Ultracold neutrons, quantum effects of gravity and the weak equivalence principle

O Bertolami¹ and  F M Nunes²

¹Departamento de Física, Instituto Superior Técnico, Av Rovisco Pais 1, 1049-001 Lisboa, Portugal
²Universidade Fernando Pessoa, Praça 9 de Abril, 4200 Porto, Portugal

Email: filomena@wotan.ist.utl.pt and and orfeu@cosmos.ist.utl.pt

Received 16 October 2002
Published 14 February 2003

PACS numbers: 03.65.Ta, 28.20. - v

Quite recently, Nesvizhevsky and collaborators have reported on the evidence of quantization of the vertical motion of neutrons bound by the gravitational field of the Earth [1]. Even though the conceptual aspects of the experiment were discussed back in the 1970s [2], concrete steps towards the goal of the final experiment were only realized more recently [3]. The experiment consists in allowing ultracold neutrons generated by a source at the Institute Laue-Langevin reactor (Grenoble) to fall towards a horizontal mirror due to the influence of the Earth's gravitational field. This potential confines the motion of the neutrons which no longer move continuously in the vertical direction, but rather jump from one height to another, as predicted by quantum mechanics. This impressive experiment complements the Collela, Overhauser and Werner experiment [4], where neutrons were split and allowed to interfere with the gravitational field, even though in that situation the neutrons were not quantum states. In the Grenoble experiment [1], the minimum energy of the detected neutrons is 1.4 × 10 ^- 12 eV, corresponding to a vertical velocity of 1.7 cm s ^- 1. In that work, it is stated that a more intense beam and an enclosure mirrored on all sides could lead to an improved energy resolution, down to 10 ^- 18 eV, if the neutron can be kept confined for all of its lifetime. In [1], besides the ground state, three excited states were determined, although with a reduced accuracy.

An earlier theoretical work [5] suggested the use of a version of the Collela, Overhauser and Werner experiment for establishing bounds on the parameters of a possible new interaction of nature. In this paper, however, we shall show that an upgraded version of the experiment of Nesvizhevsky and collaborators does not yield any bound on the strength of a new interaction of nature and explore various implications of a highly resolved neutron spectrum.

The Schrödinger problem of a neutron submitted to the local gravitational potential, V = mgx has well-known solutions (here m is the neutron's gravitational mass and x the vertical height). The eigen-solutions of can be expressed in terms of Airy functions, Φ, [6]:

with eigenvalues determined by the roots of the Airy functions:

The variable z has a simple relation to the physical height x which for the Nesvizhevsky et al experiment is of order μm:

The normalization of the wavefunction A_n can be explicitly determined in terms of the integral

First of all, directly from equation (2), one can extract a precision for the measurement of the local gravity of Δg =  6 × 10 ^- 9 m s ^- 1, under the assumption that ΔE =  10 ^- 18 eV. We will show that other effects superpose this uncertainty.

Let us then establish to what extent the weak equivalence principle can be tested once the energy resolution of the Nesvizhevsky et al experiment is improved. For that, one has to realize that in the neutron's Hamiltonian two masses are present: the neutron's inertial mass, m_i, in the kinetic term and the gravitational mass in the gravitational potential. Distinguishing these two masses implies that relationship (2) is slightly altered as m must be replaced by m²/m_i. It then follows that the corresponding uncertainty in the energy due to the difference between inertial and gravitational masses is given by

Hence, for the ground state, E₀ =  1.4 peV [3] and α₀ =  2.338, and if ΔE =  10 ^- 18 eV is attained, one can establish a somewhat modest bound:

The situation is somewhat different in what concerns a new force of nature. The exciting prospect of a new fundamental interaction, beyond the four interactions already known, sparked in 1986 from the claim that the original Ëotvös experiment, designed to verify the equality of gravitational and inertial masses, revealed evidence of a new force with sub-gravitational strength (see [7], and references therein for an exhaustive discussion). Despite the fact that Ëotvös data turned out to be inconclusive, the claim has stimulated a great deal of theoretical discussion (see, e.g., [8] for a complete set of references) as well as the repetition of old experiments using new technology. Actually, the most stringent limit on the equality of inertial and gravitational masses (of Cu and Pb) has arisen in this context and with an accuracy of 5 × 10 ^- 13 [9].

