Probing deformed commutators with macroscopic harmonic oscillators

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.

The emergence of a minimal observable length, at least as small as the Planck length L P = G/c 3 = 1.6 × 10 −35 m, is a general feature of different quantum gravity models [1,2].In the framework of quantum mechanics, the measurement accuracy is at the heart of the Heisenberg relations, that, however, do not imply an absolute minimum uncertainty in the position.An arbitrarily precise measurement of the position of a particle is indeed possible at the cost of our knowledge about its momentum.This consideration motivated the introduction of generalized uncertainty principles (GUPs) [1][2][3][4][5][6][7][8], such as Eq. 1 implies indeed a nonzero minimal uncertainty ∆q min = √ β 0 L P .The dimensionless parameter β 0 is usually assumed to be around unity, in which case the corrections are negligible unless energies (lengths) are close to the Planck energy (length).However, since there are no theories supporting this assumption, the deformation parameter has necessarily to be bound by the experiments.Any experimental limit β 0 > 1 would set a new physical length scale, √ β 0 L P , below which some new physics could come into play [9].
A direct consequence of relation ( 1) is an increase of the ground state energy E min of an harmonic oscillator.Recently, an upper limit to E min has been placed by analysing the residual motion of the first longitudinal mode of the bar detector of gravitational waves AURIGA [10,11].Although the imposed bound, β 0 < 10 33 , is extremely far from the Planck scale, it provides a first measurement just below the electroweak scale (corresponding to 10 17 L P ).
To the GUP (1) it is possible to associate a modified canonical commutator [1,3,4]: Its introduction represents a further conceptual step, as it defines the algebraic structure from which the GUP should follow, and it implies changes in the whole energy spectrum of quantum systems, as well as in the time evolution of a given observable.
Because of its importance as a prototype system, several studies have been focused on harmonic oscillators.Modifications of stationary states are calculated in Refs.[12][13][14].Approaches to construct generalized coherent states are proposed in Refs.[15,16].The modified time evolution and expectation values of position and momentum operators are discussed in [17][18][19], while in Ref.
[20] Chen et al. calculate the temporal behaviour of the position and momentum uncertainties in a coherent state, finding an unexpected squeezing effect.
In spite of this huge theoretical interest, the subject suffers from the complete lack of dedicated experiments and so far limits to the deformation parameters have been extrapolated from indirect measurements [9,21,22].It has recently been proposed that the effect of a modified commutator could be revealed studying the opto-mechanical interaction of macroscopic mechanical oscillators [23].Here we elaborate a different experimental protocol and describe a set of dedicated experiments with state-of-the-art micro-and nano-oscillators.
The basic idea of our analysis is assuming that the commutation relations between the operator q describing a measured position in a macroscopic harmonic oscillator, and its conjugate momentum p, are modified with respect to their standard form.In other words, and more generally, we suppose that the deformed commutator should be applied to any couple of position/momentum conjugate observables that are treated in a quantum way in experiments and standard theories.At the same time, we keep the validity of the Heisenberg equations for the temporal evolution of an operator Ô, i.e. d Ô/dt = [ Ô, H]/i , where H is the Hamiltonian.For an oscillator with mass m and resonance angular frequency ω 0 , we also assume that the Hamiltonian maintains its classical form H = mω 2 0 q 2 /2 + p 2 /2m.Such hypotheses are also underlying the proposal of Ref. [23].We first define the usual dimensionless coordinates Q and P, according to q = √ /(mω 0 ) Q and p = √ mω 0 P. The Hamiltonian is now written in the standard form H = ω 0 2 Q 2 + P 2 and the commutator of Eq. (2) becomes where β = β 0 mω 0 /m 2 P c 2 is a further dimensionless parameter that we assume to be small (β 1).Such assumption will have to be consistent with the experimental results.
