Broadband Thermal Noise Correlations Induced by Measurement Back-Action

Abstract

Modern mechanical sensors increasingly measure motion with precision sufficient to resolve the fundamental thermal noise floor over a broad band. Compared to traditional sensors -- achieving this limit only near resonance -- this capability provides massive gains in acquisition rates along with access to otherwise obscured transient signals. However, these stronger measurements of motion are naturally accompanied by increased back-action. Here we show how resolving the broadband thermal noise spectrum reveals back-action-induced correlations in the noise from many mechanical modes, even those well-separated in frequency. As a result, the observed spectra can deviate significantly from predictions of the usual single-mode and (uncorrelated) multimode models over the broad band, notably even at the mechanical resonance peaks. This highlights that these effects must be considered in all systems exhibiting measurement back-action, regardless of whether the resonances are spectrally isolated or the readout noise is high enough that the noise peaks appear consistent with simpler models. Additionally, these correlations advantageously allow the thermal noise spectrum to reach a minimum -- equivalent to that of a single mode -- in a band far from the resonance peak, where the mechanical susceptibility is comparatively stable against frequency noise.

Figures (3)

• Figure 1: Model for back-action-induced thermal noise correlations. (a) Canonical optomechanical system with a multimode mechanical system for one mirror. (b) Feedback loop describing (a), with one row per mechanical mode, each converting forces (including uncorrelated thermal noise $\tilde{F}_j$) to displacements $x_j$ through mechanical susceptibilities $\chi_{\text{m},j}$. The optomechanical coupling $G$ converts the total $\tilde{x}$ to detuning $\tilde{\Delta}$, and the cavity's detuning susceptibility $\mathcal{D}$ (just a constant in the fast-cavity limit) converts this to photon number $\tilde{n}_\text{cav}$, which finally applies radiation force $-\hbar G n_\text{cav}$ to all modes simultaneously, correlating displacements $\tilde{x}_j$. (c) Plot of expected thermal displacement noise amplitude spectrum $\sqrt{S_x}$ including $N$=1,001 modes (black curve) of an ideal string with a cavity mode positioned at its center (frequencies $\omega$ normalized by $\Omega_1$). For this example, the cavity is driven by a red-detuned laser, applying anti-spring and shifting all modes to lower frequency. The blue dash curve includes only the fundamental ($j=1$), which exhibits less shift. The red curve shows the result treating all $N$ modes independently, but increasing $G$ to match the fundamental resonance frequency, highlighting major broadband and (inset) resonant discrepancies. (d) Dividing $\sqrt{S_x}$ by the total susceptibility to external forces (at the cavity spot) yields a force noise floor $\sqrt{S_F}$ (black). The dashed blue lines represent the (structural) force noise saulson1990thermal for each individual mode. Importantly, the noise at the shifted resonance frequency (red dashed line) is significantly different than expected for a single mode. Moreover, the single-mode limit is approached at the bare mechanical frequency, far from the modified resonance, where the spectrum is comparatively insensitive to frequency noise artifacts.
• Figure 2: Membrane-cavity system and measurement setup. (a) Optomechanical setup. Up to 2 mW of 1550 nm light is stabilized to near the shot noise limit by intensity modulator (IM) feedback dumont2023high and / or 300 $\upmu$W of 1310 nm light drive an 80-$\upmu$m-long fiber cavity with a $\mathrm{Si_3N_4}$ "trampoline" reinhardt2016ultralownorte2016mechanical aligned near the center (inset image). The mirror coatings achieve finesse $\sim$7,000 at 1550 nm, while the 1310 nm light, outside the high-reflectivity band of the mirrors, generates roughly sinusoidal (very low finesse) fringes with negligible back-action. Reflected light from either beam is collected by a photodiode (PD), and the 1550 nm light is fed back to sheer piezos under the fiber mirrors to stabilize $\bar{\Delta}$ during measurement. In this case, the feedback gain is reduced until the feedback bandwidth $<$300 Hz, such that this does not influence the noise measurements over the bands plotted in Fig. \ref{['fig:3']}; for measurements with 1310 nm light, feedback is not necessary. (b) "Bare" mechanical spectrum (using 1310 nm light), showing the fundamental "symmetric" mode $\mathrm{s_0}$ at 37.7 kHz, along with several higher-order symmetric (torsional) modes s$_j$ (t$_j$) and antisymmetric modes a$_j$reinhardt2016ultralow. Magenta vertical lines show mode frequencies predicted by COMSOL, and gray peaks correspond to high-mass chip modes and electronic noise peaks that do not play a role.
• Figure 3: Displacement (a)-(b) and force (c)-(d) spectra measured (dark blue) with the cavity driven by 1550 nm laser at detuning (incident power) $\bar{\Delta}=-0.26\kappa$ (22 $\upmu$W) and $\bar{\Delta}=0.27\kappa$ (12 $\upmu$W) for the left and right plots, respectively. In (a) and (b), the black solid curve shows a fit to the BANC model using Eq. \ref{['eq:S_x_th_approximate']}, assuming bare mechanical frequency $\Omega_1/2\pi = 37.7$ kHz, quality factor 13.2 $\times10^6$, temperature T=295 K, susceptibility tail $\chi_\text{R}=0.457\pm0.003$ as fixed, independently measured parameters (see main text), and background noise tail $A_\text{T}$ = $4.95\times 10^{-26}$ m$^2$, $\Omega_{\text{eff}}/2\pi$ = 22.3 kHz (41.8 kHz) and $\Gamma_{\text{eff}}/2\pi$ = 1145 Hz (49 Hz) as free parameters for red (blue) detuning. The light blue curve shows the contribution from the first mode ($j=1$), and the orange shows that of the higher-order modes ($j>1$). Shaded regions represent the uncertainties, dominated by those of the quality factor and input coupling rate $\kappa_{\text{ex}}$ calibrated from measured cavity and membrane parameters (see main text). Dashed (solid) gray lines show contributed shot noise and QRFN (measured laser phase noise). (c) and (d) present force noise spectra from Eq. \ref{['eq:S_F']}, with $\chi_m$ estimated from the fit parameters. Red lines in (c)-(d) show force noise estimated by summing all modes independently (no correlations), as in Fig. \ref{['fig:1']}(c). Pink (grey) shading indicates bands where correlations suppress (enhance) the noise relative to this naive expectation. Single-mode sensitivity is approached at the bare mechanical frequency, despite thermal intermodulation noise (TIN) peaks, rather than at the resonance peak (red dashed line).