Variance as a sensitive probe of correlations enduring the infinite particle limit

Bose-Einstein condensates made of ultracold trapped bosonic atoms have become a central venue in which interacting many-body quantum systems are studied. The ground state of a trapped Bose-Einstein condensate has been proven to be 100% condensed in the limit of infinite particle number and constant interaction parameter [Lieb and Seiringer, Phys. Rev. Lett. {\bf 88}, 170409 (2002)]. The meaning of this result is that properties of the condensate, noticeably its energy and density, converge to those obtained by minimizing the Gross-Pitaevskii energy functional. This naturally raises the question whether correlations are of any importance in this limit. Here, we demonstrate both analytically and numerically that even in the infinite particle limit many-body correlations can lead to a substantial modification of the \textit{variance} of any operator compared to that expected from the Gross-Pitaevskii result. The strong deviation of the variance stems from its explicit dependence on terms of the reduced two-body density matrix which otherwise do not contribute to the energy and density in this limit. This makes the variance a sensitive probe of many-body correlations even when the energy and density of the system have already converged to the Gross-Pitaevskii result. We use the center-of-mass position operator to exemplify this persistence of correlations. Implications of this many-body effect are discussed.Comment: 20 pages, 6 figure

Impact of the range of the interaction on the quantum dynamics of a bosonic Josephson junction

The out-of-equilibrium quantum dynamics of a bosonic Josephson junction (BJJ) with long-range interaction is studied in real space by solving the time-dependent many-body Schr\"odinger equation numerically accurately using the multiconfigurational time-dependent Hartree method for bosons. Having the many-boson wave-function at hand we can examine the impact of the range of the interaction on the properties of the BJJ dynamics, viz. density oscillations and their collapse, self trapping, depletion and fragmentation, as well as the position variance, both at the mean-field and many-body level. Explicitly, the frequency of the density oscillations and the time required for their collapse, the value of fragmentation at the plateau, the maximal and the minimal values of the position variance in each cycle of oscillation and the overall pace of its growth are key to our study. We find competitive effect between the interaction and the confining trap. The presence of the tail part of the interaction basically enhances the effective repulsion as the range of the interaction is increased starting from a short, finite range. But as the range becomes comparable with the trap size, the system approaches a situation where all the atoms feel a constant potential and the impact of the tail on the dynamics diminishes. There is an optimal range of the interaction in which physical quantities of the junction are attaining their extreme values.Comment: Contribution to the Special Issue of Chemical Physics dedicated to Professor Hans-Dieter Meyer on the occasion of his 70th birthday; few typos correcte

Many-Body Quantum Dynamics of a Bosonic Josephson Junction with a Finite-Range Interaction

The out-of-equilibrium quantum dynamics of a Bose gas trapped in an asymmetric double well and interacting with a finite-range interaction has been studied in real space by solving the time-dependent many-body Schr\"odinger equation numerically accurately using the multiconfigurational time-dependent Hartree method for bosons (MCTDHB). We have focused on the weakly interacting limit where the system is essentially condensed. We have examined the impact of the range of the interaction on the dynamics of the system, both at the mean-field and many-body levels. Explicitly, we have studied the maximal and the minimal values of the many-body position variance in each cycle of oscillation, and the overall pace of its growth. We find that the range of the interaction affects the dynamics of the system differently for the right well and the left well. We have also examined the infinite-particle limit and find that even there, the impact of the range of the interaction can only be described by a many-body theory such as MCTDHB

Condensates in annuli: Dimensionality of the variance

Static and dynamic properties of Bose-Einstein condensates in annular traps are investigated by solving the many-boson Schr\"odinger equation numerically accurately using the multiconfigurational time-dependent Hartree for bosons method. We concentrate on weakly-interacting bosons exhibiting low depletion. Analysis of the mean-field position variance, which accounts for the shape of the density only, and the many-body position variance, which incorporates a tiny amount of excitations through the reduced two-particle density matrix, shows that the former behaves essentially as a quasi-one-dimensional quantity whereas the latter as a two-dimensional quantity. This brings another dimension to the physics of bosons in ring-shaped traps.Comment: 24 pages, 8 figure

Morphology of an interacting three-dimensional trapped Bose-Einstein condensate from many-particle variance anisotropy

The variance of the position operator is associated with how wide or narrow a wave-packet is, the momentum variance is similarly correlated with the size of a wave-packet in momentum space, and the angular-momentum variance quantifies to what extent a wave-packet is non-spherically symmetric. We examine an interacting three-dimensional trapped Bose-Einstein condensate at the limit of an infinite number of particles, and investigate its position, momentum, and angular-momentum anisotropies. Computing the variances of the three Cartesian components of the position, momentum, and angular-momentum operators we present simple scenarios where the anisotropy of a Bose-Einstein condensate is different at the many-body and mean-field levels of theory, despite having the same many-body and mean-field densities per particle. This suggests a way to classify correlations via the morphology of 100\% condensed bosons in a three-dimensional trap at the limit of an infinite number of particles. Implications are briefly discussed.Comment: 23 pages, 3 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