arXiv subjects
Publications and source records attributed to V. I. Man'ko.
q-deformed nonlinear field equations are constructed including Klein-Gordon and Maxwell equations. The q-deformation is interpreted as mathematical structure describing specific nonlinearity for which frequency of vibration exponentially depends on the energy
The geometrical phase is shown to be integral of motion. Deformation of particle distribution function corresponding to nonstationary Casimir effect is expressed in terms of multivariable Hermite polynomials. Correction to Planck distribution due to q--nonlinearity is discussed.
Time--dependent integrals of motion which are linear forms in position and momentum are discussed for Husimi parametric forced oscillator. Generalization of these integrals of motion for q--oscillator is presented. Squeezing and quadrature correlation phenomena are discussed on the base of Schrödinger uncertainty relation. The properties of the generalized correlated states, squeezed states, even and odd coherent states (the Schrödinger cat states) are reviewed. The relation of the constructed nonclassical states to representations of the symplectic symmetry group and finite symmetry groups is discussed.
Particle distributions in squeezed states, even and odd coherent states are given in terms of multivariable Hermite polynomials. The Q--function and Wigner function for nonclassical field states are discussed.
We investigate the general case of the photon distribution of a two-mode squeezed vacuum and show that the distribution of photons among the two modes depends on four parameters: two squeezing parameters, the relative phase between the two oscillators and their spatial orientation. The distribution of the total number of photons depends only on the two squeezing parameters. We derive analytical expressions and present pictures for both distributions.
We study the Photon statistics in the superpositions of coherent states $|α\rangle$ and $|α^*\rangle$ named ``Schrödinger real and imaginary cat states''. The oscillatory character of photon distribution function (PDF) emerging due to the quantum interference between the two components is shown, and the phenomenon of the quadrature squeezing is observed for the moderate values of $|α|\sim 1$. Despite the quantity ${\langle\triangle n^2\rangle}/{\langle n\rangle}$ tends to the unit value (like in the Poissonian PDF) at $|α|\gg 1$, the photon statistics is essentially non-Poissonian for all values of $|α|$. The factorial moments and cumulants of the PDF are calculated, and the oscillations of their ratio are demonstrated.
It is shown that the even and odd coherent light and other nonclassical states of light like superposition of coherent states with different phases may replace the squeezed light in interferometric gravitational wave detector to increase its sensitivity. (Contribution to the Second Workshop on Harmonic Oscillator, Cocoyoc, Mexico, March 1994)
Photon distribution function, means and dispersions are found explicitly for the nonclassical state of light which is created from the photon--added coherent state $\vert α,m \rangle$ due to a time--dependence of the frequency of the electromagnetic field oscillator. Generating function for factorial momenta is obtained. The Wigner function and Q--function are constructed explicitly for the excited photon--added coherent state of light. Influence of added photons on known oscillations of photon distribution function for squeezed light is demonstrated. oscillator are considered.
The cumulants and factorial moments of photon distribution for squeezed and correlated light are calculated in terms of Chebyshev, Legendre and Laguerre polynomials. The phenomenon of strong oscillations of the ratio of the cumulant to factorial moment is found.
The infinite number of time-dependent linear in field and conjugated momenta invariants is derived for the scalar field using the Noether's theorem procedure.
It is shown that linear time-dependent invariants for arbitrary multi\-dimensional quadratic systems can be obtained from the Lagrangian and Hamiltonian formulation procedures by considering a variation of coordinates and momenta that follows the classical trajectory and defines a noetherian symmetry transformation.
A two-dimensional generalized oscillator with time-dependent parameters is considered to study the two-mode squeezing phenomena. Specific choices of the parameters are used to determine the dispersion matrix and analytic expressions, in terms of standard hermite polynomials, of the wavefunctions and photon distributions. (to be publish in the Third Workshop on Squeezed States and Uncertainty Relations, Baltimore, USA, (August 1993))
The time dependent-integrals of motion, linear in position and momentum operators, of a quantum system are extracted from Noether's theorem prescription by means of special time-dependent variations of coordinates. For the stationary case of the generalized two-dimensional harmonic oscillator, the time-independent integrals of motion are shown to correspond to special Bragg-type symmetry properties. A detailed study for the non-stationary case of this quantum system is presented. The linear integrals of motion are constructed explicitly for the case of varying mass and coupling strength. They are obtained also from Noether's theorem. The general treatment for a multi-dimensional quadratic system is indicated, and it is shown that the time-dependent variations that give rise to the linear invariants, as conserved quantities, satisfy the corresponding classical homogeneous equations of motion for the coordinates.
The multimode even and odd coherent states (multimode Schroedinger cat states) are constructed for polymode parametric oscillators of the electromagnetic field. The evolution of the photon distribution function is evaluated explicitly. The distribution function is expressed in terms of the multivariable Hermite polynomials, its means and dispersions are calculated. The conditions for the existence of squeezing are formulated. The correlations among the different modes of Schroedinger cat states are studied.
The generalised quasienergy states are introduced as eigenstates of the new integral of motion for periodically and nonperiodically kicked quantum systems.The photon distribution function of polymode generalised correlated light expressed in terms of multivariable Hermite polynomials is discussed and the relation of its properties to Schrodinger uncertainty relation is given.
The effective formulas reducing the two-dimensional Hermite polynomials to the classical (one-dimensional) orthogonal polynomials are given. New one-parameter generating functions for these polynomials are derived. Asymptotical formulas for large values of indices are found. The applications to the squeezed one-mode states and to the time-dependent quantum harmonic oscillator are considered.
The oscillations of photon distribution function for squeezed and correlated light are shown to decrease when the temperature increases.The influence of the squeezing parameter and photon quadrature correlation coefficient on the photon distribution oscillations at nonzero temperatures is studied. The connection of deformation of Planck distribution formula with oscillations of distribution for squeezed and correlated light is discussed.
The review of the following results of the Refs. \cite{Sem} - \cite{Ans} is presented: For mixed state light of $N$-mode electromagnetic field described by Wigner function which has generic Gaussian form the photon distribution function is obtained and expressed expliciltly in terms of Hermite polynomials of $2N$-variables.The momenta of this distribution are calculated and expressed as functions of matrix invariants of the dispersion matrix.The role of new uncertainty relation depending on photon state mixing parameter is elucidated.New sum rules for Hermite polynomials of several variables are found.The photon statistics of polymode even and odd coherent light and squeezed polymode Schrödinger cat light is qiven explicitly.Photon distribution for polymode squeezed number states expressed in terms of multivariable Hermite polynomials is discussed.