Trust account school

Meeting proceedings of a seminar by the same name, presented by Seth T. Pruden and Mark A. Carnell, held February 26, 2020

Magnetic Field Dependence and Efimov Resonance Broadening in Ultracold Three-Body Recombination

We derive an analytic formula which describes the final bound state dependence in ultracold three-body recombination. Using an energy-dependent loss parameter, the recently observed broad resonance in an ultracold gas of $^{6}$Li atoms [T. B. Ottenstein {\it et al.}, Phys. Rev. Lett. 101, 203202 (2008)l J. H. Huckans {\it et al.}, Phys. Rev. Lett. 102, 165302 (2009)] is described quantitatively. We also provide an analytic and approximation for the three-body recombination rate which encapsulates the underlying physics of the universal three-body recombination process.Comment: 4 pages, 4 figure

{SETH}-Based Lower Bounds for Subset Sum and Bicriteria Path

Subset-Sum and k-SAT are two of the most extensively studied problems in computer science, and conjectures about their hardness are among the cornerstones of fine-grained complexity. One of the most intriguing open problems in this area is to base the hardness of one of these problems on the other. Our main result is a tight reduction from k-SAT to Subset-Sum on dense instances, proving that Bellman's 1962 pseudo-polynomial $O^{*}(T)$-time algorithm for Subset-Sum on $n$ numbers and target $T$ cannot be improved to time $T^{1-\varepsilon}\cdot 2^{o(n)}$ for any $\varepsilon>0$, unless the Strong Exponential Time Hypothesis (SETH) fails. This is one of the strongest known connections between any two of the core problems of fine-grained complexity. As a corollary, we prove a "Direct-OR" theorem for Subset-Sum under SETH, offering a new tool for proving conditional lower bounds: It is now possible to assume that deciding whether one out of $N$ given instances of Subset-Sum is a YES instance requires time $(N T)^{1-o(1)}$. As an application of this corollary, we prove a tight SETH-based lower bound for the classical Bicriteria s,t-Path problem, which is extensively studied in Operations Research. We separate its complexity from that of Subset-Sum: On graphs with $m$ edges and edge lengths bounded by $L$, we show that the $O(Lm)$ pseudo-polynomial time algorithm by Joksch from 1966 cannot be improved to $\tilde{O}(L+m)$, in contrast to a recent improvement for Subset Sum (Bringmann, SODA 2017)

Tight Approximation Algorithms for Bichromatic Graph Diameter and Related Problems

Some of the most fundamental and well-studied graph parameters are the Diameter (the largest shortest paths distance) and Radius (the smallest distance for which a "center" node can reach all other nodes). The natural and important ST-variant considers two subsets S and T of the vertex set and lets the ST-diameter be the maximum distance between a node in S and a node in T, and the ST-radius be the minimum distance for a node of S to reach all nodes of T. The bichromatic variant is the special case in which S and T partition the vertex set. In this paper we present a comprehensive study of the approximability of ST and Bichromatic Diameter, Radius, and Eccentricities, and variants, in graphs with and without directions and weights. We give the first nontrivial approximation algorithms for most of these problems, including time/accuracy trade-off upper and lower bounds. We show that nearly all of our obtained bounds are tight under the Strong Exponential Time Hypothesis (SETH), or the related Hitting Set Hypothesis. For instance, for Bichromatic Diameter in undirected weighted graphs with m edges, we present an O~(m^{3/2}) time 5/3-approximation algorithm, and show that under SETH, neither the running time, nor the approximation factor can be significantly improved while keeping the other unchanged

Comparison of Absorption, Fluorescence, and Polarization Spectroscopy of Atomic Rubidium

An ongoing spectroscopic investigation of atomic rubidium utilizes a two-photon, single-laser excitation process. Transitions accessible with our tunable laser include 5P1/2 (F ′ ) ← 5S1/2 (F) and 5P3/2 (F ′ ) ← 5S1/2 (F). The laser is split into a pump and probe beam to allow for Doppler-free measurements of transitions between hyperfine levels. The pump and probe beams are overlapped in a counter-propagating geometry and the laser frequency scans over a transition. Absorption, fluorescence and polarization spectroscopy techniques are applied to this basic experimental setup. The temperature of the vapor cell and the power of the pump and probe beams have been varied to explore line broadening effects and signal-to-noise of each technique. This humble setup will hopefully grow into a more robust experimental arrangement in which double resonance, two-laser excitations are used to explore hyperfine state changing collisions between rubidium atoms and noble gas atoms. Rb-noble gas collisions can transfer population between hyperfine levels, such as 5P3/2 (F ′ = 3) Collision ←− 5P3/2 (F ′ = 2), and the probe beam couples 7S1/2 (F ′′ = 2) ← 5P3/2 (F ′ = 3). Polarization spectroscopy signal depends on the rate of population transfer due to the collision as well as maintaining the orientation created by the pump laser. Fluorescence spectroscopy relies only on transfer of population due to the collision. Comparison of these techniques yields information regarding the change of the magnetic sublevels, mF , during hyperfine state changing collisions

Three-body rf association of Efimov trimers

We present a theoretical analysis of rf association of Efimov trimers in a 2-component Bose gas with short-range interactions. Using the adiabatic hyperspherical Green's function formalism to solve the quantum 3-body problem, we obtain universal expressions for 3-body rf association rates as a function of the s-wave scattering length $a$. We find that the association rates scale as $a^{-2}$ in the limit of large $a$, and diverge as $a^3 a_{ad}^{3}$ whenever an Efimov state crosses the atom-dimer threshold (where $a_{ad}$ stands for the atom-dimer scattering length). Our calculations show that trimer formation rates as large as $\sim10^{-21}$ cm$^6$/s can be achieved with rf Rabi frequencies of order 1 MHz, suggesting that direct rf association is a powerful tool of making and probing few-body quantum states in ultracold atomic gases.Comment: 4 pages, 2 figure