# Controllable spin-Hall and related effects of light in an atomic medium via coupling fields

## TL;DR

This paper demonstrates the controllable spin-Hall effect of light in an atomic medium with tunable birefringence, enabling precise measurement of light shifts at the quantum noise limit using balanced homodyne detection.

## Contribution

It introduces a method to induce and control SHEL in atomic media via electromagnetically induced transparency and proposes a quantum-limited measurement scheme.

## Key findings

- Spin-Hall effect of light observed with transverse shifts of about wavelength.
- Shifts can be increased by raising atomic density.
- Measurement scheme can reach nanometer precision at quantum noise limit.

## Abstract

We show the existence of spin-Hall effect of light (SHEL) in an atomic medium which is made anisotropic via electromagnetically induced transparency. The medium is made birefringent by applying an additional linearly polarized coupling light beam. The refractive index and the orientation of the optics axis are controlled by the coupling beam. We show that after transmitting the atomic medium, a linearly polarized probe light beam splits into its two spin components by opposite transverse shifts. With proper choice of parameters and atomic density of about $2.5\times10^{17}\,\mathrm{m}^{-3}$, the shifts are about the order of wavelength and can be larger than the wavelength by increasing the atomic density. We propose a novel measurement scheme based on a balanced homodyne detection (BHD). By properly choosing the polarization, phase, and transverse mode of the local oscillator of the BHD, one can independently measure (i) the SHEL shifts of the two spin components; (ii) the spatial and angular shifts; (iii) the transverse and longitudinal shifts. The measurement can reach the quantum limit of precision by detecting signals at the modulation frequency of the electro-optic modulator used to modulate the input probe beam. The precision is estimated to be at the nanometer level limited by the quantum noise.