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Physical Review A

Multipath interference in a multistate Landau-Zener-type model

Yu. N. Demkov, Valentin N. Ostrovsky

Published Feb 10, 2000• 69 citations
Publisher

Abstract

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 $\mathrm{ET}$ (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.

DOI
10.1103/physreva.61.032705

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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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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 \ensuremathε\ensuremath{\varepsilon}\ensuremathε and an additional state. It includes the conventional bow-tie model as a particular limiting case when \ensuremathε\ensuremath→0.\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\varepsilon}}0.\ensuremathε\ensuremath→​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 \ensuremathε.\ensuremath{\varepsilon}.\ensuremathε. 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 \ensuremathε.\ensuremath{\varepsilon}.\ensuremathε. 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.
The generalized bow-tie model (Yu N Demkov and V N Ostrovsky 2000 Phys. Rev. A 61 32705) is a particular generalization of the famous two-state Landau-Zener model widely used in atomic physics and beyond. It comprises an arbitrary number of states; the diabatic-potential curves are linear functions of time whereas the coupling matrix elements are constant. We derive a rigorous solution of the model by the contour integral method. The complete set of transition amplitudes is obtained by considering solution asymptotes for t →± ∞ . It agrees with the transition probabilities evaluated earlier by heuristically appealing but non-rigorous reduction of the model to the sequence of two-state transitions. An unusual quasi-factorization property of the transition amplitude matrix is established. The entire matrix is expressed via a single complex-valued vector.
The non-stationary Schrödinger equation in a finite basis of states is considered for the Hamiltonian matrix linearly depending on time. Exact analytical solutions of asymptotic transition probabilities are obtained for a bow-tie model, in which an arbitrary number of linear time-dependent diabatic potential curves cross at one point and only a particular horizontal curve has interactions with the others. Based on the contour integral method used, some mathematical aspects such as a possible generalization of the Whittaker functions are also briefly discussed.
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.
For a Landau-Zener transition in a two-level system, the probability for a particle, initially in the first level, to survive the transition and to remain in the first level, does not depend on whether or not the second level is broadened [V. M. Akulin and W. P. Schleich, Phys. Rev. A 46, 4110 (1992)]. In other words, the seminal Landau-Zener result applies regardless of the broadening of the second level. The same question for the multistate Landau-Zener transition is addressed in the present paper. While for integrable multistate models, where the transition does not involve interference of the virtual paths, it can be argued that the independence of the broadening persists, we focus on nonintegrable models involving interference. For a simplest four-state model, which allows an analytical treatment, we demonstrate that the decay of the excited states affects the survival probability provided that the widths of the final states are different.
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.
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.