Covariant eigenmode overlap formalism for gravitational wave signals in electromagnetic cavities

Abstract

We develop a coordinate invariant formalism which describes the mechanical and electromagnetic interaction of gravitational waves (GWs) with a wide class of resonant detectors. We solve the GW-modified equations of electrodynamics and elasticity with dynamic boundary conditions using an eigenmode expansion. Furthermore, we take damping effects and electromagnetic back-action on mechanical systems covariantly into account. The resulting coupling coefficients are particularly useful for high-frequency gravitational wave experiments using microwave cavities and allow a straightforward numerical implementation for arbitrary detector geometries.

Figures (7)

• Figure 1: Illustration of the setup described in this work. Electromagnetic fields such as $\bar{\bm{B}}$ with cavity boundary conditions are being monitored by an observer like an antenna. If a GW perturbs the metric of flat space $g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$, both the background field, as well as the mechanical structure get perturbed. The displacement of the boundary $\delta\bm{x}$ and rotation of the surface normal $\bar{\bm{N}}$ and tangent $\bar{\bm{T}}$ causes electric fields $\delta\bm{E}^\text{bdy}$ to be emitted at the boundary. Furthermore, the perturbation of the background field causes electric fields $\delta\bm{E}^\text{bulk}$ to be emitted throughout the cavity volume. The total signal $\delta\bm{E}^\text{obs}$ measured by the antenna is coordinate invariant and the quantity we calculate in this work.
• Figure 2: The mechanical displacement of the wall of a cylindrical microwave cavity as calculated using TT and PD coordinates evaluated near the inner radius of the cavity. The GW has a wave vector $\bm{k}_g=\omega_g\hat{\bm{x}}$ and is plus polarized with amplitude $h$. The cavity has an inner radius and length $R=L=1\,\text{m}$. Further details on the numerical calculation can be found in section \ref{['sec:details_on_examples']}. The polar plot on the right is showing the normalized mechanical response from Eqs. \ref{['eq:mech_coupling_coefficients']} for different angles with the cylinder axis for the mechanical mode with the lowest resonant frequency. The total PD and TT curves overlap up to numerical accuracy.
• Figure 3: An evaluation of the dimensionless geometric factors $\eta^\mathrm{bulk/bdy}_{n}$ both in PD and in TT frames respectively from Eqs. \ref{['eq:eta_PD_magnetostatic']} and \ref{['eq:eta_TT_magnetostatic']}, for a static magnetic field in the direction shown by the black arrow for different angles of a GW around the cylinder axis. We assume that the GW frequency $\omega_g$ is the same as the resonant frequency of the eigenmode, such that we can consider freely falling cavity walls. We show the absolute value of the coupling and add the couplings for degenerate mode polarizations in quadrature and do not include $(\eta_n^\text{bdy})^\text{TT}=0$. The total PD and TT curves overlap up to numerical accuracy.
• Figure 4: Magnetostatic experiment. The observed electric field perturbation and its contributions in a cylindrical microwave cavity in a static magnetic field $\bar{\bm{B}}=\bar{B}\hat{\bm{z}}$ aligned with the cylinder axis. The GW has a wave vector $\bm{k}_g=\omega_g\hat{\bm{x}}$ and is plus polarized with amplitude $h$. The field is measured using a short pin antenna oriented along the $y$ axis, attached to the wall at $x=z=0$. The mechanical response is calculated as described in section \ref{['sec:details_on_examples']} including the lowest three mechanical modes and lowest four EM modes with non-zero coupling. The cylinder has equal length and radius of $1\,\text{m}$ and $Q_n=10^6$ for all modes.
• Figure 5: Heterodyne experiment. The observed electric field perturbation and its contributions in two orthogonal coupled cylindrical microwave cavities loaded in phase with the TM$_{010}$ mode. The GW has a wave vector $\bm{k}_g=\omega_g\hat{\bm{x}}$ and is plus polarized with amplitude $h$. The field is measured using a short pin antenna oriented along the $z$ axis of one cavity at the center of mass of one cylinder where $\delta\bm{E}_\text{ant}^\text{PD}=0$. The response is calculated as described in section \ref{['sec:details_on_examples']}, including the out of phase oscillation of the lowest three mechanical modes and lowest four EM modes with non-zero coupling, including the out-of-phase oscillation of the TM$_{010}$ mode, taken to be $10\,\text{kHz}$ away from the symmetric oscillation. The cylinder has equal length and radius of $1\,\text{m}$ and $Q_n=10^{10}$ for all modes. Mechanical resonances are marked by vertical dashed lines.