Neutrino mass and nature through its mediation in atomic clock interference Open Access

Neutrino flavor oscillations¹ prove that neutrinos have mass and mixing among the three flavors. The aim of future oscillation experiments is the search for CP Violation in the lepton sector as well as the mass ordering of neutrino states via propagation in matter.^2,3 These experiments are blind to fundamental neutrino properties, like the absolute scale of neutrino mass and the nature of massive neutrinos, be it either Dirac or Majorana. The aforementioned neutrino properties are being studied, respectively, in the KATRIN experiment⁴ through tritium beta decay and in a plethora of neutrinoless double beta decay experiments^5,6 with improving sensitivity to this $Δ L = 2$ transition, where L is the global lepton number. As neutrino masses have a scale below 1 eV, the difficulty of these searches at nuclear release energies, or in other proposed x-ray transitions like neutrinoless double-electron capture⁷ where the stimulated enhancement has been considered,⁸ is almost insurmountable.

These neutrino mass dependencies become apparent for non-relativistic neutrinos—this is to say the Dirac or Majorana nature becomes distinguishable through the mass terms. Is it possible to descend, experimentally, to energy scales where neutrinos are non-relativistic? The only source existing in nature for such low-energy neutrinos is the relic neutrino background of the universe,¹⁰ but a conceptual basis for observing it has yet to be determined. Recently, the effective potential between aggregate matters mediated by the exchange of two non-relativistic virtual neutrinos has been derived⁹ near its range with all ingredients for the neutrino properties included—it is consistent with and predicted by the Standard Model of Particle Physics. This is a coherent $Δ L = 0$ interaction of matter with a weak charge proportional to the number of constituents, implying a violation of the equivalence principle with different weight of electrons, protons, and neutrons with respect to the mass of matter. As the Compton wavelength of neutrinos of mass 0.1 eV or less is beyond 1 μm, this regime gives the range of the interaction. At these distances, the exchanged neutrinos are non-relativistic, and so, this weak potential is sensitive to the absolute neutrino mass and the Dirac/Majorana distinction, see Fig. 1. Contrary to the concept of $Δ L = 2$ transitions existing for Majorana neutrinos only, the $Δ L = 0$ interaction discussed here exists for both Dirac and Majorana neutrinos, and is distinguishable by mass. Additionally, and unlike the case of $Δ L = 2$ searches, the $Δ L = 0$ interaction is not heavily dependent on the Pontecorvo–Maki–Nakagawa–Sakata (PNMS) mixing parameters or the ordering of neutrino masses. With this interaction, it now becomes conceptually feasible to determine both the mass and nature of neutrinos in a single experiment, which sets it apart from alternative approaches reliant on beta decay kinematics and neutrinoless double beta decay searches separately.

In this article, we explore the prospect of observing this weak, repulsive potential generated by the non-relativistic, virtual exchange of two neutrinos—the so-called $2 ν$ potential. For this goal, we maximize the $2 ν$ potential on an atomic probe P, from geometric considerations, for a distributed and extended source material S; we are led to a cylindrical source with an axial bore. In Sec. II, we discuss the distance behavior of the potential and its weak charges for P and S, in the limit of massless neutrinos, for finding the optimal thickness of S where the effective potential saturates. Section III extends this analysis to massive neutrinos, them being either Dirac or Majorana particles, which changes the distance scaling and the weak charges of the six pairs of neutrinos with definite mass and flavor mixing. The dependence of the resulting potential between the extended source S and the atomic probe P on the absolute mass and the nature of the mediating neutrinos is a demonstration of the conceptual basis of this work. Given the range of microns for the potential and the corresponding scenario where these sensitivities are apparent, in Sec. IV, we consider how previous low-energy atomic and neutron experiments, in the micron regime from a surface, place bounds on this potential. This has been considered previously,^11–15 but new experiments and associated theoretical work offer significant promise for improvement. In Sec. V, we address dominating background potentials from gravitational to the Casimir–Polder effects: utilization of different strategies for the construction of the source material and spatial scaling allows the $2 ν$ potential to be isolated. Section VI follows by proposing a schematic experiment, with the expectation that future experimental progress could open the window to a feasible tabletop experiment to explore the potential generated by the $2 ν$ exchange interaction, and so the absolute mass and the Dirac or Majorana particulate nature of the neutrino. We then present the conclusions and outlook projected by this work.

