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:cond-mat/0012303

On the generalized Landau-Zener problem

V. L. Pokrovsky, Nikolai A. Sinitsyn

Published Dec 15, 2000• 1 citation
arXivHTMLPDF

Abstract

During the adiabatic time evolution levels crossing violates the adiabaticity and makes transitions between levels possible. Conventionally only two energy levels cross simultaneously. The transition probabilities for this case were found by Landau and Zener. However, the multilevel crossing happens systematically rather than occasionally if the Hamiltonian possesses a special symmetry. The simplest physical realization of the multilevel crossing are the Zeeman multiplet in a varying magnetic field and an electron in one-dimensional chain driven by the time-dependent electric field. We present asymptotics of the transition amplitudes for these kinds of the multilevel crossing. They are based on an exact solution for a model n-state, time-dependent Schrödinger equation.

DOI
10.48550/arxiv.cond-mat/0012303

Related papers

Landau-Zener transitions in a multilevel system: An exact result

A. V. Shytov

We study the $S$ matrix for the transitions at an avoided crossing of several energy levels, which is a multilevel generalization of the Landau-Zener problem. We demonstrate that, by extending the Schr\"odinger evolution to complex time, one can obtain an exact answer for some of the transition amplitudes. Similar to the Landau-Zener case, our result covers both the adiabatic (slow evolution) and the diabatic (fast evolution) regimes. The form of the exact transition amplitude coincides with that obtained in a sequential pairwise level crossing approximation, in accord with the conjecture of Brundobler and Elser [J. Phys. A 26, 1211 (1993)].

arXiv:cond-mat/0312011•Physical Review A•Nov 17, 2004•115 citations

S-matrix for generalized Landau-Zener problem

Stefan Brundobler, Veit Elser

The Schrodinger equation i Psi =(A+Bt) Psi with constant Hermitian matrices A and B is studied. In the form where B is diagonalized, this equation is a generalization of the Landau-Zener problem to an arbitrary number of crossing energy levels. An approach-the independent crossing approximation-leading to a partial understanding of the general case in terms of the two-level problem is introduced. It is found that certain S-matrix elements (and thus the corresponding transition probabilities) for the general problem are exactly given by formulae of unexpected simplicity, suggesting that some kind of general analytic solution might exist. The full S-matrix is calculated for a solvable special case of the problem. The solution of another special case, previously discovered for systems with three states, is generalized to any number of states. The asymptotic behaviour of the system is discussed in general and given explicitly to lowest order in 1/t.

Journal of Physics A Mathematical and General•Mar 7, 1993•148 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

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

Intramanifold level mixing by time-dependent electric fields: Multilevel Landau-Zener effect

David A. Harmin

The mixing of substates of an isolated Rydberg manifold (constant n,m) is studied using time-evolution matrices U(t,t0). Coherent mixing is effected by a linearly ramped electric field, F(t)=F\ifmmode \dot{}\else \.{}\fi{}t. The spherical l basis, diabatic (hydrogenic) p basis, and adiabatic (nonhydrogenic) q basis are considered. In the linear Stark regime, the diabatic levels are coupled by F(t) through the atomic core, parametrized by quantum defects. The resulting adiabatic levels EqF in this model group into a submanifold degenerate at F=0 plus shifted levels split off from the adiabatic manifold. Analytical expressions for the q levels Eq\ensuremath{\tau} and their coupling \ensuremath{\Gamma}qq\ensuremath{'}\ensuremath{\tau} are derived as a function of rescaled field or time \ensuremath{\tau}. The time evolution of the levels' populations is studied first for the two-level case---the usual Landau-Zener effect (LZE)---which is generalized to a multilevel Landau-Zener effect (MLZE) for any n manifold. The matrices U(t,t0) are constructed in the Riemann product representation, including Magnus corrections up to fourth order. The probability Pq\ensuremath{\rightarrow}q\ensuremath{'}(Vl) of making a transition from state q to q\ensuremath{'} upon a one-way pass across the manifold depends on a single parameter Vl=[2\ensuremath{\mu}l2/(3F\ifmmode \dot{}\else \.{}\fi{}n9)]1/2 (when one \ensuremath{\mu}l\ensuremath{\ne}0). Numerical results for l=m=0 indicate that either diabatic (p\ensuremath{\rightarrow}p\ensuremath{'}=p) or adiabatic (q\ensuremath{\rightarrow}q\ensuremath{'}=q) transitions always predominate, no matter what the ramp rate F\ifmmode \dot{}\else \.{}\fi{}. Strong evidence supports the conjecture that outer diabatic transitions between edge states of a manifold obey Pndiab(V)=exp(-\ensuremath{\beta}n\ensuremath{\pi}V2) for l=m=0, with \ensuremath{\beta}n=1 for n=2 (the LZE) and \ensuremath{\beta}n\ensuremath{\approx}lnn for n\ensuremath{\gg}2 (the MLZE), and that a similar analytical result holds for arbitrary {\ensuremath{\mu}l} and m.

