Quantum information of optical magnetometry: Semiclassical Cramer-Rao bound violation and Heisenberg scaling

TL;DR

This work analyzes optical magnetometry from a quantum-information perspective, contrasting a semiclassical independent-atom model with a collective-spin description. It shows that the semiclassical model can transiently violate the quantum Cramer-Rao bound, while the collective model preserves the bound and reveals Heisenberg scaling of the quantum Fisher information with atom number, I_X^{(Q)} ∝ N^2, at zero longitudinal field. The authors employ quantum trajectories augmented with full-counting statistics to connect time-integrated photonic observables to ensemble-level information. The finding that measurement-induced correlations in a dissipative, stationary macroscopic system can yield Heisenberg scaling constitutes a new paradigm for quantum sensing and provides a potential experimental testbed for foundational questions in quantum mechanics. These insights have implications for designing high-precision sensors and for exploring quantum-classical boundaries in large atomic ensembles.

Abstract

Optical magnetometers use the rotation of linearly polarized laser light induced by the Faraday effect for high precision magnetic field measurements. Here, we carry out an in-depth quantum information investigation, deploying two distinct models: The first, semiclassical model can violate the quantum Cramer-Rao bound by several orders of magnitude for weak dissipation and large atom numbers, invalidating the semiclassical approach in this parameter regime. The second model, describing the atoms as a collective spin, respects the Cramer-Rao bound for all parameters. Interestingly, the collective model also predicts Heisenberg scaling for the quantum Fisher information. The comparison of both models shows that Heisenberg scaling is a result of measurement-induced quantum correlation in an otherwise non-interacting quantum system. As the Heisenberg scaling appears in a stationary state of a macroscopic quantum system, it can be thus viewed as a new paradigm in quantum sensing. Intriguingly, the comparison of both models with experimental data can constitute a test for the foundations of quantum mechanics in a macroscopic ensemble of atoms.

Figures (5)

• Figure 1: (a) Sketch of the optical magnetometer, which utilizes the rotation of linearly polarized laser light induced by the light-matter interaction with an atomic vapor to estimates the strength of the magnetic field $B_{\text{z}}$. (b) Four-level scheme of the atoms in the vapor in the z basis, including the atomic transitions induced by the linearly-polarized laser. (c) Fisher information calculated for the ground-state manifold in the semiclassical model (colored lines). For large atom numbers, the signal-to-noise ratio (SNR) of the measured probability distribution can exceed the quantum Fisher Information (QFI), giving rise to a semiclassical Cramer-Rao bound (CRB) violation. The predictions are confirmed by the calculations in the semiclassical four-level system (dashed black). (d) Fisher information for the collective model, in which the Cramer-Rao bound is maintained. Overall parameters are explained in Sec. \ref{['sec:modelAnalysis']}, and $B_{\text{z} }=0$.
• Figure 2: Investigation of the semiclassical model. (a) Expectation value of the spin operators as a function of $B_{\text{z}}$ in the stationary state of the master equation in Eq. \ref{['eq:masterEquation']}. (b) Expectation value of the polarization rotation in Eq. \ref{['eq:semiclassicalPolarizationRotation']}. (c) Variance of $\hat{A}_{\text{rot} }(\tau)$ as given in Eq. \ref{['eq:semiclassicalVariance']}. (d) The quantum Fisher information [QFI, blue, expression below Eq. \ref{['eq:semiclassicalVariance']}] exceeds the signal-to-noise ratio [SNR, red, Eq. \ref{['eq:semiclassicalVariance']}] for almost all $B_{\text{z} }$ except for $\left|B_{\text{z} }\right| \approx 0$, where the Cramer-Rao bound is violated. (e) The violation becomes more pronounced in the weak pump regime. The colored curves depict the results in the ground-state manifold master equation in \ref{['eq:masterEquation']}, while the black (dashed) line shows the calculation using the master equation for the four-level system for validation. Overall parameters are explained in Sec. \ref{['sec:modelAnalysis']}, and $N=8\cdot 10^{10}$ atoms.
• Figure 3: Investigation of the collective model. (a) Expectation value of the spin operators as a function of $B_{\text{z}}$ in the stationary state of the master equation in Eq. \ref{['eq:collectiveMasterEquation']}. (b) Expectation value of the measured mean photon number difference Eq. \ref{['eq:collectivePolarizationBias']}. The scaling by $\tau N$ allows us to compare it to the scaled polarization rotation depicted by the black dashed line (c) Variance of $\hat{n}_{\text{rot},- }(\tau)$ defined in Eq. \ref{['eq:collectiveNoise']}. (d) The quantum Fisher information [QFI, blue] exceeds the signal-to-noise ratio [SNR, red] everywhere, such that the Cramer-Rao bound is respected. (e) The Cramer-Rao bound is maintained in all pump regimes due to the microscopic derivation of the master equation. Overall parameters are explained in Sec. \ref{['sec:modelAnalysis']}. We consider $N=8\cdot 10^{10}$ and a scaled $\gamma_{\text{P} } = 30\text{kHz}/N$ to mimic a non-collective pump.
• Figure 4: Benchmark calculations of the collective model in Eq. \ref{['eq:effectiveMasterEquation']}. (a) Expectation value of $\hat{S}_{\text{z}}$ as a function of $h_{\text{z}}$ for four different atom numbers $N=10,20,40,80$ and fixed $\kappa_{\text{z}} = 0$ and $\kappa_{\text{D} } =0.01 h_{\text{x} }$. The solid black line depicts the mean-field result using the asymptotic cumulant-generating function in Eq. \ref{['eq:masterEquation:HolsteinPrimakoffMeanfield']}. (b) shows the same as (a), but for the quantum Fisher information.
• Figure 5: Expectation value of $\hat{S}_{\text{x}}$ as a function of $\kappa_{\text{z}}$ for $h_{\text{z} } = 0$, $\kappa_{\text{P} } = 0.1 h_{\text{x} }$ and atom numbers $N=10,20,40,80$. The solid black line depicts the mean-field result representing the thermodynamic limit (TDL). (b) Quantum Fisher information for the finite-size system (colored lines) and in the thermodynamic limit given by Eq. \ref{['eq:quantumFisherInfoCollective']} (black). (c) Deviation between the finite-size and thermodynamic-limit calculations in (b) in the crossover regime.