We theoretically investigate a driven oscillator with the superconducting inductance subject to quantum phase slips (QPS). We find uncommon nonlinearities in the proposed device: they oscillate as a function of the number of photons $N$ with a local period of the order of $\sqrt{N}$. We prove that such nonlinearities result in multiple metastable states encompassing few photons and study oscillatory dependence of various responses of the oscillator. Such nonlinearities enable new possibilities for quantum manipulation of photon states and very sensitive measurements to confirm the coherence of phase slips.

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