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Physical Review A

Adiabaticity near a continuum threshold: An exactly solvable model

D. Sokolovski, M. Ll. Pons, Juan Gonzalo Muga

Published Apr 30, 2014• 1 citation
Publisher

Abstract

We consider a potential well whose shape changes with time, so that a bound state is brought closer to the edge of the continuum. We then ask the following: how slow should the evolution be for the adiabatic approximation to hold? We answer the question by considering, in a Sturmian representation, linear evolution of a $\ensuremath{\delta}$-function potential well. The general applicability and accuracy of the obtained adiabatic criterion is discussed.

DOI
10.1103/physreva.89.042125

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We consider a potential well whose shape changes with time, so that a bound state is brought closer to the edge of the continuum. We then ask the following: how slow should the evolution be for the adiabatic approximation to hold? We answer the question by considering, in a Sturmian representation, linear evolution of a \ensuremathδ\ensuremath{\delta}\ensuremathδ-function potential well. The general applicability and accuracy of the obtained adiabatic criterion is discussed.
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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.
The adiabatic theorem states that an initial eigenstate of a slowly varying Hamiltonian remains close to an instantaneous eigenstate of the Hamiltonian at a later time. We show that a perfunctory application of this statement is problematic if the change in eigenstate is significant, regardless of how closely the evolution satisfies the requirements of the adiabatic theorem. We also introduce an example of a two-level system with an exactly solvable evolution to demonstrate the inapplicability of the adiabatic approximation for a particular slowly varying Hamiltonian.
Adiabaticity occurs when, during its evolution, a physical system remains in the instantaneous eigenstate of the Hamiltonian. Unfortunately, existing results, such as the quantum adiabatic theorem based on a slow down evolution $[H(ϵt),ϵ\ensuremath{\rightarrow}0]$, are insufficient to describe an evolution driven by the Hamiltonian H(t)H(t)H(t) itself. Here we derive general criteria and exact bounds, for the state and its phase, ensuring an adiabatic evolution for any Hamiltonian H(t)H(t)H(t). As a corollary, we demonstrate that the commonly used condition of a slow Hamiltonian variation rate, compared to the spectral gap, is indeed sufficient to ensure adiabaticity but only when the Hamiltonian is real and nonoscillating (for instance, containing exponential or polynomial but no sinusoidal functions).
A new and intuitive perturbative approach to time-dependent quantum mechanics problems is presented, which is useful in situations where the evolution of the Hamiltonian is slow. The state of a system which starts in an instantaneous eigenstate of the initial Hamiltonian is written as a power series which has a straightforward diagrammatic representation. Each term of the series corresponds to a sequence of ``adiabatic'' evolutions, during which the system remains in an instantaneous eigenstate of the Hamiltonian, punctuated by transitions from one state to another. The first term of this series is the standard adiabatic evolution, the next is the well known first correction to it, and subsequent terms can be written down essentially by inspection. Although the final result is perhaps not terribly surprising, it seems to not be widely known, and the interpretation is new, as far as we know. Application of the method to the adiabatic approximation is given, and some discussion of the validity of this approximation is presented.