[1]Joseph Smolsky [1,2]Kyle G Leach

1]Department of Physics, Colorado School of Mines, 1500 Illinois St, Golden,
80401, Colorado, USA

2]Facility for Rare Isotope Beams, Michigan State University, 640 S Shaw Lane,
East Lansing, 48824, MI, USA

3]TRIUMF, 4004 Wesbrook Mall, Vancouver, BC V6T 2A3, Canada 4]LIBPhys-UNL,
Departamento de Física, Faculdade de Ciências e Tecnologia, NOVA FCT,
Universidade Nova de Lisboa, 2829-516 Caparica, Portugal 5]Université de
Strasbourg, CNRS, Institut de Physique et Chimie des Matériaux de
Strasbourg,UMR 7504, F-67000 Strasbourg, France 6] Lawrence Livermore National
Laboratory, Livermore, CA 94550, USA 7]STAR Cryoelectonics LLC, Santa Fe, NM
87508, USA 8]Shared Instrumentation Facility, Colorado School of Mines,
Golden, CO 80401, USA 9]XIA LLC, Hayward, CA 94544, USA 10]LPC Caen, ENSICAEN,
Université de Caen, CNRS/IN2P3, Caen, France 11]Department of Physics and
Astronomy, McMaster University, Hamilton, Ontario L8S 4M1, Canada
12]Université Paris-Saclay, CEA, List, Laboratoire National Henri Becquerel
(LNE-LNHB), F-91120, Palaiseau, France 13]Pacific Northwest National
Laboratory, Richland, WA 99354, USA

# Direct Experimental Constraints on the Spatial Extent of a Neutrino
Wavepacket

<EMAIL_ADDRESS><EMAIL_ADDRESS>Ryan Abells  Pedro Amaro  Adrien
Andoche  Keith Borbridge  Connor Bray  Robin Cantor  David Diercks  Spencer
Fretwell  Stephan Friedrich  Abigail Gillespie  Mauro Guerra  Ad Hall  Cameron
N Harris  Jackson T Harris  Calvin Hinkle  Amii Lamm  Leendert M Hayen  Paul-
Antoine Hervieux  Geon-Bo Kim  Inwook Kim  Annika Lennarz  Vincenzo Lordi
Jorge Machado  Andrew Marino  David McKeen  Xavier Mougeot  Francisco Ponce
Chris Ruiz  Amit Samanta  José Paulo Santos  Caitlyn Stone-Whitehead  John
Taylor  Joseph Templet  Sriteja Upadhyayula  Louis Wagner  William K Warburton
[  [  [  [  [  [  [  [  [  [  [  [  [

###### Abstract

Despite their high relative abundance in our Universe, neutrinos are the least
understood fundamental particles of nature. They also provide a unique system
to study quantum coherence and the wavelike nature of particles in fundamental
systems due to their extremely weak interaction probabilities. In fact, the
quantum properties of neutrinos emitted in experimentally relevant sources are
virtually unknown and the spatial extent of the neutrino wavepacket is only
loosely constrained by reactor neutrino oscillation data with a spread of 13
orders of magnitude. Here, we present the first direct limits of this quantity
through a new experimental concept to extract the energy width,
$\sigma_{\textrm{N},E}$, of the recoil daughter nucleus emitted in the nuclear
electron capture (EC) decay of 7Be. The final state in the EC decay process
contains a recoiling 7Li nucleus and an electron neutrino ($\nu_{e}$) which
are entangled at their creation. The 7Li energy spectrum is measured to high
precision by directly embedding 7Be radioisotopes into a high resolution
superconducting tunnel junction that is operated as a cryogenic sensor. The
lower limit on the spatial uncertainty of the recoil daughter was found to be
$\sigma_{\textrm{N},x}\geq 6.2$ pm, which implies the final-state system is
localized at a scale more than a thousand times larger than the nucleus
itself. From this measurement, the first direct lower limits on the spatial
extent of the neutrino wavepacket were extracted using two different
theoretical methods. These results have wide-reaching implications in several
areas including the nature of spatial localization at sub-atomic scales,
interpretation of neutrino physics data, and the potential reach of future
large-scale experiments.

