Search of high-frequency variations of fundamental constants using spin-based quantum sensors

Precision searches of fundamental constant variations significantly enhance our understanding of the natural world. Earlier research predominantly utilized atomic spectroscopy [1] and optomechanical systems [2] to search variations below 0.1 GHz. However, investigating high-frequency variations in fundamental constants remains an experimental challenge, despite its significance for many fundamental physics questions. We propose and implement an experiment harnessing spin quantum sensors to search high-frequency variations in fundamental constants, encompassing the fine-structure constant and electron mass, in the previously uncharted frequency domain from 0.1 to 12 GHz. This approach yields constraints on their relative variations as low as 5 parts per million (ppm) and 8 ppm, respectively. Furthermore, based on these results, we establish stringent upper limits on the coupling constants associated with scalar field dark matter from 0.4 to 50 |$\mu$|eV. Our research highlights the potential of spin quantum sensors to investigate fundamental physics at high frequencies. Although our sensitivity requires further enhancement, as it lags in the low-frequency range by 10 orders of magnitude, this indicates the increasing challenge with rising frequency. Our research highlights the potential of spin quantum sensors to investigate fundamental physics at high frequencies.

The investigation into the nature of fundamental constants and their potential variations across space and time constitutes a captivating and multifaceted domain within modern physics, as evidenced by a body of literature [3,4]. The prospect of variability in fundamental constants is grounded in factors such as the evolution of background fields or the presence of extra dimensions, concepts often proposed within the realm of particle physics beyond the standard model and string theory. Among the various parameters governing particle physics and gravitational interactions, special attention has been directed toward the fine-structure constant (⁠|$\alpha$|⁠) and the electron mass (⁠|$m_e$|⁠) in this sphere of investigation. Precise searches conducted on Earth consistently refine our comprehension of potential variations, narrowing down the permissible limits for such deviations [5].

The time-varying fundamental constants manifest in the presence of a background scalar field [6]. This scalar field interacts with the standard model field through the coupling with the electromagnetic field tensor |$F_{\mu \nu }$|⁠, proportional to |$\phi F_{\mu \nu } F^{\mu \nu }$|⁠, and the coupling with the standard model electron field |$\psi _e$|⁠, proportional to |$\phi m_e \bar{\psi }_e \psi _e$|⁠. These interactions lead to variations in the fine-structure constant |$\alpha$| and the electron mass |$m_e$|⁠, as

Here, |$\alpha _0$| and |$m_{e, 0}$| denote the bare terms, while |$\Lambda _{\gamma }$| and |$\Lambda _{e}$| parameterize the respective couplings.

Parameters |$\alpha$| and |$m_e$| possess the potential to evolve over cosmological timescales. Ultralight bosons serve as natural dark matter candidates [7]. With sub-electronvolt masses, they exhibit substantial occupation numbers, resembling coherent oscillating waves characterized by the frequency |$\omega _{\phi }$|⁠, leading to a scalar field |$\phi \propto \cos (\omega _\phi t)$|⁠. Enhanced field values may emerge when these bosonic waves aggregate into localized compact structures or are emitted from transient astrophysical events [8]. Additionally, topological defects with spatially varying bosonic field values can produce detectable signals when passing through Earth-based detectors [9].

Temporal variations are characterized prominently by frequency. Signals originating from bosonic fields can cover a broad frequency spectrum, ranging from nanohertz to petahertz for wave-like dark matter. Astrophysical observations, including cosmological assessments and investigations into deviations in strong gravitational potentials, predominantly focus on linear and constant-order variations, probing the lower end of the frequency ranges. Existing terrestrial searches for variations in fundamental constants, by utilizing techniques such as atomic spectroscopy [1] and optomechanical systems [2], have predominantly explored frequencies below 0.1 GHz. However, there is currently a dearth of search techniques available for higher frequency ranges.

This study introduces an experiment (Fig. 1a) employing a spin quantum sensor to investigate potential variations in the fine-structure constant |$\alpha$| and electron mass |$m_e$| using the quantum mixing method [10]. The spin quantum sensor utilized is the single nitrogen-vacancy (NV) electron spin in diamond (the detailed characteristics of the quantum sensor can be found in Method II.C within the online supplementary material), comprising two strongly coupled electron spins with an effective distance proportional to the cubic lattice spacing |$a_0$|⁠, which is related to |$\alpha$| and |$m_e$|⁠.

Figure 1.

