Theory of a nonstationary quantum oscillator

V. V. Dodonov, Olga V. Man’ko, Vladimir I. Man’ko

Vladimir I. Man’ko

A. Angelow

It has been found a solution of the equation for classical nonstationary harmonic oscillator, with time dependent frequency in integral form. It has been shown, that the quantum optical problem for propagation of electromagnetic waves in nonlinear waveguide is related to the solutions of classical nonstationary harmonic oscillator. This method could be used for obtaining other solutions of the classical harmonic oscillator, with other time dependent frequency, if we have already determined the fluctuations σq, σp and covariance cqp by some different methods. There have been found the substitutions, which allow to establish a connection between the general method of linear invariants [Malkin et.al., Phys Rev.2D (1970) 1371] and some particular cases in later papers. This gives the opportunity to present the problem with equation of two- dimensional nonstationary harmonic oscillator, which is physically more informative. 1

Omar Cherbal, M. Drir, Mustapha Maamache +1 more

The most general form of Hamiltonian that preserves fermionic coherent states stable in time is found in the form of nonstationary fermion oscillator. Invariant creation and annihilation operators and related Fock states and coherent states are built up for the more general system of nonstationary forced fermion oscillator.

Kevin Zelaya, Oscar Rosas-Ortiz

Abstract We consider the relations between nonstationary quantum oscillators and their stationary counterpart in view of their applicability to study particles in electromagnetic traps. We develop a consistent model of quantum oscillators with time-dependent frequencies that are subjected to the action of a time-dependent driving force, and have a time-dependent zero point energy. Our approach uses the method of point transformations to construct the physical solutions of the parametric oscillator as mere deformations of the well known solutions of the stationary oscillator. In this form, the determination of the quantum integrals of motion is automatically achieved as a natural consequence of the transformation, without necessity of any ansätz. It yields the mechanism to construct an orthonormal basis for the nonstationary oscillators, so arbitrary superpositions of orthogonal states are available to obtain the corresponding coherent states. We also show that the dynamical algebra of the parametric oscillator is immediately obtained as a deformation of the algebra generated by the conventional boson ladder operators. A number of explicit examples is provided to show the applicability of our approach.

Carlo Heissenberg, Augusto Sagnotti