Nonlinear optical behavior of confined electrons under torsion and magnetic fields

TL;DR

This work analyzes a nonrelativistic electron confined by a geometry-induced potential from uniform torsion in a cylindrically symmetric medium, under a perpendicular magnetic field and an Aharonov–Bohm flux. By deriving an exact radial equation in a torsion-bearing metric and solving it with confluent hypergeometric functions, the authors obtain closed-form eigenstates and energies that reveal how torsion $\tau$, AB phase $l$, and magnetic quantization $\omega_c$ shape linear and third-order nonlinear optical responses as well as the photoionization cross-section. Torsion consistently blueshifts resonances and reduces dipole overlaps, enabling tunable optical switching at high intensity; AB flux lifts degeneracies and introduces asymmetries between $\Delta m=\pm1$ channels, offering a topological knob for optical control. The results provide a coherent framework for geometric/topological engineering of light–matter interactions in mesoscopic rings and dots and suggest experimental routes to isolate torsion effects via $k_z$-dependent signatures.

Abstract

In this work, we investigate the influence of torsion, Aharonov-Bohm flux, and external magnetic fields on the linear and nonlinear optical properties of a confined quantum system. The confinement potential is not assumed a priori, but emerges as a radial effective potential, analogous to a quantum dot, geometrically induced by the torsion of the material. Starting from an effective radial equation derived in a nontrivial geometric background, we analytically solve for the energy spectrum and wave functions. These solutions are then employed to evaluate the optical absorption coefficients and refractive index changes, including both linear and third-order nonlinear contributions. The formalism incorporates the electric dipole approximation and accounts for intensity-dependent effects such as saturation and spectral shifts. Our results reveal that torsion and topological parameters significantly modify the optical response, leading to tunable resonances and nontrivial dispersive behavior. This work highlights the potential of geometric and topological engineering in low-dimensional systems to control and enhance nonlinear optical phenomena.

Figures (11)

• Figure 1: Schematic illustration of the effect of torsion density ($\tau$) on the geometry of the system. (a) The case without torsion ($\tau=0$) corresponds to a two-dimensional flat disk representing Euclidean space. (b) A finite torsion ($\tau>0$) introduces a helical distortion in the surface, resulting in a non-Euclidean geometry. In both configurations, the system is subjected to a uniform magnetic field $\boldsymbol{B}$, applied perpendicular to the plane (indicated by the red arrows), and an Aharonov--Bohm flux $l$, confined along the central axis.
• Figure 2: Radial effective potential $V_{\mathrm{eff}}(\rho)$ for $m = +1$ and different values of the torsion parameter $\tau$. Colored curves correspond to the case without a magnetic field ($B = 0$), whereas the semi-transparent black dashed curve represents $B = 5\,\mathrm{T}$, illustrating the influence of an external magnetic field on the effective potential.
• Figure 3: Energy eigenvalues for the ground state ($n=0$, red) and the first radially excited state ($n=1$, blue) as a function of the torsion density $\tau$. The calculations are performed for a fixed azimuthal quantum number $m=0$, Aharonov--Bohm flux $l=0.1$ ($h/e$), and magnetic field $B=5$ T. The plot shows that the energy of both levels increases with torsion. The energy gap between the two states also widens, indicating a stronger torsional influence on the excited state. The linear increase in energy demonstrates the role of torsion as an effective confining potential.
• Figure 4: Energy eigenvalues as a function of the magnetic quantum number $m$ ($m=-2,-1,0,1,2$), for three different values of torsion density: (a) $\tau = 0$, (b) $\tau = 5.0 \times 10^6\,\mathrm{m}^{-1}$ and (c) $\tau = 15.0 \times 10^6\,\mathrm{m}^{-1}$. It can be seen that the energy values increase with increasing $\tau$.
• Figure 5: Normalized radial probability density for the $n=0$ (solid lines) and $n=1$ (dashed lines) states, calculated for a fixed flux $l=0.1$ ($h/e$) and magnetic field $B=5$ T. The colors correspond to different values of the torsion density $\tau$. All curves have been normalized by the peak value of the torsion-free ground state ($n=0$, $\tau=0$, solid black line). The plot highlights the suppression of the probability density due to both radial excitation (lower peaks for $n=1$) and increasing torsion (lower peaks for colored curves).