We now apply the transform discussed, e.g., in Ref. [22].To our purpose, P is just an auxiliary operator, we do not need to decide if either P or P corresponds to the classical momentum.Q and P obey the (non deformed) canonical commutation relation [Q, P] = i.At the first order in β, the Hamiltonian can now be written as The Heisenberg evolution equations for Q and P, using the Hamiltonian (5) read The coupled relations (6) are formally equivalent to the equations describing the evolution of a free anharmonic oscillator with position − P (Q is its conjugate momentum), in a potential The Poincaré's solution [24], for initial conditions − P(0) = A and Ṗ(0) = 0, is − P(t) = A (1 − /32) cos( ωt) + ( /32) cos(3 ωt) where = −4A 2 β/3 and ω = 1 − 3 8 ω 0 .The solution is valid at the first order in , and implies two relevant effects with respect to the harmonic oscillator: the appearance of the third harmonic and, less obvious, a dependence of the oscillation frequency on the amplitude (more precisely, a quadratic dependence of the frequency shift on the oscillation amplitude).
Using again Eq. (6b) to find Q(t), keeping the first order in βQ 2 0 where Q 0 is the oscillation amplitude for Q, we obtain where P. Pedram calculates in Ref. [19] the evolution of an harmonic oscillator with an Hamiltonian deformed according to the GUP considered in this work, and finds a frequency modified as (in our notation) ω = ω 0 1 + βQ 2 0 .Such expression is equivalent to Eq. ( 8) in the limit of small (βQ 2 0 ), satisfied in the present work.
We have performed the experiments with highly isolated oscillators, i.e., with a high mechanical quality factor Q m ω 0 τ, where τ is a long but nonetheless finite relaxation time, responsible for an additional term −P/τ in the right hand side of Eq. (6b).Damping has a twofold effect: (i) an exponential decay of the oscillation amplitude; (ii) a nontrivial time-dependence of the phase.In the limit Q m 1, the dynamics is described by a modified version of Eq. ( 7) with the replacements ωt → Φ(t), implying ω(t) = dΦ/dt, and We have examined three kinds of oscillators, with masses of respectively ≈ 10 −4 kg, ≈ 10 −7 kg, and ≈ 10 −11 kg.The measurements are performed by exciting an oscillation mode and monitoring a possible dependence of the oscillation frequency and shape (i.e., harmonic contents) on its amplitude, during the free decay.In order to keep a more general analysis, we will consider both indicators independently.
The first device is a "double paddle oscillator" (DPO) [25] made from a 300 µm thick silicon plate (Fig. 1a).Thanks to its shape, for two particular balanced oscillation modes, the Antisymmetric torsion modes (AS), the oscillator is supported by the outer frame with negligible energy dissipation and it can therefore be considered as isolated from the background [26].Vibrations are excited and detected capacitively, thanks to two gold electrodes evaporated over the oscillator, and two external electrodes.The sample is kept in a vacuum chamber, and its temperature is stabilized at 293 K within 2 mK, a crucial feature to maintain a constant resonance frequency during the measurements.We have monitored the AS2 mode, with a resonance frequency of 5636 Hz and a mechanical quality factor of 1.18 × 10 5 (at room temperature).The overall center-of-mass (c.m.) of the oscillator remains at rest during the AS motion.To our purpose, we consider the positions of the couple of c.m.'s corresponding to the two half-oscillators that move symmetrically around the oscillator rest plane (a deeper discussion of this issue is reported in Ref. [10]).The meaningful mass is the reduced mass of the couple of half-oscillators, that is calculated by FEM simulations and is m = 0.033 g.
For the measurements at intermediate mass we have used a silicon wheel oscillator, made on the 70 µm thick device layer of a SOI wafer and composed of a central disk kept by structured beams [27], balanced by four counterweights on the beams joints that so become nodal points (Fig. 1b) [28].Finally, the lighter oscillators is a L = 0.5 mm side, 30 nm thick, square membrane of stoichiometric silicon nitride, grown on a 5 mm × 5 mm, 200 µm thick silicon substrate [29].Thanks to the high tensile stress, the vibration can be described by standard membrane modes, the lowest one (monitored in this work) with shape z(x, y) = A cos(πx/L) cos(πy/L) where (x, y) are the coordinates measured from the membrane center, along directions parallel to its sides (Fig. 1c).The physical mass of the membrane is 20 ng, respectively, and the c.m. is at the position (0, 0, z cm ) with z cm = 4A/π 2 (the central position A is the monitored observable).We have performed the measurements in a cryostat at the temperature of 65 K and pressure of 10 −4 Pa, where the oscillation frequency is 747 kHz and the quality factor is 8.6 × 10 5 .Excitation and readout are performed as in the experiment with the wheel oscillators.
