# Entropy of a Turbulent Bose-Einstein Condensate

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Experimental Procedure

## 3. Extracting the Entropy from Experimental Data

#### 3.1. Classical Treatment

#### 3.2. Quantum Treatment

## 4. Results

#### 4.1. Momentum Distribution

#### 4.2. Entropy

## 5. Discussion and Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

BEC | Bose-Einstein condensate |

QUIC | Quadrupole-Ioffe configuration |

TOF | Time-of-flight |

## References

- Tsatsos, M.C.; Tavares, P.E.; Cidrim, A.; Fritsch, A.R.; Caracanhas, M.A.; dos Santos, F.E.A.; Barenghi, C.F.; Bagnato, V.S. Quantum turbulence in trapped atomic Bose–Einstein condensates. Phys. Rep.
**2016**, 622, 1–52. [Google Scholar] [CrossRef] [Green Version] - Madeira, L.; Caracanhas, M.; dos Santos, F.; Bagnato, V. Quantum Turbulence in Quantum Gases. Annu. Rev. Condens. Matter Phys.
**2020**, 11, 37–56. [Google Scholar] [CrossRef] - Henn, E.A.L.; Seman, J.A.; Roati, G.; Magalhães, K.M.F.; Bagnato, V.S. Emergence of Turbulence in an Oscillating Bose-Einstein Condensate. Phys. Rev. Lett.
**2009**, 103, 045301. [Google Scholar] [CrossRef] [PubMed] - Henn, E.A.L.; Seman, J.A.; Roati, G.; Magalhães, K.M.F.; Bagnato, V.S. Generation of Vortices and Observation of Quantum Turbulence in an Oscillating Bose-Einstein Condensate. J. Low Temp. Phys.
**2010**, 158, 435–442. [Google Scholar] [CrossRef] [Green Version] - Thompson, K.J.; Bagnato, G.G.; Telles, G.D.; Caracanhas, M.A.; dos Santos, F.E.A.; Bagnato, V.S. Evidence of power law behavior in the momentum distribution of a turbulent trapped Bose–Einstein condensate. Laser Phys. Lett.
**2014**, 11, 015501. [Google Scholar] [CrossRef] - Navon, N.; Gaunt, A.L.; Smith, R.P.; Hadzibabic, Z. Emergence of a turbulent cascade in a quantum gas. Nature
**2016**, 539, 72–75. [Google Scholar] [CrossRef] [Green Version] - Baggaley, A.W.; Barenghi, C.F.; Sergeev, Y.A. Three-dimensional inverse energy transfer induced by vortex reconnections. Phys. Rev. E Stat. Nonlinear Soft Matter Phys.
**2014**, 89, 013002. [Google Scholar] [CrossRef] [Green Version] - Marino, Á.V.M.; Madeira, L.; Cidrim, A.; dos Santos, F.E.A.; Bagnato, V.S. Momentum distribution of Vinen turbulence in trapped atomic Bose-Einstein condensates. arXiv
**2020**, arXiv:2005.11286. [Google Scholar] - Navon, N.; Eigen, C.; Zhang, J.; Lopes, R.; Gaunt, A.L.; Fujimoto, K.; Tsubota, M.; Smith, R.P.; Hadzibabic, Z. Synthetic dissipation and cascade fluxes in a turbulent quantum gas. Science
**2019**, 366, 382–385. [Google Scholar] [CrossRef] [Green Version] - Daniel García-Orozco, A.; Madeira, L.; Galantucci, L.; Barenghi, C.F.; Bagnato, V.S. Intra-scales energy transfer during the evolution of turbulence in a trapped Bose-Einstein condensate. EPL Europhys. Lett.
**2020**, 130, 46001. [Google Scholar] [CrossRef] - Pinkse, P.W.H.; Mosk, A.; Weidemüller, M.; Reynolds, M.W.; Hijmans, T.W.; Walraven, J.T.M. Adiabatically Changing the Phase-Space Density of a Trapped Bose Gas. Phys. Rev. Lett.
**1997**, 78, 990–993. [Google Scholar] [CrossRef] [Green Version] - Stamper-Kurn, D.M.; Miesner, H.J.; Chikkatur, A.P.; Inouye, S.; Stenger, J.; Ketterle, W. Reversible formation of a Bose-Einstein condensate. Phys. Rev. Lett.
**1998**, 81, 2194–2197. [Google Scholar] [CrossRef] [Green Version] - Olshanii, M.; Weiss, D. Producing Bose-Einstein Condensates Using Optical Lattices. Phys. Rev. Lett.
**2002**, 89, 090404. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Gattobigio, G.L.; Couvert, A.; Jeppesen, M.; Mathevet, R.; Guéry-Odelin, D. Multimode-to-monomode guided-atom lasers: An entropic analysis. Phys. Rev. A
**2009**, 80, 041605. [Google Scholar] [CrossRef] [Green Version] - Carr, L.D.; Shlyapnikov, G.V.; Castin, Y. Achieving a BCS Transition in an Atomic Fermi Gas. Phys. Rev. Lett.
**2004**, 92, 150404. