Couplings to classical and non-classical squeezed white noise as stationary Markov processes

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 Ito 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 Ito table for squeezed white noise is derived. From the latter non-classicality of the squeezed white noise process can now be checked by inspection.

[1]  Barchielli.  Measurement theory and stochastic differential equations in quantum mechanics. , 1986, Physical review. A, General physics.

[2]  G. Pedersen.  C-Algebras and Their Automorphism Groups , 1979 .

[3]  M. Scully,et al.  The Quantum Theory of Light , 1974 .

[4]  R. Streater,et al.  Quasi-free quantum stochastic integrals for the CAR and CCR , 1983 .

[5]  D. W. Robinson.  The ground state of the Bose gas , 1965 .

[6]  Arimitsu.  A canonical formalism of dissipative quantum systems. Non-equilibrium thermofield dynamics , 1994 .

[7]  W. Stinespring,et al.  States of the Clifford Algebra , 1964 .

[8]  T. Arimitsu,et al.  A system of quantum stochastic differential equations in terms of non-equilibrium thermo field dynamics , 1997 .

[9]  B. Kümmerer.  Markov dilations on W∗-algebras , 1985 .

[10]  Jeffrey H. Shapiro,et al.  Optical communication with two-photon coherent states-Part I: Quantum-state propagation and quantum-noise , 1978, IEEE Trans. Inf. Theory.

[11]  D. Walls.  Squeezed states of light , 1983, Nature.

[12]  M. Takesaki.  Theory of Operator Algebras II , 1979 .

[13]  M. Sentís.  Quantum theory of open systems , 2002 .

[14]  Ma,et al.  Multimode squeeze operators and squeezed states. , 1990, Physical review. A, Atomic, molecular, and optical physics.

[15]  A. Rieckers,et al.  Squeezing Bogoliubov transformations on the infinite mode CCR‐algebra , 1996 .

[16]  A. Verbeure,et al.  Quasi-free states of the C.C.R.—Algebra and Bogoliubov transformations , 1968 .

[17]  A. Rieckers,et al.  Squeezing operations in Fock space and beyond , 1997 .

[18]  B. Kümmerer.  Survey on a theory of non-commutative stationary markov processes , 1988 .

[19]  H. Yuen.  Two-photon coherent states of the radiation field , 1976 .

[20]  A. Rieckers,et al.  Squeezing of Optical States on the CCR-Algebra , 1997 .

[21]  Luigi Accardi,et al.  Quantum Stochastic Processes , 1982 .

[22]  Collett,et al.  Input and output in damped quantum systems: Quantum stochastic differential equations and the master equation. , 1985, Physical review. A, General physics.

[23]  R. Hudson,et al.  Uses of non-Fock quantum Brownian motion and a quantum martingale representation theorem , 1985 .

[24]  B. Kümmerer,et al.  Elements of quantum probability , 1996 .

[25]  Debashish Goswami,et al.  Hilbert Modules and Stochastic Dilation of a Quantum Dynamical Semigroup on a von Neumann Algebra , 1999 .

[26]  V. Gorini,et al.  Quantum detailed balance and KMS condition , 1977 .

[27]  E. Lance.  Hilbert C*-Modules , 1995 .

[28]  R. Honegger,et al.  Rigorous derivation of Planck’s law in the thermodynamic limit , 1992 .

[29]  Kumar,et al.  Photon-counting statistics of multimode squeezed light. , 1989, Physical review. A, General physics.

[30]  E. Hewitt,et al.  Abstract Harmonic Analysis , 1963 .

[31]  Dénes Petz,et al.  An invitation to the algebra of canonical commutation relations , 1990 .

[32]  R. Honegger.  Decomposition of Positive Sesquilinear Forms and the Central Decomposition of Gauge-Invariant Quasi-Free States on the Weyl-Algebra , 1990 .

[33]  Robin L. Hudson,et al.  Quantum Ito's formula and stochastic evolutions , 1984 .

[34]  Hilbert Modules in Quantum Electro Dynamics and Quantum Probability , 1998 .

[35]  C. Gardiner.  Handbook of Stochastic Methods , 1983 .

[36]  Alberto Barchielli.  Direct and heterodyne detection and other applications of quantum stochastic calculus to quantum optics , 1990 .

[37]  O. Bratteli.  Operator Algebras And Quantum Statistical Mechanics , 1979 .

[38]  R. Kadison,et al.  Fundamentals of the Theory of Operator Algebras , 1983 .

[39]  Claus Köstler,et al.  Stationary Quantum Markov processes as solutions of stochastic differential equations , 1998 .

[40]  H. Araki,et al.  On Quasifree States of the Canonical Commutation Relations (II) , 1971 .

[41]  E. J. Woods,et al.  REPRESENTATIONS OF THE CANONICAL COMMUTATION RELATIONS DESCRIBING A NONRELATIVISTIC INFINITE FREE BOSE GAS , 1963 .

[42]  Edward Nelson.  Notes on non-commutative integration , 1974 .

[43]  H. Yuen.  Generalized coherent states and the statistics of two-photon lasers☆ , 1975 .

[44]  R. Honegger,et al.  Limiting Gibbs states and dynamics for thermal photons , 1992 .

[45]  Ekert,et al.  Relationship between semiclassical and quantum-mechanical input-output theories of optical response. , 1991, Physical Review A. Atomic, Molecular, and Optical Physics.

[46]  The Wigner semi-circle law in quantum electro dynamics , 1996 .

[47]  John T. Lewis,et al.  Dilations of Irreversible Evolutions in Algebraic Quantum Theory , 1977 .