Introduction

The study of nonlinear dynamics across various physical systems has profoundly transformed our understanding of fundamental processes and enabled important applications. The boundary between stability and chaos in nonlinear dynamical systems is far more intricate than initially envisioned. A paradigmatic example is the Fermi-Pasta-Ulam-Tsingou (FPUT) problem1,2, where instead of rapid thermalization, the system exhibits long-lived recurrences and quasi-integrable dynamics, even at energies where chaos might have been expected. Despite decades of investigation, the precise mechanisms governing the onset of chaos, the conditions under which recurrences persist, and the generality of these features across different physical platforms remain unresolved. These open questions have far-reaching implications: they challenge our understanding of thermalization in isolated systems, test the limits of predictability, and impact fields ranging from condensed matter physics to quantum information science. In optical systems, investigations into nonlinear wave propagation and soliton formation have directly led to the development of optical frequency combs3,4, which are now essential tools in precision metrology5, spectroscopy6, and timekeeping7,8,9. In quantum systems, nonlinear interactions underpin phenomena such as photon blockade10, quantum squeezing11, and many-body effects12 with no classical analogs.

Mechanical oscillators can also exhibit strong nonlinearity, leading to rich dynamical behavior and routes to thermalization that differ significantly from their linear counterparts. However, nonlinear mechanical systems have not been as extensively explored, in part due to the technical challenges associated with precisely engineering and measuring nonlinear potentials, especially in macroscopic or low-dissipation regimes. Recent advances in levitation-based platforms13,14,15 and micro- and nanoscale resonators16,17 offer new opportunities to probe these systems, enabling controlled studies of nonlinear physics in both classical and quantum settings.

Diamagnetic levitation of macroscopic objects has recently emerged as a promising platform for precision sensing18,19, quantum-limited measurements20, and experimental tests of fundamental theories21,22. Diamagnetic potentials offer stable three-dimensional trapping of macroscopic objects with no external drive, no feedback, and no contact loss channels, even at room temperature.

While previous work has focused on nanogram-scale dielectric particles15 or non-dielectric materials such as composite graphene19,23,24,25–where the substantially larger mass and size of the levitated object enhance inertial stability and reduce noise and dissipation–this work aims to combine the strengths of both approaches. Specifically, we employ a large-mass dielectric while completely eliminating eddy currents inside the levitated object by exploiting the intrinsic properties of dielectrics together with tailored magnetic-field engineering. Building on early studies of field localization for dielectric levitation26,27, we implement strong field localization to levitate dielectrics with masses in excess of 1 mg in such configurations. The resulting combination of low dissipation and large-mass levitation enables the study of nonlinear mode interactions in these systems. Our system comprises a millimeter-scale dielectric particle of asymmetric shapes levitated above a permanent magnet array. We observe and characterize multiple underdamped vibrational modes, with lifetimes  > 104 seconds, and track their long-time evolution under small perturbations. The dynamics exhibit features of FPUT problem, including persistent nonlinear intermodal coupling, long-lived energy recurrences, incomplete thermalization over experimentally accessible timescales, and coherent higher harmonic generation.

Results

Diamagnetic trap

Diamagnetic levitation is the stable suspension of an object in a magnetic field due to its diamagnetic response. Diamagnetism arises from the induced magnetic moments in materials that oppose the applied magnetic field, resulting in a repulsive force.

For a diamagnetic material with volume V, magnetic susceptibility χ < 0, and permeability of free space μ0, the magnetic potential energy is: \(U=-\frac{\chi V}{2{\mu }_{0}}| \vec{B}{| }^{2}\). Stable levitation is achieved when the corresponding magnetic force balances gravity and provides a restoring force in all directions. For a particle of mass m under gravity g, the vertical equilibrium condition becomes: \(\frac{\chi V}{{\mu }_{0}}(\frac{d{B}^{2}}{dz})=mg\). The trap frequency ω0 is determined by the curvature of the magnetic field at the equilibrium position. Given that the density and magnetic susceptibility are different from one material to another, the trap can levitate different materials with different levels of ease and at certain equilibrium heights. Table 1 shows the density and magnetic susceptibility of several materials and the levitation parameter (\(B\frac{dB}{dz}\)), which corresponds to the field strength and its gradient needed to trap various materials. A larger levitation parameter corresponds to increased difficulty in achieving stable trapping. Another factor in choosing the levitation material is low conductivity to reduce eddy-current damping. As it can be seen, quartz and N-BK7 have negligible conductivity but are among the most challenging materials to stably levitate diamagnetically. Using a tailored configuration of permanent magnets, we successfully levitated macroscopic (mm-scale) quartz and N-BK7 objects, attaining ultra-low dissipation rates that reveal the intrinsic nonlinear dynamics of the trapping potential.

