Measuring the Optical Concurrence of Vector Beams with an Atomic-State Interferometer

We investigate the transmission of vector beams, correlated in their polarization and spatial degrees of freedom, through cold atoms in the presence of a transverse magnetic coupling field. The resulting phase-dependent dynamics allow us to imprint the spatially varying polarization of a vector beam onto atomic spin polarizations, thereby establishing a direct link between optical space-polarization correlations and atomic-state interference. We find that the resulting absorption profiles show interference fringes whose modulation strength is given by the squared concurrence of the vector beam, letting us identify optical concurrence from a single absorption image. We expect impact across a diverse range of applications, including spintronics, quantum memories, metrology, and clocks.

Figure 1

(a) Schematic illustration of the HOPS with ${\mathit{\sigma }}_{\pm }{e}^{\mp i2\phi }$ forming the poles and simulated vector beams located along the red line with $2\psi =\pi$, and $2\chi =0$, $\pi /10$, $7\pi /30$, $11\pi /30$, and $\pi /2$. (b) Corresponding experimentally generated vector beams. The inset on the top left shows our color encoding for the different polarization states.

Figure 2

Simplified experimental setup. Optical elements for vector beam generation and analysis: HWP, half wave plate; QWP, quarter wave plate; QP, q plate; L, lens; B, magnetic field; WP, Wollaston prism; CCD, charge-coupled device. The bottom left inset shows the atomic level scheme indicating the $\mathrm{\Lambda }$ transition driven by the vector beam and transverse magnetic coupling.

Figure 3

(a) Measured and predicted peak-normalized OD images ($0.8\times 0.8{\mathrm{mm}}^2$) for selected beams with increasing concurrence [see Fig. 1], with high (low) OD corresponding to beam areas of large (small) absorption. (b) The equivalence between the squared concurrence of the vector beam and the modulation depth $\mathcal{M}$ of the atomic absorption profile as a function of $2\chi$. The black line shows $\mathcal{M}$ according to the PT model, which is identical to $C^2$ of the target vector beams. The measured $C^2$ values are shown in blue, and the measured $\mathcal{M}$ in red, with error bars representing the standard deviation of five runs. The dashed red line indicates $\mathcal{M}$ according to the OB model.