Probing deformed commutators with macroscopic harmonic oscillators

Abstract

A minimal observable length is a common feature of theories that aim to merge quantum physics and gravity. Quantum mechanically, this concept is associated with a nonzero minimal uncertainty in position measurements, which is encoded in deformed commutation relations. In spite of increasing theoretical interest, the subject suffers from the complete lack of dedicated experiments and bounds to the deformation parameters have just been extrapolated from indirect measurements. As recently proposed, low-energy mechanical oscillators could allow to reveal the effect of a modified commutator. Here we analyze the free evolution of high-quality factor micro- and nano-oscillators, spanning a wide range of masses around the Planck mass mP (≈ 22 μg). The direct check against a model of deformed dynamics substantially lowers the previous limits on the parameters quantifying the commutator deformation.

Figure 1. Oscillating devices.

Finite elements simulation of the shapes of the oscillation modes investigated in this work (a, b, c), phase (d, e, f) and amplitude (g, h, i) of the oscillation during a free decay, obtained by phase-sensitive analysis of the measured position. In the left panels, the colour scale represents the relative magnitude of the displacement for each modal shape, decreasing from red to blue. Red solid lines: linear and exponential fits respectively to the phase (blue dots) and the amplitude (green dots) experimental data. Graphs (a, d, g) refer to the DPO oscillator, consisting of two inertial members, head and a couple of wings, linked by a torsion rod (the neck) and connected to the outer frame by a leg. The displayed AS mode consist of a twist of the neck around the symmetry axis and a synchronous oscillation of the wings. The elastic energy is primarily located at the neck, where the maximum strain field occurs during the oscillations, while the leg remains at rest and the foot can be supported by the outer frame with negligible energy dissipation. Graphs (b, e, h) refer to the balanced wheel oscillator. The central disk has a diameter of 0.54 mm, and the shape of the beams maintain it flat during the motion (as shown by its homogeneous colour) reducing the dissipation on the 0.4-mm diameter optical coating. The four paddles are carefully sized in order to balance the stress induced by the strain of the beams on the supporting wheel, such that the joints correspond to nodal points. An additional external wheel further improves the isolation from the background. Graphs (c, f, i) refer to the L=0.5-mm side, 30-nm thick, square membrane of stoichiometric SiN membrane. Its high stress increases the mechanical quality factor thanks to the dilution effect.

Figure 2. Residual angular frequency fluctuations as a function of the oscillation amplitude.

The fluctuations Δω are measured during the free decay, for the DPO (a), wheel (b) and membrane (c) oscillators. On the upper axes, the same oscillation amplitudes are normalized to the respective oscillator ground state wavefunction width . Red solid lines are the fits with Equation 8, dashed lines report the 95% confidence area. In the inset, we report the values of the quadratic coefficient b measured for the membrane oscillator at different excitation amplitudes, with their 95% confidence error bars (for appreciating the improvement in the accuracy, we just show the positive vertical semi-axis in logarithmic scale). For the two points at highest amplitude, the measured b is significantly different from zero. The green lines show the interval of b calculated from the nonlinear behaviour observed in the frequency domain for stronger excitation.

Figure 3. Upper limits to the deformed commutator.

The parameter β₀ quantifies the deformation to the standard commutator between position and momentum, or the scale below which new physics could come into play. Full symbols reports its upper limits obtained in this work, as a function of the mass. Red dots: from the dependence of the oscillation frequency from its amplitude; magenta stars: from the third harmonic distortion. In the former data set, for the intermediate mass range (10–100 μg), we report the results obtained with two different oscillators. Light blue shows the area below the electroweak scale, dark blue the area that remains unexplored. Dashed lines report some previously estimated upper limits, obtained in mass ranges outside this graph (as indicated by the arrows). Green: from high-resolution spectroscopy on the hydrogen atom, considering the ground state Lamb shift (upper line) and the 1S–2S level difference (lower line). Magenta: from the AURIGA detector. Yellow: from the lack of violation of the equivalence principle. The vertical line corresponds to the Planck mass (22 μg).