Fermat's Spiral-Based Characterization of Squeezed Nonlinear Motional States of Levitated Nanoparticles

Abstract

Abstract A nanoparticle optically levitated in vacuum represents a macroscopic mechanical oscillator that can be virtually decoupled from its thermal environment, providing a unique testbed for exploring the transition between classical and quantum dynamical regimes. It is generally assumed that such a levitated particle is confined and oscillates in a parabolic optical potential where its state is described by a Gaussian distribution in the phase space. However, real-world experiments are performed in optical potentials with weak nonlinear distortions that drive the motion into a non-Gaussian state. Such a state requires additional parameters for the full characterization, and direct quantitative comparison with the ideal Gaussian states is not straightforward. Here, we focus on the most common weak nonlinearity of Duffing type, in which the oscillation frequency decreases with the square of the amplitude of nanoparticle's oscillations. This dependence gives rise to spiral-shaped, squeezed non-Gaussian states. Given the particular observed symmetry of the phase-space distributions, we use the concept of Fermat's spiral and propose a nonlinear transformation of normalized phase-space coordinates that transforms this spiral state into a squeezed Gaussian-like state, which can be directly compared with the states of a stochastic harmonic oscillator. For comparison, we evaluate a nonlinear squeezing metric that can be used also in quantum regimes. The proposed approach can be extended to other nonlinearities of the confining potential with an explicit dependence of the oscillation frequency on the amplitude of the motion.