# Faraday and Resonant Waves in Dipolar Cigar-Shaped Bose-Einstein Condensates

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Variational Approach

## 3. Faraday Waves in Chromium, Erbium, and Dysprosium Condensates

## 4. Interaction Effects and Properties of Faraday Waves

## 5. Resonant Waves

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Time evolution of the integrated density profile variation $\delta n(x,t)$ in the weak-confinement direction for a BEC of chromium ${}^{52}$Cr (

**top**), erbium ${}^{168}$Er (

**middle**), and dysprosium ${}^{164}$Dy (

**bottom**), for the modulation frequency ${\omega}_{m}=200\times 2\pi $ Hz and amplitude $\u03f5=0.2$, and the system parameters given in Section 2.

**Figure 2.**The Fourier spectrum in the time-frequency domain of the integrated 1D density profile variations of Faraday waves at the trap center $\delta n(x=0,t)$ in x direction (

**left column**), $\delta n(y=0,t)$ in y direction (

**middle column**), and $\delta n(z=0,t)$ in z direction (

**right column**) for a BEC of chromium ${}^{52}$Cr (

**top row**), erbium ${}^{168}$Er (

**middle row**), and dysprosium ${}^{164}$Dy (

**bottom row**). Vertical blue lines represent theoretical predictions, where ${\omega}_{m}/2$ corresponds to Faraday waves, ${\omega}_{m}$, $3{\omega}_{m}/2$, and $2\phantom{\rule{0.166667em}{0ex}}{\omega}_{m}$ to resonant waves, and ${\omega}_{B}$ is the variational result for the breathing mode frequency, which is obtained by linearization of the equations of motion from Section 2.

**Figure 3.**The Fourier spectrum in the spatial-frequency domain of the integrated 1D density profile variations of Faraday waves in x direction $\delta n(x,t=272\phantom{\rule{3.33333pt}{0ex}}\mathrm{ms})$ for ${}^{52}$Cr (

**left**), $\delta n(x,t=225\phantom{\rule{3.33333pt}{0ex}}\mathrm{ms})$ for ${}^{168}$Er (

**middle**), and $\delta n(x,t=193\phantom{\rule{3.33333pt}{0ex}}\mathrm{ms})$ for ${}^{164}$Dy (

**right**) BECs with $N={10}^{4}$ atoms. The corresponding density profile variations are shown in Figure 1. Vertical blue lines represent theoretical predictions for the wave vector ${k}_{F}$ of the Faraday waves, Equation (23).

**Figure 4.**Emergence of Faraday waves for different strengths of the contact interaction: ${a}_{s}=60\phantom{\rule{0.166667em}{0ex}}{a}_{0}$ (

**top**), ${a}_{s}=80\phantom{\rule{0.166667em}{0ex}}{a}_{0}$ (

**middle**), and ${a}_{s}=150\phantom{\rule{0.166667em}{0ex}}{a}_{0}$ (

**bottom**) for a BEC of ${}^{52}$Cr. We observe that Faraday waves emerge faster as the contact interaction strength increases.

**Figure 5.**Wave vector of the Faraday waves ${k}_{F}$ as a function of the contact interaction strength for a BEC of ${}^{52}$Cr (

**left**), ${}^{168}$Er (

**middle**), and ${}^{164}$Dy (

**right**), for a fixed DDI strength. Red upper triangles were numerically obtained values using the FFT analysis as in Figure 3, and blue lines are the variational results according to Equation (23).

**Figure 6.**Wave vector of the Faraday waves ${k}_{F}$ as a function of the DDI strength for a BEC of ${}^{52}$Cr (

**left**), ${}^{168}$Er (

**middle**), and ${}^{164}$Dy (

**right**), for a fixed contact interaction strength. Red upper triangles represent numerically obtained values using the FFT analysis as in Figure 3, and blue lines are the variational results according to Equation (23).

**Figure 7.**Time evolution of the integrated density profile variation in the weak confinement direction for a BEC of ${}^{168}$Er, with the modulation frequency equal to the weak confinement frequency, ${\omega}_{m}={\Omega}_{0}$. We observe resonant behavior corresponding to the first harmonic of the resonant frequency ${\Omega}_{0}$, which sets in after around 55 ms.

**Figure 8.**The Fourier spectrum of the integrated 1D density profile variations $\delta n(x,t)$ at the trap center in the time-frequency domain (

**left**), and of the density profile variations in x direction $\delta n(x,t=68\phantom{\rule{3.33333pt}{0ex}}\mathrm{ms})$ in the spatial-frequency domain (

**right**) of resonant waves for a BEC of ${}^{168}$Er. Vertical blue line in the left panel represents the modulation frequency ${\omega}_{m}$, while in the right panel it corresponds to the theoretical prediction for the wave vector ${k}_{R}$ of the resonant waves, Equation (24).

**Figure 9.**Wave vector of the resonant waves ${k}_{R}$ as a function of the contact (

**left**) and the DDI (

**right**) strength for a BEC of ${}^{168}$Er. The results in the left panel are obtained for a fixed DDI strength, and similarly in the right panel a fixed contact interaction strength is used. In both panels, red upper triangles represent numerically obtained values using the FFT analysis as in the right panel of Figure 8, and blue lines are the variational results according to Equation (24).

**Figure 10.**Time evolution of the integrated density profile variation in the weak confinement direction for a BEC of ${}^{168}$Er. The modulation frequency is equal to twice the weak confinement frequency, ${\omega}_{m}=2{\Omega}_{0}$. We observe resonant behavior corresponding to the second harmonic of the resonant frequency ${\Omega}_{0}$, which sets in more quickly than the first harmonic, already after around 30 ms.

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**MDPI and ACS Style**

Vudragović, D.; Balaž, A.
Faraday and Resonant Waves in Dipolar Cigar-Shaped Bose-Einstein Condensates. *Symmetry* **2019**, *11*, 1090.
https://doi.org/10.3390/sym11091090

**AMA Style**

Vudragović D, Balaž A.
Faraday and Resonant Waves in Dipolar Cigar-Shaped Bose-Einstein Condensates. *Symmetry*. 2019; 11(9):1090.
https://doi.org/10.3390/sym11091090

**Chicago/Turabian Style**

Vudragović, Dušan, and Antun Balaž.
2019. "Faraday and Resonant Waves in Dipolar Cigar-Shaped Bose-Einstein Condensates" *Symmetry* 11, no. 9: 1090.
https://doi.org/10.3390/sym11091090