Table 1 Comparison of relevant diamagnetic materials
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Experiment

A set of eight N42 neodymium magnets are assembled as shown in Fig. 1a and b to focus the magnetic field in a mm-region above the magnet. A metallic rod and disk enable focusing the field in the gap above the center of the magnets where a strong field gradient enables 3D diamagnetic trapping of mm-scale dielectrics. Using this trap, we have achieved stable levitation of different weakly diamagnetic dielectrics (silica, N-BK7 glass, and crystalline quartz) of symmetric and asymmetric shapes (sphere, hemisphere, cylinder, and cube).

Fig. 1: Simulation of the trap.
Fig. 1: Simulation of the trap.
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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 close-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 ≃ 0.3 mg) plotted for coordinates x, z and angular displacement around x and y axes (ϕx&ϕy), respectively.

Figure 1c–e show the total magnetic and gravitational potential calculated for a cubic volume of quartz (side=0.5mm, mass ≃ 0.3 mg). In Fig. 1c the potential is calculated as the position of the center of mass (CoM) of the cube has been moved along the z axis starting from 250 μm above the magnets. As this simulation shows, this potential has a minimum at  ~ 850 μm above the magnets. This agrees with our experimental observation. The simulated trap along this axis shows a stiffness of about 1.2 × 10−3 N/m and an associated frequency of about 10.3 Hz. Figure 1d shows the trap potential in lateral directions (x or y). The trap frequency and stiffness along the horizontal plane are 7.1 Hz and 5.5 × 10−4 N/m respectively. Figure 1e shows the effect of the rotation of the cube around x and y axis and its ensuing change in potential. As this figure shows, the stable position is when the cube is sitting with its face parallel to the horizontal plane. Depending on the rotation around x and y, this two-dimensional trap can have different depths along the y and x axes, leading its rotational motion to assume frequencies ranging from 0 Hz up to 0.6 Hz near the stable arrangement. Also, it is worth noting that The rotational trapping is the result of asymmetry in the objects’ shape, its asymmetric magnetic susceptibility and broken rotational symmetry of the field in the trapping region (due to non-ideal magnets). As can be seen below, these predictions are in good agreement with the experimental observations.

The levitation setup is placed inside a vacuum chamber reaching pressures as low as 10−8 Torr. Images of levitated objects are shown in Fig. 2 together with typical vibrational spectra measured at the ambient pressure. Several complementary methods are used to measure the motion of the levitated cube. Depending on the experiment, the displacement can be recorded using one of these approaches. In one method, a quadrant photodiode is used to monitor small deflections of a reflected probe beam. This method is most sensitive to vibrations in the plane perpendicular to the probe light. This method is specifically used when we have levitated hemispheres. In an interferometric detection scheme, a weak probe reflected from the levitated object is interfered with a local oscillator. This method can provide high detection sensitivity; however, it suffers from slight misalignment when the levitated object drifts. We also use video analysis to monitor vibrations in all directions. In this case, three high-speed cameras record scattered light from the levitated cube. Video analysis is then employed to extract the frequencies of different modes while tracking the object’s position relative to the edges of the trap. Pixels in an active window are divided into 4 groups and are treated similar to a quadrant photo detector. For the key measurements shown in this work, we use a non-interferometric single-pixel detection scheme. In this approach, a weak probe beam is directed toward the levitated cube and reflected by a thin quartz plate placed beneath it on the magnet surface, so that the beam passes through the particle twice before being collected on a fast photodiode. The transmitted/reflected intensity encodes the particle’s motion via small shadowing and scattering variations, providing a linear readout of displacement. This approach is robust to beam-pointing fluctuations and offers high bandwidth. The signal is calibrated by comparing the measured spectral amplitudes to the expected thermal motion of the translational modes. The raw time series data for single-pixel detection and video analysis methods are provided in the Supplementary Note 1, Figs. S1, S2, and S3. The measurement of vibration spectrum of a quartz cube of 0.5 mm side length (see Fig. 2) reveal the main vibrational frequencies, in agreement with calculated values from the potentials in Fig. 1. Below we focus on the quartz cube as the levitated object due to its ease of access to various modes, relatively large levitation gaps from the metal surfaces (for reduced eddy current), and our ability to arbitrarily shape and size such particles using a laser cutter. The spectrum shows two main translational modes along z and x to be respectively at 10.1 Hz and 7.3 Hz. Two distinct lower frequency modes at 0.3 Hz and 1.4 Hz are rotation around a horizontal axis and z axis respectively. As mentioned before, the confined rotational mode around z is due to rotational asymmetry in the field profile.

