Eisenstein Series and String Thresholds

We investigate the relevance of Eisenstein series for representing certain $G(Z)$-invariant string theory amplitudes which receive corrections from BPS states only. $G(Z)$ may stand for any of the mapping class, T-duality and U-duality groups $Sl(d,Z)$, $SO(d,d,Z)$ or $E_{d+1(d+1)}(Z)$ respectively. Using $G(Z)$-invariant mass formulae, we construct invariant modular functions on the symmetric space $K\backslash G(R)$ of non-compact type, with $K$ the maximal compact subgroup of $G(R)$, that generalize the standard non-holomorphic Eisenstein series arising in harmonic analysis on the fundamental domain of the Poincar\'e upper half-plane. Comparing the asymptotics and eigenvalues of the Eisenstein series under second order differential operators with quantities arising in one- and $g$-loop string amplitudes, we obtain a manifestly T-duality invariant representation of the latter, conjecture their non-perturbative U-duality invariant extension, and analyze the resulting non-perturbative effects. This includes the $R^4$ and $R^4 H^{4g-4}$ couplings in toroidal compactifications of M-theory to any dimension $D\geq 4$ and $D\geq 6$ respectively.Comment: Latex2e, 60 pages; v2: Appendix A.4 extended, 2 refs added, thms renumbered, plus minor corrections; v3: relation (1.7) to math Eis series clarified, eq (3.3) and minor typos corrected, final version to appear in Comm. Math. Phys; v4: misprints and Eq C.13,C.24 corrected, see note adde

Comment on the Adiabatic Condition

The experimental observation of effects due to Berry's phase in quantum systems is certainly one of the most impressive demonstrations of the correctness of the superposition principle in quantum mechanics. Since Berry's original paper in 1984, the spin 1/2 coupled with rotating external magnetic field has been one of the most studied models where those phases appear. We also consider a special case of this soluble model. A detailed analysis of the coupled differential equations and comparison with exact results teach us why the usual procedure (of neglecting nondiagonal terms) is mathematically sound.Comment: 9 page

Trajectories in a space with a spherically symmetric dislocation

We consider a new type of defect in the scope of linear elasticity theory, using geometrical methods. This defect is produced by a spherically symmetric dislocation, or ball dislocation. We derive the induced metric as well as the affine connections and curvature tensors. Since the induced metric is discontinuous, one can expect ambiguity coming from these quantities, due to products between delta functions or its derivatives, plaguing a description of ball dislocations based on the Geometric Theory of Defects. However, exactly as in the previous case of cylindric defect, one can obtain some well-defined physical predictions of the induced geometry. In particular, we explore some properties of test particle trajectories around the defect and show that these trajectories are curved but can not be circular orbits.Comment: 11 pages, 3 figure

Atomic detection in microwave cavity experiments: a dynamical model

We construct a model for the detection of one atom maser in the context of cavity Quantum Electrodynamics (QED) used to study coherence properties of superpositions of electromagnetic modes. Analytic expressions for the atomic ionization are obtained, considering the imperfections of the measurement process due to the probabilistic nature of the interactions between the ionization field and the atoms. Limited efficiency and false counting rates are considered in a dynamical context, and consequent results on the information about the state of the cavity modes are obtained.Comment: 12 pages, 1 figur

Atomic Focusing by Quantum Fields: Entanglement Properties

The coherent manipulation of the atomic matter waves is of great interest both in science and technology. In order to study how an atom optic device alters the coherence of an atomic beam, we consider the quantum lens proposed by Averbukh et al [1] to show the discrete nature of the electromagnetic field. We extend the analysis of this quantum lens to the study of another essentially quantum property present in the focusing process, i.e., the atom-field entanglement, and show how the initial atomic coherence and purity are affected by the entanglement. The dynamics of this process is obtained in closed form. We calculate the beam quality factor and the trace of the square of the reduced density matrix as a function of the average photon number in order to analyze the coherence and purity of the atomic beam during the focusing process.Comment: 10 pages, 4 figure