Optical phase estimation via the coherent state and displaced-photon counting

We consider the phase sensing via a weak optical coherent state at quantum limit precision. A detection scheme for the phase estimation is proposed, which is inspired by the suboptimal quantum measurement in coherent optical communication. We theoretically analyze a performance of our detection scheme, which we call the displaced-photon counting, for phase sensing in terms of the Fisher information and show that the displaced-photon counting outperforms the static homodyne and heterodyne detections in a wide range of the target phase. The proof-of-principle experiment is performed with linear optics and a superconducting nanowire single-photon detector. The result shows that our scheme overcomes the limit of the ideal homodyne measurement, even under practical imperfections.

Figure 1

Schematic of general phase estimation.

Figure 2

Schematic of the phase estimation for the coherent state measured by the displaced-photon counting.

Figure 3

Fisher information for the displaced-photon counting, the homodyne detection, and the heterodyne detection. The vertical axis is normalized by the quantum Fisher information of the coherent state. Thus the maximum value is 1, where the FI corresponds to the QFI.

Figure 4

Normalized FI of the displaced-photon counting with the various imperfect conditions and fixed dark count of ${10}^{-5}$ counts per pulse. The mean photon number of the signal state is (a)  ${\left|\alpha \right|}^2=0.10$ and (b)  ${\left|\alpha \right|}^2=10$.

Figure 5

Experimental setup. AOM: acousto-optic modulator, BS: beamsplitter, EOM: electro-optic modulator, PZT: piezotransducer.

Figure 6

Experimental results for (a)  the variance of the estimator and (b)  the expectation value of the estimator (red dot) with the data points generated from numerical simulation (black dot). The phase shift value is fixed to $\varphi \sim 1.00$, and the parameters for the experiment are ${\left|\alpha \right|}^2=0.100$, ${\left|\beta \right|}^2=0.101$, $\xi =0.993$, $\eta =60.2\%$, and $\nu =1.13\times {10}^{-4}$.

Figure 7

Inverse of the experimentally obtained variance of the estimator multiplied the number of measurements $M$. $M$ is set to $1.0\times {10}^3$ (red circle), $1.0\times {10}^4$ (blue square), and $9.0\times {10}^5$ (green triangle). Black solid and dashed lines represent the theoretical FI of the displaced-photon counting and the homodyne detection for the experimental condition $\eta =0.602,\nu =1.13\times {10}^{-4}$, and $\xi =0.993$. Red and black solid lines are the theoretical FI of the displaced-photon counting and the homodyne detection in the ideal case.