Ameur Dhahri

THE DECOHERENCE-FREE SUBALGEBRA OF A QUANTUM MARKOV SEMIGROUP WITH UNBOUNDED GENERATOR

Let be a quantum Markov semigroup on with a faithful normal invariant state ρ. The decoherence-free subalgebra of is the biggest subalgebra of where the completely positive maps act as homomorphisms. When is the minimal semigroup whose generator is represented in a generalised GKSL form , with possibly unbounded H, Ll, we show that coincides with the generalised commutator of under some natural regularity conditions. As a corollary we derive simple sufficient algebraic conditions for convergence towards a steady state based on multiple commutators of H and Ll. We give examples of quantum Markov semigroups , with h infinite-dimensional, having a non-trivial decoherence-free subalgebra.

The quadratic Fock functor

We construct the quadratic analogue of the boson Fock functor. While in the first order case all contractions on the 1--particle space can be second quantized, the semigroup of contractions that admit a quadratic second quantization is much smaller due to the nonlinearity. Within this semigroup we characterize the unitary and the isometric elements.

Quadratic exponential vectors

We give a necessary and sufficient condition for the existence of a quadratic exponential vector with test function in L2(Rd)∩L∞(Rd). We prove the linear independence and totality, in the quadratic Fock space, of these vectors. Using a technique different from the one used by Accardi et al. [Quantum Probability and Infinite Dimensional Analysis, Vol. 25, p. 262, (2009)], we also extend, to a more general class of test functions, the explicit form of the scalar product between two such vectors.

A Lindblad model for a spin chain coupled to heat baths

We study an XY model which consists of a spin chain coupled to heat baths. We give a repeated quantum interaction Hamiltonian describing this model. We compute the explicit form of the associated Lindblad generator in the case of the spin chain coupled to one, two and several heat baths. We further study the properties of the quantum master equation such as approach to equilibrium, local equilibrium states, entropy production and quantum detailed balance condition.

Markovian properties of the spin-boson model

We systematically compare the Hamiltonian and Markovian approaches of quantum open system theory, in the case of the spin-boson model. We first give a complete proof of the weak coupling limit and we compute the Lindblad generator of this model. We study properties of the associated quantum master equation such as decoherence, detailed quantum balance and return to equilibrium at inverse temperature 0 < β ≤ ∞. We further study the associated quantum Langevin equation, its associated interaction Hamiltonian. We finally give a quantum repeated interaction model describing the spin‐boson system where the associated Markovian properties are satisfied without any assumption.

Polynomial Extensions of the Weyl C*-Algebra

We introduce higher order (polynomial) extensions of the unique (up to isomorphisms) nontrivial central extension of the Heisenberg algebra, which can be concretely realized as sub-Lie algebras of the polynomial algebra generated by the creation and annihilation operators in the Schrodinger representation. The simplest nontrivial of these extensions (the quadratic one) is isomorphic to the Galilei algebra, widely studied in quantum physics. By exponentiation of this representation we construct the corresponding polynomial analogue of the Weyl C*-algebra and compute the polynomial Weyl relations. From this we deduce the explicit form of the composition law of the associated nonlinear extensions of the 1-dimensional Heisenberg group. The above results are used to calculate a simple explicit form of the vacuum characteristic functions of the nonlinear field operators of the Galilei algebra, as well as of their moments. The corresponding measures turn out to be an interpolation family between Gaussian and Meixner, in particular Gamma.

ON THE QUADRATIC FOCK FUNCTOR

We prove that the quadratic second quantization of an operator p on L2(ℝd) ∩ L∞(ℝd) is an orthogonal projection on the quadratic Fock space if and only if p is a multiplication operator by a characteristic function χI, I ⊂ ℝd.

On the multi-dimensional Favard lemma

Abstract We prove some properties of the Jacobi sequences and the creator operators, on the d commuting indeterminates polynomial algebra. Moreover, we prove that the matrix representations of the Jacobi sequences associated to product probability measures on ℝ d {\mathbb{R}^{d}} with finite moments of any order, are diagonal in the basis introduced by the tensor product of the orthogonal polynomials of the factor measures. Finally, we give a characterization of the atomic probability measures on ℝ d {\mathbb{R}^{d}} with finite number of atoms.

Identification of the theory of orthogonal polynomials in d -indeterminates with the theory of 3 -diagonal symmetric interacting Fock spaces on ℂd

The identification mentioned in the title allows a formulation of the multidimensional Favard lemma different from the ones currently used in the literature and which parallels the original 1-dimensional formulation in the sense that the positive Jacobi sequence is replaced by a sequence of positive Hermitean (square) matrices and the real Jacobi sequence by a sequence of positive definite kernels. The above result opens the way to the program of a purely algebraic classification of probability measures on ℝd with moments of any order and more generally of states on the polynomial algebra on ℝd. The quantum decomposition of classical real-valued random variables with all moments is one of the main ingredients in the proof.

Characterization of Product Probability Measures on ℝd in Terms of Their Orthogonal Polynomials

In paper [1] the d-dimensional analogue of the Jacobi parameters has been individuated in a pair of sequences ((a.|n0),(Ω∼n)), where (a.|n0) is a sequence of Hermitean matrices and Ω∼n(n ∈ ℕ) a positive definite kernel with values in the linear operators on the n-th space of the orthogonal gradation. In this paper we prove that product measures on ℝd are characterized by the property that the (a.|n0) are diagonal and the (Ω∼n) quasidiagonal (see Definition 2 below) in the orthogonal polynomial basis.