The simplest way a new interaction or a fifth force could arise would be through the exchange of a light boson coupled to matter with sub-gravitational strength. This could originate from various physical models at the Planck scale such as the extended supergravity theories after dimensional reduction [8, 10], the compactification of five-dimensional generalized Kaluza-Klein theories that include gauge interactions at higher dimensions [11] and also from string/M-theory. A common feature of these schemes is the appearance of a new Yukawa-type modification in the interaction energy, V(r), between two point masses m₁ and m₂:

where is the distance between the masses, G_∞ is the gravitational coupling for r→∞, α₅ and λ₅ are the strength and the range of the new interaction. Of course, G_∞ is related to the Newtonian gravitational constant. Indeed, the force associated with equation (7) is given by

where

A particularly interesting implication of the mentioned approaches to Planck-scale physics is that the coupling α₅ is not a universal constant, but instead a parameter dependent on the chemical composition of the test masses as first pointed out in [12]. This dependence comes about if one assumes that the new bosonic field couples to the baryon number B = Z + N (the sum of protons and neutrons) and would imply a clear violation of the weak equivalence principle.

Several experiments have been performed in order to establish the parameters of a new interaction based on the idea of a composition-dependence differential free fall of bodies (see [7, 8] for discussions on those leading to the most stringent limits). The current data are entirely compatible with predictions of Newtonian gravity from both composition-independent or composition-dependent experiments. The bounds on parameters α₅ and λ₅ can be summarized as follows:

• laboratory experiments devised to measure deviations from the inverse-square law are sensitive to the range 10 ^- 2 m < λ₅ 1 m and constrain α₅ to be smaller than 10 ^- 4;
• nucleosynthesis bounds imply that α₅ 4 × 10 ^- 1 for λ₅ 1 m [13];
• gravimetric experiments sensitive in the range of 10 m λ₅ 10³ m suggest α₅ < 10 ^- 3;
• satellite tests probing ranges about 10⁵ m λ₅ 10⁷ m indicate that α₅ < 10 ^- 5;
• radiometric data of the Pioneer 10/11, Galileo and Ulysses spacecrafts suggest the presence of a new force with parameters α₅ =  - 10 ^- 3 and λ₅≊ 4 × 10¹³ m [14].

It is striking that, for λ₅ < 10 ^- 3 m and λ₅ > 10¹³ m, α₅ is essentially unconstrained. The former range arises in higher-dimensional superstring-motivated cosmological solutions, where matter fields that are related to open string modes lie on a lower-dimensional brane, while gravity propagates in the bulk [15]. In these scenarios, the d extra dimensions are not restricted to be small [16] and the fundamental D-dimensional scale, M_D, with D =  4  + d can be considerably smaller than the four-dimensional Planck scale. Assuming that these quantum gravity effects are just beyond current detection capability in accelerators, M_D few TeV and hence modifications to Newtonian gravity will occur in the short-range region, λ₅ < 10 ^- 3 m. This range also emerges if one assumes that scalar [17] or vector/tensor [18] excitations are associated with the observed vacuum energy density.

As a matter of fact, the millimeter range has been recently available for experimental verification and as a consequence the effects associated with d extra dimensions, which imply that α₅ = d +  1 (2d) for a circle (torus) topology [19], have been ruled out down to λ₅ 0.2 mm yielding, for two extra dimensions, the bound on the typical energy scale, M₆≥ 3.5 TeV [20].

It should be mentioned that cosmological considerations tend to yield higher bounds for M_D. For instance, requiring that nucleosynthesis yields are not affected by brane-related effects implies that M₅≊ 10 TeV [21] and hence λ₅ < 10 ^- 4 m. On the other hand, matching inflationary observable quantities in the context of supergravity models on the brane yield much higher bounds for M₅, typically M₅ 10¹³ GeV for supergravity mass term chaotic inflation [22] and M₅ 10¹⁶ GeV for supergravity inflationary models, where the potential has the form V = V₀[1  + c_n(/M_P)ⁿ], the first term being dominant [23].