The first step in the data analysis is applying to the data stream q(t) a numerical lock-in: the two quadratures X(t) and Y(t) are calculated by multiplying the data respectively by sin(ω 0 t) and cos(ω 0 t), where ω 0 is the oscillation angular frequency of the acquired time series, estimated preliminarily from a spectrum, and applying appropriate low-pass filtering.The oscillation amplitude is calculated as q 0 (t) = √ X 2 + Y 2 and the phase as Φ(t) = arctan(Y/X).For the DPO oscillator, this process is directly performed by the hardware lock-in amplifier, that is also used to frequency down-shift the signal of the wheel oscillator at cryogenic temperature before its acquisition.q 0 (t) is fitted with an exponential decay (examples are shown in Fig. 1) while Φ(t), that always remains within ±π rad, is fitted with a linear function that gives the optimal frequency and phase with respect to the preliminary tries ω 0 and Φ(0) = 0.The residuals ∆Φ of the fit are differentiated to estimate the fluctuations ∆ω in the oscillation frequency.In Fig. 2 we show ∆ω as a function of q 0 , together with its fit with the function ∆ω = a + bq 2 0 .The derived value and uncertainty in the quadratic coefficient b are the meaningful quantities that can be used to establish upper limits to the  deformation parameter β 0 .The background mechanical noise in all the experiments is dominated by the oscillator thermal noise (as verified with spectra taken without excitation).The consequent statistical uncertainty in the calculated ∆ω is inversely proportional to the amplitude q 0 , and such a weight is indeed used in the fitting procedures.
In the case of the DPO oscillator (Fig. 2a), ∆ω vs q 0 has a clear shape that is given by the intrinsic oscillator non-linearity.A similar, weaker effect is observed for the wheel oscillator at cryogenic temperature and for the membrane, at the largest excitation amplitudes (Fig. 2b-c).The quadratic coefficient for the membrane is in agreement with its calculation based on the nonlinear behaviour observed for larger amplitudes in the frequency domain [30].The meaningful quantity to calculate an upper limit to β 0 is the mean value of b plus its uncertainty.The latter is calculated from the standard deviation on several independent measurements, and it is in agreement with the error estimated from the residuals of each fit, after decimation of the data sets to obtain uncorrelated data points.The experiment has been repeated for several excitation levels, finding the expected improvements in the upper limit to b at increasing amplitudes (inset of Fig. 2c).
As previously discussed, a further useful indicator is the amplitude of the third harmonic component, also extracted with a lock-in procedure.The value of β inferred from such parameter using Eq. ( 7), is to ascribe to the relatively poor linearity of the readout, and is therefore considered as an upper limit to possible quantum gravity effects.the electroweak scale, dark blue the area that remains unexplored.Dashed lines reports 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) [21] and the 1S-2S level difference (lower line) [22].Magenta: from the AURIGA detector [10,11].Yellow: from the lack of violation of the equivalence principle [39].
A model-independent constraint to possible effects of a deformed commutator can be derived from the residual frequency fluctuations ∆ω, considered as a function of the oscillation amplitude (reported in Fig. 2).To this purpose, we summarize in Table I the maximum relative frequency shift and the maximum dimensionless oscillation amplitude Q 0 (0), for the different oscillator masses examined in this work.These data can be used to test any modified dynamics and provide the consequent upper limits to the involved parameters.
For a more accurate and specific bound, we focus on the model described by Eqs.(7)(8).The values and uncertainties in β and β 0 are obtained from b and from the third harmonic distortion, using the oscillator parameters (namely, its mass and frequency).In Table I we summarize our results for the different upper limits, given at the 95% confidence level.The results for β 0 are also displayed in Fig. 3 as a function of the oscillator mass, and compared with some previously existing limits.We have achieved a significant improvement, by many orders of magnitude, working on systems with disparate mass scales and considering different measured observables.
