Quantum metrology leverages superposition and entanglement for enhanced precision beyond classical methods. Conventional schemes involve preparing a probe state, evolving it under a parameter-encoding dynamics, and estimating the parameter from the final state. Entangled probes enable Heisenberg scaling (N⁻¹ or T⁻¹ precision, where N is the probe number and T is evolution time), surpassing the classical shot-noise limit (N⁻¹/² or T⁻¹/²). However, maintaining long coherence is crucial. Critical quantum metrology offers intrinsic robustness through adiabatic evolution near a quantum phase transition (QPT). Previous protocols often used continuous QPTs, requiring thermodynamic limits and fixed energy gaps, hindering experimental realization. This work presents an adiabatic scheme using a perturbed two-spin Ising model with a first-order QPT. A small transverse magnetic field encodes the unknown parameter, controls the energy gap, and thus the adiabatic evolution time. Nuclear magnetic resonance (NMR) experiments demonstrate Heisenberg scaling, showcasing the scheme's advantages in ease of implementation, decay robustness, and tunable energy gaps.

Quantum metrology has been extensively studied, with significant advancements in achieving Heisenberg scaling using various techniques and entangled probes. However, maintaining coherence and overcoming noise remain significant challenges. Critical quantum metrology has emerged as a promising approach, leveraging the robustness of adiabatic evolution near critical points. Existing theoretical proposals often focus on continuous QPTs, which are challenging to implement experimentally. This study builds upon these existing works by proposing and experimentally demonstrating a scheme based on a first-order QPT, addressing the limitations of previous approaches.

The study employs a two-spin-1/2 Ising model with Hamiltonian H_Ising = B_z(σ_z¹ + σ_z²) + Jσ_x¹σ_x², where B_z is the unknown longitudinal magnetic field to be estimated. A first-order QPT occurs at B_z = ±1. To lift the degeneracy and enable parameter encoding, a small transverse magnetic field B_x is introduced: H_Ising = B_z(σ_z¹ + σ_z²) + B_x(σ_x¹ + σ_x²) + Jσ_z¹σ_z². The energy gap is tuned using B_x, controlling the adiabatic evolution time. The quantum Fisher information (QFI) quantifies the precision, with F_Q ≈ 2B_x² near the critical point. The experiment uses a two-spin NMR system (¹³C and ¹H in ¹³C-labeled chloroform). The adiabatic evolution is implemented using a numerically optimized path, generated by iteratively increasing a control field while maintaining a high fidelity between the evolved state and the ground state. The optimal measurement is implemented using a unitary transformation to map the optimal observable onto a local observable measurable in NMR. The QFI is experimentally determined through repeated measurements with varied evolution times (controlled via B_x) and parameter shifts. Numerical simulations are used to validate the experimental results and assess the robustness against decay.

The experiment successfully demonstrates Heisenberg scaling of the QFI with evolution time (F_Q ∝ T²). The QFI near the critical point is significantly higher than away from it, demonstrating the advantage of critical quantum metrology. The experimental results show good agreement with numerical simulations, with a total relative deviation of about 8.8% for QFI and 5.1% for the Heisenberg scaling analysis. The adiabatic scheme shows inherent robustness against decay, surpassing standard quantum metrology schemes in noisy environments.

The findings confirm the feasibility and advantages of the proposed adiabatic critical quantum metrology scheme. The Heisenberg scaling achieved experimentally validates the theoretical predictions. The robustness of the adiabatic approach compared to standard quantum metrology in the presence of noise is a significant advantage. The tunable energy gap allows for a trade-off between precision and bandwidth. The use of a first-order QPT in a small quantum system simplifies implementation, making this approach promising for various physical systems.

This work presents a successful theoretical proposal and experimental demonstration of critical quantum metrology using a first-order QPT in a two-spin system. The achieved Heisenberg scaling and robustness against decay highlight the potential of this approach for practical quantum sensing applications. Future research could explore larger systems, explore different QPTs, and investigate alternative adiabatic evolution schemes.

The study focuses on local parameter estimation, assuming prior knowledge of the parameter's approximate value. While a two-step adaptive method is proposed for general unknown parameters, it was not experimentally implemented in this study. The experimental implementation is limited by the coherence times of the NMR system, which could be improved using more advanced techniques. The use of a small quantum system limits the potential scaling of the metrological advantage to larger systems.