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

TL;DR

This paper rigorously analyzes the weak coupling limit for infinite-dimensional quantum systems with unbounded, weakly commuting operators, deriving the limiting dynamics and modified Hamiltonian including Lamb shift effects.

Contribution

It provides a rigorous derivation of the weak coupling limit for unbounded, weakly commuting system operators, including explicit forms of the limiting dynamics and Hamiltonian modifications.

Findings

Convergence of the reduced dynamics to unitary evolution with a Lamb shift.

Explicit form of the modified Hamiltonian in the weak coupling limit.

Estimates on the convergence rate to the limiting dynamics.

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.