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Physical Review A

Quasiadiabatic solution to the two-level problem

R. T. Robiscoe

Published Feb 1, 1982• 6 citations
Publisher

Abstract

We consider a two-level system where a pair of discrete quantum states of arbitrary widths and separation are coupled by a general time-dependent external field with a pulselike matrix element, $V(t)$. Using a transformation suggested by Robinson [Phys. Rev. A 24, 2239 (1981)] and applying the WKB approximation, a quasiadiabatic solution is constructed for the system's transition amplitude. This solution is useful in estimating transition amplitudes in the asymptotic region for a wide variety of smoothly varying pulses $V(t)$.

DOI
10.1103/physreva.25.1178

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Two-level transition probabilities for asymmetric coupling pulses

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Physical Review A•Oct 1, 1981•30 citations
We consider a two-level system where a pair of discrete quantum states of arbitrary widths and separation are coupled by a general time-dependent external field with a pulselike matrix element, V(t)V(t)V(t). Using a transformation suggested by Robinson [Phys. Rev. A 24, 2239 (1981)] and applying the WKB approximation, a quasiadiabatic solution is constructed for the system's transition amplitude. This solution is useful in estimating transition amplitudes in the asymptotic region for a wide variety of smoothly varying pulses V(t)V(t)V(t).
We consider the time-dependent two-level problem of quantum mechanics, where the levels are coupled by a radio-frequency pulse with an arbitrary time-dependent envelope V(t)V(t)V(t). We derive an approximate solution for the system's transition amplitude $P(\ensuremath{\infty})$ which is correct to the third order of perturbation theory, and which applies to all pulses V(t)V(t)V(t) with finite first and second moments which obey the following: limt3V(t)=0\mathrm{lim} {t}^{3}V(t)=0limt3V(t)=0, as $t\ensuremath{\rightarrow}\ensuremath{\infty}$. Our form of solution for $P(\ensuremath{\infty})$ provides a criterion for judging the validity of a solution previously conjectured by Rosen and Zener, and it is generally useful for providing line-shape details in many cases of practical interest.
This thesis discusses different aspects of time-dependent two-level quantum systems and their coherent dynamics. In the introductory chapters, we present some basic models used to study them and some basic results related to the level-crossing problems generally. As these systems are only rarely analytically solvable, some emphasis is put on the existing approximative theories. In particular, the generalization of the method of Dykhne, Davis and Pechukas (DDP), originally developed for problems in near-adiabatic region, is studied further. In the DDP theory, the final populations of different states are determined by the complex zero points of the eigenenergies. Superparabolic level-crossing models are introduced in this thesis and they prove to be simple but versatile test models for these studies. It is shown that, by considering all of the zero points, one can obtain accurate approximations also for highly non-adiabatic regions. In Chapter 4, a general differential geometric framework for time-dependent level-crossing models is developed and its basic character is discussed. A natural way of associating a plane curve to a time-dependent two-level model is introduced. This association allows one to use all of the mathematical results concerning plane curves to study these models. As an example, we use the Four-vertex theorem to discuss the adiabatic limit of certain type of systems. We also consider the prospect of enhancing the population transfer by transforming the time-dependent coupling pulse into a zero-area coupling. This is done either smoothly or by an abrupt jump in the phase of the coupling. It has been shown earlier, that one can obtain a complete population inversion (CPI) in a robust way with strong, non-resonant zero-area pulses. We study this in the more general time-dependent setting in Chapter 5, where also the energy levels are driven in a time-dependent fashion. This allows to transport the CPI phenomenon from the strong coupling region towards more moderate couplings. As an example of the smooth case, we consider the SechTanh-model, and show that even the nearest-zero DDP approximation can be used, despite the highly non-adiabatic origin of the phenomenon. We develop an approximative scheme also for the phase-jump case, using the parabolic model as an example. As a limiting case, we derive an approximative formula for the transition probability, which has some universal character and which proves to be useful in the most interesting case where the robust CPI is obtained. It is shown that even the highly suppressed transitions in tunnelling cases can be strongly enhanced in the phase-jump scenario.
We solve the problem of two discrete quantum levels which are coupled by a time-dependent radio-frequency pulse $W(t)=V(t) {e}^{i\ensuremath{\nu}t}$, where the envelope function is of a form suggested by Rosen and Zener: $V(t)={V}_{0}sech(\frac{\ensuremath{\pi}t}{T})$. When a level damping constant $\ensuremath{\gamma}$ is included, in the manner of Bethe-Lamb theory, the solutions show new features which are not expected on the basis of a sudden-approximation theory, where V(t)=constV(t)=\mathrm{const}V(t)=const over the pulse duration TTT. Various transient effects such as "ringing" are not present in the extended Rosen-Zener solution; these effects are related to the large impulsive forces at the step discontinuities in the sudden approximation. The final-state level amplitudes can be quite different depending on the size of the pulse rise time TTT as compared with the system Bohr period $\frac{1}{\ensuremath{\omega}}$. Our results allow a continuous and quantitatively exact comparison between the extremes of the sudden ($\ensuremath{\omega}T\ensuremath{\ll}1$) and adiabatic ($\ensuremath{\omega}T\ensuremath{\gg}1$) approximations. A model of a "quasisudden" step function is also constructed, and remarks are made on the validity of a certain conjecture by Rosen and Zener.
The problem of a two-level atom interacting with a radiation pulse is studied in the limit that the atom-field detuning times the pulse duration is much greater than unity. Owing to the large atom-field detuning, transitions result from nonadiabatic coupling of the states by the field. The transition probability for the atom to be excited following the pulse is studied as a function of field strength for five different pulse shapes: hyperbolic secant, Lorentzian, hyperbolic secant squared, Lorentzian squared, and Gaussian. It is shown that the behavior of the transition probability differs qualitatively for these pulses. An explanation of this qualitative difference is given in terms of the Massey parameter. Numerical solutions are compared with asymptotic solutions and several anomalies are noted. In the limit of large field strength, a universal expression for the transition probability is found. An interesting feature of the solutions is that, in the limit of very large field strengths, the transition probability for a Gaussian pulse can approach unity despite the fact that the pulse has an exponentially small Fourier amplitude at the atom-field detuning. This apparent violation of the energy-time uncertainty principle is explained in terms of the nonlinear atom-field interactions.
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.
In a recent paper, Bambini and Berman [A. Bambini and P. R. Berman, Phys. Rev. A 23, 2496 (1981)] presented analytic solutions to a certain family of coherent-coupling pulses for a two-level system. They show, for nonresonant temporally asymmetric members of the class, that there are no solutions corresponding to vanishing transition probabilities. In this Comment, we examine the problem in greater generality and demonstrate that this property is the norm for asymmetric pulses, and that a vanishing transition probability is possible only if severely overdetermined conditions are satisfied.