Old experiments in new light: Young's double-slit and Stern-Gerlach experiments in liquid crystal microcavities

Spin-orbit interactions which couple spin of a particle with its momentum degrees of freedom lie at the center of spintronic applications. Of special interest in semiconductor physics are Rashba and Dresselhaus spin-orbit coupling (SOC). When equal in strength, the Rashba and Dresselhaus fields result in SU(2) spin rotation symmetry and emergence of the persistent spin helix (PSH) only investigated for charge carriers in semiconductor quantum wells. Recently, a synthetic Rashba-Dresselhaus Hamiltonian was shown to describe cavity photons confined in a microcavity filled with optically anisotropic liquid crystal. In this work, we present a purely optical realisation of two types of spin patterns corresponding to PSH and the Stern-Gerlach experiment in such a cavity. We show how the symmetry of the Hamiltonian results in spatial oscillations of the spin orientation of photons travelling in the plane of the cavity.

polarization into resonance in this highly anisotropic cavity, their mixing becomes described by an effective equal Rasha-Dresselhaus SOC, resulting in a dispersion with strong valley polarization 17 . Our experimental observations and analytical calculations demonstrate that the strong polarization-valley coupling in this simple system directly results in the appearance of long-range polarization [or (pseudo)spin] textures of the in-plane traveling photons with potential application for valley-optronic devices [18][19][20] alongside gapped Dirac materials.
We use a microcavity filled with a liquid crystalline medium shown schematically in Fig. 1a. The cavity is based on two SiO 2 /TiO 2 distributed Bragg reflectors, with maximum reflectance at 1.65 eV (750 nm). Approximately 3.5 µm space between the DBRs is filled with a nematic liquid crystal of high birefringence (∆n = 0.41 21,22 ), which acts as an optically uniaxial medium inside a multimode cavity. By tuning an external voltage applied to transparent ITO electrodes, we can control the anisotropy direction in x-z plane (Fig. 1b), which changes the effective refractive index, thus cavity mode energy for light polarised in x direction, whereas modes of perpendicular polarization are unaffected 23 . If the refractive indices for perpendicular linear polarizations are different, then degeneration of two modes with different numbers is possible. When two modes of opposite parities are degenerate they couple via Rashba-Dresselhaus SOC term and the dispersion can be described by an effective Hamiltonian written in the photon circular polarization basis (i.e., spin-up and spin-down states) 17 Fig. 1c.
Interestingly, one can consider the equal Rashba-Dresselhaus SOC system as a spin realisation of Young's double slit experiment in a reciprocal space. The role of the two slits discriminating the position of an incident scalar plane wave in the real space-resulting in the well known double-slit spatial interference pattern in the far field-is instead played by the two cavity dispersion spin valleys in reciprocal space discriminating the momenta of light through an optical mode (Fig. 1d). In the double-slit experiment a plane wave passing through two slits (separated by distance d) produces spherical waves that interfere, giving an image of the intensity oscillating in space with a period proportional to 1/d (Fig. 2a).
Analogously, for a homogeneous occupation of the two isoenergy spin circles in the reciprocal space, separated by the vector Q, we would obtain in real space a polarization interference image producing the persistent spin helix with a period proportional to L = 2π/Q where Q = | Q|. In other words, in the Young double slit experiment, one obtains in the far field the Fourier transform of the two slits, which is a periodic interference pattern. Here, we obtain in the near field the Fourier transform of the the two polarised circles in momentum space, which is a polarization interference pattern. Based on the fitted dispersion in Fig. 1c this leads to a helix period of L = 3.8 µm.
This change of the polarization with propagation of photons in the plane of cavity is observed experimentally in our system. An incident linearly antidiagonally polarised laser beam is tightly focused with a microscope objective on the sample, providing homogeneous occupation of photons on both spin circles in reciprocal space. The laser energy is set resonant with the cavity modes at normal incidence as marked by dashed horizontal line in Fig. 1c. Transmitted light is collected by another microscope objective, polarization resolved and imaged on a CCD camera. This allows us to map out the spatial distributions of the Stokes parameters, corresponding to intensities of horizontal (I X ), vertical (I Y ), diagonal (I d ), antidiagonal (I a ), right-hand circular (I σ + ) and left-hand circular (I σ − ) polarised light.
The measured S 1 and S 2 parameters are plotted in Fig. 2c,f, which clearly show periodic oscillations with π/2 phase shift between the two Stokes parameters. Spatial period of the oscillations is estimated as L = 4.7 µm. The phase of the PSH depends on the polarization of incident light, as described in the Supplemental Material 26 .
