Enhancement in phase sensitivity in displacement-assisted SU(1,1) interferometer via photon recycling

TL;DR

This work addresses the limits of phase estimation in displacement-assisted SU(1,1) interferometers by introducing photon recycling (PR) in a DSU(1,1) architecture and analyzing both SID and HD measurement schemes. The authors formulate a PR-DSU(1,1) model with iterative recycling, derive the corresponding input-output relations and phase-sensitivity expressions, and compute the QCRB for pure Gaussian outputs. They quantify improvements over the conventional DSU(1,1) using enhancement factors for phase sensitivity (Σ) and QCRB (Ξ), showing that both exceed unity in wide parameter regimes, especially for higher recycling transmission $T$, larger OPA gain $g$, squeezing $r$, and displacement strength $| ext{γ}|$, with HD outperforming SID. Additionally, they demonstrate that the phase sensitivity relative to the SNL (Γ) and the QCRB relative to the SNL (Λ) can surpass classical limits, particularly when photon recycling is combined with LDO, suggesting a practical pathway toward quantum-enhanced metrology in SU(1,1) interferometers.

Abstract

We propose a novel method for enhancing phase estimation in the displacement-assisted SU(1,1) (DSU(1,1)) interferometer by incorporating the photon recycling technique, evaluated under single-intensity detection (SID) and homodyne detection (HD) schemes. Our analysis showed that utilizing the photon recycling technique, the photon-recycled DSU(1,1) interferometer performs better than the conventional DSU(1,1) interferometer for some conditions. We also showed that this improvement is possible in both SID and HD schemes. In addition, to discuss the maximum sensitivity achieved by our proposed model, we have calculated the quantum Cramér-Rao bound (QCRB) within the framework and found that our proposed model approaches the QCRB. Therefore, we believe that our findings offer a promising new approach to improve phase sensitivity through photon recycling.

Figures (11)

• Figure 1: Scheme for phase estimation with photon recycling technique: (a) Schematic diagram of a DSU(1,1) interferometer with the output beam "a" disregarded. (b) The modified scheme with the output mode "a" re-injected into the input mode "a" (PR-DSU(1,1) interferometer). We are assuming that the output beam experiences a phase shift $\theta$ and photon loss of $\sqrt{1-\text{T}}$ before re-injection.
• Figure 2: The iterative-structure of series of DSU(1,1) interferometers where $(k-1)^{th}$ output mode "a" is injected into $k^{th}$ input mode "a". This model is equivalent to the scheme in Fig. 1(b).
• Figure 3: The enhancement factor $\Sigma$ as a function of T. Other parameters are $\text{g}=1$, $\text{r}=0.5$, $|\gamma|=1$.
• Figure 4: The enhancement factor $\Xi$ as a function of T. Other parameters are $\text{g}=1$, $\text{r}=0.5$, $|\gamma|=1$.
• Figure 5: The enhancement factor $\Gamma$, plotted as a function of phase angles $\theta$ and $\phi$ for (A) single-intensity detection and (B) homodyne detection. The plots are shown for transmission coefficients (a) $\text{T}=0.75$, (b) $\text{T}=0.85$, and (c) $\text{T}=0.95$, with the remaining parameters fixed at $\text{g}=1.2$, $\text{r}=0.5$, and $|\gamma|=1$.