Fig. 2: Levitation of different milligram dielectric objects.
Fig. 2: Levitation of different milligram dielectric objects.
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a–c Top view images of levitated quartz cube (side = 0.5 mm, mass ≃ 0.3 mg), hollow quartz cylinder (diameter = 0.5 mm, mass ≃ 2 mg), and N-BK7 hemisphere (diameter = 1.5 mm, mass ≃ 2.2 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. Each of the three dominant peaks were identified directly from the recorded motion of the particle using a fast camera: the translational modes appear as linear oscillations along x and z, and the rotational mode is visible as rocking of the cube. The additional unmarked peaks are not independent eigenmodes; instead, they arise from weak hybridization of the primary modes. Data was recorded using the single-pixel photo-detector.

Dissipation and noise limits

In practical settings, the trap is associated with damping that arises from two main mechanisms. At low air pressures P, the residual gas damping rate in the free-molecular regime for a spherical object of cross section A and mass m is approximately given by γ ≃ αP/m28, where α = A/vth is the damping coefficient and vth is the thermal velocity of gas. At P = 10−7 Torr and including the squeezed film correction, the damping rate of a millimeter-scale sphere only limited by gas damping is expected to be below 0.1 nHz. Achieving such low vacuum levels typically requires annealing of the vacuum chamber. However, caution must be exercised when annealing in the presence of rare-earth magnets, as some of these magnets can lose significant magnetic strength with even modest temperature increases–sometimes as little as 10 ∘C above room temperature.This effect is particularly significant in our experiment because magnets at the very center are experiencing very strong magnetic flux not aligned with their initial magnetization direction.

Eddy currents induced in conductive materials, e.g., magnets’ surfaces (nickel copper nickel coating) or support structures (metallic disk and rod used for field confinement) can also contribute to energy dissipation. For simplicity, consider a spherical dielectric particle of volume V levitated a distance d above a magnet, where the magnetic field is confined to a cylindrical region of radius a directly above the magnet. The vertical damping rate due to eddy currents can be approximated29 by \(2{\gamma }_{z}=2\frac{V}{\pi {a}^{3}}\frac{d}{{l}_{z}}\frac{{\chi }_{m}g}{{l}_{z}},\) where lz is the magnetic field penetration depth and χm is the diamagnetic susceptibility. For typical metallic surfaces (e.g., iron) surrounding the levitated particle to focus the magnetic field, the eddy current damping rate is estimated to be on the order of 5 × 10−4 Hz. In our experiment, we use Permendur, which has a resistivity approximately 20 times higher than that of iron, to reduce eddy current losses (i.e., increase lz). With this material, we estimate a minimum eddy-current-limited damping rate of about 7 × 10−7 Hz for the lowest mode (rotation around z axis) of the cube. The eddy current can further be suppressed by reducing the metal surfaces and their geometries30 (e.g., using laminated layers). We note that for the hemisphere, the symmetry of the object suppresses eddy currents, allowing dissipation in the rotational mode about the vertical axis to approach the air-damping limit ( ~ nHz at pressures obtain here), in the absence of mode coupling.

Figure 3a and b show the measurement result of damping or dissipation rate at different vacuum pressures. Each ring down measurement was obtained after the particle was first excited either magnetically, using a small drive coil placed above the trap, or mechanically, by tapping the vacuum chamber. The choice of excitation method depends on the mode being probed and is stated in the relevant figure captions. After excitation is stopped, the motion is monitored using the single-pixel detector.

Fig. 3: Characterization of damping and sensing.
Fig. 3: Characterization of damping and sensing.
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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. The segmented blue lines are the amplitude-time series data of the signal extracted by processing 1-min video segments. Each blue line corresponds to ≥100 oscillations with near constant amplitude. This gated analysis was employed to circumvent the heavy computational cost of going through all the recorded videos. The error in the time constant is the confidence interval obtained from Monte Carlo method as the Jacobian method fails indicating the time scale is not well constrained by the data. Inset shows another example of ring down measurement where a long time constant can be inferred. b Damping rate measured for several vacuum levels. The purple points in (b) represent the damping rate corresponding to the effective amplitude envelope of the coupled motion, not a single isolated mode. Due to energy exchange between modes with different intrinsic damping, the apparent decay rate is intermediate between the translational and rotational damping rates. The error bars for these measurements are smaller than the data points. 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  ~ 0.6 mHz limited by the resolution bandwidth from 2000 s-long data. d Allan deviation of results in (c) showing a plateau region near 50 s of integration time (τ). The decay shown in (a) as well as the results shown in (b) are measured using camera analysis methods, while data presented in (c, d) are based on single-pixel photo-detector.

