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arXiv:1506.04019 · Physical Review A

Adiabaticity in a time-dependent trap: The passage near a continuum threshold

D. Sokolovski, M. Ll. Pons

Published Oct 22, 2015• 2 citations
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Abstract

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.

DOI
10.1103/physreva.92.042121

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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 Pstay{P}^{\mathrm{stay}}Pstay 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, Pstay{P}^{\mathrm{stay}}Pstay 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 Pstay{P}^{\mathrm{stay}}Pstay tends to a universal value of about 3838%38, for a broad class of potentials. In the rapid-passage limit, Pstay{P}^{\mathrm{stay}}Pstay 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.
We consider a time-dependent trap externally manipulated in such a way that one of its bound states is brought into an instant contact with the continuum threshold and then down again. It is shown that, in the limit of slow evolution, the probability to remain in the bound state, Pstay{P}^{\mathrm{stay}}Pstay 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 Pstay{P}^{\mathrm{stay}}Pstay 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.
ADIABATIC passage through an almost degenerate intermediate state is a common experimental technique.It is ba sed on the Ehrenfest principle which states that if the external conditions for a system are changed slowly enough, the change of the system can be predicted by computing qu asi statically the modification of its initial quantum state.The Ehrenfest principle is not a law of na ture beca use the system has an alternate path it can follow, which might appropriately be termed 'inertial*.This path is favored by a rapid change of the external variables.Among the two paths, the adiabatic one has a privileged position.Thus, it has been shown re cently that if the external change is measured by a "slowness parameter" inversely proportional to the time ta ken to effect the change then the Ehrenfest principle is valid to all orders in th is para meter [l].The following lines are to derive a rigorous formula for the probability that the adiabatic path is not followed in the simple case of the near crossing of two quantum levels.
We consider a reduced two-channel model of an atom consisting of a quantum dot coupled to an open scattering channel described by a three-dimensional Laplacian. We are interested in the survival probability of a bound state when the dot energy varies smoothly and adiabatically in time. The initial state corresponds to a discrete eigenvalue which dives into the continuous spectrum and re-emerges from it as the dot energy is varied in time and finally returns to its initial value. Our main result is that for a large class of couplings, the survival probability of this bound state vanishes in the adiabatic limit. At the end of the paper, we present a short outlook on how our method may be extended to cover other classes of Hamiltonians; details will be given elsewhere.
We investigate the capture of a quantum particle in free space by a moving trapping potential. The capture is investigated for various initial conditions. We examine the dependence of the probability of capture on the shape and depth of the trapping potential, initial speed and mass of the particle, and other parameters. We take the trapping potential to move with an initial speed v 0 and to decelerate with constant acceleration a c until the well stops moving. We study the motion during the time the well is moving and after the well has stopped. We compare the probability of capture to the probability that the particle is in a stationary state (while the well is moving) and the probability the particle is in a bound state (after the well's motion has stopped). Our work could be of interest to instructors and students in upper-year undergraduate quantum mechanics courses.