Hermite Coherent States for Quadratic Refractive Index Optical Media

Gress, Z.; Cruz, Sara Cruz y

doi:10.1007/978-3-030-20087-9_14

Cited by 6 publications

(18 citation statements)

References 19 publications

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“…We can appreciate that ϕ 0 (x, t) is a localized wave-packet that spreads out during a finite interval of time, then it is squeezed up to it recovers its initial configuration. Such an oscillatory property is relevant in the paraxial approximation of electromagnetic signals, for it is associated with self-focusing beams in varying media [82][83][84][85]. For higher eigenfunctions there is a definite number of nodes, the position of which varies in time.…”

mentioning

confidence: 99%

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

Zelaya

Rosas-Ortiz

2020

Phys. Scr.

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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.

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mentioning

confidence: 99%

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

Zelaya

Rosas-Ortiz

2020

Phys. Scr.

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“…Remark that the frequency of V 1 (x, t) is exactly the same as the constant frequency ω 0 of the stationary oscillator (14). That is, the time-dependence of the new potential (49) arises from the additive term included by the Darboux transformation. In this respect, the nonstationary oscillators represented by such a potential increases the number of exactly solvable time-dependent oscillators already reported in the literature, where it is usual to find oscillators with time-dependent frequency that are acted by a driving force which also depends on time.…”

Section: Nonstationary Oscillatorsmentioning

confidence: 96%

“…To get more insights about the new set of functions (53) we have to emphasize that they are not eigenfunctions of the Hamiltonian defined by the time-dependent potential (49). The reason is that such a Hamiltonian is not an integral of motion.…”

Section: Solutions and Dynamical Algebra For The Time-dependent Oscilmentioning

confidence: 99%

“…It is clear that the coherent states ψ z (x, t) are not invariant under time evolution (the time-dependence is not defined by the complex eigenvalue z, as in the previous cases). Nevertheless, they form an overcomplete set in the space of states of the nonstationary oscillators V 1 (x, t) defined in (49).…”

Section: Coherent Statesmentioning

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%

See 2 more Smart Citations

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

Cruz¹,

Razo²,

Rosas-Ortiz

et al. 2020

Phys. Scr.

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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.

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Laguerre–Gaussian Wave Propagation in Parabolic Media

Cruz

Gress

Jiménez-Macías

et al. 2020

Trends in Mathematics

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We report a new set of Laguerre-Gaussian wave-packets that propagate with periodical self-focusing and finite beam width in weakly guiding inhomogeneous media. These wave-packets are solutions to the paraxial form of the wave equation for a medium with parabolic refractive index. The beam width is defined as a solution of the Ermakov equation associated to the harmonic oscillator, so its amplitude is modulated by the strength of the medium inhomogeneity. The conventional Laguerre-Gaussian modes, available for homogenous media, are recovered as a particular case.

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Hermite Coherent States for Quadratic Refractive Index Optical Media

Cited by 6 publications

References 19 publications

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

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

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

Laguerre–Gaussian Wave Propagation in Parabolic Media

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