The ring down result does not always show a clear decay and in some cases leads to long-term oscillation as shown in the inset of Fig. 3a. This is rooted in the nonlinear mode coupling and FPUT recurrence as discussed below. Results indicate that at pressures below 10−4 Torr, the air damping is insignificant and the primary source of damping is eddy current. The longest decay time measured is 8 × 104 s, limited primarily by the accuracy of amplitude decay measurements in the presence of nonlinear mode coupling and recurrences (see below). We expect the decay time for the lowest vibrational mode to be around 106 s, at an elevated (by 50%) levitation height, which can be achieved by slightly stronger magnets (see Methods). The points in Fig. 3b does not correspond to a particular mode; instead, they correspond to the effective amplitude envelope of all modes as energy repeatedly cycles between the translational and rotational modes, each of which has a different intrinsic damping rate. Because the rotational mode is significantly less dissipative than the translational modes19,25, the intermodal exchange produces an effective decay rate that lies between the individual γ- of the participating modes.

We note that the high sensitivity of our system stems primarily from its low dissipation rate. This low dissipation, combined with the system’s intrinsic nonlinearity, gives rise to rich nonlinear dynamics, as discussed below. Such nonlinear behavior can either limit sensitivity–by driving the system toward the chaotic regime–or enhance it, for example through multimode coupling or frequency-comb-based sensing. Consequently, understanding the fundamental sensing limits is inseparable from analyzing the underlying nonlinear dynamics, particularly when evaluating the dynamic range of the sensor. The acceleration sensitivity is proportional to square root of the ratio between the dissipation rate and mass, γ/m. The lowest values of this ratio have been reported for suspended mirrors31, diamagnetic levitation systems15,23, and nano-particle Paul traps32, with reported values ranging from 1010(2π)Hz/kg for nano-particle Paul traps32 to less than 100(2π)Hz/kg for composite graphite in diamagnetic traps23,24. In our experiment, we measure γ ≈ 4 μHz from the ring down data and using a mass of  ~ 0.3 mg, we obtain γ/m≤84(2π)Hz/kg, surpassing cryogenic traps15,33.

The vibrational spectra can be calibrated using the equipartition theorem to find displacement (SX) and force (SF) sensitivity, where \({S}_{X}={S}_{F}/m\sqrt{{({\omega }^{2}-{\omega }_{0}^{2})}^{2}+{\gamma }^{2}{\omega }^{2}}\). At vacuum pressures of 8 × 10−6 Torr, the fitted linewidth (limited by integration time and frequency drift) obtained was 6 × 10−4 Hz (see Fig. 3(c)). We observe a displacement noise floor of \(3\times 1{0}^{-12}\,m/\sqrt{Hz}\). From the ring down measurement, we have observed a dissipation rate (full-width half maximum) of γ/2π ≃ 4 μHz which is insensitive to the frequency drift (see below). We predict a rate of < 1 μHz after re-magnetization of the magnets (see Methods). At 1 μHz, the best gravitational acceleration sensitivity that can be reached (at room temperature) is \(\sqrt{4\gamma {k}_{B}T/m}\simeq 62\,pg/\sqrt{Hz}\). Application of the Allan deviation revealed a clear plateau near an averaging time of approximately 50 s (see Fig. 3d). This plateau corresponds to the timescale where the drift and correlated noise begin to dominate the stability.

Nonlinear dynamics

The absence of mechanical clamping and the ability to tune trap stiffness and achieve low damping enable clean access to the nonlinear regime. Nonlinear coupling arises naturally from geometric confinement, magnetic field gradients, and particle asymmetry. As the object moves in a specific direction, it experiences a magnetic field whose magnitude is now different in all directions. This gives rise to mode coupling. The nonlinearity in the system arises from the non-quadratic shape of the trapping potential, leading to nonlinear mode coupling. For a system with generalized coordinates xi (corresponding to different modes), the dynamics can be described by coupled differential equations:

$${\ddot{x}}_{i}+{\omega }_{i}^{2}{x}_{i}+\sum _{j,k}{\alpha }_{ijk}{x}_{j}{x}_{k}+\sum _{j,k,l}{\beta }_{ijkl}{x}_{j}{x}_{k}{x}_{l}+\cdots =0$$

where ωi are the linear resonance frequencies, αijk and βijkl represent the quadratic and cubic nonlinear couplings of different vibrational mode, respectively. Quadratic and cubic nonlinearities result in mode coupling, frequency shifts (e.g., Duffing behavior), harmonic generation, and inter-modal energy transfer. Depending on the symmetry and boundary conditions, energy initially localized in one mode can be transferred to others over time, potentially leading to equipartition or recurrence (FPUT physics). FPUT observed that rather than approaching thermal equilibrium, energy initially placed in a single mode undergoes quasiperiodic recurrence, a phenomenon now understood in the context of near-integrability, nonlinear normal modes, and Kolmogorov-Arnold-Moser (KAM) theory34.

