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Journal of Physics A Mathematical and General

Full asymptotic expansion of transition probabilities in the adiabatic limit

Alain Joye, C. -E. Pfister

Published Feb 21, 1991• 42 citations
Publisher

Abstract

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.

DOI
10.1088/0305-4470/24/4/012

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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.
The authors consider the evolution of a two-level system driven by a nonself-adjoint Hamiltonian H( in t) and treat the adiabatic limit in to 0. While adiabatic theorem-like results do not hold true in general for this case, they prove that they are still valid for the subspace corresponding to the eigenvalue having the largest imaginary part (least dissipative eigenvalue). The theory gives the full asymptotic expansion of the evolution restricted to this subspace. The first correction beyond Berry's phase is to their best knowledge given explicitly for the first time.
Non-adiabatic transitions in two-level systems are investigated theoretically for a random time dependence of ħω, the energy difference, between the levels. We assumed that ω = ω(x) and the coordinate x = x(t) is a random function of time. Diffusion and Poisson processes (both homogeneous and with a source) for x(t) were assumed. The cases of linear crossing terms (ω = γx) and non-linear terms (ω = ω e exp (- αx) + ω0) were considered. Values of the non-adiabatic transition probability per unit time were obtained by perturbation theory for ω1τ c ≪ 1 where τ c is the correlation time and ω1 is the off-diagonal matrix element.
The time of Zener tunneling, ${\ensuremath{\tau}}_{z}$, is calculated for a general two-level quantum-mechanical system by two methods. In the first we determine the width of the transition profile in time. In the second we apply an oscillating perturbation and examine how the final transition probability depends upon the perturbation frequency and phase. Both methods show, given that the coupling energy between the diabatic levels is \ensuremath{\Delta} and the time rate of change of the energy is \ensuremath{\alpha}, that in the adiabatic limit ${\ensuremath{\tau}}_{z}$ scales as \ensuremath{\Delta}/\ensuremath{\alpha} and in the sudden limit ${\ensuremath{\tau}}_{z}$ scales as (\ensuremath{\Elzxh}/\ensuremath{\alpha})1/2{)}^{1/2})1/2. We calculate the approximate transition probability at finite times for the two limits. The results can be applied to specific mesoscopic electronic systems.
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.
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.