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arXiv:2311.05871 · Physical Review A

Exact WKB analysis for adiabatic discrete-level Hamiltonians

Takayuki Suzuki, Eiki Taniguchi, Kaito Iwamura

Published Feb 20, 2024• 7 citations
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Abstract

The dynamics of quantum systems under the adiabatic Hamiltonian has attracted attention not only in quantum control but also in a wide range of fields from condensed-matter physics to high-energy physics because of its nonperturbative behavior. Here we analyze the adiabatic dynamics in two-level systems and multilevel systems using exact WKB analysis, which is one of the nonperturbative analysis methods. As a result, we obtain a formula for the transition probability, which is similar to the known formula in the two-level system. Although nonperturbative analysis in the adiabatic limit has rarely been studied for multilevel systems, we show that the same analysis can be applied and also provide a concrete example. The results will serve as a basis for the application of exact WKB analysis in various fields of physics.

DOI
10.1103/physreva.109.022225

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The dynamics of quantum systems under the adiabatic Hamiltonian has attracted attention not only in quantum control but also in a wide range of fields from condensed-matter physics to high-energy physics because of its nonperturbative behavior. Here we analyze the adiabatic dynamics in two-level systems and multilevel systems using exact WKB analysis, which is one of the nonperturbative analysis methods. As a result, we obtain a formula for the transition probability, which is similar to the known formula in the two-level system. Although nonperturbative analysis in the adiabatic limit has rarely been studied for multilevel systems, we show that the same analysis can be applied and also provide a concrete example. The results will serve as a basis for the application of exact WKB analysis in various fields of physics.
A method of fundamental solutions has been used to investigate transitions in two-energy-level-systems with no level crossing in a real time. Compact formulas for transition probabilities have been found in their exact form as well as in their adiabatic limit. No interference effects resulting from many-level complex crossings as announced by Joye, Mileti, and Pfister [Phys. Rev. A 44, 4280 (1991)] have been detected in either case. It is argued that these results of this work are incorrect. However, some effects of Berry's phases are confirmed.
Nonadiabatic transitions in multilevel systems appear in various fields of physics, but it is not easy to analyze their dynamics in general. In this paper, we propose to extend the adiabatic impulse approximation to multilevel systems. This approximation method is shown to be equivalent to a series of unitary evolutions and facilitates to evaluate the dynamics numerically. In particular, we analyze the dynamics of the Landau-Zener grid model and the multilevel Landau-Zener-St\"uckelberg-Majorana interference model, and confirm that the results are in good agreement with the exact dynamics evaluated numerically. We also derive the conditions for destructive interference to occur in the multilevel system.
Synopsis: It is rare that exact solutions for physical quantum systems are available, and likely that one or more types of perturbation theories need to be applied. Three time-independent, or static, methods are discussed. Using exact energy shift formulas, a result correct to second order for the perturbed system energy is derived and applied to a two-state system and a harmonic oscillator example. The WKB semiclassical approximation is motivated, and applications to barrier penetration and various types of bound-state problems are made. The third class of methods gives upper limit estimates of system ground-state properties based on energy functional variations.
The quantum nonperturbative analysis of Kroll and Watson for a two-level system of near-adiabatic atom-atom scattering in an intense laser-field mode is extended to treat a general multilevel system interacting with intense single- or many-field modes. A procedure for rigorously solving the adiabatic eigenvalue problem for the whole charge-field system is given. A new transition-probability formula is derived. Cross sections are calculated for the processes $\mathrm{Li}+\mathrm{H}(X^{1}\ensuremath{\Sigma}^{+})+\ensuremath{\nu}\ensuremath{\hbar}\ensuremath{\omega}\ensuremath{\rightarrow}\mathrm{Li}+\mathrm{H}(A^{1}\ensuremath{\Sigma}^{+} or B^{1}\ensuremath{\Pi})$ where $\ensuremath{\nu}\ensuremath{\ge}1$. Analysis of transition in an atom due to intensity variation of a laser pulse shows that desired transition probability per pulse (e.g., \textonehalf{}) may be achieved by varying the pulse parameters. For this, numerical results of $\mathrm{Na}(3s)+2\ensuremath{\hbar}\ensuremath{\omega}\ensuremath{\rightarrow}\mathrm{Na}(5s)$ and of $\mathrm{Li}(2s)+8\ensuremath{\hbar}\ensuremath{\omega}\ensuremath{\rightarrow}\mathrm{Li}(3s)$ are given.
Our aim in this paper is to study classical dynamics in two-level molecular systems. We first derive, through the Wigner phase-space transform, a classical limit that reduces unfortunately to the adiabatic approximation. This lead us to develop a one-center quantal approximation whose variables can be interpreted in a fully classical Hamiltonian scheme. As a prominent feature, this Hamiltonian couples polarization and spatial motion. We apply it successfully to the Rosenthal-St\"uckelberg oscillations. We analyze one-dimensional diffusion similar to a molecular reaction, it appears to exhibit chaotic behavior.