A Levitated-Magnet Vector Force Sensor for Spin-Dependent Exotic Interactions

Technical Summary: A Levitated-Magnet Vector Force Sensor for Spin-Dependent Exotic Interactions

Problem Statement
The search for spin-dependent exotic interactions offers a sensitive low-energy window into physics beyond the Standard Model, potentially mediated by light bosons. A central challenge in these searches is "selective discrimination": the ability to distinguish among different fermion-pair channels and effective potentials. In macroscopic experiments, source and sensor bodies contain ensembles of various fermion species, causing multiple interaction channels to contribute simultaneously. Furthermore, modern theoretical frameworks indicate that a single mediator (e.g., a $Z'$ boson) can generate several distinct effective potentials for a given coupling type. For instance, parity-violating axial-vector–vector (AV) couplings induce both $V_{11}$ and $V_{12\pm13}$ potentials. Current experimental approaches often rely on simplifications that neglect specific potential terms or assume certain couplings can be ignored, limiting the ability to resolve the multiple effective potentials associated with a single coupling class.

Methodology
The authors propose a magnetically levitated ferromagnetic vector force sensor designed to resolve these directional signatures. The system utilizes two levitated ferromagnets: one acting as a driving spin source and the other as a spin sensor.

• Vector Sensing Mechanism: The core innovation is mapping forces along different spatial axes onto distinct translational mechanical modes of the sensor, each with a unique resonance frequency. By orienting the spins of the driver ($\hat{s}'_e = \hat{x}$) and sensor ($\hat{s}_e = \hat{y}$) orthogonally and driving the source sinusoidally along the x-axis, the system generates three distinct force components:

  • Gravitational force ($F_g$) along $\hat{x}$.
  • Velocity-dependent exotic force ($F_{12\pm13}$) along $\hat{y}$.
  • Spin-dependent exotic force ($F_{11}$) along $\hat{z}$.
    Because these forces excite orthogonal modes with different resonance frequencies ($\omega_x, \omega_y, \omega_z$), the sensor can selectively amplify and distinguish interaction channels within the same coupling class by tuning the drive frequency.

• Experimental Setup: The design is based on an existing superconducting levitation platform (Ref. [30]) using a milligram-scale Nd$_2$Fe$_{14}$B permanent magnet. The setup operates in a liquid helium bath at 4 K to minimize thermal noise and utilizes Type-I superconducting traps for electromagnetic shielding and magnetic levitation. The authors analyze two configurations: a "demonstrated" setup (mass $\approx 0.356$ mg, trap axes $\approx 2.25$ mm) and a "future" optimized setup (mass $\approx 31.8$ mg, trap axes $\approx 3.0$ mm).

• Analysis: Sensitivity projections are derived using a matched-filter likelihood analysis, which is optimal for stationary Gaussian noise. The analysis assumes an Asimov dataset (no signal observed) to compute median expected sensitivities at a 95% confidence level. The thermal force noise power spectral density is calculated using the fluctuation-dissipation theorem, and systematic uncertainties are treated via a pull parameter.

Key Contributions

1. Selective Channel Discrimination: The paper demonstrates that a levitated ferromagnet can function as a vector force sensor capable of separating interaction channels (e.g., isolating pure electron-electron contributions from mixed electron-nucleon contributions) by exploiting the directional structure of spin-dependent forces.
2. Direct Macroscopic Force Probe: Unlike traditional parity-violation experiments that infer interactions from energy shifts or transition amplitudes in atomic systems, this approach probes parity-violating AV interactions directly as a macroscopic force signal in a solid-state system.
3. Resolution of Effective Potentials: The work establishes a design paradigm for resolving multiple effective potentials (such as $V_{11}$ and $V_{12\pm13}$) generated by a single mediator, challenging the common assumption that pure electron-electron contributions can be neglected in macroscopic searches.

Results

• Projected Sensitivities: Using the parameters of an already demonstrated experiment, the authors project sensitivity to the pure electron-electron coupling $|g_e^e A g_e^V|$ in the force range $\lambda \lesssim 1$ cm (mediator masses $M_{Z'} \gtrsim 10^{-5}$ eV/$c^2$). This regime was previously inaccessible to existing torsion-balance or geoelectron-based searches.
• Competitiveness: For the mixed electron-nucleon coupling $|g_e^e A g_{e+N}^V|$, the projected sensitivity reaches $\sim 10^{-26}$, which is competitive with the best results from recent diamagnetically levitated particle experiments.
• Optimization Potential: The authors show that with near-future optimizations (larger magnets and traps), the sensitivity reach could be extended by up to two orders of magnitude in the range $1 \text{ mm} \lesssim \lambda \lesssim 1 \text{ cm}$.
• Scaling Laws: The paper provides analytical scaling laws for sensitivity with respect to particle radius and trap size, indicating that while larger particles generally improve sensitivity, smaller traps are advantageous until the resonator can fully ring up within the observation time.

Significance
The paper claims that levitated ferromagnets represent a promising platform for millimeter-scale searches for spin-dependent fifth forces. By enabling the selective resolution of interaction channels, this platform addresses a key limitation in current exotic interaction searches: the inability to disentangle multiple effective potentials arising from a single coupling class. The authors argue that their results necessitate a re-evaluation of existing fifth-force experiments, suggesting that bounds derived from $V_{12\pm13}$ searches should be interpreted as constraints on combined electron-electron and electron-nucleon interactions rather than purely electron-nucleon ones. Furthermore, the ability to directly probe the pure electron-electron contribution ($g_e^e A g_e^V$) opens a previously inaccessible parameter space for testing models of new physics, such as those involving spin-1 $Z'$ bosons.