Heisenberg evolution of quantum observables represented by unbounded operators
Consider a small quantum system with state space \(\mathfrak{h}\) weakly coupled to a heat bath. In a Markovian setting, an observable \(A\) of the small system evolves in accordance with the adjoint quantum master equation \[ \frac{d}{dt}\mathcal{T}_t(A)=\mathcal{T}_t(A)G+G^*\mathcal{T}_t(A)+\sum_{k=1}^{\infty}L_k^*\mathcal{T}_t(A)L_k, \quad \mathcal{T}_0(A)=A, \] where \(G, L_1, L_2, \dots \) are linear operators in \(\mathfrak{h}\) satisfying some certain conditions. For bounded observables \(A\) (i.e. \(A\) is a bounded operator in \(\mathfrak{h}\)), the existence and uniqueness of solutions for the equations are established in \textit{A. M. Chebotarev} [Theor. Math. Phys. 80, No. 2, 804--818 (1989); translation from Teor. Mat. Fiz. 80, No. 2, 192--211 (1989; Zbl 0694.47022)] by semigroup methods. This paper deals with the adjoint quantum master equations with initial conditions given by unbounded operators, such as the position and momentum operators of quantum oscillators and the occupation number operator in many-body particle systems. It shows the existence and uniqueness of solutions of the operator equations governing the motion of unbounded observables by developing the relation between operator evolution equations arising in quantum mechanics and stochastic evolution equations of Schrödinger type. It also explores quantum dynamical semigroup properties of the Heisenberg evolutions of unbounded observables in a general setting.
scientific article; zbMATH DE number 5505593
scientific article