To model these systems, a simplified set of coupled Duffing oscillators can be considered:

$${\ddot{x}}_{i}+{\gamma }_{i}{\dot{x}}_{i}+{\omega }_{i}^{2}{x}_{i}+\sum _{j}{k}_{ij}{x}_{i}{x}_{j}+{\lambda }_{i}{x}_{i}^{3}={F}_{i}(t)$$

Here, kij models mode-mode coupling, λi represents self-nonlinearity, and Fi(t) is an external drive. Numerical integration can reveal energy redistribution, frequency shifts, and recurrence indicative of FPUT behavior.

Using time-domain displacement measurements, we identify multiple mechanical modes with frequencies below 11 Hz (see Fig. 4a). When investigating the mode more closely, we observe a slow drift in resonant frequency of different modes. Figure 4b shows mode dynamics for the three main vibrational modes of the system. We have recorded the temperature, background magnetic field, optical power and vacuum pressure over the course of the measurement and observed no correlations between these experimental parameters and the frequency changes observed.

Fig. 4: Evolution of vibrational spectrum of the levitated cube over time.
Fig. 4: Evolution of vibrational spectrum of the levitated cube over time.
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a Thermally excited displacement amplitude noise (x(ω)) plotted over several hours. b The zoomed-in spectra for three main modes, Modes A, B, and C,(respectively highlighted in red, blue and orange) correspond to the three principal motions of the particle: (A) the small tilting motion around a line in x-y plane for the cube, (B) the rotational mode around the z axis, and (C) the translational mode in the x-y plane. 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 β = 1 s−2m−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 × 10−7 and 4.1 × 10−7 Torr, respectively. The spectrum shown here is based on the data collected by single-pixel photo-detector.

Spectral analysis reveals evidence of nonlinear mode coupling, manifested partly in the generation of higher harmonics and the appearance of long-lived oscillations in non-fundamental modes. Figure 4c shows a vibrational spectra under strong initial drive where sideband and frequency shift are evident at early times after the external drive is stopped. The inset in Fig. 4c shows how a nonlinear oscillator develops small sidebands and slight spectral broadening as the amplitude of the drive increases. This schematic is included to help interpret the experimentally observed amplitude spectrum, where similar features arise at initial times when the excitation amplitude is high.

Phase analysis using the Hilbert transform uncovers partial phase coherence (see Fig. 5a) between the fundamental and its higher harmonic, with the histogram of phase differences peaking near π, suggesting that energy transfer between modes retains a degree of coordinated timing. Cross-correlation analysis of the time series supports this view (see Fig. 5b), showing oscillatory features in intermodal interactions and delayed thermalization.

Fig. 5: Fundamental modes and higher harmonics.
Fig. 5: Fundamental modes and higher harmonics.
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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, VY. 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 νx. 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 × 10−5 Torr. Data is collected using the single-pixel photo-detector.

As seen in Fig. 5c, in certain experimental conditions such as vacuum pressure (leading to drift of particle in the trap), we observe that the fifth harmonic of ωθ mode matches the ωx mode’s frequency. Under this condition we observe enhanced mode coupling where the low-frequency environmental excitations (near ωθ) gives rise to persistent energy deposited and confined in the system over many hours. The demonstration of coherence between higher harmonics and enhanced parametric mode coupling shown in Fig. 5 are other signatures of nonlinear dynamics, also observed in optical systems in the context of frequency comb generation6,35.

We examined their trajectories in a two-dimensional phase-space projection. By plotting the displacement of the x − mode (matching the 5th harmonics) against the fundamental mode (\({\nu }_{{\phi }_{z}}\)), we observed evolving Lissajous-like figures (see inset of Fig. 5c). These patterns indicate phase-locked motion at short timescales, but exhibit slow distortions and rotations over longer durations, consistent with weak nonlinear coupling and gradual energy exchange between the modes. The distortion of the classical Lissajous figures is a signature of underlying mode coupling and nonlinearity, leading to a slow drift in relative phase and amplitude. Such evolving trajectories provide visual evidence of the anharmonic and weakly interacting nature of the vibrational modes, which are expected in high-Q macroscopic levitated systems where even subtle nonlinear effects are preserved over long coherence times.

