Greenberger-Horne-Zeilinger–type and $W$-type entangled coherent states: Generation and Bell-type inequality tests without photon counting

We study Greenberger-Horne-Zeilinger–type (GHZ-type) and $W$-type three-mode entangled coherent states. Both types of entangled coherent states violate Mermin’s version of the Bell inequality with threshold photon detection (i.e., without photon counting). Such an experiment can be performed using linear optics elements and threshold detectors with significant Bell violations for GHZ-type entangled coherent states. However, to demonstrate Bell-type inequality violations for $W$-type entangled coherent states, additional nonlinear interactions are needed. We also propose an optical scheme to generate $W$-type entangled coherent states in free-traveling optical fields. The required resources for the generation are a single-photon source, a coherent state source, beam splitters, phase shifters, photodetectors, and Kerr nonlinearities. Our scheme does not necessarily require strong Kerr nonlinear interactions; i.e., weak nonlinearities can be used for the generation of the $W$-type entangled coherent states. Furthermore, it is also robust against inefficiencies of the single-photon source and the photon detectors.

Figure 1

An example of the generation of a GHZ-type three-mode ECS in free-traveling fields from a CSS using two beam splitters, BS1 with $r_1=1∕\sqrt{3}$ and BS2 with $r_2=1∕\sqrt{2}$, where $r_1$ and $r_2$ are reflectivities of the beam splitters. Note that ${\alpha }^{\prime }=\sqrt{3}\alpha$.

Figure 2

Violations of the Bell-Mermin inequality for the GHZ-type ECS of amplitude $\alpha$ using parity measurements. ${\mathrm{BM}}_{\Pi }$ represents the Bell-Mermin function defined in the text which is supposed to be not greater than 2 by a local hidden-variable theory. (a) As $\alpha$ increases the violation increases and saturates at $\approx 3.6$ for the GHZ-type ECS’s of $c_1=c_2$ (solid line) and $c_1=-c_2$ (dashed line). (b) Even when $\alpha$ is extremely small the GHZ-type ECS of $c_1=-c_2$ (dashed line) still violates the Bell-Mermin inequality while the GHZ-type ECS of $c_1=c_2$ does not.

Figure 3

Violations of Bell-Mermin inequality for a GHZ-type ECS of amplitude $\alpha$ using threshold photodetectors. ${\mathrm{BM}}_A$ represents the Bell-Mermin function defined in the text, Eq. (14), which is supposed to be less than or equal to 2 by a local hidden-variable theory. As $\alpha$ increases the violation first decreases and then ceases starting from $\alpha \approx 0.6$.

Figure 4

A schematic of our scheme to generate a $W$-type ECS. A $W$-type ECS is generated when detector B or detector C clicks.

Figure 5

Violations of the Bell-Mermin inequality for a $W$-type ECS of amplitude $\alpha$ using threshold photodetectors and nonlinear interactions. The Bell-Mermin function ${\mathrm{BM}}_{\mathcal{A}}$ is larger than the classical bound 2 for $\alpha \gtrsim 1.49$.