Fig. 1: Schematics for retrodiction beyond the Heisenberg uncertainty relation.

a Experiment schematics. A paraffin coated 20mm × 7mm × 7mm rectangular vapor cell at 53 °C resides inside a 4-layer magnetic shielding. A coherent spin state (CSS) is created along the quantization axis x by optical pumping, with a pump laser tuned to the Rb D1 transition \(5{S}_{1/2},F=2\to 5{P}_{1/2},F^{\prime} =2\) and a repump laser beam stabilized to the Rb D2 transition \(5{S}_{1/2},F=1\to 5{P}_{3/2},F^{\prime} =2\), sharing the same circular polarization. A magnetic field of 0.71G induces a ground-state Zeeman splitting of about Ω_L = 2π × 500 kHz and stabilizes the collective spin component along the x direction. A linearly polarized off-resonance D2 laser beam, propagating in the z direction, probes the quantum fluctuations of the spin. The Stokes component S_y that characterizes the linear polarization in the  ± 45^∘ basis is measured using a balanced polarimetry scheme. A Lock-in amplifier (LIA) extracts the signal at the Lamor frequency. b Pulse sequence. The pump lasers prepare the atoms in the CSS. The pump lasers are turned off and the probe laser is turned on to interact with the atoms. The probe pulse sequence is divided in four parts, measuring the quadratures indicated in the figure. c Theoretical polar plot of the variance of the atomic oscillator quadratures conditioned on prior (blue curve) and on prior and posterior measurements (red curve). The plot assumes ideal experiments without decoherence and decay, probing strengths \({\kappa }_{1}^{2}=1.7\), \({\kappa }_{3}^{2}=3.3\) and \({\kappa }_{4}^{2}=2.2\) and a duty factor of 14%, for details see Supplementary Note 4. The radial distance of 0.5 in the polar plot represents the SQL, and the retrodicted variances of the horizontal and vertical \({\hat{p}}_{\text{A}}\) and \({\hat{x}}_{\text{A}}\) quadratures are both reduced in violation of the HUR.