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Position-dependent mass oscillators and coherent states

The solution of the Schrodinger equation for a position-dependent mass quantum system is studied in two ways. First, the interaction is found which must be applied to a mass m(x) in order to supply it with a particular spectrum of energies. Second, given a specific potential V(x) acting on the mass m(x), the related spectrum is found. The method of solution is applied to a wide class of position-dependent mass oscillators and the corresponding coherent states are constructed. The analytical expressions of such position-dependent mass coherent states preserve the functional structure of the Glauber states.

A Primer on Resonances in Quantum Mechanics

After a pedagogical introduction to the concept of resonance in classical and quantum mechanics, some interesting applications are discussed. The subject includes resonances occurring as one of the effects of radiative reaction, the resonances involved in the refraction of electromagnetic waves by a medium with a complex refractive index, and quantum decaying systems described in terms of resonant states of the energy (Gamow‐Siegert functions). Some useful mathematical approaches like the Fourier transform, the complex scaling method and the Darboux transformation are also reviewed.

Dynamical Equations, Invariants and Spectrum Generating Algebras of Mechanical Systems with Position-Dependent Mass ?

We analyze the dynamical equations obeyed by a classical system with position- dependent mass. It is shown that there is a non-conservative force quadratic in the velocity associated to the variable mass. We construct the Lagrangian and the Hamiltonian for this system and find the modifications required in the Euler{Lagrange and Hamiltons equations to reproduce the appropriate Newtons dynamical law. Since the Hamiltonian is not time invariant, we get a constant of motion suited to write the dynamical equations in the form of the Hamiltons ones. The time-dependent first integrals of motion are then obtained from the factorization of such a constant. A canonical transformation is found to map the variable mass equations to those of a constant mass. As particular cases, we recover some recent results for which the dependence of the mass on the position was already unnoticed, and find new solvable potentials of the Poschl{Teller form which seem to be new. The latter are associated to either the su(1;1) or the su(2) Lie algebras depending on the sign of the Hamiltonian.

SU( 1, 1) Coherent States for Position-Dependent Mass Singular Oscillators

The Schrödinger equation for position-dependent mass singular oscillators is solved by means of the factorization method and point transformations. These systems share their spectrum with the conventional singular oscillator. Ladder operators are constructed to close the su(1,1) Lie algebra and the involved point transformations are shown to preserve the structure of the Barut-Girardello and Perelomov coherent states.

Leaky Modes of Waveguides as a Classical Optics Analogy of Quantum Resonances

A classical optics waveguide structure is proposed to simulate resonances of short range one-dimensional potentials in quantum mechanics. The analogy is based on the well known resemblance between the guided and radiation modes of a waveguide with the bound and scattering states of a quantum well. As resonances are scattering states that spend some time in the zone of influence of the scatterer, we associate them with the leaky modes of a waveguide, the latter characterized by suffering attenuation in the direction of propagation but increasing exponentially in the transverse directions. The resemblance is complete since resonances (leaky modes) can be interpreted as bound states (guided modes) with definite lifetime (longitudinal shift). As an immediate application we calculate the leaky modes (resonances) associated with a dielectric homogeneous slab (square well potential) and show that these modes are attenuated as they propagate.

Negative Time Delay for Wave Reflection from a One-dimensional Semi-harmonic Well

It is reported that the phase time of particles which are reflected by a one-dimensional semi-harmonic well includes a time delay term which is negative for definite intervals of the incoming energy. In this interval, the absolute value of the negative time delay becomes larger as the incident energy becomes smaller. The model is a rectangular well with zero potential energy at its right and a harmonic-like interaction at its left.

SU(1,1) and SU(2) approaches to the radial oscillator: Generalized coherent states and squeezing of variances

It is shown that each one of the Lie algebras su(1,1) and su(2) determine the spectrum of the radial oscillator. States that share the same orbital angular momentum are used to construct the representation spaces of the non-compact Lie group SU(1,1). In addition, three different forms of obtaining the representation spaces of the compact Lie group SU(2) are introduced, they are based on the accidental degeneracies associated with the spherical symmetry of the system as well as on the selection rules that govern the transitions between different energy levels. In all cases the corresponding generalized coherent states are constructed and the conditions to squeeze the involved quadratures are analyzed.

Factorization Method and the Position-dependent Mass Problem

The dynamics of position-dependent mass systems is considered from both, classical and quantum mechanical points of view, by means of the factorization method. Some examples are presented, with particular choices of the mass function, for the harmonic oscillator in order to illustrate our results. In the quantum regime, new isospectral position-dependent mass potentials are also constructed by the intertwiningte chnique.

Time delay in the reflection of particles by semi-harmonic wells

The phase time of wave reflection from a one-dimensional semi-harmonic well includes a delay term which can be negative according to the incoming energy and the well parameters in a quantum scattering process. The model is a rectangular well which is embedded in an environment composed by a zero potential energy to its right and a harmonic-like potential to its left.

Simulation of non-resonant quantum control protocols for a single qubit

The resonant control techniques are susceptible of assisting error accumulation in quantum control protocols consisting of large sequences of external fields. It is shown that for some algorithms the information about the final state of the system may be partially or completely lost. Non trivial operations in the non resonant regime are also discussed.