Rydberg blocking and quantum gates

The fourth lesson described experiments exploiting the Rydberg blockade, mentioned in the first lesson, to achieve entanglement and quantum gating between atoms. An essential prerequisite for these experiments is the ability to precisely control the position of the atoms. We began by recalling the general methods for manipulating laser-cooled atoms, which are the necessary first step in all these experiments. The atoms are cooled in optical molasses or in a magneto-optical trap (MOT) set up with resonant or quasi-resonant laser beams on a transition between the fundamental level of the atoms and an optical resonance level. A configuration formed by pairs of counter-propagating lasers in all three spatial directions is chosen to confine the atoms in all directions. In the case of the MOT trap, magnetic field gradients are added to ensure spatial confinement of the atoms in a relatively large volume. Atoms are detected by the fluorescence light they emit, at a frequency close to that of lasers, in the optical molasses or in the MOT.

In order to isolate only two atoms at a well-defined distance from each other, two "optical tweezers" are superimposed on the molasses or MOT, consisting of two highly focused laser beams detuned towards the red of the atoms' optical transition. This creates two very deep optical traps, exploiting the dipolar dispersive force exerted on the atoms by the gradients of non-resonant light intensity. Some of the atoms in the MOT or molasses fall into these traps, at random. If more than one atom falls into the trap, the collisions assisted by the resonant detection laser light very quickly carry them into non-trapping levels, and they are excluded from the trap. In the presence of detection light, the trap can only contain zero or one atom. This light, originating from the MOT or molasses, can be switched off once the atoms have been captured, and relit at will to detect them at a later date. In this way, we can isolate random events in which no atoms are trapped, those in which only one of the two traps is powered, and finally those in which each trap contains an atom - a situation required to operate a quantum gate.

Instead of creating a pair of traps, we can also form a whole network of them, using counter-propagating beams of strongly detuned lasers to form an interference pattern with light minima and maxima distributed periodically in space. If the depth of the traps is sufficient, this creates a "Mott insulator" in which the atoms are frozen in the periodic traps, with a well-defined number of atoms depending on the region (in general, in a "circular cake" of atoms, a wedding cake structure is observed, with the same number of atoms per trap in the center, one atom less per trap on a ring around the center, then two atoms less on a second ring and so on). Here again, the detection light that must be switched on to detect the atoms by fluorescence causes pairs of atoms to be ejected, so that traps containing (in the absence of detection light) an even number of atoms appear empty and do not fluoresce, while those initially containing an odd number of atoms appear on detection to contain just one. In other words, optical fluorescence detection measures the parity of the number of atoms per trap. These experiments with optical tweezers or optical trap arrays require high numerical aperture optics, necessary to form micron-sized traps and to be able to separate the fluorescence of atoms placed at a distance of the order of the optical wavelength.

After this general introduction to atomic trapping methods, we turned to a detailed analysis of the seminal papers from the early 2000s that set out the principle of entanglement and quantum gate experiments between two atoms held at a fixed distance from each other. As briefly mentioned in the first lesson, the idea is to play on the fact that two simultaneously excited atoms disturb each other by strongly shifting their energy levels. Depending on whether the laser used to excite the Rydberg states corresponds to a Rabi frequency that is large or small in relation to these frequency shifts, different scenarios of coupling between atoms can be envisaged. The simplest to explain is when the level shift is resolved in the excitation laser spectrum. In this case, the excitation of one atom (which we'll call the "target atom") in the presence of another already excited atom ("control atom") is impossible. This phenomenon, which prevents a second Rydberg atom from being excited in the presence of an already-excited one, is known as "Rydberg blocking". It takes place within a well-defined volume of radius, called the "blocking radius", which depends on the quantum number of the Rydberg state and the Rabi frequency of the laser excitation to that state. Excitation of the target is thus conditional on non-excitation of the control atom. If we encode a qubit in each atom in two sublevels of its ground state, we show that we can thus realize a conditional gate (phase gate or CNOT gate). In the case where the Rabi frequency of the laser exciting the Rydbergs exceeds the level shift due to the interaction between excited atoms, there is no blockage and both atoms can be simultaneously excited. In this case, however, we can play on the phase accumulated in the excited states (which is highly dependent on the position of the atoms) to create a conditional phase gate. This is less practical to use, however, since its characteristics depend very much on the exact geometry of the two-atom system.

The rest of the lesson described experiments demonstrating the effects predicted by these theoretical papers. These experiments, carried out with rubidium atoms by two research groups, one in Palaiseau and the other in Madison (USA), use three sets of lasers: a first set, quasi-resonant with rubidium's main optical resonance transition (in the near infrared), performs cooling and trapping (molasses and MOT) as well as fluorescence detection of the atoms and their optical pumping to a sublevel of the ground state, to enable laser cooling on a closed transition and also prepare the initial state of the two qubits. A second set of intense infrared lasers, strongly detuned from the atomic resonance transition, is used to create the optical tweezers. Finally, a third set of lasers, usually intense blue light, provides two-photon excitation to the Rydberg level (with a principal quantum number of the order of 50 to 100). These lasers can be switched on and off at will in the successive stages of the experiment (system preparation, excitation to the Rydberg state and detection). Without going into detail here, these devices have enabled us to demonstrate precisely the blocking effect (impossibility of exciting an atom in the presence of an already excited atom if the two atoms are close enough) and the possibility of entangling the two atoms and realizing a quantum gate of moderate fidelity between them. Finally, by analyzing in detail the shape of the Rabi oscillation of two atoms as a function of their distance, it has been possible to measure quantitatively the interaction energy of two atoms as a function of their distance, and to recover with good precision the n11/r6 lawof the van der Waals interaction between two Rydberg atoms.