###### keywords:

quantum decoherence, neutrino, wavepacket, superconducting sensing, nuclear
electron capture

Quantum mechanics is based on the concept that the microscopic scale of our
Universe contains inherent uncertainties in position and momentum that are
fundamentally connected via $\sigma_{x}\sigma_{p}\geq\hbar/2$ [1]. Using this
theoretical description, precise knowledge of the momentum width of the
particle, $\sigma_{p}$, implies a corresponding uncertainty on the extent of
its spatial wavefunction, $\sigma_{x}$. When the object loses its ability to
maintain coherence through interactions with the complex environmental bath,
its position is localized. This spatial localization of quantum objects
depends heavily on the specific environment they are subjected to [2].

Recent efforts have primarily focused on constructing increasingly large
coherent quantum systems [3, 4] and other highly delocalized objects to push
measurement and other applications to, and beyond, the standard quantum limit
[5, 6]. Such experiments are at the frontiers of quantum science and
engineering and aim, in part, to probe the interface between classical and
quantum mechanics. Here, we present a new measurement concept on the other
side of the coin; using highly localized unstable systems embedded in a
complex solid-state material at low temperature as a precision laboratory to
probe quantum properties of subatomic particles. We show that this concept is
particularly powerful for investigating properties of systems that are
otherwise not directly accessible due to weak couplings in the Standard Model
of particle physics (SM), in particular the neutrino ($\nu$).

Neutrinos are light, neutral leptons, and the only particles in the SM that
have an intrinsic chirality, in that they only interact via left-handed
currents of the weak interaction through three eigenstates, $\nu_{e}$,
$\nu_{\mu}$, and $\nu_{\tau}$. Neutrino oscillation experiments over the last
30 years [7, 8] have also indicated that these flavour states include at least
two non-zero mass eigenstates. This observation of non-zero neutrino masses
currently provides the only confirmed violation of the SM as it was originally
constructed [9, 10], for which the Nobel Prize in physics was awarded in 2015
[11]. The flavour oscillation effect for neutrinos arises from the fact that
the mass eigenstates and weak-interaction eigenstates are not equal to each
other, and due to their small interaction probabilities with the weak and
gravitational forces, are able to maintain coherence over long distances.
Thus, even for a neutrino created with a definite flavour – the electron
neutrino ($\nu_{e}$) in weak nuclear decay, for example – a coherent
superposition of states exists within the neutrino wavepacket that affects the
probability for observing a particular flavour as a function of time or
distance from the source.

Since neutrinos only interact weakly with matter, they are notoriously
difficult to detect111For example, a $\nu_{e}$ with 1 MeV kinetic energy has
an interaction cross section of roughly $10^{-42}$ cm2. [12]. This fact also
makes neutrinos attractive systems to study fundamental properties of quantum
mechanics and exotic new physics [13], since they are able to maintain
coherence over long distances. Historically, observing these so-called “ghost
particles” from various sources requires large volume detectors with limited
detection precision that run for several years [14]. These detectors rely on
neutral current (NC) and charged current (CC) interactions with an atomic
nucleus or electron within these large volumes [12]. The final state particles
are used to reconstruct the incoming neutrino energy and interaction point,
along with neutrino flavour for CC interactions. A deficit or excess flux of
any flavour in the detector, compared to the flavour ratios at the source, is
used to determine the oscillation probabilities as a function of energy and
distance. Global data from neutrino oscillation experiments have been used to
determine the mass-splittings and mixing parameters for the three SM neutrinos
[15]. Data from the Daya Bay [16, 17], RENO [18, 19], and KamLAND [20]
experiments currently also provide loose constraints on the spatial extent of
$\overline{\nu}_{e}$ wavepackets from $\beta^{-}$ decay (reactor sources) of
$2.1\times 10^{-13}$ m $\leq\sigma_{\nu,x}\leq 2$ m [21, 22, 23]. If the
$\overline{\nu}_{e}$ and $\nu_{e}$ wavepacket spatial widths are indeed near
the low end of these limits, it has a significant impact in the neutrino
physics landscape, including alleviating tension in models for sterile
neutrinos with eV-scale masses through model-dependent fits to available data
[24, 25].

