Approaching the Limit in Multiparameter AC Magnetometry with Quantum Control

Abstract

Simultaneously estimating multiple parameters at the ultimate limit is a central challenge in quantum metrology, often hindered by inherent incompatibilities in optimal estimation strategies. At its most extreme, this incompatibility culminates in a fundamental impossibility when the quantum Fisher information matrix (QFIM) becomes singular, rendering joint estimation unattainable. This is the case for a canonical problem: estimating the amplitude and frequency of an AC magnetic field, where the generators are parallel to each other. Here, we introduce a quantum control protocol that resolves this singularity. Our control protocol strategically engineers the sensor's time evolution so the generators for the two parameters become orthogonal. It not only removes the singularity but also restores the optimal scaling of precision with interrogation time for both parameters simultaneously. We experimentally validate this protocol using a nitrogen-vacancy center in diamond at room temperature, demonstrating the concurrent achievement of the optimal scaling for both parameters under realistic conditions.

Figures (7)

• Figure 1: Schematic illustration of the role of the control Hamiltonian. For the target Hamiltonian $H_\theta = \gamma B \cos(\omega t)\sigma_x$, the generators associated with $B$ and $\omega$ are commuting and parallel, resulting in a singular QFIM. Introducing the control Hamiltonian $H_c$ to optimize the time evolution makes the generators orthogonal, thereby restoring optimal scaling with respect to the sensing duration $T$.
• Figure 1: $\omega$ dependence of the quantum Fisher information matrix (QFIM) elements for $T = 1$, $5$, and $10$. The parameter values are $B = 1$ and $\gamma = 1$.
• Figure 2: (a) Schematic diagram of the experiment using a single NV center. The electronic spin serves as the sensor qubit, while the nitrogen nuclear spin is utilized as an ancilla qubit. The experimental sequence consists of three steps: Bell-state preparation, sensing, and Bell-state measurement. (b) Schematic diagram of the sensing sequence. The target magnetic field (blue) and control magnetic field (green) alternately interact with the sensor. Dynamical decoupling is implemented by applying $\pi$ pulses to cancel the Ising-type interaction between the sensor and ancilla qubits.
• Figure 2: Relative error of the generators (a) and the QFIM elements (b) as a function of $\omega T$.
• Figure 3: Normalized measured signals, corresponding to $1-p_i$ ($i=1,2$; $i=1$ in blue and $i=2$ in red), as functions of the amplitude $B$ (a,c) and frequency $\omega$ (b,d) of the target microwave field for two different numbers of repetitions, $N=1$ (a,b) and $N=8$ (c,d). Here, the control field parameters were set as follows: amplitude $B_c = 5.65\ \text{G}$ and frequency $\omega_c = (2\pi)\times1870\ \text{MHz}$. The dashed lines represent the model that accounts for SPAM errors due to laser irradiation. Error bars represent the standard deviation of the signal. Each displayed point is a result of $n=3\times10^6$ averages of sequences.