Precision measurement is fundamental to scientific advancement and technological development. Quantum metrology offers the potential to surpass the limitations of classical methods by exploiting quantum phenomena to achieve higher precision. The Heisenberg limit (HL) represents a fundamental bound on the precision achievable in quantum metrology, offering a significant improvement over the standard quantum limit (SQL). While entanglement-based approaches can achieve HL scaling, they are experimentally challenging to implement in large systems. Alternatively, using highly non-classical states in a single bosonic mode provides a hardware-efficient route to quantum-enhanced metrology. Previous work has demonstrated advantages with small photon numbers (N typically on the order of 10), but realizing the full potential of this approach requires generating and manipulating much larger Fock states. This paper addresses this challenge by presenting a method for generating and utilizing large Fock states to achieve significantly enhanced metrological performance.

Numerous studies have explored quantum-enhanced metrology using various non-classical states. Entangled states like Greenberger-Horne-Zeilinger (GHZ) states, NOON states, and spin-squeezed states have been investigated, but their manipulation in large systems remains difficult. Single-mode bosonic systems offer a hardware-efficient alternative. Schrödinger cat states, squeezed states, and maximum variance states have shown promise, but the demonstrated metrological advantages have been limited to relatively small excitation numbers. The use of Fock states, energy eigenstates with a definite photon number, has also been explored in phononic and photonic systems, but achieving high photon numbers has been a significant experimental hurdle. This study aims to overcome this limitation and demonstrate the benefits of using large Fock states in quantum metrology.

The researchers developed a programmable photon number filter (PNF) to efficiently generate large Fock states in a superconducting microwave cavity. This PNF leverages the photon-number-dependent response of an ancillary superconducting qubit dispersively coupled to the cavity. The PNF acts as a programmable filter, allowing for the selection of specific photon numbers. A sinusoidal PNF is used, with the possibility of including a Gaussian PNF to concentrate on the subspace around the target Fock state. This parameterized state preparation method enables the generation of arbitrary Fock states in logarithmic steps. The generated Fock states were characterized using qubit spectroscopy, measuring the photon number distribution via qubit frequency shifts. Further characterization was done using Ramsey experiments. For displacement and phase sensing, specific quantum circuits were implemented. For displacement sensing, a displacement operation was applied to the Fock state, followed by parity measurement. For phase sensing, a displaced Fock state was used to avoid the limitations imposed by Fock state rotational symmetry. Fisher information was calculated from the experimental data to quantify the metrological gain. Additionally, a photon-number-resolved quantum metrology scheme was implemented using multiple PNFs to achieve a deterministic measurement.

The researchers successfully generated Fock states with up to 100 photons in a superconducting microwave cavity. The programmable PNF allowed for efficient generation of these large Fock states. In displacement sensing experiments using these states, the team achieved a metrological gain of 14.8 ± 0.2 dB at N=40, demonstrating a precision scaling of N^-0.35, approaching the Heisenberg scaling of N^-1. In phase sensing, a maximum metrological gain of 12.3 ± 0.5 dB was achieved at ñ=60 (ñ=2N is the mean photon number), with sensitivity scaling of ñ^-0.87, again approaching the Heisenberg limit. The photon-number-resolved scheme showed a maximum Fisher information gain of 5.04 ± 0.06 dB for an initial coherent state. The success probability for Fock state generation scales as 1/√N, but even with this post-selection, a scaling enhancement was observed. Analysis indicates that an adaptive control scheme could achieve a N^1/2 scaling enhancement.

The results demonstrate the feasibility of achieving Heisenberg-limited quantum metrology using large Fock states in a hardware-efficient manner. The use of a programmable PNF provides a significant advancement in generating these states, circumventing the difficulties associated with other non-classical states. The near-Heisenberg scaling achieved in both displacement and phase sensing showcases the potential of this approach for surpassing classical limits. The post-selection nature of the Fock state generation does not fully negate the quantum advantage, with a demonstrable scaling improvement. The study opens up avenues for enhanced weak force detection and dark matter searches. The potential for improvement with larger photon numbers (N > 1000) is significant.

This paper demonstrates significant advancements in quantum metrology by achieving efficient generation and utilization of large Fock states. The near-Heisenberg scaling observed in both displacement and phase measurements underscores the potential of this approach for surpassing classical limits. Future work could focus on improving the quality of the cavity and qubit to enhance the metrological gain further. The methodology is also adaptable to other platforms, opening up exciting prospects for quantum sensing across various systems.

The generation of Fock states using the PNF is probabilistic, with the success probability decreasing with increasing photon number. While the study demonstrates a quantum advantage even with post-selection, fully deterministic generation would further improve the performance. The experimental setup and results are specific to a superconducting microwave cavity; extending the methodology to other platforms will require further investigation and optimization. Decoherence effects could limit the achievable metrological gain at even higher photon numbers.