Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of Physics B Atomic Molecular and Optical Physics

Multistate curve crossing: an exactly soluble model with degeneracy

Vladimir Yurovsky, A. Ben‐Reuven

Published Jan 14, 1998• 23 citations
Publisher

Abstract

The linear curve-crossing problem involving a slanted potential and a set of horizontal ones as an exactly soluble quantum-mechanical problem, is re-examined. Special attention is paid to the questions of degeneracy of the horizontal-potential channels. It is shown, by explicit derivation of an asymptotic expansion of the scattering amplitudes, that the degeneracy limit is a singular point. An explicit exact solution is derived for the degenerate case, resulting in a non-Landau-Zener saturated behaviour of the transition probabilities at high coupling intensities. Estimates are derived for the transition region as a function of the energy gap, and its divergence on approach to degeneracy. Relevance to the optical shielding of ultracold atom collisions is pointed out, with particular reference to the phenomenon of `counterintuitive' transitions which were discussed recently.

DOI
10.1088/0953-4075/31/1/005

Related papers

Saturated transitions in exactly soluble models of two-state curve crossing with time-dependent potentials

Vladimir Yurovsky, A. Ben‐Reuven

Two exactly soluble two-state curve crossing models, with potential matrices formed by a linear combination of time-independent terms and (I) inversely proportional $(\ensuremath{\sim}{t}^{\ensuremath{-}1})$ or (II) exponential time-dependent terms, are considered here. It is shown that the two models are related to each other by a simple transformation of the time variable. These models can be further transformed to a simpler one (the ``reduced'' model), in which the two crossing potentials are a constant (horizontal) one and an inversely proportional time-dependent one, and the interaction coupling them is time-independent. Analysis of the exact solution of the reduced model discloses that the nonadiabatic transition probability saturates at increasingly large coupling strengths without vanishing (in contradiction of the linear Landau-Zener theory and various adiabatic theories). Similar saturation can also occur in more general models, within certain ranges of the potential parameters involved.

Physical Review A•Dec 1, 1999•19 citations

Curve crossing in linear potential grids: The quasidegeneracy approximation

Vladimir Yurovsky, A. Ben‐Reuven

The quasidegeneracy approximation [V. A. Yurovsky, A. Ben-Reuven, P. S. Julienne, and Y. B. Band, J. Phys. B 32, 1845 (1999)] is used here to evaluate transition amplitudes for the problem of curve crossing in linear potential grids involving two sets of parallel potentials. The approximation describes phenomena, such as counterintuitive transitions and saturation (incomplete population transfer), not predictable by the assumption of independent crossings. Also, a new kind of oscillations due to quantum interference (different from the well-known St\"uckelberg oscillations) is disclosed, and its nature discussed. The approximation can find applications in many fields of physics, where multistate curve crossing problems occur.

arXiv:quant-ph/0008087•Physical Review A•Mar 6, 2001•24 citations

Exact analytical solution of theN-level Landau - Zener-type bow-tie model

Valentin N. Ostrovsky, Hiroki Nakamura

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.

Journal of Physics A Mathematical and General•Oct 7, 1997•84 citations

No-go theorem for bands of potential curves in multistate Landau–Zener model

M. V. Volkov, Valentin N. Ostrovsky

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.

Journal of Physics B Atomic Molecular and Optical Physics•Mar 16, 2005•40 citations

Complete solutions of the Landau-Zener-Stueckelberg curve crossing problems, and their generalizations and applications

Hiroki Nakamura

The compact analytical complete solutions recently obtained for the two-state Landau-Zener-Stueckelberg problems are reviewed and explained. The theory covers both Landau-Zener (LZ) type in which the two diabatic potential curves cross with the same sign of slopes and the nonadiabatic tunneling (NT) type in which the potentials cross with different signs of slopes. The theory is applicable virtually in the whole range of energy and coupling strength and is convenient for practical use. The new theory for time-dependent nonadiabatic transition can be formulated from the time-independent theory of the LZ-type. The utilizability of the theory to various multi-channel problems and also to multi-dimensional problems is demonstrated and explained. The intriguing phenomenon of complete reflection which appears in the NT-case is explicitly utilized to propose a new type of molecular switching and to control molecular processes such as molecular photodissociation. Also proposed is a new way of controlling molecular processes by using time-dependent external fields. Finally, a trial to formulate a unified analytical theory to cover both Landau-Zener-Stueckelberg and Rosen-Zener-Demkov types of nonadiabatic transitions is briefly touched upon.

