Single-Mode Displacement Sensor

TL;DR

This paper demonstrates that using grid states as sensors allows for highly accurate measurement of displacement parameters on an oscillator, with precision improving as the number of photons increases, surpassing other states like the quantum compass state.

Contribution

It introduces a phase estimation protocol for preparing grid states that enhances displacement sensing accuracy beyond previous bounds, with detailed analysis of performance under realistic conditions.

Findings

Displacement accuracy scales inversely with the square root of photon number.

Grid states outperform quantum compass states in information gain as photon number increases.

Numerical simulations show robustness of the protocol against photon loss and nonlinearities.

Abstract

We show that one can determine both parameters of a displacement acting on an oscillator with an accuracy which scales inversely with the square root of the number of photons in the oscillator. Our results are obtained by using a grid state as a sensor state for detecting small translations in phase space (displacements). Grid states were first proposed in (see https://doi.org/10.1103/PhysRevA.64.012310 ) for encoding a qubit into an oscillator: an efficient preparation protocol of such states, using a coupling to a qubit, was developed in (see https://doi.org/10.1103/PhysRevA.93.012315 ). We compare the performance of the grid state with the quantum compass or cat code state and place our results in the context of the two-parameter quantum Cram\'er-Rao lower bound on the variances of the displacement parameters. We show that the accessible information about the displacement for a grid state increases with the number of photons in the state when we measure and prepare the state using a phase estimation protocol. This is in contrast with the accessible information in the quantum compass state which we show is always upper bounded by a constant, independent of the number of photons. We present numerical simulations of a phase estimation based preparation protocol of a grid state in the presence of photon loss, nonlinearities and qubit measurement, using no post-selection, showing how the two effective squeezing parameters which characterize the grid state change during the preparation. The idea behind the phase estimation protocol is a simple maximal-information gain strategy.