Phase Detection at the Quantum Limit with Multiphoton Mach-Zehnder Interferometry

We study a Mach-Zehnder interferometer fed by a coherent state in one input port and vacuum in the other. We explore a Bayesian phase estimation strategy to demonstrate that it is possible to achieve the standard quantum limit independently from the true value of the phase shift and specific assumptions on the noise of the interferometer. We have been able to implement the protocol by using parallel operation of two photon-number-resolving detectors and multiphoton coincidence logic electronics at the output ports of a weakly illuminated Mach-Zehnder interferometer. This protocol is unbiased, saturates the Cramer-Rao phase uncertainty bound, and, therefore, is an optimal phase estimation strategy.

Figure 1

(a) Schematic of a Mach-Zehnder interferometer. A phase sensitive measurement is provided by the detection of the number of particles $N_c$ and $N_d$ at the two output ports. (b) Pulse height distribution for a VLPC used in the experiment. The power incident on the detector is 144 fW at a wavelength of 780 nm. The vertical lines show the decision thresholds [23].

Figure 2

Phase distribution $P(\varphi |N_c,N_d)$, for: (a) $N_c=0$, $N_d=0$; (b) $N_c=0$, $N_d=1$; (c) $N_c=1$, $N_d=0$; (d) $N_c=1$, $N_d=1$; (e) $N_c=0$, $N_d=2$; (f) $N_c=2$, $N_d=0$. The circles are the experimental data collected in the calibration part of the experiment, and the solid line is the ideal distribution equation (4).

Figure 3

Phase distribution $P_{\mathrm{fit}}(\varphi |\{N_c^{(i)},N_d^{(i)}{\}}_{i=1\dots p})$ obtained after $p$ independent experimental measurements: (a) $p=1$; (b) $p=10$; (c) $p=1000$. The true value of the phase shift is $\theta /\pi =0.25$, shown by the vertical dashed line. In (d), we plot $\Delta \Theta$ as a function of the number of measurements $p$. Circles are experimental results (averaged over 20 replicas of $p$ measurements, where error bars are mean square fluctuations), and the solid line is $\Delta \theta =1/\sqrt{p}$, Eq. (3). The preasymptotic behavior ($p\lesssim 20$) is due to a biased estimation.

Figure 4

Phase sensitivity as a function of the true value of the phase shift. Circles are obtained from the Bayesian distributions equation (5), with $p=10000$. The error bars give the fluctuations of $\sqrt{p}\Delta \Theta$ obtained with 20 independent replicas of the experiment. The solid black line is the theoretical prediction equation (3). The dashed blue line is the CRLB calculated with the experimental distributions. Squares are the uncertainty obtained with a generalization of the estimator equation (1) taking into account the experimental imperfections, while the dotted red line is the phase sensitivity predicted by Eq. (2). Inset: Difference between the mean value of the phase estimator $⟨{\Theta }_{\mathrm{est}}⟩$ obtained after 20 replicas of $p=10000$ independent measurements and the true value of the phase shift $\theta$. The vertical bars are the mean square fluctuations ${\sigma }_{\mathrm{est}}^2=⟨({\Theta }_{\mathrm{est}}-⟨{\Theta }_{\mathrm{est}}⟩{)}^2⟩$. The ${\sigma }_{\mathrm{est}}\gg (\theta -⟨{\Theta }_{\mathrm{est}}⟩)$ provides the evidence that our result is unbiased.