Practical resources and measurements for lossy optical quantum metrology

We study the sensitivity of phase estimation in a lossy Mach-Zehnder interferometer (MZI) using two general, and practical, resources generated by a laser and a nonlinear optical medium with passive optimal elements, which are readily available in the laboratory: One is a two-mode separable coherent and squeezed vacuum state at a beam splitter and the other is a two-mode squeezed vacuum state. In view of the ultimate precision given by quantum Fisher information, we show that the two-mode squeezed vacuum state can achieve a lower bound of estimation error than the coherent and squeezed vacuum state under a photon-loss channel. We further consider practical measurement schemes, homodyne detection and photon number resolving detection (PNRD), to characterize the accuracy of phase estimation in reality and find that the coherent and squeezed vacuum state largely achieves a lower bound than the two-mode squeezed vacuum in the lossy MZI while maintaining quantum enhancement over the shot-noise limit. By comparing homodyne detection and PNRD, we demonstrate that quadrature measurement with homodyne detection is more robust against photon loss than parity measurement with PNRD. We also show that double homodyne detection can provide a better tool for phase estimation than single homodyne detection against photon loss.

Figure 1

Lossy Mach-Zehnder interferometer for phase estimation. (a) Photon loss in all possible paths and (b) simplified photon-loss model. Here $\eta$ is the transmittance of a fictitious beam splitter and ${\eta }_a={\eta }_p{\eta }_{a_2}{\eta }_{a_3}{\eta }_d$ and ${\eta }_b={\eta }_p{\eta }_{b_2}{\eta }_{b_3}{\eta }_d$, where ${\eta }_{a_1}={\eta }_{b_1}={\eta }_p$ and ${\eta }_{a_4}={\eta }_{b_4}={\eta }_d$; BS stands for a 50:50 beam splitter and $\varphi$ a phase shift under the unitary action ${\widehat{U}}_{\varphi }=e^{-i\varphi {\widehat{a}}^{†}\widehat{a}}$.

Figure 2

Quantum Cramér-Rao bound for CSV (brown dashed curve), TMSV (red solid curve), and coherent (green dotted curve) states, respectively, as a function of loss rate $1-\eta$ at $\overline{n}=10$ (a) under symmetric photon loss (${\eta }_a={\eta }_b=\eta$) and (d) under photon loss in one arm (${\eta }_a=\eta$ and ${\eta }_b=1$) and as a function of $\overline{n}$ at a moderate loss rate $1-\eta =0.2$ (b) under symmetric photon loss and (e) under photon loss in one arm. The shot-noise limit (blue dot-dashed lines) is given by ${\mathrm{\Delta }}^2{\varphi }_{\text{SNL}}=1/2\overline{n}$. An optimal ratio $\mu$, which represents a portion of a single-mode squeezed vacuum state in the CSV state, is given as a function of loss rate $1-\eta$ at $\overline{n}=1$ (black solid curve), $\overline{n}=10$ (black dashed curve), and $\overline{n}=100$ (black dotted curve) (c) under symmetric photon loss and (f) under photon loss in one arm. In (f), the dashed and dotted curves are overlapped. In (c) and (f), the dashed (dotted) vertical line represents the loss rate to beat the SNL using the CSV state (TMSV state).

Figure 3

Phase sensitivity via parity measurement with photon number resolving detection, as a function of loss rate $1-\eta$ at $\overline{n}=10$ (a) under symmetric photon loss (${\eta }_a={\eta }_b=\eta$) and (c) under photon loss in one arm (${\eta }_a=\eta$ and ${\eta }_b=1$) and as a function of $\overline{n}$ at a loss rate $1-\eta =0.2$ (b) under symmetric photon loss and (d) under photon loss in one arm, using CSV (brown dashed curve), TMSV (red solid curve), and coherent (green dotted curve) states. The shot-noise limit (blue dot-dashed line and curve) is given by ${\mathrm{\Delta }}^2{\varphi }_{\text{SNL}}=1/2\overline{n}$.

Figure 4

Phase sensitivity via quadrature measurement with single homodyne detection as a function of loss rate $1-\eta$ at $\overline{n}=10$ (a) under symmetric photon loss (${\eta }_a={\eta }_b=\eta$) and (c) under photon loss in one arm (${\eta }_a=\eta$ and ${\eta }_b=1$) and as a function of $\overline{n}$ at a loss rate $1-\eta =0.2$ (b) under symmetric photon loss and (d) under photon loss in one arm, using CSV (brown dashed curve), TMSV (red solid curve), and coherent (green dotted curve) states. The shot-noise limit (blue dot-dashed line and curve) is given by ${\mathrm{\Delta }}^2{\varphi }_{\text{SNL}}=1/2\overline{n}$.

Figure 5

Phase sensitivity via quadrature measurement with double homodyne detection as a function of loss rate $1-\eta$ at $\overline{n}=10$ (a) under symmetric photon loss (${\eta }_a={\eta }_b=\eta$) and (c) under photon loss in one arm (${\eta }_a=\eta$ and ${\eta }_b=1$) and as a function of $\overline{n}$ at a loss rate $1-\eta =0.2$ (b) under symmetric photon loss and (d) under photon loss in one arm, using CSV (brown dashed curve), TMSV (red solid curve), and coherent (green dotted curve) states. The shot-noise limit (blue dot-dashed line and curve) is given by ${\mathrm{\Delta }}^2{\varphi }_{\text{SNL}}=1/2\overline{n}$.

Figure 6

Comparison of phase sensitivity via the quantum Cramér-Rao bound (red solid curve), parity measurement with photon number resolving detection (brown dashed curve), quadrature measurement with single homodyne detection (orange double-dot–dashed curve) and double homodyne detection (blue dotted curve) at $\overline{n}=10$, using a CSV state (a) under symmetric photon loss and (b) under photon loss on one arm and using a TMSV state (c) under symmetric photon loss and (d) under photon loss on one arm. The green dot-dashed curve represents the quantum Cramér-Rao bound for a coherent state.

Figure 7

Phase sensitivities for CSV (brown dashed curve), TMSV (red solid curve), and coherent (green dotted curve) states, respectively, as a function of loss rate $1-\eta$ at $\overline{n}=7$ under symmetric photon loss via (a) the quantum Cramér-Rao bound, (b) parity measurement with PNRD, (c) quadrature measurement with single homodyne detection, and (d) quadrature measurement with double homodyne detection. The shot-noise limit (blue dot-dashed line) is given by ${\mathrm{\Delta }}^2{\varphi }_{\text{SNL}}=1/2\overline{n}$.

Figure 8

Phase sensitivities for CSV (brown dashed curve), TMSV (red solid curve), and coherent (green dotted curve) states, respectively, as a function of loss rate $1-\eta$ at $\overline{n}=7$ under photon loss in one arm via (a) the quantum Cramér-Rao bound, (b) parity measurement with PNRD, (c) quadrature measurement with single homodyne detection, and (d) quadrature measurement with double homodyne detection. The shot-noise limit (blue dot-dashed line) is given by ${\mathrm{\Delta }}^2{\varphi }_{\text{SNL}}=1/2\overline{n}$.