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

  • Reinhard Honegger

    Universität Tübingen, Germany
  • Alfred Rieckers

    Universität Tübingen, Germany
  • Jürgen Hellmich

    Universität Tübingen, Germany
  • Claus Köstler

    Universität Tübingen, Germany
  • Burkhard Kümmerer

    Universität Stuttgart, Germany


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.

Cite this article

Reinhard Honegger, Alfred Rieckers, Jürgen Hellmich, Claus Köstler, Burkhard Kümmerer, Couplings to Classical and Non-Classical Squeezed White Noise as Stationary Markov Processes. Publ. Res. Inst. Math. Sci. 38 (2002), no. 1, pp. 1–31

DOI 10.2977/PRIMS/1145476415