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Journal of Physics B Atomic Molecular and Optical Physics

Exact results for survival probability in the multistate Landau–Zener model

M. V. Volkov, Valentin N. Ostrovsky

Published Oct 13, 2004• 58 citations
Publisher

Abstract

An exact formula is derived for survival probability in the multistate Landau–Zener model in the special case where the initially populated state corresponds to the extremal (maximum or minimum) slope of a linear diabatic potential curve. The formula was originally guessed by S Brundobler and V Elzer ( 1993 J. Phys. A: Math. Gen. 26 1211 ) based on numerical calculations. It is a simple generalization of the expression for the probability of diabatic passage in the famous two-state Landau–Zener model. Our result is obtained via analysis and summation of the entire perturbation theory series.

DOI
10.1088/0953-4075/37/20/003

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Comment on ‘Exact results for survival probability in the multistate Landau–Zener model’

B E Dobrescu, N A Sinitsyn

We correct the proof of the Brundobler–Elser formula (BEF) provided in Volkov and Ostrovsky ( 2004 J. Phys. B: At. Mol. Opt. Phys. 37 4069 ) and continued in the appendix of Volkov and Ostrovsky ( 2005 J. Phys. B: At. Mol. Opt. Phys. 38 907 ). After showing that some changes of variables employed in these articles are used erroneously, we propose an alternative change of variables which solves the problem. In our proof, we reveal the connection between the BEF for a general N -level Landau–Zener system and the exactly solvable bow-tie model. The special importance of the diabatic levels with maximum/minimum slope is emphasized throughout.

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An exact formula is derived for survival probability in the multistate Landau–Zener model in the special case where the initially populated state corresponds to the extremal (maximum or minimum) slope of a linear diabatic potential curve. The formula was originally guessed by S Brundobler and V Elzer ( 1993 J. Phys. A: Math. Gen. 26 1211 ) based on numerical calculations. It is a simple generalization of the expression for the probability of diabatic passage in the famous two-state Landau–Zener model. Our result is obtained via analysis and summation of the entire perturbation theory series.
We correct the proof of the Brundobler–Elser formula (BEF) provided in Volkov and Ostrovsky ( 2004 J. Phys. B: At. Mol. Opt. Phys. 37 4069 ) and continued in the appendix of Volkov and Ostrovsky ( 2005 J. Phys. B: At. Mol. Opt. Phys. 38 907 ). After showing that some changes of variables employed in these articles are used erroneously, we propose an alternative change of variables which solves the problem. In our proof, we reveal the connection between the BEF for a general N -level Landau–Zener system and the exactly solvable bow-tie model. The special importance of the diabatic levels with maximum/minimum slope is emphasized throughout.
In the multistate Landau–Zener model all diabatic potential curves are linear functions of time. We consider the case where there is a band of parallel potential curves with slope larger (smaller) than any of the other slopes in the system. In such a situation transitions from a lower (higher) lying state within the band to any upper (lower) state are counterintuitive, since in the simple semiclassical picture they are possible only via propagation backwards in time. We rigorously prove that the probabilities of such transitions are exactly zero. In other words the energy of states within the band can only decrease (increase). The theoretical method employed is based on analysis of perturbation theory series to arbitrary order.
We identify a nontrivial multistate Landau–Zener (LZ) model for which transition probabilities between any pair of diabatic states can be determined analytically and exactly. In the semiclassical picture, this model features the possibility of interference of different trajectories that connect the same initial and final states. Hence, transition probabilities are generally not described by the incoherent successive application of the LZ formula. We discuss reasons for integrability of this system and provide numerical tests of the suggested expression for the transition probability matrix.
Abstract The Landau–Zener formula describes the diabatic transition probability of a two-level system under linear driving. Its rigorous derivation typically relies on sophisticated mathematical tools, such as special functions, Laplace transforms, or contour integrals. In this work, we present a derivation of the Landau–Zener transition probability using a fundamentally different approach via functional equations. By leveraging integrability, we prove that this transition probability satisfies a functional equation, whose solutions establish the exponential form of the formula. The coefficient in the exponent is then determined through a lowest-order perturbation calculation. This derivation is rigorous and does not involve any sophisticated mathematics. Our work provides insights into the origin of the exponential form of the Landau–Zener transition probability, and shows that the Landau–Zener formula can be viewed as a consequence of integrability, though the two-level Landau–Zener Hamiltonian itself does not satisfy the integrability conditions.