# Morphology of an Interacting Three-Dimensional Trapped Bose–Einstein Condensate from Many-Particle Variance Anisotropy

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## Abstract

**:**

## 1. Introduction

## 2. Theory

## 3. Applications

#### 3.1. Position and Momentum Variances in an Out-of-Equilibrium Dynamics of a Three-Dimensional Trapped Bose–Einstein Condensate

#### 3.2. Angular-Momentum Variance in the Ground State of a Three-Dimensional Trapped Bose–Einstein Condensate

## 4. Summary

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Translated Angular-Momentum Variances in Three Spatial Dimensions at the Limit of an Infinite Number of Particles

## References

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**Figure 1.**Many-particle position ($\widehat{X}$, $\widehat{Y}$, and $\widehat{Z}$; in red, green, and blue) variance per particle as a function of time computed at the limit of an infinite number of particles within many-body (dashed lines) and mean-field (solid lines) levels of theory in an interaction-quench scenario. The harmonic trap is 10% anisotropic in panels (

**a**,

**c**,

**e**,

**g**) and 20% anisotropic in panels (

**b**,

**d**,

**f**,

**h**). The coupling constant g is indicated in each panel. Different anisotropy classes of the position variance emerge with time. See the text for more details. The quantities shown are dimensionless.

**Figure 2.**Many-particle momentum (${\widehat{P}}_{X}$, ${\widehat{P}}_{Y}$, and ${\widehat{P}}_{Z}$; in red, green, and blue) variance per particle as a function of time computed at the infinite-particle-number limit within many-body (dashed lines) and mean-field (solid lines) levels of theory in an interaction-quench scenario. The harmonic trap is 10% anisotropic in panels (

**a**,

**c**,

**e**,

**g**) and 20% anisotropic in panels (

**b**,

**d**,

**f**,

**h**). The coupling constant g is indicated in each panel. Different anisotropy classes of the momentum variance emerge with time. See the text for more details. The quantities shown are dimensionless.

**Figure 3.**Many-particle angular-momentum (${\widehat{L}}_{X}$, ${\widehat{L}}_{Y}$, and ${\widehat{L}}_{Z}$; in red, green, and blue) variance per particle as a function of the interaction parameter $\Lambda $ computed at the limit of an infinite number of particles within many-body (dashed lines) and mean-field (solid lines) levels of theory for the ground state of the three-dimensional anisotropic harmonic-interaction model. The frequencies of the trap are ${\omega}_{x}=0.7$, ${\omega}_{y}=5.0$, and ${\omega}_{z}=10.5$. Results at several translations ${\mathbf{r}}_{0}$ of the center of the trap are shown in the panels: (

**a**) ${\mathbf{r}}_{0}=(0,0,0)$; (

**b**) ${\mathbf{r}}_{0}=(0.25,0,0)$; (

**c**) ${\mathbf{r}}_{0}=(0,0.25,0)$; (

**d**) ${\mathbf{r}}_{0}=(0.25,0.25,0)$. Different anisotropy classes of the angular-momentum variance emerge with the interaction parameter. See the text for more details. The quantities shown are dimensionless.

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**MDPI and ACS Style**

Alon, O.E.
Morphology of an Interacting Three-Dimensional Trapped Bose–Einstein Condensate from Many-Particle Variance Anisotropy. *Symmetry* **2021**, *13*, 1237.
https://doi.org/10.3390/sym13071237

**AMA Style**

Alon OE.
Morphology of an Interacting Three-Dimensional Trapped Bose–Einstein Condensate from Many-Particle Variance Anisotropy. *Symmetry*. 2021; 13(7):1237.
https://doi.org/10.3390/sym13071237

**Chicago/Turabian Style**

Alon, Ofir E.
2021. "Morphology of an Interacting Three-Dimensional Trapped Bose–Einstein Condensate from Many-Particle Variance Anisotropy" *Symmetry* 13, no. 7: 1237.
https://doi.org/10.3390/sym13071237