Abstract
Different approaches to quantum gravity, such as string theory^{1,2} and loop quantum gravity, as well as doubly special relativity^{3} and gedanken experiments in blackhole physics^{4,5,6}, all indicate the existence of a minimal measurable length^{7,8} of the order of the Planck length, . This observation has motivated the proposal of generalized uncertainty relations, which imply changes in the energy spectrum of quantum systems. As a consequence, quantum gravitational effects could be revealed by experiments able to test deviations from standard quantum mechanics^{9,10,11}, such as those recently proposed on macroscopic mechanical oscillators^{12}. Here we exploit the submillikelvin cooling of the normal modes of the tonscale gravitational wave detector AURIGA, to place an upper limit for possible Planckscale modifications on the groundstate energy of an oscillator. Our analysis calls for the development of a satisfactory treatment of multiparticle states in the framework of quantum gravity models.
Main
General relativity and quantum physics are expected to merge at the Planck scale, defined by distances of the order of ∼L_{p} and/or extremely high energies of the order of ∼E_{p} = cℏ/L_{p} = 1.2×10^{19} GeV. Therefore, present approaches to test quantum gravitational effects are mainly focused on highenergy astronomical events^{13,14,15}, which allowed stringent limits to the predicted breaking of Lorentz invariance at the Planck scale to be put in place^{16}. On the other hand, the emergence of a minimal length scale can result in relevant consequences also for lowenergy quantum mechanics experiments. The Heisenberg relation states that the uncertainties in the measurements of a position Δx and its conjugate momentum Δp are related by ΔxΔp≥ℏ/2; that is, the position and the momentum of a particle cannot be determined simultaneously with arbitrarily high accuracy. However, an arbitrarily precise measurement of only one of the two observables, say position, is still possible at the cost of our knowledge about the other (momentum), a fact that is obviously incompatible with the existence of a minimal observable distance. This consideration motivates the introduction of generalized Heisenberg uncertainty principles^{1,2,3,4,5,6,7}. As a consequence, an alternative way to check quantum gravitational effects would be to perform highsensitivity measurements of the uncertainty relation, to reveal any possible deviation from predictions of standard quantum mechanics^{9,10,11}.
In this framework, an optical experiment has recently been proposed to test quantum gravitational modifications of the canonical commutator for the variables associated with the centre of mass of a macroscopic object^{12}. The generalization from the position/momentum of a single particle to that defining a collective motion is well described in standard quantum mechanics. The canonical commutator [x,p] = iℏ remains valid for any Lagrangian coordinate x and its conjugate momentum p, including the coordinates describing the centre of mass or a normal vibrational mode of a macroscopic object. Recent experiments have indeed cooled down mechanical modes of microoscillators to their quantum ground state^{17,18,19}, and in this condition the oscillators indeed exhibited peculiar quantum properties^{17,20}. This clearly proves that a mechanical normal mode can be described by quantummechanical observables. On the other hand, if a modified commutator is considered, a selfconsistent description in terms of macroscopic coordinates is not straightforward. Some recent works have tackled the problem, proposing solutions that imply a strong suppression of the expected effect of Planckscale physics when probed by multiparticle objects^{21,22}. However, none of such approaches is fully satisfactory. An experimental analysis is therefore highly desirable, in particular if performed on macroscopic variables that exhibit quantum properties.
