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

Physical Review A

Saturated transitions in exactly soluble models of two-state curve crossing with time-dependent potentials

Vladimir Yurovsky, A. Ben‐Reuven

Published Dec 1, 1999• 19 citations
Publisher

Abstract

Two exactly soluble two-state curve crossing models, with potential matrices formed by a linear combination of time-independent terms and (I) inversely proportional $(\ensuremath{\sim}{t}^{\ensuremath{-}1})$ or (II) exponential time-dependent terms, are considered here. It is shown that the two models are related to each other by a simple transformation of the time variable. These models can be further transformed to a simpler one (the ``reduced'' model), in which the two crossing potentials are a constant (horizontal) one and an inversely proportional time-dependent one, and the interaction coupling them is time-independent. Analysis of the exact solution of the reduced model discloses that the nonadiabatic transition probability saturates at increasingly large coupling strengths without vanishing (in contradiction of the linear Landau-Zener theory and various adiabatic theories). Similar saturation can also occur in more general models, within certain ranges of the potential parameters involved.

DOI
10.1103/physreva.60.4561

Related papers

Multistate curve crossing: an exactly soluble model with degeneracy

Vladimir Yurovsky, A. Ben‐Reuven

The linear curve-crossing problem involving a slanted potential and a set of horizontal ones as an exactly soluble quantum-mechanical problem, is re-examined. Special attention is paid to the questions of degeneracy of the horizontal-potential channels. It is shown, by explicit derivation of an asymptotic expansion of the scattering amplitudes, that the degeneracy limit is a singular point. An explicit exact solution is derived for the degenerate case, resulting in a non-Landau-Zener saturated behaviour of the transition probabilities at high coupling intensities. Estimates are derived for the transition region as a function of the energy gap, and its divergence on approach to degeneracy. Relevance to the optical shielding of ultracold atom collisions is pointed out, with particular reference to the phenomenon of `counterintuitive' transitions which were discussed recently.

Journal of Physics B Atomic Molecular and Optical Physics•Jan 14, 1998•23 citations

Nonstationary multistate Coulomb and multistate exponential models for nonadiabatic transitions

Valentin N. Ostrovsky

The nonstationary Schr\"odinger equation is considered in a finite basis of states. The model Hamiltonian matrix corresponds to a single diabatic potential curve with a Coulombic $\ensuremath{\sim}1/t$ time dependence. An arbitrary number of other diabatic potential curves are flat, i.e., time independent and have arbitrary energies. Related states are coupled by constant interactions with the Coulomb state. The resulting nonstationary Schr\"odinger equation is solved by the method of contour integral. Probabilities of transitions to any other state are obtained as $\stackrel{\ensuremath{\rightarrow}}{t}\ensuremath{\infty}$ in a simple analytical form for the case when the Coulomb state is populated initially (at instant of time $\stackrel{\ensuremath{\rightarrow}}{t}+0).$ The formulas apply both to the cases when a horizontal diabatic potential curve is crossed by the Coulomb one and to a noncrossing situation. In the limit of weak coupling, the transition probabilities are interpreted in terms of a sequence of pairwise Landau-Zener-type transitions. Mapping of the Coulomb model onto an exactly solvable exponential multistate model is established. For the special two-state case, the well-known Nikitin model is recovered.

Physical Review A•Jul 15, 2003•16 citations

Analytic solution of two-state time-independent coupled Schrödinger equations in an exponential model

Vladimir I. Osherov, Hiroki Nakamura

Quantum mechanically exact analytical solutions are obtained for a two-state exponential model, in which the exponent of diabatic coupling is one-half of that of the diabatic potential curve. A very simple and accurate semiclassical formula is found for the nonadiabatic transition probability. This gives a direct generalization of the Landau-Zener and Rosen-Zener formulas.

Physical Review A•Mar 1, 1999•16 citations

Two-state linear curve crossing problems revisited. IV. The best analytical formulas for scattering matrices

Chaoyuan Zhu, Hiroki Nakamura

Based on the achievements of the previous three papers of this series, the best working formulas for scattering matrix are obtained for both the Landau–Zener (LZ) and the nonadiabatic tunneling (NT) case: two formulas valid at b2≥0 and b2≤0 in the LZ case, and three formulas valid atb2≤−1, −1≤b2≤1 and b2≥1 in the NT case, where b2 represents the effective energy. Simple and compact formulas which work far better than the LZ formula are proposed for nonadiabatic transition probability by one passage of crossing point for both the LZ and NT cases. Furthermore, compact expressions are derived, for the first time, for the nonadiabatic tunneling probability at b2≤1, i.e., at energies lower than the bottom of the upper adiabatic potential. All the formulas proposed here can be usefully utilized at any coupling strength, namely the validity range has been very much expanded compared to the previous formulas by employing certain empirical corrections. Besides, these formulas have convenience to enable an extension to general curved potentials.

