# Heisenberg limited single-mode quantum metrology

## TL;DR

This paper demonstrates a single-mode quantum phase estimation method that approaches the Heisenberg limit, achieving significant precision enhancement over the shot-noise limit using microwave cavity states.

## Contribution

It provides the first experimental demonstration of near-Heisenberg-limited phase estimation with a single bosonic mode, utilizing high-fidelity state manipulation in a microwave cavity.

## Key findings

- Achieved phase estimation precision scaling as ~N^{-0.94}
- Realized a 9.1 dB enhancement over the shot-noise limit at N=12
- Approached the Heisenberg limit within 1.7 dB in a microwave cavity system

## Abstract

Two-mode interferometers, such as Michelson interferometer based on two spatial optical modes, lay the foundations for quantum metrology. Instead of exploring quantum entanglement in the two-mode interferometers, a single bosonic mode also promises a measurement precision beyond the shot-noise limit (SNL) by taking advantage of the infinite-dimensional Hilbert space of Fock states. However, the experimental demonstration still remains elusive. Here, we demonstrate a single-mode phase estimation that approaches the Heisenberg limit (HL) unconditionally. Due to the strong dispersive nonlinearity and long coherence time of a microwave cavity, quantum states of the form $\left(\left|0\right\rangle +\left|N\right\rangle \right)/\sqrt{2}$ are generated, manipulated and detected with high fidelities, leading to an experimental phase estimation precision scaling as $\sim N^{-0.94}$. A $9.1$~$\mathrm{dB}$ enhancement of the precision over the SNL at $N=12$, which is only $1.7$~$\mathrm{dB}$ away from the HL, is achieved. Our experimental architecture is hardware efficient and can be combined with the quantum error correction techniques to fight against decoherence, thus promises the quantum enhanced sensing in practical applications.