Electrodynamique des systèmes simples - Lab. Kastler Brossel - Génération d’une paire EPR atomique

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Accueil du site > Atomes, cavités et photons > Information quantique avec des atomes et des cavités. > Génération d’une paire EPR atomique

Génération d’une paire EPR atomique

We create an entangled atomic pair of the EPR type. The two atoms never interact directly. The entanglement is mediated by their successive interactions with the cavity mode.

The experiment [1] uses two atoms crossing the cavity successively. The first, $A_1$ , is prepared in $e$ and interacts with the cavity for a $\pi/2$ quantum Rabi pulse, creating the atom-cavity entangled state $|e,0\rangle+|g,1\rangle$ (we omit trivial normalizations).

This atom-cavity entanglement cannot be directly probed, since we have no access to the cavity field. We copy instead the cavity state onto a second atom $A_2$ . Prepared in $g$ , it interacts with the cavity mode for a $\pi$ quantum Rabi pulse. It copies the cavity state and hence its entanglement with the first atom. The final two-atom entangled state is

$|\Psi_{EPR}\rangle=\frac{1}{\sqrt 2}(|e,g\rangle-|g,e\rangle)\ ,$

and the cavity ends up in its initial vacuum state. It plays here the role of a catalyst of the entanglement between the two atoms.

In terms of the fictitious spins associated to the two-level atoms, the state $|\Psi_{EPR}\rangle$ is the rotation invariant spin-singlet state. The two atomic spins should always been found pointing in opposite directions, whatever the detection axis chosen for them. A check of this anti-correlation along two orthogonal detection directions (corresponding to non-orthogonal bases) is necessary to check the atom-atom entanglement.

Longitudinal entanglement test

We first perform the trivial check of the atomic energies anti-correlation, corresponding to a detection of both spins along the $Oz$ axis. We expect to find the atoms either in $e,g$ or $g,e$ configurations, with equal probabilities, but never in $e,e$ or $g,g$ . The table below summarizes the expected and observed probabilities.

The main channels correspond to the expected ones. We nevetheless observe that a significant population is lacking in the $g,e$ channel, most of it being found in $g,g$ . This is a direct consequence of cavity relaxation. The $g,e$ channel corresponds to a situation in which the atomic excitation is transiently stored in the cavity mode, prone to relaxation. When the photon is lost between the two atom-cavity interactions, the second atom remains in $g$ , populating spuriously the $g,g$ channel. The observed probabilities are in good agreement with our expectations. Note that this early experiment has been performed with a small cavity damping time (160 $\mu$ s only).

Transverse entanglement test

In a second experiment, we test the spin along detection axes in the equatorial plane of the Bloch sphere. For this purpose, the atoms experience two independent $\pi/2$ classical pulses in $R_2$ before being detected. Adjusting the phase of these pulses, we can adress any detection direction in the equatorial plane.

Instead of using the same axis for the two atoms, we measure the correlation :

$S_{b,\phi}=\langle \sigma_{x,1}\sigma_{phi,2}\rangle\ ,$

where the $\sigma$ ’s are Pauli matrices. This signal corresponds to a detection of the spin of $A_1$ along the $Ox$ axis, and along an axis at an angle $\phi$ with $Ox$ for $A_2$ . This correlation signal is at the heart of the Bell inequality tests.

We expect $S_b$ to be $-1$ when $\phi=0$ , since the detection outcomes on the same axis are always anti-correlated, and $S_b=1$ when $\phi=\pi$ . The correlation signal is ideally expected to vary sinusoidally between -1 and 1.

The figure above presents the observed correlation signal (dots) with a sine fit (solid line). The non-vanishing correlation proves that we do not prepare a statistical mixture of states, but an imperfect quantum superposition.

The fidelity of the state preparation is obviously limited. The main imperfections are the cavity damping and the quality of the analyzing Ramsey pulses. The limited contrast of this early experiment is not large enough for a direct atomic test of the Bell inequality violation. It nevertheless shows that the quantum Rabi oscillation is a powerful means to entangle two atoms.

[1] E. Hagley, X. Maître, G. Nogues, C. Wunderlich, M. Brune, J.M. Raimond, S. Haroche, Phys. Rev. Lett. 79, 1 (1997) : "Generation of Einstein-Podolsky-Rosen pairs of atoms"

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