Single-mode displacement sensor

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 D. Gottesman et al., Phys. Rev. A 64, 012310 (2001) for encoding a qubit into an oscillator: an efficient preparation protocol of such states, using a coupling to a qubit, was later developed B. M. Terhal and D. Weigand, Phys. Rev. A 93, 012315 (2016). 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.