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Chinese Journal of Quantum Electronics

Quantum adiabatic approximation condition and quantum geometric potential

YI Xue-hu

Published Jan 1, 2014• 0 citations
OpenAlex

Abstract

The quantum traditional adiabatic theorem was present and traditional approximation conditions were given.Based on existence of insufficence about quntum traditional adiabatic approximation condition,the new adiabatic theory and approximation condition were proposed and discussed by means of perturbative approach and U(1)-invariant expasion.The properties of quantum geometric potential contained in the new adiabatic condition were analyzed and discussed.

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The quantum traditional adiabatic theorem was present and traditional approximation conditions were given.Based on existence of insufficence about quntum traditional adiabatic approximation condition,the new adiabatic theory and approximation condition were proposed and discussed by means of perturbative approach and U(1)-invariant expasion.The properties of quantum geometric potential contained in the new adiabatic condition were analyzed and discussed.
In this paper, we present a U(1)\text{U}(1)U(1)-invariant expansion theory of the adiabatic process. As its application, we propose and discuss different sufficient adiabatic approximation conditions. In these conditions, we find an interesting invariant quantity referred to as the quantum geometric potential (QGP) contained in all time-dependent processes. Furthermore, we also give detailed discussion and analysis on the properties and effects of QGP.
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.
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 quantum adiabatic theorem is the basis of adiabatic quantum computation. However, the exact necessary and sufficient conditions for adiabatic evolution are still under debate. We discuss the adiabatic condition of a system undergoing a special evolution route, and obtain an explicit formula that is necessary and sufficient for the adiabatic evolution in this route. Based on this formula, we find that the traditional adiabatic condition is neither sufficient nor necessary. Finally, we show that no adiabatic process can occur even the evolution speed goes to 0 in some examples, which is surprising since the adiabatic theorem states that if the evolution of a system is slow enough, the adiabatic process could occur.