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.

Journal of Physics B Atomic Molecular and Optical PhysicsMar 24, 202324 citations

Landau-Zener problem with decay and dephasing

Y. Avishai, Y. B. Band

arXiv:1311.3919Physical Review ASep 30, 201422 citations

arXiv:2502.04033 · Physical Review Research

Riccati equation perspective on Landau-Zener transitions

Eric P. Glasbrenner, Yannik Gerdes, Sándor Varró, Wolfgang P. Schleich

Published Jul 30, 2025• 2 citations

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Abstract

DOI: 10.1103/nqgc-w1ns

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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.

Nov 24, 20250 citations

The analytical structure of nonadiabatic transitions : asymptotics, Riccati equations and phase singularities in the Landau-Zener model

Eric P. Glasbrenner

The control of nonadiabatic quantum transitions is a requirement for the development of modern quantum technologies. The paradigm to understand these dynamics is the Landau-Zener model. However, for nearly a century, the exact transition probability of this model has been derived using special functions or semiclassical approximations. These conventional methods obscure the underlying physical mechanisms and transition dynamics near the avoided crossing. This cumulative thesis, resulting from the author's research activities at the Intitute of Quantum Physics at Ulm University under the supervision of Prof. Dr. Wolfgang P. Schleich, establishes an analytical framework to study the underlying structure of two-level crossing dynamics. In contrast to the standard techniques, the framework developed in this thesis is based on elementary concepts combined with complex analysis. First, we map the Landau-Zener equations onto an integro-differential equation to decouple them and formulate the Markov approximation which reduces the complicated integrals to the evaluation of Fresnel integrals. This technique reproduces in the asymptotic limit the exact probability amplitude of the Landau-Zener model. Next, to uncover the origin behind the success of the Markov approximation, we analyze the Landau-Zener equations in terms of their underlying nonlinear dynamics. First, we introduce an amplitude-and-phase approach, which explains in a clear way the emergence of the Stueckelberg oscillations through the dynamics of the phase velocity. Second, we map the two-level dynamics onto a nonlinear Riccati differential equation and show that the Markov approximation is equivalent to a linearization of the derived Riccati equation. Moreover, our approach helps to identify the nonlinear term as the reason for the breakdown of the Markov approximation close to the avoided crossing. Finally, we derive exact asymptotic expressions for the transition amplitudes using elementary functions for both negative and positive times. Connecting these two regimes via the analytical continuation of the logarithmic phase reveals that the Landau-Zener transition is mainly governed by a logarithmic phase singularity. Based on this logarithmic phase singularity, we introduce a new time scale to map the system dynamics as a path via a logarithmic phase ansatz in the complex plane. Evaluating this path via the Cauchy’s integral formula yields the exact transition probability which demonstrates that the exact solution emerges from the underlying phase singularity rather than properties of special functions.The control of nonadiabatic quantum transitions is a requirement for the development of modern quantum technologies. The paradigm to understand these dynamics is the Landau-Zener model. However, for nearly a century, the exact transition probability of this model has been derived using special functions or semiclassical approximations. These conventional methods obscure the underlying physical mechanisms and transition dynamics near the avoided crossing. This cumulative thesis, resulting from the author's research activities at the Intitute of Quantum Physics at Ulm University under the supervision of Prof. Dr. Wolfgang P. Schleich, establishes an analytical framework to study the underlying structure of two-level crossing dynamics. In contrast to the standard techniques, the framework developed in this thesis is based on elementary concepts combined with complex analysis. First, we map the Landau-Zener equations onto an integro-differential equation to decouple them and formulate the Markov approximation which reduces the complicated integrals to the evaluation of Fresnel integrals. This technique reproduces in the asymptotic limit the exact probability amplitude of the Landau-Zener model. Next, to uncover the origin behind the success of the Markov approximation, we analyze the Landau-Zener equations in terms of their underlying nonlinear dynamics. First, we introduce an amplitude-and-phase approach, which explains in a clear way the emergence of the Stueckelberg oscillations through the dynamics of the phase velocity. Second, we map the two-level dynamics onto a nonlinear Riccati differential equation and show that the Markov approximation is equivalent to a linearization of the derived Riccati equation. Moreover, our approach helps to identify the nonlinear term as the reason for the breakdown of the Markov approximation close to the avoided crossing. Finally, we derive exact asymptotic expressions for the transition amplitudes using elementary functions for both negative and positive times. Connecting these two regimes via the analytical continuation of the logarithmic phase reveals that the Landau-Zener transition is mainly governed by a logarithmic phase singularity. Based on this logarithmic phase singularity, we introduce a new time scale to map the system dynamics as a path via a logarithmic phase ansatz in the complex plane. Evaluating this path via the Cauchy’s integral formula yields the exact transition probability which demonstrates that the exact solution emerges from the underlying phase singularity rather than properties of special functions.