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Williams, J.E.; Nygaard, N.; Clark, C.W. Phase diagrams for an ideal gas mixture of fermionic atoms and bosonic molecules. New J. Phys.
**2004**, 6, 123. [Google Scholar] [CrossRef] - Bourdel, T.; Khaykovich, L.; Cubizolles, J.; Zhang, J.; Chevy, F.; Teichmann, M.; Tarruell, L.; Kokkelmans, S.J.; Salomon, C. Experimental study of the BEC-BCS crossover region in lithium 6. Phys. Rev. Lett.
**2004**, 93, 050401. [Google Scholar] [CrossRef] [Green Version] - Partridge, G.B.; Strecker, K.E.; Kamar, R.I.; Jack, M.W.; Hulet, R.G. Molecular probe of pairing in the BEC-BCS crossover. Phys. Rev. Lett.
**2005**, 95, 020404. [Google Scholar] [CrossRef] [Green Version] - Zwierlein, M.W.; Schunck, C.H.; Stan, C.A.; Raupach, S.M.; Ketterle, W. Formation dynamics of a fermion pair condensate. Phys. Rev. Lett.
**2005**, 94, 180401. [Google Scholar] [CrossRef] [Green Version] - Seman, J.A.; Henn, E.A.; Shiozaki, R.F.; Roati, G.; Poveda-Cuevas, F.J.; Magalhães, K.M.; Yukalov, V.I.; Tsubota, M.; Kobayashi, M.; Kasamatsu, K.; et al. Route to turbulence in a trapped Bose-Einstein condensate. Laser Phys. Lett.
**2011**, 8, 691–696. [Google Scholar] [CrossRef] [Green Version] - Shiozaki, R.; Telles, G.; Yukalov, V.; Bagnato, V. Transition to quantum turbulence in finite-size superfluids. Laser Phys. Lett.
**2011**, 8, 393–397. [Google Scholar] [CrossRef] [Green Version] - Pitaevskii, L.; Stringari, S. Bose-Einstein Condensation and Superfluidity; Oxford University Press: Oxford, UK, 2016. [Google Scholar] [CrossRef]
- Kagan, Y.; Surkov, E.L.; Shlyapnikov, G.V. Evolution of a Bose-condensed gas under variations of the confining potential. Phys. Rev. A At. Mol. Opt. Phys.
**1996**. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Castin, Y.; Dum, R. Bose-einstein condensates in time dependent traps. Phys. Rev. Lett.
**1996**, 77, 5315–5319. [Google Scholar] [CrossRef] [PubMed] - Dalfovo, F.; Minniti, C.; Stringari, S.; Pitaevskii, L. Nonlinear dynamics of a Bose condensed gas. Phys. Lett. Sect. A Gen. At. Solid State Phys.
**1997**. [Google Scholar] [CrossRef] [Green Version] - Qu, C.; Pitaevskii, L.P.; Stringari, S. Expansion of harmonically trapped interacting particles and time dependence of the contact. Phys. Rev. A
**2016**, 94, 063635. [Google Scholar] [CrossRef] [Green Version] - Lovas, I.; Dóra, B.; Demler, E.; Zaránd, G. Quantum-fluctuation-induced time-of-flight correlations of an interacting trapped Bose gas. Phys. Rev. A
**2017**, 95, 023625. [Google Scholar] [CrossRef] [Green Version] - Caracanhas, M.; Fetter, A.L.; Baym, G.; Muniz, S.R.; Bagnato, V.S. Self-similar Expansion of a Turbulent Bose-Einstein Condensate: A Generalized Hydrodynamic Model. J. Low Temp. Phys.
**2013**, 170, 133–142. [Google Scholar] [CrossRef] - Kleinert, H. Path Integrals In Quantum Mechanics, Statistics, Polymer Physics, And Financial Markets, 5th ed.; World Scientific Publishing Company: Singapore, 2009. [Google Scholar]
- Hickstein, D.D.; Gibson, S.T.; Yurchak, R.; Das, D.D.; Ryazanov, M. A direct comparison of high-speed methods for the numerical Abel transform. Rev. Sci. Instrum.
**2019**, 90, 065115. [Google Scholar] [CrossRef] - Nazarenko, S. Wave Turbulence; Springer Science & Business Media: Berlin/Heidelberg, Germany, 2011; Volume 825. [Google Scholar]
- Kirkpatrick, T.R.; Dorfman, J.R. Transport coefficients in a dilute but condensed Bose gas. J. Low Temp. Phys.
**1985**, 58, 399–415. [Google Scholar] [CrossRef] - Kirkpatrick, T.R.; Dorfman, J.R. Time correlation functions and transport coefficients in a dilute superfluid. J. Low Temp. Phys.
**1985**, 59, 1–18. [Google Scholar] [CrossRef] - Kirkpatrick, T.R.; Dorfman, J.R. Transport in a dilute but condensed nonideal Bose gas: Kinetic equations. J. Low Temp. Phys.
**1985**, 58, 301–331. [Google Scholar] [CrossRef] - Tsubota, M.; Fujimoto, K.; Yui, S. Numerical Studies of Quantum Turbulence. J. Low Temp. Phys.
**2017**, 188, 119–189. [Google Scholar] [CrossRef] [Green Version]

