# Analysis of a Trapped Bose–Einstein Condensate in Terms of Position, Momentum, and Angular-Momentum Variance

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## Abstract

**:**

## 1. Introduction

## 2. The Anisotropic Harmonic-Interaction Model

## 3. Bosons in an Annulus Subject to a Tilt

## 4. Summary and Outlook

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Variances and Translations

## References

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**Figure 1.**Center-of-mass dynamics following a potential quench. The mean-field ($M=1$ time-adaptive orbitals) and many-body (using $M=3$, 5, 7, 10, 12, 14, and 15, 16 time-adaptive orbitals) time-dependent expectation value of the center-of-mass, $\frac{1}{N}\langle \Psi |\widehat{X}|\Psi \rangle (t)$, of $N=10$ bosons in the annuli with barrier heights and interaction strengths: (

**a**) ${V}_{0}=5$, ${\lambda}_{0}=0.02$; (

**b**) ${V}_{0}=5$, ${\lambda}_{0}=0.04$; (

**c**) ${V}_{0}=10$, ${\lambda}_{0}=0.02$; and (

**d**) ${V}_{0}=10$, ${\lambda}_{0}=0.04$ following a sudden potential tilt by $0.01x$. The corresponding depletions are plotted in Figure 2 and the respective position, momentum, and angular-momentum variances in Figure 3, Figure 4 and Figure 5. See the text for more details. The quantities shown are dimensionless.

**Figure 2.**Depletion dynamics following a potential quench. The time-dependent total number of depleted particles, $N-{n}_{1}(t)$, of $N=10$ bosons following a sudden potential tilt by $0.01x$ for annuli with barrier heights and interaction strengths (

**a**) ${V}_{0}=5$, ${\lambda}_{0}=0.02$; (

**b**) ${V}_{0}=5$, ${\lambda}_{0}=0.04$; (

**c**) ${V}_{0}=10$, ${\lambda}_{0}=0.02$; and (

**d**) ${V}_{0}=10$, ${\lambda}_{0}=0.04$. $M=3$, 5, 7, 10, 12, 14, and 15, 16 time-adaptive orbitals are used. The respective position, momentum, and angular-momentum variances are plotted in Figure 3, Figure 4 and Figure 5. See the text for more details. The quantities shown are dimensionless.

**Figure 3.**Position variance dynamics following a potential quench. The mean-field ($M=1$ time- adaptive orbitals) and many-body (using $M=3$, 5, 7, 10, 12, 14, and 15, 16 time-adaptive orbitals) time-dependent position variances per particle, $\frac{1}{N}{\Delta}_{\widehat{X}}^{2}(t)$ [left column, panels (

**a**–

**d**)] and $\frac{1}{N}{\Delta}_{\widehat{Y}}^{2}(t)$ [right column, panels (

**e**–

**h**)], of $N=10$ bosons in the annuli with barrier heights and interaction strengths (

**a**,

**e**) ${V}_{0}=5$, ${\lambda}_{0}=0.02$; (

**b**,

**f**) ${V}_{0}=5$, ${\lambda}_{0}=0.04$; (

**c**,

**g**) ${V}_{0}=10$, ${\lambda}_{0}=0.02$; and (

**d**,

**h**) ${V}_{0}=10$, ${\lambda}_{0}=0.04$ following a sudden potential tilt by $0.01x$. The respective depletions are plotted in Figure 2. See the text for more details. The quantities shown are dimensionless.

**Figure 4.**Momentum variance dynamics following a potential quench. The mean-field ($M=1$ time-adaptive orbitals) and many-body (using $M=3$, 5, 7, 10, 12, 14, and 15, 16 time-adaptive orbitals) time-dependent momentum variances per particle, $\frac{1}{N}{\Delta}_{{\widehat{P}}_{X}}^{2}(t)$ [left column, panels (

**a**–

**d**)] and $\frac{1}{N}{\Delta}_{{\widehat{P}}_{Y}}^{2}(t)$ [right column, panels (

**e**–

**h**)], of $N=10$ bosons in the annuli with barrier heights and interaction strengths (

**a**,

**e**) ${V}_{0}=5$, ${\lambda}_{0}=0.02$; (

**b**,

**f**) ${V}_{0}=5$, ${\lambda}_{0}=0.04$; (

**c**,

**g**) ${V}_{0}=10$, ${\lambda}_{0}=0.02$; and (

**d**,

**h**) ${V}_{0}=10$, ${\lambda}_{0}=0.04$ following a sudden potential tilt by $0.01x$. The respective depletions are plotted in Figure 2. See the text for more details. The quantities shown are dimensionless.

