Precise frequency or time measurement is crucial in various scientific fields. The Heisenberg uncertainty principle sets a fundamental limit on the precision of such measurements, inversely proportional to the observation time, typically constrained by the probe's coherence time. This paper explores methods to overcome this limitation. The authors focus on minimizing statistical uncertainty, which can be reduced by averaging over many measurements and increasing the observation time. The study aims to experimentally demonstrate frequency measurement precision scaling faster than the inverse linear scaling with observation time, using a single laser-cooled barium ion as a quantum probe and a coherent control feedback strategy. The core idea involves acquiring phase information at a faster rate, surpassing the Heisenberg limit, specifically for noise frequency estimation in a clock transition. This approach has potential applications in searching for exotic particles interacting with atomic electrons in a time-dependent manner.

The paper reviews existing literature on theoretical bounds on frequency estimation, highlighting the Cramér-Rao bound (CRB) and the quantum Fisher information (QFI) as key measures of estimation precision. It discusses different strategies to surpass the Shot Noise Limit (SNL), including the use of entangled probes. The authors emphasize that while the Heisenberg bound can be achieved for time-independent Hamiltonians using proper quantum strategies, surpassing it requires exploiting nonlinear Hamiltonian responses or implementing adaptive feedback protocols. Previous works have explored these approaches using various quantum systems, such as trapped ions and superconducting qubits, but with limitations in terms of accessible frequency ranges or coherence times. This paper builds on these efforts, aiming to improve precision and extend the measurable frequency range.

The experiment utilizes a single ¹³⁸Ba⁺ ion as a quantum probe, employing its electronic ground state and a metastable state as a qubit. A narrow linewidth laser, phase-locked to an ultra-stable optical cavity, prepares the ion in a coherent superposition. A time-dependent Hamiltonian is generated by intensity modulating an off-resonant laser, introducing a modulation frequency (ω) and depth (Ω). The goal is to estimate ω and Ω. The authors employ a quantum feedback strategy involving preparation, evolution under the external time-dependent field, application of an optimal level-crossing Hamiltonian (OLCH) for enhanced sensitivity, and measurement of the phase gain. The OLCH is applied at specific time instances to maximize the QFI. The sensitivity is determined by measuring the change in phase as a function of frequency or amplitude. The experiment involves performing controlled and uncontrolled measurements, comparing the results to assess the effectiveness of the control Hamiltonian. Statistical analysis, including least-squares fitting, is used to determine the scaling exponent of sensitivity with observation time. The method's applicability to searching for axion-like dark matter particles is also explored by evaluating the sensitivity to the amplitude of the interaction Hamiltonian.

The key findings demonstrate that the frequency uncertainty in the experiment scales as 1/T^1.75±0.03 when applying the optimal control Hamiltonian, surpassing the Heisenberg limit of 1/T for uncontrolled measurements (scaling as 1/T^0.87±0.02). This improvement is observed up to the qubit's decoherence time of ~80 µs. The accessible frequency range extends to kHz, representing a two-order of magnitude improvement compared to previous work. The experiment also shows a weak radiative time-dependent coupling of the probe to the environment, extending the sensing capability. Regarding dark matter search, the experiment establishes a bound on the coupling strength of a 50 kHz mass axion-like particle (ALP) to the atomic electron, finding it to be below 400 GeV^-1. This is a direct measurement, despite being weaker than astrophysical bounds. The 2D plots of Quantum Fisher Information (QFI) as a function of frequency and phase of the control Hamiltonian reveal a clear peak at the applied frequency, confirming accurate frequency estimation. The analysis of the reduced chi-squared values confirms the observed scaling exponents, attributing deviations from the ideal theoretical bounds to imperfections in state preparation and finite control pulse duration.

The results confirm the possibility of surpassing the Heisenberg scaling for parameter estimation with a time-dependent Hamiltonian using a single-atom probe and quantum coherent control protocols. The observed scaling of 1/T^1.75±0.03, while not reaching the theoretical maximum of 1/T², represents a significant improvement over the Heisenberg limit. The extension of the measurable frequency range to the kHz regime is particularly valuable for sensing applications. The application to the search for light mass ALPs showcases the method's potential in fundamental physics. While the current bound on ALP coupling strength is weaker than astrophysical limits, it represents a direct experimental test, which is novel. Future improvements, such as employing a buffer gas cell with a larger number of atoms, could significantly enhance sensitivity, potentially enabling exploration of a wider range of ALP masses and coupling strengths.

This paper successfully demonstrates surpassing the Heisenberg limit in frequency estimation using a single ion. The achieved 1/T^1.75±0.03 scaling extends the accessible frequency range to the kHz domain and showcases the technique's potential in precise noise measurement and dark matter searches. Future work could focus on optimizing experimental parameters to reach the theoretical limit and applying the technique to other systems for enhanced sensitivity in different applications. Further development could focus on searching for dark matter candidates using atomic probes in buffer gas cells to improve the current ALP coupling strength estimate.

The main limitation is the decoherence of the qubit, which restricts the maximum observation time to approximately 80 µs. Imperfections in the preparation of the initial equal superposition state and the finite duration of the control pulses also contribute to deviations from the ideal theoretical scaling. Improvements in experimental techniques, such as phase synchronization of the experimental cycle to the line frequency and intensity stabilization of the qubit laser, could potentially mitigate these limitations. The current setup also only allows for estimating the coupling strength at a single ALP mass.