arXiv:math-ph/9807031

Exponential Estimates in Adiabatic Quantum Evolution

Alain Joye, Charles‐Edouard Pfister

Published Jul 24, 1998• 18 citations

Abstract

We review recent results concerning the exponential behaviour of transition probabilities across a gap in the adiabatic limit of the time-dependent Schrödinger equation. They range from an exponential estimate in quite general situations to asymptotic Landau-Zener type formulae for finite dimensional systems, or systems reducible to this case.

DOI: 10.48550/arxiv.math-ph/9807031

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arXiv:math-ph/9807031

Exponential Estimates in Adiabatic Quantum Evolution

Alain Joye, Charles‐Edouard Pfister

Published Jul 24, 1998• 18 citations

arXiv HTML PDF

Abstract

DOI: 10.48550/arxiv.math-ph/9807031

Related papers

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V_0

such that the system stays at the initial state in its entirety when

V_0

= 0 while it is divided equally in both states when

V_0 \rightarrow {\infty}

. For a finite coupling the probability branching at the crossing is not even and thus a complete probability transfer at

t \rightarrow {\infty}

is not achieved in the linear potential crossing problem. The Landau-Zener formula for transition probability at

t \rightarrow {\infty}

is expressed in terms of the probabilities at the crossing.

\hat{H}_0

\hat{H}_1

in a total evolution-time

τ

, and our goal is to determine how the final state of the system depends on

τ

. If the system is initialized in a non-degenerate ground state, the adiabatic theorem says that in the limit of large

τ

, the system will stay in the ground state. We examine the near-adiabatic limit where the system evolves slowly enough that most but not all of the final state is in the ground state, and we find that the probability of leaving the ground state oscillates in

τ

with a frequency determined by the integral of the spectral gap along the trajectory of the Hamiltonian, so long as the gap is big. If the gap becomes exceedingly small, the final probability is the sum of oscillatory behavior determined by the integrals of the gap before and after the small gap. We confirm these analytic predictions with numerical evidence from barrier tunneling problems in the context of quantum adiabatic optimization.