Quantum feedback control of a two-atom network closed by a semi-infinite waveguide

TL;DR

This study analyzes delay-dependent coherent feedback in a two-atom network closed by a semi-infinite waveguide, with loop delays $\tau_j=\frac{2 z_j}{c}$ set by atom–mirror distances. It employs a dual-domain approach, deriving a quasi-polynomial from Laplace transforms and a Markovian master equation to capture delay-induced effects, while also modeling explicit wavepacket propagation in the spatial domain. The results show that tuning delays and chiral couplings can yield zero-, one-, or two-photon states in the waveguide, and that large delays can sustain excited atomic states, revealing non-Markovian dynamics absent in small-delay limits. A comparison with cavity-QED systems highlights richer steady-state and non-exponential behaviors in waveguide-based feedback, and the spatial-domain analysis provides concrete insights into photonic distributions within quantum networks.

Abstract

The purpose of this paper is to study the delay-dependent coherent feedback dynamics by focusing on one typical realization, i.e., a two-atom quantum network whose feedback loop is closed by a semi-infinite waveguide. In this set-up, an initially excited two-level atom can emit a photon into the waveguide, where the propagating photon can be reflected by the terminal mirror of the waveguide or absorbed by the other atom, thus constructing various coherent feedback loops. We show that there can be two-photon, one-photon or zero-photon states in the waveguide, which can be controlled by the feedback loop length and the coupling strengths between the atoms and waveguide. The photonic states in the waveguide are analyzed in both the frequency domain and the spatial domain, and the transient process of photon emissions is better understood based on a comprehensive analysis using both domains. Interestingly, we clarify that this quantum coherent feedback network can be mathematically modeled as a linear control system with multiple delays, which are determined by the distances between atoms and the terminal mirror of the semi-infinite waveguide. Therefore, based on time-delayed linear control system theory, the influence of delays on the stability of the quantum state evolution and the steady-state atomic and photonic states is investigated, for both small and large delays.

Figures (7)

• Figure 1: A quantum coherent feedback control network where two atoms are coupled to a semi-infinite waveguide.
• Figure 2: The evolution of two atoms coupled to a waveguide evaluated by the populations of excited atomic states in (a), the populations of quantum states with one photon in the waveguide in (b), and the amplitude of quantum state with two photons in the waveguide in (c).
• Figure 3: Control performance for the generation of single-photon states evaluated by the populations of excited atomic states in (a), the populations of quantum states with one photon in the waveguide in (b), and the amplitude two-photon states in the waveguide in (c).
• Figure 4: Comparison of the excited populations of two atoms influenced by atoms' positions.
• Figure 5: Comparisons on the atom's excited populations (a) and right-propagating photon wave packet (b) influenced by the coherent feedback control relied on chiral and nonchiral couplings between one atom and the semi-infinite waveguide.

Theorems & Definitions (35)

• Theorem 1
• Proof
• Theorem 2
• Proof
• Theorem 3
• Proof
• Remark 1
• Theorem 4
• Proof
• Remark 2