Estimation of a small displacement in both quadratures of an harmonic oscillator using a single-mode stabilized GKP state.

We theoretically show how to estimate isotropic displacements [1] surpassing the Gaussian limits of sensing [2], using single-mode stabilized GKP states [3]. Concretely, we show how the quantum-error-correction stabilization protocol implemented in Refs. [4, 5] can be adapted to sense weak displacements. At all times during the metrology protocol the photon number is bounded, thereby improving over a previous proposal [1] using GKP states to beat the gaussian limits of sensing at the cost of an exponential increase in the photon number with time. Crucially, the fact that the photon number remains bounded in our protocol leads to a mean-square-error scaling approaching, but not saturating, the O(1/n) ideal limit. We provide numerical simulations of both bayesian and frequentist approaches including relevant noise sources, and analytical intuition on the efficacy of the protocol.In summary, we present a displacement sensing protocol that is nonadaptive, entanglement-free, beats the Gaussian limits without postselection, is robust against noise sources, and can be implemented in current architectures.

[1] Duivenvoorden, K., Terhal, B. M., & Weigand, D. PRA 95(1), 012305, (2017).

[2] Royer, B., Singh, S., & Girvin, S. M. PRL 125(26), 260509 (2020).

[3] Hanamura, F., et al. PRA 104(6), 062061 (2021).

[4] Sivak, V. V., et al. Nature 616.7955 (2023).

[5] De Neeve, Brennan, et al. Nature Physics 18.3, 296-300 (2022)