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arXiv:1004.3100 · Physical Review Letters

Quantitative Condition is Necessary in Guaranteeing the Validity of the Adiabatic Approximation

D. M. Tong

Published Mar 25, 2010• 98 citations
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Abstract

The quantitative condition has been widely used in the practical applications of the adiabatic theorem. However, it had never been proved to be sufficient or necessary before. It was only recently found that the quantitative condition is insufficient, but whether it is necessary remains unresolved. In this Letter, we prove that the quantitative condition is necessary in guaranteeing the validity of the adiabatic approximation.

DOI
10.1103/physrevlett.104.120401

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The quantitative condition has been widely used in the practical applications of the adiabatic theorem. However, it had never been proved to be sufficient or necessary before. It was only recently found that the quantitative condition is insufficient, but whether it is necessary remains unresolved. In this Letter, we prove that the quantitative condition is necessary in guaranteeing the validity of the adiabatic approximation.
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Abstract The adiabatic theorem was proposed about 90 years ago and has played an important role in quantum physics. The quantitative adiabatic condition constructed from eigenstates and eigenvalues of a Hamiltonian is a traditional tool to estimate adiabaticity and has proven to be the necessary and sufficient condition for adiabaticity. However, recently the condition has become a controversial subject. In this paper, we list some expressions to estimate the validity of the adiabatic approximation. We show that the quantitative adiabatic condition is invalid for the adiabatic approximation via the Euclidean distance between the adiabatic state and the evolution state. Furthermore, we deduce general necessary and sufficient conditions for the validity of the adiabatic approximation by different definitions.
The condition for adiabatic approximation are of basic importance for the applications of the adiabatic theorem. The traditional quantitative condition was found to be necessary but not sufficient, but we do not know its physical meaning and the reason why it is necessary from the physical point of view. In this work, we relate the adiabatic theorem to the uncertainty relation, and present a clear physical picture of the traditional quantitative condition. It is shown that the quantitative condition is just the amplitude of the probability of transition between two levels in the time interval which is of the order of the time uncertainty of the system. We also present a new sufficient condition with clear physical picture.