To extract physics from the oscillation data, however, interpretation within
the lepton mixing framework of the SM is required [15]. Given the current
pervading anomalies in the neutrino physics landscape [26], there is a need
for new experimental techniques that can provide model-independent access to
the neutrino. Clever new experimental concepts have been presented to address
this need using efficient, small scale setups [27, 28, 29] that are less
model-dependent and not limited by low neutrino interaction cross sections.
These experiments aim to precisely measure the low-energy recoiling atoms in
nuclear electron capture (EC) decay [30] to access information of the neutrino
directly through its entanglement in the final state of radioactive decay. Of
these new concepts, the beryllium electron capture in superconducting tunnel
junctions (BeEST) experiment [27] is currently the only one to practically
employ the concept of direct EC daughter recoil detection for neutrino physics
[31] and astrophysics [32]. In this paper, we report the first direct limit on
the spatial width of a neutrino wavepacket from the energy width of the
recoiling nucleus in EC decay, $\sigma_{\textrm{N},E}$, through their mutual
entanglement.

Nuclear EC decay is a radioactive decay mode that results from the capture of
an $s$-wave orbital electron by a proton within the nuclear volume [30]. At
the fundamental level, this weak interaction process transmutes an up quark to
a down quark through the exchange of a $W$ boson within the nucleon and
conserves lepton number through the emission of $\nu_{e}$. Within the
experimental context of the work presented here, the observable final state
only contains two products: the recoiling heavy daughter system and $\nu_{e}$.
This simple two-body system that we observe is entangled at its creation, and
thus a precision measurement of the daughter recoil provides direct access to
information on the neutrino.

With seven nucleons and four electrons, the neutron deficient nucleus
beryllium-7 (7Be) is the simplest pure EC decaying system in the nuclear
landscape. 7Be decays to lithium-7 (7Li) and $\nu_{e}$ with a half-life of
$T_{1/2}=53.22(6)$ days [33] and a total decay energy of $Q_{EC}=861.963(23)$
keV [34]. A small branch of 10.44(4)% results in the population of a short-
lived excited nuclear state in 7Li ($T_{1/2}=72.8(20)$ fs) [33] that de-
excites via emission of a 477.603(2) keV $\gamma$-ray [35]. In the EC process,
the electron can be captured either from the $1s$ shell ($K$-capture) or the
$2s$ shell ($L$-capture) of Be. For $K$-capture, the binding energy of the
$1s$ hole is subsequently liberated by emission of an Auger electron [36].
Since the nuclear decay and subsequent atomic relaxation occur on short time
scales, a direct measurement produces a spectrum with four peaks: two for
$K$-capture and two for $L$-capture into the ground state (GS) and the excited
state (ES) of 7Li.

Figure 1:  A schematic image of the STJ sensor showing the five layers
described in the text. The 7Be radioactive source is sparsely implanted into
the Ta lattice of the absorber layer. The upper right zoom-in shows the final
state after EC decay including the 7Li recoil and $\nu_{e}$.

To measure the 7Be EC decay spectrum to high precision, we directly embedded
the 7Be into the top tantalum film of a superconducting tunnel junction (STJ)
pixel that was used as a high energy-resolution sensor [37]. The single STJ
pixel used in this experiment was part of a 36-pixel array where each sensor
had a surface area of $208\times 208$ $\mu$m2 and consisted of five layers
(from top to bottom): Ta (165 nm)—Al (50 nm)—Al2O3 (1 nm)—Al (50 nm)—Ta (265
nm), shown in Fig. 1, that were fabricated by photolithography at STAR
Cryoelectronics LLC [38]. STJs exploit the small energy gap in superconducting
Ta ($\Delta_{Ta}\approx 0.7$ meV) to provide $\sim 30\times$ higher energy
resolution than conventional Si or Ge detectors, thus allowing for a precision
measurement of the 7Li recoil energies.

