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Accueil du site > Atomes, cavités et photons > Information quantique avec des atomes et des cavités. > Une porte quantique atome/cavité

Une porte quantique atome/cavité

The quantum Rabi oscillation, for a full period of the coherent atom-cavity energy exchange, realizes a quantum phase gate, which can be easily turned into a CNOT gate by means of additional Ramsey pulses. Ths gate can in principle be used to prepare complex entangled states. It also leads to a quantum non demolition measurement of a single photon.

The EPR pair generation procedure, using the cavity as an entanglement catalyst, is obviously limited to two atoms. To knit more complex entangled states, we need a full-fledged universal quantum gate. The full $2\pi$ quantum Rabi oscillation provides naturally the mechanism of a phase gate.

Principle of operation

In this experiment [1] the control qubit is the cavity field, in a superposition of its $|1\rangle$ and $|0\rangle$ states, and the target is an atom. The atomic qubit is coded on the levels $g$ (logical value $1$ ) and $i$ , coding for zero. The level $i$ is the circular state with a principal qantum number 49. It is widely off resonance from the cavity mode and is thus not affected at all by the cavity field.

An atom in level $g$ entering the empty cavity does not evolve, since there is no energy to exchange between the atom and the field. When the atom-cavity interaction time is tuned for a $2\pî$ quantum Rabi pulse, an atom in $g$ entering the cavity with one photon undergoes a full Rabi swing, going transiently to $e$ and returning to $g$ at the end of the interaction. In the process, the atom-cavity state accumulates a $\pi$ phase shift.

Finally the ``truth table" of the atom-cavity interaction is :

$|i,0\rangle\rightarrow|i,0\rangle\quad|i,1\rangle\rightarrow|i,1\rangle$

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

This is precisely that of a quantum phase gate, in which the two qubit state undergoes a sign change when and only when the two qubits are in their logical state $1$ .

We have tested two complementary aspects of the coherent gate operation. In a first experiment, we have checked that the cavity qubit in one of its logical state acts as predicted on a superposition of atomic states, the phase of the $i/g$ coherence being shifted by $\pi$ when the cavity contains a photon. In a second test, we have checked the complementary effect in which the phase of a $0/1$ field coherence is shifted by $\pi$ by an atom in $g$ .

Operation on an atomic coherence

In this experiment, the cavity qubit is either in its ground state, $|0\rangle$ , or prepared in state $|1\rangle$ by a preparation atom, entering the cavity in $e$ and undergoing a $\pi$ quantum Rabi pulse.

The atomic qubit is first prepared in $R_1$ (tuned in this experiment on the $i/g$ transition) into a coherent superposition $(|i\rangle+|g\rangle)$ . After interaction with the cavity mode, the atomic coherence is probed by another $\pi/2$ pulse in $R_2$ . Scanning the relative phase of the $R_1$ and $R_2$ pulses, we observe Ramsey fringes.

The gate operation appears as a $\pi$ phase shift of the Ramsey fringes, conditioned to the presence of a single photon in $C$ . The figure above presents the experimental fringes when the cavity is empty (blue squares) or contains one photon (magenta diamonds). The two fringe patterns are clearly in phase opposition, revealing the conditional phase-shift of the atomic coherence. Noticeably, the contrast of the fringes, mainly limited by the quality of the Ramsey pulses, is not altered by the full Rabi swing undergone by the atom.

Application : QND detection of a single photon

Quite remarkably, this phase gate leads naturally to a quantum non demolition (QND) measurement of a single photon stored in the cavity [2]. When the Ramsey interferometer phase is set at a fringe extremum, the atomic state, at least in an ideal experiment, should be directly correlated to the photon number (0 or 1). For instance, for a proper phase, the atom should be certainly found in $i$ when the cavity is empty and in $g$ when it contains a photon.

In other words, the atomic state tells out the photon number in the cavity (0 or 1). This is a photodetection process, but it is very different from usual ones. In most photodetectors, the photon is destroyed while being detected. In the present situation the photon is still there after the measurement, ready to be measured again. This is a Quantum Non Demolition (QND) detection, implementing in this simple case the ideal detection process of quantum mechanics textbooks.

The figure above shows the operation of this QND procedure. We start with an initial thermal field containining on the average 0.3 photons. We measure this field with an atom, initially in $g$ and undergoing a $\pi$ quantum Rabi pulse. In this ordinary, absorptive detection process, the final probability for having the probe atom in $e$ measures the probability for having a photon in the cavity (red triangles).