The Journal of Chemical Physics•Sep 15, 1994•66 citations

Nonadiabatic decay of metastable states on coupled linear potentials

Alisher Duspayev, Ansh Shah, Georg A. Raithel

Abstract Avoided crossings of level pairs with opposite slopes can form potential-energy minima for the external degree of freedom of quantum particles, giving rise to metastable states on the avoided crossings (MSACs). Nonadiabatic decay of MSACs is studied by solving the two-component Schrödinger equation in diabatic and adiabatic representations. Non-perturbative lifetime values are found by evaluating wave function flux and scattering phases of time-independent solutions, as well as wave-function decay of time-dependent solutions. The values from these methods generally agree well, validating the utilized approaches. As the adiabaticity parameter, V, of the system is increased by about a factor of ten across the mixed diabatic/adiabatic regime, the MSAC character transitions from marginally to highly stable, with the lifetimes increasing by about ten orders of magnitude. The dependence of MSAC lifetime on the vibrational quantum number, ν, is discussed for several regimes of V. Time-dependent perturbation theory yields lifetimes that deviate by ≲30% from non-perturbative results, over the range of V and ν studied, while a semi-classical model based on Landau–Zener tunneling is up to a factor of twenty off. The results are relevant to numerous atomic and molecular systems with metastable states on intersecting, coupled potential energy curves.

arXiv:2201.12388•New Journal of Physics•May 1, 2022•3 citations

Complete solutions of the Landau-Zener-Stueckelberg curve crossing problems, and their generalizations and applications

Hiroki Nakamura

The compact analytical complete solutions recently obtained for the two-state Landau-Zener-Stueckelberg problems are reviewed and explained. The theory covers both Landau-Zener (LZ) type in which the two diabatic potential curves cross with the same sign of slopes and the nonadiabatic tunneling (NT) type in which the potentials cross with different signs of slopes. The theory is applicable virtually in the whole range of energy and coupling strength and is convenient for practical use. The new theory for time-dependent nonadiabatic transition can be formulated from the time-independent theory of the LZ-type. The utilizability of the theory to various multi-channel problems and also to multi-dimensional problems is demonstrated and explained. The intriguing phenomenon of complete reflection which appears in the NT-case is explicitly utilized to propose a new type of molecular switching and to control molecular processes such as molecular photodissociation. Also proposed is a new way of controlling molecular processes by using time-dependent external fields. Finally, a trial to formulate a unified analytical theory to cover both Landau-Zener-Stueckelberg and Rosen-Zener-Demkov types of nonadiabatic transitions is briefly touched upon.

