Extended Data Fig. 3: Cavity Wigner and PReSPA Ramsey measurements.

a, Experimental Wigner function W(α) (colour scale, dimensionless) of \(|{0}_{{\rm{L}}}\rangle \), acquired by applying a cavity displacement operation \({\hat{D}}_{\alpha }=\exp (\alpha {\hat{a}}^{\dagger }-{\alpha }^{\ast }\hat{a})\) with variable complex amplitude α followed by an ancilla-assisted photon-number-parity measurement (which is composed of two π/2 pulses of the ancilla and a delay time of π/χ_q and an ancilla readout^37,58). The Wigner function rotates around the origin over time at a rate proportional to the frequency difference between \(|1\rangle \) and \(|5\rangle \) in the rotating frame of the experiment. b, Measured Wigner function values at a fixed phase-space position (as indicated by the cross in a, at α = 0.75) as a function of time under PReSPA. Analogous to a qubit Ramsey measurement, this cavity PReSPA Ramsey experiment can be used to efficiently track the phase evolution of any two-component superposition states using the interference effect enabled by the coherent cavity displacement (\({\hat{D}}_{\alpha }\)) before readout. The exponential envelope of the sinusoidal fit indicates the rate of decay for the coherence between \(|1\rangle \) and \(|5\rangle \) under the correction of PReSPA. Similar measurements are applied to various superposition states to provide direct calibration of the frequencies and phases of these states under PReSPA. PReSPA enhances the ability to use such Ramsey measurements at high photon numbers because it approximately preserves photon number distributions in the cavity.