Nonlinear Dynamics and Fermi-Pasta-Ulam-Tsingou Recurrences in Macroscopic Ultra-low Loss Levitation

Abstract

Macroscopic systems, when governed by nonlinear interactions, can display rich behavior from persistent oscillations to signatures of ergodicity breaking. Nonlinearity, long regarded as a nuisance in precision systems, is increasingly recognized as a gateway to new physical regimes. While such dynamics have been extensively studied in optics and atomic physics, macroscopic systems are rarely associated with long-lived coherence and nonlinear control and remain an untapped platform for probing the fundamental nonlinear processes. Here, we report the observation of long-lived oscillatory dynamics in millimeter-scale levitated dielectric quartz particles exhibiting clear signatures of nonlinear mode coupling, a positive largest Lyapunov exponent of 0.0095 s^-1, and partial energy recurrences-phenomena strongly reminiscent of the Fermi-Pasta-Ulam-Tsingou physics. We observe dissipation rates below 4*10E-6 Hz, limited by our ability to measure dissipation in presence of nonlinear dynamics. We estimate an intrinsic acceleration sensitivity of 62*10E-12 g/sqrt(Hz), at room temperature. The magnetic trap is constructed from a static arrangement of permanent magnets, requiring no external power or active feedback. Our findings open a path toward leveraging nonlinear dynamics for novel applications in sensing, signal processing, and statistical mechanics.

Figures (7)

• Figure 1: (a) 3D illustration of magnetic assembly consisting of eight triangle-shaped magnets held together to create a radial magnet. A metallic core and a metallic disk are used to focus the field at the center above the magnet. (b) COMSOL field simulation shown in 2D for the whole magnet assembly and a closed up view near the levitation point. A cylindrical opening (about 2 mm in diameter) at the center of the top metallic disk guides the magnetic flux from the metal core and provides space for levitation. (c)-(e) Sum of magnetic and gravitational potentials integrated over a cubic volume made of quartz (side=0.5 mm, mass$\simeq$0.3 mg) plotted for coordinates $x$, $z$ and angular displacement around $x$ and $y$ axes ($\phi_x~\&~\phi_y$), respectively.
• Figure 2: (a)-(c) Top view images of levitated quartz cube (side=0.5 mm, mass$\simeq0.3$ mg), hollow quartz cylinder (diameter= 0.5 mm, mass$\simeq2$ mg), and N-BK7 hemisphere (diameter=1.5 mm, mass$\simeq$2.5 mg), respectively. (d) Side view image of levitated hemisphere. (e) A typical vibrational spectrum for levitated cube measured using non-interferometric single-pixel detection, with main vibrational modes indicated.
• Figure 3: (a) Ring down measurement of vibrational amplitude noise after mechanical excitation of levitated cube's modes. The data was obtained from a side camera detecting laser scattered light and analyzed similar to a quadrant detector. Inset shows another example of ring down measurement where a long time constant can be inferred. (b) Damping rate measured for several vacuum (circle symbols). The solid lines are the theoretical expectation of air damping in the free-molecular regime for two different modes of vibration (translation and rotation). The effect of squeezed-film damping is negligible. The horizontal lines are the estimated eddy current damping limit for two different modes. (c) Power spectral density (PSD) of levitated quartz cube calibrated using the equipartition theorem. The dashed line is a Lorentzian fit with FWHM of $\sim$0.6 mHz limited by the resolution bandwidth from 2000s-long data. (d) Allan deviation of results in (c) showing a plateau region near 50 s of integration time ($\tau$).
• Figure 4: (a) Thermally excited vibrational noise amplitude spectrum ($x(\omega)$) plotted over several hours. (b) The zoomed-in spectra for three main modes (A, B, and C as shown in (a)) and integrated amplitude over each frequency window are plotted as a function of time. (c) Another example of the vibrational spectra recorded after initial excitations to perform ring down measurement. Strong initial drive gives rise to a modulated and shifting spectra. Inset shows nonlinear theory prediction of the fundamental mode behavior under various excitation strength (leading to modulation akin to experimental observations). An arbitrary cubic nonlinearity of $\beta=1$ s$^{-2}$m$^{-2}$ is considered leading to creation of sidebands and frequency shift around the resonance, for a given excitation strength. Vacuum pressure for (a) and (c) was $1.8\times10^{-7}$ and $4.1\times10^{-7}$ Torr, respectively.
• Figure 5: (a) Histogram of phase difference between the fundamental and the 2nd harmonic mode indicates coherence between higher harmonics. Inset shows the 3D phase space plot of position of the two modes, $X$ and $Y$ (fundamental and 2nd harmonic) and velocity of the 2nd harmonic, $V_Y$. (b) Cross correlation of amplitude of two modes shows a peak near zero time lag, another indication of coherence between the two modes. (c) Enhanced parametric mode coupling is observed when a higher harmonic of one mode matches another mode's frequency, in this case the 5th harmonic of 1.4 Hz coincides with $\nu_x$. Inset shows the phase space plot of the two amplitudes (fundamental and higher harmonic modes) where the Lissajous-like figures indicates phase locked motion. Vacuum pressure used was $1.3\times10^{-5}$ Torr.