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

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

M. V. Volkov, Valentin N. Ostrovsky

Published Mar 16, 2005• 40 citations
Publisher

Abstract

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.

DOI
10.1088/0953-4075/38/7/011

Related papers

Counterintuitive transitions in the multistate Landau–Zener problem with linear level crossings

Nikolai A. Sinitsyn

We generalize the Brundobler–Elser hypothesis in the multistate Landau–Zener problem to the case when instead of a state with the highest slope of the diabatic energy level there is a band of states with an arbitrary number of parallel levels having the same slope. We argue that the probabilities of counterintuitive transitions among such states are exactly zero.

arXiv:quant-ph/0403113•Journal of Physics A Mathematical and General•Oct 21, 2004•48 citations

Analytical results for state-to-state transition probabilities in the multistate Landau-Zener model by nonstationary perturbation theory

M. V. Volkov, Valentin N. Ostrovsky

Multistate generalizations of Landau-Zener model are studied by summing entire series of perturbation theory. A technique for analysis of the series is developed. Analytical expressions for probabilities of survival at the diabatic potential curves with extreme slope are proved. Degenerate situations are considered when there are several potential curves with extreme slope. Expressions for some state-to-state transition probabilities are derived in degenerate cases.

arXiv:cond-mat/0611046•Physical Review A•Feb 20, 2007•17 citations

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

M. V. Volkov, Valentin N. Ostrovsky

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.

Journal of Physics B Atomic Molecular and Optical Physics•Oct 13, 2004•58 citations

Nonstationary multistate Coulomb and multistate exponential models for nonadiabatic transitions

Valentin N. Ostrovsky

The nonstationary Schr\"odinger equation is considered in a finite basis of states. The model Hamiltonian matrix corresponds to a single diabatic potential curve with a Coulombic $\ensuremath{\sim}1/t$ time dependence. An arbitrary number of other diabatic potential curves are flat, i.e., time independent and have arbitrary energies. Related states are coupled by constant interactions with the Coulomb state. The resulting nonstationary Schr\"odinger equation is solved by the method of contour integral. Probabilities of transitions to any other state are obtained as $\stackrel{\ensuremath{\rightarrow}}{t}\ensuremath{\infty}$ in a simple analytical form for the case when the Coulomb state is populated initially (at instant of time $\stackrel{\ensuremath{\rightarrow}}{t}+0).$ The formulas apply both to the cases when a horizontal diabatic potential curve is crossed by the Coulomb one and to a noncrossing situation. In the limit of weak coupling, the transition probabilities are interpreted in terms of a sequence of pairwise Landau-Zener-type transitions. Mapping of the Coulomb model onto an exactly solvable exponential multistate model is established. For the special two-state case, the well-known Nikitin model is recovered.

Physical Review A•Jul 15, 2003•16 citations

The exact solution of the multistate Landau-Zener type model: the generalized bow-tie model

Yu. N. Demkov, Valentin N. Ostrovsky

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.

Journal of Physics B Atomic Molecular and Optical Physics•Jun 7, 2001•67 citations

Perturbative approach to time-dependent quantum systems and applications to one-crossing multistate Landau-Zener models

Rongyu Hu, Chen Sun

We formulate a perturbative approach for studying a class of multilevel time-dependent quantum systems with constant off-diagonal couplings and diabatic energies being odd functions of time. Applying this approach to a general multistate Landau-Zener (MLZ) model with all diabatic levels crossing at one point (named the one-crossing MLZ model), we derive analytical formulas of all its transition probabilities up to fourth order in the couplings. For one-crossing MLZ models it is difficult to obtain such analytical results by other kinds of approximation methods; thus, these perturbative results can serve as reliable benchmarks for future studies of any one-crossing MLZ models that have not been exactly solved.

arXiv:2407.06832•Physical Review A•Nov 21, 2024•0 citations
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.
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.
The nonstationary Schr\"odinger equation is considered in a finite basis of states. The model Hamiltonian matrix corresponds to a single diabatic potential curve with a Coulombic $\ensuremath{\sim}1/t$ time dependence. An arbitrary number of other diabatic potential curves are flat, i.e., time independent and have arbitrary energies. Related states are coupled by constant interactions with the Coulomb state. The resulting nonstationary Schr\"odinger equation is solved by the method of contour integral. Probabilities of transitions to any other state are obtained as $\stackrel{\ensuremath{\rightarrow}}{t}\ensuremath{\infty}$ in a simple analytical form for the case when the Coulomb state is populated initially (at instant of time $\stackrel{\ensuremath{\rightarrow}}{t}+0).$ The formulas apply both to the cases when a horizontal diabatic potential curve is crossed by the Coulomb one and to a noncrossing situation. In the limit of weak coupling, the transition probabilities are interpreted in terms of a sequence of pairwise Landau-Zener-type transitions. Mapping of the Coulomb model onto an exactly solvable exponential multistate model is established. For the special two-state case, the well-known Nikitin model is recovered.
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.
We formulate a perturbative approach for studying a class of multilevel time-dependent quantum systems with constant off-diagonal couplings and diabatic energies being odd functions of time. Applying this approach to a general multistate Landau-Zener (MLZ) model with all diabatic levels crossing at one point (named the one-crossing MLZ model), we derive analytical formulas of all its transition probabilities up to fourth order in the couplings. For one-crossing MLZ models it is difficult to obtain such analytical results by other kinds of approximation methods; thus, these perturbative results can serve as reliable benchmarks for future studies of any one-crossing MLZ models that have not been exactly solved.