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

arXiv:0912.2569 · Europhysics Letters (EPL)

Generalized formula for the Landau-Zener transition in interacting Bose-Einstein condensates

Artur Mikhailovich Ishkhanyan

Published May 1, 2010• 12 citations
arXivHTMLPDF

Abstract

We present a rigorous analysis of the generalized Landau-Zener problem for the two-level interacting Bose-Einstein condensates. We show that the dynamics of the system is accurately, in detail, described by a two-term variational ansatz that is valid for the whole time domain and is applicable for any set of involved parameters. Applying an exact third-order nonlinear differential equation we construct an advanced fifth-order polynomial equation for the final transition probability serving as a highly accurate generalized Landau-Zener formula.

DOI
10.1209/0295-5075/90/30007

Related papers

Weak coupling regime of the Landau-Zener transition for association of an atomic Bose-Einstein condensate

N. Sahakyan, H. H. Azizbekyan, H. A. Ishkhanyan +2 more

In the framework of a basic semiclassical time-dependent nonlinear two-state problem, we study the weak coupling limit of the nonlinear Landau-Zener transition at coherent photo- and magneto-association of an atomic Bose-Einstein condensate. Using an exact third-order nonlinear differential equation for the molecular state probability, we develop a variational approach which enables us to construct an accurate analytic approximation describing time dynamics of the coupled atom-molecular system for the case of weak coupling. The approximation is written in terms of the solution to an auxiliary linear Landau-Zener problem with some effective Landau-Zener parameter. The dependence of this effective parameter on the input Landau-Zener parameter is found to be unexpected: as the generic Landau-Zener parameter increases, the effective Landau-Zener parameter first monotonically increases (starting from zero), reaches its maximal value and then monotonically decreases again reaching zero at some point. The constructed approximation quantitatively well describes many characteristics of the time dynamics of the system, in particular, it provides a highly accurate formula for the final transition probability to the molecular state. The present result for the final transition probability improves the accuracy of the previous approximation by Ishkhanyan et al. [Phys. Rev. A 69 , 043612 (2004); J. Phys. A 38 , 3505 (2005)] by order of magnitude.

arXiv:0909.0834•Laser Physics•Dec 7, 2009•10 citations

Towards a generalized Landau-Zener formula for an interacting Bose-Einstein condensate in a two-level system

Dirk Witthaut, Eva-Maria Graefe, H. J. Korsch

We consider the Landau-Zener problem for a Bose-Einstein condensate in a linearly varying two-level system, for the full many-particle system as well as in the mean-field approximation. Novel nonlinear eigenstates emerge in the mean-field description, which leads to a breakdown of adiabaticity: The Landau-Zener transition probability does not vanish even in the adiabatic limit. It is shown that the emergence of nonlinear eigenstates and thus the breakdown of adiabaticity corresponds to quasi-degenerate avoided crossings of the many-particle levels. The many-particle problem can be solved approximately within an independent crossings approximation, which yields an explicit generalized Landau-Zener formula. A comparison to numerical results for the many-particle system and the mean-field approximation shows an excellent agreement.

arXiv:quant-ph/0602163•Physical Review A•Jun 7, 2006•86 citations

Landau-Zener transition in quadratic nonlinear two-state systems

Artur Mikhailovich Ishkhanyan

A comprehensive theory of the Landau-Zener transition in quadratic nonlinear two-state systems is developed. A compact analytic formula involving elementary functions only is derived for the final transition probability. The formula provides a highly accurate approximation for the whole rage of the variation of the Landau-Zener parameter.

arXiv:0912.4763•Physical Review A•May 5, 2010•7 citations

Variational ansatz for the nonlinear Landau–Zener problem for cold atom association

Artur Mikhailovich Ishkhanyan, B Joulakian, Kalle‐Antti Suominen

We present a rigorous analysis of the Landau–Zener linear-in-time term crossing problem for quadratic-nonlinear systems relevant to the coherent association of ultracold atoms in degenerate quantum gases. Our treatment is based on an exact third-order nonlinear differential equation for the molecular state probability. Applying a variational two-term ansatz, we construct a simple approximation that accurately describes the whole-time dynamics of the coupled atom–molecular system for any set of involved parameters. Ensuring an absolute error of less than 10 −5 for the final transition probability, the resultant solution improves by several orders of magnitude the accuracy of the previous approximations by A Ishkhanyan et al developed separately for the weak coupling ( 2005 J. Phys. A: Math. Gen. 38 3505 ) and strong interaction ( 2006 J. Phys. A: Math. Gen. 39 14887 ) limits. In addition, the constructed approximation covers the whole moderate-coupling regime, providing this intermediate regime with the same accuracy as the two mentioned limits. The obtained results reveal the remarkable observation, that for the strong-coupling limit the resonance crossing is mostly governed by the nonlinearity, while the coherent atom–molecular oscillations arising soon after the resonance has been crossed are basically of a linear nature. This observation is supposed to be of a general character, due to the basic attributes of the resonance-crossing processes in the nonlinear quantum systems of the discussed type of involved quadratic nonlinearity.

