Unification of Thermal and Quantum Noises in Gravitational-Wave Detectors

Contemporary gravitational-wave detectors are fundamentally limited by thermal noise—due to dissipation in the mechanical elements of the test mass—and quantum noise—from the vacuum fluctuations of the optical field used to probe the test-mass position. Two other fundamental noises can in principle also limit sensitivity: test-mass quantization noise due to the zero-point fluctuation of its mechanical modes and thermal excitation of the optical field. We use the quantum fluctuation-dissipation theorem to unify all four noises. This unified picture shows precisely when test-mass quantization noise and optical thermal noise can be ignored.

Figure 1

A qualitative depiction of noise terms arising from the general fluctuation-dissipation theorem, coupling into each of the mechanical modes $\widehat{x}$ and optical modes $\widehat{E}$. At low frequencies—up to $\sim \mathrm{kHz}$—contributions to the mechanical motion of the test masses are plotted as power spectra. At these frequencies, thermal noise (red) dominates over the zero-point fluctuations of the test masses (purple). At high frequencies—in the $\sim \mathrm{THz}$ range—the optical power spectrum is shown. Here, the optical vacuum fluctuations (purple) are the relevant effect; the thermal occupation of optical modes (red) is exponentially suppressed. For each noise curve, the relevant prefactors to the susceptibilities $\mathrm{Im}\chi$ [from Eqs. (5), (6), and (8)] are also shown. Gray shows the optical sidebands due to mechanical motion.

Figure 2

Gray shows the design sensitivity of Advanced LIGO, which is dominated by mechanical thermal noise (red solid line) up to $\sim 200\mathrm{Hz}$ and by optical quantum noise (purple solid line) above that frequency [21]. Orange dashed line is the predicted mechanical quantum noise in a simplified model of the test-mass pendulum [Eq. (12)], while purple dashed line shows the prediction [Eq. (9)] for the full mechanical degree of freedom. The difference in shape between the test-mass quantum and thermal noises is a result of the frequency-dependent factor in Eq. (9). Red dashed line is the optical thermal noise predicted from the known optical quantum noise using Eq. (16).