Radioactive 7Be was implanted directly into the top tantalum film of the STJ
pixel through apertures in a Si collimator that was situated $\sim 100~{}\mu$m
above the chip. The 7Be was produced and delivered to the implantation chamber
end-station at the TRIUMF-ISAC facility in Vancouver, Canada [39] at an energy
of 30 keV. The isotopically pure $(\sim 90\%)$ 7Be+ beam was produced using
the isotope separation on-line (ISOL) technique [40] via spallation reactions
from a 10 $\mu$A, 480-MeV proton beam incident on a stack of thin uranium
carbide targets. Once released from the target through diffusion, the created
beryllium atoms were selectively ionized using the ion guide laser ion source
(IG-LIS) [41] operated in suppression mode [42]. The resulting beam of 7Be+
ions was implanted at a rate of $6.2\times 10^{6}$ $s^{-1}$ for a period of 25
hours to generate an initial per-pixel activity in the STJs of roughly 50 Bq.
Following implantation, the chip was cleaned and processed at TRIUMF to remove
loose and surface-deposited activity and subsequently shipped to Lawrence
Livermore National Laboratory (LLNL).

The 7Li recoil spectrum from the decay of 7Be was measured at LLNL with the
STJ detector at a temperature of $\sim$0.1 K in a two-stage adiabatic
demagnetization refrigerator (ADR) with liquid nitrogen and liquid helium
precooling. Signal traces were read out continuously at a rate of 1.25 MSa/s
at a 16 bit resolution using a PXIe-6356 ADC. For real-time in-situ monitoring
of the response and energy calibration, the STJs were simultaneously exposed
to 3.49865(15) eV photons from a pulsed, 355 nm frequency tripled Nd:YVO4
laser triggered at a rate of 100 Hz that was fed into the cold stage of the
ADR through a fiber to illuminate the pixels. The laser intensity was adjusted
such that multi-photon absorption provided a comb of peaks over the energy
range from $20-120$ eV.

Data were acquired for $\sim$20 hours on November 6, 2022. Voltage traces from
the STJ array were continuously recorded along with the laser and processed
using a trapezoidal filter. The pulsed laser was used for energy calibration
and drift correction over the period of data collection. The laser events were
selectively identified in the data stream via their coincidence with a logic
pulse that was used to trigger the laser, and the comb of peaks is shown in
Fig. 2. The laser signal was fit to a superposition of individual Gaussian
functions, each corresponding to an integer multiple of the single photon
energy. The peaks were used to generate a quadratic calibration function which
was applied to the corresponding 7Be spectrum. The energy resolution of the
laser peaks was $1.91(2)$ eV at 105 eV with residuals of $\sim$0.01 eV in the
region of interest [43, 44].

Figure 2: The measured 7Li recoil spectrum with the four peaks described in
the text for 20 hours of data from the single STJ pixel shown in the inset.
The measured $\sigma_{\textrm{N},E}$ of the K-GS peak is shown, and is
conservatively extracted as the upper limit on the energy width of the recoil.
The calibration laser spectrum (violet) is also shown for comparison.

The final 7Be decay spectrum shown in Fig. 2 includes events from a single
pixel that were not in coincidence with the laser logic pulse or simultaneous
signals in other pixels. The latter of these two criteria was imposed to
remove the large, low-energy background from 478 keV $\gamma$-ray Compton
scattering interactions in the 0.5 mm Si substrate below the STJ array which
generated signals in many pixels simultaneously. The resulting spectrum
contains four main peaks resulting from the two atomic and two nuclear
processes described above. The two recoil peaks from the excited state decay
of 7Li were Doppler-broadened from $\gamma$-decay in flight [32, 31].

