Single arm interferometry to probe the scalar field dark matter

1. Problem Statement

The nature of the dark sector, specifically Scalar Dark Matter (SDM), remains a major unsolved problem in cosmology. SDM candidates are ultra-light scalar particles (mass $m_\phi \ll 1$ eV) that may form a coherent, oscillating classical field permeating the galaxy.

• The Challenge: Standard interferometric techniques (like Michelson-Morley type double-arm interferometers used in LIGO) are ineffective for detecting SDM-photon interactions. Because the interaction Lagrangian affects both arms of a standard interferometer identically, the resulting phase shifts cancel out in differential measurements.
• The Goal: The authors propose a novel detection method to reveal the coupling between photons and the scalar dark matter field $\phi$, which is currently unobservable via standard phase-difference analysis.

2. Methodology

The authors propose a single-arm interferometric scheme utilizing non-classical states of light (squeezed states) to break the symmetry that causes cancellation in standard setups.

A. Theoretical Framework

• Interaction Model: The interaction is described by an effective coupling term in the Lagrangian density:
  $$\mathcal{L}_{int} = \frac{1}{4} \frac{\phi}{\Lambda_\gamma} F_{\mu\nu}F^{\mu\nu}$$
  where $\phi$ is the scalar field, $F_{\mu\nu}$ is the electromagnetic field tensor, and $\Lambda_\gamma$ parametrizes the interaction strength.
• Field Dynamics: The SDM field is modeled as a coherently oscillating classical field: $\phi(t) \approx \phi_0 \cos(\omega t + \tilde{\theta})$, with $\omega \approx m_\phi$.
• Hamiltonian: Upon quantization, the interaction leads to a Hamiltonian operator proportional to the photon number operator $\hat{N}$:
  $$\hat{H}_{int}(t) = -\frac{\phi(t)}{\Lambda_\gamma} \int d^3k \sum_\lambda |\vec{k}| \hat{N}_{\vec{k}\lambda}$$
  This interaction induces a time-dependent phase shift $\delta(t)$ on the photon state but does not change the photon number in a simple coherent state evolution.

B. Experimental Setup (Single-Arm Interferometer)

The proposed setup consists of a single optical path of length $L$ (e.g., inside a Fabry-Perot cavity) with the following sequence:

1. Input: A coherent laser beam $|\alpha\rangle$.
2. Squeezing: A squeezing operator $\hat{S}(r)$ is applied to the beam, creating a squeezed state $|\Phi(r)\rangle = \hat{S}(r)|\alpha\rangle$.
3. Propagation: The beam propagates through the vacuum over length $L$, interacting with the SDM field. This imparts a phase shift $\delta(L)$ dependent on the SDM parameters.
4. Anti-Squeezing: An inverse squeezing operator $\hat{S}^{-1}(r)$ is applied.
5. Detection: A photodetector measures the output photon number $N_{out}$.

Key Mechanism:
In the absence of SDM interaction, the sequence $\hat{S}^{-1}\hat{S}$ returns the state to the original $|\alpha\rangle$. However, if SDM interaction occurs, the phase shift $\delta$ accumulates between the squeezing and anti-squeezing operations. Due to the non-commutativity of the phase shift operator and the squeezing operator (specifically the Bogoliubov transformation), the final state differs from the initial state, resulting in a measurable change in the output photon number ($N_{out} \neq N_{in}$).

3. Key Contributions

• Novel Detection Strategy: The paper introduces the first proposal to use a single-arm interferometer combined with squeezed light to detect scalar dark matter. This bypasses the cancellation problem inherent in double-arm interferometers.
• Analytical Derivation: The authors derive the explicit relationship between the SDM parameters (mass $m_\phi$, coupling $g_\gamma = 1/\Lambda_\gamma$) and the observable signal (change in photon number $\Delta N$).
• Demonstration of Necessity of Squeezing: The analysis proves that without the Bogoliubov transformation (squeezing/anti-squeezing), the interaction term $\propto \hat{N}$ would only induce a global phase shift, which is unobservable in photon counting statistics. Squeezing converts this phase shift into an amplitude (intensity) modulation.

4. Results

The authors performed a numerical analysis using parameters compatible with existing gravitational wave detectors (LIGO, VIRGO) and optical cavities (PVLAS).

• Parameters:

  • Local Dark Matter Density: $\rho_{local} \approx 4 \times 10^5 \, \text{GeV/m}^3$.
  • Coupling constant range: $g_\gamma \in [10^{-35}, 10^{-28}] \, \text{eV}^{-1}$.
  • Scalar mass range: $m_\phi \in [10^{-22}, 10^{-12}] \, \text{eV}$ (covering fuzzy cold dark matter).
  • Effective Arm Length: $L = 950 \, \text{km}$ (simulating LIGO's effective path).
  • Squeezing: $r = 0.6$ (LIGO standard) and $r = 1.7$ (15 dB squeezed states).

• Signal-to-Noise Ratio (SNR):
  The signal is defined as the deviation in photon number $\Delta N(L) = N_{out} - N_{in}$.
  $$\frac{\Delta N(L)}{N_{in}} \propto \sin^2(\delta(L)) \left[ \left(\frac{1}{|\alpha|^2} + 2\right)\sinh^2(2r) + \sinh(4r) \right]$$
  The analysis shows that for realistic parameters:

  • $\Delta N > \sqrt{N_{out}}$ (the Poissonian shot noise), making the signal detectable.
  • The signal scales strongly with the squeezing parameter $r$.
  • The method is sensitive to a wide range of SDM masses, including the "fuzzy" dark matter regime ($m_\phi \sim 10^{-22}$ eV).

• Comparison with Existing Limits:
  The proposed method can probe regions of the parameter space ($g_\gamma$ vs. $m_\phi$) that overlap with and extend beyond current constraints derived from LIGO data (specifically the analysis by Göttl et al., 2024).

5. Significance and Limitations

• Significance:

  • This approach offers a complementary avenue to existing searches (axion haloscopes, atomic spectroscopy, and gravitational wave detectors) for ultra-light scalar dark matter.
  • It leverages existing infrastructure (LIGO, PVLAS) by repurposing them for single-arm operation with squeezed light, potentially avoiding the need for entirely new hardware.
  • It highlights the utility of quantum metrology (squeezed states) in fundamental physics searches beyond just improving sensitivity to gravitational waves.

• Limitations & Future Work:

  • Coherence: The model assumes a perfectly coherent, monochromatic SDM field. Realistic galactic velocity dispersion may cause decoherence, reducing the signal for very long baselines.
  • Noise: The current analysis uses an idealized noise model. Real-world factors like optical losses, residual phase noise, and technical noise in squeezing generation must be rigorously budgeted to confirm feasibility.
  • Specificity: The signal (phase shift/intensity variation) is generic to any scalar field coupling to electromagnetism. Distinguishing SDM from other scalar backgrounds would require analyzing temporal modulations linked to Earth's motion through the galactic halo.

Conclusion:
The paper successfully demonstrates that a single-arm interferometer utilizing squeezed light can detect the interaction between scalar dark matter and photons via intensity variations, a signal that is invisible to standard double-arm interferometry. This provides a promising, experimentally viable path to constrain or discover ultra-light scalar dark matter.