Atoms that interact strongly with each other over long distances

The third lesson looked at the study of the interaction between Rydberg atoms. The excited electrons of two atoms "feel" each other at a great distance via the dipole-dipole interaction. This coupling is analogous to that between two atoms in their ground state at a great distance relative to the size of the atoms ("tail" of the molecular potential in the case of coupling between the two atoms of a diatomic molecule, for example). But whereas in the latter case, the interaction takes place at just a few angstroms, i.e. a few atomic distances, it takes place between Rydberg atoms at quasi-macroscopic distances of several microns, of the order of a hundred times the size of the electron clouds of the excited atoms. It is the wide range and high intensity of these interactions between Rydberg atoms that makes these systems so interesting for experiments in quantum information or non-linear optics.

The lesson began with a reminder of the van der Waals interaction between two hydrogen atoms in their ground state, at a distance r from each other. Each atom carries an electric dipole that is zero on average, but exhibits fluctuations. At zero order, these random fluctuations are uncorrelated on both atoms. They are correlated at order one of the interaction, leading to a 1/r3 correction of the system state of the two atoms and a 1/r6 correction of their energy at order two. The interaction energy is negative, leading to an attractive force in 1/r7 between the two atoms. The situation is different if one of the two atoms is excited, in an optical resonance level, and the other in its ground state. In this case, there are two states of equal energy in the system of the two atoms, corresponding to an exchange of excitation between them. The dipole-dipole interaction then acts on the system's energies to first order, leading to a lifting of the degeneracy in 1/r3. If one of the atoms is initially excited and the other in its ground state, and if their distance is fixed, this lifting of degeneracy corresponds to an oscillation between the two atoms, which periodically exchange their excitation with a period proportional to r3.

These two situations (interactions in 1/r3 and 1/r6) are found in the case of interaction between two Rydberg atoms, albeit with very different orders of magnitude. If the two atoms are initially in the same quantum state, and if there are no degenerate Rydberg level pairs with the combined state of the two atoms, their coupling is second-order, but in n11/r6. The enormous amplification factor n11 compared to the coupling of two atoms in the ground state is due, on the one hand, to the large size of the electric dipole matrix elements involved, by the square of their product (factor n8), and on the other hand to the close proximity in energy of pairs of atomic levels very close to the initial state (the inverse of the energy difference varying in n3). If, due to the particular values of the quantum defects, there accidentally exists a couple of levels degenerate with the initial state of the two atoms (or different in energy from this state by an amount smaller than the dipole-dipole matrix element that couples them), the interaction between the atoms acts to first order on the energies, leading to a lifting of the degeneracy and a resonant exchange of excitation in n4/r3 - again a strongly exalted effect (factor in n4) compared to what it is for two weakly excited atoms.

The existence of a pair of levels degenerate with respect to the initial state and the corresponding periodic exchange of energy (in the case of two atoms at a fixed distance) is called a Förster resonance. This can either occur accidentally (coincidence of levels linked to the specific values of quantum defects) or be induced by the application of an electric field bringing the pairs of levels into resonance by playing on the difference in their Stark effects. All the above analysis is valid only for an interatomic distance that is neither too small (the electron orbitals must not penetrate each other, i.e. r must be greater than a value proportional to n2), nor too large (radiation propagation delay effects must be negligible for the electrostatic approximation implicitly admitted above to be valid). This last condition implies that r must be less than the characteristic microwave wavelengths that Rydberg atoms are likely to emit or absorb (which are proportional to n3). Finally, r must be of the order of a fraction of a micron to around 10 to 100 microns for n of the order of 50 to 100.

The interaction effects between Rydberg atoms were first observed in gases or jets of atoms at ordinary temperature, moving at speeds of the order of several hundred meters per second, so that the effects in 1/r3 or 1/r6 were averaged by the motion of the atoms and the observations relatively qualitative. The lesson described some of these experiments carried out in the 1980s. A Förster resonance experiment on an atomic jet of sodium atoms demonstrated the resonant exchange of excitation between atoms, initially both in the same |ns>state, when these atoms were brought into the pair of |n'p, n''p>states degenerated from |ns, ns>by the application of an electric field. By measuring the rate of transfer to the p states as a function of the electric field and the density of the excited jet of atoms, we were able to deduce the effective cross-section of the resonant transfer and show that it was indeed explained by a Förster interaction between the two atoms. In another experiment, we measured the width of the excitation laser lines of cesium Rydberg atoms in a high-density jet and observed a broadening of the lines, which can be explained by the simultaneous two-photon excitation of pairs of atoms whose energy levels are displaced by the quasi-resonant interaction in n4/r3.

We concluded the lesson by studying a different, albeit related, situation: the interaction of a Rydberg atom with its images in a cavity formed by two parallel mirrors, as it passes at equal distance d between the two surfaces. The fluctuations of the electric dipoles of the atom and its images are then automatically correlated, so that the interaction modifies the energies of the Rydberg states to first order, by a quantity in n4/d3. We have described a laser spectroscopy experiment of Rydberg atoms between mirrors, carried out in the early 1990s, which quantitatively measured this level-shifting effect for sodium atoms with n varying from 10 to 13 and d from 0.3 to 0.7 microns. The agreement between the experiment and theoretical predictions was excellent.