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Publications of the Research Institute for Mathematical Sciences

Couplings to Classical and Non-Classical Squeezed White Noise as Stationary Markov Processes

Jürgen Hellmich, Reinhard Honegger, Claus Köstler, Burkhard Kümmerer, Alfred Rieckers

Published Mar 31, 2002• 17 citations
Publisher

Abstract

The squeezed white noise states of the quantum optics literature are identified with specific quasifree states on the bosonic C *-Weyl (resp. CCR) algebra, which are not gauge invariant. Their properties are discussed, and concrete realizations of their GNS-representations are given. The squeezed white noise states are obtained from the chaotic temperature (or white noise states) by the application of squeezing Bogoliubov transformations, which may lead to a noise reduction in selected modes. A squeezing strength is determined, below which the squeezed white noise states remain classical and above which they are rendered non-classical. We further couple additional systems to the squeezed white noise, the quantum character of which are leading to a large class of stationary quantum Markov processes with related quantum stochastic calculus. Their interactions are described in terms of unitary cocycles, which arise from adapted additive cocycles by means of solutions of stochastic differential equations resp. by stochastic Itô integrals. The restriction of the Markovian dynamics to the sub-system leads to a quantum semi-group, for which the Lindblad generator is of detailed balance type. Finally the stochastic Itô table for squeezed white noise is derived. From the latter non-classicality of the squeezed white noise process can now be checked by inspection.

DOI
10.2977/prims/1145476415

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The squeezed white noise states of the quantum optics literature are identified with specific quasifree states on the bosonic C *-Weyl (resp. CCR) algebra, which are not gauge invariant. Their properties are discussed, and concrete realizations of their GNS-representations are given. The squeezed white noise states are obtained from the chaotic temperature (or white noise states) by the application of squeezing Bogoliubov transformations, which may lead to a noise reduction in selected modes. A squeezing strength is determined, below which the squeezed white noise states remain classical and above which they are rendered non-classical. We further couple additional systems to the squeezed white noise, the quantum character of which are leading to a large class of stationary quantum Markov processes with related quantum stochastic calculus. Their interactions are described in terms of unitary cocycles, which arise from adapted additive cocycles by means of solutions of stochastic differential equations resp. by stochastic Itô integrals. The restriction of the Markovian dynamics to the sub-system leads to a quantum semi-group, for which the Lindblad generator is of detailed balance type. Finally the stochastic Itô table for squeezed white noise is derived. From the latter non-classicality of the squeezed white noise process can now be checked by inspection.
In this thesis we investigate the convergence of various quantum random walks to quantum stochastic cocycles defined on a Bosonic Fock space. We prove a quantum analogue of the Donsker invariance principle by invoking the so-called semigroup representation of quantum stochastic cocycles. In contrast to similar results by other authors our proof is relatively elementary. We also show convergence of products of ampliated random walks with different system algebras; in particular, we give a sufficient condition to obtain a cocycle via products of cocycles. The CCR algebra, its quasifree representations and the corresponding quasifree stochastic calculus are also described. In particular, we study in detail gauge-invariant and squeezed quasifree states. We describe repeated quantum interactions between a `small' quantum system and an environment consisting of an infinite chain of particles. We study different cases of interaction, in particular those which occur in weak coupling limits and low density limits. Under different choices of scaling of the interaction part we show that random walks, which are generated by the associated unitary evolutions of a repeated interaction system, strongly converge to unitary quantum stochastic cocycles. We provide necessary and sufficient conditions for such convergence. Furthermore, under repeated quantum interactions, we consider the situation of an infinite chain of identical particles where each particle is in an arbitrary faithful normal state. This includes the case of thermal Gibbs states. We show that the corresponding random walks converge strongly to unitary cocycles for which the driving noises depend on the state of the incoming particles. We also use conditional expectations to obtain a simple condition, at the level of generators, which suffices for the convergence of the associated random walks. Limit cocycles, for which noises depend on the state of the incoming particles, are also obtained by investigating what we refer to as `compressed' random walks. Lastly, we show that the cocycles obtained via the procedure of repeated quantum interactions are quasifree, thus the driving noises form a representation of the relevant CCR algebra. Both gauge-invariant and squeezed representations are shown to occur.
Squeezing processes are commonly described in terms of quadratic Hamiltonians, which generate unitary implementations of Bogoliubov transformations of the quantized electromagnetic field. Here the behaviour of the quasifree, the classical, and the coherent photon states under general squeezing Bogoliubov transformations is investigated. It is found that there is a great variety of mixed classical states, which remain classical under the squeezing operation, whereas each pure classical state becomes non-classical. Especially, some classical, microscopic first order coherent states remain classical and coherent of first order under one-mode squeezing. This contrasts squeezing of macroscopic coherent states.
The mathematical model describing a light beam prepared in an arbitrary quantum optical state is a quasifree quantum stochastic process on the C* algebra of the canonical commutatation relations. For such quantum stochastic processes the concept of stochastic states is introduced. Stochastic quantum states have a classical analog in the following sense: If the light beam is prepared in a stochastic state, one can construct a generalized classical stochastic process, such that the distributions of the quantum observables and the classical random variables coincide. A sufficient algebraic condition for the stochasticity of a quantum state is formulated. The introduced formalism generalizes the Wigner representation from a single field mode to a continuum of modes. For the special case of a single field mode the stochasticity condition provides a new criterion for the positivity of the Wigner function related to the given state. As an example the quantized eletromagnetic field in empty space at temperature T=0 is discussed. It turns out that the corresponding classical stochastic process is not a white noise but a colored noise with a linearly increasing spectrum.
A model of a system driven by quantum white noise with singular quadratic self- interaction is considered and an exact solution for the evolution operator is found.It is shown that the renormalized square of the squeezed classical white noise is equivalent to the quantum Poisson.process.A convenient regularization of singular quantum differential equations for the evolution operator is suggested.We describe how equations driven by nonlinear functionals of white noise can be derived in nonlinear quantum optics by using the stochastic limit.