Microspheres with Atomic-Scale Tolerances Generate Hyperdegeneracy

Jacob Kher-Alden , Shai Maayani, Leopoldo L. Martin, Mark Douvidzon, Lev Deych, and Tal Carmon Department of Mechanical Engineering, Technion-Israel Institute of Technology, 3200003 Haifa, Israel Departamento de Fisica and IUdEA, Universidad de La Laguna (ULL) 38200, Tenerife, Spain Physics Department, Queens College of CUNY, Flushing, New York 11367, USA Graduate Center of CUNY, 365 5th Ave, New York 10016, USA

Degeneracies, including exceptional points [1][2][3][4], where level coalescence with only one eigenmode, play a crucial role in multiple fields ranging from sensing of nanoparticles, biological, and chemical analytes [1,2,[5][6][7][8][9][10][11] to gyroscopes and other navigation systems [12,13]. Often, optical resonators are being used to generate degeneracies that are typically twofold. The general problem of experimentally demonstrating the highest degree of degeneracy relates to the need for resonators to possess the highest degree of nontrivial symmetry (e.g., perfect spheres)-as theory suggests [14]. These resemble cases such as in cesium, where the spherically symmetric Coulomb potential supports hyperfine splitting. Yet, making such perfect spherical resonators, to host hyperdegeneracy, is impossible using current fabrication techniques or even 3D printers. This is because even a stem to hold the sphere breaks the symmetry. Here, we fabricate optically tweezed [15,16] contactless resonators, spherical to within fabrication tolerances of less than the Bohr radius, in the vicinity of tapered-fiber couplers.
Giving devices levity, while seemingly depriving them of their gravity [17] was always igniting the imagination. Such a levitation here permits almost perfect microspheres since in small droplets, surface tension dominates over gravity to minimize surface area toward the smallest possible area (per a given volume) that spheres exhibit. In this regard, the optical tweezers apply a force comparable to gravity in size, but in the opposite direction. Our vision includes better sensitivity using highest degeneracy orders, and scalability of the levitating resonator to optical circuits made of many levitators, by using currently available holographic-tweezers technology [1]. A longerterm vision relates to the fact that the degenerate modes here are orbital angular-momentum modes, which can be utilized to construct high-dimensional modes of photons called qudits [18]. Combining degenerate qudits with low absorption [19] might impact time-complexity reduction in problem solving [20]. In more detail, perturbed symmetry in nature is commonly related to degeneracy. For example, hyperfine splitting in alkali atoms represents degeneracy that is lifted by perturbations of the atom spherically symmetric Coulomb potential by merely intranucleus inhomogeneities. Inspired by this symmetry-degeneracy relation, we extend this principle from atoms to classicalwave resonators [14,21], with lifted degeneracies due to the perturbation by the tweezer pressure and gravity. Such resonators can host light, sound, or other waves in nature. Generally speaking, the highest nontrivial symmetry in nature is spherical-invariance with respect to rotations by any angle around any axis passing through the center of the corresponding sphere. The highest symmetry of spheres results in modes with the highest degree of degeneracy. To give it a scale, the order of degeneracy in our microspheres is more than 100 times larger when compared to a ring. Realization of optical resonators with shapes only weakly deviating from spherical is of great interest. For example, as it has been shown in Ref. [22], the sensitivity of the optical spectra of WGM resonators to the presence of various analytes improves dramatically when deviation from sphericity of a resonator drops below a critical value [23]. Observation of spherical-symmetry degeneracy in microresonators is challenging due to fabrication difficulties. Even if planar lithography could produce smooth microspheres with a sufficiently small deviation from sphericity, their support will break the symmetry. We overcome this difficulty using optical levitation, which does not require any symmetry-breaking support. Contactless devices are also almost perfectly proof against contaminations, as demonstrated on silicon wafers [24], by keeping them cleanest and unscratched even at the atomic level. Levitating devices can hence benefit from both worlds-highest degree of cleanness to support lowest optical losses and symmetry to support highest degrees of degeneracy.
