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Journal of Physics B Atomic Molecular and Optical Physics

Reply to comment on ‘Exact results for survival probability in the multistate Landau–Zener model’

M. V. Volkov, Valentin N. Ostrovsky

Published Feb 15, 2006• 8 citations
Publisher

Abstract

After using a corrected change of variables, Dobrescu and Sinitsyn (2006 J. Phys. B: At. Mol. Opt. Phys. 39 1253) subsequently circumvent the direct calculation of the perturbation theory expansion by comparing the problem to the bow-tie model that was solved exactly previously (Ostrovsky and Nakamura 1997 J. Phys. B: At. Mol. Opt. Phys. 30 6939). We argue that the direct calculations are feasible and interesting: for instance, they are able to enlighten the specific structure of the survival probability in the bow-tie model, as illustrated by some examples.

DOI
10.1088/0953-4075/39/5/n02

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

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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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After using a corrected change of variables, Dobrescu and Sinitsyn (2006 J. Phys. B: At. Mol. Opt. Phys. 39 1253) subsequently circumvent the direct calculation of the perturbation theory expansion by comparing the problem to the bow-tie model that was solved exactly previously (Ostrovsky and Nakamura 1997 J. Phys. B: At. Mol. Opt. Phys. 30 6939). We argue that the direct calculations are feasible and interesting: for instance, they are able to enlighten the specific structure of the survival probability in the bow-tie model, as illustrated by some examples.
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.
The bow-tie model is a particular generalization of the famous two-state Landau-Zener model to an arbitrary number of states. Recently, solution of the bow-tie model was found by the contour integral method [V. N. Ostrovsky and H. Nakamura, J. Phys. A 30, 6939 (1997)]. However, its physical interpretation remained unclear, because the model implies simultaneous strong interaction of all the states apparently irreducible to any simpler pattern. We introduce here a generalized bow-tie model that contains an additional parameter $\ensuremath{\varepsilon}$ and an additional state. It includes the conventional bow-tie model as a particular limiting case when $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\varepsilon}}0.$ In the generalized model all the state-to-state transitions are reduced to the sequence of pairwise transitions and the two-state Landau-Zener model is applied to each of them. Such a reduction is well justified at least in the opposite limit of large parameter $\ensuremath{\varepsilon}.$ Several paths connect initial and final states; the contributions of different paths are summed up coherently to obtain the overall transition probability. The special ET\mathrm{ET}ET (energy and time) symmetry intrinsic for the generalized bow-tie model results in a particular property of interference phases: only purely constructive or purely destructive interference is operative. The complete set of transition probabilities is obtained in a closed form. Importantly, the results do not depend on the parameter $\ensuremath{\varepsilon}.$ In the limit of conventional bow-tie model all previously derived results are reproduced. This amounts to rationalization of the bow-tie model by its interpretation in terms of multipath successive two-state transitions.
Abstract The concept of quantum integrability has been introduced recently for quantum systems with explicitly time-dependent Hamiltonians (Sinitsyn et al 2018 Phys. Rev. Lett . 120 190402). Within the multistate Landau–Zener (MLZ) theory, however, there has been a successful alternative approach to identify and solve complex time-dependent models (Sinitsyn and Chernyak 2017 J. Phys. A: Math. Theor . 50 255203). Here we compare both methods by applying them to a new class of exactly solvable MLZ models. This class contains systems with an arbitrary number of interacting states and shows quick growth with N number of exact adiabatic energy crossing points, which appear at different moments of time. At each N , transition probabilities in these systems can be found analytically and exactly but complexity and variety of solutions in this class also grow with N quickly. We illustrate how common features of solvable MLZ systems appear from quantum integrability and develop an approach to further classification of solvable MLZ problems.