Robust continuous-variable multipartite entanglement in circular arrays of nonlinear waveguides

Abstract

Encoding continuous-variable quantum information in the optical domain has recently enabled the generation of large entangled states, yet robust implementation remains a challenge. Here, we present a straightforward protocol for generating multipartite entanglement based on spontaneous parametric down-conversion in a circular array of quadratic nonlinear waveguides. We provide a rigorous theoretical framework, including comprehensive derivations of the propagation equations and the identification of regimes where analytical solutions are possible. Crucially, our approach identifies the pump and detection configurations required to sustain and measure multipartite full inseparability across arbitrary propagation distances and for any number of waveguides $N=4 n$. This regime, elusive to standard numerical methods, represents a key requirement for scalable quantum protocols. Our scheme is inherently robust as it relies on phase-matched propagation eigenmodes, making it resilient against variations in sample length, coupling, and nonlinearity.

Figures (4)

• Figure 1: Sketch of a nonlinear circular array made up of eight waveguides working in a SPDC configuration, pumping all the waveguides. Nearest-neighbor waveguides are evanescently coupled through the coupling constant $J$. Quantum noise variances and correlations can be measured by multimode balanced homodyne detection. $z$ is the propagation direction.
• Figure 2: Covariance matrices V(z) in the individual mode basis for an eight-waveguide circular array with homogeneous coupling profile. (a) Quantum correlations for a flat pump profile with uniform amplitude and uniform phase (r = 0). For a flat pump profile with an alternative $\pi$ phase, we obtained a block diagonal matrix as displayed in (b). Quantum correlations obtained for a flat pump profile with an alternative $\pi/2$ phase are shown in (c). We chopped absolute values lower than $10^{-2}$ from covariance matrices for exposition. The coupling and nonlinearity parameters as follows: J = 0.45 $\text{mm}^{-1}$, $\eta$ = 0.015 $\text{mm}^{-1}$, and z = 20 mm.
• Figure 3: Multipartite entanglement generation in the individual mode basis versus propagation for four and eight waveguides and a flat pump profile ($\eta_{j} = \vert \eta \vert$). VLF inequalities for a typical value of coupling constant $J = 0.45 ~ mm^{-1}$ are shown in red. For comparison, VLF inequalities in the unphyisical limit of infinite coupling are shown in dashed black. Simultaneous values under the threshold value VLF = 4 (dot dashed), imply CV multipartite entanglement. $\eta = 0.015 ~ mm^{-1}$.
• Figure 4: VLF inequalities in the limit of infinite coupling for a large number of involved modes: $N=40, 60$ and $80$. Simultaneous values under the threshold value, VLF = 4, imply CV multipartite entanglement. $J = 100 ~ mm^{-1}.$$\eta = 0.015 ~ mm^{-1}$.