Phase sensitivity of lossy Mach-Zehnder interferometer via photon addition operation

TL;DR

This work investigates enhancing phase estimation in a lossy Mach-Zehnder interferometer by applying photon addition operations to a coherent plus squeezed input. It compares two PA schemes, one at the input port (Scheme A) and one inside the interferometer (Scheme B), using intensity difference and homodyne detections and analyzes both phase sensitivity and quantum Fisher information (QFI) along with the Quantum Cramér-Rao Bound (QCRB). The results show that PA inside the MZI (Scheme B) coupled with homodyne detection yields the best phase sensitivity and highest QFI, with greater robustness to internal losses; performance improves with the added photon number m and with the input parameters α and r. In the small squeezing or lossy regimes, certain PA configurations can approach the Heisenberg limit or surpass the standard quantum limit, providing a practical method for quantum precision measurements in realistic settings."

Abstract

Photon addition operations applied to squeezed states have been shown to significantly enhance phase sensitivity. In this study, we extend this approach by applying photon addition not only to coherent states but also within a Mach--Zehnder interferometer setup, using coherent and squeezed vacuum states as input. Both intensity-difference and homodyne detection are used to evaluate photon addition schemes, and their phase sensitivities are compared under ideal and lossy conditions, respectively. We also analyze the quantum Fisher information of these two schemes. Results show both schemes improve phase sensitivity, quantum Fisher information, and loss resistance. In particular, photon addition within the interferometer performs better. Homodyne detection outperforms intensity difference detection under photon losses. Notably, each scheme has different parameter dependencies, making them suitable for different application scenarios. When the squeezing parameter is small, photon addition employed at the coherent input with intensity difference detection can approach the Heisenberg limit in ideal conditions and can exceed the standard quantum limit in high-loss conditions. Our proposed scheme represents a valuable method for quantum precision measurements.

Figures (11)

• Figure 1: Schematic diagram of a MZI using a coherent state $\left \vert \alpha \right \rangle _{a}$ and a squeezed state $\left \vert r\right \rangle _{b}$. (a) The ideal standard MZI. (b) The lossy MZI with PA at the input port. (c) The lossy MZI with PA at the interior. $BS$ is the beam splitter, $U_{ \phi }$ is the phase shifter, and $D_{a}$ ($D_{b}$) is the detector. The fictitious BSs inside the MZI model photon loss in modes $a$ and $b$, respectively. $a_{v}$ and $b_{v}$ are vacuum modes.
• Figure 2: The phase sensitivity as a function of $\phi$ with $\alpha =1$ and $r=1$: (a) for intensity difference detection; and (b) homodyne detection. Here, the solid line represents Scheme A, and the dashed line represents Scheme B.
• Figure 3: The optimal phase sensitivity based on intensity difference detection as a function of (a) $\alpha$, with $r=1$, (b) $r$ with $\alpha =1$. The optimal phase sensitivity based on homodyne detection as a function of (c) $\alpha$ with $r=1$, (d) $r$ with $\alpha =1$. The solid line is Scheme A and the dashed line is Scheme B. $\phi$ is optimized.
• Figure 4: The standard deviation $\sigma (O_{i})$$= \sqrt{\left \langle O_{i}^{2}\right \rangle -\left \langle O_{i}\right \rangle ^{2}}$ and the slope $\left \langle O_{i}\right \rangle _{ \phi }=\left \vert \partial \left \langle O_{i}\right \rangle /\partial \phi \right \vert$ as a function of (a) $\alpha$, with $r=1$, (b) $r$ with $\alpha =1$ for intensity difference detection; (c) $\alpha$, with $r=1$, (d) $r$ with $\alpha =1$ for homodyne detection. $\phi$ is optimized.
• Figure 5: The optimal phase sensitivity as a function of $T$ with fixed $\alpha =1$ and $r=1$, (a) for intensity difference detection; (b) for homodyne detection. $\phi$ is optimized. Here, the solid line represents Scheme A, and the dashed line represents Scheme B.