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

. I. Komech, E. A. Kopylova

Abstract

We consider damped driven Maxwell-Bloch equations which are finite-dimensional approximation of the damped driven Maxwell-Schrödinger equations. The equations describe a single-mode Maxwell field coupled to a two-level molecule. 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 equations in the interaction picture. We calculate all harmonic states and analyse their stability. The calculations rely on the Bloch-Feynman gyroscopic representation of von Neumann equation for the density matrix. The asymptotics follow by application of the averaging theory of the Bogolyubov type.

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

Abstract

We consider damped driven Maxwell-Bloch equations which are finite-dimensional approximation of the damped driven Maxwell-Schrödinger equations. The equations describe a single-mode Maxwell field coupled to a two-level molecule. 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 equations in the interaction picture. We calculate all harmonic states and analyse their stability. The calculations rely on the Bloch-Feynman gyroscopic representation of von Neumann equation for the density matrix. The asymptotics follow by application of the averaging theory of the Bogolyubov type.
Paper Structure (15 sections, 3 theorems, 100 equations)

This paper contains 15 sections, 3 theorems, 100 equations.

Table of Contents

  1. Introduction
  2. The Bloch-Feynman "gyroscopic" representation
  3. Dynamics in the interaction picture
  4. The averaging
  5. Stationary states for the averaged dynamics (harmonic states)
  6. Spectra of linearised equations at the harmonic states
  7. Attraction to stable submanifold
  8. Single-frequency asymptotics
  9. KBM vector field
  10. The asymptotics in the interaction picture
  11. The a priori bounds and well-posedness
  12. The Maxwell--Bloch equations for pure and mixed states
  13. The Maxwell--Bloch equations for pure states
  14. The von Neumann equation for mixed states
  15. On possible treatment of the laser action

Key Result

Theorem 1.1

Let $\Omega= \omega$ and the pumping be quasiperiodic. Then for any $r>0$, the following asymptotics hold. i) Let the initial state $X(0)=({\bf M},{\bf S})\in Z^r$. Then the corresponding solutions admit the following adiabatic asymptotics: ii) Let $cr>|{\bf A}^e|$, and $D^r_ d$ denote a suitable subset of the tubular $d$-neighborhood of the stable submanifold $Z^r_+$ with sufficiently small $d

Theorems & Definitions (12)

  • Theorem 1.1
  • Remark 1.2
  • Remark 3.1
  • Remark 5.1
  • Remark 5.2
  • Lemma 7.1
  • proof
  • Remark 7.2
  • Remark 8.1
  • Lemma A.1
  • ...and 2 more