Experimental generation of four-mode continuous-variable cluster states

Continuous-variable Gaussian cluster states are a potential resource for universal quantum computation. They can be efficiently and unconditionally built from sources of squeezed light using beam splitters. Here we report on the generation of three different kinds of continuous-variable four-mode cluster states. In our realization, the resulting cluster-type correlations are such that no corrections other than simple phase-space displacements would be needed when quantum information propagates through these states. At the same time, the inevitable imperfections from the finitely squeezed resource states and from additional thermal noise are minimized, as no antisqueezing components are left in the cluster states.

Figure 1

The created four-mode cluster states. Each cluster node, corresponding to an optical mode, is represented by a circle. Neighboring nodes are connected by lines.

Figure 2

The schematic of the optical setups to create the linear cluster state (top) and $\mathsf{T}$-shape cluster state (bottom). Four squeezed states are generated by OPOs (optical parametric oscillators). HBS is half beam splitter and 20% is 1:4 beam splitter. Boxes including $i$ are Fourier transforms (90° rotations in phase space), and $-i$ is a $-90°$ rotation. LO is local oscillator for homodyne detection of the output states.

Figure 3

In all graphs, the measurement variances without squeezing (upper) and with squeezing (lower) are shown. The graphs of the first row are the results of $⟨{\left[\Delta ({\widehat{p}}_{\mathrm{L}1}-{\widehat{x}}_{\mathrm{L}2})\right]}^2⟩$ and $⟨{\left[\Delta ({\widehat{p}}_{\mathrm{L}2}-{\widehat{x}}_{\mathrm{L}1}-{\widehat{x}}_{\mathrm{L}3})\right]}^2⟩=⟨{\left[\Delta ({\widehat{p}}_{\mathrm{S}3}-{\widehat{x}}_{\mathrm{S}1}-{\widehat{x}}_{\mathrm{S}2})\right]}^2⟩$. Those of the second row are for $⟨{\left[\Delta ({\widehat{p}}_{\mathrm{L}3}-{\widehat{x}}_{\mathrm{L}2}-{\widehat{x}}_{\mathrm{L}4})\right]}^2⟩=⟨{\left[\Delta ({\widehat{p}}_{\mathrm{S}2}-{\widehat{x}}_{\mathrm{S}3}-{\widehat{x}}_{\mathrm{S}4})\right]}^2⟩$ and $⟨{\left[\Delta ({\widehat{p}}_{\mathrm{L}4}-{\widehat{x}}_{\mathrm{L}3})\right]}^2⟩$, and those of the third row are for $⟨{\left[\Delta ({\widehat{p}}_{\mathrm{S}1}-{\widehat{x}}_{\mathrm{S}3}-{\widehat{x}}_{\mathrm{S}4})\right]}^2⟩$ and $⟨{\left[\Delta ({\widehat{p}}_{\mathrm{S}4}-{\widehat{x}}_{\mathrm{S}1}-{\widehat{x}}_{\mathrm{S}2})\right]}^2⟩$. The measurement frequency is $1\mathrm{MHz}$, resolution bandwidth is $30\mathrm{kHz}$, and video bandwidth is $300\mathrm{Hz}$. All results are obtained with 20 times averaging.

Figure 4

Upper graphs are $⟨{\left[\Delta ({\widehat{p}}_{\mathrm{T}1}-{\widehat{x}}_{\mathrm{T}2}-{\widehat{x}}_{\mathrm{T}3}-{\widehat{x}}_{\mathrm{T}4})\right]}^2⟩$ and $⟨{\left[\Delta ({\widehat{p}}_{\mathrm{T}2}-{\widehat{x}}_{\mathrm{T}1})\right]}^2⟩$. Bottom graphs are $⟨{\left[\Delta ({\widehat{p}}_{\mathrm{T}3}-{\widehat{x}}_{\mathrm{T}1})\right]}^2⟩$ and $⟨{\left[\Delta ({\widehat{p}}_{\mathrm{T}4}-{\widehat{x}}_{\mathrm{T}1})\right]}^2⟩$. The conditions of the measurements are the same as in Fig. 3.