Quantum-controlled cluster states

Quantum optical cluster states have been increasingly explored in the light of their importance for measurement-based quantum computing. Here we set forth a method for generating quantum controlled cluster states: pumping an optical parametric oscillator with spatially structured light. We show that state-of-the-art techniques for producing clusters in the spectral and temporal domains are improved by a structured light pump, which manipulates the spatial mode couplings in the parametric interaction. We illustrate the method considering a second-order pump structure and show that a simple mode rotation yields different cluster states, including but not limited to the two-dimensional square and hexagonal lattices. We also introduce a novel technique for generating nonuniform cluster states and propose a simple setup for its implementation.

Figure 1

Peres-Horodecki (PPT) values obtained for a ${\mathrm{HG}}_{11}$ pump mode rotated clockwise by $\pi /4-\theta$. Each curve corresponds to a different value of $\theta$, and we considered $\gamma =0.1$.

Figure 2

Cluster states produced by dual-frequency spatially structured pump. In (a) we show the considered pump modes, which are rotated petal modes. The produced cluster states are shown for $p_1=1$ and $p_2=3$, and the following combinations of spatial modes: (b) ${\theta }_1={\theta }_2=\pi /8$, (c) ${\theta }_1={\theta }_2=0$, and (d) ${\theta }_1={\theta }_2=\pi /4$. Although in (b) the cluster has a dual-rail structure, in (c) and (d) it is dismantled in two independent single-rail clusters. The numbers in (b)–(d) are the weights of the cluster edges.

Figure 3

(a) Experimental apparatus for the production of a bilayer 2D lattice with structured light. The detailed cluster graph is shown in (b). In panels (c)–(f), we show the resulting clusters for different pump structures, defined by the the rotation angles ${\theta }_1$ and ${\theta }_2$ indicated in each panel. The filled and unfilled squares indicate frequency indices $n$ with different parities, and the shaded regions in (c)–(e) highlight the 2D structures achieved in each case.

Figure 4

Graphs for cluster states produced by a pump structure $\{{\theta }_1,{\theta }_2\}$ varying between (a) $\{0,\pi /4\}$ and $\{\pi /4,0\}$ and (b) $\{0,\pi /4\}$ and $\{\pi /4,\pi /4\}$.

Figure 5

Simplified experimental arrangement to produce separate time-varying spatial structures for the two pump frequency components. The pump frequency modes $p_1$ and $p_2$ carry second-order vector vortices in their transverse structures, and the spatial mode switching is achieved by passing each of them through a polarization EOM, a nonpolarizing beam splitter (BS), and a polarizer (Pol).

Figure 6

Proposed experimental scheme for the production of 2D cluster states with structured light.

Figure 7

Graph of the 2D cluster state defined by the nullifiers (C3) and (C4).