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Quantum Probability and Infinite Dimensional Analysis

A LIMIT THEOREM FOR CONDITIONALLY INDEPENDENT BEAM SPLITTINGS

Karl‐Heinz Fichtner, Volkmar Liebscher, Masanori Ohya

Published Jan 1, 2005• 2 citations
Publisher

Abstract

Consider a Boson Fock space Γ(L 2 (G, µ)). Then the coherent state φ f to a wave function f describes a system of particles in the same one-particle state f / �f � in the space G. For testing whether two coherent states φf and
DOI
10.1142/9789812702104_0015

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We consider a mean-field model to describe the dynamics of [Formula: see text] bosons of species one and [Formula: see text] bosons of species two in the limit as [Formula: see text] and [Formula: see text] go to infinity. We embed this model into Fock space and use it to describe the time evolution of coherent states which represent two-component condensates. Following this approach, we obtain a microscopic quantum description for the dynamics of such systems, determined by the Schrödinger equation. Associated to the solution to the Schrödinger equation, we have a reduced density operator for one particle in the first component of the condensate and one particle in the second component. In this paper, we estimate the difference between this operator and the projection onto the tensor product of two functions that are solutions of a system of equations of Hartree type. Our results show that this difference goes to zero as [Formula: see text] and [Formula: see text] go to infinity.
This paper is the second in a series of papers considering symmetry properties of bosonic quantum systems over 2D graphs, with continuous spins, in the spirit of the Mermin-Wagner theorem. In the model considered here the phase space of a single spin is ℋ1=L2(M), where M is a d -dimensional unit torus M=ℝd/ℤd with a flat metric. The phase space of k spins is ℋk=L2sym(Mk) , the subspace of L2(Mk) formed by functions symmetric under the permutations of the arguments. The Fock space H =⊕k=0,1,… ℋk yields the phase space of a system of a varying (but finite) number of particles. We associate a space H≃H(i) with each vertex i∈Γ of a graph (Γ,ℰ) satisfying a special bidimensionality property. (Physically, vertex i represents a heavy “atom” or “ion” that does not move but attracts a number of “light” particles.) The kinetic energy part of the Hamiltonian includes (i) -Δ/2 , the minus a half of the Laplace operator on M , responsible for the motion of a particle while “trapped” by a given atom, and (ii) an integral term describing possible “jumps” where a particle may join another atom. The potential part is an operator of multiplication by a function (the potential energy of a classical configuration) which is a sum of (a) one-body potentials U(1)(x) , x∈M , describing a field generated by a heavy atom, (b) two-body potentials U(2)(x,y) , x,y∈M , showing the interaction between pairs of particles belonging to the same atom, and (c) two-body potentials V(x,y) , x,y∈M , scaled along the graph distance d(i,j) between vertices i,j∈Γ , which gives the interaction between particles belonging to different atoms. The system under consideration can be considered as a generalized (bosonic) Hubbard model. We assume that a connected Lie group G acts on M , represented by a Euclidean space or torus of dimension d'≤d , preserving the metric and the volume in M . Furthermore, we suppose that the potentials U(1) , U(2) , and V are G -invariant. The result of the paper is that any (appropriately defined) Gibbs states generated by the above Hamiltonian is G -invariant, provided that the thermodynamic variables (the fugacity z and the inverse temperature β ) satisfy a certain restriction. The definition of a Gibbs state (and its analysis) is based on the Feynman-Kac representation for the density matrices.