Electrodynamique des systèmes simples - Lab. Kastler Brossel - Intrication par collision coherente dans la cavité

Aller au menu
Aller à la recherche
Aller au contenu
Aller à la navigation

Cavity quantum electrodynamics - Laboratoire Kastler Brossel

Partenaires

Annuaire

Rechercher

Accueil du site > Atomes, cavités et photons > Information quantique avec des atomes et des cavités. > Intrication par collision coherente dans la cavité

Intrication par collision coherente dans la cavité

Quantum gates based on the resonant atom-field interaction are limited by the cavity losses. We present here a direct entanglement generation between two atoms. The atoms interact simultaneously with the detuned cavity mode. The mode enhances the van der Waals interaction between the atoms and results in a coherent cavity-mediated ’collision’ generating entanglement. The scheme is nearly insensitive to cavity losses.

Atom-atom entanglement in free space

Atom-atom entanglement can, in principle, be created without a cavity, in free space, during a ’collision’ between two Rydberg atoms, initially prepared in the state $|e,g\rangle$ . The two atoms move at a constant velocity $v$ along straight line trajectories, with a minimal inter-atomic distance $b$ (impact parameter). Note that the interaction of the atoms for a medium range $b$ value is much lower than their kinetic energies at thermal velocities. The deflection of the atomic trajectories due to the interaction is thus negligible.

The intial state is degenerate with $|g,e\rangle$ and these two states are coupled via the exchange of a virtual photon, ’emitted’ by the first atom and ’absorbed’ by the second. The dipole-dipole or van der Waals interaction in free space thus creates a two-atom entangled state as the output of the collision process, of the form :

$\cos\theta_0|e,g\rangle+\sin\theta_0 e^{i\Phi}|g,e\rangle\ .$

The mixing angle $\theta$ can be easily estimated. The van der Waals coupling varies as $1/r^3$ , where $r$ is the interatomic distance. It is thus only important when $r$ is near its minimum value $b$ . We thus get :

$\theta_0\sim\alpha\frac{c}{v}\left(\frac{a_0n^2}{b}\right)^2\ ,$

where $\alpha$ is the fine structure constant, $c$ the speed of light, $a_0$ the Bohr radius and $n$ the principal quantum number of state $e$ . The mixing angle is thus a simple combination of the geometric size of the atom (compared to the impact parameter) with the ratio $c/v$ .

A maximally entangled state is reached when $\theta_0=\pi/4$ . For thermal velocities and $n=51$ , the required impact parameter is about 10 $\mu$ m. This is clearly much larger than the atomic size (0.2 $\mu$ m), validating the dipole approximation for the interatomic interaction. This large distance illustrates the large dipoles of the Rydberg states.

The dipole-dipole interaction has thus been proposed as a mechanism for an efficient quantum gate between two cold atoms. However, the interatomic distance can only been accurately controlled with cold atoms, a rather delicate situation. It is thus not an easy task to produce entanglement in free space. Experiments are in progress to implement this mechanism, using for instance individual atoms in optical tweezers. A more straightforward route to the interatomic entanglement is to return to the cavity context.

A cavity-induced collision

The two atoms are now simultaneously coupled to the cavity mode, detuned from the atomic transition by a large amount $\delta$ . There are thus no direct atom-cavity energy exchanges. The virtual photon exchange that mediates the van der Waals interaction can nevertheless take advantage of the coupling with the empty cavity mode.

The transition from $|e,g,0\rangle$ to the final state $|g,e,0\rangle$ (the last symbol in the ket gives the photon number in the cavity) thus occurs through a virtual transition to the intermediate $|g,g,1\rangle$ state, detuned by $\delta$ from the degenerate intial and final states.

The mixing angle can be estimated in these conditions. We get :

$\theta_c=\alpha\frac{\omega}{\delta}\frac{c}{v}\left(\frac{a_0n^2}{b_c}\right)^2\ ,$

where $v$ is the common velocity of the two atoms crossing the cavity mode, $\omega$ the atomic transition frequency and $b_c$ an effective ’impact parameter’, of the order of the cavity dimensions, given by :

$b_c=\left(Lw/\sqrt{2\pi}\right)^{1/2}$

( $L$ cavity length, $w$ mode waist).

The mixing angle is thus that of a collsion in free space at millimeter-sized distance, multiplied by a factor $\omega/\delta$ . Since the cavity mode has a relatively small spectral width, this factor can be huge in the non-resonant regime. The atom-atom van der Waals coupling is thus greatly enhanced by the detuned cavity mode.

The main asset of this entangling interaction is that it is, to first order, insensitive to the cavity losses, since the photon emission is only virtual. In loose terms, the photon is stored in the cavity for a time interval of the order of $1/\delta$ only. Thus, $\delta$ should be much larger than the cavity width. The enhancement factor $\eta=\omega/\delta$ can be at most of the order of the cavity quality factor.

Another asset is that the mixing angle is also insensitive to the initial cavity state. When the cavity contains $N$ photons, two quantum processes contribute to the coupling of $|e,g,N\rangle$ with $|g,e,N\rangle$ , one in which a photon is virtually emitted by the first atom as above, the other in wich the second atom first absorbs a photon. These two proceses interfere and the final mixing angle is found to be independent upon $N$ . This scheme, proposed for the first time by Zheng and Guo [1], is thus highly promising for experiments with moderate finesse cavities at finite temperatures.

Experimental implementation

We prepare two atoms in the states $e$ and $g$ [2]. Prepared at different times, these atoms have slightly different velocities, so that they cross the cavity axis at the same time and are finally separately detected. They thus interact nearly simultaneously with the cavity mode.

We first measure the final populations of the four possible final states ( $e,e$ , $e,g$ , $g,e$ and $g,g$ ) as a function of the atom-cavity detuning measured by the enhancement factor $\eta=\omega/\delta$ (we have to take care in the data analyzis that the atoms are dispersively coupled to the two non-degenerate cavity mode).

This figure presents (dots) the measured populations as a function of $\eta$ . We clearly observe the quantum Rabi oscillation between $|e,g\rangle$ (blue) and $|g,e\rangle$ (red). The spurious channels (magenta) have a small $<10$ % population. The lines are fits on the simple model above in its validity range (solid lines) and on a numerical simulation of the experiment (dotted lines).

In order to check the coherence of the process, we have tested, as in the EPR pair experiment, the transverse correlations of the atom-atom entangled state produced at $\theta=\pi/4$ (crossing of the blue and red curves above). The two atoms undergo two separate $\pi/2$ pulses in $R_2$ after their common interaction with the mode before being detected. We reconstruct a ’Bell signal’ as a function of the relative phase $\phi$ of the two Ramsey pulses. It should oscillate between $+1$ and $-1$ . The experimental signal is plotted above, together with a sine fit. The contrast is limited, mainly by the cross talk between the two Ramsey pulses performed when the atoms are at a small mutual distance. The oscillation nevertheless reveals the quantum coherence of the prepared state.

[1] S. B. Zheng and G. C. Gu, Phys. Rev. Lett. 85, 2392 (2000)

[2] S. Osnaghi, P. Bertet, A. Auffeves, P. Maioli, M. Brune, J.M. Raimond, S. Haroche, Phys. Rev. Lett. 87, 037902 (2001) “Coherent control of an atomic collision in a cavity”

Dans la même rubrique :

Partenaires

Menu

Rechercher

Intrication par collision coherente dans la cavité

Navigation