All-optical spin-orbit coupling of light using electromagnetically induced transparency

Analysis indicates that tunable spin-orbit coupling (SOC) can be introduced into the paraxial Schrödinger-type equation for a spinor image traveling through a rotating electromagnetically induced transparency (EIT) medium. As a fundamental example, we show that, due to a radial gradient of large group refractive index, the SOC can lead to spatial separation of the oppositely oriented spinor states in the probe image, mimicking the fine-structure splitting of a spin-$\frac{1}{2}$ quantum harmonic oscillator described by the Pauli equation. Such a result enables a direct manipulation and visualization of spin-orbit physics by all-optical means based on EIT slow-light media. Possible experimental implementations of our model with ultracold atoms are discussed.

Figure 1

(a) Experimental schematic to show the SOC of light via EIT. The spinor image strikes the medium of a thickness $d$ at its waist in the input plane, then propagates a distance $D$ to the detection plane. The medium can rotate at a frequency $-\mathrm{\Omega }$ around the $z$ axis. The strong coupling field has an inhomogeneous profile, e.g., a Gaussian beam. (b) A typical $\mathrm{\Lambda }$-level EIT system without rotation, where $|1\rangle$ and $|2\rangle$ are the ground states and $|3\rangle$ is the excited state. (c) Upon rotation, the medium interacts with the two LG modes acquiring different geometric phases in the rotating frame, particularly related to the opposite rotational Doppler shifts $\pm \ell \mathrm{\Omega }$.

Figure 2

(a) Rotation-mediated all-optical SOC splitting of the spinor image at different propagation distances, where the SOC can give rise to spatial separation for the spin $|{+}_z\rangle$ (solid) and $|{-}_z\rangle$ (dashed) states. The inset is a closeup showing the small splitting at $D=5z_R$. As an example, the resulting 2D intensity profile of the spinor image at $D=5z_R$ is presented in (b), where the equal-intensity contours are superimposed. The outer (inner) white circle corresponds to the peak position of the spin $|{+}_z\rangle$ ($|{-}_z\rangle$) wave function in (a). As seen, beyond the rotary photon drag, the small SOC splitting can break the mirror symmetry of the 2D image and thus lead to a visible azimuthal distortion for the lobes in contrast to the nonrotating case shown in (c), i.e., screw-propeller shaped in (b) vs petal shaped in (c).

Figure 3

Proposed experimental schematic using rotating ultracold ${}^{87}\mathrm{Rb}$ atoms in a MI phase. The four 844-nm beams can be rotated by modulating the dual-axis acousto-optic deflectors and intersected near the focus of an objective lens to generate a rotating 2D lattice in the $x\text{−}y$ plane [39, 40]. The lattice in the $z$ direction can be created by retroreflecting the 765-nm laser beam on a coated window. Due to the cylindrical symmetry of the setup, the probe and coupling beams can copropagate along the $z$ axis. Note that, for simplicity, we here only show a basic configuration. In practice, other custom-made devices are also needed to guarantee the functionality, such as the vacuum chamber, the magneto-optical trap, other lenses in the objective, etc. To measure the weak probe image, high-sensitive photon detector (PD) (e.g., a scanning optical fiber with an avalanche photodiode or an electron multiplying CCD [14, 38]) can be used in the imaging system.

Figure 4

(a) Rotation-mediated all-optical SOC splitting for the spin $|{+}_z\rangle$ (solid) and $|{-}_z\rangle$ (dashed) states at different propagation distances, where $n_a\approx {10}^{12}{\mathrm{cm}}^{-3}$. The inset is a closeup showing the small splitting at $D=5z_R$. As an example, the intensity profile superimposed by the equal-intensity contours at $D=5z_R$ are shown in (b). The outer (inner) white circle corresponds to the peak position of the spin $|{+}_z\rangle$ ($|{-}_z\rangle$) wave function in (a). As seen again, the very small SOC splitting can break the mirror symmetry of the spinor image in contrast to the nonrotating case in (c). All the results are consistent with those in Fig. 2 in the main text.

Figure 5

The characteristic parameter $\varsigma$ can be used to estimate the Kerr nonlinearity in the three-level $\mathrm{\Lambda }$-type EIT system. Both curves depict the variation of $\varsigma$ in a radial direction through one of the intensity maxima in the probe image (e.g., at $\varphi =0$). Namely, the peak values in the curves correspond to the maximal Kerr nonlinearity, where curves I and II depend on the EIT parameters used in Figs. 2 and 4, respectively. As seen, we have $\varsigma \ll 1$ in the entire transverse range. This means that, compared with the linear susceptibility, the nonlinearity associated with the spatial variation of the light fields is negligible in our present scheme.