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We report the development of a superconducting acousto-optic phase modulator fabricated on a lithium niobate substrate. A titanium-diffused optical waveguide is placed in a surface acoustic wave resonator, where the electrodes for mirrors and an interdigitated transducer are made of a superconducting niobium titanium nitride thin film. The device performance is evaluated as a substitute for the current electro-optic modulators, with the same fiber coupling scheme and comparable device size. Operating the device at a cryogenic temperature (T=8K), we observe the length-half-wave-voltage (length-$V_pi$) product of 1.78 V$cdot$cm. Numerical simulation is conducted to reproduce and extrapolate the performance of the device. An optical cavity with mirror coating on the input/output facets of the optical waveguide is tested for further enhancement of the modulation efficiency. A simple extension of the current device is estimated to achieve an efficient modulation with $V_pi=$ 0.27 V.
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