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Discrete degrees of freedom, such as spin and orbital, can provide intriguing strategies to manipulate electrons, photons, and phonons. With a spin degree of freedom, topological insulators supporting backscattering-immune edge states have stimulated intense interests in condensed-matter physics 1 , optics 2,3 , acoustics 4 , and mechanics 5-7 . However, orbital as another fundamental attribute in crystals has seldom been investigated in topological insulators. Here, we invent a new type of topological insulators with both spin and auxiliary orbital degrees of freedom on a nanomechanical platform. Specifically, we experimentally demonstrated spin-polarized topological edge states in a nanomechanical crystal whose orbitals can arbitrarily be manipulated by the crystal configuration. Harnessing this unique feature, we further experimentally realized adiabatic transition between distinct topological phases without closing the energy bandgap. The auxiliary orbital degree of freedom has unveiled unprecedented strategies to manipulate topological phase transitions and to study topological phases of matter, which enable new generations of integrated devices and circuits that may lead to future quantum computers.

Motivated by applications in mobile optical sensing, ultracompact high‐resolution integrated spectrometers have attracted much interest. Here, a high‐resolution integrated speckle spectrometer, comprising a linear coherent network formed by mutually coupled Mach–Zehnder interferometers and nonidentical microring resonators, is proposed and demonstrated. Deep‐etched grating lines used as mirrors on the edges of the coherent network increase the effective optical path lengths. The speckle spectrometer is realized on a silicon nitride platform, operating at 776 nm central wavelength. The eight‐in−eight‐out linear coherent network provides 64 physical channels. Fine spectral lines separated by 20 pm are experimentally resolved within a device footprint of 520 µm × 220 µm. Compressive sensing is achieved for sparse spectra over a wide optical bandwidth. Up to 600 distinctive wavelength channels can be reconstructed from the 64 physical channels, giving 12 nm operating bandwidth. Both sparse spectra and continuous spectra are well reconstructed experimentally. The integrated speckle spectrometer has great potential for use in future biosensing and bioimaging applications where high spectral resolution is desired.

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