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Geometric noise spectrum in interferometers

Laurent Freidel, Robin Oberfrank

Abstract

We study the power spectral density of time delay fluctuations in an interferometer as a potential low-energy quantum gravitational observable. We derive a general expression for the spectrum in terms of the Wightman function of linear metric perturbations, which we then apply to a variety of cases. We analyze the intrinsic graviton fluctuations in the vacuum, thermal, and squeezed states, as well as the fluctuations induced by the vacuum stress-energy of a massless scalar field. We find that the resulting spectra are free of ultraviolet divergences and that, while thermal and squeezed states provide a natural amplification mechanism, the spectra remain suppressed by the Planck scale.

Geometric noise spectrum in interferometers

Abstract

We study the power spectral density of time delay fluctuations in an interferometer as a potential low-energy quantum gravitational observable. We derive a general expression for the spectrum in terms of the Wightman function of linear metric perturbations, which we then apply to a variety of cases. We analyze the intrinsic graviton fluctuations in the vacuum, thermal, and squeezed states, as well as the fluctuations induced by the vacuum stress-energy of a massless scalar field. We find that the resulting spectra are free of ultraviolet divergences and that, while thermal and squeezed states provide a natural amplification mechanism, the spectra remain suppressed by the Planck scale.
Paper Structure (35 sections, 252 equations, 3 figures)

This paper contains 35 sections, 252 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Conventions and notation
  3. Power spectrum of light ray quantum fluctuations
  4. Michelson interferometer model
  5. Time delay due to linear metric perturbations
  6. Momentum space
  7. Noise power spectral density
  8. Quantum fluctuations
  9. The effect of intrinsic fluctuations
  10. Vacuum state
  11. Thermal state
  12. Planckian enhancement from thermality
  13. Squeezed state
  14. The effect of induced fluctuations
  15. Energy backreaction from massless scalar field
  16. ...and 20 more sections

Figures (3)

  • Figure 1: Spacetime diagram of a null ray crossing the interferometer in the $t-n$ plane. Straight black lines indicate the the beamsplitter $x_B$ and the mirror $x_M$ timelike geodesics. Wiggly lines correspond to the outgoing $x_+$ as well as the incoming $x_-$ null geodesics. Blue dots and labels indicate the perturbed intersection proper times.
  • Figure 2: Plot of the dimensionless function $F_h(x)$ from \ref{['eqn:Fh']} that determines the frequency-dependence of the noise spectral density according to \ref{['eqn:SDh2']}. We also show its large-frequency envelope $(3x \pi)^{-1}$.
  • Figure 3: Plot of the dimensionless function $F_H(x)$ from \ref{['eqn:FH']} that determines the frequency-dependence of the noise spectral density of the induced metric fluctuations according to \ref{['eqn:SDH']}. We also show its envelope $x(5 \cdot 1280 \pi^3)^{-1}$.