Physical Review A•Jul 1, 1991•33 citations

Landau-Zener transitions in a linear chain

V. L. Pokrovsky, Nikolai A. Sinitsyn

We present an exact asymptotic solution for electron transition amplitudes in an infinite linear chain driven by an external time-dependent electric field. This solution extends the Landau-Zener theory for the case of an infinite number of states in the discrete spectrum. In addition to the transition amplitudes we calculate the effective diffusion constant.

arXiv:cond-mat/0112419•Physical review. B, Condensed matter•Apr 4, 2002•43 citations
During the adiabatic time evolution levels crossing violates the adiabaticity and makes transitions between levels possible. Conventionally only two energy levels cross simultaneously. The transition probabilities for this case were found by Landau and Zener. However, the multilevel crossing happens systematically rather than occasionally if the Hamiltonian possesses a special symmetry. The simplest physical realization of the multilevel crossing are the Zeeman multiplet in a varying magnetic field and an electron in one-dimensional chain driven by the time-dependent electric field. We present asymptotics of the transition amplitudes for these kinds of the multilevel crossing. They are based on an exact solution for a model n-state, time-dependent Schrödinger equation.
We study the SSS matrix for the transitions at an avoided crossing of several energy levels, which is a multilevel generalization of the Landau-Zener problem. We demonstrate that, by extending the Schr\"odinger evolution to complex time, one can obtain an exact answer for some of the transition amplitudes. Similar to the Landau-Zener case, our result covers both the adiabatic (slow evolution) and the diabatic (fast evolution) regimes. The form of the exact transition amplitude coincides with that obtained in a sequential pairwise level crossing approximation, in accord with the conjecture of Brundobler and Elser [J. Phys. A 26, 1211 (1993)].
The Schrodinger equation i Psi =(A+Bt) Psi with constant Hermitian matrices A and B is studied. In the form where B is diagonalized, this equation is a generalization of the Landau-Zener problem to an arbitrary number of crossing energy levels. An approach-the independent crossing approximation-leading to a partial understanding of the general case in terms of the two-level problem is introduced. It is found that certain S-matrix elements (and thus the corresponding transition probabilities) for the general problem are exactly given by formulae of unexpected simplicity, suggesting that some kind of general analytic solution might exist. The full S-matrix is calculated for a solvable special case of the problem. The solution of another special case, previously discovered for systems with three states, is generalized to any number of states. The asymptotic behaviour of the system is discussed in general and given explicitly to lowest order in 1/t.
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.
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.
The mixing of substates of an isolated Rydberg manifold (constant n,m) is studied using time-evolution matrices U(t,t0). Coherent mixing is effected by a linearly ramped electric field, F(t)=F\ifmmode \dot{}\else \.{}\fi{}t. The spherical l basis, diabatic (hydrogenic) p basis, and adiabatic (nonhydrogenic) q basis are considered. In the linear Stark regime, the diabatic levels are coupled by F(t) through the atomic core, parametrized by quantum defects. The resulting adiabatic levels EqF in this model group into a submanifold degenerate at F=0 plus shifted levels split off from the adiabatic manifold. Analytical expressions for the q levels Eq\ensuremath{\tau} and their coupling \ensuremath{\Gamma}qq\ensuremath{'}\ensuremath{\tau} are derived as a function of rescaled field or time \ensuremath{\tau}. The time evolution of the levels' populations is studied first for the two-level case---the usual Landau-Zener effect (LZE)---which is generalized to a multilevel Landau-Zener effect (MLZE) for any n manifold. The matrices U(t,t0) are constructed in the Riemann product representation, including Magnus corrections up to fourth order. The probability Pq\ensuremath{\rightarrow}q\ensuremath{'}(Vl) of making a transition from state q to q\ensuremath{'} upon a one-way pass across the manifold depends on a single parameter Vl=[2\ensuremath{\mu}l2/(3F\ifmmode \dot{}\else \.{}\fi{}n9)]1/2 (when one \ensuremath{\mu}l\ensuremath{\ne}0). Numerical results for l=m=0 indicate that either diabatic (p\ensuremath{\rightarrow}p\ensuremath{'}=p) or adiabatic (q\ensuremath{\rightarrow}q\ensuremath{'}=q) transitions always predominate, no matter what the ramp rate F\ifmmode \dot{}\else \.{}\fi{}. Strong evidence supports the conjecture that outer diabatic transitions between edge states of a manifold obey Pndiab(V)=exp(-\ensuremath{\beta}n\ensuremath{\pi}V2) for l=m=0, with \ensuremath{\beta}n=1 for n=2 (the LZE) and \ensuremath{\beta}n\ensuremath{\approx}lnn for n\ensuremath{\gg}2 (the MLZE), and that a similar analytical result holds for arbitrary {\ensuremath{\mu}l} and m.