Noise in gravitational-wave detectors and other classical-force measurements is not influenced by test-mass quantization

It is shown that photon shot noise and radiation-pressure back-action noise are the sole forms of quantum noise in interferometric gravitational wave detectors that operate near or below the standard quantum limit, if one filters the interferometer output appropriately. No additional noise arises from the test masses’ initial quantum state or from reduction of the test-mass state due to measurement of the interferometer output or from the uncertainty principle associated with the test-mass state. Two features of interferometers are central to these conclusions: (i) The interferometer output [the photon number flux $\mathcal{N}^\mathrm{}\left(t\right)\mathrm{}$ entering the final photodetector] commutes with itself at different times in the Heisenberg picture, $\left[\mathcal{N}^\mathrm{}\right(t),\mathcal{N}^{(t}^{\prime })]=0\mathrm{}$ and thus can be regarded as classical. (ii) This number flux is linear to high accuracy in the test-mass initial position and momentum operators $x_o$ and $p_o,$ and those operators influence the measured photon flux $\mathcal{N}^\mathrm{}\left(t\right)\mathrm{}$ in manners that can easily be removed by filtering. For example, in most interferometers $x_o$ and $p_o$ appear in $\mathcal{N}^\mathrm{}\left(t\right)\mathrm{}$ only at the test masses’ $\sim 1\mathrm{Hz}$ pendular swinging frequency and their influence is removed when the output data are high-pass filtered to get rid of noise below $\sim 10\mathrm{Hz}.$ The test-mass operators $x_o$ and $p_o$ contained in the unfiltered output $\mathcal{N}^\mathrm{}\left(t\right)\mathrm{}$ make a nonzero contribution to the commutator $\left[\mathcal{N}^\mathrm{}\right(t),\mathcal{N}^{(t}^{\prime })].$ That contribution is precisely canceled by a nonzero commutation of the photon shot noise and radiation-pressure noise, which also are contained in $\mathcal{N}^\mathrm{}\left(t\right).$ This cancellation of commutators is responsible for the fact that it is possible to derive an interferometer’s standard quantum limit from test-mass considerations, and independently from photon-noise considerations, and get identically the same result. These conclusions are all true for a far wider class of measurements than just gravitational-wave interferometers. To elucidate them, this paper presents a series of idealized thought experiments that are free from the complexities of real measuring systems.