Asymmetry-enhanced phase sensing via asymmetric entangled coherent states

We study quantum phase sensing with asymmetric two-mode entangled coherent states (ECSs) in which the two local amplitudes take different values. We find the asymmetry-enhanced phase-sensing phenomenon in which the phase-sensing sensitivity is enhanced with increasing the asymmetry in the ECSs. We indicate that the phase-sensing sensitivity can attain and even surpass the Heisenberg limit in certain regimes of parameters. We further study the effect of decoherence induced by photon loss on the phase-sensing sensitivity. It is shown that the asymmetric ECSs have greater capability against decoherence compared with the symmetric ECSs. It is indicated that the asymmetric ECSs have significant advantages over the symmetric ECSs in quantum phase sensing. We also study the practical phase-sensing scheme with the intensity-difference measurement and show that the asymmetry in the asymmetric ECSs can enhance the phase sensitivity in the practical phase-measurement scheme. Our work reveals the asymmetry in the asymmetric ECSs is a resource for quantum-enhanced sensing and may be applied to ultrasensitive quantum phase sensing in the presence of photon loss.

Figure 1

Schematic of the quantum phase sensing with the Mach-Zehnder interferometer. BS, beam splitter.

Figure 2

The QFI with respect to the total mean photon number $\overline{n}$ in the absence of photon loss ($R=0$). The dotted, dash-dotted, and dashed lines correspond to $k=1,2,$ and 10, respectively. The solid line denotes the QFI at the SQL.

Figure 3

The QFI with respect to the total mean photon number for different values of the asymmetric parameter $k$ and the photon-loss rate $R$. (a) The asymmetric parameter takes $k=1,2,3$, and the photon-loss rate $R=0$. (b) The asymmetric parameter takes $k=1,2,3$, and the photon-loss rate $R=0.3$. (c) The asymmetric parameter takes $k=1,2,3$, and the photon-loss rate $R=0.5$. (d) The asymmetric parameter takes $k=1,2,3$, and the photon-loss rate $R=0.6$. The dotted, dot-dashed, and dashed lines correspond to the values of $k=1,2,$ and 3, respectively.

Figure 4

The QFI with respect to the photon-loss rate $R$ for different values of the asymmetric parameter $k$ under the condition of the same input mean photon number $\overline{n}=26$. The dotted line describes the case of the symmetric ECSs $(k,\alpha )=(1,3.60)$. The dot-dashed and dashed lines denote the asymmetric-ECS cases $(k,\alpha )=(2,2.28)$ and $(k,\alpha )=(5,1.00)$, respectively.

Figure 5

The phase sensitivity in the intensity-difference measurement scheme for different values of the asymmetric parameter $k$ and photon-loss rate $R$ under the condition of the same input mean photon number $\overline{n}=26$. (a) The case without photon loss. (b) The case in the presence of photon loss. Here the solid line denotes the corresponding quantum Cramér-Rao bound $\mathrm{\Delta }{\varphi }_{\mathrm{CR}}$ when $k=1$ and $\overline{n}=26$.

Figure 6

The classical Cramér-Rao bound of the phase sensitivity in the intensity-difference measurement scheme for different values of the asymmetric parameter $k$ under the condition of the same input mean photon number $\overline{n}=26$. The dotted line describes the case of the symmetric ECSs $(k,\alpha )=(1,3.60)$. The dot-dashed and dashed lines denote the asymmetric-ECS cases $(k,\alpha )=(2,2.28)$ and $(k,\alpha )=(5,1.00)$, respectively. Here the solid line denotes the corresponding quantum Cramér-Rao bound $\mathrm{\Delta }{\varphi }_{\mathrm{CR}}$ when $k=1$ and $\overline{n}=26$.