Laguerre asymptotic formula and stability of Landau levels influenced by an electric field

TL;DR

This paper analyzes how electric fields influence Landau levels in quantum systems, showing that high-energy levels become increasingly unstable and providing explicit transition probability predictions based on Laguerre polynomial asymptotics.

Contribution

It demonstrates that inter-Landau level transition probabilities depend on the electric field's Fourier component and links Laguerre polynomial asymptotics to Landau level stability.

Findings

Transition probability depends on electric field Fourier component

High-energy Landau levels are less stable asymptotically

Laguerre polynomial asymptotics predict transition probabilities

Abstract

Consider the quantum evolution of a charged particle subjected to a uniform magnetic field and an electric field E(t) that exists for a finite period of time. The electric field can induce intra-Landau level transitions (magnetic translations) that do not change the energy of the particle. It may also induce energy changing inter-Landau level transitions. Our purpose in this paper is two-fold: We first demonstrate that the inter-Landau level transition probability is completely determined by the Fourier component of the electric field corresponding to the cyclotron frequency. Then we point out that the Fejer asymptotic form of Laguerre polynomials implies that no matter how small the Fourier component is inter-Landau level transition probability from a fixed Landau level to other energy levels can be arbitrarily close to 1 if the original Landau energy level is high enough, i.e., influenced by a given electric field, Landau levels of higher energy are less stable asymptotically and their transition probabilities are explicitly predicted by the Fejer formula.