Atoms highly sensitive to external fields

The second lesson described the effect of external fields, dynamic or static, on atoms carried in a Rydberg state. The first focus was on the interaction of these atoms with radiation. The interaction between the ground state (or a weakly excited state) and a Rydberg state of quantum number n is described by a matrix element of the electric dipole between these two states proportional to n-3/2, which indicates the very rapid decay of the coupling with n and the need for very intense lasers to ensure efficient coupling and a Rabi frequency large enough to excite the Rydberg atoms. The radiative lifetime of Rydberg atoms, inversely proportional to the square of the electric dipole matrix element, increases as n3 and becomes of the order of a hundred microseconds for n of the order of 50.

While the dipole elements between ground state and highly excited state are very small, those between highly excited states of neighboring principal quantum numbers are enormous, varying like n2. This explains the extreme sensitivity of Rydberg atoms to millimeter waves, resonant or quasi-resonant with transitions between adjacent Rydberg levels. This extreme sensitivity to millimetre-wave radiation is not accompanied by significant spontaneous emission at these transitions. The lifetime associated with these transitions varies as n5, and becomes extremely long, of the order of 30 ms, for n = 50. This low rate of partial spontaneous emission on millimeter transitions(n2 times lower than the rate of optical transition to the deep levels of the atom) is due to the fact that the density of radiation states, proportional to the cube of the frequency, varies as n-9 and becomes extremely low for states of large n. Circular Rydberg states, with maximum angular momentum equal to(n-1)h/2π, can only radiate on millimeter transitions between neighboring levels and therefore have, despite very strong coupling to radiation, a very long radiative lifetime. It is this remarkable combination of properties - very strong coupling to millimeter waves combined with a very long radiative lifetime - that is exploited in cavity electrodynamics experiments with Rydberg atoms. All these radiative properties were described in detail in the lesson and justified by very simple calculations within the framework of the semi-classical Rydberg atom model.

The lesson continued with a brief reminder of the superradiance of ensembles of Rydberg atoms initially in an excited state. This collective emission, linked to the spontaneous phasing of the dipoles of the atoms contained in a volume proportional to Lλ2 (where L is the length of the sample and λ the wavelength of the emission), occurs preferably on microwave transitions that benefit from a collective amplification factor proportional to λ2 (or even n6), counterbalancing the unfavorable factor for spontaneous emission at long wavelengths from these atoms(n2 times less likely than optical spontaneous emission to deep levels). The superradiance threshold corresponds to a few tens of thousands of atoms in the sample for n of the order of 20 to 30. This threshold can be lowered to unity when the atoms radiate into a sufficiently large overvoltage cavity (the atom can be said to "superradiate" with its images in the cavity walls). We are then in the Purcell regime of amplification of spontaneous emission from an isolated atom. For even better cavities, we reach the regime where the photon emitted by the atom can then be absorbed by it, corresponding to the Rabi oscillation regime in the vacuum field (the so-called "strong coupling" regime of cavity electrodynamics).

We then turned our attention to the coupling of Rydberg atoms with a static electric field (Stark effect). Sub-levels of the same hydrogenoid manifold have their degeneracy lifted to first order by the electric field, forming a range of states whose energies vary linearly with the field. The slope of the levels is proportional to their electric dipole. Levels with a positive slope correspond to atoms attracted to weak fields, and those with a negative slope to strong fields(low field seekers and high field seekers). These "Stark sublevels" are identified by a "parabolic" quantum number and by the projection m of their angular momentum onto the direction of the electric field (which remains a good quantum number). A state with a given parabolic quantum number (eigenstate of the electric dipole) and a given m is a linear combination of states n, l, m of the spherical basis with the same n and m and l values between m and n-1. There remains a degeneracy of states with the same parabolic number and different m values. When the field becomes sufficiently large, Stark states with different multiplicities of n merge. The conservation of the Runge Lenz vector means that the Stark Hamiltonian in hydrogen does not couple levels with different slopes and multiplicities. These levels therefore intersect in hydrogen. Not so for alkaline atoms. Quantum defects, which reflect the non-conservation of the Runge-Lenz vector, have the effect of transforming hydrogen level crossings into anticrossings, thus qualitatively modifying the Stark spectra of alkalis compared with those of hydrogen. The lesson analyzed these effects in detail and described the wave functions of the various Stark levels whose electronic distribution is associated with the non-zero values of the electric dipoles of the different states. Laser spectroscopy experiments on hydrogen Stark states carried out in the 1970s and 1980s were described, as were more recent experiments involving the braking and trapping of Rydberg atoms in electric fields. These experiments take advantage of the giant electric dipoles of these states to manipulate their external degrees of freedom (position and velocity) in suitably arranged electric field distributions.

The final part of the lesson studied the ionization process of Rydberg atoms in an electric field. If the field becomes large enough, the excited electron is torn from the atomic core. The value of the electric field at which this ionization occurs is level-dependent, making the process a very convenient means of selectively detecting Rydberg states, widely used in many, many experiments. Understanding the ionization mechanism is therefore essential. In fact, the mechanism is quite complex: it depends on the time during which the field is applied and on whether or not the Rydberg atom is a hydrogen atom. Classically, ionization can be understood as resulting from the lowering of the atom's coulombic potential by the addition of the electric field potential, which varies linearly with the position of the excited electron along the direction of the field. The addition of this potential creates a col which lowers the total potential in the direction of the electric field. Ionization should classically occur as soon as the energy of the level displaced by the Stark effect reaches the energy of the collar, which occurs first for states with a positive slope in the electric field(low field seekers), whereas states with a negative slope(high field seekers) should ionize in a higher field. Seemingly surprisingly, the opposite is observed for hydrogen. The reason is that low field seekers correspond to a distribution of maximum electron density along the field, but in a direction opposite to that of the collar. In other words, the excited electron avoids the neck region from which it could escape, and requires a much larger field than that given by the classical model to force it to ionize. In the case of alkalis, on the other hand, if the atoms are allowed to ionize "slowly", the electron escapes according to the classical model. The perturbation linked to the quantum defect (non-conservation of the Runge-Lenz vector) is once again at work. It mixes the low and high field seekers and means that the electron can, for the duration of the slow ionization, explore the region of the collar through which it escapes. If, on the other hand, an experiment is carried out to detect alkali atoms that ionize rapidly in the electric field, it can be seen that they require the same electric field as hydrogen, as the core perturbation then has no time to mix the levels to bring the excited electron close to the collar. A late 1970s laser spectroscopy experiment demonstrating these effects, carried out in D. Kleppner's group at MIT, was described at the end of the lesson.