Quantum noise scaling in continuously operating multiparameter sensors

We experimentally investigate the quantum noise mechanisms that limit continuously operating multiparameter quantum sensors. Using a hybrid rf-dc optically pumped magnetometer, we map the photon shot noise, spin projection noise, and measurement back-action noise over an order of magnitude in probe power and a factor of three in pump power while remaining quantum-noise-limited. We observe linear, quadratic, and cubic scaling of the respective total noise powers with probe photon flux, together with a quadratic dependence of back-action on pump photon flux, in quantitative agreement with a stochastic Bloch-equation model. At higher probe powers, additional probe-induced relaxation modifies the spin-noise spectrum while preserving the integrated noise scaling. Our results reveal fundamental, resource-dependent trade-offs unique to continuously monitored multiparameter sensors and establish experimentally the quantum limits governing their optimal operation.

💡 Research Summary

This paper presents a comprehensive experimental study of the quantum‑noise mechanisms that set the performance limits of continuously operating, multiparameter quantum sensors. Using a hybrid rf‑dc optically pumped magnetometer (hOPM) based on a 87Rb vapor cell, the authors independently vary the probe‑beam power (0.5–3 mW) and the pump‑laser power (5–15 µW) while preparing the probe in three distinct quantum states: coherent, squeezed, and antisqueezed. By demodulating the polarization‑rotation signal at the Larmor frequency, they obtain noise spectra that contain a flat photon‑shot‑noise (PSN) background and Lorentzian‑shaped contributions from spin‑projection noise (SPN) and measurement back‑action noise (MBA).

The key results are quantitative scaling laws for each noise component as a function of the optical resources. PSN scales linearly with probe power (S_PS​N ∝ P_pr) and shows no dependence on pump power, as expected for quantum fluctuations of the probe’s Stokes S₂ component. SPN, originating from intrinsic spin‑fluctuations, scales quadratically with probe power (S_SP​N ∝ P_pr²) at low probe intensities; at higher probe powers the spectrum broadens due to probe‑induced relaxation (Γ_pr = α P_pr), but the integrated SPN power continues to follow the P_pr² law. MBA, which stems from fluctuations of the probe’s ellipticity (Stokes S₃) that exert an optical Zeeman torque on the polarized spin ensemble, exhibits a cubic dependence on probe power and a quadratic dependence on pump power (S_MBA ∝ P_pr³ P_pu²). The cubic term dominates over the explored probe‑power range, while a small constant offset is attributed to technical background noise.

All three scaling behaviors are in excellent quantitative agreement with a stochastic Bloch‑equation model that incorporates optical Zeeman shifts, probe‑induced relaxation, and Langevin noise sources. The model predicts exactly the observed linear, quadratic, and cubic dependencies, and also accounts for the redistribution of spin‑noise power across a broadened linewidth at high probe powers.

Importantly, the study demonstrates that in continuously monitored multiparameter sensors the measurement back‑action cannot be avoided, unlike in single‑parameter schemes where increasing measurement strength is always beneficial. Consequently, the overall sensitivity exhibits a non‑monotonic dependence on probe intensity: increasing probe power suppresses PSN but simultaneously amplifies SPN transduction and, more dramatically, MBA. The optimal operating point is therefore determined jointly by probe and pump powers, balancing high‑frequency sensitivity limited by PSN against low‑frequency sensitivity limited by MBA. The use of squeezed probe light reduces PSN but proportionally enhances MBA, illustrating the “quantum back‑action conservation” principle: quantum‑enhancement techniques do not automatically improve total sensor performance.

The authors verify that the same scaling laws hold for both the dc and rf readout channels, confirming the universality of the results across different measurement modalities. They also discuss the broader relevance of these findings to a wide class of spin‑based quantum sensors that rely on continuous optical pumping and non‑destructive readout, such as atomic magnetometers, gyroscopes, and devices searching for physics beyond the Standard Model.

In conclusion, the work provides the first experimental validation of the complete set of quantum‑noise scaling laws (linear PSN, quadratic SPN, cubic MBA with an additional quadratic pump dependence) for a continuously operating multiparameter sensor, both with and without quantum‑enhanced probing. The results furnish a clear, resource‑based roadmap for designing and optimizing such sensors, complementing earlier studies on atom‑number scaling, and establishing a full picture of the quantum‑noise landscape that governs their ultimate performance.