Rydberg blocking induces strong coupling between photons

The sixth and final lesson presented an overview of the giant non-linear optical effects produced by the Rydberg blocking effect for optical fields propagating in a dense ensemble of atoms. The effect is based on the phenomenon of electromagnetically inducedtransparency (EIT). We began by recalling what normal EIT is (i.e. for a gas of atoms evolving between moderately excited levels), before looking at its effects when one of the levels concerned is a highly excited Rydberg state.

An atomic medium strongly absorbing a probe beam resonating on an optical transition between a ground state |b> and an excited state |e> becomes transparent to this probe in the presence of a pump beam (or "control beam") satisfying a two-photon resonance condition towards a third level |a> (the three levels |b>, |a> and |e> forming a Λ or step configuration). By quantum interference, the |e> state becomes a "black state" uncoupled from the radiation. The effect occurs in a narrow window of frequency transparency around the resonance condition. The probe photons couple strongly to an atomic "spin wave" carrying the atoms in the medium into a superposition of the |a> and |b> states. The probe coupled to the spin wave becomes a "darkstate polariton" wave, propagating through the medium with a group velocity much lower than in vacuum ("slow light"). This effect is due to a very rapid variation over the width of the transparency window in the refractive index seen by the probe.

A similar dispersive effect occurs for a probe not resonating with the |b>-|e> transition, when probe and pump satisfy a two-photon resonance condition on the |b>-|a> transition. The (real) refractive index seen by the probe becomes equal to 1 within a narrow non-refraction window, with the non-refracted probe propagating slowly through the medium as a "black-state polariton". We have described in detail these electromagnetically induced transparency effects, which have been the subject of numerous theoretical and experimental studies over the last twenty years. These effects are often realized in ensembles of cold atoms. The very low velocity of the atoms means that the Doppler shifts to which the black-state polariton condition is very sensitive are negligible. What's more, despite their low absolute density, these gases can have a high optical density (in the absence of a control beam), which is a necessary condition for observing TEI. The concept of the "black-state polariton" associated with TEI is very similar to that described in the previous lesson. This polariton describes the propagation of a mixed atomic and photonic wave in the medium, and the slow light corresponds to the almost complete adiabatic transformation of the light wave into an atomic spin wave once the light has penetrated the optically dense atomic medium.

We then studied what happens when the |a> level is a Rydberg state (in a step confguration |b>, |e>, |a>). The TEI phenomenon can occur for the propagation of a single-photon wave packet, as there is then only one excited atom in the |a> state involved. Let's suppose that, in the presence of the control field, the one-photon probe packet propagates as slow light in a medium with the shape of a very thin cylindrical brush, and that an atom in a Rydberg state is excited in its midst (using a transverse laser beam). Rydberg blocking will have a spectacular effect on the electromagnetically induced transparency along the cylindrical medium. As the "one-photon polariton" approaches the excited atom, the polariton is destroyed, as two Rydberg excitations cannot coexist within a blocking radius. The medium then becomes opaque to the absorbed photon, as the |e> state is no longer a black state and rapidly scatters the photon. If the blocking atom is prepared in a superposition of Rydberg and ground states, a quantum gate is created which, with a certain probability amplitude, blocks the photon, and with a complementary amplitude lets it through.

Analogous phenomena are expected when the medium is non-resonant for the photon. In the presence of the control field, the photon propagates without dispersion at slow group velocity until it encounters the "blocking" atom. As it approaches this atom, it "sees" a "blade" whose thickness is of the order of twice the blocking radius. The non-dispersion condition is then destroyed, and the photon's passage through this "blade" is accompanied by a phase shift. This phase shift depends on the state of the blocking atom. If the blocking atom is in a superposition of its Rydberg and ground states, an entanglement situation is created, superimposing a state in which a phase-shifted photon is associated with the Rydberg state of the "blocking" atom, and a non-phase-shifted photon with the blocking atom in its ground state.

To our knowledge, these situations have not been realized in this ideal form, but experiments have recently demonstrated the interaction effects between photons in a cylindrical medium subjected to TEI. The lesson briefly described one such experiment, carried out at MIT. The first step was to observe the transmission peak electromagnetically induced by a control field as a function of probe field intensity, in the case of a step configuration in which the upper level is a highly excited Rydberg state. Transmission saturation is clearly apparent as soon as the field contains more than one photon at a time in the medium, clearly demonstrating the strong non-linearity effect described above. The experiment continued by measuring the statistics of photons transmitted by the medium subjected to the TIE. A photon-unbundling effect was observed, demonstrating that electromagnetically induced transparency effectively allows only one photon at a time to pass through the medium. The quantitative analysis of these and similar experiments is quite complex, involving the solution of a hierarchy of coupled nonlinear evolutionary equations. Such an analysis was beyond the scope of this lecture, which confined itself to a qualitative description of these effects.

The physics of interacting cold Rydberg atoms is very rich, and this lecture could only give a very partial view of it. A number of effects not described in the lecture were discussed in the accompanying seminars.