FPUT physics has been traditionally studied in optical systems36,37 and cold atoms38. The theoretical investigation of FPUT physics in graphene has been suggested2 but experimental studies in mechanical systems is rare. In our system, we are able to observe temporal recurrence in the amplitude evolution of individual modes (see Fig. 6a), indicating non-ergodic, structured energy exchange that deviates from simple thermalization. These features are hallmarks of weakly nonlinear dynamics, consistent with the onset of FPUT-like behavior.

Fig. 6: Correlation and thermalization.
Fig. 6: Correlation and thermalization.
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a Normalized kinetic energy of three main vibrational modes extracted using digital filtering shows oscillation and exchange of energy between the modes, akin to FPUT recurrence. Vacuum pressure used was 1 × 10−5 Torr. b–d Autocorrelation of strongly coupled modes at 1.4 Hz and 7 Hz and weakly coupled mode at 10.1 Hz, respectively. e, f Histogram of spectral entropy of first 10 modes for time windows immediately and 10 h after the excitation stops, respectively. The data used in this figure is collected employing the single-pixel photo-detector.

We compute the autocorrelation functions of individual mode amplitudes to probe the temporal coherence and memory effects in the system (see Fig. 6b–d). For the first and second modes, the autocorrelation exhibits a pronounced peak and decays slowly, indicating persistent correlations over long timescales. This is characteristic of low-frequency modes retaining coherence and being less susceptible to rapid energy dispersion. In contrast, the autocorrelation of higher frequency mode νz weakly coupled to others also peaks at zero lag time but shows a narrower profile. This suggests that while this mode is phase-correlated with the base mode, their coherence decays more rapidly, and they exhibit more localized temporal structure.

In addition, spectral entropy can be used to look for oscillations that can signal periodic energy spreading and refocusing. To quantify this, we compute the instantaneous Shannon entropy of the mode energy distribution at each time step. Shannon entropy has a long history as a nonlinear-dynamics metric, including its use in early synchronization studies by Tass et al.39, although here we apply it to the instantaneous distribution of modal energies rather than phase relationships. Let Ei(t) denote the instantaneous energy of the i-th mode, and let Pi(t) = Ei(t)/∑jEj(t) be the normalized energy fraction of mode i. The entropy at time t is then given by \(S(t)=-{\sum }_{i}{P}_{i}(t)\log {P}_{i}(t)\). We normalize S(t) by the maximum possible entropy \({S}_{\max }=\log (N)\), where N is the total number of modes, so that the normalized entropy \(\widetilde{S}(t)=S(t)/{S}_{\max }\) ranges from 0 (perfect energy localization) to 1 (complete equipartition). The histogram of the normalized entropy reveals that after a long evolution time the system spends a significant fraction of its time in highly ordered states, with entropy values clustered near zero(see Fig. 6e, f). As time evolves, nonlinear coupling and dispersion redistribute energy, leading to partial localization in mode space or the formation of coherent structures. The energy spectrum becomes more uneven—a few modes carry most of the energy while others are depleted - reducing the entropy toward smaller values. This indicates that the energy remains predominantly localized in a few modes rather than being evenly distributed, which is characteristic of non-ergodic or weakly thermalizing behavior.

To further study the onset of chaotic behavior, one can estimate the largest Lyapunov exponent (LLE), denoted by \({\lambda }_{\max }\), which quantifies the average exponential rate at which two initially close trajectories in a dynamical system diverge in phase space. Given two trajectories starting infinitesimally close with initial separation δx0, their separation δx(t) evolves approximately as

$$\parallel \delta {{\bf{x}}}(t)\parallel \approx \parallel \delta {{{\bf{x}}}}_{0}\parallel {e}^{{\lambda }_{\max }t}.$$

A positive value of \({\lambda }_{\max } > 0\) indicates sensitive dependence on initial conditions, a hallmark of chaotic dynamics, whereas \({\lambda }_{\max }\le 0\) corresponds to stable or periodic behavior.

As shown in Fig. 7, we estimated the LLE from the measured time-series data using phase-space reconstruction and the method of nearest-neighbor divergence. To compute the Lyapunov exponent from a single scalar observable, we reconstruct the effective state space using Taken’s delay-embedding method, where the state vector is formed as

$${{\bf{X}}}(t)=[x(t),x(t+\tau ),x(t+2\tau ),\ldots +x(t+7\tau )].$$

Here, an embedding dimension of 8 simply means that eight time-delayed copies of the measured signal are used to reconstruct the underlying dynamics. The embedding dimension was systematically increased to test the robustness of the result with LLE slope remained essentially constant at \({\lambda }_{\max }\simeq 0.0095\). The presence of a clear linear divergence region over approximately 100 s, combined with the stability of the slope across embeddings, supports the interpretation of λmax > 0 as an indicator for onset of deterministic chaos.

