# A Method for the Dynamics of Vortices in a Bose-Einstein Condensate: Analytical Equations of the Trajectories of Phase Singularities

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## Abstract

**:**

## 1. Introduction

## 2. Model and System

- Evolve each scattering mode. To this end, find the generating function (which is to evolve the fundamental mode ${\mathsf{\Phi}}_{0,0}$ in this case). Then use Equation (22) to find each evolved scattering mode.
- Use the evolved scattering mode to find the solution, via Equation (14), that is,$$\varphi (r,\theta ,t)=\sum _{\left\{\ell ,p\right\}}{t}_{\ell ,p}{\varphi}_{\ell ,p}(r,\theta ,t).$$

## 3. Some Examples in the Homogenous System

#### 3.1. Two Initial Singularities, One Positive and One Negative

#### 3.2. Two Positive Singularities and One Negative

## 4. Some Examples in the Parabolically Trapped System

## 5. Conclusions and Outlook

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**a**–

**c**) Amplitude (square root of density) of the initial condition for $N=5,20$ and 30 respectively, when there are initially two vortices of opposite charge ($q=\pm 1)$, located at $(\pm 0.5,0)$. (

**d**) corresponding phase. The phase is the same in the three cases because, due to the form of Equation (4), one decides the position of the singularities with the coordinates of the initial vortices, and therefore the phase profile. (

**e**,

**f**) amplitude and phase, respectively, for three vortices, one negatively charged ($q=-1$) in the origin and two, positively charged ($q=1$) located at $(\pm 1,0)$. In all plots, the dashed black square is the window represented in other figures in the paper.. Also notice that the computational box is much larger than the one represented in all figures.

**Figure 2.**For $N=30$ amplitude and phase for $g=0.4$ (repulsive case) after $t=1.5$ a.u. (

**a**,

**b**) and after $t=2.5$ a.u. (

**c**,

**d**). Notice we plot a box which is smaller than the computational box. For $t=1.5$ a.u. merging has not yet occurred but for $t=2.5$ a.u. the two singularities have merged leaving a phase profile without singularities. The dashed black squares mark the plotting box in other figures below.

**Figure 3.**For N = 30 amplitude and phase for g = 0.6 (repulsive case) after t = 1.5 a.u. (

**a**,

**b**) and after t = 2.5 a.u. (

**c**,

**d**). Here, merging has not occurred at $t=1.5$ a.u. nor at $t=2.5$ a.u., contrarily as in the case with $g=0.4$. The dashed black squares mark the plotting box in other figures below.

**Figure 4.**Amplitude and phase for $g=-0.5$ (attractive case) after $t=2.2$ a.u. for $N=5$ (

**a**,

**b**), $N=20$ (

**c**,

**d**), and $N=30$, (

**e**,

**f**). For $t=2.2$ a.u. and $N=5$ and $N=30$ merging has not yet occurred. However, for $N=20$ it has taken place (see Figure 4 for the cases of $N=5$ and $N=30$. We also plot in panels (

**g**,

**h**) the amplitude and phase for $N=30$ and $t=4.0$ a.u., a time in which instability has occurred and the simulation is not valid anymore.

**Figure 6.**(

**a**) Trajectories of phase singularities when the initial condition contains a phase singularity of charge $q=1$ at $\mathbf{a}=(0.5,0)$ and another phase singularity of charge $q=-1$ at $\mathbf{a}=(-0.5,0)$, when $g=0$ and $N=30$. The singularities annihilate at ${t}_{\mathrm{m}}=1.9\phantom{\rule{0.166667em}{0ex}}a.u$. (

**b**–

**d**) Trajectories of phase singularities for the same initial condition ($N=30$) for repulsive interactions given by $g=0.4,0.54$, and $g=1$, respectively. (

**e**,

**f**) Trajectories of phase singularities for an initial condition with $N=5$ in the linear case and for attractive interactions given by $g=-0.75$, respectively. For $N=5$ the merging time in the non-interacting case is ${t}_{\mathrm{m}}=3.4$ a.u. an decreases with $\left|g\right|$ (see Figure 2). Green (red) tubes represent positively (negatively) charged singularities.

**Figure 7.**(

**a**) Representation of the x position of the trajectories with time, in the repulsive case, for the same four exemplary cases as in Figure 6 ($N=30$, blue, magenta, red, green curves for $g=0,0.3,0.54$ and $g=1$, respectively). As shown, for $g=0.54$ they stay parallel for long times. (

**b**) The merging time as a function of g, for $N=5$ (upper, blue curve), $N=20$ (green middle curve), and $N=30$ (lower, orange curve) atoms in the repulsive case. For $N=5$ the merging time is increased for any value of g. On the contrary, for $N=30$, only for g larger than 0.38 the merging time starts to grow. (

**c**) same than (

**a**) for the attractive case, and three exemplary cases ($N=5$, blue, magenta, red curves for $g=0,-0.25$ and $g=-0.75$, respectively). (

**d**) The merging time as a function of $\left|g\right|$, for $N=5$ (blue curve), $N=10$ (green curve), and $N=20$ (orange curve) atoms in the attractive case. For $N=5$ the merging time decreases with $\left|g\right|$. For $N=10$ it decreases slightly and for $N=20$ it increases. In all cases we show the results before instability occurs (we do not show the case with $N=30$ as instability occurs already for $g=-0.3$).

