Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On periodic solutions and attractors for the Maxwell--Bloch equations

Alexander Komech

TL;DR

The paper proves the existence of time-periodic solutions with a $T$-periodic Maxwell field for the Maxwell–Bloch equations, modeled as a finite-dimensional approximation of semiclassical Maxwell–Schrödinger dynamics, under periodic pumping and for any $N\ge 1$ two-level molecules. It exploits the $U(1)$ gauge symmetry to reduce the system to the Hopf fibration, yielding a reduced dynamical system on $Y = R^2 \times S^2$, and uses a Poincaré map together with Lefschetz fixed-point theory on a compactified phase space to construct $T$-periodic solutions. A priori bounds and dissipativity imply a compact global attractor, and the method extends from the one-molecule case to many molecules via the product gauge group $(U(1))^N$. The results provide a rigorous foundation for periodic laser-action regimes in MB-type models and establish a general/topological framework for periodic solutions in dissipative gauge-invariant systems.

Abstract

We consider the Maxwell-Bloch system which is a finite-dimensional approximation of the coupled nonlinear Maxwell-Schrödinger equations. The approximation consists of one-mode Maxwell field coupled to two-level molecule. We construct time-periodic solutions to the factordynamics which is due to the symmetry gauge group. For the corresponding solutions to the Maxwell--Bloch system, the Maxwell field, current and the population inversion are time-periodic, while the wave function acquires a unit factor in the period. The proofs rely on high-amplitude asymptotics of the Maxwell field and a suitable extension of the Lefschetz theorem on fixed points and the Euler characteristic for noncompact manifolds. We also prove the existence of the global compact attractor.

On periodic solutions and attractors for the Maxwell--Bloch equations

TL;DR

The paper proves the existence of time-periodic solutions with a -periodic Maxwell field for the Maxwell–Bloch equations, modeled as a finite-dimensional approximation of semiclassical Maxwell–Schrödinger dynamics, under periodic pumping and for any two-level molecules. It exploits the gauge symmetry to reduce the system to the Hopf fibration, yielding a reduced dynamical system on , and uses a Poincaré map together with Lefschetz fixed-point theory on a compactified phase space to construct -periodic solutions. A priori bounds and dissipativity imply a compact global attractor, and the method extends from the one-molecule case to many molecules via the product gauge group . The results provide a rigorous foundation for periodic laser-action regimes in MB-type models and establish a general/topological framework for periodic solutions in dissipative gauge-invariant systems.

Abstract

We consider the Maxwell-Bloch system which is a finite-dimensional approximation of the coupled nonlinear Maxwell-Schrödinger equations. The approximation consists of one-mode Maxwell field coupled to two-level molecule. We construct time-periodic solutions to the factordynamics which is due to the symmetry gauge group. For the corresponding solutions to the Maxwell--Bloch system, the Maxwell field, current and the population inversion are time-periodic, while the wave function acquires a unit factor in the period. The proofs rely on high-amplitude asymptotics of the Maxwell field and a suitable extension of the Lefschetz theorem on fixed points and the Euler characteristic for noncompact manifolds. We also prove the existence of the global compact attractor.
Paper Structure (9 sections, 8 theorems, 52 equations)

This paper contains 9 sections, 8 theorems, 52 equations.

Table of Contents

  1. Introduction
  2. The contraction of high-amplitude Maxwell field and well-posedness
  3. Gauge symmetry and reduced dynamics in the Hopf fibration
  4. Periodic solutions for the reduced MB equations with one particle
  5. A priori estimate for fixed points of the Poincaré map
  6. Modified dynamics on the compactified phase space
  7. Maxwell--Bloch equations with many molecules
  8. Acknowledgements
  9. The Maxwell--Bloch equations as the Galerkin approximation

Key Result

Lemma 2.1

Let $A^e(t)\in C[0,\infty)$. Then there exists the Lyapunov function $V(A,B)$ such that and for solutions to (HMB2),

Theorems & Definitions (20)

  • Remark 1.1
  • Remark 1.2
  • Lemma 2.1
  • proof
  • Corollary 2.2
  • Remark 2.3
  • Remark 2.4
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • ...and 10 more