Fig. 7: The onset of chaos.
Fig. 7: The onset of chaos.
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Mean log divergence of nearby phase space trajectories as a function of time, estimated from the reconstructed phase space of the measured time series using embedding dimension 8, time delay 0.5 s, and Theiler window of 200 s. The divergence shows a clear linear growth region over 100 seconds, with a fitted slope corresponding to a largest Lyapunov exponent λmax ≃ 0.0095. Inset shows the median frequency of the 7 Hz mode as a function of time obtained from 30 s-long data segments. For this dataset, the cube was excited magnetically using the coil.

We also observe discrete jumps in the 7 Hz mode’s instantaneous frequency (see inset of Fig. 7). The observed sudden frequency jumps reflects the complex nonlinear dynamics underlying the system signifying transitions towards non-integrable dynamics and the onset of chaotic behavior. The results are consistent with the theoretical framework of KAM and associated nonlinear dynamics. Moreover, the evidence of chaotic behavior is illustrated in Supplementary Video 1, where excess excitation causes the particle to escape the trap.

Discussion

The results described in this work rely on the low dissipation rate and high intrinsic nonlinearity of diamagnetic trapping. The low dissipation rate arises from the suppression of eddy currents within the levitated object and from operating at low vacuum pressures. The nonlinearity originates from the trap geometry: as the object moves, it samples magnetic-field gradients whose magnitudes differ along each spatial direction, giving rise to nonlinear mode coupling. This behavior reflects the inherently non-quadratic shape of the trapping potential. These unique characteristics establish macroscopic diamagnetic levitation as an exceptionally low-dissipation and highly controllable platform for exploring nonlinear dynamics, advancing sensing applications, and probing fundamental physics. Several avenues can be pursued to further enhance performance and broaden scientific relevance.

Although our current system demonstrates exceptionally low mechanical dissipation, further reductions appear within reach. A simple re-magnetization of the magnets can increase the trapping height leading to four fold improvement (see “Methods”) to reach dissipation rates about 1 μHz. Eddy current damping–arising from field-induced currents in nearby conductive materials–can be mitigated by substituting metallic components with layered or patterned high-resistivity materials and carefully engineering magnetic field confinement. Simulations indicate that certain modes and optimized geometries may reduce dissipation rates to the nanohertz regime, approaching the fundamental limit set by residual gas damping.

In the low-dissipation regime and with reduced nonlinearities, the system could approach force sensitivities on the order of few \(pg/\sqrt{Hz}\). To the best of our knowledge, this performance exceeds that of state-of-the-art atomic interferometers and nanomechanical resonators, but in a much simpler, room-temperature setting. Furthermore, by inducing and stabilizing rotational motion of the levitated particle, the system could serve as an ultra-low-power gyroscope19 or accelerometer, potentially useful in portable or space-based inertial navigation systems.

In preliminary experiments, we have shown that the particle can be electrically driven (magnetically using a coil above the disk or electrostatically using voltages applied to the disk and the magnets) with high precision. This same interface could be used to implement feedback cooling24, dynamically reducing the effective temperature and suppressing unwanted nonlinear effects, or alternatively to enhance specific nonlinearities for controlled studies of energy transfer and chaos.

Moreover, the nonlinear characteristics of the trap–such as the relative strengths of quadratic and cubic terms in the restoring force can be finely tuned by adjusting the magnetic field landscape. This opens the door to engineering Duffing-like or parametrically coupled oscillators, where energy exchange between modes can be selectively enhanced or suppressed. Independent control of these terms would enable detailed studies of bifurcations, synchronization, and dynamical phase transitions in classical systems.

Conclusion

We have demonstrated that diamagnetic levitation of mm-scale dielectric objects can be achieved with ultra-low dissipation rates of about 4 μHz. Such low dissipation rates and nonlinear mode interactions in the system enable us to study various nonlinear effects including parametric mode coupling, higher harmonic generation, FPUT recurrence and positive Lyapunov exponents.

The platform studied here offers high tunability, e.g., via particle shape, magnetic trap geometry, or external perturbation, enabling accurate sensing and controlled studies of: KAM transitions, resonance overlap and chaos, prethermal plateaus, weak turbulence and thermalization scaling laws. This work lays the foundation for an experimental paradigm in classical nonlinear physics–one where macroscopic coherence, strong nonlinearity, and ultra-low damping coexist, opening a path to development of novel multimodal sensors and study of nonlinear phenomena long considered purely theoretical and numerical.

