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Limits of Perturbation Theory for Multimode Light Propagation in Dispersive Optical Cavities

K. S. Tikhonov, D. M. Malyshev, V. A. Averchenko

TL;DR

The paper analyzes the limits of perturbation theory for group velocity dispersion (GVD) in synchronously pumped dispersive optical cavities, focusing on multimode quantum light propagation. It develops both a rigorous steady-state solution and a perturbative Heisenberg-Langevin approach, revealing a shared matrix structure governed by the dispersive coupling matrix $O_{nm}$ and the perturbation parameter $(N_gamma/N_D)$. Perturbation theory remains valid only when $(N_gamma/N_D) O_{nn}$ stays below unity, but high-order modes have increasing $O_{nn}$, leading to a mode-dependent breakdown at a threshold $n_{lim}$ that also hinges on mode-dependent decay rates. The results provide concrete criteria for when PT methods are reliable and offer design guidance for leveraging GVD in multimode quantum-light generation and intracavity spectroscopy.

Abstract

Temporal modes of quantum light pulses is a promising resource for modern quantum technologies, driving advancements in quantum computing, communication, and metrology. Precise control and manipulation of these modes remain critical challenges, particularly in systems where nonlinear multimode dynamics interact with dispersion effects. In this work, we focus on the role of group velocity dispersion (GVD) within optical cavities - a phenomenon traditionally viewed as detrimental but increasingly recognized as a versatile tool for quantum light manipulation. We present a perturbation-theory-based approach to analyze GVD effects in a synchronously pumped dispersive cavity. By comparing perturbative solutions to rigorous steady-state results, we establish the validity region of the perturbative approach and assess its limitations in multimode systems. Our study identifies key parameters governing the breakdown of perturbation theory, such as mode order, dispersion strength, and cavity decay rates.

Limits of Perturbation Theory for Multimode Light Propagation in Dispersive Optical Cavities

TL;DR

The paper analyzes the limits of perturbation theory for group velocity dispersion (GVD) in synchronously pumped dispersive optical cavities, focusing on multimode quantum light propagation. It develops both a rigorous steady-state solution and a perturbative Heisenberg-Langevin approach, revealing a shared matrix structure governed by the dispersive coupling matrix and the perturbation parameter . Perturbation theory remains valid only when stays below unity, but high-order modes have increasing , leading to a mode-dependent breakdown at a threshold that also hinges on mode-dependent decay rates. The results provide concrete criteria for when PT methods are reliable and offer design guidance for leveraging GVD in multimode quantum-light generation and intracavity spectroscopy.

Abstract

Temporal modes of quantum light pulses is a promising resource for modern quantum technologies, driving advancements in quantum computing, communication, and metrology. Precise control and manipulation of these modes remain critical challenges, particularly in systems where nonlinear multimode dynamics interact with dispersion effects. In this work, we focus on the role of group velocity dispersion (GVD) within optical cavities - a phenomenon traditionally viewed as detrimental but increasingly recognized as a versatile tool for quantum light manipulation. We present a perturbation-theory-based approach to analyze GVD effects in a synchronously pumped dispersive cavity. By comparing perturbative solutions to rigorous steady-state results, we establish the validity region of the perturbative approach and assess its limitations in multimode systems. Our study identifies key parameters governing the breakdown of perturbation theory, such as mode order, dispersion strength, and cavity decay rates.

Paper Structure

This paper contains 7 sections, 33 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Propagation of a Classical Optical Pulse in the Dispersive Cavity
  3. Rigorous steady-state solution
  4. Perturbation theory solution of Heisenberg-Langevin equation
  5. Assessing the Limits of Perturbation Theory
  6. Conclusion
  7. The solution of Sellmeier equation

Figures (2)

  • Figure 1: A ring cavity of length $L$ with a dispersing medium and a light pulse $E_i(t, z)$ propagating in it. The cavity is pumped with a pulsed light field $E_{in}(t,0)$ through the input-output mirror with reflection coefficient $\sqrt{\mathcal{R}}$ and transmission coefficient $\sqrt{\mathcal{T}}$. The pulse undergoes a round-trip within the cavity, with a time $T_R$.
  • Figure 2: a) The examined dependencies of $\gamma_n$; b) The non-zero coefficient of the coupling matrix O. c-f), the orange-shaded regions indicate where the inequality is violated. The solid line marks the boundary where the $(N_\gamma/N_D)O_{nn}$ ratio equals 1. The dashed, dotted, and solid curves correspond to boundaries where the first term of the perturbation theory is twice, five times, and ten times larger than the second term, respectively.