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On dynamical semigroup for damped driven Jaynes-Cummings equations

A. I. Komech, E. A. Kopylova

Abstract

The article addresses the damped driven Jaynes-Cummings for quantised one-mode Maxwell field coupled to a two-level molecule. We consider a broad class of damping and pumping which are polynomial in the creation and annihilation operators. Our main result is the construction of a contraction dynamical semigroup in the Hilbert space of Hermitian Hilbert-Schmidt operators in the case of a nonpositive dissipation operator and time-independent pumping. All trajectories of the semigroup are generalised solutions to the Jaynes-Cummings equations. As a key example, we prove nonpositivity for the basic dissipation operator of Quantum Optics.

On dynamical semigroup for damped driven Jaynes-Cummings equations

Abstract

The article addresses the damped driven Jaynes-Cummings for quantised one-mode Maxwell field coupled to a two-level molecule. We consider a broad class of damping and pumping which are polynomial in the creation and annihilation operators. Our main result is the construction of a contraction dynamical semigroup in the Hilbert space of Hermitian Hilbert-Schmidt operators in the case of a nonpositive dissipation operator and time-independent pumping. All trajectories of the semigroup are generalised solutions to the Jaynes-Cummings equations. As a key example, we prove nonpositivity for the basic dissipation operator of Quantum Optics.
Paper Structure (7 sections, 3 theorems, 26 equations)

This paper contains 7 sections, 3 theorems, 26 equations.

Table of Contents

  1. Introduction
  2. Notations and main results
  3. Nonpositivity of the dissipation operator and of its adjoint
  4. Semigroup and generalised solutions
  5. Acknowledgements
  6. Conflict of interest
  7. Data availability statement

Key Result

Lemma 2.2

All conditions H1--H3 hold for the dissipation operator $D=D_1$.

Theorems & Definitions (10)

  • Definition 1.1
  • Definition 1.2
  • Remark 1.3
  • Remark 2.1
  • Lemma 2.2
  • Definition 2.3
  • Theorem 2.4
  • Remark 3.1
  • Lemma 4.1
  • proof