We use the computed potentials to optimize the geometry of the source: inner radius r₁, outer radius r₂, and length L, see Fig. 2(a). We fix the inner radius r₁ = 1 μm, where the interaction departs from the massless neutrino case, see Fig. 1, and study the behavior of the weak potential benchmarked against the gravitational potential from the source as function of $r 2 − r 1$⁠, Fig. 2(b), and L, Fig. 2(c). The 2ν potential decreases with distance faster than the gravitational potential; consequently, the ratio between them for the interaction between the source and the probe decreases. The potential increases with the volume of the source until it saturates at about 4 μm due to its fast decrease with distance compared to the increasing volume, proportional to $L r 2 2$⁠. This saturation behavior is in stark contrast with that for the gravitational potential, and so fixes the appropriate choice of r₂ of the cylinder. The scale differs between Figs. 2(b) and 2(c), due to this differing dependency on the volume of the source in L and r₂. We conclude that a finite cylinder thickness contributes to an enhancement of the 2ν exchange potential up to values reached at about 4 μm. At these distances, the effect of the neutrino mass and its Dirac/Majorana nature will emerge,⁹ as seen in Fig. 1.

Along the axis of the source, see the solid red line of Fig. 2(a), the 2ν potential has a weaker curvature than that of gravity, Fig. 2(d). When considered radially, Fig. 2(e), the weak curvature is evident as well, with the addition of a gravitational instability (a discontinuity) at the center. This could prove useful, as a way of distinguishing the 2ν potential from the gravity of the source.

The results obtained for the atom probe-cylindrical material source at micron distances show the expected behavior of the decrease in the 2ν potential with increasing inner radius of the cylinder and the increase in the sensitivity to the neutrino absolute mass and its nature. Conceptually, there exists the possibility to explore these properties using the approach shown. It is a novel method, entirely independent of the $Δ L = 2$ experiments that allow for Majorana neutrinos only, and it moves the burden of proof to a fundamentally lower energy scale where the neutrino mass term emerges.

Is there a plausible detection scenario? The $2 ν$ potential is minute, at micron distances being orders of magnitude weaker (⁠ $10 − 15$⁠) than that of gravity between the probe and the source, with the magnitude being $10 − 56$ J at 1 μm—see Fig. 3. This is well below the direct detection limit of existing low-energy particle and atomic physics experiments. The atom–volume interaction, along with improvements from considering larger weak charge probes (see  Appendixes), are enhancements that make it worthwhile to consider previous experiments while also provoking thought experiments that could lead to a future bespoke low-energy experiment. We start by considering a selection of prior experiments utilizing probe particles near surfaces.

A series of experiments in the micron regime have been performed in atomic and neutron physics, often studying Casimir–Polder interactions¹⁶ and limits on new Yukawa-type short-range forces.¹⁷ The geometries employed in these experiments involved planar sources, which are considered unfavorable despite their analytical solubility. The calculation of the Casimir force between plates for neutrinos with two specific masses in the loop has been previously conducted.¹⁸ However, resolution of the $2 ν$ potential in these experiments would necessitate a sensitivity of approximately $10 − 60$ J at a distance of 1 μm. We focus on experiments in the range 0.5–10 μm as a region of interest and convert results from prior works to approximate equivalent potentials for the continued illustration of our point: An atomic beam of neutral ²³Na passing between two parallel plates with a variable spacing between 0.7 and 7 μm resolved the Casimir–Polder potential to $1.7 × 10 − 28$ J—and was found to be in complete agreement with theory.¹⁹ Modern ultracold bouncing neutron experiments²⁰ demonstrating gravitational resonance spectroscopy resolved energy differences of $3 × 10 − 33$ J in the tens of μm regime. Neutrons are a notable option as a probe, despite a low weak charge, as they are essentially undisturbed by residual electromagnetic forces, such as the Casimir–Polder interaction, given that its electric polarizability is small. Measurements of the change in induced oscillation frequencies in magnetically trapped Bose–Einstein condensates of alkali atoms 6–9 μm above a surface²¹ realized a sensitivity nearing $5 × 10 − 37$ J. Atom interferometry on a chip²² observed $3 × 10 − 38$ J about 10 μm above a surface, with one hour of integration and with 3 μm resolution. State-of-the-art atom interferometry, run in free space and as a differential experiment,²³ is sensitive enough to be only one order of magnitude away from observing QED corrections, a promising lead. An atomic clock trapped on a chip²⁴ provides bounds in the region 0.6–10 μm to $7 × 10 − 40$ J. All of these experiments are critically constrained by sub-optimal planar geometry, source/probe composition, and the effects of surface physics, which was often the point of interest in these works, and so demonstrate a lack of sensitivity by orders of magnitude by comparison to cylindrical or spherical geometries. There are no experiments that place favorable bounds from low energy on the $2 ν$ potential.

