Probing the nonlocality of Landau levels in GaAs quantum wells through modified Purcell factors, Lamb shifts and dipole emitted spectra

Technical Summary: Probing the Nonlocality of Landau Levels in GaAs Quantum Wells

Problem Statement
In two-dimensional electron gases (2DEGs) subjected to strong perpendicular magnetic fields, electrons occupy quantized cyclotron orbits known as Landau levels (LLs). While these levels are often treated using local dielectric models where the polarization response $P(r)$ depends only on the local electric field $E(r)$, this approximation fails when the electric field varies significantly over the spatial extent of the cyclotron orbits. The radius of these orbits (the Larmor radius, $r_c$) can span tens of nanometers for fields of a few Tesla. In state-of-the-art nanostructured terahertz (THz) architectures, where electric field variations occur on sub-micrometer scales, the intrinsic nonlocality of the LL response becomes significant. Existing theories often rely on local approximations or treat nonlocality only in the context of polaritonic resonator geometries, lacking a microscopic description of the intrinsic single-particle electronic nonlocality required to interpret recent experimental observations of spectral features near $1.6\omega_c$.

Methodology
The authors develop a microscopic theory of the nonlocal susceptibility for a Landau-quantized 2DEG in a GaAs quantum well.

• Hamiltonian and Wavefunctions: The system is modeled using a non-interacting 2DEG Hamiltonian with a perpendicular magnetic field in the symmetric gauge. The wavefunctions separate into in-plane Landau states and out-of-plane confinement envelopes. Correlated phases (e.g., fractional quantum Hall effects) and spin degrees of freedom are excluded to focus on single-particle nonlocality.
• Nonlocal Susceptibility: The authors derive a fully retarded, nonlocal surface susceptibility $\chi(q, \omega)$ that connects the surface dipole density to the electric field. This susceptibility is a tensor containing longitudinal ($L$), transverse ($T$), and Hall ($H$) components, dependent on the in-plane wavevector $q$. Unlike local models, this formulation accounts for spatial dispersion where $q \neq 0$.
• Green's Function (GF) Formalism: Using the derived susceptibility, the authors construct a self-consistent, fully retarded Green's function (GF) for a layered environment containing the 2DEG. This GF is calculated beyond the Markov and rotating-wave approximations.
• Observables: The GF is used to calculate three key electromagnetic observables for a proximal point dipole emitter:
  1. Purcell Factor ($F_P$): The enhancement of spontaneous emission.
  2. Lamb Shift ($\Delta\omega$): The environment-induced shift in the emitter's resonance frequency.
  3. Emission Spectra ($S$): The frequency distribution of emitted light detected in the far-field.

Key Contributions and Results
The study demonstrates that nonlocal effects fundamentally alter the electromagnetic response of the 2DEG, particularly in the near-field regime (distances up to several hundred nanometers).

1. Brightening of Dipole-Forbidden Transitions:

   • In a local model, a spatially homogeneous field drives only dipole-allowed transitions ($\ell \to \ell+1$ at $\omega_c$).
   • The nonlocal theory reveals that the evanescent near-field of a dipole, containing large in-plane wavevectors, relaxes selection rules. This activates "dipole-forbidden" transitions at multiples of the cyclotron frequency ($2\omega_c, 3\omega_c$).
   • Result: For emitter-2DEG separations up to several hundred nanometers, the Purcell factor for the $2\omega_c$ transition in the nonlocal model exceeds local predictions by up to two orders of magnitude.

2. Modification of the Dipole-Allowed Transition:

   • Even for the primary dipole-allowed transition ($\omega_c$), nonlocal corrections become significant at distances comparable to the Larmor radius of the highest occupied Landau level ($r_c(\ell_F)$).
   • For typical experimental parameters ($B = 1-5$ T), this corresponds to distances of approximately 25–140 nm.

3. Sign Reversal of the Lamb Shift:

   • The environment-induced Lamb shift at the $2\omega_c$ frequency exhibits a qualitative difference between local and nonlocal models.
   • Result: While the local model predicts a shift based on the off-resonant tail of the $\omega_c$ transition, the nonlocal model predicts a resonant coupling channel at $2\omega_c$, causing the Lamb shift to change sign (e.g., from positive to negative) compared to local predictions.

4. Collective Magnetoplasmon Resonance:

   • The self-consistent treatment, which includes electron-electron interactions via the screened susceptibility denominator, reveals a collective magnetoplasmon resonance near $1.5-1.6\omega_c$.
   • Result: This feature is absent in local LL approximations but appears in the nonlocal spectra for small emitter distances. The authors note this is consistent with recent experimental observations in subwavelength THz cavities that were previously unexplained by local theories.

Significance
The paper establishes that a local material model is inadequate for describing electromagnetic interactions in Landau-quantized 2DEGs when the electric field varies over the spatial extent of the cyclotron orbits. This length scale (tens to hundreds of nanometers) is typical of modern nanostructured THz architectures and is significantly larger than the few-nanometer scales where plasmonic nonlocality usually becomes relevant in visible-range systems.

The authors claim that their microscopic theory provides a firm footing for interpreting recent experiments involving Landau level polaritons and ultrastrong light-matter coupling. Specifically, the theory explains:

• The emergence of spectral features at multiples of $\omega_c$ that are locally forbidden.
• The existence of collective resonances near $1.6\omega_c$ in ultrastrongly coupled cavities.
• The necessity of nonlocal descriptions for accurate predictions of Purcell factors and Lamb shifts in the THz regime, particularly in the ultrastrong-coupling limit where off-resonant transitions cannot be neglected.

The work suggests that intrinsic electronic nonlocality is a dominant factor in the design and analysis of next-generation THz quantum devices, distinct from and more extensive than previously understood plasmonic nonlocal effects.