Single-Shot Single-Mode Optical Two-Parameter Displacement Estimation beyond Classical Limit

Uncertainty principle prohibits the precise measurement of both components of displacement parameters in phase space. We have theoretically shown that this limit can be beaten using single-photon states, in a single-shot and single-mode setting [F. Hanamura et al., Estimation of gaussian random displacement using non-gaussian states, Phys. Rev. A 104, 062601 (2021).]. In this Letter, we validate this by experimentally beating the classical limit. In optics, this is the first experiment to estimate both parameters of displacement using non-Gaussian states. This result is related to many important applications, such as quantum error correction.

Figure 1

A schematic figure of the system to estimate displacements. After the displacement is applied to the input probe single-photon state, dual-homodyne measurement with an ancillary single-photon state is performed, obtaining the measurement outcome $y_x$, $y_p$.

Figure 2

Bayesian estimation of displacement. The posterior distribution of $\xi$, $\eta$ after obtaining the measurement outcome is expressed as the product of the prior distribution and the likelihood function.

Figure 3

Experimental setup. OPO, optical parametric oscillator; SNSPD, superconducting nanostrip single-photon detector; EOM, electro-optic modulator; AWG, arbitrary waveform generator.

Figure 4

Experimental results. (a) The simultaneous distribution of measurement outcomes $y_x$, $y_p$ when no displacement is applied. (b) Blue dot: radial profile of the distribution of (a). Black solid line: theoretical prediction when imperfect single-photon states with 25% 0-photon component and 2% 2-photon component are used. (c) Estimation error $v^{\prime }$ normalized by the classical limit $v_C^{\prime }$ as a function of the prior variance of displacement $v$, with $r=0.2$. $v$ corresponding to the red points are actually measured, and the blue points are calculated by postselecting the events so that the prior distribution of displacements has the desired $v$. A theoretical curve for the same assumption as (b) is shown together. (d) Estimation error $v^{\prime }$ normalized by the classical limit $v_C^{\prime }$ as a function of the range of postselection $r$, with $v=0.34$, which is near the optimal point of (c). A theoretical curve for the same assumption as (b) is shown together.