arXiv subjects
Publications and source records attributed to Mathieu Giroux.
In this talk, we review a loop-by-loop approach used to generate differential equations for multi-scale (dual) Feynman integrals. We illustrate the method on a well-established example: the unequal mass elliptic sunrise.
We present a loop-by-loop method for computing the differential equations of Feynman integrals using the recently developed dual form formalism. We give explicit prescriptions for the loop-by-loop fibration of multi-loop dual forms. Then, we test our formalism on a simple, but non-trivial, example: the two-loop three-mass elliptic sunrise family of integrals. We obtain an epsilon-form differential equation within the correct function space in a sequence of relatively simple algebraic steps. In particular, none of these steps relies on the analysis of $q$-series. Then, we discuss interesting properties satisfied by our dual basis as well as its simple relation to the known epsilon-form basis of Feynman integrands. The underlying K3-geometry of the three-loop four-mass sunrise integral is also discussed. Finally, we speculate on how to construct a "good" loop-by-loop basis at three-loop.
Free-standing silicon nitride (SiN) mechanical resonators are of central interests in applications such as temperature and mass sensing, and for fundamental optomechanical reasearch. Understanding thermal coupling between a membrane resonator and its environment is required for predicting thermal noise, frequency noise, as well as sensors responses to temperature changes. In this work, we provide closed-form derivations of intrinsic thermal coupling quantities in free-standing thin films, namely total thermal conductance with the surroundings, thermal response time, and the relative contribution of thermal radiation. Our model is valid for any free-standing thin film anchored on all sides, although we particularly emphasize the specific case of SiN for which spectral emissivity is thoroughly investigated as a function of thickness and temperature. We find that radiative heat exchanges can play a non-negligible role, and even dominate thermal coupling for membranes of sizes commonly employed in optomechanics experiments. We experimentally confirm the validity of our model by measuring radiative thermal coupling between a SiN mechanical resonator and a ceramic heater in high vacuum.
Near-field radiative heat transfer (NFHT) research currently suffers from an imbalance between numerous theoretical studies, as opposed to experimental reports that remain, in proportion, relatively scarce. Existing experimental platforms all rely on unique custom-built devices on which it is difficult to integrate new materials and structures for studying the breadth of theoretically proposed phenomena. Here we show high-resolution NFHT measurements using, as our sensing element, silicon nitride (SiN) freestanding nanomembranes$-$a widely available platform routinely used in materials and cavity optomechanics research. We measure NFHT by tracking the high mechanical quality (Q) factor ($>2\times10^6$) resonance of a membrane placed in the near-field of a hemispherical hot object. We find that high Q-factor enables a temperature resolution ($1.2\times10^{-6} \ \mathrm{K}$) that is unparalleled in previous NFHT experiments. Results are in good agreement with a custom-built model combining heat transport in nanomembranes and the effect of non-uniform stress/temperature on the resonator eigenmodes.
In this paper, we shall compute the chain complex and the corresponding homology of some Morse function $f$ over integer coefficients. The definition of the correct boundary operator requires a careful construction of moduli space of (pseudo)gradient flow lines orientations. We will then apply this construction in the computation of these homology groups on 4-manifolds.