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Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences

Quantum electrodynamics of an atom in front of a non–dispersive dielectric half–space. II. Effects of finite temperature

Shin-Tza Wu, Claudia Eberlein

Published Aug 8, 2000• 19 citations
Publisher

Abstract

The energy–level shifts of an atom located at a distance Z outside a dielectric half–space are calculated at finite temperatures. The refractive index of the medium is real and frequency independent; the limiting case of perfect reflectors is also considered. The analytic expressions for the energy–level shifts are analysed in various regimes of temperature and of the distance of the atom from the surface. The significance of thermal corrections in comparison with zero–temperature effects is analysed in each regime and discussed for recent experiments.

DOI
10.1098/rspa.2000.0595

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The energy–level shifts of an atom located at a distance Z outside a dielectric half–space are calculated at finite temperatures. The refractive index of the medium is real and frequency independent; the limiting case of perfect reflectors is also considered. The analytic expressions for the energy–level shifts are analysed in various regimes of temperature and of the distance of the atom from the surface. The significance of thermal corrections in comparison with zero–temperature effects is analysed in each regime and discussed for recent experiments.
The energy–level shifts and the change in the rate of spontaneous emission are calculated for an atom located at a distanceZ from a dielectric half–space. The dielectric is a non–dispersive and non–absorbing medium characterized by a constant real refractive indexn. The explicit analytic formulae derived are applicable for arbitrary values ofn. All results are analysed in the non-retarded and retarded limits, which apply to the atom being close to or far from the interface, respectively, i.e. toZ being small or large on the scale of a wavelength of a typical atomic transition. For ground-state atoms, the energy–level shift varies as 1/Z3 in the non–retarded regime and as 1/Z4 in the retarded regime, which agrees with the Casimir–Polder result for the limitn→α. For excited–state atoms, the energy–level shifts receive additional contributions that oscillate with the distanceZ from the interface.
Quantum electrodynamic theory (QED) in the vicinity of macroscopic structures has achieved new importance due to its applicability, particularly in nanotechnology. There are many powerful methods for studying QED near media with diverse properties and geometries. However, applying them to a particular problem generally necessitates extensive numerical calculations. This is not the case for simple systems of high symmetry, in which the electromagnetic field can be quantised by explicit mode expansion, allowing exact analytic calculations. In the present thesis, we calculate the energy-level shift of a ground state atom near a non-dispersive and non-dissipative dielectric slab. The shift is due to the interaction of the atom with electromagnetic field fluctuations, which in turn are affected by the presence of the slab. Thus, a quantisation of the electromagnetic field in the presence of a layered system is required. We derive the field modes, which comprise of a continuous set of travelling modes (with incident, reflected and transmitted parts) and trapped modes, subject to repeated total internal reflection and emerge as an evanescent field outside the slab, they only exist at certain discrete frequencies. The shift is obtained by means of second-order perturbation theory. It splits up naturally into two contributions, due to the different nature of the modes, and a problem arises when we have to add them all. We have come up with a convenient method of summing over all modes, and its validity has been demonstrated by proving the completeness. The calculation of the shift follows as an application of our method. The result is analysed asymptotically for various regions, reducing to simple formulas that can be utilised in recent experiments, in which the thickness of the substrate matters.
We use the master equation approach to calculate the energy level shifts of an atom in the presence of a general dielectric semi-infinite medium characterized by a dielectric constant ε(ω)ε(ω)ε(ω). Particularly, we analyze the case of a non-dispersive medium for which we obtain a general expression for the interaction as well as the asymptotic behaviors for k0z≪1k_0 z \ll 1k0​z≪1 (non-retarded regime) and k0z≫1k_0 z \gg 1k0​z≫1 (retarded regime), where ω0=k0cω_0 = k_0 cω0​=k0​c is the main transition frequency of the atom. The limiting cases ε≃1ε\simeq 1ε≃1 and ε≫1ε\gg 1ε≫1 are discussed for both retarded and non-retarded limits. For the retarded limit, we compute the non-additivity contribution of van der Waals forces.
We investigate the influence of the presence of a dielectric medium on the energy shift for a uniformly moving atom in blackbody radiation. The atomic energy-level shift usually depends on the temperature of thermal radiation, the atomic velocity, the refractive index of the medium, and the atomic polarizability. Analytical and numerical analyses indicate that the presence of the dielectric medium has profoundly disturbed the behavior of the atomic energy shifts, especially when the atomic speed is greater than the speed of light in the medium. Our work further suggests that the combination of a medium and atomic relativistic center-of-mass motion can provide insight into the radiative properties of atom-field coupling system.
We study the thermalization of an elementary quantum system modeled by a two-level atom interacting with stationary electromagnetic fields out of thermal equilibrium near a dielectric slab. The slab is held at a temperature different from that of the region where the atom is located. We find that when the slab is nonabsorbing and nondispersive, out-of-thermal-equilibrium effects exist only when its thickness is infinite. In other words, no out-of-thermal-equilibrium effects appear for a real dielectric slab of a finite thickness ddd. Furthermore, a finite thick dielectric slab with a tiny imaginary part in the relative permittivity $\mathrm{Im}\ensuremath{\epsilon}$ behaves like a half-space dielectric substrate if $\frac{\mathrm{Im}\ensuremath{\epsilon}}{\sqrt{\mathrm{Re}\ensuremath{\epsilon}\ensuremath{-}1}}\frac{d}{{\ensuremath{\lambda}}_{0}}>1$ is satisfied, where ${\ensuremath{\lambda}}_{0}$ is the transition wavelength of the atom. This condition can serve as a guide for an experimental verification, using a dielectric substrate of a finite thickness, of the effects that arise from out-of-thermal-equilibrium fluctuations with a half-space (infinite thickness) dielectric.