Quantum mechanics places a fundamental limit on the precision of continuous measurements. The Heisenberg uncertainty principle dictates that as the precision of a measurement of an observable (for example, position) increases, back action creates increased uncertainty in the conjugate variable (for example, momentum). In interferometric gravitational-wave detectors, higher laser powers reduce the position uncertainty created by shot noise (the photon-counting error caused by the quantum nature of the laser) but necessarily do so at the expense of back action in the form of quantum radiation pressure noise (QRPN)1. Once at design sensitivity, the gravitational-wave detectors Advanced LIGO2, VIRGO3 and KAGRA4 will be limited by QRPN at frequencies between 10 hertz and 100 hertz. There exist several proposals to improve the sensitivity of gravitational-wave detectors by mitigating QRPN5,6,7,8,9,10, but until now no platform has allowed for experimental tests of these ideas. Here we present a broadband measurement of QRPN at room temperature at frequencies relevant to gravitational-wave detectors. The noise spectrum obtained shows effects due to QRPN between about 2 kilohertz and 100 kilohertz, and the measured magnitude of QRPN agrees with our model. We now have a testbed for studying techniques with which to mitigate quantum back action, such as variational readout and squeezed light injection7, with the aim of improving the sensitivity of future gravitational-wave detectors.
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The data pertaining to this study are available from the corresponding authors upon reasonable request.
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J.C. and T.C. are supported by National Science Foundation grants PHY-1150531 and PHY-1806634. N.A., A.L. and N.M. are supported by National Science Foundation grants PHY-1707840 and PHY-1404245. A portion of this work was performed in the UCSB Nanofabrication Facility. J.C. and T.C. thank D. McClelland, R. Ward and M. J. Yap for comments and discussion. The authors thank C. Wipf and V. Sudhir for comments during the manuscript preparation. This document has been assigned the LIGO document number LIGO-P1800033.
Nature thanks Warwick Bowen, Yuta Michimura and the other anonymous reviewer(s) for their contribution to the peer review of this work.
The authors declare no competing interests.
Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Extended data figures and tables
For the measurement with 220 mW circulating power, each noise source that contributed to the sum of subdominant noises in Fig. 2 is shown (see key). The narrow peaks in the displacement noise measurement are a result of parametric nonlinear coupling between various mechanical modes, and this coupling is negligible at low circulating powers.
Extended Data Fig. 2 Dependence of thermal noise on circulating power as caused by change in beam position.
a, The total displacement noise around the yaw mechanical mode for each of the four circulating power levels (see key). b, As a but centred on the pitch mechanical mode. In these measurements, thermal noise is the dominant noise source at frequencies near the mechanical resonances. The thermal noise around the pitch mode decreases from 73 mW to 110 mW of circulating power, and then increases at 220 mW. This change is consistent with the cavity mode passing through the nodal point of this mode at an intermediate power level. Each panel includes an image from the finite element model depicting the motion associated with the mechanical mode. In both images, the blue portion represents a positive displacement from equilibrium (thin black outline), and the red area denotes a negative displacement. The nodal line for the mechanical modes is drawn in white.
The effect of the change in beam position is seen in the change of height of the peaks in the displacement spectrum at the frequencies of the higher-order mechanical modes. The blue curve is taken with 10 mW of circulating power with a cavity mode alignment similar to the QRPN measurement with 220 mW circulating power. The green curve is for an alignment to minimize the coupling of the pitch and yaw modes at 10 mW circulating power. The red curve is based on a model that sets the modal mass of the higher-order modes so that the peaks match those in the displacement measurement shown in Fig. 2.
Modelled quantum displacement noise at 20 kHz and 220 mW of circulating power is shown as a function of intracavity loss and cavity detuning: the optical-spring frequency, which has been precisely measured, is held constant in the model. To be conservative, the range in values for the cavity loss and detuning in this figure are much larger than the constraints obtained by measurements of the optical spring.
This contour plot shows what fraction of the total measured displacement noise power spectral density (PSD) is contributed by QRPN, as a function of measurement frequency and circulating power. The quantity shown on the colour scale at right is the ratio of the PSDs of the QRPN model to the total measured noise. Whereas in the rest of this Letter we present the data as amplitude spectral densities in order to put them in the perspective of GW measurements, we use PSDs to calculate percentage and ratios, and to make this figure, because all the noises are added in quadrature to make up the total noise. We interpolate the data between the measurements at 73 mW, 110 mW, 150 mW and 220 mW. The vertical stripe at 876 Hz is an artefact of the fundamental resonance not being perfectly resolved in the measurement. The blue vertical stripes at 3.7 kHz, 15 kHz and 28 kHz are higher-order mechanical modes of the microresonator. The contours are at a spacing of 0.05 (5%).
Shown is the measured QRPN level obtained by multiplying the transfer function measurement TFAM by the shot noise of the effective cavity input power and applying the calibration procedure, as described in Methods. The result of this calculation agrees with the modelled QRPN.
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Cripe, J., Aggarwal, N., Lanza, R. et al. Measurement of quantum back action in the audio band at room temperature. Nature 568, 364–367 (2019). https://doi.org/10.1038/s41586-019-1051-4
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