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Physical Review Letters

Experimental Study of the Validity of Quantitative Conditions in the Quantum Adiabatic Theorem

Jiangfeng Du, Ling-Zhi Hu, Ya Wang, Jianda Wu, Meisheng Zhao, Dieter Suter

Published Aug 7, 2008• 73 citations
Publisher

Abstract

The quantum adiabatic theorem plays an important role in quantum mechanics. However, counter-examples were produced recently, indicating that their transition probabilities do not converge as predicted by the adiabatic theorem [K. P. Marzlin et al., Phys. Rev. Lett. 93, 160408 (2004); D. M. Tong et al., Phys. Rev. Lett. 95, 110407 (2005)]. For a special class of Hamiltonians, we examine the standard criterion for adiabatic evolution experimentally and theoretically, as well as three newly suggested adiabatic conditions. We show that the standard criterion is neither sufficient nor necessary.

DOI
10.1103/physrevlett.101.060403

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The quantum adiabatic theorem plays an important role in quantum mechanics. However, counter-examples were produced recently, indicating that their transition probabilities do not converge as predicted by the adiabatic theorem [K. P. Marzlin et al., Phys. Rev. Lett. 93, 160408 (2004); D. M. Tong et al., Phys. Rev. Lett. 95, 110407 (2005)]. For a special class of Hamiltonians, we examine the standard criterion for adiabatic evolution experimentally and theoretically, as well as three newly suggested adiabatic conditions. We show that the standard criterion is neither sufficient nor necessary.
The quantum adiabatic theorem is a widely applied result in quantum mechanics. However, recently, counterexamples were produced, indicating that the transition probabilities in some systems do not converge as predicted by the adiabatic theorem [K. P. Marzlin and B. C. Sanders, Phys. Rev. Lett. 93, 160408 (2004)]. In this paper, we examine a proof of the adiabatic theorem, paying specific attention to the validity of the assumptions made in the proof. We show that the convergence of the transition probabilities is path dependent when the change in the Hamiltonian is large. Finally, we consider the application of this reformulated form of the adiabatic theorem to the adiabatic evolution paradigm in quantum computation.
A consensus that questions the perfunctory use of the quantum adiabatic theorem has emerged since Marzlin and Sanders [Phys. Rev. Lett. 93, 160408 (2004)] showed the existence of an inconsistency in the applicability of the theorem. Further analysis proved that the inconsistency may arise from the existence of resonant terms in the Hamiltonian, but recent work indicates that the debate about the full extent of the problem remains open. Here, we first show that key premises required in the standard demonstration of the theorem do not hold for a dual Hamiltonian involved in the Marzlin-Sanders inconsistency. Also, we show that two simple conditions can identify systems for which the adiabatic approximation fails, in spite of satisfying traditional quantitative conditions that were believed to guarantee its validity. Finally, we prove that the inconsistency only arises for Hamiltonians that contain resonant terms whose amplitudes go asymptotically to zero.
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.
We analyze the validity of the adiabatic approximation, and in particular the reliability of what has been called the ``standard criterion'' for validity of this approximation. Recently, this criterion has been found to be insufficient. We will argue that the criterion is sufficient only when it agrees with the intuitive notion of slowness of evolution of the Hamiltonian. However, it can be insufficient in cases where the Hamiltonian varies rapidly but only by a small amount. We also emphasize the distinction between the adiabatic theorem and the adiabatic approximation, two quite different, although closely related, ideas.