We consider a dynamical semigroup for unbounded Kossakowski-Lindblad-Davies generator corresponding to evolution of an open system for a tuned repeated harmonic perturbation. For this evolution, we prove the existence of uniquely determined minimal trace-preserving strongly continuous dynamical semigroups on the space of states. The corresponding dual W*-dynamical system is shown to be unital quasi-free and completely positive automorphisms of the canonical commutation relation-algebra. © 2016 AIP Publishing LLC.
We prove new a priori estimates for the resolvent of a class of minimal Quantum Dynamical Semigroup (QDS).These estimates simplify the proof of the unital property of QDS and suggest a continuity condition for time-dependent in- finitesimal generators to ensure existence and conservativity of the Markov master evolution equation. \S 1. INTRODUCTION.The theory of the Markov
mas ter equation has been intensively studied dur- ing the recent years [1][2][3][4][5][6][7][8][9][10].Important applications of this theory were contributed to qu.antum chemistry [3] and quantum optics [4-6].Numerical Monte-Carlo and Runge-Kutta algorithms for these equations are discussed in [7-8].One of efficient analytical tools in the theory of master equation is the interaction representation technique [6-7], where the Markov
m..aster equations with time-dependent coeffi- cients arise in a natural way.To study Markov
mas ter equations with unbounded time-dependent coefficients describing evolution of observables from the algebra
B(H) of bounded operators or states from the algebra
T(H) of operators with finite trace, we introduce the continuity conditions for the generator.We recall that one parameter contraction semigroup
Pt(⋅) acting in
B(H) is called a Quantum Dynamical Semigroup (QDS) if it is completely positive, normal, conservative, and ultraweak continuous
[2,11] .Normal and ultraweak continuity properties mean respectively that l.u.b.
Pt(xn)=Pt(1.u.b.xn) and
Tr{ρ(Pt(B)− B)}arrow0 as
tarrowO for any
ρ∈T(H) and
B∈B(H) .In the Heisenberg rep- resentation, the conservative (or unital) property means the conservation of the unit operator
I in the algebra
B(H) of observables:
Pt(I)≡I∀t≥0 ; in the Schr\"odinger representation, it means the conservation of the trace of an initial state
ρ∈T(H) during the evolution:
TrTt(β)≡bρt≥0 , where
Tt :
T(H)arrowT(H) and
Pt :
B(H)arrowB(H) are dual semigroups:
Tr{τt(ρ)B}=Tr{ρPt(B)} .The semigroup
Tt=Pt1 is called the predual of the semigroup
Pt .The dagger "\dagger '' is used to denote predual operators and maps.The map.
P(⋅) with the predual
P↑(⋅) is referred to as a completely positive
(CP) if
∑i,jTr{P(Bi∗Bj)σjσi}∗≥0 or equivalently
∑i,jTr{B∗iBjPt(σjσi)∗}≥0 (1.1)
We extend Howland time-independent formalism to the case of completely positive and trace preserving dynamics of finite-dimensional open quantum systems governed by periodic, time-dependent Lindbladian in Weak Coupling Limit, expanding our result from previous papers. We propose the Bochner space of periodic, square integrable matrix-valued functions, as well as its tensor product representation, as the generalized space of states within the time-independent formalism. We examine some densely defined operators on this space, together with their Fourier-like expansions and address some problems related to their convergence by employing general results on Banach space-valued Fourier series, such as the generalized Carleson–Hunt theorem. We formulate Markovian dynamics in the generalized space of states by constructing appropriate time-independent Lindbladian in standard (Lindblad–Gorini–Kossakowski–Sudarshan) form, as well as one-parameter semigroup of bounded evolution maps. We show their similarity with Markovian generators and dynamical maps defined on matrix space, i.e. the generator still possesses a standard form (extended by closed perturbation) and the resulting semigroup is also completely positive, trace preserving and a contraction.
Dynamical semigroups have become the key structure for describing open system dynamics in all of physics. Bounded generators are known to be of a standard form, due to Gorini, Kossakowski, Sudarshan and Lindblad. This form is often used also in the unbounded case, but rather little is known about the general form of unbounded generators. In this paper we first give a precise description of the standard form in the unbounded case, emphasizing intuition, and collecting and even proving the basic results around it. We also give a cautionary example showing that the standard form must not be read too naively. Further examples are given of semigroups, which appear to be probability preserving to first order, but are not for finite times. Based on these, we construct examples of generators which are not of standard form.
Sufficient conditions for a quantum dynamical semigroup (QDS) to be unital are proved for a class of problems in quantum optics with Hamiltonians which are self-adjoint polynomials of any finite order in creation and annihilation operators. The order of the Hamiltonian may be higher than the order of completely positive part of the formal generator of a QDS.The unital property of a minimal quantum dynamical semigroup implies the uniqueness of the solution of the corresponding Markov master equation in the class of quantum dynamical semigroups and, in the corresponding representation, it ensures preservation of the trace or unit operator. We recall that only in the unital case the formal generator of MME determines uniquely the corresponding QDS.