arXiv:0909.0536•Journal of Physics B Atomic Molecular and Optical Physics•Oct 29, 2009•20 citations

Landau-Zener Transition for Association of an Atomic Bose-Einstein Condensate With Inter-Particle Elastic Interactions Included

N. Sahakyan, H. H. Azizbekyan, Ruzan Sokhoyan +3 more

We study strong coupling limit of the non-linear Landau-Zener transition at coherent photo- and magneto-association of cold atoms, focusing on the role of the elastic interspecies scatterings. Using an exact third-order nonlinear differential equation for the molecular state probability, we develop a nontrivial version of the strained parameter method which enables us to construct a highly accurate and simple analytic approximation describing time dynamics of the coupled atom-molecular system.

•Oct 14, 2008•0 citations

Derivation of the Landau–Zener formula via functional equations

Chen Sun

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.

arXiv:2504.02576•Journal of Physics A Mathematical and Theoretical•Sep 2, 2025•2 citations
We present a rigorous analysis of the generalized Landau-Zener problem for the two-level interacting Bose-Einstein condensates. We show that the dynamics of the system is accurately, in detail, described by a two-term variational ansatz that is valid for the whole time domain and is applicable for any set of involved parameters. Applying an exact third-order nonlinear differential equation we construct an advanced fifth-order polynomial equation for the final transition probability serving as a highly accurate generalized Landau-Zener formula.
In the framework of a basic semiclassical time-dependent nonlinear two-state problem, we study the weak coupling limit of the nonlinear Landau-Zener transition at coherent photo- and magneto-association of an atomic Bose-Einstein condensate. Using an exact third-order nonlinear differential equation for the molecular state probability, we develop a variational approach which enables us to construct an accurate analytic approximation describing time dynamics of the coupled atom-molecular system for the case of weak coupling. The approximation is written in terms of the solution to an auxiliary linear Landau-Zener problem with some effective Landau-Zener parameter. The dependence of this effective parameter on the input Landau-Zener parameter is found to be unexpected: as the generic Landau-Zener parameter increases, the effective Landau-Zener parameter first monotonically increases (starting from zero), reaches its maximal value and then monotonically decreases again reaching zero at some point. The constructed approximation quantitatively well describes many characteristics of the time dynamics of the system, in particular, it provides a highly accurate formula for the final transition probability to the molecular state. The present result for the final transition probability improves the accuracy of the previous approximation by Ishkhanyan et al. [Phys. Rev. A 69 , 043612 (2004); J. Phys. A 38 , 3505 (2005)] by order of magnitude.
We consider the Landau-Zener problem for a Bose-Einstein condensate in a linearly varying two-level system, for the full many-particle system as well as in the mean-field approximation. Novel nonlinear eigenstates emerge in the mean-field description, which leads to a breakdown of adiabaticity: The Landau-Zener transition probability does not vanish even in the adiabatic limit. It is shown that the emergence of nonlinear eigenstates and thus the breakdown of adiabaticity corresponds to quasi-degenerate avoided crossings of the many-particle levels. The many-particle problem can be solved approximately within an independent crossings approximation, which yields an explicit generalized Landau-Zener formula. A comparison to numerical results for the many-particle system and the mean-field approximation shows an excellent agreement.
We present a rigorous analysis of the Landau–Zener linear-in-time term crossing problem for quadratic-nonlinear systems relevant to the coherent association of ultracold atoms in degenerate quantum gases. Our treatment is based on an exact third-order nonlinear differential equation for the molecular state probability. Applying a variational two-term ansatz, we construct a simple approximation that accurately describes the whole-time dynamics of the coupled atom–molecular system for any set of involved parameters. Ensuring an absolute error of less than 10 −5 for the final transition probability, the resultant solution improves by several orders of magnitude the accuracy of the previous approximations by A Ishkhanyan et al developed separately for the weak coupling ( 2005 J. Phys. A: Math. Gen. 38 3505 ) and strong interaction ( 2006 J. Phys. A: Math. Gen. 39 14887 ) limits. In addition, the constructed approximation covers the whole moderate-coupling regime, providing this intermediate regime with the same accuracy as the two mentioned limits. The obtained results reveal the remarkable observation, that for the strong-coupling limit the resonance crossing is mostly governed by the nonlinearity, while the coherent atom–molecular oscillations arising soon after the resonance has been crossed are basically of a linear nature. This observation is supposed to be of a general character, due to the basic attributes of the resonance-crossing processes in the nonlinear quantum systems of the discussed type of involved quadratic nonlinearity.
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.