Quantum noise scaling in continuously operating multiparameter sensors

TL;DR

The paper addresses quantum-noise limits in continuously monitored multiparameter spin sensors, focusing on a hybrid rf–dc optically pumped magnetometer. Using controlled pump and probe powers and polarization-squeezed readout, they map PSN, SPN, and MBA and show distinct scaling laws: $\mathcal{S}^{\mathrm{PSN}} \propto P_{\mathrm{pr}}$, $\mathcal{S}^{\mathrm{SPN}}_{\mathrm{tot}} \propto P_{\mathrm{pr}}^{2}$, and $\mathcal{S}^{\mathrm{MBA}}_{\mathrm{tot}} \propto P_{\mathrm{pr}}^{3} P_{\mathrm{pu}}^{2}$. A stochastic Bloch-equation model captures the spin dynamics and readout noise, and the data reveal a fundamental trade-off between high-frequency sensitivity limited by PSN and low-frequency sensitivity limited by MBA, implying an optimal operating point. The results generalize to other continuously monitored spin ensembles and provide practical guidance for designing quantum-noise-limited, multiparameter sensors.

Abstract

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.

Figures (3)

• Figure 1: Quantum noise characterization scheme and methodology. (a) Schematic of the experimental setup. $^{87}$Rb vapor is optically pumped in the Bell--Bloom configuration. A dc magnetic field $\mathbf{B}_{dc}$ applied in the $x$--$z$ plane at $45^{\circ}$ with respect to the pump--probe direction enables hybrid dc/rf magnetometry LipkaPRAppl2024. The polarization rotation of the probe beam, prepared in a squeezed (black), coherent (red), or antisqueezed (blue) state, is detected using a Wollaston prism (WP) and a balanced photodetector. HWP - half wave plate, LIA - lock-in amplification. (b) Representative time-domain polarization-rotation signals for the three probe states. (c) Measured polarization noise spectra with a zoomed inset around the Larmor frequency for the three probe states, polarized atoms. PSD - power spectral density. Probe power 2.8mW, pump power 10µW. (d) Demodulated polarization noise PSD of, obtained by a digital lock-in with the pump power as phase reference LipkaPRAppl2024. (e) Fits to the demodulated noise spectra for all three probe states. PSN, SPN and MBA are indicated only for the antisqueezed probe state, for clarity. Shaded regions indicate $\pm1\sigma$ uncertainty obtained by bootstrapping.
• Figure 2: Quantum noise scaling with probe and pump power. Measurements for the dc (squares, solid fits) and rf (circles, dashed fits) channels using coherent (red), squeezed (black), and antisqueezed (blue) probe light. Error bars show $\pm 1 \sigma$ uncertainty obtained by bootstrapping. Left: PSN power spectral density $\mathcal{S}^{\mathrm{PSN}}$, exhibiting linear scaling with probe power, $\mathcal{S}^{\mathrm{PSN}} = a_0 + a_1 P_{\mathrm{pr}}$, and no dependence on pump power. Middle: Total SPN power for an unpolarized ensemble, showing quadratic scaling with probe power, $\mathcal{S}^{\mathrm{SPN}}_{\mathrm{tot}} = a_0 + a_2 P_{\mathrm{pr}}^{2}$, and no dependence on pump power. Right: Total MBA noise power for polarized ensembles, displaying cubic scaling with probe power, $\mathcal{S}^{\mathrm{MBA}}_{\mathrm{tot}} = a_0 + a_3 P_{\mathrm{pr}}^{3}$, and a quadratic dependence on pump power. Fit parameters for all panels are given in End Matter.
• Figure 3: SPN power spectral density and total SPN power as a function of probe power for an unpolarized atomic ensemble, probed with coherent (red), squeezed (black), and antisqueezed (blue) light, shown for both dc and rf measurement channels. Dashed lines indicate quadratic fits of the form $\mathcal{S}^{\mathrm{SPN}} = a_2 P_{\mathrm{pr}}^{2} + a_0$ to the probe-power dependence; the corresponding fit parameters are summarized in Table \ref{['tab:spn_combined_fits']}. The data point at $P_{\mathrm{pr}}=3$ mW is excluded from the quadratic fit to $\mathcal{S}^{\mathrm{SPN}}$ due to probe-induced linewidth broadening but remains consistent with the quadratic scaling of the total SPN noise power.