Nonclassicality of bright Greenberger-Horne-Zeilinger–like radiation of an optical parametric source

With the emerging possibility to obtain emissions of triples of Greenberger-Horne-Zeilinger–entangled photons via direct parametric generation, we study here bright emissions of this kind which involve higher-order emissions of two triples, three triples, etc. Such states would constitute a natural generalization of the four-mode (two beams plus polarization) squeezed vacuum. We show how to avoid technical difficulties related to straight-ahead generalization of the usual approximate description of parametric down-conversion which uses a classical pump field. Using Padé approximation, we turn the first terms of the nonconverging perturbation expansion into elements of a converging series. This allows us to study nonclassicality of the new bright generalized Greenberger-Horne-Zeilinger states. We present both violations of local realism by such a radiation and simple entanglement indicators tailored for this case.

Figure 1

Probability $p\left(k\right)$ of observing $k$ triplets of photons for $k=0,\cdots ,5$ as a function of the amplification gain $\mathrm{\Gamma }$. The probability of a vacuum event decreases when $\mathrm{\Gamma }$ increases. Also, all probabilities approach zero when the number of triples goes to infinity: $p(k\to \infty )\to 0$. This tendency is typical also for a two-mode squeezed vacuum.

Figure 2

Left-hand side of inequality (19) as a function of amplification gain $\mathrm{\Gamma }$ (${\mathrm{\Gamma }}_t$). The threshold value of $\mathrm{\Gamma }$, such that for all $\mathrm{\Gamma }<{\mathrm{\Gamma }}_t$ inequality (19) is violated, is ${\mathrm{\Gamma }}_t=0.77$.

Figure 3

Threshold efficiency ${\eta }_t$ as a function of amplification gain $\mathrm{\Gamma }$. As expected, the value of ${\eta }_t$ increases with $\mathrm{\Gamma }$. Note that for small values of amplification gain $(\mathrm{\Gamma }\to 0)$ threshold efficiency ${\eta }_t=0.79$, and it conforms with ${\eta }_t$ for qubits given in [3]. That is because for $\mathrm{\Gamma }\to 0$ our $|\text{GBHZ}\rangle$ effectively becomes a superposition of $|\text{GHZ}\rangle$ and vacuum.

Figure 4

Expectation value of entanglement indicator (23) for $|\text{BGHZ}\rangle$ as a function of amplification gain $\mathrm{\Gamma }$. The lower blue line is for state ${\rho }^{\prime }$, from which we removed the vacuum contribution and renormalized it, (24). This procedure allows one to see better violations for low $\mathrm{\Gamma }$, as in the original state in such a case the vacuum term is dominant. The value $-1$ points to the theoretical value for the three-qubit entangled GHZ state and the original entanglement witness (22). This is because, for a very small $\mathrm{\Gamma }$, the state $|\text{BGHZ}\rangle$ is effectively just a superposition of vacuum and a single three-photon GHZ emission. Removal of vacuum and renormalization leaves just the GHZ state.

Figure 5

Comparison of the violation of (32) for $\widehat{\rho }$ and ${\widehat{\rho }}^{\prime }$ as a function of amplification gain $\mathrm{\Gamma }$. See also the caption of Fig. 4. Note that in the case of indicator (32) we seem to have slightly more robust violations of the separability threshold than for (23).