Optimal Gain Sensing of Quantum-Limited Phase-Insensitive Amplifiers

Phase-insensitive optical amplifiers uniformly amplify each quadrature of an input field and are of both fundamental and technological importance. We find the quantum limit on the precision of estimating the gain of a quantum-limited phase-insensitive amplifier using a multimode probe that may also be entangled with an ancilla system. In stark contrast to the sensing of loss parameters, the average photon number $N$ and number of input modes $M$ of the probe are found to be equivalent and interchangeable resources for optimal gain sensing. All pure-state probes whose reduced state on the input modes to the amplifier is diagonal in the multimode number basis are proven to be quantum optimal under the same gain-independent measurement. We compare the best precision achievable using classical probes to the performance of an explicit photon-counting-based estimator on quantum probes and show that an advantage exists even for single-photon probes and inefficient photodetection. A closed-form expression for the energy-constrained Bures distance between two product amplifier channels is also derived.

Figure 1

A general ancilla-assisted parallel strategy for sensing the gain $G$ of a QLA ${\mathcal{A}}_G$ (dashed box). Each of $M$ signal ($S$) modes (one of which is shown) of a probe ${|\psi ⟩}_{AS}$ possibly entangled with an ancilla system $A$ is subject to a two-mode squeezing interaction ${\widehat{H}}_I=i\hslash \kappa (\widehat{a}\widehat{e}-{\widehat{a}}^{†}{\widehat{e}}^{†})$ between the $S$ mode (annihilation operator $\widehat{a}$) and an environment ($E$) mode (annihilation operator $\widehat{e}$) initially in the vacuum state. An estimate $\check{G}$ of $G={\cosh }^2\kappa t$ is obtained using the optimal joint measurement on the output of $AS$.

Figure 2

The optimal quantum (blue) [Eq. (6)] and classical QFI (red) [Eq. (7)] for $N=6$ and $M=9$. Also shown are the FI of homodyne (purple dash-dotted), heterodyne detection (green dotted), and the inverse MSE of the photodetection-based estimator [Eq. (8)] (yellow dashed) for a coherent-state probe.

Figure 3

Gain estimation under inefficient detection. Each mode of a product signal-only probe ${\otimes }_{m=1}^M|{\psi }_m⟩$ passes through a QLA ${\mathcal{A}}_G$. Detection with quantum efficiency ${\eta }_d$ is modeled by a beam splitter with mode $\widehat{f}$ in vacuum and output mode $\widehat{b}$ that is measured using an ideal photodetector $D$, resulting in photon count $Y_m$.

Figure 4

Performance of single-photon probes with inefficient detection. Left and center: QCRBs of multimode single-photon (blue solid) and coherent-state (red solid) probes along with the MSE of $\check{G}$ of Eq. (9) for single-photon (blue dashed) and coherent-state (red dashed) probes for ${\eta }_d=0.7$ (left) and ${\eta }_d=0.9$ (center) with $M=N=20$. Right: the threshold gain beyond which single-photon probes and photon counting beat the coherent-state QCRB.