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

Europhysics Letters (EPL)

Quantization rule for transition suppression in dynamic two-level system

Er’el Granot

Published Oct 16, 2024• 2 citations
Publisher

Abstract

Abstract The transmission probability in a generic two-quasi-state system is addressed. An approximate expression for the transmission probability between the two quasi-states is derived for the case where the energy gap between the two states varies. Unlike the Landau-Zener formula, the temporal change can have a generic form and can vanish several times. In particular, when the energy gap vanishes twice, an analytical quantization rule for the suppression of transmission between two quasi-states is derived.

DOI
10.1209/0295-5075/ad877a

Related papers

Dynamic two-level system in the presence of a delta-function pulse: Exactly solvable dynamic system

Er’el Granot

Abstract The Landau-Zener formula represents the only case where a dynamic quantum system was solved exactly analytically. There is no model of a system, which is perturbed by a temporal pulse, and can be solved exactly. It is suggested to use a delta-function perturbation, a model which has an exact analytical solution, to model the effect of a finite period pulse perturbation. Not only can this model be solved exactly and mimic the effect of a very narrow pulse, but it also presents a quantization rule for suppressed transmission ( i.e. , no change in initial state), which resembles qualitatively a similar quantization rule that appears due to the presence of a finite period pulse. Nevertheless, despite its simplicity, the model exhibits non-trivial dynamics that challenge approximation schemes. Specifically, it shows that the suppressed transmission criterion substantially changes due to an arbitrarily small change in potential value.

Europhysics Letters (EPL)•Jan 1, 2026•1 citation

Landau-Zener problem with waiting at the minimum gap and related quench dynamics of a many-body system

Uma Divakaran, Amit Dutta, Diptiman Sen

We discuss a technique for solving the Landau-Zener (LZ) problem of finding the probability of excitation in a two-level system. The idea of time reversal for the Schr\"odinger equation is employed to obtain the state reached at the final time and hence the excitation probability. Using this method, which can reproduce the well-known expression for the LZ transition probability, we solve a variant of the LZ problem, which involves waiting at the minimum gap for a time ${t}_{w}$; we find an exact expression for the excitation probability as a function of ${t}_{w}$. We provide numerical results to support our analytical expressions. We then discuss the problem of waiting at the quantum critical point of a many-body system and calculate the residual energy generated by the time-dependent Hamiltonian. Finally, we discuss possible experimental realizations of this work.

arXiv:0910.5548•Physical Review B•Feb 19, 2010•9 citations

Analytic approach to the Landau–Zener problem in bounded parameter space

Felipe Matus, Jan Střeleček, Pavel Cejnar

Abstract Three analytic solutions to the Schrödinger equation for the time-dependent Landau–Zener (LZ) Hamiltonian are presented. They correspond to specific finite-time driving paths in a bounded parameter space of a two-level system. Two of these paths go through the avoided crossing of levels, either with a constant speed or with variable speed that decreases in the region of reduced energy gap, the third path bypasses the crossing such that the energy gap remains constant. The solutions yield exact time dependencies of the excitation probability for the system evolving from the ground state of the initial Hamiltonian. The LZ formula emerges as an approximation valid within a certain interval of driving times for the constant-speed driving through the avoided crossing. For long driving times, all solutions converge to the prediction of the adiabatic perturbation theory. The excitation probability vanishes at some discrete time instants.

arXiv:2302.11372•Journal of Physics A Mathematical and Theoretical•Apr 21, 2023•3 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

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.

arXiv:1311.3919•Physical Review A•Sep 30, 2014•22 citations

A simple approach to the Landau–Zener formula

Amar C. Vutha

The Landau–Zener formula provides the probability of non-adiabatic transitions occurring when two energy levels are swept through an avoided crossing. The formula is derived here in a simple calculation that emphasizes the physics responsible for non-adiabatic population transfer.

arXiv:1001.3322•European Journal of Physics•Feb 11, 2010•23 citations
Abstract The transmission probability in a generic two-quasi-state system is addressed. An approximate expression for the transmission probability between the two quasi-states is derived for the case where the energy gap between the two states varies. Unlike the Landau-Zener formula, the temporal change can have a generic form and can vanish several times. In particular, when the energy gap vanishes twice, an analytical quantization rule for the suppression of transmission between two quasi-states is derived.
Abstract The Landau-Zener formula represents the only case where a dynamic quantum system was solved exactly analytically. There is no model of a system, which is perturbed by a temporal pulse, and can be solved exactly. It is suggested to use a delta-function perturbation, a model which has an exact analytical solution, to model the effect of a finite period pulse perturbation. Not only can this model be solved exactly and mimic the effect of a very narrow pulse, but it also presents a quantization rule for suppressed transmission ( i.e. , no change in initial state), which resembles qualitatively a similar quantization rule that appears due to the presence of a finite period pulse. Nevertheless, despite its simplicity, the model exhibits non-trivial dynamics that challenge approximation schemes. Specifically, it shows that the suppressed transmission criterion substantially changes due to an arbitrarily small change in potential value.
We discuss a technique for solving the Landau-Zener (LZ) problem of finding the probability of excitation in a two-level system. The idea of time reversal for the Schr\"odinger equation is employed to obtain the state reached at the final time and hence the excitation probability. Using this method, which can reproduce the well-known expression for the LZ transition probability, we solve a variant of the LZ problem, which involves waiting at the minimum gap for a time tw{t}_{w}tw​; we find an exact expression for the excitation probability as a function of tw{t}_{w}tw​. We provide numerical results to support our analytical expressions. We then discuss the problem of waiting at the quantum critical point of a many-body system and calculate the residual energy generated by the time-dependent Hamiltonian. Finally, we discuss possible experimental realizations of this work.
Abstract Three analytic solutions to the Schrödinger equation for the time-dependent Landau–Zener (LZ) Hamiltonian are presented. They correspond to specific finite-time driving paths in a bounded parameter space of a two-level system. Two of these paths go through the avoided crossing of levels, either with a constant speed or with variable speed that decreases in the region of reduced energy gap, the third path bypasses the crossing such that the energy gap remains constant. The solutions yield exact time dependencies of the excitation probability for the system evolving from the ground state of the initial Hamiltonian. The LZ formula emerges as an approximation valid within a certain interval of driving times for the constant-speed driving through the avoided crossing. For long driving times, all solutions converge to the prediction of the adiabatic perturbation theory. The excitation probability vanishes at some discrete time instants.
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+T+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.