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Accueil du site > Atomes, cavités et photons > Information quantique avec des atomes et des cavités. > Trois points pour ’tricoter’ un état intriqué

Trois points pour ’tricoter’ un état intriqué

The resonant quantum Rabi oscillation is at the heart of quantum information processing with atoms and cavities. Selected interaction times provide the basic stitches that create and process atom-field entanglement. The stitches can be knitted in more complex sequences for the generation of multi-qubit entangled states.

This figure presents the first period of the quantum Rabi oscillation signal. Let us recall that the atom enters the empty cavity in the upper state $e$. The probability for finding it in the same state $e$ after the effective interaction time $t_r$ is then an oscillatory function.

At most times in this evolution, the atom and the cavity are in an entangled state, quantum superposition of $|e,0\rangle$ (initial state) and $|g,1\rangle$ . Some specific interaction times realize interesting operations, which are the basic stitches of our quantum information experiments. These times are highlighted on the figure above.

$\pi/2$ quantum Rabi pulse : entanglement generation

An effective interaction time of a quarter of period of the vacuum Rabi oscillation ( $\pi/2$ pulse) performs the transformation :

$|e,0\rangle\rightarrow\frac{1}{\sqrt 2}(|e,0\rangle+|g,1\rangle\ .$

The resulting state is a maximally entangled Bell state of the atom and of the cavity mode, here considered as two qubits (the cavity mode evolves in these experiments only in the $|0\rangle,|1\rangle$ subspace). This stitch efficiently creates atom-field entanglement. This entanglement will survive as long as the shortest lived qubit (cavity mode in most experiments so far), much longer than the mere 5 $\mu$ s needed to generate the entangled state.

$\pi$ quantum Rabi pulse : qubit transfer

An effective interaction time corresponding to a half period of the quantum Rabi oscillation ( $\pi$ pulse) produces the transformations :

$|e,0\rangle\rightarrow|g,1\rangle\ ,$

and

$|g,1\rangle\rightarrow-|e,0\rangle\ ,$

the minus sign in the latter transformation resulting from the unitarity of the evolution. It is not a surprise that the atom and the cavity exchange a quantum of energy in the process.

The situation is more interesting when the atom is initially in a superposition of its state. The transformation then reads :

In quantum information terms, the qubit carried by the atom has been copied onto the cavity state. This operation thus allow us to prepare the cavity in an arbitary quantum state (the input atomic state being in turn generated in the Ramsey zone $R_1$ before interaction with the cavity).

This operation is obviously reversible. An atom entering in $g$ in the cavity and undergoing the same $\pi$ pulse leads to the transformation :

Within a sign change, the cavity state is faithfully copied onto the atom. This allows us to extract quantum information from the cavity (once again the cavity photons cannot be directly detected - all information we get on them has to be carried away by atoms). These qubit copy information play a very important role in all our experiments, and in particular in the quantum memory operation.

$2\pi$ quantum Rabi pulse : conditional logic

The full period of the quantum Rabi oscillation can be used to implement a conditional quantum dynamics, at the heart of our quantum phase gate. The atom-field system then undergoes the transformations :

$|e,0\rangle\rightarrow-|e,0\rangle\qquad|g,1\rangle\rightarrow-|g,1\rangle\ .$

Note that $|g,0\rangle$ does not evolve in the same time, since there is no energy to exchange between the atom and the field.

The energy of both system returns to its initial value, but the quantum state accumulates an overall $\pi$ phase shift, reminiscent of the $\pi$ phase shift for a spin state undergoing a $2\pi$ rotation in real space.

This phase shift can easily be turned into a conditional logic operation. Let us code the atomic qubit on the levels $g$ and $i$ ( $i$ being the circular state with a principal quantum number 49). Since the $i\rightarrow g$ transition is far off resonance, the $|i,n\rangle$ state ( $n=0,1$ ) does not evolve. The same applies to $|g,0\rangle$ . Finally, there is an evolution (a global $\pi$ phase shift) only when the atom is initially in $g$ and when the cavity contains one photon. This is the qubit-qubit conditional dynamics of a quantum phase gate, the non-trivial building block of a quantum information network.

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