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Journal of Physics Conference Series

Dynamics of Schrödinger cat states

Octavio Castaños, Julio A. López-Saldívar

Published Aug 24, 2012• 14 citations
Publisher

Abstract

A review of the definition of Crystallized Schrödinger cat states and their statistical properties is presented. The associated Wigner and Husimi quasi-probability distribution functions are also calculated together with the second moment of the Husimi function. The Hamiltonian formulation of the Nöther's theorem is used to get the linear time dependent invariants of until general quadratic Hamiltonians in the quadratures of the electromagnetic field. By means of these invariants, the evolution of Crystallized Schrödinger cats is studied, thus we have explicit time dependent expressions for the means and dispersions of the quadratures of the electromagnetic field together with the corresponding Wigner and Husimi functions. By means of these results, we establish some conditions on the parameters of the Crystallized cat states to have the description of non-classical light.

DOI
10.1088/1742-6596/380/1/012017

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A review of the definition of Crystallized Schrödinger cat states and their statistical properties is presented. The associated Wigner and Husimi quasi-probability distribution functions are also calculated together with the second moment of the Husimi function. The Hamiltonian formulation of the Nöther's theorem is used to get the linear time dependent invariants of until general quadratic Hamiltonians in the quadratures of the electromagnetic field. By means of these invariants, the evolution of Crystallized Schrödinger cats is studied, thus we have explicit time dependent expressions for the means and dispersions of the quadratures of the electromagnetic field together with the corresponding Wigner and Husimi functions. By means of these results, we establish some conditions on the parameters of the Crystallized cat states to have the description of non-classical light.
The two-mode even and odd coherent Schrödinger cat states are introduced for the generalized stationary and non-stationary harmonic oscillator. The expected values of positions and momenta and their dispersion matrices are calculated for the Schrödinger cat states. The quadrature squeezing and correlation phenomenon are studied and the Wigner and Husimi functions are constructed explicitly for these states. The photon statistics of the two-mode even and odd coherent states of the generalized oscillator is described in terms of the photon distribution function, which is expressed in terms of Hermite polynomials. The photon number, means and dispersions show oscillations which are characteristics of squeezing and correlation phenomenon.
The theory of quantum propagator and time‐dependent integrals of motion in quantum optics is reviewed as well as the properties of the Wigner function, Q‐function, and coherent state representation. Propagators and wave functions of the free particle, the harmonic oscillator, and the oscillator with varying frequency are studied using time‐dependent integrals of motion linear in position and momentum. Such nonclassical states of light (of quantum systems) as squeezed states, correlated states, even and odd coherent states (Schrödinger cat states) are considered. Photon distribution functions of Schrödinger cat male and female states are given, and the photon distribution function of squeezed vacuum is derived using the theory of the oscillator with varying parameters. Properties of multivariable Hermite polynomials used for the description of the multimode squeezed and correlated light and polymode Schrödinger cats are studied.
New states of electromagnetic field, i.e. even and odd negative binomial states are introduced here. These states interpolate between the even (odd) coherent states and the even (odd) quasi-thermal states. Various statistical properties of these states such as the mean photon number, second-order coherence function, normal squeezing are calculated. The Wigner function for these states are also discussed. These results may be useful for carrying out systematic study of the non-classical properties of `Schrödinger cat states' as one moves from even (odd) coherent state to even (odd) quasi-thermal state.
Integrals of motion and statistical properties of quantized electromagnetic field (e.-m. field) in time-dependent linear dielectric and conductive media are considered, using Choi-Yeon quantization, based on Caldirola-Kanai type Hamiltonian. Eigenstates of quadratic and linear invariants are constructed, the solutions being expressed in terms of a complex parametric function that obeys classical oscillator equation with time-varying frequency. The time evolutions of initial Glauber coherent states and Fock states are considered. The medium conductivity and the time-dependent electric permeability are shown to generate squeezing and non-vanishing covariances. In the time-evolved coherent and squeezed states all the second statistical moments of the electric and magnetic field components are calculated and shown to mminimize the Robertson-Schrödinger uncertainty relation.
The Wigner distribution function for the time-dependent quadratic Hamiltonian system in the coherent Schrodinger cat state is investigated. The type of state we consider is a superposition of two coherent states, which are by an angle of π out of phase with each other. The exact Wigner distribution function of the system is evaluated under a particular choice of phase, δc, q. Our development is employed for two special cases, namely, the Caldirola–Kanai oscillator and the frequency stable damped harmonic oscillator. On the basis of the diverse values of the Wigner distribution function that were plotted, we analyze the nonclassical behavior of the systems.