The skeleton of the Jacobian, the Jacobian of the skeleton, and lifting meromorphic functions from tropical to algebraic curves

Let K be an algebraically closed field which is complete with respect to a nontrivial, non-Archimedean valuation and let \Lambda be its value group. Given a smooth, proper, connected K-curve X and a skeleton \Gamma of the Berkovich analytification X^\an, there are two natural real tori which one can consider: the tropical Jacobian Jac(\Gamma) and the skeleton of the Berkovich analytification Jac(X)^\an. We show that the skeleton of the Jacobian is canonically isomorphic to the Jacobian of the skeleton as principally polarized tropical abelian varieties. In addition, we show that the tropicalization of a classical Abel-Jacobi map is a tropical Abel-Jacobi map. As a consequence of these results, we deduce that \Lambda-rational principal divisors on \Gamma, in the sense of tropical geometry, are exactly the retractions of principal divisors on X. We actually prove a more precise result which says that, although zeros and poles of divisors can cancel under the retraction map, in order to lift a \Lambda-rational principal divisor on \Gamma to a principal divisor on X it is never necessary to add more than g extra zeros and g extra poles. Our results imply that a continuous function F:\Gamma -> R is the restriction to \Gamma of -log|f| for some nonzero meromorphic function f on X if and only if F is a \Lambda-rational tropical meromorphic function, and we use this fact to prove that there is a rational map f : X --> P^3 whose tropicalization, when restricted to \Gamma, is an isometry onto its image.Comment: 21 pages, 1 figur

The Feasibility of a Fully Miniaturized Magneto-Optical Trap for Portable Ultracold Quantum Technology

Experiments using laser cooled atoms and ions show real promise for practical applications in quantum- enhanced metrology, timing, navigation, and sensing as well as exotic roles in quantum computing, networking and simulation. The heart of many of these experiments has been translated to microfabricated platforms known as atom chips whose construction readily lend themselves to integration with larger systems and future mass production. To truly make the jump from laboratory demonstrations to practical, rugged devices, the complex surrounding infrastructure (including vacuum systems, optics, and lasers) also needs to be miniatur- ized and integrated. In this paper we explore the feasibility of applying this approach to the Magneto-Optical Trap; incorporating the vacuum system, atom source and optical geometry into a permanently sealed micro- litre system capable of maintaining $10^{-10}$ mbar for more than 1000 days of operation with passive pumping alone. We demonstrate such an engineering challenge is achievable using recent advances in semiconductor microfabrication techniques and materialsComment: 23 pages, 10 figure

On the structure of nonarchimedean analytic curves

Let K be an algebraically closed, complete nonarchimedean field and let X be a smooth K-curve. In this paper we elaborate on several aspects of the structure of the Berkovich analytic space X^an. We define semistable vertex sets of X^an and their associated skeleta, which are essentially finite metric graphs embedded in X^an. We prove a folklore theorem which states that semistable vertex sets of X are in natural bijective correspondence with semistable models of X, thus showing that our notion of skeleton coincides with the standard definition of Berkovich. We use the skeletal theory to define a canonical metric on H(X^an) := X^an - X(K), and we give a proof of Thuillier's nonarchimedean Poincar\'e-Lelong formula in this language using results of Bosch and L\"utkebohmert.Comment: 23 pages. This an expanded version of section 5 of arXiv:1104.0320 which appears in the conference proceedings "Tropical and Non-Archimedean Geometry