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Journal of Physics A Mathematical and General

Non-trivial prefactors in adiabatic transition probabilities induced by high-order complex degeneracies

Alain Joye

Published Nov 21, 1993• 30 citations

Abstract

The adiabatic limit of the transition probability for two-level systems driven by real symmetric time-dependent Hamiltonians is considered. When the Hamiltonian depends analytically on time, the transition probability is governed, in simple cases, by a complex eigenvalue crossing point which is, generically, a square root branching point for the eigenvalues. In such a case, the transition probability is given by the Dykhne formula, the recently discovered geometric prefactor being equal to one. In this paper we deal with the general situation where the relevant eigenvalue crossing point is a branching point of order n/2, n>or=1, for the eigenvalues and a zero of order m>or=0 for the Hamiltonian itself. The analysis shows that the Dykhne formula must be completed by a novel prefactor which depends on both n and m. In particular, this prefactor can take the value zero, in contrast to the geometrical prefactor. We also consider the case where the transition probability is governed by N complex eigenvalue crossing points of different orders n j and m j , j=1, ..., N. The end result displays an interference phenomenon between the individual prefactors similar to the case of generic eigenvalue crossing points n j =1 and m j =0 considered earlier.

DOI: 10.1088/0305-4470/26/22/045

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The adiabatic limit of the transition probability for two-level systems driven by real symmetric time-dependent Hamiltonians is considered. When the Hamiltonian depends analytically on time, the transition probability is governed, in simple cases, by a complex eigenvalue crossing point which is, generically, a square root branching point for the eigenvalues. In such a case, the transition probability is given by the Dykhne formula, the recently discovered geometric prefactor being equal to one. In this paper we deal with the general situation where the relevant eigenvalue crossing point is a branching point of order n/2, n>or=1, for the eigenvalues and a zero of order m>or=0 for the Hamiltonian itself. The analysis shows that the Dykhne formula must be completed by a novel prefactor which depends on both n and m. In particular, this prefactor can take the value zero, in contrast to the geometrical prefactor. We also consider the case where the transition probability is governed by N complex eigenvalue crossing points of different orders n j and m j , j=1, ..., N. The end result displays an interference phenomenon between the individual prefactors similar to the case of generic eigenvalue crossing points n j =1 and m j =0 considered earlier.

We consider the transition probability for two-level quantum-mechanical systems in the adiabatic limit when the Hamiltonian is analytic. We give a general formula for the leading term of the transition probability when it is governed by N complex eigenvalue crossings. This leading term is equal to a decreasing exponential times an oscillating function of the adiabaticity parameter. The oscillating function comes from an interference phenomenon between the contributions from each complex eigenvalue crossing, and when N=1, it reduces to the geometric prefactor recently studied.

We study the Landau–Zener problem for a decaying two-level system described by a non-Hermitian Hamiltonian, depending analytically on time. Use of a super-adiabatic basis allows us to calculate the survival probability P in the slow-sweep limit, without specifying the Hamiltonian explicitly. It is found that P consists of a 'dynamical' and a 'geometrical' factor. The former is determined by the complex adiabatic eigenvalues E ± ( t ), only, whereas the latter solely requires the knowledge of α ± ( t ), the ratio of the components of each of the adiabatic eigenstates. Both factors can be split into a universal one, depending only on the complex level crossing points, and a nonuniversal one, involving the full time dependence of E ± ( t ) and α ± ( t ). This general result is applied to the Akulin–Schleich model where the initial upper level is damped with damping constant γ. For analytic power-law sweeps we find that Stückelberg oscillations of P exist for γ smaller than a critical value γ c and disappear for γ > γ c . A physical interpretation of this behaviour will be presented by use of a damped harmonic oscillator.

Using a comparison equation method rooted in the Landau-Zener model, we report results for the probability of transition and survival in two-state models for a crossing of levels having some width, to account for dissipation. The theory allows arbitrary forms for the state energies, their rate of decay, and coupling, subject only to the limitation inherent in the Landau-Zener class (coupling with infinite ``pulse area''). We show how the strong decay result of adiabatic elimination can be recovered, but with important restrictions as to its usability. We also confirm the validity of the well-known Dykhne formula for crossings of real energy levels, and use the new theory to extend the Dykhne result to a class of dissipative crossings we call pseudoreal. Finally, we examine the overall probability for transition at two successive crossings, as occurs in atomic collisions. Here, interference of the crossing states leads to oscillations in the transition probability with collision time. For slow collisions we find that even a modest level of dissipation heavily damps these oscillations, but at the same time enhances the average probability of transition.