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Physical Review A

New results for transition probabilities in two-level systems: The large-detuning regime

Edward J. Robinson, P. R. Berman

Published Feb 1, 1983• 19 citations

Abstract

The problem of calculating transition probabilities in two-level systems is studied in the limit where the detuning is large compared to the inverse duration of the interaction. Coupling potentials whose Fourier transforms $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{V}(\ensuremath{\omega})$ are of the form $f(\ensuremath{\omega}){e}^{\ensuremath{-}(|b\ensuremath{\omega}|)}$ for large frequencies give rise to solutions which may be classified into families according to the form of $f(\ensuremath{\omega})$. Within each family transition probabilities may be calculated from formulas that differ only in the numerical value of a scaling parameter. In cases where the coupling function has a pole in the complex time plane, the families are identified with the order of this singularity. In particular, for poles of first order, a connection with the Rosen-Zener solution can be made. The analysis is performed via high-order perturbation expansions which are shown to always converge for two-level systems driven by coupling potentials of finite pulse area.

DOI: 10.1103/physreva.27.1022

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The problem of calculating transition probabilities in two-level systems is studied in the limit where the detuning is large compared to the inverse duration of the interaction. Coupling potentials whose Fourier transforms

\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{V}(\ensuremath{\omega})

are of the form

f(\ensuremath{\omega}){e}^{\ensuremath{-}(|b\ensuremath{\omega}|)}

for large frequencies give rise to solutions which may be classified into families according to the form of

f(\ensuremath{\omega})

. Within each family transition probabilities may be calculated from formulas that differ only in the numerical value of a scaling parameter. In cases where the coupling function has a pole in the complex time plane, the families are identified with the order of this singularity. In particular, for poles of first order, a connection with the Rosen-Zener solution can be made. The analysis is performed via high-order perturbation expansions which are shown to always converge for two-level systems driven by coupling potentials of finite pulse area.

The problem of the evolution of a two-level quantum system in a radiation field of Gaussian shape with detuning is addressed by seeking an approximate expression for the transformation function governing the interaction. The transformation operator requires the evaluation of two independent matrix elements: the former is obtained by the Rosen-Zener conjecture; the latter is based on an expansion using the main parameters involved in the first one. The expressions obtained, which are supposed to hold in the case of small detunings, have been tested by comparison with a numerical integration. Then the solutions are modified to take into account decay phenomena to other levels. As a final example, the expressions dealing with the transition probability in the case of two separated oscillatory fields are given.

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}

. We calculate the approximate transition probability at finite times for the two limits. The results can be applied to specific mesoscopic electronic systems.

The time-dependent behavior of a two-level system interacting with a quantum oscillator system is analyzed in the case of a coupling larger than both the energy separation between the two levels and the energy of the quantum oscillator ($\ensuremath{\Omega}<\ensuremath{\omega}<\ensuremath{\lambda}$, where $\ensuremath{\Omega}$ is the frequency of the transition between the two levels, $\ensuremath{\omega}$ is the frequency of the oscillator, and $\ensuremath{\lambda}$ is the coupling between the two-level system and the oscillator). Our calculations show that the amplitude of the expectation value of the oscillator coordinate decreases as the two-level system undergoes the transition from one level to the other, while the transfer probability between the levels is staircaselike. This behavior is explained by the interplay between the adiabatic and nonadiabatic regimes encountered during the dynamics with the system acting as a quantum counterpart of the Landau-Zener model. The transition between the two levels occurs as long as the expectation value of the oscillator coordinate is driven close to zero. On the contrary, if the initial conditions are set such that the expectation values of the oscillator coordinate are far from zero, the system will remain locked on one level.

Three ingredients of the elementary Landau-Zener problem determine the familiar expression

a_{LZ}\equiv\exp\left[-π/(2ε)\right]

for the asymptotic value of the probability amplitude for remaining in the initial level: (i) A wave whose phase is determined by the product of a contour integral over a simple pole at the origin of the complex plane and the inverse of twice the scaled chirp parameter

ε

. (ii) An asymptotic limit of the associated path connecting the points

\pm 1

along the real axis and circumventing the pole in the upper half-plane, and (iii) a half-circle in the lower half plane enclosing together with the asymptotic path the pole. The Cauchy theorem immediately provides us with the value $\iiπ$ of the asymptotic contour, and thus with

a_{LZ}

. Our analysis demonstrates not only that

a_{LZ}

is the consequence of a logarithmic phase singularity but also explains why the Markov approximation also leads to

a_{LZ}

.