Ponderomotive energy shifts experienced by Rydberg atoms in optical fields are known to be well approximated by the classical quiver energy of a free electron. We examine such energy shifts quantum mechanically and elucidate how they relate to the ponderomotive shift of a free electron in off-resonant fields. We derive and evaluate corrections to the ponderomotive free-electron polarizability in the length and velocity (transverse or Coulomb) gauges, which agree exactly as mandated by the gauge invariance. We also show how the free electron value emerges from the Dirac equation through summation over the Dirac sea states. We find that the free-electron ac Stark shift comes as an expectation value of a term proportional to the square of the vector potential in the velocity gauge. On the other hand, the same dominant contribution can be obtained to first order via a series expansion of the exact energy shift from the second-order perturbation theory in the length gauge. Finally, we numerically examine the validity of the free-electron approximation. The correction to the free-electron value becomes smaller with increasing principal quantum number, and it is well below a percent for 60$s$ states of Rb and Sr away from the resonances.
Ponderomotive energy shifts experienced by Rydberg atoms in optical fields are known to be well approximated by the classical quiver energy of a free electron. We examine such energy shifts quantum mechanically and elucidate how they relate to the ponderomotive shift of a free electron in off-resonant fields. We derive and evaluate corrections to the ponderomotive free-electron polarizability in the length and velocity (transverse or Coulomb) gauges, which agree exactly as mandated by the gauge invariance. We also show how the free electron value emerges from the Dirac equation through summation over the Dirac sea states. We find that the free-electron ac Stark shift comes as an expectation value of a term proportional to the square of the vector potential in the velocity gauge. On the other hand, the same dominant contribution can be obtained to first order via a series expansion of the exact energy shift from the second-order perturbation theory in the length gauge. Finally, we numerically examine the validity of the free-electron approximation. The correction to the free-electron value becomes smaller with increasing principal quantum number, and it is well below a percent for 60s states of Rb and Sr away from the resonances.
We develop a theoretical understanding of the trapping of divalent Rydberg atoms in optical lattices. Because the size of the Rydberg electron cloud can be comparable to the scale of spatial variations of laser intensity, we pay special attention to averaging optical fields over the atomic wave functions. The optical potential is proportional to the ac Stark polarizability. We find that in the independent-particle approximation for the valence electrons, this polarizability breaks into two contributions: the singly ionized core polarizability and the contribution from the Rydberg electron. Unlike the usually employed free-electron polarizability, the Rydberg contribution depends both on the laser intensity profile and on the rotational symmetry of the total electronic wave function. We focus on the J=0 Rydberg states of Sr and evaluate the dynamic polarizabilities of the 5sns(1S0) and 5snp(3P0) Rydberg states. We specifically chose the Sr atom for its optical-lattice clock applications. We find that there are several magic wavelengths in the infrared region of the spectrum at which the differential Stark shift between the clock states [5s2(1S0) and 5s5p(3P0)] and the J=0 Rydberg states [5sns(1S0) and 5snp(3P0)] vanishes. We tabulate these wavelengths as a function of the principal quantum number n of the Rydberg electron. We find that because the contribution to the total polarizability from the Rydberg electron vanishes at short wavelengths, magic wavelengths below $\ensuremath{\sim}$1000 nm are ``universal'' as they do not depend on the principal quantum number n.
In this thesis, we examine the ponderomotive interaction between an applied optical field and a highly excited Rydberg electron. An atom in a Rydberg state is essentially composed of an electron loosely bound, at a relatively large radial separation, to a positive ionic core. As such, the ponderomotive interaction for the Rydberg electron is similar to the interaction with free charges, which has been studied in context of plasma physics, new particle accelerator techniques, ion trapping, and electron diffraction among others. We are focused on using the ponderomotive interaction with the Rydberg electron to exert control over both the center-of-mass and electronic states of translationally cold Rydberg atoms. These capabilities can be adapted as tools for application in many experiments in areas such as atomic spectroscopy and quantum information processing. Our theoretical investigations have provided a well-defined parameter space for our experimental work and have allowed us to develop experimental methods appropriate for studying Rydberg atoms in ponderomotive potentials. In dense gases of cold Rydberg atoms, rich dynamics stem from electric multipole interactions among the Rydberg atoms. For example, interatomic forces between Rydberg atoms cause state-changing collisions which can significantly increase the kinetic energy of the colliding atoms. In addition to the studies of Rydberg atoms in ponderomotive potentials, we discuss collisions of cold Rydberg atoms in which internal energy of the Rydberg atoms is converted into kinetic energy.
Accurate data on electric-dipole transition matrix elements (EDTMs) for bound-bound Rydberg-atom transitions have become increasingly important in science and technology. Here we compute radial EDTMs of rubidium in length, velocity, and acceleration forms for electric-dipole-allowed transitions between states with principal and angular-momentum quantum numbers n and $\ensuremath{\ell}$ ranging from 15 to 100. Wave functions are computed based upon model potentials from [Phys. Rev. A 49, 982 (1994)]. Length-gauge EDTMs of strong low-$\ensuremath{\ell}$ Rydberg transitions, often employed in research and technology, are found to deviate from the fundamentally more accurate velocity-gauge form by approximately-n-independent and series-dependent shifts. The shift amount peaks at about 0.34ea0 for the Rb $nD\ensuremath{\leftrightarrow}\phantom{\rule{4pt}{0ex}}(n+1)P$ series, a particularly strong Rydberg-transition series in Rb. The shift corresponds to relative EDTM corrections of up to $\ensuremath{\approx}{10}^{\ensuremath{-}3}$, which can be of concern in high-precision applications. We discuss the physical reasons for the observed gauge differences, explain the conditions for applicability of the velocity- and length-gauge forms for different transition series, and present a decision tree of how to choose EDTMs.
