Figure 1

(a) Abstract illustration and (c) experimental setup of one-mode LUBO transformations using a four-mode linear cluster state. There is a 1-to-1 correspondence between (a) and (c). Squeezed vacuum states are generated by subthreshold optical parametric oscillators containing periodically poled ${\mathrm{KTiOPO}}_4$ crystals as nonlinear media. (b) Phase space representations of quantum states in each step of the Fourier transformation (i) and the 10 dB $x$-squeezing operation (ii), starting with a vacuum state input ($\alpha$) and an $x$-coherent state input ($\beta$) Disp.: displacement in phase space, Tele.: teleportation, Op: operation, EOM: electro-optical modulator, HBS: half beam splitter.Reuse & Permissions

Figure 2

Fourier transformation operation; (a) Measurement results of the input state. Trace (i) shows the shot noise level (SNL) and (ii) shows the phase scan of the input state. (b) Measurement results of the output state. Trace (i) shows the SNL, (ii) shows the phase scan of the output state, and (iii) shows the measurement result of the $x$ quadrature with a vacuum input. The measurement quadrature angle is determined through the relative phase between the signal beam and the local oscillator beam. The measurement frequency is 1 MHz and the resolution and video bandwidths are 30 kHz and 300 Hz, respectively. Traces a(i), b(i), and b(iii) are averaged 20 times.Reuse & Permissions

Figure 3

Squeezing operations; (a),(b) 10 dB $x$-squeezing operation with an $x$-coherent input (a) and a vacuum input (b). Trace (i): shot noise level, trace (ii): phase scan of the output state, trace (iii): measurement of $x$, trace (iv): measurement of $p$, dotted line (v): $x$ of the input, and dotted line (vi): $p$ of the input. The measurement settings are the same as in Fig. 2. Traces (i), (iii), and (iv) are averaged 20 times. (c) experimental results (dots) and theoretical calculation (solid curves) of 3, 6, and 10 dB $x$-squeezing operations. Traces (i) and (ii) correspond to a $p$ measurement with $p$-coherent input, and $x$ measurement with $x$-coherent input, respectively, traces (iii) and (iv) correspond to a $p$ measurement with vacuum input, and $x$ measurement with vacuum input, respectively. Each data point has an error of about $\pm 0.2\mathrm{dB}$.Reuse & Permissions