Geometrical phase and Hall effect associated with the transverse spin of light

Abstract

By analyzing the vectorial Helmholtz equation within the thin-layer approach, we find that light acquires a novel geometrical phase, in addition to the usual one (the optical Berry phase), during the propagation along a curved path. Unlike the optical Berry phase, the novel geometrical phase is induced by the transverse spin along the binormal direction and associated with the curvature of the curve. Furthermore, we show a novel Hall effect of light induced by the torsion of the curve and associated with the transverse spin along the binormal direction, which is different from the usual spin Hall effect of light. Finally, we demonstrate that the usual and novel geometrical phase phenomena are described by different geometry-induced U(1) gauge fields in different adiabatic approximations. In the nonadiabatic case, these gauge fields are united in one effective equation by SO(3) group.

Geometry-induced Monopole Magnetic Field and Quantum Spin Hall Effect

“…This equation is key in the present paper, which condenses the initial spirit of the thin-layer quantization formalism [4]. Interestingly, it is can be extended to vector fields, such as electromagnetic field [13].…”

“…The scalar geometric potential has been proved that can construct a topological band structure for periodically minimal surfaces [8], can generate bound states for spirally rolled-up nanotubes [9], can eliminate the reflection for bent waveguides [10], can provide the transmission gaps for periodically corrugated thin layers [11,12] and so on. The geometric momentum and the geometric angular momentum can additionally contribute [13] and modify the spin-orbit coupling [6,14,15]. As empirical evidences, the scalar geometric potential has been realized by an optical analogue in a topological crystal [16], and the geometric momentum was observed to affect the propagation of surface plasmon polaritons on metallic wires [17].…”

Effective dynamics for a spin-1/2 particle constrained to a curved layer with inhomogeneous thickness

“…[13] The geometric potential is attractive and leads to the formation of bound states, opening up new possibilities for constructing quantum dots [14,15] and quantum waveguides. [16][17][18] Inspired by these applications, numerous researchers have extended this method to other scenarios, including a charge particle in electric and magnetic fields, [19][20][21] a Dirac particle, [22][23][24] a spin-1/2 particle, [25][26][27][28][29][30] higher-dimensional induced gauge potential, [31][32][33][34][35] quantum scattering, [36] photons, [37][38][39] magnetism, [40,41] and quantum many-body systems. [42,43] It has been revealed that the effective dynamics exhibit additional geometric effects associated with the internal degrees of freedom and properties of the confined particle.…”

Quantum mechanics of fermion confined to a curved surface in Foldy-Wouthuysen representation

“…Specifically, the curvature effects were widely investigated in thin magnetic shells [5,6], nematic shells [7], titania single crystals [8], smectic liquid crystals [9], quantum spin Hall system [10], photonic crystal fiber [11], domain wall pinning [12], domain wall motion [13], antiferromagnets [14] and so on. For two-dimensional (2D) curved surfaces, the thin-layer quantization scheme [15][16][17] is an effective and suitable method, which has been successfully employed to deduce effective Maxwell's equation [18][19][20], effective Schrödinger equation [21], effective Pauli equation [22][23][24] and effective Dirac equation [25][26][27][28][29][30][31] for electromagnetic field and electrons confined to a 2D curved surface. There are two important results, geometric potential [15,16] and geometric momentum [32,33], which have been proved experimentally in topological crystal [34] and plasmon polarization [35], respectively.…”