Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On single-frequency asymptotics for the Maxwell-Bloch equations: pure states

A. I. Komech, E. A. Kopylova

Abstract

We consider damped driven Maxwell-Bloch equations for a single-mode Maxwell field coupled to a two-level molecule. The equations are used for semiclassical description of the laser action. Our main result is the construction of solutions with single-frequency asymptotics of the Maxwell field in the case of quasiperiodic pumping. The asymptotics hold for solutions with harmonic initial values which are stationary states of averaged reduced equations in the interaction picture. We calculate all harmonic states and analyse their stability. Our calculations rely on the Hopf reduction by the gauge symmetry group U(1). The asymptotics follow by an extension of the averaging theory of Bogolyubov--Eckhaus--Sanchez-Palencia onto dynamical systems on manifolds.

On single-frequency asymptotics for the Maxwell-Bloch equations: pure states

Abstract

We consider damped driven Maxwell-Bloch equations for a single-mode Maxwell field coupled to a two-level molecule. The equations are used for semiclassical description of the laser action. Our main result is the construction of solutions with single-frequency asymptotics of the Maxwell field in the case of quasiperiodic pumping. The asymptotics hold for solutions with harmonic initial values which are stationary states of averaged reduced equations in the interaction picture. We calculate all harmonic states and analyse their stability. Our calculations rely on the Hopf reduction by the gauge symmetry group U(1). The asymptotics follow by an extension of the averaging theory of Bogolyubov--Eckhaus--Sanchez-Palencia onto dynamical systems on manifolds.
Paper Structure (21 sections, 10 theorems, 111 equations, 1 figure)

This paper contains 21 sections, 10 theorems, 111 equations, 1 figure.

Table of Contents

  1. Introduction
  2. $U(1)$-symmetry and the Hopf fibration
  3. Resolution of singularity and the reduced dynamics
  4. Averaging of dynamics in the interaction picture
  5. Stationary states for averaged dynamics (harmonic states)
  6. Zero population inversion
  7. Nonzero population inversion
  8. Fully inverted states are not harmonic
  9. Summary
  10. Stability of harmonic states
  11. Zero population inversion.
  12. Nonzero population inversion
  13. Single-frequency asymptotics
  14. KBM vector field
  15. Single-frequency asymptotics in the interaction picture
  16. ...and 6 more sections

Key Result

Theorem 1.1

Let $\Omega= \omega$ and the pumping be quasiperiodic. Then for any $r> 0$, the following asymptotics hold. i) Let $X(0)=X^r$ be a harmonic state of (HMB2), and ${\bf Y}^r=({\bf M}^r,{\bf Q}^r):=\Pi X^r$ be the corresponding stationary solution to (aver). Then for solutions $(M(t),Q(t))=\Pi X(t)$ t ii) Let, additionally, ${\bf Y}^r=({\bf M}^r,{\bf Q}^r)$ be an asymptotically stable stationary so

Figures (1)

  • Figure 1: Stereografic projections

Theorems & Definitions (27)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Lemma 2.1
  • proof
  • Remark 2.2
  • Remark 2.3
  • Remark 3.1
  • Remark 4.1
  • Remark 4.2
  • ...and 17 more