The dominant long-range interaction of an alkali-metal atom with charged particles at low energies is given for large separation R by the dispersive potential $W\ensuremath{\simeq}\ensuremath{-}{\ensuremath{\alpha}}_{d}{/R}^{4}$ in terms of the dipole polarizability ${\ensuremath{\alpha}}_{d}.$ For atoms prepared initially in Rydberg states of quantum numbers (n, l), the potential assumes a more complicated form due to the complete or near degeneracy of the n manifold. Contributions to the polarizability are treated in two parts, (a) one for the nondegenerate states and (b) the other for the degenerate or near degenerate cases. It is shown that ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\alpha}}}_{d}^{(b)}$ for case (b) is in general R dependent, and in the limit of complete degeneracy, diverges as R2. That is, for a small energy gap \ensuremath{\Delta} between a pair of nearly degenerate states which are dipole coupled, the dispersion potential ${W}^{(b)}\ensuremath{\simeq}{D/R}^{2}$ for R<Rx and ${W}^{(b)}\ensuremath{\simeq}\ensuremath{-}{\ensuremath{\alpha}}_{d}^{(b)}{/R}^{4}$ for R>Rx, where D is the dipole moment, ${R}_{x}=[2|D/\ensuremath{\Delta}|{]}^{1/2},$ and ${\ensuremath{\alpha}}_{d}^{(b)}\ensuremath{\simeq}{D}^{2}/\ensuremath{\Delta}.$ They may also compete with a 1/R3 potential for Rydberg atoms with l>0. The total ${\ensuremath{\alpha}}_{d}$ can be very large in magnitude for small \ensuremath{\Delta} and even assume negative values, but the corresponding Rx also increases as \ensuremath{\Delta} decreases. The validity region in R of the ${R}^{\ensuremath{-}4}$ behavior of the potential recedes to larger R as the polarizability grows. A general formula for ${\ensuremath{\alpha}}_{d}$ is given, taking into account the effects of fine-structure splitting, the Lamb shift, and quantum defects.
Neutral-atom quantum simulation is susceptible to entanglement between the atom's internal electronic state and its center-of-mass position. In many alkali Rydberg platforms, the 'spin-motion coupling' is exacerbated by the free expansion required to avoid ponderomotive anti-trapping from optical fields. A recent proposal (arXiv:2505.01071) claims sufficiently excited Rydberg states could be trapped in a monochromatic, far-detuned, circularly polarized optical field by harnessing a large vector polarizability. We disprove the proposal through analytic calculation and measurement of the vector polarizability of the 54S, 54P, and 53D orbitals of Cesium. Regarding the optical angular frequency ω, we analytically derive that the scalar, vector, and tensor polarizabilities scale as ω−2, ω−3, and ω−4, as opposed to the proposed scaling of ω−2, ω−1, and ω−2. We refine the sum-over-states expression for vector and tensor polarizability to be numerically stable and predict negligible vector and tensor polarizabilities far detuned from resonances, in agreement with our measurements. However, we find vector polarizability can enhance a recent proposal for near-detuned attractive trapping. Furthermore, we evaluate the breakdown of the electric-dipole approximation and derive no effect stronger than ponderomotive repulsion. We conclude that an attractive, monochromatic, far-detuned optical trap for alkali Rydberg states is not possible, regardless of the beam geometry.
A formulation of quantum electrodynamics is given that applies to atoms in a strong laser field by perturbation theory in a non-relativistic regime. Dipole approximation is assumed. The dual Dyson series, here discussed by referring it to the Birkhoff theorem for singularly perturbed linear differential equations, can be applied and a perturbation series obtained transforming the Hamiltonian by a Pauli-Fierz transformation. But, if just few photons are present high-order harmonics cannot be generated. So, it is proven that odd high-order harmonics only appear when the laser field is intense and one can substitute the creation and annihilation operators by the square root of the mean number of photons taken to be huge, the field retaining its coherency property as observed experimentally for harmonics. In this case, the Hamiltonian for perturbation theory comes to the Kramers-Henneberger form. The theory has a dipolar contribution when the free-electron quiver motion amplitude is larger than the atomic radius. For a Coulomb potential one has that the outer electron is periodically kicked, and so a prove is given that the same should happen to Rydberg atoms in intense microwave fields. The distribution representing the kicking has a Fourier series with just odd terms. Using a modified Rayleigh-Schrödinger perturbation theory, it is shown that under the same condition of validity of the quiver motion amplitude to atomic radius ratio, the atomic wave function is only slightly modified by the laser field due to the way the energy levels rearrange themselves. This gives a prove of stabilization in the limit of laser frequency going to infinity. Then, perturbation theory can be applied when the Keldysh parameter becomes smaller with respect to the shifted distance between the energy levels of the atom.