**Figure 8.**For $N=150$, amplitude and phase for $g=0.3$ after $t=1$ a.u. (

**a**,

**b**) and after $t=4$ a.u. (

**c**,

**d**). In (

**d**) we see that merging of two of the singularities has occurred, leaving only one on-axis singularity. (

**e**–

**h**), same for $g=0.6$. Now, at $t=4$ a.u. merging has not yet occurred. Also two pairs of singularities have appeared far from origin. The dashed black squares mark the plotting box in Figure 9.

**Figure 9.**(

**a**) Trajectories of phase singularities when the initial condition contains a phase singularity of charge $q=-1$ at the origin, and two singularities with $q=1$ each one, at $\mathbf{a}=(\pm ,1,0)$, when (

**a**) $g=0$, (

**b**) $g=0.3$, (

**c**) $g=0.45$ and (

**d**) $g=0.6$, when $N=150$. For the non-interacting case, the merging point coincides with the analytically calculated. The interactions move the merging time (

**b**). There is also a re-appearance of the singularities (see panels (

**b**,

**c**)). For large enough singularities, these singularities do not merge again for very long simulations (see panel (

**d**)). There is also one more effect visible in panel (

**d**): the generation of vortex-antivortex pairs. Green (red) tubes represent positively (negatively) charged singularities.

**Figure 10.**(

**a**) Trajectory of a single phase singularity in a parabolic trap, initially off-axis, i.e., located at $\mathbf{a}=(1,0)$, with $N=12$ atoms. (

**b**) Trajectories of two phase singularities in a parabolic trap, one positively charged initially at $\mathbf{a}=(1,0)$ and one negatively charged initially at $\mathbf{b}=(-1,0)$, with $N=30$ atoms. In this case, the singularities merge at a ${t}_{\mathrm{m}}$ which we obtain numerically. (

**c**) Trajectories of a negatively charged phase singularity initially at the center of the trap, and two positively charged phase singularities initially at $\mathbf{a}=(1.5,0)$ and ${\mathbf{a}}^{\prime}=(-1.5,0)$, respectively, in a potential parabolic trap, with $N=150$ atoms. The two positively charged singularities tend to the center of the trap, where one of them annihilates the central negative phase singularity leaving only one positively charged singularity which stays there for the rest of the evolution. In all cases, $g=0$. In all panels, time is represented with a color gradient from blue to red. Green (red) circles represent positively (negatively) charged singularities. Also, we plot on top of each panel the evolution in time, for better visualization.

**Figure 11.**Trajectory of two phase singularities in a parabolic trap, one positively charged initially at $\mathbf{a}=(0.5,0)$ and one negatively charged initially at $\mathbf{b}=(-0.5,0)$, for $N=30$ atoms, and (

**a**) $g=0$ and (

**b**) $g=0.3$. In the linear case the singularities merge at a merging time of ${t}_{}=1.2$ a.u. In the interacting case, the singularities tend to the origin but do not merge. Instead, after colliding at the center of the trap they perform an intricate dynamics. Green (red) tubes represent positively (negatively) charged singularities.

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## Share and Cite

**MDPI and ACS Style**

De María-García, S.; Ferrando, A.; Conejero, J.A.; De Córdoba, P.F.; García-March, M.Á.
A Method for the Dynamics of Vortices in a Bose-Einstein Condensate: Analytical Equations of the Trajectories of Phase Singularities. *Condens. Matter* **2023**, *8*, 12.
https://doi.org/10.3390/condmat8010012

**AMA Style**

De María-García S, Ferrando A, Conejero JA, De Córdoba PF, García-March MÁ.
A Method for the Dynamics of Vortices in a Bose-Einstein Condensate: Analytical Equations of the Trajectories of Phase Singularities. *Condensed Matter*. 2023; 8(1):12.
https://doi.org/10.3390/condmat8010012

**Chicago/Turabian Style**

De María-García, Sergi, Albert Ferrando, J. Alberto Conejero, Pedro Fernández De Córdoba, and Miguel Ángel García-March.
2023. "A Method for the Dynamics of Vortices in a Bose-Einstein Condensate: Analytical Equations of the Trajectories of Phase Singularities" *Condensed Matter* 8, no. 1: 12.
https://doi.org/10.3390/condmat8010012