Quantum coherent feedback control of an N-level atom with multiple excitations

TL;DR

This work analyzes a quantum coherent feedback network in which a ladder-type $N$-level atom couples to a cavity that is connected to one or more waveguides, forming delay-based feedback loops. The authors cast the quantum dynamics as a linear time-delay system and use Lyapunov–Krasovskii and LMIs to link exponential stability to photon generation, deriving delay-differential equations for the state amplitudes and examining detuned (time-varying) cases. They extend the single-waveguide model to multiple parallel waveguides, detailing photon distributions across waveguides via a structured state space and providing simulation results that illustrate photon routing and oscillatory exchange, dependent on loop length and inter-waveguide couplings. Across resonant and detuned regimes, the framework demonstrates how to engineer quantum states and photonic distributions in cavity–QED networks using delay control, with potential applications in deterministic multi-photon generation and photon routing.

Abstract

The purpose of this paper is to study the dynamics of a quantum coherent feedback network, where an $N$-level atom is coupled with a cavity and the cavity is also coupled with single or multiple parallel waveguides. When the atom is initialized at the highest energy level, it can emit multiple photons into the cavity, and the photons can be further transmitted to the waveguides and re-interact with the cavity quantum electrodynamics (cavity-QED) system. The transmission of photons in the waveguide can construct a feedback channel with a delay determined by the length of the waveguide. We model the dynamics of the atomic and photonic states of the cavity-QED system as a linear control system with delay. By tuning the control parameters such as the coupling strengths among the atom, cavity and waveguide, the eigenstates of the quantum system can be exponentially stable or unstable, and the exponential stability of the linear quantum control system with delay is related with the generation of single- and multi-photon states. Besides, when the cavity-QED system is coupled with multiple parallel waveguides, the emitted photons oscillate among different waveguides and the stability of quantum states is influenced by the feedback loop length and coupling strengths among waveguides.

Figures (6)

• Figure 1: Schematic of a ladder-type $N$-level atom inside a cavity (designated by the two red bars) coupled with a waveguide (designated by the blue band).
• Figure 2: The populations of the zero-, one- and two-photon states in the waveguide.
• Figure 3: Feedback control performance when the four-level atom is resonant (the solid lines) or non-resonant (the dashed lines) with the cavity.
• Figure 4: An $N$-level atom in a cavity coupled with multiple parallel waveguides.
• Figure 5: Coherent feedback control of a two-level atom in a cavity coupled with three parallel waveguides.

Theorems & Definitions (23)

• Proposition 1
• proof
• Proposition 2
• proof
• Proposition 3
• proof
• Proposition 4
• proof
• Remark 1
• Definition 1