Before the absorptive measurement, we insert an atom performing the QND field detection and we plot the final probability for having a photon in the cavity (measured by the absorbing atom) as a function of the Ramsey interferometer setting. This probability is clearly modulated. This modulation reveals a measurement-induced modification of the field.

When the QND atom is found in $g$ and when the Ramsey phase is such that $g$ correlates to one photon (summits of blue curve), it is more likely to find a photon in the cavity than before the measurement. Having more photons after the measurement than before is an unusual situation, clearly demonstrating the QND nature of the process.

When the interferometer phase is such that $i$ and $g$ are detected with equal probabilities whatever the photon number, the QND process does not bring any information and the final photon number is unchanged.

We have performed a thorough analyzis of the imperfections of the method. There is a 20% probability that the photon is spuriously absorbed in the $2\pi$ Rabi rotation and a similar probability for erroneous photon number measurement if it is not the case.

The spurious absorption rate, due to the imperfections of the $2\pi$ quantum Rabi pulse, limits the repeatability of the QND measurement to one or two successive measurements, while it should in principle be possible to perform many measurements, all giving the same outcome.

Since this early experiment, we made considerable progresses. The dispersive atom-field interaction allows now for a nearly ideal QND measurement of a single photon and for many repeated measurements. We have recently observed in this way the quantum jumps of light for the first time.

Operation on a field coherence

We have also tested the action of our phase gate on a field state superposition. As a reasonable approximation of a $0/1$ superposition, we prepare in the cavity a coherent state with a small amplitude $|\alpha\rangle$ (about 0.2 photons on the average). Without loss of generality, we assume here that $\alpha$ is real. Whe send then an atom in the cavity, tuned for the $2\pi$ quantum Rabi pulse.

When the atom is in $i$ , the field is unchanged. We perform a second field injection, with an amplitude $\alpha \exp(i\phi)$ . The resulting field amplitude, $\alpha(1+\exp(i\phi))$ should cancel for $\phi=\pm\pi$ . We observe this cancellation by probing the cavity field with an atom prepared in $g$ and undergoing a $\pi$ quantum Rabi pulse.

The probability for detecting the atom in $e$ , revealing the final photon number, is plotted below (blue dots on left, the line is a theoretical fit) as a function of $\phi$ . We observe a near cancellation of the field at $\phi=\pm\pi$ , as expected.

When the gate atom has crossed the cavity, it phase-shfits by $\pi$ the $|g,1\rangle$ state. This corresponds to a $\pi$ phase shift of the classical coherent field amplitude, when neglecting the many-photon states contributions : $|\alpha\rangle\rightarrow|-\alpha\rangle$ . The final probability for getting a photon in the field (and, accordingly, for detecting the probe atom in $e$ ) must now have a minimum for $\phi=0$ . This is effectively observed (figure above, red dots on the right). This experiment thus tests another aspect of the coherent quantum phase gate operation.

Tuning of the phase rotation

At exact resonance, for a $2\pi$ quantum Rabi pulse, the phase accumulated by the $|g,1\rangle$ state is exactly $\pi$ . This phase can be tuned by changing the atom-cavity resonance condition.

When the atom and the cavity are non resonant, the exact $2\pi$ pulse condition is no longer met. Numerical simulations show, however, that the $|g,1\rangle$ state almost exactly returns to the initial condition, with an extra phase factor that varies over $2\pi$ when the atom-cavity detuning $\delta$ is tuned across resonance. The adiabatic variation of the atom-field coupling plays a major role again here. The worst spurious transfer from $g$ to $e$ is below 3%, much lower than other experimental imperfections.

This curve shows the accumulated phase as a function of $\delta$ . We do observe that it can be tuned over a wide range, offering additional flexibility for quantum information manipulations in this cavity QED system.

[1] A. Rauschenbeutel, G. Nogues, S. Osnaghi, P. Bertet, M. Brune, J.M. Raimond et S. Haroche, Phys. Rev. Lett., 83, 5166 (1999) : "Coherent operation of a tunable quantum phase gate in Cavity QED"

[2] G. Nogues, A. Rauschenbeutel, S. Osnaghi, M. Brune, J.M. Raimond, S. Haroche, Nature, 400, 239 (1999) : "Seeing a single photon without destroying it"

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