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arXiv:quant-ph/0207107 · Acta Physica Polonica B

Adiabatic limit interference effects for two energy level transition amplitudes and Nikitin - Umanskii formula studied by fundamental solution method

Stefan Giller

Published Jul 18, 2002• 3 citations
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Abstract

A method of fundamental solutions has been used to study adiabatic transition amplitudes in two energy level systems for a class of Hamiltonians allowing some simplifications of Stokes graphs corresponding to such transitions. It has been shown that for simplest such cases the amplitudes take the Nikitin - Umanskii form but for more complicated ones they are formed by a sum of terms strictly related to a structure of Stokes graph corresponding to such cases. This paper corrects our previous one [Phys. Rev. A, 63 052101 (2001)] and its results are in a full agreement with the ones of Joye, Mileti and Pfister [Phys. Rev. A, 44 4280 (1991)].

DOI
10.48550/arxiv.quant-ph/0207107

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A method of fundamental solutions has been used to study adiabatic transition amplitudes in two energy level systems for a class of Hamiltonians allowing some simplifications of Stokes graphs corresponding to such transitions. It has been shown that for simplest such cases the amplitudes take the Nikitin - Umanskii form but for more complicated ones they are formed by a sum of terms strictly related to a structure of Stokes graph corresponding to such cases. This paper corrects our previous one [Phys. Rev. A, 63 052101 (2001)] and its results are in a full agreement with the ones of Joye, Mileti and Pfister [Phys. Rev. A, 44 4280 (1991)].
A method of fundamental solutions has been used to investigate transitions in two-energy-level-systems with no level crossing in a real time. Compact formulas for transition probabilities have been found in their exact form as well as in their adiabatic limit. No interference effects resulting from many-level complex crossings as announced by Joye, Mileti, and Pfister [Phys. Rev. A 44, 4280 (1991)] have been detected in either case. It is argued that these results of this work are incorrect. However, some effects of Berry's phases are confirmed.
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.
This paper is concerned with the problem of calculating amplitudes for nonadiabatic transitions induced by a time-dependent Hamiltonian, in the semiclassical limit h/→0, with emphasis on questions relevant to semiclassical theories of electronically inelastic scattering. For this problem the semiclassical limit is mathematically equivalent to the adiabatic limit, and the adiabatic theorem says that all these transition amplitudes vanish in the limit; the question is, what is the asymptotic form of the nonadiabatic amplitudes, as they go to zero? We consider Hamiltonia that are analytic matrix functions of time. We prove a generalization of the adiabatic theorem to the complex time plane; paradoxically, the adiabatic theorem in the complex plane gives us directly the nonadiabatic amplitudes along the real time axis. We derive Dykhne’s remarkable formula for the two-state case, which says that the limiting form of the transition amplitude depends only on the energy curves of the two states, not on the nonadiabatic coupling which is responsible for transition between them. We discuss the three-state problem at length and show that the obvious generalization of the Dykhne formula is sometimes true, sometimes false. To indicate the scope of methods based on the adiabatic theorem in the complex plane, we give an elementary proof of the semiclassical formula for above-barrier reflection of a one-dimensional particle.
A semiclassical theory for inelastic scattering of a particle by a two-state system in the near-adiabatic limit is developed. The exponentially small transition amplitude is calculated. The theory is an extension of Pokrovskii and Khalatnikov's theory for above-barrier reflection (Zh. Eksp. Teor. Fiz. 40, 1713 (1961) [Sov. Phys.---JETP 13, 1207 (1961)]). The analysis involves studies of the WKB solutions in the complex coordinate plane along certain contours. The region of validity of the theory is established. Our result has the same form as the Landau-Zener-Stueckelberg formula; however, our theory is applicable to more general systems. Numerical comparisons with exact solutions are presented; the differences between our results and exact solutions become negligible as the adiabatic limit is approached.
The authors consider a two-level quantum mechanical system driven by an analytic time-dependent Hamiltonian of the form H( in t). In the adiabatic limit, in <<1, the transition probability P(+,-) from one energy level (labelled by -) at time t=- infinity to the other (labelled by +) at time t=+0OI is known to behave as P(+,-)=exp(- alpha -1 in -1 ) exp( alpha 0 )(1+O( in )). Using a simple iterative procedure generating Hamiltonians H 0 =H,H 1 ,. . .H N+1 , they compute the full asymptotic expansion of the transition probability ln P(+,-)=- alpha -1 in -1 + alpha 0 + j=0 N alpha j in j +O ( in N+1 ) For all N>or=0.
We study the SSS matrix for the transitions at an avoided crossing of several energy levels, which is a multilevel generalization of the Landau-Zener problem. We demonstrate that, by extending the Schr\"odinger evolution to complex time, one can obtain an exact answer for some of the transition amplitudes. Similar to the Landau-Zener case, our result covers both the adiabatic (slow evolution) and the diabatic (fast evolution) regimes. The form of the exact transition amplitude coincides with that obtained in a sequential pairwise level crossing approximation, in accord with the conjecture of Brundobler and Elser [J. Phys. A 26, 1211 (1993)].