Fluctuations and correlations in rotating Bose-Einstein condensates

We investigate the effects of correlations on the properties of the ground state of the rotating harmonically-trapped Bose gas by adding Bogoliubov fluctuations to the mean-field ground state of an $N$-particle single-vortex system. We demonstrate that the fluctuation-induced correlations lower the energy compared to that of the mean-field ground state, that the vortex core is pushed slightly away from the center of the trap, and that an unstable mode with negative energy (for rotations slower than a critical frequency) emerges in the energy spectrum, thus, pointing to a better state for slow rotation. We construct mean-field ground states of 0-, 1-, and 2-vortex states as a function of rotation rate and determine the critical frequencies for transitions between these states, as well as the critical frequency for appearance of a metastable state with an off-center vortex and its image vortex in the evanescent tail of the cloud.Comment: Added a paragraph to Section III; Revised arguments in Section III.A, results unchanged; Added reference

Properties of rotating ultracold bosonic quantum gases

In this dissertation, we study rotational properties of ultracold bosonic quantum gases in two trapped configurations, a quasi-two-dimensional gas in a harmonic trap and a quasi-one-dimensional gas in a toroidal trap. First, we investigate the effects of correlations on the properties of the ground state of the rotating harmonically-trapped Bose gas by adding Bogoliubov fluctuations to the mean-field ground state of an N-particle single-vortex system. We demonstrate that the fluctuation-induced correlations lower the energy compared to that of the mean-field ground state, that the vortex core is pushed slightly away from the center of the trap, and that an unstable mode with negative energy (for rotations slower than a critical frequency) emerges in the energy spectrum, thus, pointing to a better state for slow rotation. We construct mean-field ground states of 0-, 1-, and 2-vortex states as a function of rotation rate and determine the critical frequencies for transitions between these states, as well as the critical frequency for appearance of a metastable state with an off-center vortex and its image vortex in the evanescent tail of the cloud. Then, we show how the configuration-space form of the above-mentioned Bogoliubov ground-state wave function of a bosonic condensate with a single vortex in a harmonic trap can be described in terms of bosonic Jastrow correlations. We then generalize this result to study the effects of such correlations on a mean-field vortex lattice state and show that the included correlations lower the energy below that of the mean-field state. Although the reduction is relatively small, it is a precursor of the more general expected effect of correlations in describing the melting of the vortex lattice at a high angular momentum per particle. Finally, we study the stability and dynamics of an ultracold bosonic gas trapped in a toroidal geometry and driven by rotation in the absence of dissipation. We first delineate, via the Bogoliubov mode expansion, the regions of stability and the nature of instabilities of the system for both repulsive and attractive interaction strengths. To study the response of the system to variations in the rotation rate, we introduce a “disorder” potential, breaking the rotational symmetry. We demonstrate the breakdown of adiabaticity as the rotation rate is slowly varied and find forced tunneling between the eigenstates of the system. The nonadiabaticity is signaled by the appearance of a swallow-tail loop in the lowest-energy level, a general sign of hysteresis. Then, we show that this system is in one-to-one correspondence with a trapped gas in a double-well potential and thus exhibits macroscopic quantum self-trapping. Finally, we show that self-trapping is a direct manifestation of the behavior of the lowest-energy level

Business Closures and (Re)Openings in Real-Time Using Google Places: Proof of Concept

We present a new estimation of business opening and closure rates using data from Google Places—the data set behind the Google Maps service. Our algorithm, through a bisection routine, counts the appearance and disappearance of “pins” that represent unique businesses. As a proof of concept, we compute business opening and closure rates for the city of Ottawa during the reopening phase of the COVID-19 pandemic in mid-2021. The lifting of restrictions coincides with a wave of re-entry of temporarily closed businesses, suggesting that government support may have facilitated the survival of hibernating businesses. Our entry estimates are validated by a survey of new businesses. This methodology allows policymakers to monitor business dynamics in quasi-real-time during rapidly unfolding crises

The Great Migration and African-American Genomic Diversity

We present a comprehensive assessment of genomic diversity in the African-American population by studying three genotyped cohorts comprising 3,726 African-Americans from across the United States that provide a representative description of the population across all US states and socioeconomic status. An estimated 82.1% of ancestors to African-Americans lived in Africa prior to the advent of transatlantic travel, 16.7% in Europe, and 1.2% in the Americas, with increased African ancestry in the southern United States compared to the North and West. Combining demographic models of ancestry and those of relatedness suggests that admixture occurred predominantly in the South prior to the Civil War and that ancestry-biased migration is responsible for regional differences in ancestry. We find that recent migrations also caused a strong increase in genetic relatedness among geographically distant African-Americans. Long-range relatedness among African-Americans and between African-Americans and European-Americans thus track north-and west-bound migration routes followed during the Great Migration of the twentieth century. By contrast, short-range relatedness patterns suggest comparable mobility of similar to 15-16km per generation for African-Americans and European-Americans, as estimated using a novel analytical model of isolation-by-distance

Pairwise genetic relatedness across US census regions among (A) African-Americans, (B) European-Americans, and (C) African-Americans and European-Americans.

<p>(D) Census-based prediction for African-Americans (see <a href="http://www.plosgenetics.org/article/info:doi/10.1371/journal.pgen.1006059#sec008" target="_blank">Materials and Methods</a>). On each map, the line connecting two regions shows the average relatedness between individuals in those regions, and the thickness and opacity of the lines are on a linear scale between the minimum and maximum values shown above the map. Relatedness between regions with fewer than 10,000 possible pairs of individuals is not shown (see <a href="http://www.plosgenetics.org/article/info:doi/10.1371/journal.pgen.1006059#sec008" target="_blank">Materials and Methods</a> for details). All numbers are in units of cM. (E) Decay of average IBD (shown in logarithmic scale) as a function of distance using IBD segments of length 18cM or longer from HRS (dots), compared to the analytical model (lines).</p

Admixture times and proportions of ancestral populations for SCCS in (A) the model with two pulses of admixture and (B) the model with three pulses of admixture.

<p>Because the model features a continuous time parameter but discrete generation times, a single pulse occurring at a fractional time contributes migrants to the two adjacent discrete generation times. African, European, and Native American ancestries are displayed respectively in blue, red, and yellow. Rectangles show the proportion of each ancestry at each generation. Pie charts represent migrations, with the size of the pie representing the amounts of migrants at a given generation and the sectors representing the proportion of migrants coming from each source population. (C) Distribution of continuous ancestry tract lengths (dots) compared with predictions from the best-fit model (lines) for SCCS. Points in the shaded area are within one standard deviation of the predicted result. Kinks in the distribution are due to the finite length of chromosomes [<a href="http://www.plosgenetics.org/article/info:doi/10.1371/journal.pgen.1006059#pgen.1006059.ref016" target="_blank">16</a>]. (D) Inferred time to admixture and African ancestry proportions as functions of birth year in HRS African-Americans. (E) Proportions of African ancestry in African-Americans within the North, South, and West using region of birth, region of residence, and migration status; bootstrap <i>p</i>-values are calculated between disjoint sets of individuals.</p