Quantized photonic spin Hall effect in graphene

We examine the photonic spin Hall effect (SHE) in a graphene-substrate system with the presence of an external magnetic field. In the quantum Hall regime, we demonstrate that the in-plane and transverse spin-dependent splittings in the photonic SHE exhibit different quantized behaviors. The quantized SHE can be described as a consequence of a quantized geometric phase (Berry phase), which corresponds to the quantized spin-orbit interaction. Furthermore, an experimental scheme based on quantum weak value amplification is proposed to detect the quantized SHE in the terahertz frequency regime. By incorporating the quantum weak measurement techniques, the quantized photonic SHE holds great promise for detecting quantized Hall conductivity and the Berry phase. These results may bridge the gap between the electronic SHE and photonic SHE in graphene.

Figure 1

Schematic illustrating the wave reflection at a graphene-substrate interface in a Cartesian coordinate system. The graphene sheet is placed on the top of a homogeneous and isotropic substrate. An imposed static magnetic field $\mathbf{B}$ is applied perpendicular to the interface of the graphene-substrate system. On the reflecting surface, the photonic SHE occurs which manifests itself as in-plane and transverse splittings of two spin components.

Figure 2

Quantized (a) in-plane and (b) transverse spatial shifts for the graphene-substrate system as a function of the magnetic field and Fermi energy. We assume the incident wave packet has $|\mathrm{H}\rangle$ polarization, incident angle ${\theta }_i={60}^{\circ }$, and $\omega /2\pi =1\mathrm{THz}$. The refractive index of undoped Si in the terahertz range is $n_{Si}=3.415$. We choose a carrier density of $4\times {10}^{16}{\mathrm{cm}}^{-3}$ and a mobility of $1500{\mathrm{cm}}^2/\mathrm{Vs}$. The temperature is chosen as $T=4\mathrm{K}$. These parameters are the same as in Ref. [30].

Figure 3

Quantized (a) in-plane and (b) transverse angular shifts in the photonic SHE as a function of the magnetic field and Fermi energy. The beam waist is chosen as $w_0=1\mathrm{mm}$. Other parameters are the same as in Fig. 2.

Figure 4

Representation of the preselection and postselection states for $|{\psi }_i\rangle$ and $|{\psi }_f\rangle$ on the Poincaré sphere. A middle state $|{\psi }_m\rangle$ is also introduced due to the Kerr rotation. The quantum state is chosen as $\mathrm{\Theta }=\pi /2$. For $|H\rangle$ input polarization $\mathrm{\Phi }=0$, while for $|V\rangle$ input polarization $\mathrm{\Phi }=\pi$. The angle $\mathrm{\Delta }$ gives an origin to the real and imaginary parts of the weak value.