Table 2 Notations and conventions, used in this review
From: Advanced quantum techniques for future gravitational-wave detectors
Notation and value | Comments |
|---|---|
L | Length of the arms of the interferometer |
\(\tau =L/c\) | Light travel time at distance L |
\(\omega \) | Optical frequencies |
\(\omega _0\) | Interferometer resonance frequency |
\(\omega _p\) | Optical pumping frequency (laser frequency) |
\(\varOmega = \omega -\omega _p\) | Modulation sideband frequency w.r.t. laser frequency \(\omega _p\) |
\(\varDelta =\omega _p-\omega _0\) | Optical pump detuning from the cavity resonance frequency \(\omega _0\) |
\(\mathcal {E}_{0} = \sqrt{\dfrac{4\pi \hbar \omega _p}{\mathcal {A}c}}\) | Normalisation constant of the second quantisation of a monochromatic light beam |
\(A^{in} = \sqrt{\dfrac{2P^{in}}{\hbar \omega _p}}\) | Classical quadrature amplitude of the incident light beam with power \(P^{in}\) |
\(T\,(R)\) | Power transmissivity (reflectivity) of the mirror |
\(\gamma _{\mathrm{arm}} = cT/4L\) | Arm cavity half-bandwidth for input mirror transsmissivity T and perfect end mirror |
\(\delta _{\mathrm{arm}}\) | Arm cavity detuning/differential detuning of the arms of Fabry–Perot–Michelson interferometer |
\(\gamma \) | Interferometer effective half-bandwidth |
\(\beta (\varOmega )\) | Phase shift acquired by sidebands in the interferometer |
\(\mathcal {K}(\varOmega )\) | Optomechanical coupling factor (Kimble factor) of the interferometer |
\(P^{in}\) | Incident light beam power |
\(P_c = 2P_{\mathrm{arm}}\) | Total power, circulating in both arms of the interferometer (at the test masses) |
M | Mass of the mirror |
m | Reduced mass of the signal mechanical mode of the interferometer (e.g., dARM mode)\(^\mathrm{a}\) |
\(\varTheta = \dfrac{4\omega _p P_c}{mcL}\) | Normalised intracavity power |
\(h_{\mathrm{SQL}} = \sqrt{\dfrac{8\hbar }{mL^2\varOmega ^2}}\) | Standard quantum limit of a free mass for GW strain |
\(x_{\mathrm{SQL}} = \sqrt{\dfrac{2\hbar }{m\varOmega ^2}}\) | Standard quantum limit of a free mass for displacement |