We study the applicability of the Zhu-Nakamura theory to a class of time-dependent quantum mechanical level-crossing models called superparabolic level-glancing models. The phenomenon of a level glancing, being on the borderline between a proper crossing of energy levels and an avoided crossing, is also an important special case between the two different approximative expressions in the Zhu-Nakamura theory. It is seen that the application of the theory to these models is not straightforward. We discuss some possible causes of these difficulties and also compare the approximative formulas of Zhu-Nakamura theory to those obtained by the generalization of the Dykhne-Davis-Pechukas theory.
We study the applicability of the Zhu-Nakamura theory to a class of time-dependent quantum mechanical level-crossing models called superparabolic level-glancing models. The phenomenon of a level glancing, being on the borderline between a proper crossing of energy levels and an avoided crossing, is also an important special case between the two different approximative expressions in the Zhu-Nakamura theory. It is seen that the application of the theory to these models is not straightforward. We discuss some possible causes of these difficulties and also compare the approximative formulas of Zhu-Nakamura theory to those obtained by the generalization of the Dykhne-Davis-Pechukas theory.
Level crossing models for two-state quantum systems are applicable to a wide variety of physical problems. We address the special case of level glancing, i.e., when energy levels reach a degeneracy at a specific point of time, but never actually cross. The simplest model with such behavior is the parabolic model, and its generalizations, which we call superparabolic models. We discuss their basic characteristics, complementing the previous work on the related nonlinear crossing models [Vitanov and Suominen, Phys. Rev. A 59, 4580 (1999)].
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.
Level crossing models for two-state quantum systems are applicable to a wide variety of physical problems. We address the special case of level glancing, i.e., when energy levels reach a degeneracy at a specific point of time, but never actually cross. The simplest model with such behavior is the parabolic model, and its generalizations, which we call superparabolic models. We discuss their basic characteristics, complementing the previous work on the related nonlinear crossing models [Vitanov and Suominen, Phys. Rev. A 59, 4580 (1999)].
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 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 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. We suggest experimental situations where the predicted deviations can be observed.
We study the dynamics of a two-level crossing model with a cubic modification of the linear Landau-Zener tunneling, and express the non-adiabatic transition amplitudes in terms of the bi-confluent Heun functions which are generalizations of the confluent hypergeometric functions. We find a closed-form series expression of the transition probability at the long time limit, and derive tractable approximate formulas to the state populations by use of simple functions in a large part of the parameter space. The analytical approximations are validated by comparison with numerical results.
Abstract We study the dynamics of a two-level crossing model with a parabolic separation of the diabatic energies. The solutions are expressed in terms of the tri-confluent Heun equations—the generalization of the confluent hypergeometric equations. We obtain analytical approximations for the state populations in terms of Airy and Bessel functions. Applicable expressions are derived for a large part of the parameter space. We also provide simple formulas which connect local solution in different time regimes. The validity of the analytical approximations is shown by comparing them to numerical simulations.