**Figure 1.**Momentum distribution of an unperturbed cloud ($A=0$) held in the trap for a short time (${t}_{\mathrm{hold}}=$ 20 ms) obtained through an optical absorption image following the release of the trap. The normalization is such that the integration over the plane yields one.

**Figure 2.**Momentum distribution of an unperturbed cloud ($A=0$) held in the trap for different times (${t}_{\mathrm{hold}}=$) from 20 to 115 ms. We show both (

**a**) the angular average using the absorption images, and (

**b**) the three-dimensional reconstruction of the momentum distributions using the inverse Abel transform, Equation (21). The profiles for different hold times are very similar, with small differences at high-momenta due to heating in the system.

**Figure 3.**Momentum distribution of the cloud for the excitation amplitudes $A=0.20$, 0.25, 0.30, 0.40, 0.50, and 0.60${\mathsf{\mu}}_{0}$, panels (

**a**–

**f**) respectively, and holding times ranging from 20 to 90 ms. Increasing the excitation amplitude corresponds to larger energy input to the BEC, thus driving the system toward higher-momenta regions. For strong enough excitations, $A\u2a7e$ 0.50${\mathsf{\mu}}_{0}$ in this experimental setting, the system enters a turbulent regime with a particle cascade characterized by a power-law $n\left(k\right)\propto {k}^{-\delta}$. In panels (

**e**,

**f**), we plot a line corresponding to $\propto {k}^{-2.3}$ to guide the eye.

**Figure 4.**Entropy per particle calculated using Equation (16) for several excitation amplitudes as a function of the time held in the trap. The unperturbed BEC corresponds to a constant entropy per particle over time. Increasing the amplitude corresponds to higher values of the entropy per particle until the turbulent regime is reached, $A\u2a7e$ 0.50${\mathsf{\mu}}_{0}$. The particle cascade that occurs at ≈ 35 ms (see Section 4.1) is accompanied by a sudden increase in the entropy per particle, as seen for ${t}_{\mathrm{hold}}\u2a7e$ 45 ms.

**Figure 5.**Entropy per particle as a function of the momentum for an unperturbed BEC (

**a**) and a turbulent cloud (

**b**). Notice that the same scale was employed in both plots. The turbulent BEC experiences a sudden increase in the entropy per particle, for ${t}_{\mathrm{hold}}\u2a7e$ 45 ms, and it reaches thermal equilibrium for long times, corresponding to a more flat distribution of entropy among the momentum classes.

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Madeira, L.; García-Orozco, A.D.; dos Santos, F.E.A.; Bagnato, V.S.
Entropy of a Turbulent Bose-Einstein Condensate. *Entropy* **2020**, *22*, 956.
https://doi.org/10.3390/e22090956

**AMA Style**

Madeira L, García-Orozco AD, dos Santos FEA, Bagnato VS.
Entropy of a Turbulent Bose-Einstein Condensate. *Entropy*. 2020; 22(9):956.
https://doi.org/10.3390/e22090956

**Chicago/Turabian Style**

Madeira, Lucas, Arnol Daniel García-Orozco, Francisco Ednilson Alves dos Santos, and Vanderlei Salvador Bagnato.
2020. "Entropy of a Turbulent Bose-Einstein Condensate" *Entropy* 22, no. 9: 956.
https://doi.org/10.3390/e22090956