Acta Polytechnica

Time-dependent step-like potential with a freezable bound state in the continuum

Izamar Gutiérrez Altamirano, Alonso Contreras-Astorga, Alfredo Raya Montaño

Published Feb 28, 2022• 1 citation

Abstract

In this work, we construct a time-dependent step-like potential supporting a normalizable state with energy embedded in the continuum. The potential is allowed to evolve until a stopping time ti, where it becomes static. The normalizable state also evolves but remains localized at every fixed time up to ti. After this time, the probability density of this state freezes becoming a Bound state In the Continuum. Closed expressions for the potential, the freezable bound state in the continuum, and scattering states are given.

DOI: 10.14311/ap.2022.62.0056

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Acta Polytechnica

Time-dependent step-like potential with a freezable bound state in the continuum

Izamar Gutiérrez Altamirano, Alonso Contreras-Astorga, Alfredo Raya Montaño

Published Feb 28, 2022• 1 citation

Publisher

Abstract

DOI: 10.14311/ap.2022.62.0056

Related papers

General Theory of Constructing Potential with Bound States in the Continuum

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Abstract We present a general theory of potentials that support bound states at positive energies (bound states in the continuum). On the theoretical side, we prove that, for systems described by nonlocal potentials of the form $V(r,r^{\prime })$ , bound states at positive energies are as common as those at negative energies. At the same time, we show that a local potential of the form $V(r)$ rarely supports a positive-energy bound state. On the practical side, we show how to construct a (naturally nonlocal) potential that supports an arbitrary normalizable state at an arbitrary positive energy. We demonstrate our theory with numerical examples both in momentum and coordinate spaces with emphasis on the important role played by nonlocal potentials. Finally, we discuss how to observe bound states at positive energies, and where to search for nonlocal potentials that may support them.Abstract We present a general theory of potentials that support bound states at positive energies (bound states in the continuum). On the theoretical side, we prove that, for systems described by nonlocal potentials of the form $V(r,r^{\prime })$, bound states at positive energies are as common as those at negative energies. At the same time, we show that a local potential of the form $V(r)$ rarely supports a positive-energy bound state. On the practical side, we show how to construct a (naturally nonlocal) potential that supports an arbitrary normalizable state at an arbitrary positive energy. We demonstrate our theory with numerical examples both in momentum and coordinate spaces with emphasis on the important role played by nonlocal potentials. Finally, we discuss how to observe bound states at positive energies, and where to search for nonlocal potentials that may support them.

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V(r,r^{\prime })

, bound states at positive energies are as common as those at negative energies. At the same time, we show that a local potential of the form

V(r)

rarely supports a positive-energy bound state. On the practical side, we show how to construct a (naturally nonlocal) potential that supports an arbitrary normalizable state at an arbitrary positive energy. We demonstrate our theory with numerical examples both in momentum and coordinate spaces with emphasis on the important role played by nonlocal potentials. Finally, we discuss how to observe bound states at positive energies, and where to search for nonlocal potentials that may support them.

We consider a time-dependent trap externally manipulated in such a way that one of its bound states is brought up towards the continuum threshold and then down again. We evaluate the probability

{P}^{\mathrm{stay}}

of a particle, initially in a bound state of the trap, continuing in it at the end of the passage. We use the Sturmian representation, whereby the problem is reduced to evaluating the reflecting coefficient of an absorbing potential. In the slow-passage limit,

{P}^{\mathrm{stay}}

goes to 1 for a state turning before reaching the continuum threshold and vanishes if the bound state crosses into the continuum. For a slowly moving state, just ``touching'' the threshold

{P}^{\mathrm{stay}}

tends to a universal value of about

38%

, for a broad class of potentials. In the rapid-passage limit,

{P}^{\mathrm{stay}}

depends on the choice of the potential. Various types of trapping potentials are considered, with an analytical solution obtained in the special case of a zero-range well.

{P}^{\mathrm{stay}}

tends to a universal limit and is determined only by the manner in which the adiabatic bound state approaches and leaves the threshold. The task of evaluating

{P}^{\mathrm{stay}}

in the adiabatic limit can be reduced to studying the loss from a zero-range well and is performed numerically. Various types of trapping potentials are considered. Applications of this theory to cold atoms in traps and to propagation of transversal modes in tapered waveguides are proposed.