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Physical Review A

Level shifts of simple atoms between parallel mirrors

Wonho Jhe

Published Nov 1, 1991• 20 citations
Publisher

Abstract

We have made a quantitative study of quantum electrodynamic corrections to the atomic energy levels between parallel conducting mirrors. Within the framework of simple two-level atoms, we evaluate the level shifts numerically and discuss the van der Waals, Casimir, and cavity-resonant radiative shifts as their asymptotic forms. In particular, we find that the level shifts of an excited state are substantially different not only from those of a classical dipole radiator but also from those of an excited atom near a single mirror. For the ground state, however, the shifts are qualitatively similar to those near a single mirror. We also apply the results to some simple atomic systems that may be used to test the cavity QED theory.

DOI
10.1103/physreva.44.5932

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We have made a quantitative study of quantum electrodynamic corrections to the atomic energy levels between parallel conducting mirrors. Within the framework of simple two-level atoms, we evaluate the level shifts numerically and discuss the van der Waals, Casimir, and cavity-resonant radiative shifts as their asymptotic forms. In particular, we find that the level shifts of an excited state are substantially different not only from those of a classical dipole radiator but also from those of an excited atom near a single mirror. For the ground state, however, the shifts are qualitatively similar to those near a single mirror. We also apply the results to some simple atomic systems that may be used to test the cavity QED theory.
We calculate the energy-level shifts of an atom near a dielectric microsphere. With the quantum electrodynamic linear-response formalism and the exact field susceptibility satisfying the boundary conditions, the energy shifts of a ground-state atom are calculated numerically. From the numerical results, in particular, we are able to identify the short-range instantaneous van der Waals interaction energy and the long-range Casimir-Polder retarded interaction energy between two polarizable objects (atom and cavity).
We investigate the quantum electrodynamic (QED) properties of an atomic electron close to the focus of a spherical mirror. We first show that the spontaneous emission and excited-state level shift of the atom can be fully suppressed with mirror-atom distances of many wavelengths. A three-dimensional theory predicts that the spectral density of vacuum fluctuations can indeed vanish within a volume ${\ensuremath{\lambda}}^{3}$ around the atom, with the use of a far-distant mirror covering only half of the atomic emission solid angle. The modification of these QED atomic properties is also computed as a function of the mirror size, and large effects are found for only moderate numerical apertures. We also evaluate the long-distance ground-state energy shift (Casimir-Polder shift) and find that it scales as $(\ensuremath{\lambda}/R){}^{2}$ at the focus of a hemispherical mirror of radius RRR, as opposed to the well-known $(\ensuremath{\lambda}/R){}^{4}$ scaling law for an atom at a distance RRR from an infinite plane mirror. Our results are relevant for investigations of QED effects as well as free-space coupling to single atoms using high-numerical-aperture lenses.
We compute the energies of the interaction between two atoms placed in the middle of a perfectly reflecting planar cavity, in the weak-coupling nondegenerate regime. Both inhibition and enhancement of the interactions can be obtained by varying the size of the cavity. We derive exact expressions for the dyadic Green's function of the cavity field which mediates the interactions and apply time-dependent quantum perturbation theory in the adiabatic approximation. We provide explicit expressions for the van der Waals potentials of two polarizable atomic dipoles and the electrostatic potential of two induced dipoles. We compute the van der Waals potentials in three different scenarios: two atoms in their ground states, two atoms excited, and two dissimilar atoms with one of them excited. In addition, we calculate the phase-shift rate of the two-atom wave function in each case. The effect of the two-dimensional confinement of the electromagnetic field on the dipole-dipole interactions is analyzed. This effect depends on the atomic polarization. For dipole moments oriented parallel to the cavity plates, both the electrostatic and the van der Waals interactions are exponentially suppressed for values of the cavity width much less than the interatomic distance, whereas for values of the width close to the interatomic distance, the strength of both interactions is higher than their values in the absence of cavity. For dipole moments perpendicular to the plates, the strength of the van der Waals interaction decreases for values of the cavity width close to the interatomic distance, while it increases for values of the width much less than the interatomic distance with respect to its strength in the absence of cavity. We illustrate these effects by computing the dipole-dipole interactions between two alkali atoms in circular Rydberg states.
I revisit the problem of the interaction between two dissimilar atoms with one atom in an excited state, recently addressed by Berman [Phys. Rev. A 91, 042127 (2015)], Donaire et al. [Phys. Rev. Lett. 115, 033201 (2015)], and Milonni and Rafsanjani [Phys. Rev. A 92, 062711 (2015)], for which precedent approaches have given conflicting results. In the first place, I discuss to what extent these works provide equivalent results. I show that the phase-shift rate of the two-atom wave function computed by Berman, the van der Waals potential of the excited atom by Donaire et al., and the level shift of the excited atom by Milonni and Rafsanjani possess equivalent expressions in the quasistationary approximation. In addition, I show that the level shift of the ground-state atom computed by Milonni and Rafsanjani is equivalent to its van der Waals potential. A diagrammatic representation of all those quantities is provided. The equivalences among them are, however, not generic. In particular, it is found that for the case of the interaction between two identical atoms excited, the phase-shift rate and the van der Waals potentials differ. Concerning the conflicting results of previous approaches in regards to the spatial oscillation of the interactions, I conclude, in agreement with Berman and with Milonni and Rafsanjani, that they refer to different physical quantities. The impacts of free-space dissipation and finite excitation rates on the dynamics of the potentials are analyzed. In contrast with Milonni and Rafsanjani, the oscillatory versus monotonic spatial forms of the potentials of each atom are found not to be related to the reversible versus irreversible nature of the excitation transfer involved.
We discuss the radiative level shifts of a uniformly accelerated atom moving between two infinite reflecting plates and interacting with a massless scalar field in the vacuum state. The atom, supposed to be a two-level system, accelerates in a direction parallel to the conducting plates. We evaluate separately the contributions of vacuum fluctuations and radiation reaction field to the energy shift of the atomic levels, and discuss their dependence on acceleration, atomic position and cavity length.
Abstract We consider the resonant van der Waals (vdW) interaction between two correlated identical two-level atoms (at least one of which being excited) within the framework of macroscopic cavity quantum electrodynamics in linear, dispersing and absorbing media. The interaction of both atoms with the body-assisted electromagnetic field of the cavity is assumed to be strong. Our time-independent evaluation is based on an extended Jaynes–Cummings model. For a system prepared in a superposition of its dressed states, we derive the general form of the vdW forces, using a Lorentzian single-mode approximation. We demonstrate the applicability of this approach by considering the case of a planar cavity and showing the position dependence of Rabi oscillations. We also show that in the limiting case of weak coupling, our results reproduce the perturbative ones for the case where the field is initially in vacuum state while the atomic state is in a superposition of two correlated states sharing one excitation.