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arXiv:2607.18333

A First Bound on the Moffat Energy and Thorium--229 Clock as a Probe of the Nonlocal Time-Energy Structure

Arvin Kouroshnia, J. W. Moffat

quant-phhep-exhep-phhep-thnucl-thSubmitted Jul 19, 2026• Announced Jul 21, 2026• 0 citations
v1 · Jul 19, 2026v2 · Jul 22, 2026v3 · Jul 23, 2026v4 · Aug 2, 2026v5 · Aug 13, 2026v6 · Aug 23, 2026
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Abstract

The new Thorium--229 nuclear clocks give us a new laboratory system in which to test if modification of the Time-Energy uncertainty principle is correct. In this paper we derive the nonlocal Time-Energy uncertainty principle and then apply the published $^{229}$Th nuclear clock data as a first probe of the nonlocality scale \(E_M\). We note that since these experiments were not designed to test nonlocal quantum field theory the constraints are interpreted as motivating bounds and not definitive exclusions. By using the direct clock-energy channel, we find conservative lower bounds on \(E_M\) at the tens of MeV scale, with optimistic present-data estimates reaching the hundred MeV scale. Including the known nuclear sensitivity enhancement of the $^{229}$Th transition gives us stronger model dependent bounds in the GeV range, while nuclear-scale reference-energy scenarios can reach the TeV range. The conclusion we draw from this is that nuclear clocks already provide an experimental route from nonlocal time--energy uncertainty to measurable laboratory bounds on non-Planckian nonlocality.

