Consistency of the Adiabatic Theorem

Abstract

The adiabatic theorem provides the basis for the adiabatic model of quantum computation. Recently the conditions required for the adiabatic theorem to hold have become a subject of some controversy. Here we show that the reported violations of the adiabatic theorem all arise from resonant transitions between energy levels. In the absence of fast driven oscillations the traditional adiabatic theorem holds. Implications for adiabatic quantum computation is discussed.A statement of the traditional adiabatic theorem [1,2,3], as described in most recent publications, is as follows: Consider a system with a time dependent Hamiltonian H(t) and a wave function |ψ(t) , which is the solution of the Schrödinger equation (h = 1)Let |E n (t) be the instantaneous eigenstates of H(t) with eigenvalues E n (t). If at an initial time t = 0 the system starts in an eigenstate |E n (0) of the Hamiltonian H(0), it will remain in the same instantaneous eigenstate, |E n (t) , at a later time t = T , as long as the evolution of the Hamiltonian is slow enough to satisfywhereThe adiabatic theorem has recently gained renewed attention as it provides the basis for one of the important schemes for quantum computation, i.e., adiabatic quantum computation [4,5]. Recently, the adiabatic condition (2) has become a subject of controversy. It was first shown by Marzlin and Sanders [6] and then by Tong et al. [7] that if a first system with Hamiltonian H(t) follows an adiabatic evolution, a second system defined by Hamiltoniancannot have an adiabatic evolution unlesseven if both systems satisfy the same condition (2). Here, T denotes time ordering operator. Recently, the validity of the adiabatic theorem was experimentally examined [8], and (2) was reported to be neither sufficient nor necessary condition for adiabaticity. These inconsistencies have created debates among researchers [9,10,11,12] and motivated a search for alternative conditions [13,14,15,16,17,18], reexamination of some adiabatic algorithms [19], or generalizations of the adiabatic theorem to open quantum systems [20].While it is valuable to find new conditions that guarantee adiabaticity in general, it is important to understand why the traditional adiabatic condition (2) is sufficient for some Hamiltonians, but not for others. Moreover, from the practical point of view it is much easier to work with a simple condition like (2) than some other more sophisticated ones. In this letter, we relate the reported violations of the traditional adiabatic theorem to resonant transitions between energy levels. We further show that in the absence of such resonant oscillations, the traditional adiabatic condition is sufficient to assure adiabaticity. Our line of thought is close to that of Duki et al. [9], but largely extended with rigorous mathematical proofs.It is well known that fast driven oscillations invalidate the adiabatic theorem [21]. Consider a simple example of a two-state system driven by an oscillatory force:We take V to be a small positive number. The exact instantaneous eigen...

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