Rankin-Cohen brackets and formal quantization

In this paper, we use the theory of deformation quantization to understand Connes' and Moscovici's results \cite{cm:deformation}. We use Fedosov's method of deformation quantization of symplectic manifolds to reconstruct Zagier's deformation \cite{z:deformation} of modular forms, and relate this deformation to the Weyl-Moyal product. We also show that the projective structure introduced by Connes and Moscovici is equivalent to the existence of certain geometric data in the case of foliation groupoids. Using the methods developed by the second author \cite{t1:def-gpd}, we reconstruct a universal deformation formula of the Hopf algebra \calh_1 associated to codimension one foliations. In the end, we prove that the first Rankin-Cohen bracket $RC_1$ defines a noncommutative Poisson structure for an arbitrary \calh_1 action.Comment: 21 pages, minor changes and typos correcte

A long-lived spin-orbit-coupled degenerate dipolar Fermi gas

We describe the creation of a long-lived spin-orbit-coupled gas of quantum degenerate atoms using the most magnetic fermionic element, dysprosium. Spin-orbit-coupling arises from a synthetic gauge field created by the adiabatic following of degenerate dressed states comprised of optically coupled components of an atomic spin. Because of dysprosium's large electronic orbital angular momentum and large magnetic moment, the lifetime of the gas is limited not by spontaneous emission from the light-matter coupling, as for gases of alkali-metal atoms, but by dipolar relaxation of the spin. This relaxation is suppressed at large magnetic fields due to Fermi statistics. We observe lifetimes up to 400 ms, which exceeds that of spin-orbit-coupled fermionic alkali atoms by a factor of 10-100, and is close to the value obtained from a theoretical model. Elastic dipolar interactions are also observed to influence the Rabi evolution of the spin, revealing an interacting fermionic system. The long lifetime of this weakly interacting spin-orbit-coupled degenerate Fermi gas will facilitate the study of quantum many-body phenomena manifest at longer timescales, with exciting implications for the exploration of exotic topological quantum liquids.Comment: 11 pages, 8 figures, one appendi

Anisotropic collisions of dipolar Bose-Einstein condensates in the universal regime

We report the measurement of collisions between two Bose-Einstein condensates with strong dipolar interactions. The collision velocity is significantly larger than the internal velocity distribution widths of the individual condensates, and thus, with the condensates being sufficiently dilute, a halo corresponding to the two-body differential scattering cross section is observed. The results demonstrate a novel regime of quantum scattering, relevant to dipolar interactions, in which a large number of angular momentum states become coupled during the collision. We perform Monte-Carlo simulations to provide a detailed comparison between theoretical two-body cross sections and the experimental observations.Comment: 10 pages, 5 figure

Methods for comparative ChIA-PET and Hi-C data analysis.

The three-dimensional architecture of chromatin in the nucleus is important for genome regulation and function. Advanced high-throughput sequencing-based methods have been developed for capturing chromatin interactions (Hi-C, genome-wide chromosome conformation capture) or enriching for those involving a specific protein (ChIA-PET, chromatin interaction analysis with paired-end tag sequencing). There is widespread interest in utilizing and interpreting ChIA-PET and Hi-C. We review methods for comparative ChIA-PET and Hi-C data analysis and visualization. The topics reviewed include: downloading ChIA-PET and Hi-C data from the ENCODE and 4DN portals; processing ChIA-PET data using ChIA-PIPE; processing Hi-C data using Juicer or distiller and cooler; viewing 2D contact maps using Juicebox or Higlass; viewing peaks, loops, and domains using BASIC Browser; annotating convergent and tandem CTCF loops

Anisotropic expansion of a thermal dipolar Bose gas

We report on the anisotropic expansion of ultracold bosonic dysprosium gases at temperatures above quantum degeneracy and develop a quantitative theory to describe this behavior. The theory expresses the post-expansion aspect ratio in terms of temperature and microscopic collisional properties by incorporating Hartree-Fock mean-field interactions, hydrodynamic effects, and Bose-enhancement factors. Our results extend the utility of expansion imaging by providing accurate thermometry for dipolar thermal Bose gases, reducing error in expansion thermometry from tens of percent to only a few percent. Furthermore, we present a simple method to determine scattering lengths in dipolar gases, including near a Feshbach resonance, through observation of thermal gas expansion.Comment: main text and supplement, 11 pages total, 4 figure