Spontaneous Excitation And Lamb Shift Of An Atom In Curved Spacetime

Spontaneous emission and the Lamb shift are two important radiative prop-erties of atoms. Theoretically, they can be attributed to vacuum fluctuations, radiation reaction or a combination of them. In the1980s, Dalibard, Dupont-Roc and Cohen-Tannoudji (DDC) introduced, in their studies of the interaction between an atom and the radiation fields, a symmetric ordering between the oper-ators of the atom and the fields, which makes vacuum fluctuations and radiation reaction possess distinct physical meanings, and thus established a method, which we call DDC formalism, to study the interaction between the atom and the field. This formalism was then used to study the average rate of change of the atomic energy in Minkowski spacetime, and it is found that a ground-state atom in uni-form acceleration in vacuum would spontaneously excite. From then on, the DDC formalism was widely used to study the radiative properties of atoms in various cases in a flat spacetime. Since matter and motion are indispensable in our Uni-verse, spacetime is generally not flat but curved, and this renders the vacuum fluctuations different from those in a flat spacetime. Will these differences affect the radiative properties of an atom? and can the corrections thus induced be potentially observable? In this dissertation, we try to answer these questions by studying some of the radiative properties of an atom, i.e., spontaneous excitation and the Lamb shift, in curved spacetimes using the DDC formalism.The DDC formalism was firstly used by Audretsch and Miiller to study the evolution of an atom in interaction with vacuum scalar field fluctuations, then Passante generalized it to the case of an atom in interaction with vacuum electro- magnetic fluctuations. In these two cases, the coupling between the atom and the field are both linear, thus the evolution of the atom boils down to the contribu-tions of vacuum fluctuations and radiation reaction. For a static atom initially in its ground state, the contribution of vacuum fluctuations and radiation reaction to the average rate of change of its energy cancel completely, and so the atom is stable. However, for a uniformly accelerated atom initially in its ground state, the contribution of vacuum fluctuations and radiation reaction to the average rate of change of its energy can no longer cancel, as a result, the atom excites spon-taneously. In this dissertation, we generalize the DDC formalism to the case of an atom in interaction with vacuum Dirac field fluctuations and study concretely the average rate of change of atomic energy of a uniformly accelerated atom in Minkowski spacetime. Some interesting features are found. Since we can only introduce a nonlinear interaction between the atom and the Dirac field, the evo-lution of the atom is now attributed to the contributions of vacuum fluctuations, radiation reaction of the atom and the cross term of vacuum fluctuations and ra-diation reaction, and furthermore, the contributions of the three parts are not of the same order because the contribution of radiation reaction is of higher order than the others. For a static atom initially in its ground state, it is the complete cancelation of the contribution of vacuum fluctuations and that of the cross term that assures the stability of the atom. For a ground-state atom in uniform acceler-ation, as the contribution of vacuum fluctuations and the contribution of the cross term no longer cancel, it would spontaneously excite.In a flat spacetime, the non-inertial motion and the appearance of boundaries may intrigue changes in vacuum fluctuations and radiation reaction. Consider an atom in a curved spacetime, for one thing, the modes of the field would be affected by the spacetime curvature which is some what similar to the case of modes reflected by boundaries in a flat spacetime; for another, according to the equivalence principal, the physical effects of the gravitational field and of non-inertial force are locally indistinguishable, thus a static atom in the gravitational field easily reminds us of an atom in uniform acceleration in a flat spacetime and we anticipate that some similarities between them may be found.Schwarzchild spacetime is a typical curved spacetime. By using the DDC formalism, we studied the radiative properties of a static atom in this spacetime in interaction with Boulware, Unruh and Hartle-Hawking vacuum scalar field and electromagnetic fluctuations separately, and we found that the spacetime curvature and Hawking radiation of black holes all induce corrections to the radiative prop-erties of the atom. The results we obtained essentially give a new derivation of the Hawking radiation. We found that the ground state of a static atom in interaction with Boulware vacuum fluctuations is stable, while a static ground-state atom in interaction with Unruh vacuum fluctuations would spontaneously excite just as if there were Hawking radiation emanating from the horizon of the black hole, and when it is interacting with Hartle-Hawking vacuum fluctuations, the ground-state atom would also spontaneously excite just as if there were Hawking radiation ema-nating from the event horizon of the black hole and there were thermal radiation at infinity which is in equilibrium with Hawking radiation. The static atom in Unruh and Hartle-Hawking vacua would feel thermal radiation and the temperature it feels satisfies the Tolman relation. In the case of a two dimensional Schwarzchild spacetime, since there is no backscattering of the field modes off the spacetime cur-vature, the thermal radiation an atom feels in Hartle-Hawking vacuum is twice of that in Unruh vacuum. This conclusion isn’t valid for the case of four dimensional Schwarzchild spacetime in which the field modes would be backscattered by the spacetime curvature. The effect of backscattering of the spacetime curvature on the field modes is distinctive for the scalar field and the electromagnetic field. Due to the backscattering of the field modes off the spacetime curvature, the thermal flux would diminish in the process of propagation. Similarly, the Lamb shift also gets corrected by these peculiar properties of the Schwarzchild spacetime.Though Hawking radiation induces corrections to the atomic Lamb shift, the corrections are rather difficult to observe in experiment. What about the correction intrigued by spacetime curvature? By exploiting the numerical method proposed by Jenson.et al, we numerically compute the atomic Lamb shift for a static atom at a arbitrary position in the exterior region of a spherical massive body. Our results give the revision induced by the spacetime curvature for the atomic Lamb shift. Due to the spacetime curvature, the Lamb shift of a static atom outside the massive body would always be smaller than that in Minkowski spacetime, and the revision induced by spacetime curvature is potentially observable. So our research in principle provides us possible way for verifying quantum effects in curved spacetime through astronomical observations.De Sitter spacetime is another typical curved spacetime. We study the ra-diative properties of atoms in this spacetime and give a new derivation for the Gibbons-Hawking effect. We demonstrate that a freely falling atom would spon-taneously excite as if it was immersed in a thermal bath at the Gibbons-Hawking temperature. Compared with the Lamb shift of an inertial atom in Minkowski spacetime, the Lamb shift of the freely falling atom in de Sitter spacetime gets revised by a term which is the same as that induced by a thermal bath with the Gibbons-Hawking temperature in Minkowski spacetime. The static atom also feels thermal radiation in de Sitter spacetime and the temperature it feels is Ts, the square of which can be expressed as the sum of the squared Gibbons-Hawking temperature and the squared Unruh temperature, and as a result, the static would spontaneously excite as if it was immersed in a thermal bath at the temperature Ts. Compared with the Lamb shift of an inertial atom in a flat spacetime, the Lamb shift of a static atom in de Sitter spacetime gets revised by a new term the same as that induced by a thermal bath at the temperature Ts in Minkowski spacetime.