Optical resonances of spherical dielectrics [see Figs. 1(a)-1(d) and Supplemental Material Sec. SM1 and Ref. [14] in [25]] are characterized by the number of wavelengths that can fit along the equator of a resonator (l) and the number of the intensity maxima between their north and south pole (l − m þ 1). For a given l, m equals −l; −l þ 1; −l þ 2; …; l, and the modes are therefore 2l þ 1 degenerate with respect to azimuthal (magnetic in the quantum context) number m. For instance, for a resonance characterized by l ¼ 113 one can expect 227 degenerate modes, which is typical to our experiment. Since eigenfrequencies of the degenerate modes are the same, they generate a standing interference structure, of the type that one can see in Figs. 1(e) and 1(f). This degeneracy can be qualitatively understood [14] by noting that modes having a smaller m travel at larger inclinations with respect to the equator, which is compensated for by shorter circumference of circles lying above or below the equator [Figs. 1(a)-1(d), right]. This qualitative description is consistent with the characteristic equation for the frequencies of the resonances [23], where r and n are the sphere's radius and refractive index, k is the wave vector, and j l ðzÞ and y l ðzÞ are the spherical Bessel functions of the first and second kind. Importantly, Eq. (1) does not contain m, making its solutions independent of this quantum number. When a spherical resonator is deformed into a spheroid, the degeneracy is lifted and new eigenfrequencies can be found at [23] [25] to represent the location where the electric field falls to half of its maximum (minimum). Green represents an equal intensity surface. The polarization is TE, the radial number is 1, the refractive index is 1.4, and the sphere's circumference is 11 optical wavelengths.
where ν ð0Þ l is the unperturbed frequency of the spherical resonator and η is the shape deformation parameter defined in Supplemental Material, Sec. SM2 [25]. Such a split of the resonances was indeed experimentally observed in resonators with large deviations from sphericity [21,28].
In our experiment (Fig. 2), we make droplets from silicone oil of 10 cSt viscosity. It is helpful here that unlike solids, the liquid droplet does not contain any dislocation, inclinations, and thermally induced stresses, which are typical for solid resonators and reduce their optical quality [29]. The sphericity of the droplet, held by our tweezers, is perturbed by the combination of radiative pressure [30] and gravity.
In our experimental setup, a tunable laser is evanescently coupled via a tapered fiber [31] to an optically tweezed [15,16] levitating droplet of silicone oil, and excites its resonances. This approach distinguishes our experiment from earlier works, in which resonances of levitating [16] or falling [32] droplets were excited by free-propagating light beams, where large phase mismatch could be solved by designing the cavity shape [33]. In our approach, phase matching is significantly improved by realizing contactless coupling to the standard fiber, tapered until its effective refractive index matches the resonator's effective index [31]. Resonances are observed as Lorentzian dips of the transmission through the fiber as we tune the frequency of the incident laser light.
Characterizing the linewidths of the dips reveals an optical quality factor of the resonances to be 1.2 × 10 9 with finesse of 11.6 × 10 6  Analyzing the family of degenerate modes by their density of modes is expected to reveal that density goes down with frequency as theoretically derived in Eq. (SM12) [25]. Indeed, as one can qualitatively see [Figs. 4(a) and 4(b)], the resonances turn less dense as frequency goes up. Going now to a quantitative analysis, we fit our experimentally measured density of modes [Fig 4(c), blue] to its theoretically predicted [Fig 4(c), orange] value [Eq. (SM12)], using the spectral span of the hyperfine family Δ l as a free parameter (Supplemental Material Sec. SM2 [25]). Our experimental parameters reflect angular momentum of l ¼ 113 and degeneracy order of 227. The theoretical model assumes that only one family of modes is presented in the frequency span that we analyze [ Fig. 4(c)]. Yet, we expect that other families of modes, e.g., with higher radial order, might contribute stray modes with frequencies penetrating the spectral band that we measure. These stray modes are most likely to explain the fact that our measured density of mode [ Fig. 4(c), blue] is typically above the corresponding theoretical model line that assumes one family of modes [ Fig. 4(c), orange]. With this circumstance taken into account, we can conclude that the fit presented in Fig. 4(c) with Δ l ¼ 1.66 GHz agrees reasonably well with our experimental data. This value of Δ l corresponds to the record low deformation parameter η ≃ 5.42 × 10 −6 , which translates into the horizontal stretch of the droplet diameter from its ideal sphere value by merely 0.98 Å. The evanescent coupling by the tapered fiber could also induce the perturbation to the eigenmodes and break the spherical symmetry of the system. Yet, calculating the taper's contribution to degeneracy in our case reveals that its effect is smaller by more than one order of magnitude, when compared to asphericity. In conclusion, except for the hyperdegeneracy and ultrahigh finesse that levitators permit, our vision includes their scalability to optical circuits that will be made of many levitating devices. This can be done using the impressive boost in the resolution and refresh rate of a spatial-light modulator, suggesting that many optical devices, similar to the ones reported here, can levitate while one controls their positions [15] and shape [34]. In more detail, spatial-light modulators with 10 × 10 7 pixels are nowadays available, suggesting that levitators can be deformed upon request and then positioned by holographic tweezers, in 3D, upon need. In the past, using optical tweezers allowed deforming droplets to shapes such as triangles [34], suggesting that a variety of different devices might be possible. Continuously transforming multiple droplets, using dynamical traps, can give rise to photonics where the devices can also change while operating, a freedom that is challenging using current fabrication technologies. Such devices that change