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Springer eBooks

Zernike System Stems from Free Motion on the 3-Sphere

Kurt Bernardo Wolf, Natig M. Atakishiyev, GEORGE S. POGOSYAN, Alexander Yakhno

Published Aug 8, 2020• 0 citations
Publisher
DOI
10.1007/978-3-030-55777-5_16

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Abstract Classical or quantum systems that stem from a basic symmetry are seen to be special in having several important properties. The harmonic oscillator and the Bohr system are such. Recent research into the Zernike system provides reasons to include it in this privileged class. Here we show that free motion on the 3-sphere can be projected down to produce classical orbits or complete and orthogonal bases for wavefronts in a circular pupil. This line of inquiry has been pursued in company with N.M. Atakishiyev, G.S. Pogosyan, C. Salto-Alegre, and A. Yakhno.

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Mechanical tuning of whispering gallery modes over a 05 THz tuning range with MHz resolution in a silica microsphere at cryogenic temperatures

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We experimentally demonstrate the mechanical tuning of whispering gallery modes in a 40 μm diameter silica microsphere at 10K, over a tuning range of 450 GHz and with a resolution less than 10 MHz. This is achieved by mechanically stretching the stems of a double-stemmed silica microsphere with a commercially available piezo-driven nano-positioner. The large tuning range is made possible by the millimeter long slip-stick motion of the nano-positioner. The ultrafine tuning resolution, corresponding to sub-picometer changes in the sphere diameter, is enabled by the use of relatively long and thin fiber stems, which reduces the effective Poisson ratio of the combined sphere-stem system to approximately 0.0005. The mechanical tuning demonstrated here removes a major obstacle for the use of ultrahigh Q-factor silica microspheres in cavity QED studies of solid state systems and, in particular, cavity QED studies of nitrogen vacancy centers in diamond.

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Free Particle Motion in Three Dimensions

S. Brandt, Hans Dieter Dahmen

Springer eBooks•Jan 1, 1992•0 citations
Abstract Classical or quantum systems that stem from a basic symmetry are seen to be special in having several important properties. The harmonic oscillator and the Bohr system are such. Recent research into the Zernike system provides reasons to include it in this privileged class. Here we show that free motion on the 3-sphere can be projected down to produce classical orbits or complete and orthogonal bases for wavefronts in a circular pupil. This line of inquiry has been pursued in company with N.M. Atakishiyev, G.S. Pogosyan, C. Salto-Alegre, and A. Yakhno.
We analyze the recently proposed classical/quantum mechanical interpretation of Zernike system and establish its equivalence to the Higgs oscillator on sphere or pseudosphere (Lobachevsky plane). We show that the non-reality of the classical Zernike Hamiltonian is an insignificant artifact of imaginary gauge and can be eliminated with a canonical transformation. The quantum counterpart of this canonical transformation is a similarity transformation mapping the system to the quantum Higgs oscillator with integration measure depending on [Formula: see text] parameters. When [Formula: see text], it results in the Hermitian Hamiltonian describing a free particle on (pseudo)sphere, while deviation from this point leads to a pseudo-Hermitian system.
We experimentally demonstrate the mechanical tuning of whispering gallery modes in a 40 μm diameter silica microsphere at 10K, over a tuning range of 450 GHz and with a resolution less than 10 MHz. This is achieved by mechanically stretching the stems of a double-stemmed silica microsphere with a commercially available piezo-driven nano-positioner. The large tuning range is made possible by the millimeter long slip-stick motion of the nano-positioner. The ultrafine tuning resolution, corresponding to sub-picometer changes in the sphere diameter, is enabled by the use of relatively long and thin fiber stems, which reduces the effective Poisson ratio of the combined sphere-stem system to approximately 0.0005. The mechanical tuning demonstrated here removes a major obstacle for the use of ultrahigh Q-factor silica microspheres in cavity QED studies of solid state systems and, in particular, cavity QED studies of nitrogen vacancy centers in diamond.