Nonlinear phase estimation: Parity measurement approaches the quantum Cramér-Rao bound for coherent states

Quantum-enhanced phase estimation paves the way for precision measurements and is of great realistic significance. In this paper we theoretically investigate the second-order nonlinear phase estimation using a coherent state and parity measurement. A numerical expression of the parity signal is derived, and its visibility is analyzed. The resolution and sensitivity of the signal are compared to a linear phase estimation protocol with the same input and measurement. Additionally, by using the phase-averaging approach, we make an attempt at unveiling the lowdown on a conclusive sensitivity limit without external phase reference. Finally, the effects of several realistic scenarios on the resolution and sensitivity are studied, including photon loss, imperfect detector, those which are a combination thereof, and unbalanced beam splitter.

Figure 1

Schematic diagram of the measuring protocol for nonlinear phase estimation. L, laser; BS, beam splitter; RM, reflection mirror; KM, Kerr medium; PNRD, photon-number-resolving detector.

Figure 2

The signal with parity measurement as a function of the nonlinear phase, where the red dash-dotted, blue dotted, and black solid lines correspond to the scenarios of $N=2$, 5, and 10, respectively.

Figure 3

(a) The FWHMs of the nonlinear and linear phase protocols as functions of the mean photon number, where the blue dotted and red solid lines represent linear and nonlinear phase protocols, respectively. The inset shows the details of the linear and nonlinear signals with $N=10$. (b) The coefficient, quotient of the FWHM of the linear phase signal and that of the nonlinear one, as a function of the mean photon number, where the red solid line is a reference line with $C=N$.

Figure 4

The optimal sensitivity with parity measurement as a function of the mean photon number. Where the red dotted line, green solid line, and blue dashed line scale with $N^{-1}, N^{-3/2}$, and $N^{-2}$, respectively.

Figure 5

The optimal sensitivity with parity measurement and quantum Cramér-Rao bound as functions of the mean photon number.

Figure 6

(a) The FWHM with parity measurement under the photon loss as a function of the losses of two paths in the interferometer, where $N=10$. The color bar on the right of the two-dimensional planar forms indicates the corresponding value. (b) The broadening coefficient as a function of mean photon number in the case of $L_A=L_B=L$, where $L$ takes on the values of 0.4, 0.3, 0.2, and 0.1.

Figure 7

The optimal sensitivity with parity measurement under the photon loss as a function of the losses of two paths in the interferometer, where $N=10$. The color bar on the right of the two-dimensional planar forms indicates the corresponding value.

Figure 8

Schematic of the effect of response-time delay on the measuring outcomes.

Figure 9

The optimal sensitivity with parity measurement as a function of the rate of dark counts, where $N$ takes on the values of 10, 15, and 20.

Figure 10

The optimal sensitivity with parity measurement as a function of both total loss and the rate of dark counts, where $N=10$. The color bar on the right of the 2D planar form indicates the corresponding value.

Figure 11

(a) The FWHM with parity measurement as a function of the transmissivity of the second beam splitter, where $N=3$. The two insets are parity signals with $T\left(\theta \right)=0.4$ and $T\left(\theta \right)=0.6$, respectively. (b) The sensitivity with parity measurement as a function of the transmissivity of the second beam splitter.

Figure 12

A fit to the CFI of parity measurement.