**Figure 5.**Angular-momentum variance dynamics following a potential quench. The mean-field ($M=1$ time-adaptive orbitals) and many-body (using $M=3$, 5, 7, 10, 12, 14, and 15, 16 time-adaptive orbitals) time-dependent angular-momentum variance per particle, $\frac{1}{N}{\Delta}_{{\widehat{L}}_{Z}}^{2}(t)$, of $N=10$ bosons in the annuli with barrier heights and interaction strengths (

**a**) ${V}_{0}=5$, ${\lambda}_{0}=0.02$; (

**b**) ${V}_{0}=5$, ${\lambda}_{0}=0.04$; (

**c**) ${V}_{0}=10$, ${\lambda}_{0}=0.02$; and (

**d**) ${V}_{0}=10$, ${\lambda}_{0}=0.04$ following a sudden potential tilt by $0.01x$. The respective depletions are plotted in Figure 2. See the text for more details. The quantities shown are dimensionless.

**Figure 6.**Depletion dynamics following a potential quench en route to the particle limit. The time-dependent total number of depleted particles, $N-{n}_{1}(t)$, of $N=10$, $N=100$, and $N=1000$ bosons with interaction parameter $\Lambda ={\lambda}_{0}(N-1)=0.36$ for an annulus with barrier height ${V}_{0}=10$ following a sudden potential tilt by $0.01x$. The number of time-adaptive orbitals is $M=3$. The respective position, momentum, and angular-momentum variances along with the expectation value of the center-of-mass are plotted in Figure 7. See the text for more details. The quantities shown are dimensionless.

**Figure 7.**Position, momentum, and angular-momentum variance dynamics following a potential quench en route to the particle limit. The mean-field ($M=1$ time-adaptive orbitals) and many-body (using $M=3$ time-adaptive orbitals) time-dependent position variances per particle, (

**a**) $\frac{1}{N}{\Delta}_{\widehat{X}}^{2}(t)$ and (

**b**) $\frac{1}{N}{\Delta}_{\widehat{Y}}^{2}(t)$, momentum variances per particle, (

**c**) $\frac{1}{N}{\Delta}_{{\widehat{P}}_{X}}^{2}(t)$ and (

**d**) $\frac{1}{N}{\Delta}_{{\widehat{P}}_{Y}}^{2}(t)$, and angular-momentum variance per particle, (

**f**) $\frac{1}{N}{\Delta}_{{\widehat{L}}_{Z}}^{2}(t)$, of $N=10$, $N=100$, and $N=1000$ bosons with interaction parameter $\Lambda ={\lambda}_{0}(N-1)=0.36$ for an annulus with barrier height ${V}_{0}=10$ following a sudden potential tilt by $0.01x$. (

**e**) The time-dependent expectation value of the center-of-mass, $\frac{1}{N}\langle \Psi |\widehat{X}|\Psi \rangle (t)$. The respective depletions are plotted in Figure 6. See the text for more details. The quantities shown are dimensionless.

© 2019 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Alon, O.E. Analysis of a Trapped Bose–Einstein Condensate in Terms of Position, Momentum, and Angular-Momentum Variance. *Symmetry* **2019**, *11*, 1344.
https://doi.org/10.3390/sym11111344

**AMA Style**

Alon OE. Analysis of a Trapped Bose–Einstein Condensate in Terms of Position, Momentum, and Angular-Momentum Variance. *Symmetry*. 2019; 11(11):1344.
https://doi.org/10.3390/sym11111344

**Chicago/Turabian Style**

Alon, Ofir E. 2019. "Analysis of a Trapped Bose–Einstein Condensate in Terms of Position, Momentum, and Angular-Momentum Variance" *Symmetry* 11, no. 11: 1344.
https://doi.org/10.3390/sym11111344