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Physical Review A

Interferences in adiabatic transition probabilities mediated by Stokes lines

Alain Joye, G. Mileti, Ch.-Ed. Pfister

Published Oct 1, 1991• 44 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.44.4280

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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.
A two-state quantum system whose Hamiltonian varies adiabatically has a weak probability of making transitions between its instantaneous energy eigenstates. If the system is started out in an eigenstate and its evolution tracked using suitably-defined 'superadiabatic' states, the transition probability exhibits the error-function growth characteristic of the birth of subdominant exponentials across Stokes lines. Here we study model Hamiltonians associated with pairs of Stokes lines in the complex time-plane. The overlapping of Stokes crossing phenomena influences the final transition probability and also leads to interference effects in the transition history. These effects can be adjusted by turning parameters in the models.
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.
In a quantum system with a smoothly and slowly varying Hamiltonian, which approaches a constant operator at times $\stackrel{\ensuremath{\rightarrow}}{t}\ifmmode\pm\else\textpm\fi{}\ensuremath{\infty},$ the transition probabilities between adiabatic states are exponentially small. They are characterized by an exponent that depends on a phase integral along a path around a set of branch points connecting the energy-level surfaces in complex time. Only certain sequences of branch points contribute. We propose that these sequences are determined by a topological rule involving the Stokes lines attached to the branch points. Our hypothesis is supported by theoretical arguments and results of numerical experiments.
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.