Further and contrary to mass density- and number density-dependent potentials, which govern, respectively, the gravitational and weak interaction, the Casimir–Polder interaction is controlled by the electronic structure of the material source through its electric polarizability.⁹ Most prior experiments in the micron regime occurred near metallic surfaces (gold being a common surface substrate, over silicon). Choice of a pure non-polar covalent material in a high-quality detector grade crystal, like diamond, silicon, or germanium, allows for a suppression of the Casimir–Polder effects. For our proposed experiment, we choose germanium versus similar materials based on it having a high weak charge for each atom and given that it can be grown with a low impurity level—this can be accomplished by re-growing several times with Czochralski's method.^25,26 The isotopic disorder of germanium crystals can be taken into account by using the average weak charge and the number density of atoms—this approximation demonstrates a near-zero electronic polarizability of the entire source in the region of interest. The absence of electric polarizability for the material source, as expected from its tetrahedral crystalline structure, would be confirmed with Raman and other infrared spectroscopic techniques. Another option consists of materials constructed from dielectric nanotubes, expected to have low values of the transverse component of the electric polarizability.^27,28 We summarize values of the various potentials for combinations of probe and source elements in Table I, found in  Appendix.

The gravitational interaction cannot be suppressed. However, in an interferometric experiment, we can try to suppress the effect by compensation within the arms of the interferometer without affecting the $2 ν$ potential; the differing spatial dependence of the two interactions provides a conceptual solution. By example, we take two cylinders of the same density and length, one with an inner radius of 10 μm and another with 1 μm. The weak potential is negligible for the large radius cylinder, but the gravitational potential at the center can be made equal to that of the smaller radius cylinder by compensating the longer distance with a larger mass at a magic width, common to the two cylinders—see Fig. 4. This allows for a pathway to disentangle local source gravity from the $2 ν$ potential, in addition to using a combination of different properties for pattern generation,²⁹ like spatial dependence and different atom probes looking for a violation of the equivalence principle.

With this last consideration, we turn our attention to recent developments in atomic physics, especially techniques from metrology, as a starting point for a possible experiment. The physics of the $2 ν$ potential gives a series of additional directives for the experiment: heavy, neutral atoms, and so with an appreciable weak charge, as the probe and a dense covalent source material, so as to suppress electromagnetic effects and that can be precisely constructed via crystal growth. Given this, we chose ¹⁷⁴Yb as the probe and ⁷⁴Ge as the source for all the work shown in this article—see Table I ( Appendix B) for different combinations of probe and source species. Such an experiment as this is sensitive to the positioning of the probe with respect to the axis of the cylindrical source, as is demonstrable from Fig. 2; utilizing an ultracold quantum gas in a magnetic waveguide (MWG) is a plausible probe scenario, to keep the experiment in the interaction volume at the point of highest symmetry. An advantage of ultracold gases is the ability to generate unique states of matter based on dimensionality, scattering length, and temperature; a preparation as one-dimensional hard bosons³⁰ would be ideal, with minimal atom–atom (probe–probe) interactions. We envision a chain of probes numbering $N atoms ≤ 100$ atoms in the two arms of an atomic clock interferometer,^31,32 see Fig. 5. The two clocks, initialized at the same time and running parallel to each other, are held in free space above a magnetic waveguide, and/or within a series of cylindrical masses like that of Fig. 2 (the main experiment) and modified like that of Fig. 4 (systematics, source gravity effects). These clocks are functioning as local atomic potentiometers, held for a long time between the sets of pulses of the interferometer. Such an experiment would be enhanced with detection utilizing single-atom methods^33,34 and spin-squeezing.³⁵ 

Designing and constructing an experiment to start with exploring source gravity physics at micron distances should be made. Moving forward is to involve new results from atomtronics,³⁷ entanglement protocols,^35,38 cryogenic cooling of the source for improved vacuum and reduced blackbody and Casimir effects, and high-fidelity single-atom detection,^33,39 to constitute a new state-of-the-art experiment in an attempt to explore the two-neutrino exchange potential from low energy with precision atomic physics.