Let us examine to what extent the Grenoble experiment can allow imposing limits on the parameters of a putative new interaction of nature once the energy resolution of the experiment is increased to ΔE =  10 ^- 18 eV.

In order to study possible limits on a fifth force which is sensitive to baryon number, we assume that its effects can be treated as a perturbation of the local gravitational potential. In this way, first-order perturbation theory provides a correction to the energy of the nth level defined by

where Ψ_n is the unperturbed eigenstate and the perturbation potential is given by V(x)  = α₅ fmgx exp( - x/λ₅), where f is the fraction of the source (Earth) that contributes to the force acting on the neutrons. Since the neutrons are constrained vertically in the length scale x≈μm, the bounds on α₅ f become interesting for, say λ 10 μm. In this range, the exponential in V(x) can be expanded around x =  0. Retaining the first two terms in the expansion, we derive an upper bound for α₅ f in terms of the energy resolution ΔE of the experiment

Here, we have used the numerical values for the following integrals:

Directly from equation (11), we conclude that, within the range where no constraints exist, 10 μm λ≤ 1 cm, the upper limit on α₅ f would be of the order of 1.3 × 10 ^- 2 as long as the energy resolution of ΔE =  10 ^- 18 eV is achieved. Since the tested region for the range is quite small, f 1, implying that there is no limit for α₅ as the effect of a possible new force is far too small to affect neutrons in the experiment. On the other hand, if the new force were of long range so that f =  1, still no bound on α₅ would arise given the relationship between Newton's constants, G_N and G_∞, namely G_N = G_∞(1  + α₅). We mention that the same conclusions can be drawn for the so-called massive version of the Brans-Dicke theory [24], where λ₅ = m ^- 1 and α₅ =  1/2ω +  3, m being the scalar field mass and ω the Brans-Dicke coupling parameter.

We now turn to the issue of the gravitational screening. A possible absorption of the gravitational force between two bodies, when a medium is screened by another, has been the subject of a series of tests. This effect is reminiscent of the magnetic permeability of materials, and a screening or extinction coefficient, h, was proposed by Majorana [25] in 1920 in order to measure the ability of an object of dimension L with density ρ(r) to shield the gravitational force between masses m₁ and m₂:

Naturally, h must be quite small. Several attempts to measure this constant from general principles have been made. For instance, Weber [26] has argued that quasi-static shielding could be predicted from a general relativistic analysis of tidal phenomena, and stated that the effect should be extremely small. More recently, it has been shown that a lunar laser-ranging experiment can set the impressive limit, h≤ 1.0 × 10 ^- 21 cm² g ^- 1 [27]. The most stringent laboratory limit on the gravitational shielding constant has its origin in a recent measurement of Newton's constant carried out at the Physik-Institut, Universität Zürich, which yields h≤ 4.3 × 10 ^- 14 cm² g ^- 1 [28].

In order to use an improved version of the Grenoble experiment for obtaining a limit on the gravitational screening, one considers the effect of the Moon on the local gravity acceleration. This effect is about g_M≊ 3 × 10 ^- 6 g. Of course, an experiment of this nature implies accurate estimates of various local gravitational perturbations related to human activity and geophysical nature. Comparing the contribution of the Moon when Moon, Earth and the experiment are in alignment and the former is `screened' by Earth, g_M2 =  3.3 × 10 ^- 6 g, from which it follows that an effect on the neutron spectra will be felt only if

Here we have assumed that Earth's density is constant over its diameter: ρ_⊕ =  5.51 g cm ^- 3. The result in equation (16) is much less stringent than the above-mentioned bounds. Similar analysis using the Sun leads to a limit that is about four times greater.

Naturally, any actual experiment to measure gravitational origin effects on ultracold neutrons requires, besides the six orders of magnitude improvement in the energy resolution of the neutron spectrum, a series of precautions in order to eliminate local gravitational perturbations and asymmetries.

Acknowledgments

The authors would like to thank Clovis de Matos for discussions and Professor Edward Witten for valuable comments.

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