A gravitational analogue of Brownian motion could now make it possible to investigate Planck-scale physics using the latest quantum technology.

Peut-on atteindre l’échelle de Planck par l’expérience?

Malgré le triomphe de la relativité générale comme théorie isolée de la gravitation, il s’est avéré difficile de l’intégrer dans l’ensemble de la physique, et en particulier de la concilier avec la mécanique quantique. Faute de pouvoir s’appuyer sur des données expérimentales, le domaine de la gravité quantique est resté mal défini. Les expériences devraient pouvoir aller jusqu’à l’échelle de Planck; or, pour y arriver directement au moyen d’accélérateurs, il faudrait des machines atteignant une énergie de 1019 GeV. Toutefois, il est possible qu’à proximité de l’échelle de Planck les gravitons à l’état fondamental étirent et écrasent la géométrie de l’espace–temps, ce qui entraînerait la décohérence des ondes de matière quantique. Grâce à cet effet, on pourrait peut-ètre faire des expériences de physique à l’échelle de Planck au moyen d’un interféromètre atomique, par un procédé analogue, dans le domaine de la gravité, à l’étude du mouvement brownien.

Physics on the large scale is based on Einstein’s theory of general relativity, which interprets gravity as the curvature of space–time. Despite its tremendous success as an isolated theory of gravity, general relativity has proved problematic in its integration with physics as a whole, and in particular with the physics of the very small, which is governed by quantum mechanics. There can be no unification of physics that does not include both general relativity and quantum mechanics. Superstring theory and its recent extension to the more general theory of branes is a popular candidate for a unified theory, but the links with experiment are very tenuous. The approach known as loop quantum gravity attempts to quantize general relativity without unification, and has so far received no obvious experimental verification. The lack of experimental guidance has made the issue extremely hard to pin down.

One hundred years ago, when Max Planck introduced the constant named after him, he also introduced the Planck scales, which combined his constant with the velocity of light and Isaac Newton’s gravitational constant to give the fundamental Planck time around 10^{–43} s, the Planck length around 10^{–35} m and the Planck mass around 10^{–8} kg. Experiments on quantum gravity require access to these scales, but direct access using accelerators would require machines that reach an energy of 10^{19} GeV, well beyond the reach of any experiments currently conceivable.

For almost a century it has been widely perceived that the lack of experimental evidence for quantum gravity presents a major barrier to a breakthrough. One possible way of investigating physics at the Planck scale, however, is to use the kind of approach developed by Albert Einstein in his study of thermal fluctuations of small particles through Brownian motion, where he showed that the visible motion provided a window onto the invisible world of molecules and atoms. The idea is to access the Planck scale by observing decoherence in matter waves caused by quantum fluctuations, as first proposed using neutrons more than 20 years ago by CERN’s John Ellis and colleagues (Ellis *et al*. 1984). Since then, ultra-cold atom technologies have advanced considerably, and armed with the sensitivity of modern atomic matter-wave interferometry we are now in a position to consider using “macroscopic” instruments to access the Planck scales, a possibility that William Power and Ian Percival outlined more recently (Power and Percival 2000).

Our recent work represents a new approach to gravitationally produced decoherence near the Planck scale (Wang *et al*. 2006). It has been made possible by the recent discovery by one of us of the conformal structure – the scaling property of geometry – of canonical gravity, one of the earliest important approaches to quantum gravity. This leads to a theoretical framework in which the conformal field interacts with gravity waves at zero-point energy using a conformally decomposed Hamiltonian formulation of general relativity (Wang 2005). Working in this framework, we have found that the effects of ground-state gravitons on the geometry of space–time can lead to observable effects by causing quantum matter waves to lose coherence.

The basic scenario is that near the Planck scale, ground-state gravitons constantly stretch and squash the geometry of space–time causing conformal fluctuations in space–time. This process is analogous to the Brownian motion of a pollen particle interacting with ambient molecules of much smaller sizes. It means that information on gravitons near the Planck scale can be extracted by observing the conformal fluctuations of space–time, which can be done by analysing their blurring effects on coherent matter waves.

