Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Weak coupling limit for quantum systems with unbounded weakly commuting system operators

Ilya Lopatin, Alexander Pechen

Abstract

This work is devoted to a rigorous analysis of the weak coupling limit (WCL) for the reduced dynamics of an open infinite-dimensional quantum system interacting with electromagnetic field or a reservoir formed by Fermi or Bose particles in the dipole approximation. The free system Hamiltonian and the system part of the Hamiltonian describing interaction with the reservoir are considered as unbounded operators with continuous spectrum which are commuting in a weak sense. We derive in the weak coupling limit the reservoir statistics, which is determined by whose terms in the multi-point correlation functions of the reservoir which are non-zero in the WCL. Then we prove that the resulting reduced system dynamics converges to unitary dynamics (such behavior sometimes called as Quantum Cheshire Cat effect) with a modified Hamiltonian which can be interpreted as a Lamb shift to the original Hamiltonian. We obtain exact form of the modified Hamiltonian and estimate the rate of convergence to the limiting dynamics. For Fermi reservoir, we prove the convergence of the full Dyson series. For Bose case the convergence is understood term by term.

Weak coupling limit for quantum systems with unbounded weakly commuting system operators

Abstract

This work is devoted to a rigorous analysis of the weak coupling limit (WCL) for the reduced dynamics of an open infinite-dimensional quantum system interacting with electromagnetic field or a reservoir formed by Fermi or Bose particles in the dipole approximation. The free system Hamiltonian and the system part of the Hamiltonian describing interaction with the reservoir are considered as unbounded operators with continuous spectrum which are commuting in a weak sense. We derive in the weak coupling limit the reservoir statistics, which is determined by whose terms in the multi-point correlation functions of the reservoir which are non-zero in the WCL. Then we prove that the resulting reduced system dynamics converges to unitary dynamics (such behavior sometimes called as Quantum Cheshire Cat effect) with a modified Hamiltonian which can be interpreted as a Lamb shift to the original Hamiltonian. We obtain exact form of the modified Hamiltonian and estimate the rate of convergence to the limiting dynamics. For Fermi reservoir, we prove the convergence of the full Dyson series. For Bose case the convergence is understood term by term.
Paper Structure (8 sections, 9 theorems, 118 equations, 2 figures)

This paper contains 8 sections, 9 theorems, 118 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The Model and Some General Definitions
  3. Dyson Series for the Reduced System Dynamics
  4. Reservoir Correlation Functions in the Weak Coupling Limit
  5. Reduced System Dynamic in the Weak Coupling Limit
  6. Numerical comparison of the free and limiting dynamics
  7. Discussion
  8. Proof of Lemma \ref{['le:EstimateIntegral']}

Key Result

Lemma 3.1

Let Conditions cond:PermutForHermite and cond:Interaction hold. Then for all $X \in \mathcal{U}^S$ 1) operator series in the right hand side (r.h.s.) of Eq. eq:DysonSeriesGeneral is correctly defined and absolutely and uniformly converges on any compact for any argument $y \in \mathcal{D} \otimes \m

Figures (2)

  • Figure 1: Evolution of the Gaussian wavepacket under the free system Hamiltonian $H^{\rm S}$ (upper row) and under the modified limiting Hamiltonian with $c_1 = - c_2$ (middle row) and $c_2 = 0$ (bottom row) at four different time instants [0.0, 0.5, 1.0, 1.5].
  • Figure 2: Evolution of the Gaussian wavepacket under free Hamiltonian $H^{\rm S}$ (left) and under modified lmiting Hamiltonian $\widetilde{H}_{\lambda}$ (right) (Multimedia available online in the ancillary file EvolutionOfGaussianWavepacket.mp4).

Theorems & Definitions (25)

  • Remark
  • Example 2.1
  • Remark
  • Remark
  • Lemma 3.1
  • proof
  • Lemma 4.1
  • Lemma 4.2
  • proof
  • Remark
  • ...and 15 more