As mentioned above, this result can be understood as a consequence of an interference process between spins in different momentum valleys (i.e., valley polarization). Assuming that the cavity extends infinitely in the x-y plane with the two almost perfect mirror planes separated by distance L, we can represent the modal electric fields inside the cavity, cor- of the Hamiltonian (1) by the plane waves: Here n that incident electromagnetic waves of energy ε and momentum k will excite modes ε ± ( k). A linearly polarised incident plane wave with the polarization angle Θ = 1 2 tan −1 (S 2 /S 1 ) with respect to the x−axis will excite either e +iΘ Ψ + or e −iΘ Ψ − waves inside the cavity, provided k belongs to the red or blue circle in Fig. 1d. Therefore the transmitted light will be either right-hand or left-hand circularly polarised. If, however, the surface of the microcavity is illuminated with a focused coherent beam at normal incidence then the entire isoenergy momentum spin circles are excited and a specific polarization pattern determined by Θ, as a manifestation of the optical PSH, will emerge. Let F denote the distance between the focus of the incident beam and the illuminated surface. The electric field at this surface is a combination of plane waves (up to a common factor): where R = F/2k 0 and k 0 is equal to the light wavenumber in the vacuum. Taking into account that each of those plane waves couples either to the field e +iΘ Ψ + or e −iΘ Ψ − in-side the cavity and performing the integral with respect to k one arrives at the final expression for the electric field of the transmitted wave at the opposite surface. Denoting where J 0 is the Bessel function of zeroth order, we obtain The analytical results (5) It then becomes apparent that a family of inseparable states exist which satisfy |β + | = |β − | and β + /β − = e 2iΘ . Similar to a Michelson interferometer, Θ is an effective "pathdifference" variable which uniquely determines the location of the S 1,2 ( r) interference minima and maxima in the PSH [see SI and Eq. (5)]. This reflects the fact that for an inseparable state, any effects on one DOF (e.g., spin) will have measurable outcome in the other DOF (e.g., momentum valley) with exciting potential in optical metrology that benefits from parallelised DOF measurements 30 .
In the case of a tightly focused pump, as considered above, one has φ + ( r) φ − ( r) and In the more general case where φ + ( r) = φ − ( r) (causing the interference circles in the S 1,2 in Fig. 2d,g) the amount of inseparability is no longer a global quantity and depends on the spatial coordinate r. In this case the S 3 ( r) Stokes parameter becomes a useful measure on the amount of inseparability in space written as C 2 + S 2 3 = 1, which is an analog of complimentarity proposed by Eberly et al. 32 .
Moreover, scrutinising the S 3 parameter we can demonstrate analogy to an optical Stern-Gerlach experiment 33 using our system. The effective magnetic field of the equal Rasha-Dresselhaus SOC causes a spin-selective deflection of the cavity photons along the two opposite directions in the cavity plane defined by the valleys location ± Q/2 (Fig. 1d). This deflection appears due to the different Bessel solutions for σ + and σ − which are shifted due to the anisotropy of the dispersion. This can be evidenced for the linearly polarised case C = 1. For non-homogeneous occupation of dispersion cavity valleys (Fig. 3b) we observe that S 3 (x, y) = −S 3 (x, −y) as indicated theoretically in Fig. 3c and experimentally in Fig. 3d. Such non-homogeneous occupation of the valleys can be achieved by using a broad normal incident excitation beam which only excites a locality in reciprocal space around k = 0. In the experiment presented in Fig. 3d we used linearly polarised (diagonal) light from broadband halogen lamp transmitted through the sample by the same optical system (see Methods). Our results therefore open a new method to design an optical Stern-Gerlach experiment in the classical optics regime. The notable difference between our setup and the actual Stern-Gerlach is that there is no constant force acting on the photon pseudospins but rather, they obtain constant group velocities in opposite directions in the cavity due to the effective magnetic force gradient (yellow arrows in Fig. 1d). This result should not be confused with valley selective circular dichroism 34 as there are no absorption processes involved here, or the much weaker spin Hall effect of light coming from geometric phases 35 , or circular birefringence in chiral materials like the Fresnel triprism 36 since our system is achiral (i.e., the x-z plane is a plane of symmetry). A future path of investigation can then involve a quantum Stern-Gerlach experiment operating in the single photon regime. Due to the compact design LCMC can be easily integrated with optoelectronics devices, giving the perspective of simulating complex spin systems, studying effects that are difficult to control in condensed matter systems, and developing photonic valleytronic devices in a relatively simple setting at room temperature.

Methods
For transmission measurements laser light from Ti:Sapphire laser with energy 1.705 eV is focused with 100× microscope objective and collected by another objective with 50× magnification. Both objectives have numerical aperture equal to 0.55. Incident light polarization is linearly polarised in antidiagonal direction. Fig. 3d was obtained by polarization-resolved tomography. Linearly polarised (in diagonal direction) light from broadband halogen lamp was transmitted through the sample by the same optical system. Transmitted light was imaged by lens with 400 mm focal length on entrance slit of a monochromator equipped with a CCD camera. Tomography was obtained by motorised movement of the imaging lens in direction perpendicular to the slit. Fig. 3d presents cross section at the energy of Rashba-Deresselhaus resonance.

Data availability
The data that support the plots within this paper and other findings of this study are available from the corresponding authors upon reasonable request.