Electrodynamique des systèmes simples - Lab. Kastler Brossel - Génération pas à pas d’un état intriqué à trois qubits.

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Accueil du site > Atomes, cavités et photons > Information quantique avec des atomes et des cavités. > Génération pas à pas d’un état intriqué à trois qubits.

Génération pas à pas d’un état intriqué à trois qubits.

Using the phase gate, we prepare an entangled state of three qubits (two atoms and the cavity mode). This is a rather complex quantum information procedure, involving up to three one-qubit and three two-qubit quantum gates.

This experiment [1] makes use of three qubits, two atoms, $A_1$ and $A_2$ , and the cavity field $C$ . A third atomic qubit, $A_3$ , is used later for the cavity-state readout.

Preparation of the entangled state

The atom $A_1$ is prepared in $e$ and crosses the initially empty cavity. Tuned at resonance for a $\pi/2$ quantum Rabi pulse, it prepares the atom-cavity entangled state

$\frac{1}{\sqrt 2}(|e_1,0_c\rangle+|g_1,1_c\rangle)\ ,$

where subscripts are used to clearly identify the qubits carriers. This is also the first step in the EPR atomic pair generation. Here, instead of merely copying the cavity state onto a second atom, we operate the quantum phase gate between the cavity and $A_2$ .

The second atom $A_2$ is initially prepared in a coherent superposition of $i$ and $g$ by a $\pi/2$ pulse in $R_1$ , resonant on the $i/g$ transition. It then completes the quantum phase gate operation with the cavity. This entangles $A_2$ with $C$ , and hence with $A_1$ also and we end up in a three-qubit entangled state :

$|\Psi_{GHZ}\rangle=\frac {1}{2}[|e_1,0_c\rangle(|i_2\rangle+|g_2\rangle)+|g_1,1_c\rangle(|i_2\rangle-|g_2\rangle)]\ ,$

which can also be written as :

$|\Psi_{GHZ}\rangle=\frac {1}{2}[|i_2\rangle}(|e_1,0_c\rangle+|g_1,1_c\rangle)+|g_2\rangle}(|e_1,0_c\rangle-|g_1,1_c\rangle)]\ ,$

showing that, now, the phase of the $A_1-C$ EPR pair is controlled by the state of atom $A_2$ .

This is clearly a maximally entangled state of the three quantum systems. It can be cast in the more familiar form :

$|\Psi_{GHZ}\rangle=\frac{1}{\sqrt 2}(|0_1,0_2,0_c\rangle+}(|1_1,1_2,1_c\rangle)\ ,$

by defining

$|0(1)_1 \rangle=|e(g)_1\rangle\quad|0(1)_c\rangle=|0(1)\rangle$

and

$|0(1)_2\rangle=\frac{1}{\sqrt 2}(|g_2\rangle+(-)|i_2\rangle\ .$

This is thus a Greenberger-Horne and Zeilinger state of the three qubits. These states present interesting non-local properties, since they could, in principle, lead to a discrimination between quantum mechanics and local hidden variable theories based on a single ideal experiment.

As for the atomic EPR pair, a demonstration of the entanglement requires measurements of the three qubits in two sets of non-commuting bases. To perform these measurements, we copy the state of the cavity $C$ onto a third atom $A_3$ , via a $\pi$ quantum Rabi pulse.

Due to severe timing requirements (this experiment has been performed with a ring surrounding the cavity), this state-copy operation is performed after the coherence of the triplet has been lost due to $A_1$ ’s crossing of the stray field region in the ring exit hole. We prepare at a given time the two-atom-cavity entangled state, but we never have the three-atoms GHZ state at hand. The final detection results are, however, exactly the same as if this state was effectively available for a while.

The three atomic states are analyzed along different axes by submitting them finally to selected $\pi/2$ pulses in $R_2$ before detection of their energy.

Longitudinal correlation experiment

In a first experiment, we directly detect the three qubits in the $0/1$ basis. $A_1$ and $C$ (i.e. $A_3$ ) are directly detected in their energy basis. $A_2$ undergoes a $\pi/2$ pulse that maps $|0(1)_2\rangle$ onto $i_2$ and $g_2$ . At the end of the sequence, we should ideally detect only the two combinations $g_1,g_2,e_3$ or $e_1,i_2,g_3$ with equal 50% probabilities.

This histogram shows the probabilities of the eight possible detection channels. The expected ones (in green) are dominant. Spurious processes (and, in particular, cavity relaxation) weakly populate the other channels.

Transverse correlation experiment

In a second experiment, we have checked the phase correlation in a transverse basis for the three qubits. We now apply $\pi/2$ pulses on $A_1$ and $A_2$ (with an adjustable relative phase $\phi$ ), none on $A_3$ .

This experiment tests in fact an expression of the GHZ states given above, which shows that, in a proper basis, the GHZ state describes an $A_1-C$ (or equivalently $A_1-A_3$ ) EPR pair whose phase is correlated to the state of $A_2$ .

We derive from the experiment the Bell signal $S_b$ (see the section on the EPR atomic pair) revealing the coherence of the $A_1-A_3$ pair). It is plotted as a function of $\phi$ in the figure above, when $A_2$ is not sent in the apparatus (green dots and sinusoidal line fit). It oscillates, clearly demonstrating the $A_1-A_3$ entanglement in this simple replication of the EPR pair experiment.

When the gate atom $A_2$ is sent in the apparatus, and detected in its neutral state $i_2$ , the Bell signal (shown in blue above) is not appreciably modified. On the contrary, when this atom is detected in its active $g_2$ state (red dots), the phase of the Bell signal is reversed, demonstrating the transverse entanglement of the GHZ state.

Estimate of the state preparation fidelity

A more quantitative estimate of the state preparation fidelity can be obtained from these measurements. Assuming that imperfections do not create spurious coherences (a quite reasonable assumption), the state fidelity is given by the contrast of the Bell signals in the transverse experiment and the total population of the expected channels in the longitudinal experiment. From experimental data, we obtain a raw fidelity of 0.43. Taking into account the measured imperfections of the analyzing Ramsey pulses, we infer a state preparation fidelity :

$F=0.54\pm 0.03\ ,$

which clearly shows that we indeed prepared a three-qubit entangled state. This experiment, published in 2000, has been for a time the most complex entanglement generation sequence performed on individually addressed qubits. Ion traps make it possible now to realize somewhat more complex sequences.

Generalization of the procedure

At least in principle, a whole sequence of quantum gate atoms $A_2, A'_2....$ could be sent trough the cavity before its state read out. This sequence would prepare a multi-atom generalization of the GHZ state, a so-called Mermin state.

The experimental limitations did not allow us to go farther than the three qubit entanglement reported here. Cavity relaxation and the complex timing imposed by the cavity photon recirculation ring made it impossible to smuggle in another atom. More importantly, the limited fidelity of the individual transformations would have made the final preparation fidelity rather low.

The state-of-the-art cavity with a very long damping time and an open structure, with no limitations on the transport of atomic coherences, could make it possible to generalize the sequence to much more complex state. The QND single-photon detection process is a first indication since it (classically) entangles a few hundreds of atoms with a a single control photonic qubit.

[1] A. Rauschenbeutel, G. Nogues, S. Osnaghi, P. Bertet, M. Brune, J.M. Raimond, S. Haroche, Science, 288, 2024 (2000) : "Step by step engineered many particle entanglement"

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