# Symmetry Breaking in Interacting Ring-Shaped Superflows of Bose–Einstein Condensates

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## Abstract

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## 1. Introduction

## 2. The Model

## 3. Symmetry Breaking in Coupled Condensate Rings

#### 3.1. Spontaneous Symmetry Breaking in a Stationary Hybrid Vortex Structure

#### 3.2. Influence of the Symmetry on Dynamics of the Merging Rings

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(Color online) Hybrid vortex stationary states with hidden vorticity and zero population imbalance, $P=0$ (see Equation (13)): (

**a**) $\left({m}_{1,}{m}_{2}\right)=(+1,-1)$, (

**b**) $(+2,-2)$, (

**c**) $(+3,-3)$. Shown are density isosurfaces (the top row) and the z-component of the corresponding tunnel-flow density distribution through the barrier, ${j}_{z}(x,y,z=0)$ (the bottom row). The cores of the Josephson vortices are indicated by black lines in the top row, and by white dashed lines in the bottom one. Vertical red and blue dashed lines designate cores of the counter-propagating persistent currents in the two rings.

**Figure 2.**(Color online) The final value of the total angular momentum per particle, ${L}_{p}={L}_{z}/N$, for the merging rings with initial vorticities $({m}_{1},{m}_{2})$, as a function of initial imbalance P: (

**a**) $({m}_{1}=+1,{m}_{2}=-1)$, (

**b**) $({m}_{1}=+2,{m}_{2}=-2)$. Solid black lines with circles represent the final states for the axially symmetric trapping potential (the horizontal sheet beam, which corresponds to $\Omega =0$ in Equation (6)). Surprisingly, merging counter-rotating flows in the axially-symmetric trap never evolve towards the non-rotating ground state, with ${L}_{p}=0$, even for small imbalances, $|P|\ll 1$. The vorticity of the final state is imposed by the less populated component if $|P|<{P}_{\mathrm{cr}}\approx 0.1755$ (see Equation (19)) for initial vorticities $(+1,-1)$, and ${P}_{\mathrm{cr}}\approx 0.21$ for $(+2,-2)$ (this counter-intuitive result is explained in the main text), and by the stronger component if $|P|>{P}_{\mathrm{cr}}$. The impact of the symmetry breaking of the trapping potential is illustrated for the setup with initial vorticities $(+1,-1)$ by examples of the final states for two values of the imbalance, $P=0.06$ and $P=0.29$. For $P=0.06$, the filled black circle corresponds to the nonrotating (horizontal) barrier (with ${\Omega}_{0}=0$ in Equation (6), see Figure 3), while the blue triangle and red square correspond to the barrier rotating around the x-axis barrier with angular velocity ${\Omega}_{1}=2\pi \times 0.11$ Hz (see Figure 4) and ${\Omega}_{2}=2\pi \times 0.23$ Hz (see Figure 5), respectively. For $P=0.29$, the filled black circle corresponds to the ${\Omega}_{0}=0$, and the green diamond corresponds to ${\Omega}_{1}=2\pi \times 0.11$ Hz. Note that, above the threshold imbalance, $P>{P}_{\mathrm{cr}}$, the final state with ${L}_{p}=-1$ is never observed even for the system with broken symmetry.

**Figure 3.**(Color online) The evolution of the merging rings without symmetry breaking ($\Omega =0$ in Equation (6)). The barrier separating two rings is switched off at $t>{t}_{d}=0.015$ s. Red (blue) lines indicates positions of the vortex (antivortex) core. The population of the bottom ring, with ${m}_{1}=+1$, is slightly larger than in the top one, with ${m}_{2}=-1$ (imbalance parameter (13) is $P=0.06$). The final state has $m=-1$, as shown in Figure 2a by the filled black circle. Note that, in the course of the evolution of the merging counter-rotating flows in the axially symmetric trap, the discrete rotational symmetry is observed for the positions of the vortex cores with respect to the rotation around the z-axis by an angle of $\pi $. The symmetric drift of two diametrically opposite antivortices towards the central hole leads to subsequent annihilation of the central vortex and relaxation of the toroidal condensate into a final antivortex state.

**Figure 4.**(Color online) The evolution of the merging rings affected by symmetry breaking which is induced by slow rotation of the sheet beam (6) around the x-axis, with angular velocity ${\Omega}_{1}=2\pi \times 0.11$ Hz. Note that the system with the broken symmetry evolves towards the final topological charge (vorticity) $m=+1$, while, for the same value of imbalance (13), $P=0.06$ (the population of the bottom ring with ${m}_{1}=+1$ slightly dominates over the top one, with ${m}_{2}=-1$), the final state of the axially-symmetric system has $m=-1$ (see Figure 3).

**Figure 5.**(Color online) The evolution of the merging rings affected by stronger, than in the case of Figure 4, symmetry breaking, imposed by rotation (6) with angular velocity ${\Omega}_{2}=2\pi \times 0.23$ Hz. The population in the bottom ring with ${m}_{1}=+1$ slightly dominates over the top one, with ${m}_{2}=-1$ (the respective imbalance parameter (13) is $P=0.06$, as well as in Figure 3 and Figure 4). Being controlled by the barrier’s angular velocity, $\Omega $, the symmetry breaking drives the merging rings to final states with different topological charges (vorticities). In this case, when the axial symmetry is strongly broken, the final nonrotating state is established, with vorticity $m=0$. Surprisingly, merging counter-rotating persistent currents evolve into a nonrotating final state neither for the symmetric system (see Figure 3, where ${\Omega}_{0}=0$ and the final topological charge is $m=-1$), nor for a weakly asymmetric trapping potential (see Figure 4, where ${\Omega}_{1}=2\pi \times 0.11$ Hz and the final topological charge is $m=+1$).

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**MDPI and ACS Style**

Oliinyk, A.; Yatsuta, I.; Malomed, B.; Yakimenko, A.
Symmetry Breaking in Interacting Ring-Shaped Superflows of Bose–Einstein Condensates. *Symmetry* **2019**, *11*, 1312.
https://doi.org/10.3390/sym11101312

**AMA Style**

Oliinyk A, Yatsuta I, Malomed B, Yakimenko A.
Symmetry Breaking in Interacting Ring-Shaped Superflows of Bose–Einstein Condensates. *Symmetry*. 2019; 11(10):1312.
https://doi.org/10.3390/sym11101312

**Chicago/Turabian Style**

Oliinyk, Artem, Igor Yatsuta, Boris Malomed, and Alexander Yakimenko.
2019. "Symmetry Breaking in Interacting Ring-Shaped Superflows of Bose–Einstein Condensates" *Symmetry* 11, no. 10: 1312.
https://doi.org/10.3390/sym11101312