Results and Discussions With a large enough repeated binary-outcome photon counting measurement, it is shown that the likelihood function can be well approximated by a Gaussian function [Figs. 2 (c) and 2 (d)], where its peak determines the MLE. To confirm it, we analytically derive the approximate results of the likelihood function and the MLE [Eqs. (12)-(19)], which shows that the MLE can saturate the CRB asymptotically. The above results also hold for a combination of two binary-outcome measurements with and without an offset phase shift [Figs. 4 (b)-(e) and Eqs. (23)-(32)]. For the six-photon twin-Fock state, the divergence of the phase sensitivity at a certain phase shift can be removed by comparing Fig. 3(a) and Fig. 4(f). Therefore, the microscopy imaging with a combination of two binary-outcome measurements can avoid the imaging speckles [Fig. 3 (c) and Fig. 5 (a)]. The overall quality of the imaging in Fig. 5 (a), quantified by the root-mean-square error of the MLE, outperforms that of classical light illumination by a factor of 1.82, approaching to its theoretical prediction.

Quantum multiphoton microscopy utilizes quantum correlation effects of photons to improve the imaging quality of biological samples at low light illumination. Based on a N-photon NOON state, the microscopy imaging has been successfully demonstrated in recent two experiments, which shows the imaging quality better than that of coherent light illumination by a factor of N (N=2, 3). However, the NOON states are difficult to prepare and are easily subject to the loss-induced decoherence. Furthermore, the microscopy imaging shows speckles within a local region, due to the divergence of the phase sensitivity. The twin-Fock states of light are believed to be more robust to the decoherence. For a binary-outcome photon counting measurement, it has been shown that a better phase sensitivity can be obtained in a comparison with that of the NOON states. Therefore, it is interesting to investigate the super-sensitive microscopy using the N-photon twin-Fock states of light. Recently, it is shown that the visibility of the six-photon count rate can reach 94%, which is significantly better than that of the five-photon NOON state (42%). Here, we investigate quantum-enhanced microscopy illuminated by the twin-Fock state of the light. With a combination of two binary-outcome measurements with and without an offset phase shift, it is shown that the divergence of the phase sensitivity at certain phase shifts can be removed, which avoids the imaging speckles. We hope our observations can be helpful on the quantum-enhancement microscopy with the large-N twin-Fock states.

A binary-outcome photon counting measurement is employed in present work, where the detection event with equal number of photons is a measurement outcome. All the other detection events are treated as another outcome. Starting from general principle of quantum metrology, we first calculate the Fisher information and the Cramer-Rao lower bound (CRB) of the phase sensitivity, which determine the enhancement factor of the imaging quality for the N-photon twin-Fock states. Then, we derive the phase distribution (the likelihood function) and the maximum likelihood estimator (MLE) by considering the binary-outcome measurements. Using Monte Carlo method, we simulate the measurement probabilities of the six-photon twin-Fock state and the single-photon state, where the experimental imperfection is added artificially. The microscopy imaging is reconstructed using numerical result of the MLE. Finally, we derive the likelihood function and show the microscopy imaging for a combination of two binary-outcome measurements with and without an offset phase shift.

Regardless of the specific model, we first prove analytically that the likelihood functions of single and two groups of binary-outcome photon counting measurements can approximate a Gaussian function, the maximum likelihood estimator is asymptotically unbiased which can saturate the lower limit of phase measurement of the above two measurement schemes. Based on the six-photon twin-Fock state, this paper studies the maximum likelihood estimator and phase sensitivity of the binary-outcome photon counting measurements, and reconstructs the two-dimensional microscopy imaging of the birefringent sample with the MLE. Our results show that a combination of binary-outcome photon counting measurements can avoid the divergence of phase sensitivity at dark spots, thus overcoming the speckle problem of microscopy imaging. The maximum likelihood estimator at each pixel in the reconstructed image is close to the optimal phase working point, and the overall quality factor of the image is measured by the root-mean-square error of the estimator.