Continuous-variable spatio-spectral quantum networks in nonlinear photonic lattices

TL;DR

This work addresses the scalability bottleneck of optical quantum networks by introducing continuous-variable entanglement generated through nondegenerate SPDC in a χ(2) nonlinear photonic lattice that encodes information in both spatial and spectral degrees of freedom. The authors develop a comprehensive framework, diagonalizing the low-gain SPDC dynamics across fixels, L-fixels, and N-fixels to produce broadband, distributable squeezed modes that form 2D spatio-spectral grid cluster states, representable by a complex-weighted adjacency matrix Z. Robustness to realistic losses is analyzed via nullifier variances and van Loock-Furusawa criteria, showing spectral entanglement remains tolerant to moderate loss while spatial entanglement is more fragile but can be mitigated with higher pump gain and media optimization. The results suggest practical routes to scalable quantum networks and measurement-based quantum computing using integrated photonics, with opportunities for phase-locked distribution, spectral demultiplexing, and pump-shaping-driven control of the generated graph states.

Abstract

Multiplexing information in different degrees of freedom and use of integrated and fiber-optic components are natural solutions to the scalability bottleneck in optical quantum communications and computing. However, for bulk-optics systems, where size, cost, stability, and reliability are factors, this remains either impractical or highly challenging to implement. In this paper we present a framework to engineer continuous-variable entanglement produced through nondegenerate spontaneous parametric down-conversion in χ^(2) nonlinear photonic lattices in spatial and spectral degrees of freedom that can solve the scalability challenge. We show how spatio-spectral pump shaping produce cluster states that are naturally distributable in quantum communication networks and a resource for measurement-based quantum computing.

Figures (5)

• Figure 1: Sketch of a nonlinear photonic lattice and measurement basis in a given spatio-spectral mode ( fixel $\mathsf{j}$) associated to a given waveguide (pixel $j$) and frequency band (frexel $l$). Left: a pump pulse (blue) is coupled to the center waveguide of a $N=9$ lattice producing SPDC (red) that spreads accordingly to a coupling profile. Right: orthonormal homodyne measurement basis --frexels (rainbow)-- for a Gaussian-shaped local oscillator (decomposed in frequency bins, normalized in power --rainbow slices), and, in red, the signal marginal of the joint spatio-spectral amplitude (JSSA) (equally for idler) projected on the frexel basis. This example shows the measurement of 16 frexels corresponding to one pixel, hence $9\times16=144$ fixels. In general the system gives access to $N\times L=\mathsf{N}$ fixels with the notations of Table I.
• Figure 2: Sketch of mode bases appearing in the text. In the left we display spectral and spatial non-overlapping (individual) mode basis: frexels and pixels, respectively. On the center we show two families of spatio-spectral modes: fixels and L-fixels. Fixels are non-overlapping modes separable in spectral and spatial DOF. L-fixels are non-overlapping modes in frequency and overlapping modes (supermodes) in space, separable in spectral and spatial DOF. The dotted boxes represent single spatio-spectral modes (yellow for fixels, black for L-fixels). On the right we introduce a third family of spatio-spectral modes: N-fixels. They are overlapping modes in frequency and space, in general inseparable in spectral and spatial DOF, thus only describable in terms of fixels (or L-fixels). This basis is not accessible experimentally unless spectral and spatial DOF are decoupled. Fixels and L-fixels are independent of $z$ (but L-fixels depend on the coupling profile), while N-fixels depend on coupling and pump profiles, and on $z$. The nonlinear photonic lattice produces quantum correlations in fixels and L-fixel bases, and independent squeezing in the N-fixel basis. In the figure we have set as an example $L=7$ spectral and $N=7$ spatial modes --thus $\mathsf{N}=49$ spatio-spectral modes. Blue, black and yellow abscissas represent frequency, guided mode, and fixels, respectively. Rectangles represent relative amplitudes with respect to other basis. w.r.t. stands for with respect to.
• Figure 3: Set of $7\times 2$-grid cluster states obtained with a monochromatic pump spectrum and a flat spatial pump distribution in a weakly-coupled photonic lattice. Each grid state is composed by $\mathsf{N=}$14 fixels: a pair of frexels $l\equiv s, L+1-l \equiv i$ --symmetric with respect to $\omega_{h}/2$ (in red)-- for each of the seven pixels $j=1, \dots, 7$. Solid lines stand for spectral entanglement (horizontal) and dotted lines stand for spatial entanglement (vertical). $L/2$ ($(L-1)/2$) spectrally-shifted copies of this state are generated for an even (odd) number of frexels $L$. Each node of the cluster state is in a given spatio-spectral mode representd by gray (spatial) and blue (spectral) rectangles (cf. Fig. \ref{['F2']}). Blue and black abscissas represent frequency and guided mode, respectively.
• Figure 4: Real part ${\bf V}$ (upper figure) of the complex-weighted adjacency matrix ${\bf Z}={\bf V}+i \,{\bf U}$ obtained from Equations (\ref{['Grid1']}) and (\ref{['Grid2']}) for $L=2$, $N=7$. ${\bf V}$ is the canonical graph of the state, whereas the trace of the imaginary part ${\bf U}$ accounts for the error of the approximation. The value of Tr(${\bf U}$) for different values of nonlinearity is shown in the lower figure (blue curve). The inset shows the matrix ${\bf U}$ for a specific pump power. As it is diagonal and small, it represents the variances of the graph nullifiers. $x_{s(i),j}$ is the amplitude quadrature of a fixel $\mathsf{j}$ (pixel $j$, frexel $l$). The vacuum shot noise is set as 1. We have applied a $\pi/2$ rotation in idler-mode phase space (exchange of labels of quadratures for the idler modes), and used a homogeneous coupling profile $\vec{f}=\vec{1}$ with $C_{M}=0.01$ mm$^{-1}$, $g \sqrt{p_{h}}=0.05$ mm$^{-1}$ (upper figure, lower figure inset) and $z=20$ mm.
• Figure 5: Effect of losses on the variances of the nullifiers. Dependence of 3-node and 4-node variances with $\eta$ are in blue and red, respectively. Black horizontal lines represent the squeezing threshold per nullifier for spatial (Var$(\hat{\delta})<1/2$) and spectral (Var$(\hat{\delta})<2$) full inseparability.