We have explored the prospect of observing the weak, repulsive potential generated by the non-relativistic, virtual exchange of two neutrinos at micron distances. Within this range, the size of the potential is dependent upon the absolute mass and the fundamental particulate nature of the neutrino. We have shown an optimization of this potential on an atomic probe from geometric considerations, generating a matter source with the criterion that the potential saturates. This shows a 15 order of magnitude improvement over the atom–atom interaction previously calculated. We consider prior low-energy atomic and neutron experiments in the micron regime and find that planar geometries and insufficient precision place no meaningful bounds on such a potential. We propose to suppress residual Casimir–Polder interactions by the use of a purely covalent material like germanium. A schematic experiment is outlined to begin exploring this potential, starting with source gravity at micron distances, and that could lead to a low-energy bounding of the $2 ν$ potential probes. The gravity effect can be compensated, using an appropriate matter source in different arms as well as different probes—this differential experiment is necessary to elucidate the violation of the equivalence principle by this weak potential. This last comparison makes apparent the violation of the equivalence principle implied by the weak charge of aggregate matter compared to its mass. Despite this new understanding, distinct non-vanishing results of the potential remains out of reach for existing state-of-the-art low-energy precision measurements; future experimental progress could open this window with a low-energy precision measurement tabletop experiment.

Here, we show a new understanding of the accessibility of atomic experiments to the open problems of neutrino mass and nature. While the expected results for the detection of the weak potential between aggregate matter at micron distances remain out of reach, the complementarity of our concept, based on quantum physics, makes this approach worthwhile to pursue. Contrary to the search of $Δ L = 2$ processes for Majorana neutrinos, this $Δ L = 0$ effect would provide distinct results between Dirac and Majorana mass terms. Attaining such sensitivity to the weak potential could elucidate both the absolute neutrino mass and its nature without the need for additional assumptions, all within a single experiment.

We would like to thank Andres Cantarero and Pablo Rodríguez for enlightening discussions on the properties of dielectric materials. J.B. has been supported by MICINN/AEI, PID2020-113334GB-I00/AEI/10.13039/501100011033, and CIPROM/2021/054 (Generalitat Valenciana). F.S. thanks the Swiss National Foundation under the Grant No. 200020_204609. All authors warmly thank the Boninchi Foundation for their generous support for this project.

The authors have no conflicts to disclose.

José Bernabeu: Conceptualization (equal); Formal analysis (lead); Funding acquisition (supporting); Investigation (equal); Methodology (equal); Project administration (equal); Resources (equal); Software (lead); Validation (equal); Writing – original draft (equal); Writing – review & editing (equal). Dylan O. Sabulsky: Conceptualization (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Project administration (equal); Resources (equal); Software (equal); Validation (equal); Writing – original draft (equal); Writing – review & editing (equal). Federico Sánchez: Conceptualization (equal); Formal analysis (equal); Funding acquisition (lead); Investigation (lead); Methodology (equal); Project administration (equal); Resources (equal); Software (equal); Validation (equal); Writing – original draft (equal); Writing – review & editing (equal). Alejandro Segarra: Formal analysis (equal); Methodology (equal); Software (equal).

The data presented and analyzed in this study are available from the corresponding author upon reasonable request.

The optimal source potential is the one that optimizes $V S P 0 ρ$⁠. We have computed the quantity for several source and probe species. The relative strength to that of the gravitational force goes as $V S P 0$⁠. Values are tabulated in Table I. We can gain a factor of ∼10 in atom–atom potential strength (⁠ $V S P 0$⁠) from carbon to tungsten, and a factor of 5.5 when considering the density of the source material (⁠ $V S P 0 ρ$⁠). The loss of strength by changing the $74 184 W$ source by $32 74 Ge$ is around 3.6. We consider other aspects related to the control of the Casimir–Polder potential in the selection of the source material. We have also computed the same potential for a lighter probe atom (⁠ $19 40 K$⁠), see Table I. The $19 40 K$ potential strength versus the one with $70 174 Yb$ varies from 4.5 to 4.9 times depending on the source atom species. The same ratio is obtained for $V S P ρ$ since the density depends only on the source component. This result shows, as expected, that the optimal configuration is the one with the largest density in the source and largest number of nucleons in the source and probe species. The case of the carbon source is special since it is possible to achieve high-density material with respect the nominal atomic mass using graphite as a source material. The $V S P ρ$ strength has a similar magnitude to the one of $32 74 Ge$ despite a large difference in the atom–atom potential (V_SP) between the two source atomic species.