AIP conference proceedings•Jan 1, 2000•2 citations

Two-state linear curve crossing problems revisited. IV. The best analytical formulas for scattering matrices

Chaoyuan Zhu, Hiroki Nakamura

Based on the achievements of the previous three papers of this series, the best working formulas for scattering matrix are obtained for both the Landau–Zener (LZ) and the nonadiabatic tunneling (NT) case: two formulas valid at b2≥0 and b2≤0 in the LZ case, and three formulas valid atb2≤−1, −1≤b2≤1 and b2≥1 in the NT case, where b2 represents the effective energy. Simple and compact formulas which work far better than the LZ formula are proposed for nonadiabatic transition probability by one passage of crossing point for both the LZ and NT cases. Furthermore, compact expressions are derived, for the first time, for the nonadiabatic tunneling probability at b2≤1, i.e., at energies lower than the bottom of the upper adiabatic potential. All the formulas proposed here can be usefully utilized at any coupling strength, namely the validity range has been very much expanded compared to the previous formulas by employing certain empirical corrections. Besides, these formulas have convenience to enable an extension to general curved potentials.

The Journal of Chemical Physics•Sep 15, 1994•66 citations
The linear curve-crossing problem involving a slanted potential and a set of horizontal ones as an exactly soluble quantum-mechanical problem, is re-examined. Special attention is paid to the questions of degeneracy of the horizontal-potential channels. It is shown, by explicit derivation of an asymptotic expansion of the scattering amplitudes, that the degeneracy limit is a singular point. An explicit exact solution is derived for the degenerate case, resulting in a non-Landau-Zener saturated behaviour of the transition probabilities at high coupling intensities. Estimates are derived for the transition region as a function of the energy gap, and its divergence on approach to degeneracy. Relevance to the optical shielding of ultracold atom collisions is pointed out, with particular reference to the phenomenon of `counterintuitive' transitions which were discussed recently.
Two exactly soluble two-state curve crossing models, with potential matrices formed by a linear combination of time-independent terms and (I) inversely proportional $(\ensuremath{\sim}{t}^{\ensuremath{-}1})$ or (II) exponential time-dependent terms, are considered here. It is shown that the two models are related to each other by a simple transformation of the time variable. These models can be further transformed to a simpler one (the ``reduced'' model), in which the two crossing potentials are a constant (horizontal) one and an inversely proportional time-dependent one, and the interaction coupling them is time-independent. Analysis of the exact solution of the reduced model discloses that the nonadiabatic transition probability saturates at increasingly large coupling strengths without vanishing (in contradiction of the linear Landau-Zener theory and various adiabatic theories). Similar saturation can also occur in more general models, within certain ranges of the potential parameters involved.
The quasidegeneracy approximation [V. A. Yurovsky, A. Ben-Reuven, P. S. Julienne, and Y. B. Band, J. Phys. B 32, 1845 (1999)] is used here to evaluate transition amplitudes for the problem of curve crossing in linear potential grids involving two sets of parallel potentials. The approximation describes phenomena, such as counterintuitive transitions and saturation (incomplete population transfer), not predictable by the assumption of independent crossings. Also, a new kind of oscillations due to quantum interference (different from the well-known St\"uckelberg oscillations) is disclosed, and its nature discussed. The approximation can find applications in many fields of physics, where multistate curve crossing problems occur.
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.
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.
The compact analytical complete solutions recently obtained for the two-state Landau-Zener-Stueckelberg problems are reviewed and explained. The theory covers both Landau-Zener (LZ) type in which the two diabatic potential curves cross with the same sign of slopes and the nonadiabatic tunneling (NT) type in which the potentials cross with different signs of slopes. The theory is applicable virtually in the whole range of energy and coupling strength and is convenient for practical use. The new theory for time-dependent nonadiabatic transition can be formulated from the time-independent theory of the LZ-type. The utilizability of the theory to various multi-channel problems and also to multi-dimensional problems is demonstrated and explained. The intriguing phenomenon of complete reflection which appears in the NT-case is explicitly utilized to propose a new type of molecular switching and to control molecular processes such as molecular photodissociation. Also proposed is a new way of controlling molecular processes by using time-dependent external fields. Finally, a trial to formulate a unified analytical theory to cover both Landau-Zener-Stueckelberg and Rosen-Zener-Demkov types of nonadiabatic transitions is briefly touched upon.
Based on the achievements of the previous three papers of this series, the best working formulas for scattering matrix are obtained for both the Landau–Zener (LZ) and the nonadiabatic tunneling (NT) case: two formulas valid at b2≥0 and b2≤0 in the LZ case, and three formulas valid atb2≤−1, −1≤b2≤1 and b2≥1 in the NT case, where b2 represents the effective energy. Simple and compact formulas which work far better than the LZ formula are proposed for nonadiabatic transition probability by one passage of crossing point for both the LZ and NT cases. Furthermore, compact expressions are derived, for the first time, for the nonadiabatic tunneling probability at b2≤1, i.e., at energies lower than the bottom of the upper adiabatic potential. All the formulas proposed here can be usefully utilized at any coupling strength, namely the validity range has been very much expanded compared to the previous formulas by employing certain empirical corrections. Besides, these formulas have convenience to enable an extension to general curved potentials.