Direct Measurement of Decoherence for Entanglement between a Photon and Stored Atomic Excitation

Violations of a Bell inequality are reported for an experiment where one of two entangled qubits is stored in a collective atomic memory for a user-defined time delay. The atomic qubit is found to preserve the violation of a Bell inequality for storage times up to $21\mu \mathrm{s}$, 700 times longer than the duration of the excitation pulse that creates the entanglement. To address the question of the security of entanglement-based cryptography implemented with this system, an investigation of the Bell violation as a function of the cross correlation between the generated nonclassical fields is reported, with saturation of the violation close to the maximum value allowed by quantum mechanics.

Figure 1

(a),(b) Relevant levels and decay paths from $|e⟩$ to $|s⟩$, starting from an unpolarized Cs ensemble. The distributions $p_{m_F}^{+},p_{m_F}^{-}$ for populations in $|s⟩$ from the two possible decay paths are also shown. (c) Experimental setup. Write and read pulses are sent sequentially with 400 ns delay, until a detection in field 1 occurs. This event triggers the “memory start” circuit, which stops the write or read sequence for a programmable time $\tau$ by way of independent electro-optic Mach-Zehnder intensity modulators (IM). The read beam power is $\approx 150\mu \mathrm{W}$, while the write beam power is much weaker ($\mu \mathrm{W}$ range, see text).

Figure 2

Measurement of the Bell parameter $S$ as a function of the average value for the normalized cross-correlation function ${\overline{g}}_{12}$. The fitted $S_{\max }$ is $2.74\pm 0.04$. Inset: Measurement of the correlation function $E({\theta }_1,{\theta }_2)$ as a function of ${\theta }_2$, for an average ${\overline{g}}_{12}=57$. Solid squares shows the fringe for ${\theta }_1=0°$ ($V=0.94\pm 0.02$), while open circles correspond to ${\theta }_1=-45°$ ($V=0.90\pm 0.02$).

Figure 3

Measurement of the Bell parameter $S$ (open circles) and of $g_{12}$ for the two different polarization configurations (solid symbols) as a function of $\tau$. To increase the repetition rate, repumping light has not been used between trials and the period has been shortened to $1.45\mu \mathrm{s}$, such that 1400 trials can be performed in a measurement period.