The Quantum Noise Fraction and the addressable fraction in High-Frequency Gravitational Wave Detection
The quantum noise fraction $β$, the share of a detector's noise power that is quantum in origin, sets a ceiling $\mathcal{E}_{\mathrm{max}} = 1/\sqrt{1-β}$ on the sensitivity improvement that removing quantum-origin noise can yield. In the kHz--GHz band the readout degree of freedom is a bosonic mode and $β= 1/(2\bar{n}_{\mathrm{th}}+1)$, with $\bar{n}_{\mathrm{th}}$ the Bose--Einstein occupation. Quantum noise then dominates only on the cold side of the thermal frontier $k_BT\ln 3 = \hbarω$ (above 229 MHz at 10 mK), which governs mechanical resonators and electromagnetic cavities alike; a real readout turns it into a joint condition on the occupation and on the excess noise of the readout chain. On a multimode acoustic antenna the frontier can be observed within one device, since the logarithmic slope of $\sqrt{S_hQ}$ along the odd-overtone comb increases by one half across it whatever the dependence of the quality factor on overtone number. On this antenna, crossing the frontier costs more in classical sensitivity than quantum enhancement returns. For a GHz mode of an existing bulk acoustic wave device at 10 mK, $β= 0.984$ and $\mathcal{E}_{\mathrm{max}} = 7.8$. That ceiling cannot be reached with a linear readout, because with intrinsic damping the zero-point term is the force noise of the bath, fixed by the fluctuation--dissipation theorem. The addressable fraction $β_a$ excludes it: at the standard quantum limit the same mode has $β= 0.992$ but $β_a = 0.496$, an available improvement of 1.41 against a ceiling of 11.0. An array of $10^4$ such devices cross-correlated for one year reaches $3.6\times 10^{-24}$ Hz$^{-1/2}$ for an integration bandwidth equal to the linewidth; even granting $\mathcal{E}_{\mathrm{max}} = 7.8$ in full, a factor $1.0\times 10^{10}$ above the big-bang-nucleosynthesis bound remains, and it is classical.