A direct consequence of a modified commutator, when applied to the position and momentum of an harmonic oscillator, is a change in the oscillator groundstate energy E_{min} with respect to the usual ℏω_{0}/2. The commutators commonly proposed in quantum gravity theories^{23,24}, giving origin to a position indetermination larger than in standard quantum mechanics (gravitational induced uncertainty), translate into a larger minimal energy. Therefore, an experiment measuring a low energy level E_{exp} for a normal mode of a macroscopic system puts a straightforward upper limit to the corresponding E_{min}:
Experimental systems particularly suitable for exploiting this relation are the cryogenic Weber bars, originally conceived and still working as detectors for gravitational waves^{25}. They consist of large metallic bars, weighing several tons and having a main longitudinal mechanical mode oscillating around ω_{0}/2π≃1 kHz. For our purpose, their favourable characteristics are their very large mass (M∼10^{13} times the Planck mass M_{p} = E_{p}/c^{2} = 2.2×10^{−8} kg); the large size of the groundstate momentum wave packet associated with their main modes (where, with respect to, for example, microoscillators, the relatively low frequency is compensated by the large effective mass M_{eff}); and the low level of thermal energy that is reached experimentally.
Some Weber bars have been operated at ultracryogenic temperature^{26}. However, for our purpose we focus on the AURIGA detector, whose first longitudinal mode, starting from a background bar temperature of 4.2 K, has been further cooled using a cold damping technique down to the millikelvin regime^{27}, a level that could be reached owing to its high mechanical quality factor. We remark that modal cooling techniques have been exploited to bring mechanical modes of microoscillators to their quantum ground state and that these experiments have proved that quantum behaviour of macroscopic coordinates can be obtained in this way^{17,18,19,20}. Therefore cold damping is a valid technique for our purpose of reaching and measuring the lowest oscillator modal energy, even if such modal cooling cannot be exploited to increase the sensitivity of the oscillating system as a detector of external excitation.
In AURIGA, the modal motion is measured by coupling the bar to two further oscillators: the first is a mechanical one, having the same frequency but a lower mass, and working as a resonant mechanical amplifier read by a capacitive transducer; the second one is a nearly resonant electrical LC circuit coupled to a d.c. superconducting quantum interference device amplifier^{28,29}. All of the three oscillators have been cooled down by cold damping, to roughly the same temperature. A detailed analysis of the system shows that a minimal energy of 1.3×10^{−26} J can be attributed to the first longitudinal mode of the bar, whose resonance frequency is ω_{0}/2π = 900 Hz (ref. 27). The motion of this mode is symmetrical with respect to the plane, perpendicular to the bar axis, that bisects the bar, and implies an oscillation of the centreofmass of each halfbar. The reduced mass of this couple of centreofmasses is M_{red} = M/2 = 1.1×10^{3} kg (where M is the bar physical mass), and the energy associated with the oscillation of the centreofmasses is ∼ 80% of the total modal energy. The measured modal energy E_{exp} is therefore an upper limit also for the minimal energy E_{min} of the oscillation of the centreofmasses and it can be used in equation (1) and in the relations that follow from it (a detailed description of the AURIGA detector, of the minimal measured energy and a discussion on the appropriate value of the mass to be used in this analysis are reported in the Supplementary Information).
We start our analysis from the uncertainty relation^{1,2,7} where β_{0} is a dimensionless parameter that should be around unity if the modification is efficient at the Planck scale. If β_{0} is larger than unity, it defines a new length scale where some new physics should come into play^{9}. Normalizing the coordinate and momentum to their groundstate wavepacket size according to and , the Hamiltonian operator for a harmonic oscillator is written as and the uncertainty relation given in equation (2) becomes where β = β_{0} (ℏmω_{0}/M_{p}^{2}c^{2}). The minimal oscillator energy is found for 〈X〉 = 〈P〉 = 0 using the equality in equation (4). Extracting ΔX from equation (4) and inserting it in equation (3) we obtain and minimizing with respect to ΔP we find the minimal energy where the last approximation (which simplifies the form but is not indeed necessary) is valid for β≫1. Comparing this expression of E_{min} with the measured E_{exp}, according to equation (1), we obtain β<(2E_{exp}/ℏω_{0}), which for β_{0} can be written in the meaningful form We now come back to the first longitudinal mode of AURIGA with cold damping, and consider the oscillation of the couple of centreofmasses of the halfbars. Using the mode resonance frequency, the reduced mass M_{red} and the measured modal energy E_{exp} = 1.3×10^{−26} J, we obtain β<4.4×10^{4} and β_{0}<3×10^{33}. Our upper limit for β_{0} is still far from forbidding new physics at the Planck scale. It can be compared to similar limits of β_{0}<10^{34}, imposed by the lack of observed deviations from standard theory at the electroweak scale; β_{0}<10^{36}, calculated from the accurate measurement of the Lamb shift in hydrogen^{9}; and β_{0}<4×10^{34} from the 1S–2S level energy difference, again in hydrogen^{22}. These limits are summarized in Fig. 1, together with those obtainable with the same method from other experiments with cold macroscopic oscillators (listed in the Supplementary Information).
