Quantum trajectory of a spin oscillator, continuous retrodiction beyond the Heisenberg uncertainty relation

The paper in detail

The oscillator is the collective spin of a highly polarised atomic vapour. Through the Holstein–Primakoff mapping \(\hat x_A=\hat J_y/\sqrt{|\langle J_x\rangle|}\) and \(\hat p_A=\hat J_z/\sqrt{|\langle J_x\rangle|}\), so \([\hat x_A,\hat p_A]=i\). A 0.71 G field makes the spin precess at \(\Omega_L=2\pi\times500.35\) kHz, and a far-detuned probe reads the Faraday rotation, coupling the light to whichever transverse component \(\hat p_A^\prime\) currently lies along it. Modulating the probe at \(4\Omega_L\) samples \(\hat x_A\) and \(\hat p_A\) alternately. Integrated over 37.3 µs, this amounts to a balanced continuous measurement of both.

Conditioning on the past only, the stochastic master equation keeps the state Gaussian, and its variances settle where measurement information balances back-action and damping. With balanced strengths, perfect pumping and perfect efficiency, the predicted variance is exactly the coherent-state value 0.5. Each measurement of one quadrature unsqueezes the other, so prediction cannot go below the standard quantum limit. In the experiment it sits 0.88 dB above it.

Retrodiction adds the effect matrix \(E(t)\), propagated backwards from the identity at the end of the record. The retrodicted outcome distribution at time \(t\) uses both \(\rho(t)\) and \(E(t)\), and for Gaussian dynamics its variance is the harmonic combination of the forward and backward variances. Both quadratures then reach 1.01 dB below the limit, and the angle-resolved data in Fig. 3 show the reduction for every quadrature. Unbalancing the strengths moves squeezing between them: at a ratio of 1.64 to 0.36, \(\hat x_A\) is \(-2.88\) dB and \(\hat p_A\) is \(+1.89\) dB.

The variances do not depend on the probe duty cycle, up to fully continuous probing, so the stroboscopic sequence is a convenience rather than a requirement. The authors suggest reaching the 3 dB bound with an optical cavity or adaptive measurement, and going beyond it by entangling with a second ensemble. How best to turn sub-SQL retrodiction into a sensing advantage for an unknown signal is left open.

Transferable methods

1. Combining a forward state with a backward effect

   A guess about a measurement at time \(t\) can use data recorded after \(t\). The past quantum state formalism pairs the filtered state \(\rho(t)\) with an effect operator \(E(t)\) that carries the later record backwards. For Gaussian dynamics both are Gaussian, and their variances combine like independent estimates.

   For a measurement with operators \(\Omega_m\) at time \(t\), \[P^{\rm ret}(m)=\frac{\operatorname{Tr}\bigl(\Omega_m\rho(t)\Omega_m^\dagger E(t)\bigr)}{\sum_{m^\prime}\operatorname{Tr}\bigl(\Omega_{m^\prime}\rho(t)\Omega_{m^\prime}^\dagger E(t)\bigr)},\] where \(E\) equals the identity after the final detection and evolves backwards under the adjoint of the stochastic master equation. With no later data \(E=I\), and the Born rule is recovered.

   Let \(a=2\kappa_r^2\eta/\Gamma_{\rm tot}\) and let \(S\) be the square root in Eqs. (5) and (6). Then the steady forward and backward variances are \(\sigma_r^2=(S-1)/a\) and \(\gamma_r^2=(S+1)/a\). Combining them as \(1/\Delta_r^2=1/\sigma_r^2+1/\gamma_r^2\) gives \(\Delta_r^2=(S^2-1)/(2aS)\). Editorial check: substituting \(S^2-1\) reproduces the paper's Eq. (7), \[\Delta_r^2=\frac{4V_0+\kappa_{\bar r}^2/\Gamma_{\rm tot}}{4\sqrt{1+4V_0\kappa_r^2\eta/\Gamma_{\rm tot}+\kappa_r^2\kappa_{\bar r}^2\eta/\Gamma_{\rm tot}^2}}.\]

