A Note on the Off-Axis Gaussian Beams Propagation in Parabolic Media

2020

Phys. Scr.

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 timedependent 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.

Section: Discussionmentioning

confidence: 99%

mentioning

confidence: 99%

Quantum nonstationary oscillators: Invariants, dynamical algebras and coherent states via point transformations

Zelaya

2020

Phys. Scr.

“…Besides, they have close formal similarities with general superpotentials leading to isospectral potentials in supersymmetric quantum mechanics [63][64][65][66][67][68][69]. Recent applications include propagation of optical beams in parabolic media [171][172][173] and Kerr media [174][175][176][177] as well, studies of the geometry of the Riccati equation [178] and the fourth-order Schrödinger equation with the energy spectrum of the Pöschl-Teller system [179]. Further discussion on the subject can be found in the Schuch's book on a nonlinear perspective to quantum theory [180].…”

Section: Time-dependent Oscillator Wave Packetsmentioning

confidence: 99%

Coherent and Squeezed States: Introductory Review of Basic Notions, Properties, and Generalizations

Integrability, Supersymmetry and Coherent States

2019

A short review of the main properties of coherent and squeezed states is given in introductory form. The efforts are addressed to clarify concepts and notions, including some passages of the history of science, with the aim of facilitating the subject for nonspecialists. In this sense, the present work is intended to be complementary to other papers of the same nature and subject in current circulation.

“…It is expected that our approach can be applied to study either trapping of particles by electromagnetic fields [18][19][20][21][22][23], or the propagation of electromagnetic signals [47][48][49]. Our approach can be extended to the case of non-Hermitian Hamiltonians [38,39], for which some interesting results have been reported quite recently [46].…”

Section: Discussionmentioning

confidence: 99%

“…[45,46]. Here, we construct a series of wave-packets associated to the stationary oscillator that have the profile of the Hermite-Gauss modes of quantum optics [47][48][49]. We derive the corresponding invariant (which has the profile of the Ermakov-Lewis-Reisenfeld one), and show that such states are very useful to construct the new time-dependent oscillators as well as their solutions.…”

Section: Introductionmentioning

confidence: 99%

Coherent states for exactly solvable time-dependent oscillators generated by Darboux transformations

Cruz¹,

Razo²,

et al. 2020

Phys. Scr.

Self Cite

The Darboux method is commonly used in the coordinate variable to produce new exactly solvable (stationary) potentials in quantum mechanics. In this work we follow a variation introduced by Bagrov, Samsonov, and Shekoyan (BSS) to include the time-variable as a parameter of the transformation. The new potentials are nonstationary and define Hamiltonians which are not integrals of motion for the system under study. We take the stationary oscillator of constant frequency to produce nonstationary oscillators, and also provide an invariant that serves to define uniquely the state of the system. In this sense our approach completes the program of the BSS method since the eigenfunctions of the invariant are an orthonormal basis for the space of solutions of the related Schrödinger equation. The orthonormality holds when the involved functions are evaluated at the same time. The dynamical algebra of the nonstationary oscillators is generated by properly chosen ladder operators and coincides with the Heisenberg algebra. We also construct the related coherent states and show that they form an overcomplete set that minimizes the quadratures defined by the ladder operators. These states are not invariant under time-evolution since their time-dependence relies on the basis of states and not on the complex eigenvalue that labels them. Some concrete examples are provided.