Invitation to Quantum Dynamical Semigroups

Abstract

The theory of quantum dynamical semigroups within the mathematically rigorous framework of completely positive dynamical maps is reviewed. First, the axiomatic approach which deals with phenomenological constructions and general mathematical structures is discussed. Then basic derivation schemes of the constructive approach including singular coupling, weak coupling and low density limits are presented in their higly simplified versions. Two-level system coupled to a heat bath, damped harmonic oscillator, models of decoherence, quantum Brownian particle and Bloch-Boltzmann equations are used as illustrations of the general theory. Physical and mathematical limitations of the quantum open system theory, the validity of Markovian approximation and alternative approaches are discussed also.

Dissipative quantum dynamics with the surrogate Hamiltonian approach. A comparison between spin and harmonic baths

“…To rationalize, one should notice that the coherence in cat states composed of a superposition of two coherent states in a single mode does not represent entanglement. Therefore, this phase loss does not characterize decoherence in accordance with Alicki's notion [34]. Moreover when a pure dephasing term was added it was found that it did not erode the phase coherence between cat states [51].…”

“…The first difficulty is that there are different approaches to the definition of decoherence. Alicki [34] identifies pure decoherence (dephasing) with the decay of the off-diagonal elements of the density operator, which is not accompanied by dissipation. He then argues that dephasing cannot be caused by a harmonic oscillator bath with a coupling, which is linear in coordinates or momenta.…”

Fluctuations of Quantum Currents and Unravelings of Master Equations

“…It is widely accepted that master equations gives a good description of the degrees of freedom of a small system in certain limiting regimes. In the physics literature, this limiting regime is characterized by the Born-Markov approximation and the rotating wave approximation, see [2,12] for a review. Another name, common especially in the mathematical physics literature, is the weak coupling limit, which makes these two approximations exact.…”

Parameter estimation in atomic spectroscopy using exceptional points

“…The irreversible character of the L-GKS equation is well known and indicated by the semi-group character of the evolution operator [43,44,15,45]. The generator of the dynamics L is therefore non hermitian.…”