Specifically, nonlinear mode coupling, together with high spectral purity, can be a model for frequency comb generation in mechanical resonators40,41–a phenomenon extensively studied in ref. 3 but not yet realized in large-scale mechanical systems. The ability to generate and control mechanical frequency combs could have important implications for sensing.

On the other hand, because the levitated object is dielectric, the system can be adapted to incorporate optical trapping forces. As previously proposed42,43, it is possible to levitate a dielectric mirror and use it as the end-mirror of an optical cavity. Combining diamagnetic and “coherent" optical trapping could yield hybrid traps with greatly enhanced stiffness and mechanical frequencies, allowing access to the ultra-high-Q regime required for many sensing applications as well as tests of gravitational decoherence44, the Schrödinger-Newton equation45,46,47, and other collapse models in quantum mechanics. Such a system would enable sensitive searches for deviations from quantum linearity at macroscopic scales.

Methods

Materials

Quartz cubes were cut from a 2-inch quartz wafer (0.5 mm thick) using a CNC laser cutter (LPKF ProtoLaser R). To stabilize the wafer during the cutting process and prevent suction from the laser cutter, the quartz wafer was mounted on a 4-inch silicon wafer using a PMMA adhesive layer. The cutting was performed at a laser frequency of 300 kHz with a power of 7.3 W. To maintain vertical sidewalls and ensure precise beam focusing, the laser focus offset was adjusted every 250 μm of cube height. At each focus offset, 200 repetitions were applied using the specified parameters. Following the cutting process, PMMA was removed with acetone, and the quartz cubes were separated from the wafer.

Optical setup

The experimental setup is shown in Fig. S4. Illumination of the particle for measurement was provided by an 800 nm low-noise Ti:Sapphire laser (SolsTis; M Squared). A thin reflective glass surface placed beneath the levitated cube projected its shadow onto a single-pixel detector (BD10FS-Si; Quanfluence). The output was sampled using Liquid Instruments’ Moku:Go at 1 k Sample. Most data were collected overnight during weekends, typically over 10-h intervals. Two cameras (Thorlabs’ CS165MU), positioned above and to the side of the levitated particle, recorded the scattered light. The resulting images were analyzed to extract various vibrational modes and compared with theoretical predictions. Because this method is largely insensitive to alignment and power drift, it was used for ring-down measurements. Additional detection techniques, including interferometric and quadrant photodiode methods, were employed to validate the sensitivity and signal-to-noise ratio of the single-pixel and camera-based approaches.

To verify that particle frequency and amplitude fluctuations were not influenced by environmental factors, we monitored variation of the magnetic field (ΔB) a few centimeters above the magnet, outside the chamber, temperature (ΔT) at three locations adjacent to the chamber, air flow near the optical table, laser power, and typical vacuum pressure (P). Although variations were observed, e.g., ΔB ≃ 0.7 G, ΔT ≃ 1. 1 ∘C, P = 1.27( ± 0.01) × 10−5 Torr, over a period of 10 h in these environmental parameters, no correlation with fast particle dynamics ( < 1 h time scale) was found. The slow frequency drift over long time scales (several hours), however, may be attributed to the slow drift of the background magnetic field.

For particle loading, a vacuum tweezer was used to release the particle into the central hole of the top disk. A small mechanical perturbation caused the particle to “jump” into the trapping location. Its height and position relative to the magnet center were fine-tuned using three screws to adjust the tilt and height of the top disk. The entire setup was mounted on adjustable legs to allow coarse control of the overall alignment. The total assembly including the vacuum chamber, ion pump and the measurement optics were placed on top of a vibration isolation system (200CM-1-Minus-K).

To reach low vacuum pressures, we slightly annealed the chamber (MCF600-Kimball Physics) at about 20 degrees above the room temperature. This led to slight demagnetization of the rare-earth magnets (grade N42) lowering the trapping height of the particle. Simple re-magnetization of the magnets will regain the higher trapping height leading to reduction of the eddy current damping by a factor of three to four for all vibrational modes, with νz having the lowest dissipation.

Numerical simulation

The simulation of the field was carried out using axially symmetric modeling in COMSOL Multiphysics. The field cross section (shown in Fig. 1b) was then exported to MATLAB to be integrated over cubic boundaries. Variation of these boundaries over the allowed range of motion with respect to different axes was then employed to produced the potentials shown in Fig. 1c–e.