Optimal Quantum Phase Estimation

By using a systematic optimization approach, we determine quantum states of light with definite photon number leading to the best possible precision in optical two-mode interferometry. Our treatment takes into account the experimentally relevant situation of photon losses. Our results thus reveal the benchmark for precision in optical interferometry. Although this boundary is generally worse than the Heisenberg limit, we show that the obtained precision beats the standard quantum limit, thus leading to a significant improvement compared to classical interferometers. We furthermore discuss alternative states and strategies to the optimized states which are easier to generate at the cost of only slightly lower precision.

Figure 1

General optical interferometric setup with two arms. Channel $a$ acquires a phase $\phi$ relative to channel $b$. The two beam splitters in channel $a$ and $b$ symbolize photon losses.

Figure 2

Phase estimation precision $\delta {\phi }_{\min }$ for losses in both arms of the interferometer (a) versus photon number $N$ (${\eta }_a={\eta }_b=\eta =0.9$) and (c) versus transmissivity ${\eta }_a={\eta }_b=\eta$ ($N=20$). The precision for losses in one arm, i.e., ${\eta }_b=1$, is shown in (b) versus $N$ (${\eta }_a=\eta =0.9$) and in (d) versus ${\eta }_a=\eta$ ($N=20$). Blue, solid line: Optimal state; Red, dashed line: $N00N$ state; Green, dashed-dotted line: $N00N$ chopping strategy; Shaded area: Region between Heisenberg limit and classical limit (see text).

Figure 3

Coefficients $x_k=|{\alpha }_k{|}^2$ of the optimal state versus photon number $N$. (a) losses in both arms (${\eta }_a={\eta }_b=\eta =0.9$). (b) losses in one arm (${\eta }_a=\eta =0.9$, ${\eta }_b=1$).

Figure 4

Differential scaling of phase estimation precision $\delta {\phi }_{\min }$ with number of photons $N$ for different transmissivities. (a) Optimal state for losses in both arms (${\eta }_a={\eta }_b\equiv \eta$) and (b) for losses in one arm (${\eta }_a\equiv \eta$, ${\eta }_b=1$). The color coding is the same in both plots.