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arXiv:1909.02981 · Infinite Dimensional Analysis Quantum Probability and Related Topics

Stationary states of weak coupling limit-type Markov generators and quantum transport models

Alvaro Hernandez-Cervantes, Roberto B. Quezada

Published Mar 1, 2020• 7 citations

Abstract

We prove that every stationary state in the annihilator of all Kraus operators of a weak coupling limit-type Markov generator consists of two pieces, one of them supported on the interaction-free subspace and the second one on its orthogonal complement. In particular, we apply the previous result to describe in detail the structure of a slightly modified quantum transport model due to Arefeva et al. (modified AKV’s model) studied first in [J. C. García et al., Entangled and dark stationary states of excitation energy transport models in many-particles systems and photosynthesis, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 21(3) (2018), Article ID: 1850018, p. 21, doi:10.1142/S0219025718500182], in terms of generalized annihilation and creation operators.

DOI: 10.1142/s0219025720500034

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We prove that every stationary state in the annihilator of all Kraus operators of a weak coupling limit-type Markov generator consists of two pieces, one of them supported on the interaction-free subspace and the second one on its orthogonal complement. In particular, we apply the previous result to describe in detail the structure of a slightly modified quantum transport model due to Arefeva et al. (modified AKV’s model) studied first in [J. C. García et al., Entangled and dark stationary states of excitation energy transport models in many-particles systems and photosynthesis, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 21(3) (2018), Article ID: 1850018, p. 21, doi:10.1142/S0219025718500182], in terms of generalized annihilation and creation operators.

The fast recurrent subspace (the biggest support of all invariant states) of a weak coupling limit type quantum Markov semigroup modeling a quantum transport open system of [Formula: see text]-energy levels is determined. This is achieved by characterizing the structure of all the invariant states and their spectra in terms of a natural generalization of the discrete Fourier transform operator. Finally, the attraction domains and long-time behaviour of the evolution are studied on hereditary subalgebras where faithful invariant states exist.

Generators of Gaussian QMS have a GKLS form with noise operators that are linear in creation and annihilation operators, and Hamiltonian which is quadratic. We characterize Gaussian QMS arising in the Weak Coupling Limit (WCL) of a system interacting with some reservoirs. It turns out that their generators, under some natural assumptions, have a GKLS representation with noise operators that are either annihilation or creation operators and Hamiltonian [Formula: see text] which is diagonalizable as a quadratic operator.

We investigate the structure of quantum Markov generators that describe the reduced dynamics of a test particle interacting with a dilute Bose gas in the low density limit. These generators, called low density limit type generators (LDL), differ from the weak coupling limit type generators (WCL) studied in [4, 5] because of the presence of the T-operator that describes the change in momentum of the gas particle due to collisions with the test particle. We propose a general definition of Markov generator of stochastic limit type that includes (almost) all generators arising in the stochastic limit approach and we prove that the associated semigroups as well as their invariant states (or weights) have a very special structure which extends the explicit representation for the generic quantum Markov semigroups of weak coupling limit type, due to Accardi, Hachicha and Ouerdiane [7]. We also investigate the structure of invariant states and give conditions on the T - operator for the existence of a unique invariant state and of equilibrium states.

The generator of a Gaussian quantum Markov semigroup on the algebra of bounded operator on a [Formula: see text]-mode Fock space is represented in a generalized GKLS form with an operator [Formula: see text] quadratic in creation and annihilation operators and Kraus operators [Formula: see text] linear in creation and annihilation operators. Kraus operators, commutators [Formula: see text] and iterated commutators [Formula: see text] up to the order [Formula: see text], as linear combinations of creation and annihilation operators determine a vector in [Formula: see text]. We show that a Gaussian quantum Markov semigroup is irreducible if such vectors generate [Formula: see text], under the technical condition that the domains of [Formula: see text] 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 [Formula: see text] is fully non-commutative.