Experimental Creation of a Fully Inseparable Tripartite Continuous-Variable State

A continuous-variable tripartite entangled state is experimentally generated by combining three independent squeezed vacuum states, and the variances of its relative positions and total momentum are measured. We show that the measured values violate the separability criteria based on the sum of these quantities and prove the full inseparability of the generated state.

Figure 1

Schematic of the generation of three independent squeezed vacuum states.

Figure 2

Noise measurement results on output mode 1 alone. (i) The corresponding vacuum noise $⟨(\Delta {\widehat{x}}_1^{(0)}{)}^2⟩=\frac{1}{4}$; (ii) the noise of the $x$ quadrature $⟨(\Delta {\widehat{x}}_1{)}^2⟩$; (iii) the noise of the $p$ quadrature $⟨(\Delta {\widehat{p}}_1{)}^2⟩$; and (iv) the noise of the scanned phase. The measurement frequency is centered at 900 kHz, resolution bandwidth is 30 kHz, and video bandwidth is 300 Hz. Except for (iv), traces are averaged ten times.

Figure 3

Schematic of the measurements of the variances (a) $⟨[\Delta ({\widehat{x}}_3-{\widehat{x}}_1){]}^2⟩$ and $⟨[\Delta ({\widehat{x}}_2-{\widehat{x}}_3){]}^2⟩$ and (b) $⟨[\Delta ({\widehat{p}}_1+{\widehat{p}}_2+{\widehat{p}}_3){]}^2⟩$. BS1 and BS2 are beam splitters with $T/R$ ratios of $1/2$ and $1/1$, respectively. The ellipses illustrate the squeezed quadrature of each beam. ${\mathrm{L}\mathrm{O}}_{ix,p}$ denote local oscillator beams for homodyne detector $i$ with their phases locked at the $x$ and $p$ quadratures, respectively.

Figure 4

Noise measurement results corresponding to the variances of the lhs of inequality (3). (a) (i) is $⟨[\Delta ({\widehat{x}}_1^{(0)}-{\widehat{x}}_2^{(0)}){]}^2⟩=\frac{1}{2}$ and (ii) is $⟨[\Delta ({\widehat{x}}_1-{\widehat{x}}_2){]}^2⟩$; (b) (i) $⟨[\Delta ({\widehat{x}}_2^{(0)}-{\widehat{x}}_3^{(0)}){]}^2⟩=\frac{1}{2}$ and (ii) $⟨[\Delta ({\widehat{x}}_2-{\widehat{x}}_3){]}^2⟩$; (c) (i) $⟨[\Delta ({\widehat{x}}_3^{(0)}-{\widehat{x}}_1^{(0)}){]}^2⟩=\frac{1}{2}$ and (ii) $⟨[\Delta ({\widehat{x}}_3-{\widehat{x}}_1){]}^2⟩$; (d) (i) $⟨[\Delta ({\widehat{p}}_1^{(0)}+{\widehat{p}}_2^{(0)}+{\widehat{p}}_3^{(0)}){]}^2⟩=\frac{3}{4}$ and (ii) $⟨[\Delta ({\widehat{p}}_1+{\widehat{p}}_2+{\widehat{p}}_3){]}^2⟩$. The measurement conditions are the same as for Fig. 2 with 10 times averages.