Comments
18 pages, under review at Annalen der Physik

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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE•Aug 20, 2009•2 citations
The new Thorium--229 nuclear clocks give us a new laboratory system in which to test if modification of the Time-Energy uncertainty principle is correct. In this paper we derive the nonlocal Time-Energy uncertainty principle and then apply the published 229^{229}229Th nuclear clock data as a first probe of the nonlocality scale (E_M). We note that since these experiments were not designed to test nonlocal quantum field theory the constraints are interpreted as motivating bounds and not definitive exclusions. By using the direct clock-energy channel, we find conservative lower bounds on (E_M) at the tens of MeV scale, with optimistic present-data estimates reaching the hundred MeV scale. Including the known nuclear sensitivity enhancement of the 229^{229}229Th transition gives us stronger model dependent bounds in the GeV range, while nuclear-scale reference-energy scenarios can reach the TeV range. The conclusion we draw from this is that nuclear clocks already provide an experimental route from nonlocal time--energy uncertainty to measurable laboratory bounds on non-Planckian nonlocality.
The laser-accessible nuclear transition in the thorium-229 isotope has been identified as a promising candidate for the realization of an optical nuclear clock. Such a nuclear clock might rival or outperform current optical clocks based on electron-shell transitions in atoms or ions, is expected to be more robust against external perturbations, and provides enhanced sensitivity in clock-based tests of fundamental principles of physics. Here, we implement a thorium-229 nuclear clock by stabilizing a continuous-wave laser to the 148 nm nuclear transition with rapid feedback based on continuous absorption spectroscopy. The thorium-229 nuclei are embedded into a millimeter-sized, room temperature calcium fluoride crystal. A subharmonic of the 148 nm radiation is continuously compared to a Yb+ single-ion clock. The nuclear clock shows a simple shot-noise limited scaling of the fractional frequency instability of 3⋅10−12τ/s3\cdot 10^{-12} \sqrt{τ/\text{s}}3⋅10−12τ/s​ where τττ is the averaging time, approaching 10−1510^{-15}10−15 instabilities over 1 day of continuous operation. Improvements of the instability by several orders of magnitude can be projected for future solid-state nuclear clocks. We use the nuclear clock to constrain models of ultralight dark matter by searching for periodic fluctuations and slow drifts in the nuclear transition energy, on time scales between 20 s and 1 day. Drawing benefit from the enhanced sensitivity of the thorium-229 transition, these constraints compete with the best atomic clocks concerning dark matter coupling to photons and go beyond previous measurements regarding coupling to the strong force and quarks.
Atomic clocks achieve extraordinary precision through quantum control over large ensembles of atoms, with precision increasing and atom number advancing steadily. Emerging nuclear clock technology encodes clock states in nuclear rather than electronic transitions, offering a new frontier for precision measurement. Here we study whether a large ensemble of such clock nuclei could be used to observe their mutual gravitational interaction. We investigate the gravitational redshift induced by neighboring atoms, including nuclear energy superpositions as sources of gravity, giving rise to a genuinely quantum gravitational interaction between pairs of nuclei. We find that the resulting clock frequency shift due to mutual gravity may become non-negligible for a macroscopic number of nuclei, for instance in a solid-state thorium clock, and we propose approaches to mitigate electromagnetic field effects by exploiting spatial symmetry and high-precision spectroscopy. We further discuss how energy-induced mass superpositions could be directly probed using external gravitational field sensors. By contrast, observing gravity-induced entanglement involving clocks would require substantially greater experimental advances due to decoherence from spontaneous emission. Taken together, our results suggest that nuclear clocks, either alone or in combination with complementary sensors such as optomechanical systems, may offer a future route toward probing gravity at short distances and testing gravitating quantum source masses beyond the Newtonian limit.
Solid-state frequency standards are generally limited by strong decoherence, rendering conventional interrogation schemes inefficient. The 229^{229}229Th nuclear clock provides a unique and timely platform for solid-state optical metrology and nuclear cavity quantum electrodynamics (QED), featuring a coherence time many orders of magnitude shorter than the radiative lifetime in current experiments. Here, we propose and analyze three cavity QED-enhanced clock interrogation schemes that turn this timescale mismatch into an advantage, leveraging collective coupling of thorium nuclei to nanophotonic modes to enable fast interrogation and detection despite the long population lifetime. We reveal the central role of collective cooperativity in determining the clock frequency instability, and derive the optimal conditions (power, working-point detuning, thorium density) for clock operation.
The Th-229 nuclear isomeric state has the lowest energy of all known nuclear excited states, placing it within the reach of current table-top laser technology. This extraordinary property has made this nuclear isomer an attractive candidate for a nuclear optical clock of incredibly high precision and accuracy, both as isolated trapped Th-229 ions and embedded into solid-state platforms. Activity around Th-229 has surged in recent years, driven by breakthroughs in its direct laser excitation. The underlying nuclear physics that gives rise to this unique isomer will be elucidated, as well as the nearly half-century of efforts that led to its direct excitation. The design and systematics of a Th-229 nuclear clock will be discussed, both in ion traps and in the solid-state. These systematics, such as frequency shifts and quenching channels, can be leveraged both to probe the local chemical environment, and as a control knob during clock operation. Finally, the nuclear clock's high sensitivity to the variations of fundamental constants will be discussed.
A nuclear clock synchronized to 229^{229}229Th
Atomic clocks have made time and frequency the most precisely measured quantities in physics, progressing from microwave standards that realize the SI second to optical clocks with unprecedented precision. A nuclear clock transfers the frequency reference from an electronic to a nuclear transition, and the uniquely low-lying, laser-accessible isomeric transition in 229^{229}229Th currently offers the most practical route to compact, robust timekeeping and sensitive tests of fundamental physics. Realizing such a clock requires turning spectroscopy of the 229^{229}229Th nuclear resonance into a stable discriminator for steering a traceable oscillator. Here we demonstrate a 229^{229}229Th nuclear clock by stabilizing a continuous-wave, narrow-linewidth 148.4-nm vacuum-ultraviolet (VUV) laser to a resolved, weakly temperature-sensitive nuclear transition in 229^{229}229Th:::CaF2_22​ crystals. A 101010-μWμ\mathrm{W}μW VUV source generated by four-wave mixing in cadmium vapour and phototube-based frequency-modulation absorption readout provide a fast, high-signal-to-noise nuclear discriminator. The clock reaches a fractional frequency instability of 5×10−13/τ/s5\times10^{-13}/\sqrt{τ/\mathrm{s}}5×10−13/τ/s​ for averaging time τττ. Clock-transition frequencies measured in two independently fabricated crystals agree at the 10−1310^{-13}10−13 level and are consistent with previous VUV-comb measurements on other 229^{229}229Th:CaF2_22​ crystals. These results establish laser-addressed nuclei as operational clock references and provide a reproducible solid-state platform for compact nuclear clocks, nuclear quantum sensors and precision tests of fundamental physics.
Atomic clocks have reached the Standard Quantum Limit (SQL) of precision,1 set by the projection noise inherent in measurements on uncorrelated atoms. It is possible to overcome this limit by entangling the atoms to generate a "squeezed state" of the atomic ensemble. We use the collective interaction of an atomic ensemble with a far-detuned light field in an optical resonator to prepare squeezed states by two different methods: quantum non-demolition (QND) measurement and Hamiltonian evolution. We apply both methods to an ensemble of 5 x 104 87Rb atoms in a superposition of hyperfine clock states. We measure the suppression of projection noise and compare it to the accompanying reduction in signal, thereby quantifying the net gain in spectroscopic sensitivity. By QND measurement, with resolution up to 9 dB below the projection noise level, we achieve 3.0(8) dB of metrologically relevant squeezing. Whereas the measurement-based approach relies on knowledge of the (randomly distributed) measurement outcome to produce a conditionally squeezed state, the method of Hamiltonian evolution produces a known squeezed state independent of detector performance. We mimic the dynamics of the one-axis twisting Hamiltonian, proposed as a generator of squeezed states by Kitagawa and Ueda,2 by using the atom-induced frequency shift of the resonator mode and the corresponding resonator-field-induced shift of the atomic transition frequency to introduce an effective interaction among the atoms. The resulting deterministic squeezing is sufficient to allow a 6.0(4) dB improvement in spectroscopic sensitivity over the SQL.