Spontaneous emission of graviton rates for the quantum bouncer states are
evaluated

Boughn S

G Pignol

K Protasov

V Nesvizhevsky

Weinberg S

English

arXiv

Crossref

A note on spontaneous emission of gravitons by a quantum bouncer

International audienceSpontaneous emission of graviton rates for the quantum bouncer states are evaluated

Pignol, G.

Protasov, K.V.

Nesvizhevsky, V.V.

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A note on spontaneous emission of gravitons by aquantum bouncerG. Pignol, K.V. Protasov, V.V. NesvizhevskyTo cite this version:G. Pignol, K.V. Protasov, V.V. Nesvizhevsky. A note on spontaneous emission of gravitons bya quantum bouncer. Classical and Quantum Gravity, IOP Publishing, 2007, 24, pp.2439-2441.<10.1088/0264-9381/24/9/N02>. <hal-00136128>HAL Id: hal-00136128https://hal.archives-ouvertes.fr/hal-00136128Submitted on 12 Mar 2007HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestine´e au de´poˆt et a` la diffusion de documentsscientifiques de niveau recherche, publie´s ou non,e´manant des e´tablissements d’enseignement et derecherche franc¸ais ou e´trangers, des laboratoirespublics ou prive´s.hal-00136128, version 1 - 12 Mar 2007Spontaneous emission of graviton by a quantumbouncerG Pignol, K ProtasovLaboratoire de Physique Subatomique et de Cosmologie, CNRS/IN2P3-UJF-INPG,53, Avenue des Martyrs, Grenoble, FranceE-mail: pignol@lpsc.in2p3.frV NesvizhevskyInstitute Laue Langevin, 6, rue Jules Horowitz, Grenoble, FranceAbstract. Spontaneous emission of graviton rates for the quantum bouncer statesare evaluated.The quantum problem of a ball bouncing above a ideal mirror was considered a longtime ago, however just as a mere exercise of elementary quantum mechanics. Things didchange since the quantization of energy of Ultra Cold Neutrons (UCN) bouncing abovea mirror in the Earth’s gravitational field had been demonstrated in an experimentperformed at the Institute Laue Langevin (ILL) [1, 2, 3]. However, this effect does notdemonstrate any quantum behavior of the gravitational field itself. In analogy withelectrodynamics, the observation of spectral lines in atoms shows the quantum behaviorof electrons, but does not provide any clue concerning the possible quantization of theelectromagnetic field. What does provide a clue is the observation of spontaneous decayof an excited state, for instance, which can only be explained in terms of photon emission.So the observation of spontaneous decay of an excited state in the ILL experimentwould be of interest, since it would be a Planck-scale physics effect. Nevertheless, thedecay rate is expected to be low, and the purpose of this letter is to estimate it.First we will set notations of the quantum bouncer problem, focusing on its physicalimplementation, that is, neutrons falling on the Earth’s gravitational field above aperfect mirror. We then derive the spontaneous emission rate in a semi-classicalapproach.The stationary Schro¨dinger equation for the vertical motion (z axis) of the mass mquantum bouncer is:−~22md2ψdz2+mgz ψ = E ψ. (1)The boundary condition due to the presence of the perfect mirror at z = 0 isψ(z = 0) = 0. The characteristic length and the characteristic energy of this problemSpontaneous emission of graviton by a quantum bouncer 2are:z0 =(~22m2g)1/3= 5.87 µm (2)E0 = mgz0 = 0.60 peV. (3)The eigenproblem (1) can be solved in terms of the first Airy function Ai(X), which hasan infinite number of negative zeros, denoted by {−λ1,−λ2, . . .} in decreasing order.The energy of the stationary states is equal to En = E0λn, and the wave function of thenth state is:ψn(z) = CnAi(zz0− λn)θ(z) (4)where Cn normalizes the probability to find a neutron anywhere to 1. The sequenceof the Airy function zeros has no simple analytic expression, but using the Bohr-Sommerfield rules, we find a fairly good approximation of this sequence:λn ≈(3pi8(4n− 1))2/3, (5)this approximation is known to be very good even for the lowest states.In order to evaluate the rate for a bouncer to make a transition k → n, we willfollow the semi-classical analysis as in ref. [4]. This procedure consists in deriving theclassical gravitational power P emitted by an oscillating quadrupole Q cos(ωt), and byreplacing the classical quadrupole by the quantum quadrupole moment for the transitionk → n which readsQkn = m 〈k|zˆ2|n〉, (6)The quantum mechanical transition rate is:Γspk→n =P~ωkn=415ω5knM2P lc4Q2kn, (7)MP l is the Planck mass and ωkn = (Ek−En)/~ is the angular frequency of the transition.This formula is valid if the quadrupole approximation ωkn zk ≪ c holds, i.e. k ≪ 108.This semi-classical derivation for spontaneous emission of gravitons was recentlyconsidered in ref. [5] for atomic hydrogen. The authors provide a field theory derivationof Eq. (7) and show that it satisfies the detailed balance.Since the quadrupole matrix elements for the quantum bouncer are known explicitlyin terms of the Airy function zeros [6], we get:〈k|zˆ2|n〉 =24(−1)k−n+1(λk − λn)4z20 . (8)Or, the transition probability is equal toΓspk→n =51251(λk − λn)3(mMP l)2E50 z40 c(~c)5=5× 10−77s−1(λk − λn)3. (9)For the two lowest quantum states λ2−λ1 = 1, 75 and the probability of the spontaneousgraviton emission is as low as:Γsp2→1 ∼ 10−77s−1. (10)This effect is unfortunately experimentally too rare to observe.Spontaneous emission of graviton by a quantum bouncer 3References[1] V. V. Nesvizhevsky et al., Nature 415, 297 (2002).[2] V. V. Nesvizhevsky et al., Phys. Rev. D 67, 102002 (2003)[3] V. V. Nesvizhevsky et al., J. Phys. C 67, (2005)[4] S. Weinberg, “Gravitation and cosmology” (John Wiley, New York, 1972), chap. 10.[5] S. Boughn and T. Rothman, Class. Quant. Grav. 23, 5839 (2006)[6] D. M. Goodmanson, Am. J. Phys. 68, 866868 (2000).

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