Bell-inequality tests with entanglement between an atom and a coherent state in a cavity

We study Bell-inequality tests with entanglement between a coherent-state field in a cavity and a two-level atom. In order to detect the cavity field for such a test, photon on-off measurements and photon number parity measurements, respectively, are investigated. When photon on-off measurements are used, at least 50$\%$ of detection efficiency is required to demonstrate violation of the Bell inequality. Photon number parity measurements for the cavity field can be effectively performed using ancillary atoms and an atomic detector, which leads to large degrees of Bell violations up to Cirel'son's bound. We also analyze decoherence effects in both field and atomic modes and discuss conditions required to perform a Bell inequality test free from the locality loophole.

Figure 1

Numerically optimized values of Bell functions ${\mathcal{B}}_{\mathcal{O}}$ with displaced on-off measurements against amplitude $\alpha$ of state (1). The detection efficiency ranges in value from $\eta =0$ (lower curve) to $\eta =1$ (upper curve), with intervals of 0.1 shown by the family of curves. The horizontal line corresponds to the case of $\eta =0.5$, which coincides with the classical limit of the Bell-CHSH inequality.

Figure 2

(a) Numerically optimized values of Bell function ${\mathcal{B}}_{\mathcal{O}}$ with displaced on-off measurements against detection efficiency $\eta$. The local realistic bound, 2, is violated for $\eta ⩾0.5$. (b) Plot of optimizing values of $\left|\alpha \right|$ with respect to $\eta$.

Figure 3

Numerically optimized values of Bell function ${\mathcal{B}}_{\Pi }$ with displaced parity measurements against $\left|\alpha \right|$. The solid curve corresponds to the absolute values of the Bell function maximized over arbitrary $\zeta$, ${\zeta }^{\prime }$, $\beta$, and ${\beta }^{\prime }$, while the dashed curve corresponds to those values maximized over arbitrary $\beta$ and ${\beta }^{\prime }$ but real $\zeta$ and ${\zeta }^{\prime }$.

Figure 4

Schematic of the proposal. The horizontal arrow is to describe the entangled state (1) generation (with $R_A$ and $C$) and measurement for atom $A$. The vertical arrow depicts the indirect parity measurement of the cavity field using ancillary atom $B$.

Figure 5

Numerically optimized values of Bell function ${\mathcal{B}}_{{\rm Y}}$ with indirect measurements against $\left|\alpha \right|$. The result is found to be identical to the one using direct parity measurements shown as the solid curve in Fig. 3.

Figure 6

Sideview of the atom $A$'s path with intervals of time. Each interval denotes an amount of time required for atom $A$ to pass through the region related with atomic velocity $v$. We note that the distance $l=v\times (t_4+t_5)$, which corresponds to the length of the cylindrical cavity, is a crucial factor in a loohole-free Bell inequality test.

Figure 7

A timeline for decoherence with dynamical parameters related in each regions (from left to right). The top line is for atom $A$, the middle for cavity field $C$, and the bottom for atom $B$. The times when Ramsey pulses are applied are described as vertical dashed lines. We consider a Ramsey pulse application as an instant event as a Ramsey pulse lasts as short as 1 $\mu$s order [45]. Regions are differently hatched depending on the types of dynamics. In the diagonally hatched regions, atoms $A$ and $B$ travel in free spaces with the spontaneous emission rate ${\gamma }_0$ before and after Ramsey pulses as shown in Fig. 4. In the cross-hatched region, atom $A$ travels in a cylindrical cavity with the inhibited spontaneous emission rate ${\gamma }^{\prime }$. In the vertically hatched region, the cavity dissipation with rate $\kappa$ occurs in the cavity ($C$) field. The horizontally hatched regions correspond to the dynamics of the atom-field interaction ${\widehat{H}}_I$ in the main cavity $C$ together with spontaneous emission ${\gamma }_c$ and cavity dissipation $\kappa$. Abbreviations $C$, $D$, $D^{\prime }$, $R_A$, $R_B$, $R_D$, and $R_{D^{\prime }}$ are consistent with those in Fig. 4.

Figure 8

The Bell function under realistic conditions discussed in Sec. 5b are plotted with optimizing conditions in Eqs. (14) and (21) for several different cases of separation $l$. As the separation $l$ increases, the maximum values of the Bell function decrease. The decoherence effects become heavier as $\left|\alpha \right|$ increases.

Figure 9

Contour plots of the Bell function with respect to photon storage time $T_C$ in the main cavity and amplitude $\left|\alpha \right|$ of the entangled state. The atomic life time in cylindrical cavity $T_{\mathrm{atom}}$ is fixed at 1000, 2000, 4000, and $\infty$ (seconds). The minimum distance condition $l=52.99$ (km) for a loophole-free Bell test was assumed. Higher inhibition of spontaneous emission in the cylindrical cavity reduces the required photon storage time in the main cavity.