Casimir-Foucault interaction : Free energy and entropy at low temperature

The present thesis focuses on several topics within three separate but related branches of the overall field of dispersion forces. The three branches are: temperature corrections to the Casimir force between real materials (Part 1), explicit calculation of Casimir energy in wedge geometries (Part 2), and Casimir-Polder forces on particles out of thermal equilibrium (Part 3). Part 1 deals primarily with analysis of a previously purported thermodynamic inconsistency in the Casimir-Lifshitz free energy of the interaction of two plane mirrors – violation of the third law of thermodynamics – when the latter’s dielectric response is described with dissipative models. It is shown analytically and numerically that the Casimir entropy of the interaction between two metallic mirrors described by the Drude model does tend to zero at zero temperature, provided electronic relaxation does not vanish. The leading order terms at low temperature are found. A similar calculation is carried out for the interaction of semiconductors with small but non-zero DC conductivity. In a generalisation, it is shown that a violation of the third law can only occur for permittivities whose low-frequency behaviour is temperature dependent near zero temperature. A calculation using path integral methods shows that the low temperature behaviour of the interaction of fluctuating Foucault currents in two mirrors of Drude metal is identical to that of the full Casimir-Lifshitz free energy, reasserting a previous finding by Intravaia and Henkel that such fluctuating bulk currents are the physical reason for the anomalous entropy behaviour. In a related effort, an analysis of the frequency dependence of the Casimir force by Ford is generalised to imperfectly reflecting mirrors. A paradox is pointed out, in that the effects of a perturbation of the reflecting properties of the mirrors in a finite frequency window can be calculated in two ways giving different results. It is concluded that optimistic conclusions reached by Ford based on one of these methods, which seems to allow radically changing and tailoring the Casimir force with engineered materials, can not be realised. Part 2 presents several explicit calculations of the Casimir energy of different wedge and cylinder geometries. The Casimir energy of a perfectly conducting wedge intercut by a circularly cylindrical arc, either perfectly conducting or (magneto)dielectric, is calculated. The energy is found to include a singular and non-regularisable term due to the corners where the arc meets the wedge, whereas the finite part is an immediate generalisation of the previously known results for a circular cylinder. The energy of a magnetodielectric wedge obeying a criterion of isorefractivity (spatially uniform speed of light) superimposed coaxially on a perfectly conducting cylindrical shell is calculated. This is the first expression for the energy of a wedge which is not perfectly reflecting. Finally, the energy of the perfectly conducting wedge and arc (and, as a special case, cylinder) is extended to the case of non-zero temperatures. After a regularisation procedure making use of the Chowla-Selberg formula an analytical expression for the temperature-dependent energy at all temperatures is derived, and showed to coincide with previously calculated high-temperature asymptotics by Bordag, Nesterenko and Pirozhenko. Part 3 considers numerical and analytical studies of the Casimir-Polder forces acting on particles prepared in a given eigenstate (or superposition of such) in an environment which is otherwise at thermal equilibrium. We first consider cold polar molecules outside a metallic halfspace. It is found that the force in the near-zone (non-retarded regime) is much weaker than what would result from a naive perturbative calculation, and that in the far-zone (retarded regime) the force becomes spatially oscillatory. It is demonstrated how these spatial oscillations may be enhanced in a resonating planar cavity, although for polar molecules the resulting amplitude is still insufficient for observation. A cylindrical cavity, however, can achieve a better enhancement factor. The Casimir-Polder forces on Rydberg atoms near a surface are calculated; because of the very large transition dipole moments of Rydberg transitions, the force is enormous on an atomic scale. We show that the oscillating force on Rydberg atoms can be enhanced into the observable regime by use of a fine-tuned cylindrical cavity. A particle in an eigenstate which is in the non-retarded regime with respect to all its dominant transitions is shown to feel a Casimir-Polder force which is virtually independent of temperature from zero to room temperature and beyond. Both for cold polar molecules and Rydberg atoms, the temperature-independent regime extends to a few and hundreds of micrometers, respectively, and includes the separations generally accessed in experiments