master equations, the authors enable coherent feedback, state filtering, and delayed control in both semi-infinite and infinite waveguide geometries, including multi-point coupling. The approach reveals how non-Markovian commutators and Itô rules shape system dynamics and measurement outputs, and it provides Markovian approximations to connect with standard QSDEs. Numerical simulations illustrate filtering behavior in few-atom networks, highlighting how delays and drive fields influence entanglement, excitation transfer, and observer performance, with potential applications in robust quantum networking and quantum control.

Abstract

In this paper, we investigate non-Markovian quantum dynamics from the perspective of quantum noises in a network of atoms mediated by a waveguide. In such networks, quantum coherent feedback control becomes achievable when coherent fields (or quantum noises) in the format of photons with continuous modes propagate through the waveguide. Different from traditional Markovian quantum systems, the non-Markovian quantum network can be regarded as a quantum system interacting with multiple input quantum noise channels with different time delays, and the \rm{Itō} relationships among different quantum noise channels can be represented with a non-Markovian integral process with integral kernels determined by the distances among atoms and their coupling strengths to the waveguide. Then the non-Markovian dynamics of the quantum network can be modeled with the quantum stochastic differential equation (QSDE) containing integral kernels determined by the \rm{Itō} relationships among quantum noises. Utilizing this stochastic approach related to quantum noises, the filtering of quantum states can be modulated by parameters such as atom-waveguide coupling strengths, quantum control fields, and measurement results collected at the output end of the waveguide.
Paper Structure (33 sections, 16 theorems, 95 equations, 3 figures)

This paper contains 33 sections, 16 theorems, 95 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Non-Markovian dynamics realized by atoms coupled to a waveguide
  3. Atoms coupled to a semi-infinite waveguide
  4. Atoms coupled to an infinite waveguide
  5. Multi-point coupling between atoms and waveuide
  6. Multi-point coupling between atoms and a semi-infinite waveguide
  7. Multi-point coupling between atoms and an infinite waveguide
  8. Non-Markovian memory kernel for atom-waveguide interactions
  9. Kernel for semi-infinite waveguide case
  10. Kernel for infinite waveguide case
  11. Non-Markovian quantum noise properties based on atom-waveguide interactions
  12. Non-Markovianity of quantum noises in a semi-infinite waveguide
  13. Non-Markovianity of quantum noises in an infinite waveguide
  14. QSDE for atom-waveguide interactions
  15. QSDE for semi-infinite waveguide circumstance
  16. ...and 18 more sections

Key Result

Lemma 1

For the non-Markovian Hamiltonian in Eq. (con:HeffSemiInputFieldKappa), consider the $j$th input quantum noise channel, the commutator for the quantum noise in the time domain reads

Figures (3)

  • Figure 1: Non-Markovian multi-agent quantum network realizations: (a) Multiple two-level atoms coupled to a semi-infinite waveguide closed by the mirror. (b) Multiple two-level atoms coupled to an infinite waveguide. (c) Non-Markovian multi-point coupling between atoms and waveguide, where the waveguide can be semi-infinite as in (a) or infinite as in (b).
  • Figure 2: Comparisons of quantum filtering evaluated by $|\alpha(t)|^2$ and $|\beta(t)|^2$ when two atoms are coupled to a semi-infinite waveguide.
  • Figure 3: Quantum filtering for a three-atom network.

Theorems & Definitions (40)

  • Remark 1
  • Lemma 1
  • Proof
  • Lemma 2
  • Proof
  • Proposition 1
  • Proof
  • Lemma 3
  • Proof
  • Lemma 4
  • ...and 30 more