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Quantum electrodynamics of an atom in front of a dielectric slab

Reyes Contreras, Ana María

Published Feb 1, 2009• 0 citations
OpenAlex

Abstract

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.

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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.
The energy-level shift of a ground-state atom in front of a nondispersive dielectric half-space is calculated by quantizing the electric field by means of a normal-mode expansion and applying second-order perturbation theory to the electric-dipole Hamiltonian $\mathit{\ensuremath{\mu}}\ensuremath{\cdot}\mathbf{E}.$ It is shown that the contributions to this shift coming from traveling and from evanescent waves can be combined into a single expression which lends itself readily to asymptotic analysis for large atom-surface separations, while in the opposite asymptotic regime when the atom is close to the surface the combined expression is less convenient. Employing a Green's-function formalism instead of the normal-mode expansion leads directly to the combined formula, and in that case it is advantageous to be able to apply the same transformation backwards and split the energy shift into a sum of distinct contributions corresponding to different physical processes. The analysis serves to shed light on common sources of error in the literature and paves the way for the study of more complicated models in cavity quantum electrodynamics.
We determine the energy-level shift experienced by a neutral atom due the quantum electromagnetic interaction with a layered dielectric body. We use the technique of normal-mode expansion to quantize the electromagnetic field in the presence of a layered, nondispersive, and nonabsorptive dielectric. We explicitly calculate the equal-time commutation relations between the electric field and vector potential operators. We show that the commutator can be expressed in terms of a generalized transverse $\ensuremath{\delta}$ function and that this is a consequence of using the generalized Coulomb gauge to quantize the electromagnetic field. These mathematical tools turn out to be very helpful in the calculation of the energy-level shift of the atom, which can be in its ground state or excited. The results for the shift are then analyzed asymptotically in various regions of the system's parameter space, with a view to providing quick estimates of the influence of a single dielectric layer on the Casimir-Polder interaction between an atom and a dielectric half space. We also investigate the impact of resonances between the wavelength of the atomic transition and the thickness of the top layer.
A quantum-electrodynamic theory of spontaneous emission in presence of dielectrics and conductors is developed. The theory makes use of the master-equation techniques and the response-function formalism of part I of this series of papers. Various observable entities such as damping coefficients (lifetimes), Lamb shifts, and frequency shifts are related to the appropriate surface-dependent response functions. The results are valid for arbitrary geometries (involving linear dielectrics) and naturally contain, as a special case, the usual results of spontaneous emission in free space. As explicit examples we consider a two-level atom (also the multilevel atom) in presence of a plane dielectric interface and between two conducting plates. Formulas for the shifts and widths are given and their asymptotic behavior for large and short distances is discussed. The behavior when the atom is embedded inside the dielectric is different than when the atom is outside the dielectric. The origin of coherence in the present model is discussed and the coherence effects in this model are contrasted with those in Dicke's model. The results are also compared with those obtained by the image method. Exact expressions for the operator radiation-reaction fields are obtained in terms of the atomic polarization operators and response functions. Approximate results for such fields are also given. The far-zone behavior of the radiation fields is obtained in terms of response functions and the polarization operators. Some of the normally ordered correlation functions are also calculated. The connection of some of the theoretical results with a recent experimental work of Carniglia, Mandel, and Drexhage is discussed. The effect of anisotropy of the dielectric function on the lifetimes as well as on the far-field correlation functions is also considered. Finally the contribution to the shifts and widths due to the excitation of surface polariton modes is computed and the results are compared with those obtained by the quantization of surface polariton field. It is found that such surface modes contribute significantly to widths.
The spontaneous emission of photons by an atom placed near the dielectric slab is analyzed using global free electromagnetic modes, satisfying necessary continuity conditions at the slab boundaries. These modes include the modes extended over the whole space (traveling) and the modes trapped to a waveguide formed by the slab (trapped). To describe angular properties of spontaneous emission a parametrization of the traveling modes by the outgoing waves has been proposed. Angular characteristics of the traveling photon emission, distributions of the trapped photon emission, as well as global decay rates have been calculated using the quantum approach and standard perturbation theory. Enhancement and suppression of radiative decay rates in various configurations have been shown. A certain calculational error in an earlier paper by Khosravi and Loudon [Proc. R. Soc. London Ser. A 436, 373 (1992)] affecting thin slabs and results near waveguide-mode thresholds has been improved.
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.