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Physical Review A

Failures of the adiabatic approximation in quantum tunneling time

Zhenglu Duan, Weiping Zhang

Published Dec 10, 2012• 2 citations
Publisher

Abstract

In this Brief Report, we investigate the validity of the adiabatic approximation in the tunneling time problem of a two-level atomic wave packet transmitting through a laser beam. We calculate the tunneling time of ground-state atoms using adiabatic approximation and compare it to the exact solution. We find that the adiabatic solution misses an important interference dip in the tunneling time which indicates that the adiabatic approximation is not always valid although the adiabatic conditions are well satisfied.

DOI
10.1103/physreva.86.064101

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In this Brief Report, we investigate the validity of the adiabatic approximation in the tunneling time problem of a two-level atomic wave packet transmitting through a laser beam. We calculate the tunneling time of ground-state atoms using adiabatic approximation and compare it to the exact solution. We find that the adiabatic solution misses an important interference dip in the tunneling time which indicates that the adiabatic approximation is not always valid although the adiabatic conditions are well satisfied.
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Atoms can be prepared at low temperatures by laser cooling. The center of mass motion of the cold atoms exhibits wave nature of quantum mechanics. In this paper, we study the quantum tunneling effect of a cold atomic beam propagating through a laser beam which is blue detuned. The blue detuned laser beam is physically equal to a quantum potential barrier. According to two-level model of atom, we systematically analyze the quantum reflection and transmission of the vector atomic matter wave with internal structures propagating through a laser beam. Quantum tunneling time the atomic beam spends in transmitting the laser beam is studied in details. Because of quantum wave nature, cold atoms tunneling through a laser beam can obviously have a completely different feature from that of a classical particle (hot atom).
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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.
By stating the adiabatic theorem of quantum mechanics in a clear and rigorous way, we establish a necessary condition and a sufficient condition for its validity, where the latter is obtained employing our recently developed adiabatic perturbation theory. Also, we simplify further the sufficient condition into a useful and simple practical test at the expense of its mathematical rigor. We present results for the most general case of quantum systems, i.e., those with degenerate energy spectra. These conditions are of upmost importance for assessing the validity of practical implementations of non-Abelian braiding and adiabatic quantum computation. To illustrate the degenerate adiabatic approximation, and the necessary and sufficient conditions for its validity, we analyze in depth an exactly solvable time-dependent degenerate problem.