In analysing the consequence of generalized Heisenberg uncertainty principles on the foreseen behaviour of a macroscopic object, the preliminary assumption that the modified rules can be applied to macroscopic coordinates must be carefully taken into account. With this consideration in mind, our experimental analysis has a double interpretation. On one side, it sets interesting limits to the Planckscale physics, and helps to compare the consequences of various approaches to quantum gravity. On the other side, it strongly calls for an effort in developing theories that offer a reasonable path from basic properties of spacetime geometry and of the measurement process (that is, main issues of the theories that should merge into quantum gravity) to multiparticle and macroscopic reality. This research field, which suffers from poor experimental feedback, is likely to benefit from precise metrological systems, including for example interferometric gravitational wave detectors^{25,30,31}, whose data are already the subject of analyses focused on quantum gravity effects^{32,33}, and ultracryogenic devices that could in the near future include, owing to dedicated experiments, oscillators with mass of the order of, or larger than, M_{p} operating in their fundamental quantum state^{27}.
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Acknowledgements
F. Marin and F. Marino thank D. Seminara and M. Inguscio for useful discussions.
Author information
Affiliations
Dipartimento di Fisica, Università di Firenze, Via Sansone 1, I50019 Sesto Fiorentino (FI), Italy
 Francesco Marin
Istituto Nazionale di Fisica Nucleare (INFN), Sezione di Firenze, Via Sansone 1, I50019 Sesto Fiorentino (FI), Italy
 Francesco Marin
European Laboratory for NonLinear Spectroscopy (LENS), Via Carrara 1, I50019 Sesto Fiorentino (FI), Italy
 Francesco Marin
 & Francesco Marino
CNRIstituto dei Sistemi Complessi, Via Madonna del Piano 10, I50019 Sesto Fiorentino (FI), Italy
 Francesco Marino
Institute of Materials for Electronics and Magnetism, NanoscienceTrentoFBK Division, I38123 Trento, Italy
 Michele Bonaldi
INFN, Gruppo Collegato di Trento, I38123 Povo, Trento, Italy
 Michele Bonaldi
 , Paolo Falferi
 , Renato Mezzena
 & Giovanni A. Prodi
INFN, Sezione di Padova, via Marzolo 8, I35131 Padova, Italy
 Massimo Cerdonio
 , Livia Conti
 , Luca Taffarello
 , Gabriele Vedovato
 & JeanPierre Zendri
Istituto di Fotonica e Nanotecnologie, CNR—Fondazione Bruno Kessler, I38123 Povo, Trento, Italy
 Paolo Falferi
 & Andrea Vinante
Dipartimento di Fisica, Università di Trento, I38123 Povo, Trento, Italy
 Renato Mezzena
 & Giovanni A. Prodi
INFN, Laboratori Nazionali di Legnaro, I35020 Legnaro (PD), Italy
 Antonello Ortolan
Leiden Institute of Physics, Niels Bohrweg 2, 2333 CA Leiden, The Netherlands
 Andrea Vinante
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Contributions
F. Marin conceived the work, which was further developed with F. Marino and all other authors. F. Marin and F. Marino wrote the manuscript with input from all of the authors.
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The authors declare no competing financial interests.
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Correspondence to Francesco Marin.
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