Since the most probable 7Be decay mode is K-shell capture to the nuclear
ground state of 7Li [32], the K-GS peak provides the highest statistical
significance for a measurement of the energy width. The K-GS peak shown in
Fig. 2 was fit in the region from $100-111$ eV with a single Voigt function
[45, 46] to extract the energy uncertainty width $\sigma_{\textrm{N},E}\leq$
2.9 eV at 95% confidence level (CL) using the profile likelihood method [47].
Some of this width is due to structural and chemical variations of the Li $1s$
binding energy at different sites in the Ta lattice, which contribute
$\mathcal{O}$(1eV) to the broadening [48]. However, due to their complexity
and the fact that these effects are not yet quantified in detail for the BeEST
energy spectrum, we conservatively attributed all of the measured broadening
in the K-GS peak to quantum uncertainty. We then converted the full energy
width to a momentum width of the 7Li recoil. The uncertainty was propagated
using $\sigma_{p}=\sqrt{m/2E}\,\sigma_{E}$ and resulted in
$\sigma_{\textrm{Li},p}\leq 16$ keV/$c$. Finally, using the Heisenberg
uncertainty relation, we extracted a lower limit on the spatial width on the
nuclear recoil of $\sigma_{N,x}\geq 6.2$ pm, shown in Fig. 3 (a). These
results are based on our measurement of the K-GS peak energy width under the
experimental conditions reported here and have no underlying assumptions about
the scale of localization.

Figure 3: (a) The lower limit on the spatial width of the 7Li final-state
recoil compared to the approximate nuclear and atomic scales. (b) BeEST lower
limits on spatial width of $\nu_{e}$ using the two theoretical approaches
described in the text compared to limits extracted from reactor oscillation
experiments [21, 23]. The band on the left corresponds to neutrino wavepacket
widths that improve eV-scale $\nu_{s}$ model fits to data [24, 25], and are
excluded by both BeEST limits. The other bands show predictions based on
different methods and localization scales (widened for clarity): (i) The green
band around 2.7 nm comes from conservation of momentum with localization on
the scale of the $1s$ electron orbital [49]. (ii) The orange band around 1.4
$\mu$m comes from conservation of energy with localization caused by atomic
interactions [50]. (iii) The red band around 200 nm is based on a QFT approach
that also uses atomic interactions for localization [51].

To extract a limit on the neutrino spatial width, $\sigma_{\nu,x}$, two
approaches were used: one based on conservation of energy outlined in Ref.
[50] and the other on conservation of momentum in Ref. [52]. In the first
method, the neutrino energy width is equivalent to that of the entangled
recoil nucleus, which in our case is the upper-limit on the measured energy
width $\sigma_{\nu,E}=\sigma_{\textrm{N},E}\leq 2.9$ eV. This limit was then
converted to a momentum uncertainty using the relativistic energy-momentum
relation, $p_{\nu}=E_{\nu}/c$ and $\sigma_{p}=\sigma_{E}/c$, and the spatial
uncertainty was extracted as $\sigma_{\nu,x}\geq 35$ nm at 95% CL using the
uncertainty principle. We also note that this result is the same as a similar
approach prescribed in Ref. [53] based on environmental interactions leading
to a velocity dependent relation between the spatial width and energy width.

For the second approach reported in Ref. [52], the momentum widths of the
recoil and nucleus are equivalent, $\sigma_{\nu,p}=\sigma_{\textrm{Li},p}\leq
16$ keV/$c$, and the uncertainty principle again gives the spatial width.
Under this approach, we determine a lower limit on the spatial width of the
neutrino of $\sigma_{\nu,x}=\sigma_{N,x}\geq 6.2$ pm at 95% CL.

Since the two methods described above yield dramatically different results for
the neutrino wavepacket width, the details of which are discussed extensively
in the literature [53, 50, 54, 55, 52, 51, 49], we do not claim validity of
one over the other here, but rather discuss the potential implications of our
measurement on the neutrino wavepacket width extracted using both.