(a) The schematic diagram of the experiment. The NV center serves as a quantum mixer to mix the variation signal with the bias ac field [10]. (b) Left: diamond-based spin quantum sensor is used in our experiment. A green laser initializes the sensor, and the microwave radiation is fed through a coplanar waveguide for NV spin control. Fluorescence is measured by a confocal optical system. An external permanent magnet adjusts the Zeeman energy level. See Method II.A within the online supplementary material for details. Middle: structure of the spin quantum sensor. The NV center is formed by a nearest-neighbor pair comprising a lattice vacancy and a nitrogen atom, which substitutes for a carbon atom. Right: NV center spin energy levels. The ground state of the NV center is an |$S = 1 $| spin system, with a zero-field splitting of |$D/2{\pi} = 2.87 $| GHz. The degenerate |${m}_{s}={\pm 1} $| states are lifted up by a magnetic field, with energy |$E_{ \pm }/\hbar = D \ {\pm}\ {\gamma}_{eB_0}$|⁠. Variations in the fundamental constants will lead to changes in the spin levels. (c) We confirm the validity of the quantum mixing method by introducing a simulated signal (see Method II.B within the online supplementary material for details). The upper curve represents the spin longitude relaxation without the simulated signal, whereas the lower curve exhibits the spin longitude relaxation with the simulated signal included, revealing an accelerated decay resulting from the quantum mixing process. (d) The upper limit of the additional relaxation rate was determined with a 95% confidence level within the frequency range of 0.1 to 12 GHz. (e) The upper limits of the variations in fundamental constants. Utilizing the upper limits |${\gamma}_{\phi}^{\text{up}}$| of the additional relaxation rate in the frequency range of 0.1–12 GHz, we estimated the upper limit of the variations in fundamental constants, employing a 95% confidence level. The upper limits of variations in the fine-structure constant |$\alpha $| at different frequencies are depicted on the left y axis, while those of variations in the electron mass |$m_e$| are displayed on the right y axis. The estimated results are represented by dash–dot lines. The data from our experimental results can be found in Table S1. (f) Constraints on the coupling parameters |$\Lambda _{\gamma }^{-1}$| and |$\Lambda _{e}^{-1}$| as functions of the field mass |$m_{\phi }$| and Compton frequency for scalar field dark matter were established at a 95% confidence level. The violet region illustrates the excluded parameter space in this experiment, with the dashed violet line corresponding to simulated results from the ensemble spin sensor. Other colored regions represent excluded parameter spaces in previous experiments: GEO600, AURIGA, DAMNED, dynamical decoupling in an Sr|$^{+}$| optical clock, Cs clock in cavity, ADMX, CAST search for axions, tests of the equivalence principle and searches for ‘fifth forces’. The data from our experimental results can be found in Table S2, while references for other related data are provided in Fig. S6.

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The NV center, composed of a substitutional nitrogen atom adjacent to a vacancy (Fig. 1b, left), can be described as two holes occupying double-generated orbitals, forming a spin-1 system. The energy difference in this system is predominantly determined by the spin-spin interaction |$D=2\pi \cdot 2.87$| GHz between these two holes and the Zeeman splitting of spins under an external magnetic field |$B_0$|⁠. The zero-field splitting |$D\propto \mu _B^2a_0^{-3}$| is highly sensitive to variations in the spin magnetic moment coupling (⁠|$\mu _B^2$|⁠) and the lattice constant (⁠|$a_0$|⁠). The former, |$\mu _B^2$|⁠, is proportional to |$\alpha /m_e^2$|⁠, while the latter, |$a_0$|⁠, is proportional to the Bohr radius (⁠|$a_B=\hbar /(\alpha m_e c)$|⁠). Additionally, according to Bloch et al. [11], the external magnetic field |$B_0$| depends on the electron mass, impacting the spin dynamics of the NV center. Consequently, the energy difference between the levels of the NV center, denoted |$E_{\pm }/\hbar =D\pm \gamma _eB_0$| (Fig. 1b, right), with |$\gamma _e$| the gyromagnetic ratio of the electron, provides a valuable means for searching variations simultaneously in both the electron mass and the fine-structure constant.

In this experiment, we utilize the energy level difference |$E_-$| (Fig. 1b) to quantify variations in both the fine-structure constant |$\alpha$| and the electron mass |$m_e$| with |$B_0\approx 51$| mT as (see Method I.B within the online supplementary material for details)

Through the quantum mixing method [10], high-frequency energy level variations can be converted into transitions between NV spin levels (see Methods I.C and I.D within the online supplementary material). The validity of the method is confirmed by introducing a simulated signal in Method II.B within the online supplementary material. This transition leads to spin relaxation, as depicted in Fig. 1c. By measuring the magnitude of this relaxation rate, we can estimate the upper limits on the amplitude of the high-frequency variations (see Method II.D within the online supplementary material for details). Through adjusting the frequency of the bias ac field, we can estimate the upper limits of potential variations in fundamental constants (see Method II.E within the online supplementary material for details) ranging from 0.1 to 12 GHz. The upper limits |$\gamma ^{\text{up}}_\phi$| of the relaxation rate |$\gamma _\phi$| are plotted in Fig. 1d at a 95% confidence level (equivalent to a 2|$\sigma$| standard deviation). It is worth mentioning that our experiment covered the frequency range of |$\omega _\phi /2\pi$| from 0.1 to 12 GHz, comprising a total of 147 experimental points for different |$\omega _\phi$| frequencies. The bias ac field frequency, |$\omega _1/2\pi$|⁠, varied from approximately 0.42 to 10.58 GHz according to Equation S1, with field strength |$\Omega _1/2\pi$| about 10 MHz (see Fig. S1).

