Optimization of quantum interferometric metrological sensors in the presence of photon loss

We optimize two-mode entangled number states of light in the presence of loss in order to maximize the extraction of the available phase information in an interferometer. Our approach optimizes over the entire available input Hilbert space with no constraints, other than fixed total initial photon number. We optimize to maximize the Fisher information, which is equivalent to minimizing the phase uncertainty. We find that in the limit of zero loss, the optimal state is the maximally path-entangled (so-called N00N) state, for small loss, the optimal state gradually deviates from the N00N state, and in the limit of large loss, the optimal state converges to a generalized two-mode coherent state, with a finite total number of photons. The results provide a general protocol for optimizing the performance of a quantum optical interferometer in the presence of photon loss, with applications to quantum imaging, metrology, sensing, and information processing.

Figure 1

(Color online) Abstract interferometer condensing the input state plus the first beam splitter into the first box, followed by two propagating modes with loss modeled by additional beam splitters. The box on the right includes a beam splitter and the photon-number resolving detectors.

Figure 2

(Color online) Minimum detectable phase sensitivities calculated from the normalized Fisher information: (a) as a function of detector arm loss $\left(R_A\right)$ with three different losses in the control arm $\left(R_B\right)$ for $N=6$. (b) As a function of $R_A$ for $R_B=0$ in log scale. (c) As a function of input photon-number $N$ for $R_B=0$ with fixed $R_A$. Lines represent the results of curve-fitting using a functional form $1/N^x$. In the absence of loss, the Heisenberg limit, $x=1$, is obtained. For high loss, $x$ tends toward 0.5, approaching the shot-noise scaling.

Figure 3

(Color online) (a) Projection of the optimal state on N00N, $\left(\text{m},{\text{m}}^{\prime }\right)$, and GPCS as a function of $R_A$, for $N=4$. (b) The optimal input state composition. The vertical axis shows coefficients ${\left|c_k\right|}^2$ of the optimally entangled input state. Both figures clearly demonstrate that the optimal state changes from N00N-type to GPCS-type as loss increases, with the crossover occurring at approximately 5 dB loss.