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Physical Review A

Population transfer in a level-crossing model with two time scales

Kalle‐Antti Suominen, B. M. Garraway

Published Jan 1, 1992• 86 citations
Publisher

Abstract

We set up a level-crossing model with two time scales and compare the numerically calculated probability transfer with several analytic results. We also examine the probability transfer in the adiabatic limit and make a comparison with a calculation based on the approach of Dykhne [Sov. Phys.---JETP 11, 411 (1960); 14, 941 (1962)] and of Davis and Pechukas [J. Chem. Phys. 64, 3129 (1976)]. We find that we can get an improved agreement if we modify their approach.

DOI
10.1103/physreva.45.374

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We set up a level-crossing model with two time scales and compare the numerically calculated probability transfer with several analytic results. We also examine the probability transfer in the adiabatic limit and make a comparison with a calculation based on the approach of Dykhne [Sov. Phys.---JETP 11, 411 (1960); 14, 941 (1962)] and of Davis and Pechukas [J. Chem. Phys. 64, 3129 (1976)]. We find that we can get an improved agreement if we modify their approach.
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 examine the adiabatic limit of population transfer in two-level models driven by a chirped laser field. We show that the nonadiabatic correction is minimized when the adiabatic eigenenergies associated to the dynamics are parallel. In the diagram of the difference of the eigenenergy surfaces as a function of the parameters, this corresponds to an adiabatic passage along a level line. The analytical arguments are based on the Dykhne-Davis-Pechukas treatment. We illustrate this behavior with various examples.
We examine the effect of nonlinearity at a level crossing on the probability for nonadiabatic transitions P. By using the Dykhne-Davis-Pechukas formula, we derive simple analytic estimates for P for two types of nonlinear crossings. In the first type, the nonlinearity in the detuning appears as a perturbative correction to the dominant linear time dependence. Then appreciable deviations from the Landau-Zener probability PLZ{P}_{\mathrm{LZ}}PLZ​ are found to appear for large couplings only, when P is small; this explains why the Landau-Zener model is often seen to provide more accurate results than expected. In the second type of nonlinearity, called essential nonlinearity, the detuning is proportional to an odd power of time. Then the nonadiabatic probability P is qualitatively and quantitatively different from PLZ{P}_{\mathrm{LZ}}PLZ​ because, on the one hand, it vanishes in an oscillatory manner as the coupling increases, and, on the other, it is much larger than PLZ.{P}_{\mathrm{LZ}}.PLZ​. We suggest experimental situations where the predicted deviations can be observed.
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.