# Josephson-like Oscillations in Toroidal Spinor Bose–Einstein Condensates: A Prospective Symmetry Probe

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## Abstract

**:**

## 1. Introduction

## 2. General Modeling of Spinor BECs

## 3. Modeling the Potential Barrier and Defining Some Approximations

## 4. Current Density and the Parity of the Order Parameter

## 5. The Broken-Symmetry Spinor BEC Case

#### 5.1. Ferromagnetic States

#### 5.2. Polar State

## 6. Landau Critical Velocity, Lower Bound for R, and Experimental Protocol

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## References

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**Figure 1.**Schematic illustration of the external potentials applied to the condensate: (

**a**) effective toroidal region $\mathcal{T}$ (with average radius R) where the potential ${U}_{\mathrm{trap}}$ confines the condensate; (

**b**) repulsive barrier ${U}_{B}={U}_{B}\left(\phi \right)$, for $\phi $ the azimuthal angle in cylindrical coordinates $(r,\phi ,z)$. Here, we consider thin angular widths $2\delta \ll 2\pi $.

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**MDPI and ACS Style**

Donato, M.H.F.; Muniz, S.R.
Josephson-like Oscillations in Toroidal Spinor Bose–Einstein Condensates: A Prospective Symmetry Probe. *Symmetry* **2022**, *14*, 867.
https://doi.org/10.3390/sym14050867

**AMA Style**

Donato MHF, Muniz SR.
Josephson-like Oscillations in Toroidal Spinor Bose–Einstein Condensates: A Prospective Symmetry Probe. *Symmetry*. 2022; 14(5):867.
https://doi.org/10.3390/sym14050867

**Chicago/Turabian Style**

Donato, Mário H. Figlioli, and Sérgio R. Muniz.
2022. "Josephson-like Oscillations in Toroidal Spinor Bose–Einstein Condensates: A Prospective Symmetry Probe" *Symmetry* 14, no. 5: 867.
https://doi.org/10.3390/sym14050867