We analyze the two-photon absorption and emission process and characterize the stationary states at zero and positive temperature. We show that entangled stationary states exist only at zero temperature and, at positive temperature, there exists infinitely many commuting invariant states satisfying the detailed balance condition.

We characterize the stationary states of an excitation energy transfer model in quantum many-particle systems [Y. Aref’eva, I. Volovich and S. Kozyrev, Stochastic limit method and interference in quantum many-particles systems, Theor. Math. Phys. 183(3) (2015) 782–799] as well as the stationary states of a quantum photosynthesis model [S. Kozyrev and I. Volovich, Dark states in quantum photosynthesis, arXiv:1603.07182v1 [physics.bio-ph]] in terms of a transport operator. It turns out that, apart from the ground state, all invariant states of the excitation energy transport model are entangled. For the photosynthesis model, any invariant state in the commutant of the system Hamiltonian is a mixed bright–dark state in the sense of [S. Kozyrev and I. Volovich, Dark states in quantum photosynthesis, arXiv:1603.07182v1 [physics.bio-ph]] and it is pure dark if and only if the bright vector belongs to the kernel of this state.

ABSTRACT. We prove new a priori estimates for the resolvent of a minimal quantum dynamical semigroup.These estimates simplify well-known conditions sufficient for conservativity and impose continuity conditions on the time-dependent operator coefficients ensuring the existence of conservative solutions of the Markov evolution equations.KEY WORDS: master equation, minimal resolvent, regularity conditions, time-dependent coefficients, a priori estimates for the resolvent, von Neumann algebra, predual semigroup.We introduce operator-valued continuity conditions for infinitesimal operators of the Markov evolution equation describing the dynamics of an open quantum system in the algebra B(~) of bounded operators and in the algebra of operators T(~) with finite trace on a Hilbert space 7"l (see [1][2][3]). A contraction semigroup Pt(" ) acting in B(7"~) is said to be a quantum dynamical semigroup [3] if it is completely positive, normal, conservative, and ultraweakly continuous. Normal and ultraweak continuity properties mean respectively that l.u.b. Pt(X,,) = Pt(t.u.b. X,,) andfor any p E 7"(7"/) and B E/3(7"/). In the Heisenberg representation, the conservativity property means the conservation of the unit operator I in the algebra B('H) of observables (Pt(I) = I Vt > 0); in the SchrSdinger representation, it means the conservation of the trace of an initial state p E T(7-/) during the where CPn(~) is-the cone of completely positive normal maps of the~ von Neumann algebra /3(7"/) [3].The theory of equations with time-dependent coefficient plays an important role in problems of interaction representation.

We propose a definition of cycle representation for Quantum Markov Semigroups (qms) and Quantum Entropy Production Rate (QEPR) in terms of the ρ-adjoint. We introduce the class of circulant qms, which admit non-equilibrium steady states but exhibit symmetries that allow us to compute explicitly the QEPR, gain a deeper insight into the notion of cycle decomposition and prove that quantum detailed balance holds if and only if the QEPR equals zero.

We introduce three new principles: the nonlinear Boltzmann-Gibbs prescription, the local KMS condition and the generalized detailed balance (GDB) condition. We prove the equivalence of the first two under general conditions and we discuss a master equation formulation of the third one