   Editorial check of the limits: with \(V_0=1/2\), \(\eta=1\), balanced strengths and \(k=\kappa^2/\Gamma_{\rm tot}\), one finds \(S=1+k\), \(\sigma^2=1/2\) and \(\Delta^2=(2+k)/(4(1+k))\), which falls to \(1/4\), or \(-3\) dB. With the reported \(\Gamma_{\rm tot}=2\pi\times331\) Hz, \(\kappa^2\eta=2\pi\times1002\) Hz, \(\eta=0.801\) and \(V_0\) 9 percent above 0.5, Eq. (7) gives \(\Delta^2\approx0.341\). Applying the 0.27 dB degradation stated in the supplement gives 0.363, close to the paper's theoretical 0.37 and the measured 0.38 to 0.39. The same parameters predict a forward variance of 0.556, or \(+0.46\) dB, while the measured value is \(+0.88\) dB.

   How to reuse it

   When a monitored system is analysed after the fact, smooth rather than filter: run the estimate forwards and backwards and combine. For linear Gaussian systems this reduces to adding inverse variances, and it sharpens estimates of anything at intermediate times, such as a force or field acting at time \(t\).

2. Alternating QND measurements to watch both quadratures

   A single QND probe measures one quadrature and pushes its back-action into the conjugate one. Gating the probe at four times the Larmor frequency picks out the moments when the measured lab-frame component equals \(\hat x_A\) or \(\hat p_A\), so one beam monitors both. The cost is that each measurement heats the other quadrature.

   In the lab frame \(\hat p_A^\prime=-\hat x_A\sin\Omega_Lt+\hat p_A\cos\Omega_Lt\), and the probe couples through \(H=\kappa\hat p_A^\prime\hat p_L\). Short pulses of 0.1 µs at \(\Omega_Lt\in\{0,\pi/2,\pi,3\pi/2\}\) sample \(\pm\hat p_A\) and \(\mp\hat x_A\) in turn. The two signals are \(\pi/2\) out of phase in the detected light, so they separate cleanly.

   The conditional variances obey \[\frac{d\sigma_r^2}{dt}=-2\Gamma_{\rm tot}\sigma_r^2+2\Gamma_{\rm tot}V_0+\frac{\kappa_{\bar r}^2}{2}-2\eta\kappa_r^2\sigma_r^4,\] where measuring \(r\) removes variance through the last term and measuring the conjugate \(\bar r\) adds \(\kappa_{\bar r}^2/2\). Editorial remark: setting the derivative to zero gives a quadratic in \(\sigma_r^2\) whose positive root is Eq. (5). The back-action term is why balanced prediction cannot fall below 1/2.

   Editorial remark: in decibels, the product \(\Delta x\,\Delta p\) relative to the Heisenberg value is half the sum of the two variance figures. Balanced operation gives \((-1.01-1.01)/2=-1.01\) dB. The unbalanced setting with \(-2.88\) and \(+1.89\) dB gives about \(-0.50\) dB, so moving measurement strength towards one quadrature buys squeezing there at a net loss in the product.

   How to reuse it

   Phase-selective stroboscopic readout lets one probe track two conjugate variables of any oscillator, such as a mechanical mode or the motion of a trapped ion, and so reconstruct its phase-space trajectory. It pairs naturally with forward–backward estimation.

Source: Main text, Eqs. (1)-(7), and Supplementary Section S4. 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. Earlier work by the same groups (Bao et al., 2020) inferred \(x\) from earlier data and \(p\) from later data, which left rotated quadratures above the limit. Here both quadratures are monitored at once and every quadrature falls below it, which the authors describe as the first such demonstration. The experiment makes concrete that the uncertainty relation constrains prediction from a state, not what a full measurement record reveals about the past. It also points to multi-parameter sensing that uses data on both sides of the time of interest.

Limitations. Nothing here violates the uncertainty principle. The sub-SQL numbers are conditional variances of weak verification measurements given records on both sides, not simultaneous sharp values, and a projective measurement of one quadrature would destroy the other's squeezing. The scheme is limited to 3 dB even with ideal efficiency, and the experiment reaches about 1 dB, held back by 96.9 percent spin polarisation and 80.1 percent detection efficiency. The verification pulses add some back-action, which the Supplementary Information analyses. The paper contains no statement about AI use.

Description based on the full arXiv HTML of v1, read 2026-11-06: main text, figure captions and the supplementary sections on experimental imperfections and damping rates. The supplementary derivation of the backward equations was read but not rederived in full.