The uncertainty product of an out-of-equilibrium many-particle system

In the present work we show, analytically and numerically, that the variance of many-particle operators and their uncertainty product for an out-of-equilibrium Bose-Einstein condensate (BEC) can deviate from the outcome of the time-dependent Gross-Pitaevskii dynamics, even in the limit of infinite number of particles and at constant interaction parameter when the system becomes 100% condensed. We demonstrate our finding on the dynamics of the center-of-mass position--momentum uncertainty product of a freely expanding as well as of a trapped BEC. This time-dependent many-body phenomenon is explained by the existence of time-dependent correlations which manifest themselves in the system's reduced two-body density matrix used to evaluate the uncertainty product. Our work demonstrates that one has to use a many-body propagation theory to describe an out-of-equilibrium BEC, even in the infinite particle limit.Comment: 26 pages, 5 figure

Uncertainty product of an out-of-equilibrium Bose-Einstein condensate

The variance and uncertainty product of the position and momentum many-particle operators of structureless bosons interacting by a long-range inter-particle interaction and trapped in a single-well potential are investigated. In the first example, of an out-of-equilibrium interaction-quench scenario, it is found that, despite the system being fully condensed, already when a fraction of a particle is depleted differences with respect to the mean-field quantities emerge. In the second example, of the pathway from condensation to fragmentation of the ground state, we find out that, although the cloud's density broadens while the system's fragments, the position variance actually decreases, the momentum variance increases, and the uncertainty product is not a monotonous function but has a maximum. Implication are briefly discussed.Comment: 14 pages, 3 figure

Many-body effects in the excitation spectrum of weakly-interacting Bose-Einstein condensates in one-dimensional optical lattices

In this work, we study many-body excitations of Bose-Einstein condensates (BECs) trapped in periodic one-dimensional optical lattices. In particular, we investigate the impact of quantum depletion onto the structure of the low-energy spectrum and contrast the findings to the mean-field predictions of the Bogoliubov-de Gennes (BdG) equations. Accurate results for the many-body excited states are obtained by applying a linear-response theory atop the MCTDHB (multiconfigurational time-dependent Hartree method for bosons) equations of motion, termed LR-MCTDHB. We demonstrate for condensates in a triple well that even weak ground-state depletion of around $1\%$ leads to visible many-body effects in the low-energy spectrum which deviate substantially from the corresponding BdG spectrum. We further show that these effects also appear in larger systems with more lattice sites and particles, indicating the general necessity of a full many-body treatment

Many-body tunneling dynamics of Bose-Einstein condensates and vortex states in two spatial dimensions

In this work, we study the out-of-equilibrium many-body tunneling dynamics of a Bose-Einstein condensate in a two-dimensional radial double well. We investigate the impact of interparticle repulsion and compare the influence of angular momentum on the many-body tunneling dynamics. Accurate many-body dynamics are obtained by solving the full many-body Schr\"odinger equation. We demonstrate that macroscopic vortex states of definite total angular momentum indeed tunnel and that, even in the regime of weak repulsions, a many-body treatment is necessary to capture the correct tunneling dynamics. As a general rule, many-body effects set in at weaker interactions when the tunneling system carries angular momentum.Comment: 26 pages, 9 figure

Breaking the resilience of a two-dimensional Bose-Einstein condensate to fragmentation

A two-dimensional Bose-Einstein condensate (BEC) split by a radial potential barrier is investigated. We determine on an accurate many-body level the system's ground-state phase diagram as well as a time-dependent phase diagram of the splitting process. Whereas the ground state is condensed for a wide range of parameters, the time-dependent splitting process leads to substantial fragmentation. We demonstrate for the first time the dynamical fragmentation of a BEC despite its ground state being condensed. The results are analyzed by a mean-field model and suggest that a large manifold of low-lying fragmented excited states can significantly impact the dynamics of trapped two-dimensional BECs.Comment: 5+eps pages, 4 figure

performance: An R package for assessment, comparison and testing of statistical models

A crucial part of statistical analysis is evaluating a model’s quality and fit, or performance. During analysis, especially with regression models, investigating the fit of models to data also often involves selecting the best fitting model amongst many competing models. Upon investigation, fit indices should also be reported both visually and numerically to bring readers in on the investigative effort. The performance R-package (R Core Team, 2021) provides utilities for computing measures to assess model quality, many of which are not directly provided by R’s base or stats packages. These include measures like R2, intraclass correlation coefficient (ICC), root mean squared error (RMSE), or functions to check for vexing issues like overdispersion, singularity, or zeroinflation. These functions support a large variety of regression models including generalized linear models, (generalized) mixed-effects models, their Bayesian cousins, and many others