AIP conference proceedings•Jan 1, 2000•2 citations
Two exactly soluble two-state curve crossing models, with potential matrices formed by a linear combination of time-independent terms and (I) inversely proportional (\ensuremath∼t\ensuremath−1)(\ensuremath{\sim}{t}^{\ensuremath{-}1})(\ensuremath∼t\ensuremath−1) or (II) exponential time-dependent terms, are considered here. It is shown that the two models are related to each other by a simple transformation of the time variable. These models can be further transformed to a simpler one (the ``reduced'' model), in which the two crossing potentials are a constant (horizontal) one and an inversely proportional time-dependent one, and the interaction coupling them is time-independent. Analysis of the exact solution of the reduced model discloses that the nonadiabatic transition probability saturates at increasingly large coupling strengths without vanishing (in contradiction of the linear Landau-Zener theory and various adiabatic theories). Similar saturation can also occur in more general models, within certain ranges of the potential parameters involved.
The linear curve-crossing problem involving a slanted potential and a set of horizontal ones as an exactly soluble quantum-mechanical problem, is re-examined. Special attention is paid to the questions of degeneracy of the horizontal-potential channels. It is shown, by explicit derivation of an asymptotic expansion of the scattering amplitudes, that the degeneracy limit is a singular point. An explicit exact solution is derived for the degenerate case, resulting in a non-Landau-Zener saturated behaviour of the transition probabilities at high coupling intensities. Estimates are derived for the transition region as a function of the energy gap, and its divergence on approach to degeneracy. Relevance to the optical shielding of ultracold atom collisions is pointed out, with particular reference to the phenomenon of `counterintuitive' transitions which were discussed recently.
The nonstationary Schr\"odinger equation is considered in a finite basis of states. The model Hamiltonian matrix corresponds to a single diabatic potential curve with a Coulombic $\ensuremath{\sim}1/t$ time dependence. An arbitrary number of other diabatic potential curves are flat, i.e., time independent and have arbitrary energies. Related states are coupled by constant interactions with the Coulomb state. The resulting nonstationary Schr\"odinger equation is solved by the method of contour integral. Probabilities of transitions to any other state are obtained as $\stackrel{\ensuremath{\rightarrow}}{t}\ensuremath{\infty}$ in a simple analytical form for the case when the Coulomb state is populated initially (at instant of time $\stackrel{\ensuremath{\rightarrow}}{t}+0).$ The formulas apply both to the cases when a horizontal diabatic potential curve is crossed by the Coulomb one and to a noncrossing situation. In the limit of weak coupling, the transition probabilities are interpreted in terms of a sequence of pairwise Landau-Zener-type transitions. Mapping of the Coulomb model onto an exactly solvable exponential multistate model is established. For the special two-state case, the well-known Nikitin model is recovered.
Based on the achievements of the previous three papers of this series, the best working formulas for scattering matrix are obtained for both the Landau–Zener (LZ) and the nonadiabatic tunneling (NT) case: two formulas valid at b2≥0 and b2≤0 in the LZ case, and three formulas valid atb2≤−1, −1≤b2≤1 and b2≥1 in the NT case, where b2 represents the effective energy. Simple and compact formulas which work far better than the LZ formula are proposed for nonadiabatic transition probability by one passage of crossing point for both the LZ and NT cases. Furthermore, compact expressions are derived, for the first time, for the nonadiabatic tunneling probability at b2≤1, i.e., at energies lower than the bottom of the upper adiabatic potential. All the formulas proposed here can be usefully utilized at any coupling strength, namely the validity range has been very much expanded compared to the previous formulas by employing certain empirical corrections. Besides, these formulas have convenience to enable an extension to general curved potentials.
Abstract Avoided crossings of level pairs with opposite slopes can form potential-energy minima for the external degree of freedom of quantum particles, giving rise to metastable states on the avoided crossings (MSACs). Nonadiabatic decay of MSACs is studied by solving the two-component Schrödinger equation in diabatic and adiabatic representations. Non-perturbative lifetime values are found by evaluating wave function flux and scattering phases of time-independent solutions, as well as wave-function decay of time-dependent solutions. The values from these methods generally agree well, validating the utilized approaches. As the adiabaticity parameter, V, of the system is increased by about a factor of ten across the mixed diabatic/adiabatic regime, the MSAC character transitions from marginally to highly stable, with the lifetimes increasing by about ten orders of magnitude. The dependence of MSAC lifetime on the vibrational quantum number, ν, is discussed for several regimes of V. Time-dependent perturbation theory yields lifetimes that deviate by ≲30% from non-perturbative results, over the range of V and ν studied, while a semi-classical model based on Landau–Zener tunneling is up to a factor of twenty off. The results are relevant to numerous atomic and molecular systems with metastable states on intersecting, coupled potential energy curves.
The compact analytical complete solutions recently obtained for the two-state Landau-Zener-Stueckelberg problems are reviewed and explained. The theory covers both Landau-Zener (LZ) type in which the two diabatic potential curves cross with the same sign of slopes and the nonadiabatic tunneling (NT) type in which the potentials cross with different signs of slopes. The theory is applicable virtually in the whole range of energy and coupling strength and is convenient for practical use. The new theory for time-dependent nonadiabatic transition can be formulated from the time-independent theory of the LZ-type. The utilizability of the theory to various multi-channel problems and also to multi-dimensional problems is demonstrated and explained. The intriguing phenomenon of complete reflection which appears in the NT-case is explicitly utilized to propose a new type of molecular switching and to control molecular processes such as molecular photodissociation. Also proposed is a new way of controlling molecular processes by using time-dependent external fields. Finally, a trial to formulate a unified analytical theory to cover both Landau-Zener-Stueckelberg and Rosen-Zener-Demkov types of nonadiabatic transitions is briefly touched upon.