Jan 1, 20260 citations

Elementary asymptotic approach to the Landau-Zener problem

Eric P. Glasbrenner, Wolfgang P. Schleich

We present an asymptotic approach towards the standard Landau-Zener problem based on two linearly independent elementary waves of constant amplitude but time-dependent phase. The two contributions to this phase are quadratic and logarithmic in time and result from the linear chirp of the energies and the lowest order correction in the coupling between the two levels in the long-time limit. Indeed, our solutions subjected to initial conditions at a large but finite time in the past, are valid for large negative and large positive times. Due to their asymptotic nature they are not valid in the neighborhood of the moment when the levels cross. However, as the starting point of the dynamics moves further into the past, the time interval of the break-down of our asymptotic solutions shrinks and vanishes in the limit of the infinite past which corresponds to the standard Landau-Zener situation. Our approach explains not only every feature of the exact solution but yields deeper insights into the origin of the effects. In particular, it (i) brings to light the subtleties involved in the asymptotic limit leading to the standard expressions for the Landau-Zener transition amplitudes, (ii) identifies the logarithmic phase as the origin of the exponential transition probability amplitude, and (iii) reveals the structure of the lowest order corrections to the Landau-Zener result when the starting point is not in the infinite past.We present an asymptotic approach towards the standard Landau-Zener problem based on two linearly independent elementary waves of constant amplitude but time-dependent phase. The two contributions to this phase are quadratic and logarithmic in time and result from the linear chirp of the energies and the lowest order correction in the coupling between the two levels in the long-time limit. Indeed, our solutions subjected to initial conditions at a large but finite time in the past, are valid for large negative and large positive times. Due to their asymptotic nature they are not valid in the neighborhood of the moment when the levels cross. However, as the starting point of the dynamics moves further into the past, the time interval of the break-down of our asymptotic solutions shrinks and vanishes in the limit of the infinite past which corresponds to the standard Landau-Zener situation. Our approach explains not only every feature of the exact solution but yields deeper insights into the origin of the effects. In particular, it (i) brings to light the subtleties involved in the asymptotic limit leading to the standard expressions for the Landau-Zener transition amplitudes, (ii) identifies the logarithmic phase as the origin of the exponential transition probability amplitude, and (iii) reveals the structure of the lowest order corrections to the Landau-Zener result when the starting point is not in the infinite past.

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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

The Landau–Zener formula made simple

Eric P. Glasbrenner, Wolfgang P. Schleich

Journal of Physics B Atomic Molecular and Optical PhysicsMar 24, 202324 citations

