Taming a macroscopic free mass through continuous quantum measurement

The paper in detail

A free particle has no frequency or length scale of its own. The paper argues that continuous position measurement supplies one. Thermal and back-action noise make the state diffuse in phase space, measurement keeps narrowing it, and the two balance in a steady Gaussian with finite entropy. The time to reach that balance defines a reference frequency, which equals the frequency where the position noise falls below the measurement imprecision. Here \(\Omega_{\mathrm{ref}}/2\pi=6\) Hz and \(t_{\mathrm{ref}}=26\) ms.

The system is a 0.5 × 0.5 × 2 mm fused-silica mirror bar of 1 mg, on a fused-silica fibre 15 cm long and 1 µm thick, at below \(10^{-10}\) mbar. Its yaw angle is read by a double optical lever, which cancels input beam jitter, and is controlled by a radiation-pressure push beam. The rotor is trapped by feedback, released, and then tracked. The thermal noise is consistent with damping rate \(\gamma_{\mathrm{ref}}=2\pi\times7.3\) nHz at the reference frequency, an effective quality factor of about \(10^9\).

Purity \(\mathcal P=1/\sqrt{\det V}\), with \(V\) the conditional covariance in units of \(\hbar/2\), sets the von Neumann entropy of the Gaussian state. The paper assigns a temperature through \(T=(k_B\,\partial S/\partial E)^{-1}\), where \(E\) is the kinetic energy about the inferred centroid, and obtains \[k_BT=\tfrac32f(\mathcal P)E,\qquad f(\mathcal P)=\frac{2\mathcal P}{\ln\frac{1+\mathcal P}{1-\mathcal P}}\approx1.\] Given the rotor's isolation, the same relation predicts a lowest reachable temperature of 15 nK, set by the trade-off between imprecision and back-action.

Fitting a single free parameter, the effective imprecision \(39\pm1\) prad/\(\sqrt{\mathrm{Hz}}\), gives purities \((1.45\pm0.04)\times10^{-4}\) for causal states and \((5.8\pm0.2)\times10^{-4}\) for smoothed ones. The smoothed purity equals that of an oscillator with 860 thermal quanta. The measured noise floor excludes part of the continuous-spontaneous-localisation parameter space at a 100 nm collapse length, reaching below Adler's upper bound. The authors expect that a cavity readout with finesse about 1000 could reach purity 0.4, which by their extended criterion would suffice for gravity-mediated entanglement between two rotors, given enough coupling.

Transferable methods

1. Measurement as the trap: steady states of a monitored free mass

   Without a restoring force, a monitored particle still reaches a steady conditional state. Its covariances scale with fixed powers of the force noise, and that scaling is where the factor \(3/2\) in the temperature formula comes from.

   Take \(\dot x=p/m\), with momentum diffusion \(D\) and position measured with white imprecision \(R\). The Kalman–Bucy covariance obeys \[\dot V=AV+VA^{\mathsf T}+Q-\frac{VC^{\mathsf T}CV}{R},\qquad A=\begin{pmatrix}0&1/m\\0&0\end{pmatrix},\ Q=\begin{pmatrix}0&0\\0&D\end{pmatrix},\ C=(1\ 0),\] and its steady state is finite even though \(A\) has no restoring term. The measurement term removes the variance that the dynamics creates.

   Editorial derivation: solving it at fixed \(R\) gives \(V_{xx}\propto D^{1/4}\), \(V_{xp}\propto D^{1/2}\) and \(V_{pp}\propto D^{3/4}\). Then \(\det V\propto D\), so \(\mathcal P\propto D^{-1/2}\), while \(E=V_{pp}/2m\propto D^{3/4}\). Eliminating \(D\) gives \(\mathcal P\propto E^{-2/3}\).

   For a Gaussian state \(S=\mu_+\ln\mu_+-\mu_-\ln\mu_-\) with \(\mu_\pm=(\mathcal P^{-1}\pm1)/2\), so \(dS/d\mathcal P=-\ln\frac{1+\mathcal P}{1-\mathcal P}\big/(2\mathcal P^2)\). With \(d\mathcal P/dE=-2\mathcal P/(3E)\), \[\frac{dS}{dE}=\frac{\ln\frac{1+\mathcal P}{1-\mathcal P}}{3\mathcal PE},\qquad k_BT=\frac{3\mathcal PE}{\ln\frac{1+\mathcal P}{1-\mathcal P}}=\tfrac32f(\mathcal P)E.\] So the \(3/2\) is the exponent \(2/3\), inverted. It reflects how the measured free particle responds when its noise changes, which is how we read the paper's derivation. The derivation itself is in the supplement, which we did not read.

