Spin-Orbit Coupling for Photons and Polaritons in Microstructures

We use coupled micropillars etched out of a semiconductor microcavity to engineer a spin-orbit Hamiltonian for photons and polaritons in a microstructure. The coupling between the spin and orbital momentum arises from the polarization-dependent confinement and tunneling of photons between adjacent micropillars arranged in the form of a hexagonal photonic molecule. It results in polariton eigenstates with distinct polarization patterns, which are revealed in photoluminescence experiments in the regime of polariton condensation. Thanks to the strong polariton nonlinearities, our system provides a photonic workbench for the quantum simulation of the interplay between interactions and spin-orbit effects, particularly when extended to two-dimensional lattices.

Popular Summary

One of the most fundamental properties of electromagnetism and special relativity is the coupling between the spin of an electron and its orbital motion. Spin-orbit coupling explains fine structure in atoms and the spin Hall effect in semiconductors, and it underlies many intriguing properties of topological insulators, in particular, their chiral edge states. An outstanding question is if it is possible to synthesize spin-orbit coupling in particles without a magnetic moment, such as photons. We use coupled micropillars etched out of a semiconductor microcavity to engineer a spin-orbit Hamiltonian for photons (whose polarization degree of freedom acts as its spin) and polaritons in a microstructure.

We design a microstructure made of six micropillars, each $3\mu \mathrm{m}$ in diameter. In our structures, the coupling between the spin and orbital momentum arises from the polarization-dependent confinement and tunneling of photons between micropillars arranged in the form of a hexagonal photonic molecule, since the micropillars overlap slightly. We conduct photoluminescence experiments at 10 K to observe the wave function of polariton condensates. The engineered spin-orbit coupling results in the helical shape of the wave function, directly visible in the spatially resolved polarization patterns of the emitted light. We find that spin-orbit coupling can be affected by changing the shape of the micropillars.

The strong optical nonlinearity of polariton systems that we observe suggests exciting perspectives for using quantum fluids of polaritons for quantum simulations of the interplay between interactions and spin-orbit coupling.

Figure 1

Photonic molecular states. (a) Scanning electron microscope image of the polaritonic molecule. (b) Polarization-insensitive emission spectrum in momentum space at $k_x=0.2\mu \mathrm{m}$ and low excitation power (7 mW), showing the four molecular states $E(0),\dots ,E(3)$ arising from the coupling of the lowest energy mode of each single micropillar. (c) Scheme of the phase winding of the eigenfunctions of the $E(l)$ states without accounting for the spin. (d) Measured polarization-insensitive real-space emission. (e) Interference pattern and (f) extracted phase for each of the individual emission lines. The solid arrow points the pillar used for the phase reference in the interference experiment. (g) Measured polarization-insensitive momentum-space emission. (h) Simulated momentum-space emission obtained by Fourier transforming the tight-binding model eigenfunctions with a Gaussian distribution over each lobe.

Figure 2

Spin-orbit split states. (a) Scheme of two coupled pillars showing different tunneling probabilities for photons polarized longitudinal (light blue) and transverse (pink) to the link. (b) This results in polarization splitting of the bonding and antibonding states. (c) Scheme of the unit vectors longitudinal and transverse to each link, and radial and azimuthal on each site, used to calculate the SO-coupled eigenstates of Hamiltonian (3). (d) Polarization-dependent classification of the energy states in the absence of SO coupling ($t_L=t_T$ and $\mathrm{\Delta }E=0$) as a function of their total momentum $k=l+\sigma$. States located on each horizontal line have a well-defined orbital momentum $l$ and form degenerate multiplets. (e) Same as (d) for $t_L\gtrsim t_T$. The SO coupling is evidenced in the anticrossing of the bands, at $k=0$ and $k=3$. (f)–(g) Fine structure and polarization pattern of the molecular eigenstates corresponding to the multiplets $l=1$ (f) and $l=2$ (g), split by the SO coupling for $\hslash \mathrm{\Delta }t>\mathrm{\Delta }E$. Note that $\epsilon ,{\epsilon }^{\prime }\ll 1$ (see Sec. I of Ref. [32]). The salmon color shows the states in which condensation takes place in Figs. 4 and 5. The insets in (f) and (g) show the relative energy of the top and bottom states of each multiplet when $\mathrm{\Delta }E>0$.

Figure 3

Power dependence of the condensation process. (a)–(f) Energy- and momentum-resolved emission of the molecule as a function of the excitation power, indicated in each panel. At low power (a), the $E(0),\dots ,E(3)$ energy levels are visible along with higher energy states (above the dashed line) arising from the coupling of higher energy modes of each individual pillar [32]. When increasing the excitation power, polariton condensation occurs first in one of the states of the multiplet $E(2)$ (c) and then in one of the states of the multiplet $E(1)$ (e,f). In (d), the double line in $E(2)$ arises from fluctuations in the position of the pump spot during the measurement (see Sec. E of Ref. [32]). These fluctuations were absent in the experiments shown in Figs. 4 and 5. (g) Emitted intensity of the $E(1)$ and $E(2)$ states as a function of the excitation power. The circles indicate the power at which the experiments shown in Figs. 4 and 5 were performed.

Figure 4

Condensation in the $k=0$ state of the $E(1)$ manifold. (a,b) Real space emission in the ${\sigma }_{+}$ and ${\sigma }_{-}$ circular polarizations for a polariton condensate in the state $|{\psi }_{\text{lower}}(k=0)⟩$, at a pumping intensity of 84 mW. Each polarization component contains a vortical current in the clockwise (${\sigma }_{+}$) and counterclockwise (${\sigma }_{-}$) directions, evidenced in the forklike dislocations apparent in the interferometric measurements shown in (c) and (d), respectively, and in the extracted phase gradients in (e) and (f). The phase changes from 0 to $\mp 2\pi$ when circumventing the molecule. (g) The green traces show the plane of linear polarization of the emission measured locally, superimposed to the total emitted intensity of the molecule. The condensate shows azimuthal linear polarization. Solid lines depict the contour of the microstructure. (h) Spatially resolved emission spectrum.

Figure 5

Condensation in the $k=3$ state of the $E(2)$ manifold. (a,b) Real space emission in the ${\sigma }_{+}$ and ${\sigma }_{-}$ circular polarizations for a polariton condensate in the state $|{\psi }_{\text{lower}}(k=3)⟩$, at 25 mW. Each polarization component contains a vortical current of double phase charge in the counterclockwise (${\sigma }_{+}$) and clockwise (${\sigma }_{-}$) directions, evidenced in the double forklike dislocations apparent in the interferometric measurements shown in (c) and (d), respectively, and in the extracted phase gradients in (e) and (f). The phase changes from 0 to $\pm 4\pi$ when circumventing the molecule. (g) The green traces show the plane of linear polarization of the emission measured locally, superimposed to the total emitted intensity of the molecule. At the center of each micropillar, the condensate shows a linear polarization pattern pointing radially. Solid lines depict the contour of the microstructure. (h) Spatially resolved emission spectrum.