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Physical Review A

Landau-Zener model: Effects of finite coupling duration

Nikolay V. Vitanov, B. M. Garraway

Published Jun 1, 1996• 256 citations
Publisher

Abstract

We present the generalization of the Landau-Zener model for a constant coupling of a finite duration. The exact evolution matrix is expressed in terms of sums of by-products of parabolic cylinder functions estimated at the turn-on time and at the turn-off time of the coupling. Various approximations in terms of simpler functions are derived and applied to several physically distinct cases. They allow us to study the dependence of the transition probability on the interaction parameters: coupling strength, coupling duration, and detuning slope. Furthermore, the analytic approximations reveal the effects of the finite coupling duration as well as those caused by adding a constant detuning shift, absence of a level crossing, turn-on time or turn-off time near the crossing (``half crossing''), turn-on time and turn-off time near the crossing (``nonsubstantial crossing''). The results are used to obtain analytic approximations to the time evolution in the original Landau-Zener model. Furthermore, following related studies on other models, we define the Landau-Zener class of models that, along with the finite Landau-Zener model presented in this work, contains an infinite number of members that give the same transition probability. Comparison of this class to the Allen-Eberly class shows that the two classes contain members with the same coupling but different detuning chirps as well as members with the same chirp but different couplings. The former case reveals chirp effects while the latter demonstrates shape effects. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physreva.53.4288

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We present the generalization of the Landau-Zener model for a constant coupling of a finite duration. The exact evolution matrix is expressed in terms of sums of by-products of parabolic cylinder functions estimated at the turn-on time and at the turn-off time of the coupling. Various approximations in terms of simpler functions are derived and applied to several physically distinct cases. They allow us to study the dependence of the transition probability on the interaction parameters: coupling strength, coupling duration, and detuning slope. Furthermore, the analytic approximations reveal the effects of the finite coupling duration as well as those caused by adding a constant detuning shift, absence of a level crossing, turn-on time or turn-off time near the crossing (half crossing''), turn-on time and turn-off time near the crossing (nonsubstantial crossing''). The results are used to obtain analytic approximations to the time evolution in the original Landau-Zener model. Furthermore, following related studies on other models, we define the Landau-Zener class of models that, along with the finite Landau-Zener model presented in this work, contains an infinite number of members that give the same transition probability. Comparison of this class to the Allen-Eberly class shows that the two classes contain members with the same coupling but different detuning chirps as well as members with the same chirp but different couplings. The former case reveals chirp effects while the latter demonstrates shape effects. \textcopyright{} 1996 The American Physical Society.
The exact analytical solution of the degenerate Landau-Zener model, wherein two bands of degenerate energies cross in time, is presented. The solution is derived by using the Morris-Shore transformation, which reduces the fully coupled system to a set of independent nondegenerate two-state systems and a set of decoupled states. Due to the divergence of the phase of the off-diagonal element of the propagator in the original Landau-Zener model, not all transition probabilities exist for infinite time duration. In general, apart from some special cases, only the transition probabilities between states within the same degenerate set exist, but not between states of different sets. An illustration is presented for the transition between the magnetic sublevels of two atomic levels with total angular momenta J=2J=2J=2 and 1.
We present an asymptotic approach towards the standard Landau-Zener problem based on two linearly independent elementary waves of constant amplitude but time-dependent phase. The two contributions to this phase are quadratic and logarithmic in time and result from the linear chirp of the energies and the lowest order correction in the coupling between the two levels in the long-time limit. Indeed, our solutions subjected to initial conditions at a large but finite time in the past, are valid for large negative and large positive times. Due to their asymptotic nature they are not valid in the neighborhood of the moment when the levels cross. However, as the starting point of the dynamics moves further into the past, the time interval of the break-down of our asymptotic solutions shrinks and vanishes in the limit of the infinite past which corresponds to the standard Landau-Zener situation. Our approach explains not only every feature of the exact solution but yields deeper insights into the origin of the effects. In particular, it (i) brings to light the subtleties involved in the asymptotic limit leading to the standard expressions for the Landau-Zener transition amplitudes, (ii) identifies the logarithmic phase as the origin of the exponential transition probability amplitude, and (iii) reveals the structure of the lowest order corrections to the Landau-Zener result when the starting point is not in the infinite past.
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.
We formulate a perturbative approach for studying a class of multilevel time-dependent quantum systems with constant off-diagonal couplings and diabatic energies being odd functions of time. Applying this approach to a general multistate Landau-Zener (MLZ) model with all diabatic levels crossing at one point (named the one-crossing MLZ model), we derive analytical formulas of all its transition probabilities up to fourth order in the couplings. For one-crossing MLZ models it is difficult to obtain such analytical results by other kinds of approximation methods; thus, these perturbative results can serve as reliable benchmarks for future studies of any one-crossing MLZ models that have not been exactly solved.
We reexamine the accuracy of the Landau–Zener approximation, whose analytic simplicity has made it often cited and applied over the past 70years in describing nonadiabatic transitions. We find that this approximation may be in substantial disagreement with exact numerical results for a model physical system and a wide range of coupling strengths and relative velocities. We conclude that the Landau–Zener approximation should be used with caution for quantitative estimates of nonadiabatic transition probabilities or cross sections.
This paper presents analytic formulas for various transition times in the Landau-Zener model. Considerable differences are found between the transition times in the diabatic and adiabatic bases, and between the jump time (the time for which the transition probability rises to the region of its asymptotic value) and the relaxation time (the characteristic damping time of the oscillations which appear in the transition probability after the crossing). These transition times have been calculated by using the exact values of the transition probabilities and their derivatives at the crossing point and approximations to the time evolutions of the transition probabilities in the diabatic basis, derived earlier [N. V. Vitanov and B. M. Garraway, Phys. Rev. A 53, 4288 (1996)], and similar results in the adiabatic basis, derived in the present paper.