Quantum metrology for simultaneously estimating the linear and nonlinear phase shifts

Abstract

Quantum metrology allows the enhanced precision on estimating phase shifts in an optical interferometer to reach the $\ensuremath{\propto}1/N$ scaling of precision ($N$ is the mean photon number of the light field). But when $N$ is very large, the nonlinear optical effects may be nonnegligible. Here we investigate theoretically the quantum uncertainty bounds for simultaneously detecting the linear and nonlinear phase shifts. Focusing our attentions only on the linear and the nonlinear Kerr phase shifts, we find the NOON states can not be used to measure the two phase parameters. Using the entangled coherent states, the estimation precision is limited by the $1/{N}^{1/2}$ ($1/{N}^{3/2}$) scaling for the linear (nonlinear) phase. To increase the precisions and beat the standard quantum limit, we show that the entangled Schr\"odinger-cat states can be used to reach the $1/N$ scaling for the linear phase and $1/{N}^{2}$ scaling for the nonlinear phase.