Bipartite entanglement extracted from multimode squeezed light generated in lossy waveguides

TL;DR

The paper addresses entanglement extraction from multimode, lossy type-II PDC by identifying the measurement basis that maximizes two-mode entanglement. The authors show that entanglement in the resulting TMBS is quantified by squeezing, with the maximal entanglement corresponding to the minimal quadrature variance $\lambda_{-}(\sigma)$, which equals the symplectic value $\nu_{-}(\tilde{\sigma})$ of the partially transposed state. They introduce the MSq-basis as the optimal basis, derived from the eigenvector associated with $\lambda_{\min}$ of the full covariance $\Sigma$, and compare it to Mercer-Wolf and Williamson-Euler bases. Numerical simulations across lossless and lossy waveguides demonstrate that MSq-TMBS yields the strongest entanglement and purity, while MW can lose entanglement under unbalanced losses. The work provides practical guidance for optimizing TMBS-based quantum information tasks in realistic, multimode, lossy settings by selecting the measurement basis that aligns with maximal squeezing.

Abstract

Entangled two-mode Gaussian states constitute an important building block for continuous variable quantum computing and communication protocols. In this work, we theoretically study two-mode bipartite states which are extracted from multimode light generated via type-II parametric down-conversion (PDC) in lossy waveguides. For these states, we demonstrate that the squeezing quantifies entanglement and we construct a measurement basis which results in the maximal bipartite entanglement. We illustrate our findings by numerically solving the spatial master equation for PDC in a Markovian environment. The optimal measurement modes are compared with two widely-used broadband bases: the Mercer-Wolf basis (the first-order coherence basis) and the Williamson-Euler basis.

Figures (4)

• Figure 1: The scheme of a TMBS preparation from a multimode type-II PDC process. After the lossy PDC process, a multimode mixed state with the covariance matrix $\Sigma$ is generated. Two quantum pulse gates (QPG) in partition A with the mode $\mathbf{u}_A$ and in partition B with the mode $\mathbf{u}_B$ provide the TMBS with the covariance matrix $\sigma(\mathbf{u}_A, \mathbf{u}_B)$.
• Figure 2: Numerical results for a lossless waveguide WG0, the parameters of which are given in the main text. (a) Normalized joint spectral intensity for low-gain PDC, $\delta \omega = \omega - \omega_0$, where $\omega_0=\omega_p/2$ is the central frequency of the PDC light. (b) Dependence of the smallest quadrature variance $\braket{(\Delta \hat{p}^F)^2}=\lambda_{-}(\sigma)$ as a function of the number of photons $N_A=N_B$. The additional axis shows the corresponding logarithmic negativity $\mathcal{E}(\sigma^S)$.
• Figure 3: Numerical results for lossy waveguides: (a,c,e) WG1 and (b,d,f) WG2. The waveguide parameters are given in the main text. Different colors correspond to TMBS build with different modes: MW (blue), WE (green), and MSq (red). (a, b) The number of photons $N_A$ (solid lines) and $N_B$ (dashed lines) as a function of losses $\bar{\eta}$. (c, d) The dependencies of the smallest quadrature variance $\braket{(\Delta \hat{p}^F)^2} = \lambda_{-}(\sigma)$ for $\sigma^{MW}$, $\sigma^{WE}$ and $\sigma^{MSq}$ as a function of losses $\bar{\eta}$. Additional right axis show the logarithmic negativity $\mathcal{E}(\sigma)$. The gray area highlights the region with zero logarithmic negativity. (e, f) Purities of $\sigma^{MW}$, $\sigma^{WE}$ and $\sigma^{MSq}$ as a function of losses $\bar{\eta}$.
• Figure 4: TMBS for the lossy waveguide WG2 (unbalanced losses) with $\bar{\eta} =5$ dB and $r=\frac{1}{3}$. (a,b,c) Covariance matrices $\sigma^{MW}$, $\sigma^{WE}$ and $\sigma^{MSq}$, respectively. (d) Absolute value and (e) phase of signal modes $\mathbf{u}_A$; (f) absolute value and (g) phase of idler modes $\mathbf{u}_B$. The blue, green, and red colors correspond to the MW-, WE-, and MSq-TMBS bases, respectively. Note that the phases of modes are well defined, providing the covariance matrices (a,b,c) to be in the form Eq. \ref{['eq_cov_matrix_1x1']}.