In conclusion, we have performed an extended experimental analysis of the possible dependence of the oscillation frequency and third harmonic distortion on the oscillation amplitude in micro-and nano-oscillators, spanning a wide range of masses.Assuming that a deformed commutator between position and momentum governs the dynamics through standard Heisenberg equations, we obtain a reduction by many orders of magnitude of the previous upper limits to the parameters quantifying the commutator deformation.We remark that the measurements have been performed on state of the art oscillators, allowing low statistical uncertainty (due to the high mechanical quality factor), low background noise (thanks to the shot-noise limited detection and the cryogenic environment), and the highest excitation amplitude allowed by each oscillator.The latter condition is not commonly explored in metrological micro-and nano-oscillators [31,32], and we could indeed achieve the limit given by the intrinsic oscillators non-linearity.
Extending the use of the Heisenberg evolution equations with deformed commutators from an ideal particle to a macroscopic dynamics is not free from conceptual problems [33].In particular, it is unknown at which constituent particle level quantum gravity effects could intervene [34].
Moreover, we remark that in quantum mechanics the wavefunction associated to the c.m. can have properties that cannot be simply reduced to the coordinates of such constituent particles.A direct extrapolation from quantum to classical dynamics, discussed e.g. in Refs.[35][36][37], implies crucial consequences, the first being the violation of the equivalence principle [38][39][40].Current bounds to such violation, obtained using sensitive torsion balances [41], correspond to a deformation parameter β 0 < 10 21 [39].Our limits are substantially lower.Our model shows that remarkable deviations from classical trajectories are, in any case, expected as soon as the momentum is of the order of (or exceeds) m P c.This condition is straightforward at astronomic level [42], and even for kg scale bodies.This may either indicate the breakdown of the Eherenfest' theorem at all scales [18] (requiring to revise the rules connecting quantum to classical dynamics), or a possible mass dependence of the deformation parameter.In this context our experiments, involving a wide range of masses, taking as "natural" reference the Planck mass, become particularly meaningful.Our experimental results, when used to set limits on the deformed commutator (2), should not be simply intended as a check of possible deformations of quantum mechanics, but as a test of a 'composite' hypothesis, involving also the form of the classical limit corresponding to the modified quantum rules.
We finally remark that a clear quantum signature have been recently obtained even in "macroscopic" nano-oscillators [43][44][45][46] very similar to those exploited in this work, suggesting that well isolated mechanical oscillators are indeed privileged experimental systems to explore the classicalto-quantum transition.Since gravity effects could play a role in the wavefunction decoherence that marks such transition [47,48], and it cannot be excluded that quantum gravity is inextricably linked to peculiar quantum features, an intriguing extension of the present experiment (or a similar investigation) would naturally be performed with macroscopic oscillators in a fully quantum regime.

FIG. 1 :
FIG.1: 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.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.Graphs (c, g, i) refer to the SiN membrane.
On the surface of the central disk, a multilayer SiO 2 /Ta 2 O 5 dielectric coating forms an high reflectivity mirror.The device also includes intermediate stages of mechanical isolation.The design strategy allows to obtain a balanced oscillating mode (its resonance frequency is 141 797 Hz), with a planar motion of the central mass (significantly reducing the contribution of the optical coating to the structural dissipation) and a strong isolation from the frame.The oscillator is mechanically excited using a piezoelectric ceramic glued on the sample mount.The surface of the core of the device works as end mirror in one arm of a stabilized Michelson interferometer, that allows to measure its displacement.The quality factor surpasses 10 6 at the temperature of 4.3 K, kept during the measurements.As for the DPO, the c.m. of the oscillator remains at rest and, for the following analysis of the possible quantum gravity effects, we consider the reduced mass m = 20 µg.We have also performed room temperature measurements on a simpler device, lacking of counterweights, with an oscillating mass of 77 µg and a frequency of 128 965 Hz.

FIG. 2 :
FIG. 2: Residual angular frequency fluctuations ∆ω as a function of the oscillation amplitude, during the free decay, for the DPO (a), wheel (b) and membrane (c) oscillators.On the upper axes, the scales are normalized to the respective oscillator ground state wavefunction width √ /mω 0 .Red solid lines are the fits with Equation 8, dashed lines reports 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.

FIG. 3 :
FIG. 3: The parameter β 0 quantifies the deformation to the standard commutator between position and momentum, or the scale √ β 0 L P 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; green stars: from the third harmonic distortion.Light blue shows the area below

TABLE I :
Maximum relative frequency shifts measured for different oscillators, corresponding oscillation amplitudes, and upper limits to the deformation parameters β and β 0 obtained in this work.