Abstract:
Very large spontaneous-emission-rate enhancements (~1000) are obtained for quantum emitters coupled with tiny plasmonic resonance, especially when emitters are placed in the mouth of nanogaps formed by metal nanoparticles that are nearly in contact. This fundamental effect of light emission at subwavelength scales is well documented and understood as resulting from the smallness of nanogap modes. In contrasts, it is much less obvious to figure out whether the radiation efficiency is high in these gaps, or if the emission is quenched by metal absorption especially for tiny gaps a few nanometers wide; the whole literature only contains scattered electromagnetic calculations on the subject, which suggest that absorption and quenching can be kept at a small level despite the emitter proximity to metal. Thus through analytical derivations in the limit of small gap thickness, it is our objective to clarify why quantum emitters in nanogap antennas offer good efficiencies, what are the circumstances in which high efficiency is obtained, and whether there exists an upper bound for the maximum efficiency achievable.

Abstract:
The emerging field of on-chip integration of nanophotonic devices and cold atoms offers extremely-strong and pure light-matter interaction schemes, which may have profound impact on quantum information science. In this context, a long-standing obstacle is to achieve strong interaction between single atoms and single photons, while at the same time trap atoms in vacuum at large separation distances from dielectric surfaces. In this work, we study new waveguide geometries that challenge these conflicting objectives. The designed photonic crystal waveguide is expected to offer a good compromise, which additionally allows for easy manipulation of atomic clouds around the structure, while being tolerant to fabrication imperfections.

Abstract:
Light localization due to random imperfections in periodic media is paramount in photonics research. The group index is known to be a key parameter for localization near photonic band edges, since small group velocities reinforce light interaction with imperfections. Here, we show that the size of the smallest localized mode that is formed at the band edge of a periodic medium is driven instead by the effective photon mass, i.e. the flatness of the dispersion curve. Our theoretical prediction is supported by numerical simulations, which reveal that photonic-crystal waveguides can exhibit surprisingly small localized modes, much smaller than those observed in Bragg stacks thanks to their larger effective photon mass. This possibility is demonstrated experimentally with a photonic-crystal waveguide fabricated without any intentional disorder, for which near-field measurements allow us to distinctly observe a wavelength-scale localized mode despite the smallness ($\sim 1/1000$ of a wavelength) of the fabrication imperfections.

We give a proof in semi-group
theory based on the Malliavin Calculus of Bismut type in semi-group theory and
Wentzel-Freidlin estimates in semi-group of our result giving an expansion of
an hypoelliptic heat-kernel outside the cut-locus where Bismut’s non-degeneray condition plays a
preominent role.

Abstract:
We have used a nonlinear one-dimensional heat transfer model based on temperature-dependent blood perfusion to predict temperature distribution in dermis and subcutaneous tissues subjected to point heating sources. By using Jacobi elliptic functions, we have first found the analytic solution corresponding to the steady-state temperature distribution in the tissue. With the obtained analytic steady-state temperature, the effects of the thermal conductivity, the blood perfusion, the metabolic heat generation, and the coefficient of heat transfer on the temperature distribution in living tissues are numerically analyzed. Our results show that the derived analytic steady-state temperature is useful to easily and accurately study the thermal behavior of the biological system, and can be extended to such applications as parameter measurement, temperature field reconstruction and clinical treatment.

Abstract:
We introduce a general framework to study dipole-dipole energy transfer between an emitter and an absorber in a nanostructured environment. The theory allows us to address F\"orster Resonant Energy Transfer (FRET) between a donor and an acceptor in the presence of a nanoparticle with an anisotropic electromagnetic response. In the particular case of a magneto-optical anisotropy, we compute the generalized FRET rate and discuss the orders of magnitude. The distance dependence, the FRET efficiency and the sensitivity to the orientation of the transition dipoles orientation differ from standard FRET and can be controlled using the static magnetic field as an external parameter.

