# Characteristic Length Scale during the Time Evolution of a Turbulent Bose-Einstein Condensate

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## Abstract

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## 1. Introduction

## 2. Experimental Procedure

## 3. Momentum Distributions

## 4. The Characteristic Length Scale

## 5. Discussion and Final Remarks

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

QT | Quantum turbulence |

BEC | Bose-Einstein condensate |

QUIC | Quadrupole–Ioffe configuration |

ToF | Time-of-flight |

## Appendix A. The Abel Transform

## References

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**Figure 1.**(

**a**) Schematic representation of the excitation protocol. The experiment begins with the production of an unperturbed BEC in the trap. Then, a sinusoidal potential of amplitude A and period $\tau $ is applied during ${t}_{\mathrm{exc}}$. The system evolves during a time t, after which the trap is released, and an absorption image is taken after a time-of-flight ${t}_{\mathrm{ToF}}$. (

**b**) Absorption images for an excitation amplitude of $A=1.8$ ${\mu}_{0}$ as a function of the holding time.

**Figure 2.**Momentum distribution $n({k}_{x},{k}_{y})$ obtained from the absorption images of the cloud for an excitation of amplitude $A=1.8$ ${\mu}_{0}$ and $t=36.7\phantom{\rule{0.166667em}{0ex}}\mathrm{ms}$.

**Figure 3.**Time evolution of the momentum distributions for an excitation amplitude of A = 1.8 ${\mu}_{0}$. We present the results obtained with both (

**a**) the angular average of the absorption image and (

**b**) the three-dimensional reconstruction of the cloud using the inverse Abel transform. In both plots, we include the curve corresponding to the power-law behavior characteristic of the turbulent states as a guide to the eye.

**Figure 4.**Time evolution of the characteristic length scale for different excitation amplitudes computed with (

**a**) the two-dimensional projection of the cloud and (

**b**) its three-dimensional reconstruction using the inverse Abel transform. Although the values computed with two-dimensional profiles are higher, their qualitative behavior is the same.

**Figure 5.**(

**a**) Extrapolation of the characteristic length scale to the instant when the excitation is turned on, ${L}_{0}$, and (

**b**) the characteristic time of the particle transfer, ${t}_{0}$, as a function of the excitation amplitude. The results were obtained fitting the data to the functional form of Equation (8). Although the analysis employing two- or three-dimensional momentum distributions produces different values of ${L}_{0}$, both approaches yield the same values for the characteristic time.

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**MDPI and ACS Style**

Madeira, L.; García-Orozco, A.D.; Moreno-Armijos, M.A.; dos Santos, F.E.A.; Bagnato, V.S.
Characteristic Length Scale during the Time Evolution of a Turbulent Bose-Einstein Condensate. *Symmetry* **2021**, *13*, 1865.
https://doi.org/10.3390/sym13101865

**AMA Style**

Madeira L, García-Orozco AD, Moreno-Armijos MA, dos Santos FEA, Bagnato VS.
Characteristic Length Scale during the Time Evolution of a Turbulent Bose-Einstein Condensate. *Symmetry*. 2021; 13(10):1865.
https://doi.org/10.3390/sym13101865

**Chicago/Turabian Style**

Madeira, Lucas, Arnol D. García-Orozco, Michelle A. Moreno-Armijos, Francisco Ednilson Alves dos Santos, and Vanderlei S. Bagnato.
2021. "Characteristic Length Scale during the Time Evolution of a Turbulent Bose-Einstein Condensate" *Symmetry* 13, no. 10: 1865.
https://doi.org/10.3390/sym13101865