Single-Shot Non-Gaussian Measurements for Optical Phase Estimation

Estimation of the properties of a physical system with minimal uncertainty is a central task in quantum metrology. Optical phase estimation is at the center of many metrological tasks where the value of a physical parameter is mapped to the phase of an electromagnetic field and single-shot measurements of this phase are necessary. While there are measurements able to estimate the phase of light in a single shot with small uncertainties, demonstrations of near-optimal single-shot measurements for an unknown phase of a coherent state remain elusive. Here, we propose and demonstrate strategies for single-shot measurements for ab initio phase estimation of coherent states that surpass the sensitivity limit of heterodyne measurement and approach the Cramer-Rao lower bound for coherent states. These single-shot estimation strategies are based on real-time optimization of coherent displacement operations, single photon counting with photon number resolution, and fast feedback. We show that our demonstration of these optimized estimation strategies surpasses the heterodyne limit for a wide range of optical powers without correcting for detection efficiency with a moderate number of adaptive measurement steps. This is, to our knowledge, the most sensitive single-shot measurement of an unknown phase encoded in optical coherent states.

Figure 1

Optimized non-Gaussian estimation strategy. (a) Concept of the adaptive displaced photon counting measurement for single-shot phase estimation. (b) Simulated Holevo variance multiplied by the quantum Fisher information for coherent states $(4|\alpha {|}^2)$. Shown are strategies maximizing the sharpness $⟨S(\beta ,m)⟩$ (blue) and the mutual information $I(\beta ,m)$ (orange), and a nonoptimized strategy (gray), all with $L=30$ adaptive steps, PNR(3), and ideal efficiency. Also shown is the CRLB for coherent states $(1/4|\alpha {|}^2)$, the lower bound of a heterodyne measurement $(1/2|\alpha {|}^2)$, and the performance of the adaptive homodyne scheme termed “Mark II” from Ref. [30]. Bold lines are the average of five Monte-Carlo simulations of ${10}^3$ randomly distributed initial phases, and the shaded regions represent one standard deviation. The performance of both estimation strategies at $|\alpha {|}^2={10}^3$ for PNR(3) with $L=200$ and for PNR(12) with $L=30$ are shown for reference.

Figure 2

Experimental setup for the adaptive displaced photon counting measurement (see text for details). HeNe: helium-neon laser, AOM: acousto-optic modulator, Att.: attenuator, PM: phase modulator, AM: amplitude modulator, BS: beam splitter, APD: avalanche photodiode; FPGA: field-programmable gate array.

Figure 3

Experimental results. Experimentally obtained Holevo variance multiplied by the QFI for coherent states ($4|\alpha {|}^2$) for estimation strategies maximizing the sharpness and the mutual information as blue and orange points, respectively. Also shown are the expected variances (solid lines) accounting for experimental parameters. Included are the CRLB for coherent states $1/4|\alpha {|}^2$ (solid red line) and the ideal heterodyne limit $1/2|\alpha {|}^2$ (dashed red line), together with these bounds adjusted to the efficiency of our implementation $\eta =0.70$ (solid and dashed black lines). Note that both strategies outperform the ideal heterodyne bound from $|\alpha {|}^2\approx 20$ to $\approx 600$ and the adjusted bound from $|\alpha {|}^2\approx 6$ to $>{10}^3$. Furthermore, both estimation strategies achieve a minimum of $\approx 1.68$ times the CRLB (1.18 times adjusted) at $|\alpha {|}^2\approx 70$.