First, our extracted limits on the neutrino widths, $\sigma_{\nu,x}\geq 6.2$
pm and $\sigma_{\nu,x}\geq 35$ nm, are both much larger than the level at
which wavepacket separation in eV-scale models could be used to remove tension
between reactor experiments and the so-called “gallium anomaly” in EC decay
oscillation experiments [24, 25], as shown in Fig. 3 (b). To place our
extracted limits within the context of the broader neutrino physics landscape,
we then take the standard approach [24, 25] of assuming that the variations in
$\sigma_{\nu,x}$ between types of sources ($\beta^{\pm}$ or EC) is small
compared to variations between scales of localization. Under this treatment,
our lower-limit on the neutrino width extracted using the first approach is
more stringent than the previous experimental limits on $\sigma_{\nu,x}$ by
reactor sources by more than an order of magnitude. Additionally, our limits
using either method exclude the possibility of future experiments such as JUNO
to be able to observe wavepacket separation [56, 22]. These results have
implications beyond what is discussed here, however for clarity and model-
independence of what we report we only present the above discussion.

Beyond the implications for neutrino physics, perhaps the more important
result from our work is the potential to use nuclear EC decay as a fundamental
test of quantum mechanics at subatomic scales. Since the electron is captured
and the neutrino is produced within the nuclear volume via weak coupling at
the quark level, there is a question as to whether processes such as short-
range correlations [57] are able to decohere and localize the system via
interactions within the atomic nucleus ($<$10 fm). Our experimental
determination of the position width of the daughter recoil,
$\sigma_{\textrm{N},x}\geq 6.2$ pm, excludes this possibility by several
orders of magnitude. This general conclusion is consistent with nuclear
$\gamma$ decay, an electromagnetic process, in highly localized environments
through the Mössbauer effect, which suggests localization on the scale of 1-10
pm [58].

In summary, we present a new experimental concept using highly localized EC
decaying rare isotopes inside superconducting sensors to measure the quantum
properties of the final-state system in electroweak nuclear decay. Here, we
report an upper limit of the energy width of the recoiling 7Li system from the
EC decay of 7Be in a tantalum matrix at 0.1 K of $\sigma_{E,N}\leq 2.9$ eV
(95% CL). This limit corresponds to a limit on the spatial width of the 7Li
nuclear recoil of $\sigma_{\textrm{N},x}\geq 6.2$ pm, which suggests that
localization of the final state system occurs at a scale much larger than the
nucleus in which the interaction occurs. From this measurement, we extract
limits on the spatial width of the entangled neutrino using two methods
reported in the literature, the first such limits on an EC $\nu_{e}$ source to
date. Although the extraction of this value depends dramatically on the
theoretical prescriptions used, both methods give neutrino wavepacket sizes
that are too large to observe wavepacket separation in future large scale
oscillation experiments such as JUNO, and does not relieve the tension in 3+1
$\nu_{s}$ model fits. These results demonstrate the power of using precision
measurements of EC decay as a probe of fundamental physics, and the reported
results have implications for the details of neutrino oscillation and BSM
theories, as well as serving as a test of quantum mechanics at the subatomic
scale.

## Acknowledgements

We thank David Moore, Daniel Carney, Benjamin Jones, Carlos Argüelles-Delgado,
Joseph Formaggio, and Fred Sarazin for useful discussions. The BeEST
experiment is funded by the Gordon and Betty Moore Foundation
(10.37807/GBMF11571), the U.S. Department of Energy Office of Science, Office
of Nuclear Physics under Award Numbers DE-SC0021245 and SCW1758, the LLNL
Laboratory Directed Research and Development program through Grants No.
19-FS-027 and No. 20-LW-006, the European Metrology Programme for Innovation
and Research (EMPIR) Projects No. 17FUN02 MetroMMC and No. 20FUN09 PrimA-LTD,
and the FCT—Fundação para a Ciência e Tecnologia (Portugal) through national
funds in the framework of the projects UID/04559/2020 (LIBPhys). TRIUMF
receives federal funding via a contribution agreement with the National
Research Council of Canada. This work was performed under the auspices of the
U.S. Department of Energy by Lawrence Livermore National Laboratory under
Contract No. DE-AC52-07NA27344. Francisco Ponce is supported by Pacific
Northwest National Laboratory, which is managed for the US Department of
Energy by Battelle under contract DE-AC05-76RL01830. KGL gratefully
acknowledges support from the Facility for Rare Isotope Beams (FRIB) while on
sabbatical. FRIB is a US Department of Energy, Office of Science User Facility
under Award Number DE-SC0000661.

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