The curvature of space–time produces changes in proper time, the time measured by moving clocks. For sufficiently short time intervals, near the Planck time, proper time fluctuates strongly owing to quantum fluctuations. For longer time intervals, proper time is dominated by a steady drift due to smooth space–time. Proper time is therefore made up of the quantum fluctuations plus the steady drift. The boundary separating the shorter time-scale fluctuations from the longer time-scale drifts is marked by a cut-off time, τ_{cut-off}, which defines the borderline between semi-classical and fully quantum regimes of gravity. It is given by τ_{cut-off} = λ*T*_{Planck}, for quantum-gravity theories, where *T*_{Planck} is the Planck time, and λ is a theory-dependent parameter determined by the amplitude of zero-point gravitational fluctuations. A lower limit on λ is given by noting that the quantum-to-classical transition should occur at length scales λ*L*_{Planck} that are greater than the Planck length *L*_{Planck} by a few orders of magnitude, so we can expect λ > 10^{2}.

A matter-wave interferometer can be used to measure quantum decoherence due to fluctuations in space–time, and hence provide experimental guidance to the value of λ. In an atom interferometer an atomic wavepacket is split into two wavepackets, which follow different paths before recombining (see “Atom interferometer”). The phase change of each wavepacket is proportional to the proper time along its path, resulting in an interference pattern when the wavepackets recombine. The detection of the decoherence due to space–time fluctuations on the Planck scale would provide experimental access to quantum-gravity effects analogous to accessing to atomic scales provided by Brownian motion.

In our analysis we found an equation that gives λ (See “equation”).

*M* is the mass of the quantum particle; *T* is the separation time before two wavepackets recombine; and Δ denotes the loss of contrast of the matter wave and is a measure of the decoherence (Wang *et al*. 2006). Existing matter-wave experiments set limits on the size of λ, their sensitivity depending on both Δ and *M*. Results using caesium atom interferometers (Chu *et al*. 1997) and also from a fullerene C_{70} molecule interferometer (Hackermueller *et al*. 2004) with its larger value of *M*, both set a lower bound for λ of the order of 10^{4}, well within the theoretical limits of λ > 10^{2}. This suggests that the sensitivities of advanced matter-wave interferometers may well be approaching the fundamental level due to quantum space–time fluctuations. Investigating Planck-scale physics using matter-wave interferometry may therefore become a reality in the near future.

Further improved measurements will confirm and refine this bound on λ, pushing it to higher values. An atom interferometer in space, such as the proposed HYPER mission, could provide such improvements. However, the lower bound of λ calculated using current experimental data is already within the expected range. This is a very good sign and strongly suggests that the measured decoherence effects are converging towards the fundamental decoherence due to quantum gravity. Therefore, a space mission flying an atom-wave interferometer with significantly improved accuracy will be able to investigate Planck-scale physics.

As well as causing quantum matter waves to lose coherence at small scales, the conformal gravitational field is responsible for cosmic acceleration linked to inflation and the problem of the cosmological constant. Our formula, which relates the measured decoherence of matter waves to space–time fluctuations, is “minimum” in the sense that ground-state matter fields have not been taken on board. Their inclusion may further increase the estimated conformal fluctuations and result in an improved “form factor” in our formula. In this sense, the implications go beyond quantum gravity to more generic physics at the Planck scale. Furthermore, it opens up new perspectives of the interplay between the conformal dynamics of space–time and vacuum energy due to gravitons, as well as elementary particles. (A well known example of vacuum energy is provided by the Casimir effect.) These may have important consequences on cosmological problems such as inflation and dark energy.

### Further reading

S Chu *et al*. 1997 *Phil. Trans. R. Soc. Lond. A* **355** 2223.

J Ellis *et al*. 1984 *Nucl. Phys. B* **241** 381.

L Hackermueller *et al*. 2004 *Nature* **427** 711.

W L Power and I C Percival 2000 *Proc. Roy. Soc. Lond. A* **456** 955.

C H-T Wang, R Bingham and J T Mendonça 2006 *Class. Quantum Grav*. **23** L59.

C H-T Wang 2005 *Phys. Rev. D* **71** 124026.

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