Algebraic Exact Solution for Driven Landau Levels in Two-dimensional Electron Gases

TL;DR

The paper develops a gauge- and representation-independent, exact algebraic solution for driven Landau levels in a 2DEG under arbitrary time-dependent fields by a time-dependent displacement-operator transformation that maps to a displaced, solvable frame. The central result is |N(t)> = D(α_t^{(0)}) e^{-i ω_c (a†a+1/2) t} |N> e^{-i ∫ Δ_t^{(0)} dt/ħ}, with α_t^{(0)} governed by iħ dot α_t^{(0)} - ħ ω_c α_t^{(0)} = - z_t, enabling Floquet analysis away from resonance and revealing resonant delocalization when Ω = ω_c. The study also derives the instantaneous energy-absorption rate P(t), showing quantum interference effects that distinguish coherent from Fock/thermal initial states, and demonstrates how different driving polarizations control spectral properties in resonance. Overall, the framework offers a versatile tool for exploring driven quantum Hall systems and can be extended to include interactions, disorder, and environmental coupling, with implications for non-equilibrium phenomena and Floquet engineering in 2DEGs.

Abstract

Controlling quantum systems with time-dependent fields opens avenues for engineering novel states of matter and exploring non-equilibrium phenomena. Landau levels in two-dimensional electron gases (2DEGs), with their discrete energy spectrum and characteristic cyclotron dynamics, provide an important platform for realizing and studying such driven quantum systems. While exact solutions for driven Landau levels exist, they have been limited to specific gauges or representations. In this work, we present an algebraic, gauge- and representation-independent exact solution for driven Landau levels in 2DEGs subject to arbitrary time-dependent electromagnetic fields. Our approach, based on a time-dependent unitary transformation via the displacement operator, provides clear physical insights into the driven quantum dynamics. We apply this method to derive the exact Floquet states and quasienergies for periodically driven Landau levels, and we extend our analysis to the resonant driving regime, where the Floquet picture breaks down and the electron wavefunction exhibits unbounded spatial spreading. Furthermore, we calculate the instantaneous energy absorption rate, revealing distinct absorption behaviors between coherent states and Fock or thermal states, stemming from quantum interference effects.