Quadratic Fisher Information Achieved Via Weak-Coupling with Ancillary Qubit Systems

Scientists are continually striving to enhance the precision of quantum measurements, and new research from Peng Chen and Jun Jing, both of Zhejiang University, details a promising pathway towards achieving this goal. Their work explores a measurement-based protocol utilising a spin ensemble coupled to an ancillary qubit via a Heisenberg interaction, demonstrating how optimised weak coupling and precise timing can dramatically improve phase estimation sensitivity. This innovative approach allows the researchers to achieve quadratic scaling of Fisher information with probe size, crucially remaining robust against imperfections in encoding and coupling strength , a significant leap forward in overcoming limitations previously hindering precision measurements. By leveraging unconditional measurement on the qubit, Chen, Jing et al. suggest a viable alternative to complex resources like Greenberger-Horne-Zeilinger states and squeezing Hamiltonians, potentially unlocking the Heisenberg limit for phase sensing.

The research establishes that this unconventional measurement strategy offers a compelling alternative to traditional methods relying on complex entangled states like Greenberger-Horne-Zeilinger (GHZ) states or squeezed spin states, both of which present significant challenges in practical implementation. This innovative protocol promises to simplify the pursuit of Heisenberg-limited precision, particularly in scenarios where generating and maintaining highly entangled states proves difficult. Furthermore, the insensitivity to imprecise phase encoding directions and coupling strengths makes this approach exceptionally practical for real-world applications.

This work opens exciting possibilities for advancements in diverse fields including atomic clocks, gravitational wave detection, biological sensing, and magnetometry. By replacing the need for complex resource states and intricate Hamiltonian engineering, this measurement-based protocol offers a pathway towards more robust and scalable quantum sensors. The researchers suggest that their findings could significantly simplify the development of high-precision devices, paving the way for more sensitive and accurate measurements across a broad spectrum of scientific and technological domains.

Spin Ensemble Phase Estimation via Quadratic Scaling

This quadratic scaling remains robust even with imperfect encoding and varying coupling strengths, representing a significant methodological advance. Here, U(t1) and U(t2) denote the durations of two sequential unitary evolution stages, while Rα(θ) represents a parametric encoding of the phase into the probe spin system via a collective angular momentum operator Jα. Researchers realized this measurement through fluorescence collection from an electron spin in an NV center, a technique enabling continuous monitoring without disturbing the quantum state. To achieve this, the team engineered a spin rotation Rα(θ) using a sequence of rotations around the z and y axes, generated by dispersive coupling between the probe and the system under measurement.

The longitudinal and transversal coupling strengths were maintained at approximately 80MHz under an external magnetic field of 379 G, ensuring they remained significantly weaker than the system frequencies. This careful control allowed the researchers to satisfy the condition p g2z −g2 = (2ωA −ωP )/(N + 1), crucial for achieving Heisenberg scaling in metrological precision. Furthermore, the research demonstrates the creation of a GHZ-like state, |GHZ⟩= 1/√2 |j, ±j⟩x + e−iφ|j, ∓j⟩x, through a two-path evolution facilitated by the joint unitary evolution and the unconditional measurement. By allowing the polarized state |j, ±j⟩x to evolve along two paths, one with IN+1 and the other with e−iπJz, the team constructed a superposition, analogous to techniques used in continuous-variable systems employing ancillary two-level systems as quantum switches.

The. The unnormalized output state, ρ(θ), was then written as ρ(θ) = Uθ,+ρth P ⊗ρAU † θ,+ + Uθ,−ρth P ⊗ρAU † θ,−, with N± representing the probabilities of measurement outcomes |+⟩ and |−⟩ about the ancillary qubit. Tests prove that under optimised conditions, the normalised output state, ρ(θ), can be approximated as ρ(θ). The quantum Fisher information (QFI) was calculated, revealing a quadratic scaling law of N under large-N and low-temperature limits, as shown in Figure 0.2, with larger β values yielding behaviour closer to the Heisenberg scaling. Numerical simulations confirmed the analytical results for N ≥10, N ≥15, and N ≥70 when β = 2, β = 1, and β = 0.1, respectively.

Furthermore, the study investigated the effects of imprecise control, finding the protocol robust against small deviations in the phase encoding direction and coupling strengths. The team determined that the maximum QFI is obtained when the deviation δ is approximately zero for large N, and QFI remains larger than 0.988 even when |δ| = 0.1, demonstrating remarkable insensitivity to encoding imperfections. Measurements confirm that this asymptotic Heisenberg-scaling behaviour and insensitivity hold even when the probe system starts from a mixed state, with QFI remaining above 0.956 when |δ| = 0.1 for a thermal state with β = 1.

Heisenberg Limit Reached via Weak Coupling, demonstrating enhanced

The significance of this work lies in its ability to achieve Heisenberg-scaling precision without relying on complex resources such as Greenberger-Horne-Zeilinger-like states or squeezing Hamiltonians. The authors acknowledge a limitation in that achieving optimal precision requires careful control over the phase encoding direction and coupling strengths. Future research could explore extending this protocol to more complex systems and investigating its resilience to additional sources of noise.