Fisher information of a squeezed-state interferometer with a finite photon-number resolution

Squeezed-state interferometry plays an important role in quantum-enhanced optical phase estimation, as it allows the estimation precision to be improved up to the Heisenberg limit by using ideal photon-number-resolving detectors at the output ports. Here we show that for each individual $N$-photon component of the phase-matched coherent $\otimes$ squeezed vacuum input state, the classical Fisher information always saturates the quantum Fisher information. Moreover, the total Fisher information is the sum of the contributions from each individual $N$-photon component, where the largest $N$ is limited by the finite number resolution of available photon counters. Based on this observation, we provide an approximate analytical formula that quantifies the amount of lost information due to the finite photon number resolution; e.g., given the mean photon number $\overline{n}$ in the input state, over 96% of the Heisenberg limit can be achieved with the number resolution larger than $5\overline{n}$.

Figure 1

(a) Photon-counting measurement at output ports of the MZI fed with a coherent state $|\alpha \rangle$ and a squeezed vacuum $|\xi \rangle$, and the $N$-photon state $|{\psi }_N^{\mathrm{BS}}\rangle$, postselected by the number of photons being detected $N=N_1+N_2=2J$. (b) For a given $J=N/2=10$, the probability distribution $p_{\mu }={\left|\langle J,\mu |{\psi }_N^{\mathrm{BS}}\rangle \right|}^2$ against $\mu =(N_1-N_2)/2\in [-J,+J]$ and $x={\left|\alpha \right|}^2/tanh\left|\xi \right|$. At a certain value of the ratio $x_N^{\left(\mathrm{opt}\right)}$ (blue line), the distribution shows two symmetric peaks at $\mu =\pm J$, indicating the appearance of a path-entangled NOON state. (c) The fidelity between the $N$-photon state and an ideal NOON state and (d) the QFI of the $N$-photon state, with their values calculated at $x=x_N^{\left(\mathrm{opt}\right)}$ (blue solid line) and $x_N^{\left(\mathrm{FI}\right)}$ (red line with crosses); see text. The inset in (c) indicates $x_N^{\left(\mathrm{FI}\right)}\le x_N^{\left(\mathrm{opt}\right)}\le N/2$.

Figure 2

(a) The generation probability $G_N$, (b) the QFI of each $N$-photon state $F_{Q,N}$, (c) the weighted QFI $G_N{F}_{Q,N}$, and (d) the total QFI $F_Q(N_{\mathrm{res}},\overline{n},{\alpha }^2)$, for $\overline{n}=5$ fixed and $N_{\mathrm{res}}=3\overline{n}$ (solid circles), $5\overline{n}$ (squares), $10\overline{n}$ (open circles), and $\infty$ (red solid lines). The last case is given by Eq. (16), which predicts ${\alpha }_{\mathrm{opt}}^2\simeq \overline{n}/2$ and $F_{Q,\mathrm{opt}}^{\left(\mathrm{id}\right)}\simeq \overline{n}(\overline{n}+3/2)$. For each a given $\overline{n}\in [1,10]$, using the same values of $N_{\mathrm{res}}$ and maximizing the total QFI with respect to ${\alpha }^2$ to obtain (e) ${\alpha }_{\mathrm{opt}}^2/\overline{n}$, and (f) the associated QFI $F_{Q,\mathrm{opt}}$. In (c) the peak height of the weighted QFI is about $\overline{n}/2$. The vertical lines in (d) are the optimal value of ${\alpha }^2$ for different values of $N_{\mathrm{res}}$. The dashed line in (f) is the classical limit $F_Q=\overline{n}$.

Figure 3

Numerical results of $F_Q/F_{Q,\mathrm{opt}}^{\left(\mathrm{id}\right)}$ as a function of $N_{\mathrm{res}}/\overline{n}$ for given values of $\overline{n}$, using the optimal condition that maximizes Eq. (16). The solid line is given by our asymptotic result [Eq. (18)], and the dashed line is a fitting result for the case $\overline{n}=20$. Both of them indicate that $96\%$ of the ideal QFI can be obtained as long as $x=N_{\mathrm{res}}/\overline{n}\gtrsim 5$ (the vertical dashed lines).