Journal of Physics B Atomic Molecular and Optical Physics

The Landau–Zener formula made simple

Eric P. Glasbrenner, Wolfgang P. Schleich

Published Mar 24, 2023• 24 citations

Abstract

Abstract We employ the Markov approximation and the well-known Fresnel-integral to derive in ‘one-line’ the familiar expression for the Landau–Zener transition probability. Moreover, we provide numerical as well as analytical justifications for our approach, and identify three characteristic motions of the probability amplitude in the complex plane.

DOI: 10.1088/1361-6455/acc774

Related papers

Riccati equation perspective on Landau-Zener transitions

Eric P. Glasbrenner, Yannik Gerdes, Sándor Varró +1 more

arXiv:2502.04033Physical Review ResearchJul 30, 20252 citations

Riccati equation perspective on Landau-Zener transitions

Eric P. Glasbrenner, Yannik Gerdes, Sándor Varró +1 more

Nov 24, 20250 citations

Model study of the Landau–Zener approximation

S. Geltman, Nathan D. Aragon

American Journal of PhysicsOct 18, 20053 citations

Derivation of the Landau–Zener formula via functional equations

Chen Sun

arXiv:2504.02576Journal of Physics A Mathematical and TheoreticalSep 2, 20252 citations

A logarithmic phase singularity at the heart of Landau-Zener transitions

Eric P. Glasbrenner, David Fabian, Wolfgang P. Schleich

quant-pharXiv:2606.31568Jul 1, 20260 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:0912.4763Physical Review AMay 5, 20107 citations

Journal of Physics B Atomic Molecular and Optical Physics

The Landau–Zener formula made simple

Eric P. Glasbrenner, Wolfgang P. Schleich

Published Mar 24, 2023• 24 citations

Publisher

Abstract

DOI: 10.1088/1361-6455/acc774

Related papers

Riccati equation perspective on Landau-Zener transitions

Eric P. Glasbrenner, Yannik Gerdes, Sándor Varró +1 more

We express the dynamics of the two probability amplitudes in the elementary Landau-Zener problem in terms of the solution of the corresponding Riccati differential equation and identify three key features: (i) The solution of the Riccati equation provides the bridge between the two probability amplitudes. (ii) Neglecting the nonlinearity in the Riccati equation is equivalent to the Markov approximation which yields the exact asymptotic expression for one of the probability amplitudes. (iii) The Riccati equation identifies the origin of the failure of the Markov approximation not being able to provide us, in general, with the correct asymptotic expression of the other probability amplitude. Our approach relies on approximate yet analytical solutions of the Riccati equation in different time regimes, highlighting the impact of its nonlinear nature on the time evolution of the system.We express the dynamics of the two probability amplitudes in the elementary Landau-Zener problem in terms of the solution of the corresponding Riccati differential equation and identify three key features: (i) The solution of the Riccati equation provides the bridge between the two probability amplitudes. (ii) Neglecting the nonlinearity in the Riccati equation is equivalent to the Markov approximation which yields the exact asymptotic expression for one of the probability amplitudes. (iii) The Riccati equation identifies the origin of the failure of the Markov approximation not being able to provide us, in general, with the correct asymptotic expression of the other probability amplitude. Our approach relies on approximate yet analytical solutions of the Riccati equation in different time regimes, highlighting the impact of its nonlinear nature on the time evolution of the system.

arXiv:2502.04033Physical Review ResearchJul 30, 20252 citations

Riccati equation perspective on Landau-Zener transitions

Eric P. Glasbrenner, Yannik Gerdes, Sándor Varró +1 more

Nov 24, 20250 citations

Model study of the Landau–Zener approximation

S. Geltman, Nathan D. Aragon

We reexamine the accuracy of the Landau–Zener approximation, whose analytic simplicity has made it often cited and applied over the past 70years in describing nonadiabatic transitions. We find that this approximation may be in substantial disagreement with exact numerical results for a model physical system and a wide range of coupling strengths and relative velocities. We conclude that the Landau–Zener approximation should be used with caution for quantitative estimates of nonadiabatic transition probabilities or cross sections.We reexamine the accuracy of the Landau–Zener approximation, whose analytic simplicity has made it often cited and applied over the past 70years in describing nonadiabatic transitions. We find that this approximation may be in substantial disagreement with exact numerical results for a model physical system and a wide range of coupling strengths and relative velocities. We conclude that the Landau–Zener approximation should be used with caution for quantitative estimates of nonadiabatic transition probabilities or cross sections.

American Journal of PhysicsOct 18, 20053 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.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:2504.02576Journal of Physics A Mathematical and TheoreticalSep 2, 20252 citations

A logarithmic phase singularity at the heart of Landau-Zener transitions

Eric P. Glasbrenner, David Fabian, Wolfgang P. Schleich

Three ingredients of the elementary Landau-Zener problem determine the familiar expression $a_{LZ}\equiv\exp\left[-π/(2ε)\right]$ for the asymptotic value of the probability amplitude for remaining in the initial level: (i) A wave whose phase is determined by the product of a contour integral over a simple pole at the origin of the complex plane and the inverse of twice the scaled chirp parameter $ε$ . (ii) An asymptotic limit of the associated path connecting the points $\pm 1$ along the real axis and circumventing the pole in the upper half-plane, and (iii) a half-circle in the lower half plane enclosing together with the asymptotic path the pole. The Cauchy theorem immediately provides us with the value $\iiπ$ of the asymptotic contour, and thus with $a_{LZ}$ . Our analysis demonstrates not only that $a_{LZ}$ is the consequence of a logarithmic phase singularity but also explains why the Markov approximation also leads to $a_{LZ}$ .Three ingredients of the elementary Landau-Zener problem determine the familiar expression $a_{LZ}\equiv\exp\left[-π/(2ε)\right]$ for the asymptotic value of the probability amplitude for remaining in the initial level: (i) A wave whose phase is determined by the product of a contour integral over a simple pole at the origin of the complex plane and the inverse of twice the scaled chirp parameter $ε$. (ii) An asymptotic limit of the associated path connecting the points $\pm 1$ along the real axis and circumventing the pole in the upper half-plane, and (iii) a half-circle in the lower half plane enclosing together with the asymptotic path the pole. The Cauchy theorem immediately provides us with the value $\iiπ$ of the asymptotic contour, and thus with $a_{LZ}$. Our analysis demonstrates not only that $a_{LZ}$ is the consequence of a logarithmic phase singularity but also explains why the Markov approximation also leads to $a_{LZ}$.

quant-pharXiv:2606.31568Jul 1, 20260 citations

Landau-Zener transition in quadratic nonlinear two-state systems

Artur Mikhailovich Ishkhanyan

arXiv:0912.4763Physical Review AMay 5, 20107 citations

Three ingredients of the elementary Landau-Zener problem determine the familiar expression

a_{LZ}\equiv\exp\left[-π/(2ε)\right]

for the asymptotic value of the probability amplitude for remaining in the initial level: (i) A wave whose phase is determined by the product of a contour integral over a simple pole at the origin of the complex plane and the inverse of twice the scaled chirp parameter

ε

. (ii) An asymptotic limit of the associated path connecting the points

\pm 1

along the real axis and circumventing the pole in the upper half-plane, and (iii) a half-circle in the lower half plane enclosing together with the asymptotic path the pole. The Cauchy theorem immediately provides us with the value $\iiπ$ of the asymptotic contour, and thus with

a_{LZ}

. Our analysis demonstrates not only that

a_{LZ}

is the consequence of a logarithmic phase singularity but also explains why the Markov approximation also leads to

a_{LZ}