Violation of the Cauchy-Schwarz Inequality in the Macroscopic Regime

We have observed a violation of the Cauchy-Schwarz inequality in the macroscopic regime by more than 8 standard deviations. The violation has been obtained while filtering out only the low-frequency noise of the quantum-correlated beams that results from the technical noise of the laser used to generate them. We use bright intensity-difference squeezed beams produced by four-wave mixing as the source of the correlated fields. We also demonstrate that squeezing does not necessarily imply a violation of the Cauchy-Schwarz inequality.

Figure 1

Experimental setup. A 4WM process is used to generate quantum-correlated bright beams. $\mathrm{PBS}=\mathrm{\text{polarizing}}$ beam splitter; $\mathrm{BS}=50/50$ beam splitter.

Figure 2

Correlation functions of the fluctuations for the probe (dotted curve), conjugate (dashed curve), and cross $g^{(2)}$ (solid curve). The inset shows an expanded view of the peaks of the correlation functions. The horizontal dashed line shows the mean value of the zero-time autocorrelation functions for the probe and the conjugate. The indicated uncertainties are discussed in the text.

Figure 3

Normalized noise power spectra for the probe ($S_p$), conjugate ($S_c$), and intensity difference ($S_{\mathrm{diff}}$) for a gain of (a) 10 and (b) 2. All of the spectra are normalized to their respective SQL, represented by the dashed line.

Figure 4

Effect of frequency filtering on the violation of the CSI. Violation parameter ($V$) as a function of high-frequency cutoff for $G=2$, $G=5$, $G=8$, and $G=10$. $V<1$ indicates a violation of the CSI. The height of the rectangles represents the statistical uncertainties.