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Physics-Uspekhi

The Majorana formula and the Landau–Zener–Stückelberg treatment of the avoided crossing problem

Francesco Di Giacomo, Evgeniĭ E. Nikitin

Published May 31, 2005• 45 citations
Publisher

Abstract

The Majorana formula for the probability of spin-flipping in a time-dependent magnetic field which almost vanishes at a certain moment is discussed in connection with the celebrated work by Landau, Zener, and Stuckelberg for nonadiabatic dynamics in the avoided crossing situation.
DOI
10.1070/pu2005v048n05abeh002804

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The approach by Ettore Majorana for non-adiabatic transitions between two quasi-crossing levels is revisited and significantly extended. We rederive the transition probability, known as the Landau-Zener-Stückelberg-Majorana formula, and introduce Majorana's approach to modern readers. This result, typically referred as the Landau-Zener formula, was published by Majorana before Landau, Zener and Stückelberg. Moreover, we go well beyond previous results and we now obtain the full wave function, including its phase, which is important nowadays for quantum control and quantum information. The asymptotic wave function correctly describes the dynamics away from the avoided-level crossing, while it has limited accuracy in that region.
In the standard Landau-Zener-St\"uckelberg-Majorana (LZSM) problem, the bias sweep rate and gap are both time independent and fully characterize the LZSM problem. We consider the nonlinear LZSM problem, in which at least one of the two characteristic parameters varies as the system traverses the avoided-crossing region. This situation results in what could be thought of as a more accurate description of any realistic situation as compared with the idealized linear LZSM problem. We consider both the case of perturbative nonlinearities, where the nonlinearity adds small corrections to the linear problem, and the case of essential nonlinearities, where the sweep and/or minimum-gap functions are qualitatively different from those of the linear LZSM problem. In the case of perturbative nonlinearities, we derive analytic expressions for the LZSM transition probability based on the Dykhne-Davis-Pechukas (DDP) formula, taking into account the leading corrections to the standard LZSM formula. We compare the derived approximate expressions with numerical simulation results and comment on the validity of the approximations. In particular, if the nonlinear term is small in comparison to the linear term throughout the finite duration of the avoided crossing traversal, the perturbative approximation is valid. Our results also provide information about the validity of the DDP formula. In addition to reviewing cases of essential nonlinearity treated previously in the literature, we analyze the case of an essentially nonlinear sweep function that describes an almost square pulse.
We study the stability of counterdiabatic drive by computing the joint dependence of the probability of nonadiabatic transitions ${P}_{\ensuremath{\eta}}(\ensuremath{\delta})$ on the adiabatic parameter $\ensuremath{\delta}$ and on the normalized amplitude of the adiabatic gauge potential (AGP) $\ensuremath{\eta}$ in the Landau-Zener-St\"uckelberg-Majorana model. We show that the Dykhne-Davis-Pechukas formula cannot be readily applied since the AGP introduces a singularity in the Hamiltonian which makes the wave function multivalued in the complex-time plane. This can be understood as the non-Abelian Aharonov-Bohm phase introduced by the AGP and leads to the counterdiabatic correction of the Landau-Zener formula. In particular, it shows that, unlike the nonperturbative suppression of transitions in the adiabatic limit $\ensuremath{\delta}\ensuremath{\rightarrow}0$, the probability is only perturbatively suppressed in the counterdiabatic limit $\ensuremath{\eta}\ensuremath{\rightarrow}1$. We then consider the extension of our results to integrable time-dependent quantum Hamiltonians. We prove that the AGP satisfies the flatness constraint which characterizes integrability in these models, which allows us to derive simple expressions for the AGP and the probability of transitions ${P}_{\ensuremath{\eta}}(\ensuremath{\delta})$ near adiabatic or counterdiabatic evolution in three- and four-state integrable examples.
Since the pioneering works by Landau, Zener, Stückelberg, and Majorana (LZSM), it has been known that driving a quantum two-level system results in tunneling between its states. Even though the interference between these transitions is known to be important, it is only recently that it became both accessible, controllable, and useful for manipulating a growing number of quantum systems. Here, we systematically study various aspects of LZSM physics and review the relevant literature, significantly expanding the review article in Ref. [504].
Model calculations are presented for nonadiabatic passage of an oriented spin through a reversing solenoidal magnetic field; these are useful for comparison with experiments described elsewhere. The model corresponds closely to the well-known Landau-Zener-Stueckelberg model of a two-state curve crossing in molecular collision theory; various modifications appropriate to the Majorana problem are considered. The resulting transition amplitudes are discussed for the F=1F=1F=1 hyperfine component of the 22S122^{2}S_{\frac{1}{2}}22S21​​ metastable state of atomic hydrogen as an example.