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Two-level adiabatic transition probability for small avoided crossings generated by tangential intersections

Kenta Higuchi, Takuya Watanabe

TL;DR

This work analyzes transition probabilities in a two-level quantum system with tangential avoided crossings under a two-parameter singular limit $h\to0$, $\varepsilon\to0$. It develops a rigorous framework based on Jost/exact-WKB constructions, the method of successive approximations (MSA), and SU(2) transfer matrices to connect local solutions across multiple crossings. The main contributions are precise asymptotic expansions for the transition probability $P(\varepsilon,h)$ in the non-adiabatic regime with multiple crossings, including explicit quantum-interference terms, and a comprehensive description of regime coexistence where adiabatic and non-adiabatic crossings interact, yielding a regime-switch in $P$ controlled by the data $(m_k, t_k)$. The results generalize Landau–Zener-type formulas to tangential multi-crossings and provide a detailed mechanism for interference effects and parameter-dependent switches in transition probabilities with potential relevance to molecular and quantum-chemistry contexts.

Abstract

In this paper, the asymptotic behaviors of the transition probability for two-level avoided crossings are studied under the limit where two parameters (adiabatic parameter and energy gap parameter) tend to zero. This is a continuation of our previous works where avoided crossings are generated by tangential intersections and obey a non-adiabatic regime. The main results elucidate not only the asymptotic expansion of transition probability but also a quantum interference caused by several avoided crossings and a coexistence of two-parameter regimes arising from different vanishing orders.

Two-level adiabatic transition probability for small avoided crossings generated by tangential intersections

TL;DR

This work analyzes transition probabilities in a two-level quantum system with tangential avoided crossings under a two-parameter singular limit , . It develops a rigorous framework based on Jost/exact-WKB constructions, the method of successive approximations (MSA), and SU(2) transfer matrices to connect local solutions across multiple crossings. The main contributions are precise asymptotic expansions for the transition probability in the non-adiabatic regime with multiple crossings, including explicit quantum-interference terms, and a comprehensive description of regime coexistence where adiabatic and non-adiabatic crossings interact, yielding a regime-switch in controlled by the data . The results generalize Landau–Zener-type formulas to tangential multi-crossings and provide a detailed mechanism for interference effects and parameter-dependent switches in transition probabilities with potential relevance to molecular and quantum-chemistry contexts.

Abstract

In this paper, the asymptotic behaviors of the transition probability for two-level avoided crossings are studied under the limit where two parameters (adiabatic parameter and energy gap parameter) tend to zero. This is a continuation of our previous works where avoided crossings are generated by tangential intersections and obey a non-adiabatic regime. The main results elucidate not only the asymptotic expansion of transition probability but also a quantum interference caused by several avoided crossings and a coexistence of two-parameter regimes arising from different vanishing orders.
Paper Structure (18 sections, 13 theorems, 163 equations, 6 figures)

This paper contains 18 sections, 13 theorems, 163 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Results
  3. Assumptions and main result
  4. Coexistence of the two parameter regimes
  5. Construction of exact solutions
  6. Estimates of fundamental solutions
  7. Method of successive approximations (MSA)
  8. Connection formulas
  9. Across the vanishing point
  10. Between the vanishing points
  11. End of the proofs
  12. Proof of Theorem \ref{['mainthm']}
  13. Proof of Theorem \ref{['2ndthm']}
  14. Asymptotic behavior of the transfer matrix $T_k$ in the adiabatic regime
  15. Product of the transfer matrices
  16. ...and 3 more sections

Key Result

Theorem 1

Assume Conditions condi_1, condi_2 and $m_*\geq 2$. Then there exist $\mu_0>0$ and $h_0>0$ such that, for any $\varepsilon$ and $h$ with $\mu_*(\varepsilon,h) \in (0, \mu_0]$ and $h \in (0,h_0]$, the transition probability $P(\varepsilon, h)$ has the asymptotic expansions: where the coefficient $C_{*}(h)$ consists of the product of two factors $\gamma_{*}$ and $\delta_*(h)$, that is $C_{*}(h) = \

Figures (6)

  • Figure 1: An example of $V(t)$ and energies $E_\pm(\varepsilon,h)$
  • Figure 2: Cases (b) (left) and (c) (right) in Example \ref{['2avo']}
  • Figure 3: Cases b (above) and c (below) in Example \ref{['3avo']}
  • Figure 4: Adiabatic and non-adiabatic regimes $({\rm A})_m$ and $({\rm N})_m$ for $m=3,11$ (logarithmic scale, $10^{-150}\le\varepsilon,h\le 1$, $({\rm A})_m=\{(\varepsilon,h);\,\mu_m\ge100\}$, $({\rm N})_m=\{(\varepsilon,h);\,0<\mu_m\le0.01\}$).
  • Figure 5: Adiabatic and non-adiabatic regimes $(A)_m$ and $(N)_m$ for $m=1,2,3$.
  • ...and 1 more figures

Theorems & Definitions (34)

  • Definition 2.1
  • Remark 2.2
  • Theorem 1
  • Remark 2.3
  • Remark 2.4
  • Example 2.5: Two avoided crossings
  • Remark 2.6
  • Example 2.7: Three avoided crossings
  • Remark 2.8
  • Theorem 2
  • ...and 24 more