The tests of the relaxation rate |$\gamma _\phi$| allow us to estimate the upper limits on magnitudes of high-frequency variations in these constants. The high-frequency variations are constrained within a 95% confidence level, equivalent to a 2|$\sigma$| standard deviation. The experimental outcomes are depicted in Fig. 1e, where the jagged features in the curve result from the near-resonant response of the spin quantum sensor, scaling as |$\sim \exp (\Delta \omega ^2/2\Gamma _2^2)$| (see Method II.B within the online supplementary material for details). The results demonstrate that, within the frequency range of 0.1 to 12 GHz, the best constraint on variations in the fundamental constants lies within the range of approximately |$10^{-6}$| to |$10^{-3}$| in |$\alpha$| and |$m_e$|⁠.

The essential application of the searches for fundamental constant variations lies in the detection of dark matter. There is growing attention towards ultralight dark matter (with mass less than 1 eV). This type of dark matter is of particular interest as it can address some of the peculiarities of small-scale behavior within galaxies while still being consistent with the success of |$\Lambda$|CDM (Lambda cold dark matter) on larger scales. Scalar dark matter, a type of ultralight dark matter, can interact directly with electrons and electromagnetic fields, leading to changes in the electron mass and the fine-structure constant. The mass of the field is given by |$m_\phi =\hbar \omega _\phi /c^2$|⁠, where |$\omega _\phi$| represents the angular Compton frequency. Currently, direct searches constrain the coupling between the scalar field and standard model matter to below 0.4 |$\mu$|eV, corresponding to a frequency of 100 MHz. However, there is a current lack of searching methods for higher mass ranges. By utilizing our method to search fundamental constant variations, we established stringent upper limits on the couplings of scalar dark matter fields in the mass range from 0.4 to 50 |$\mu$|eV (0.1 to 12 GHz), as illustrated in Fig. 1f.

We have evaluated the potential impacts of employing an ensemble of NV centers, considering experimentally feasible parameters such as a 1-mm|$^3$| diamond with 10 ppm nitrogen impurities and a 10% NV yield. The simulation reveals that utilizing an ensemble of NV centers leads to an increased sensitivity in the search for coupling parameters by a factor of |$2.8\times 10^4$|⁠. The estimated results are depicted as dash–dot lines in Fig. 1f. These estimates suggest that the constraints imposed by the fifth force can be exceeded at 57 GHz (0.24 meV). While sensitivity decreases with an increase in |$\omega _\phi$|⁠, this effect can be counteracted by enhancing the intensity of the bias ac field. Improving the efficiency of the microwave waveguide enables the attainment of approximately the gigahertz level, thus mitigating the impact of the rising |$\omega _\phi$| by at least 3 orders. Furthermore, through the utilization of diamond fabrication techniques and quantum control methods, the linewidth of the NV center can be greatly reduced, potentially matching that of the dark matter scalar field, thereby enhancing sensitivity.

In conclusion, we propose and implement a spin quantum sensor based on the NV center in diamond to search the variations in fundamental constants. This system operates at significantly higher frequency ranges beyond the scope of the current gravitational wave detectors and atomic spectroscopy techniques. It enables determination of the upper limit of the variations of the fine-structure constant |$\alpha$| and electron mass |$m_e$| down to 5 ppm and 8 ppm, respectively, and addresses the search for scalar dark matter fields through their coupling with the standard model fields. Additionally, with the advancements in electron spin resonance technology, a broad frequency range from the zero field up to at least 10 THz can be essentially covered. Further developments based on similar approaches will enable extending such systems to cover an even broader frequency spectrum range.

ACKNOWLEDGEMENTS

The authors thank X. Rong for useful discussions.

FUNDING

This work was supported by the National Natural Science Foundation of China (12150011, T2388102, 12075115, 12075116 and 11890702), the Anhui Initiative in Quantum Information Technologies (AHY050000) and the Innovation Program for Quantum Science and Technology (2021ZD0302200). Y.C. is supported by VILLUM FONDEN (37766), the Danish Research Foundation, the European Union’s H2020 ERC Advanced Grant ‘Black holes: gravitational engines of discovery’ (Gravitas–101052587), the Munich Institute for Astro-, Particle and BioPhysics (MIAPbP) that is funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy–EXC-2094–390783311 and by FCT (Fundação para a Ciência e Tecnologia I.P, Portugal) under project No. 2022.01324.PTDC.

AUTHOR CONTRIBUTIONS

J.D. and P.H. supervised the project and proposed the idea. X.K., P.H. and J.D. designed the experiments. X.K. and C.J. prepared the setup. X.K., Y.Z., C.J. and S.C. performed the experiment and the simulation. Y.C. performed the theoretical analysis. All authors wrote the manuscript, analyzed the data, discussed the results and commented on the manuscript.

Conflict of interest statement. None declared.

REFERENCES

Author notes