1,720,991 research outputs found
On irreducibility of Gaussian quantum Markov semigroups
No full textThe generator of a Gaussian quantum Markov semigroup on the algebra of bounded operator on a d-mode Fock space is represented in a generalized GKLS form with an operator G quadratic in creation and annihilation operators and Kraus operators L1,... ,Lm linear in creation and annihilation operators. Kraus operators, commutators |G,L-l| and iterated commutators |G, |G,L-l||, horizontal ellipsis up to the order 2d - m, as linear combinations of creation and annihilation operators determine a vector in DOUBLE-STRUCK CAPITAL C-2d. We show that a Gaussian quantum Markov semigroup is irreducible if such vectors generate DOUBLE-STRUCK CAPITAL C-2d, under the technical condition that the domains of G and the number operator coincide. Conversely, we show that this condition is also necessary if the linear space generated by Kraus operators and their iterated commutator with G is fully non-commutative
Boson Quadratic GKLS Generators
No full textWe discuss some recent results on quantum Markov semigroups whose
GKLS (Gorini-Kossakowski-Lindblad-Sudharshan) generator is given formally by
expressions quadratic in bosonic creation and annihilation operators.We present the
construction and the characteristic property of quantum Gaussian state preservation.
We give results on existence and uniqueness of invariant densities and the longtime
behaviour. We also discuss irreducibility and the structure of the so-called
decoherence-free subalgebra. Our motivation is twofold. First we wish to develop
tools for investigating the dynamics of open quantum systems of bosons. Second
we would like to describe the structure and mathematical properties of Gaussian
quantum Markov semigroups
A characterization of quantum Markov semigroups of weak coupling limit type
Full text linkWe characterize generators of quantum Markov semigroups leaving invariant a maximal abelian purely atomic algebra and certain operator subspaces associated with it in a natural way. From this result, we also establish a characterization of generators of quantum Markov semigroups of weak coupling limit type associated with a nondegenerate Hamiltonian
Supercritical Poincaré–Andronov–Hopf Bifurcation in a Mean-Field Quantum Laser Equation
Full text linkWe deal with the dynamical system properties of a Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) equation with mean-field Hamiltonian that models a simple laser by applying a mean field approximation to a quantum system describing a single-mode optical cavity and a set of two level atoms, each coupled to a reservoir. We prove that the mean field quantum master equation has a unique regular stationary solution. In case a relevant parameter Cb, i.e., the cavity cooperative parameter, is less than 1, we prove that any regular solution converges exponentially fast to the equilibrium, and so the regular stationary state is a globally asymptotically stable quilibrium solution.
We obtain that a locally exponential stable limit cycle is born at the regular stationary state as Cb passes through the critical value 1. Then, the mean-field laser equation has a Poincar ́e-Andronov-Hopf bifurcation at Cb = 1 of supercritical-like type. Namely, we derive rigorously, at the level of density matrices, the ransition from a global attractor quantum state, where the light is not emitted, to a locally stable set of coherent quantum states producing coherent light. Moreover, we establish the local exponential stability of the limit cycle in case a relevant parameter is between the first and second laser thresholds appearing in the semiclassical laser theory. Thus,
we get that the coherent laser light persists over time under this condition. In order to prove the exponential convergence of the quantum state, as the time goes to +∞,
we develop a new technique for proving the exponential convergence in open quantum systems that is based in a new variation of constant formula, which is obtained by combining probabilistic techniques with classical arguments from the semigroup theory. Furthermore, applying our main results we find the long-time behavior of the von Neumann entropy, the photon-number statistics, and the quantum variance of the quadratures
Basic Properties of a Mean Field Laser Equation
Full text linkWe study the nonlinear quantum master equation describing a laser under the mean field approximation. The quantum system is formed by a single mode optical cavity and two level atoms, which interact with reservoirs. Namely, we establish the existence and uniqueness of the regular solution to the nonlinear operator equation under consideration, as well as we get a probabilistic representation for this solution in terms of a mean field stochastic Schr ̈odinger equation. To this end, we find a regular solution for the nonautonomous linear quantum master equation in Gorini–Kossakowski–Sudarshan–Lindblad form, and we prove the uniqueness of the solution to the nonautonomous linear adjoint quantum master equation in Gorini–Kossakowski–Sudarshan–Lindblad form. Moreover, we obtain rigorously the Maxwell–Bloch equations from the mean field laser equation
Quadratic open quantum harmonic oscillator
Full text linkWe study the quantum open system evolution described by a Gorini–Kossakowski–Sudarshan–Lindblad generator with creation and annihilation operators arising in Fock representations of the sl2 Lie algebra. We show that any initial density matrix evolves to a fully supported density matrix and converges towards a unique equilibrium state. We show that the convergence is exponentially fast and we exactly compute the rate for a wide range of parameters. We also discuss the connection with the two-photon absorption and emission process
Gaussian Quantum Markov Semigroups on a One-Mode Fock Space: Irreducibility and Normal Invariant States
No full textWe consider the most general Gaussian quantum Markov semigroup on a one-mode Fock space, discuss its construction from the generalized GKSL representation of the generator. We prove the known explicit formula on Weyl operators, characterize irreducibility and its equivalence to a Hörmander type condition on commutators and establish necessary and sufficient conditions for existence and uniqueness of normal invariant states. We illustrate these results by applications to the open quantum oscillator and the quantum Fokker-Planck model
Classical and Quantum Markov Processes Associated with q -Bessel Operators
No full textWe study the fundamental properties of classical and quantum Markov processes generated by q-Bessel operators and their extension to the algebra of all bounded operators on the Hilbert space Lq,α2. In particular, we find a suitable generalized Gorini-Kossakowski-Sudarshan-Lindblad representation for the infinitesimal generator of q-Bessel operator and show that both the classical and quantum Markov processes are transient for α 0 and recurrent for α = 0. We also show that they do not admit invariant states and, moreover that the support projection of any initial state instantaneously fills the full space
The role of the atomic decoherence-free subalgebra in the study of quantum Markov semigroups
Full text linkWe show that for a Quantum Markov Semigroup (QMS) with a faithful normal invariant state, atomicity of the decoherence-free subalgebra and environmental decoherence are equivalent. Moreover, we prove that the predual of the decoherence-free subalgebra is isometrically isomorphic to the subspace of reversible states. We also describe, in an explicit and constructive way, the relationship between the decoherence-free subalgebra and the fixed point subalgebra
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