Inverse application of the generalized Littlewood theorem concerning integrals of the logarithm of analytic functions: an easy method to establish equalities between different analytic functions

Recently, we have established and used the generalized Littlewood theorem concerning contour integrals of the logarithm of analytical function to obtain new criteria equivalent to the Riemann hypothesis. Later, the same theorem was applied to calculate certain infinite sums and study the properties of zeroes of a few analytical functions. In this Note, we discuss what, in a sense, are inverse applications of this theorem. We first prove an easy Lemma that if two meromorphic on the whole complex plane functions $f(z)$ and $g(z)$ have the same zeroes and poles, taking into account their orders, and have appropriate asymptotic for large $|z|$, then for some integer $n$, $d^n \ln(f(z)/dz^n = d^n\ln(g(z)/dz^n$. The use of this Lemma enables easy proofs of many identities between elliptic functions and their transformation rules. In particular, we show how for any complex number $a$, $\wp(z)-a$, where $\wp(z)$ is Weierstrass $\wp$-function, can be presented as a product and ratio of three elliptic $\theta_1$-functions of certain arguments. We also establish n-tuple rules for elliptic theta-functions and /rho_z(z) functions.Comment: 30 pages. Submitted to the Symmetry journa

Slowing and cooling molecules and neutral atoms by time-varying electric field gradients

A method of slowing, accelerating, cooling, and bunching molecules and neutral atoms using time-varying electric field gradients is demonstrated with cesium atoms in a fountain. The effects are measured and found to be in agreement with calculation. Time-varying electric field gradient slowing and cooling is applicable to atoms that have large dipole polarizabilities, including atoms that are not amenable to laser slowing and cooling, to Rydberg atoms, and to molecules, especially polar molecules with large electric dipole moments. The possible applications of this method include slowing and cooling thermal beams of atoms and molecules, launching cold atoms from a trap into a fountain, and measuring atomic dipole polarizabilities.Comment: 13 pages, 10 figures. Scheduled for publication in Nov. 1 Phys. Rev.

Discrete structure of ultrathin dielectric films and their surface optical properties

The boundary problem of linear classical optics about the interaction of electromagnetic radiation with a thin dielectric film has been solved under explicit consideration of its discrete structure. The main attention has been paid to the investigation of the near-zone optical response of dielectrics. The laws of reflection and refraction for discrete structures in the case of a regular atomic distribution are studied and the structure of evanescent harmonics induced by an external plane wave near the surface is investigated in details. It is shown by means of analytical and numerical calculations that due to the existence of the evanescent harmonics the laws of reflection and refraction at the distances from the surface less than two interatomic distances are principally different from the Fresnel laws. From the practical point of view the results of this work might be useful for the near-field optical microscopy of ultrahigh resolution.Comment: 25 pages, 16 figures, LaTeX2.09, to be published in Phys.Rev.

High-sensitivity diamond magnetometer with nanoscale resolution

We present a novel approach to the detection of weak magnetic fields that takes advantage of recently developed techniques for the coherent control of solid-state electron spin quantum bits. Specifically, we investigate a magnetic sensor based on Nitrogen-Vacancy centers in room-temperature diamond. We discuss two important applications of this technique: a nanoscale magnetometer that could potentially detect precession of single nuclear spins and an optical magnetic field imager combining spatial resolution ranging from micrometers to millimeters with a sensitivity approaching few femtotesla/Hz$^{1/2}$.Comment: 29 pages, 4 figure

Guiding slow polar molecules with a charged wire

We demonstrate experimentally the guiding of cold and slow ND3 molecules along a thin charged wire over a distance of ~0.34 m through an entire molecular beam apparatus. Trajectory simulations confirm that both linear and quadratic high-field-seeking Stark states can be efficiently guided from the beam source up to the detector. A density enhancement up to a factor 7 is reached for decelerated beams with velocities ranging down to ~50 m/s generated by the rotating nozzle technique

Coherent cooperative fluorescence resonance energy transfer

Cooperative fluorescence resonance energy transfer effect is experimentally demonstrated for a few crystals doped with rare-earth ions. We show that, at the liquid helium temperatures in similar crystals, coherent cooperative fluorescence resonance energy transfer, as well as an inverse coherent up-conversion process, could be observed, and briefly discuss possible applications of these effects

On equalities involving integrals of the logarithm of the Riemann zeta-function and equivalent to the Riemann hypothesis

By using the generalized Littlewood theorem about a contour integral involving the logarithm of an analytic function, we show how an infinite number of integral equalities involving integrals of the logarithm of the Riemann zeta-function and equivalent to the Riemann hypothesis can be established and present some of them as an example. It is shown that all earlier known equalities of this type, viz., the Wang equality, Volchkov equality, Balazard-Saias-Yor equality, and an equality established by one of the authors, are certain special cases of our general approach

Application of the Mathieu’s equation for a design of a photonic crystal supporting surface electromagnetic waves

Nowadays, unique characteristics of surface electromagnetic waves, particularly, surface plasmons supported by a specially designed photonic crystal find numerous applications. We propose to exploit an evident analogy between such a photonic crystal and a structure with a sine-modulated refractive index. The light propagation inside the latter is described by the famous Mathieu’s differential equation. This application of the Mathieu’s equation can be useful for a design of multilayer structures, and also for fundamental understanding of electromagnetic phenomena in inhomogeneous media