   Editorial check: integrating the Riccati equation numerically gave exponents \(0.2500\), \(0.5000\) and \(0.7500\), and a finite-difference \(dS/dE\) reproduced \(\tfrac32fE\) to 0.07%. From the stated parameters, assuming room temperature, the thermal torque noise at 6 Hz is \(1.6\times10^{-20}\) N m\(/\sqrt{\mathrm{Hz}}\), close to the stated \(1.8\times10^{-20}\). Equating the free-mass position noise with 39 prad\(/\sqrt{\mathrm{Hz}}\) puts the crossover at 5.6 Hz, against the stated 6 Hz. The quoted scales \(\theta_{\mathrm{ref}}=2.1\times10^{-12}\) rad and \(L_{\mathrm{ref}}=2.6\times10^{-23}\) J s also follow, and purity \(5.8\times10^{-4}\) does correspond to 861 thermal quanta.

   How to reuse it

   For any linearly monitored system, the steady-state Riccati solution gives the scales that a trap would otherwise supply, and how it scales with the noise strengths fixes which thermodynamic quantities can be assigned. Check those exponents before comparing figures across platforms, since an oscillator with the same purity scales differently.

2. Checking a conditional state against its smoothed version

   A conditional covariance is a model prediction, and an error in the noise model hides in it. Smoothing with future data gives a better estimate of the same past state, so the scatter between the causal and smoothed estimates can be measured and compared with the prediction.

   Let \(R\) be the causal estimate of \((\theta,L)\) at \(t_0\), from data before \(t_0\), and \(R_s\) the Rauch–Tung–Striebel smoothed estimate, which also uses data after \(t_0\). Their difference \(R_\leftrightarrow=R-R_s\) is observable, and its sample covariance \(V_\leftrightarrow\) can be computed from the record.

   Editorial derivation of Eq. (2): with \(X\) the true state, the smoothed error \(X-R_s\) is orthogonal to every function of the full record, which includes \(R-R_s\). Writing \(X-R=(X-R_s)-(R-R_s)\) and taking covariances, the cross terms vanish, so \[V=V_s+V_\leftrightarrow.\] The paper's estimator \(V^{(\mathrm{est})}=V_\leftrightarrow+V_s\) is this identity, with the theoretical \(V_s\) as a correction of 10–20%.

   Since both filters use the model, the identity holds only if the model is right. The single unknown, the imprecision level, is therefore fitted until the predicted \(V\) equals \(V_\leftrightarrow+V_s\) in every phase-space direction. The largest remaining mismatch was 1.5%. The method is self-consistent rather than independent, but a wrong noise model would leave a mismatch that the fit could not remove.

   For a free particle the smoothing gain is large: purity rises fourfold, from \(1.45\times10^{-4}\) to \(5.8\times10^{-4}\). That gain is available because the causal states are far from pure; the paper notes that near-pure states would gain little from smoothing.

   How to reuse it

   Wherever a Kalman filter infers a hidden state, the causal and smoothed difference gives a model check without a separate verifying measurement. The check needs long records on both sides of the time of interest, and it tests consistency, so it cannot rule out an error shared by both filters.

Source: Sections 2–5 (Eqs. (1)–(2), Figs. 2–4). Based on the linked paper version. Derivations marked as editorial, and suggested uses in other problems, are editorial guidance rather than results of the paper.

Context & limitations

Why it matters. Measurement replaces the trap: it sets the frequency scale and the steady state. The thermal noise matches a damping rate of only \(2\pi\times7.3\) nHz, and the experiment excludes new low-frequency parameter space for continuous spontaneous localisation. By the authors' figure of merit for gravity-mediated entanglement, it is nearly two orders of magnitude ahead of earlier systems.

Limitations. The 140 nK figure is for smoothed states, which use data recorded after the time in question, so they cannot be acted on in real time. Entropic temperature is a definition based on the conditional entropy, not the temperature of a bath. The states are still far from pure, and the figure of merit for gravity is \(0.8\times10^{-6}\), far below the value of 1 that entanglement would need. The supplementary information, which holds the filter model and the temperature derivation, was not read.

Description based on the full main text of the linked arXiv version, read 2026-11-03. The supplementary information, which the HTML version does not include, was not read; the derivations of Eq. (1), of the minimum temperature and of the gravity criterion are taken from the main text's summary of it.