Landau-Zener problem with decay and dephasing

Y. Avishai, Y. B. Band

Two aspects of the classic two-level Landau-Zener (LZ) problem are considered. First, we address the LZ problem when one or both levels decay, i.e., ${\ensuremath{\varepsilon}}_{j}(t)\ensuremath{\rightarrow}{\ensuremath{\varepsilon}}_{j}(t)\ensuremath{-}i{\ensuremath{\Gamma}}_{j}/2$. We find that if the system evolves from an initial time $\ensuremath{-}T$ to a final time $+T$ such that $|{\ensuremath{\varepsilon}}_{1}(\ifmmode\pm\else\textpm\fi{}T)\ensuremath{-}{\ensuremath{\varepsilon}}_{2}(\ifmmode\pm\else\textpm\fi{}T)|$ is not too large, the LZ survival probability of a state $|j\ensuremath{\rangle}$ can increase with increasing decay rate of the other state $|i\ensuremath{\ne}j\ensuremath{\rangle}$. This surprising result occurs because the decay results in crossing of the two eigenvalues of the instantaneous non-Hermitian Hamiltonian. On the other hand, if $|{\ensuremath{\varepsilon}}_{1}(\ifmmode\pm\else\textpm\fi{}T)\ensuremath{-}{\ensuremath{\varepsilon}}_{2}(\ifmmode\pm\else\textpm\fi{}T)|\ensuremath{\rightarrow}\ensuremath{\infty}$ as $T\ensuremath{\rightarrow}\ensuremath{\infty}$, the probability is independent of the decay rate. These results are based on analytic solutions of the time-dependent Schr\"odinger equations for two cases: (a) the energy levels depend linearly on time, and (b) the energy levels are bounded and of the form ${\ensuremath{\varepsilon}}_{1,2}(t)=\ifmmode\pm\else\textpm\fi{}\ensuremath{\varepsilon}tanh(t/\mathcal{T})$. Second, we study LZ transitions affected by dephasing by formulating the Landau-Zener problem with noise in terms of a Schr\"odinger-Langevin stochastic coupled set of differential equations. The LZ survival probability then becomes a random variable whose probability distribution is shown to behave very differently for long and short dephasing times. We also discuss the combined effects of decay and dephasing on the LZ probability.Two aspects of the classic two-level Landau-Zener (LZ) problem are considered. First, we address the LZ problem when one or both levels decay, i.e., ${\ensuremath{\varepsilon}}_{j}(t)\ensuremath{\rightarrow}{\ensuremath{\varepsilon}}_{j}(t)\ensuremath{-}i{\ensuremath{\Gamma}}_{j}/2$. We find that if the system evolves from an initial time $\ensuremath{-}T$ to a final time $+T$ such that $|{\ensuremath{\varepsilon}}_{1}(\ifmmode\pm\else\textpm\fi{}T)\ensuremath{-}{\ensuremath{\varepsilon}}_{2}(\ifmmode\pm\else\textpm\fi{}T)|$ is not too large, the LZ survival probability of a state $|j\ensuremath{\rangle}$ can increase with increasing decay rate of the other state $|i\ensuremath{\ne}j\ensuremath{\rangle}$. This surprising result occurs because the decay results in crossing of the two eigenvalues of the instantaneous non-Hermitian Hamiltonian. On the other hand, if $|{\ensuremath{\varepsilon}}_{1}(\ifmmode\pm\else\textpm\fi{}T)\ensuremath{-}{\ensuremath{\varepsilon}}_{2}(\ifmmode\pm\else\textpm\fi{}T)|\ensuremath{\rightarrow}\ensuremath{\infty}$ as $T\ensuremath{\rightarrow}\ensuremath{\infty}$, the probability is independent of the decay rate. These results are based on analytic solutions of the time-dependent Schr\"odinger equations for two cases: (a) the energy levels depend linearly on time, and (b) the energy levels are bounded and of the form ${\ensuremath{\varepsilon}}_{1,2}(t)=\ifmmode\pm\else\textpm\fi{}\ensuremath{\varepsilon}tanh(t/\mathcal{T})$. Second, we study LZ transitions affected by dephasing by formulating the Landau-Zener problem with noise in terms of a Schr\"odinger-Langevin stochastic coupled set of differential equations. The LZ survival probability then becomes a random variable whose probability distribution is shown to behave very differently for long and short dephasing times. We also discuss the combined effects of decay and dephasing on the LZ probability.

arXiv:1311.3919Physical Review ASep 30, 201422 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}

+T

such that $|{\ensuremath{\varepsilon}}_{1}(\ifmmode\pm\else\textpm\fi{}T)\ensuremath{-}{\ensuremath{\varepsilon}}_{2}(\ifmmode\pm\else\textpm\fi{}T)|$ is not too large, the LZ survival probability of a state $|j\ensuremath{\rangle}$ can increase with increasing decay rate of the other state $|i\ensuremath{\ne}j\ensuremath{\rangle}$. This surprising result occurs because the decay results in crossing of the two eigenvalues of the instantaneous non-Hermitian Hamiltonian. On the other hand, if $|{\ensuremath{\varepsilon}}_{1}(\ifmmode\pm\else\textpm\fi{}T)\ensuremath{-}{\ensuremath{\varepsilon}}_{2}(\ifmmode\pm\else\textpm\fi{}T)|\ensuremath{\rightarrow}\ensuremath{\infty}$ as $T\ensuremath{\rightarrow}\ensuremath{\infty}$, the probability is independent of the decay rate. These results are based on analytic solutions of the time-dependent Schr\"odinger equations for two cases: (a) the energy levels depend linearly on time, and (b) the energy levels are bounded and of the form ${\ensuremath{\varepsilon}}_{1,2}(t)=\ifmmode\pm\else\textpm\fi{}\ensuremath{\varepsilon}tanh(t/\mathcal{T})$. Second, we study LZ transitions affected by dephasing by formulating the Landau-Zener problem with noise in terms of a Schr\"odinger-Langevin stochastic coupled set of differential equations. The LZ survival probability then becomes a random variable whose probability distribution is shown to behave very differently for long and short dephasing times. We also discuss the combined effects of decay and dephasing on the LZ probability.