Abstract:
We define a Lie algebroid on the space of smooth 1-forms in the Nualart-Pardoux sense on the Wiener space associated to the stochastic linear Poisson structure on the Wiener space defined Léandre (2009). 1. Introduction Infinite dimensional Poisson structures play a big role in the theory of infinite dimensional Lie algebras [1], in the theory of integrable system [2], and in field theory [3]. But for instance, in [2], the test functional space where the hydrodynamic Poisson structure acts continuously is not conveniently defined. In [4, 5] we have defined such a test functional space in the case of a linear Poisson bracket of hydrodynamic type. On the other hand, it is very well known [6] that the theories of Lie groupoids and Lie algebroids play a key role in Poisson geometry. It is interesting to study a Lie algebroid for the Poisson structure [4] defined analytically in the framework of [4]. We postpone until later the study the Lie groupoid associated to the same Poisson structure but in the algebraic framework of [5]. The definition of this Lie groupoid in the framework of [4] presents, namely, some difficulties. Moreover some deformation quantizations for symplectic structures in infinite dimensional analysis were recently performed (see the review of Léandre [7] on that). The theory of groupoids is related [8] to Kontsevich deformation quantization [9]. Let us recall what a Lie algebroid is [6, 10–13]. We consider a bundle on a smooth finite dimensional manifold . is the tangent bundle of . and denote the space of smooth section of and . A Lie algebroid on is given by the following data. (i)A Lie bracket structure on has in particular to satisfy the Jacobi relation (ii)A smooth fiberwise linear map , called the anchor map, from into satisfies the relation for any smooth sections , of and any element of , the space of smooth functions on . Let us recall the definition of a Poisson structure on . It is an antisymmetric -bilinear map from into , which is a derivation on each components, vanishes on the constant and satisfies the Jacobi relation A Poisson structure is given by a bivector , that is, an element of as where and can be seen as 1-forms and the dual of is . denotes the interior product of the bivector by the 1-form . If is a bivector, then we can define a fiberwise linear smooth map from , the cotangent bundle of , into , called . If is a smooth section of , then This allows us to define a Lie algebroid structure on [11, 14–18] as follows. (i)The Bracket is defined by where is the usual Lie derivative of a 1-form: if is a vector field and

Abstract:
We translate into the language of semi-group theory Bismut's Calculus on boundary processes (Bismut (1983), Lèandre (1989)) which gives regularity result on the heat kernel associated with fractional powers of degenerated Laplacian. We translate into the language of semi-group theory the marriage of Bismut (1983) between the Malliavin Calculus of Bismut type on the underlying diffusion process and the Malliavin Calculus of Bismut type on the subordinator which is a jump process. 1. Introduction Let be vector fields on with bounded derivatives at each order. Let be an Hoermander's type operator on . Let be a second Hoermander's operator on . Bismut [1] considers the generator and the Markov semi-group . This semi-group has a probabilistic representation. We consider a Brownian motion independent of the others Brownian motions . Bismut introduced the solution of the stochastic differential equation starting at in Stratonovitch sense: where are independent Brownian motions. Let us introduce the local time associated with and its right inverse (see [2, 3]). Then, Such operator is classically related to the Dirichlet Problem [3]. Classically [4], where is the solution of the Stratonovitch differential equation starting at : The question is as following: is there an heat-kernel associated with the semi-group ? This means that There are several approaches in analysis to solve this problem, either by using tools of microlocal analysis or tools of harmonic analysis. Malliavin [5] uses the probabilistic representation of the semi-group. Malliavin uses a heavy apparatus of functional analysis (number operator on Fock space or equivalently Ornstein-Uhlenbeck operator on the Wiener space, Sobolev spaces on the Wiener space) in order to solve this problem. Bismut [6] avoids using this machinery to solve this hypoellipticity problem. In particular, Bismut’s approach can be adapted immediately to the case of the Poisson process [7]. The main difficulty to treat in the case of a Poisson process is the following: in general the solution of a stochastic differential equation with jumps is not a diffeomorphism when the starting point is moving (see [8–10]). The main remark of Bismut in [1] is that if we consider the jump process , then it is a diffeomorphism almost surely in . So, Bismut mixed the tools of the Malliavin Calculus for diffusion (on the process ) and the tools of the Malliavin Calculus for Poisson process (on the jump process ) in order to show that this isthe problem if Developments on Bismut’s idea was performed by Léandre in [9, 11]. Let us remark that