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PHYSICS feature of turbulence. Although the full characterization of turbulent regimes is still under investigation in our laboratory, it seems to depend on the condensed fraction, but not on the exact excitation conditions, provided that a stationary state was established. The typical results determined from the free-fall expansion of the turbulent and the nonturbulent BECs are presented in Fig. 1, where the clouds’ typical sizes are plotted as a function of the elapsed TOF. For a standard (nonturbulent) BEC, the expected aspect ratio inversion of the Thomas–Fermi radii during the free-fall expansion takes place at about 8.5 ms TOF, as shown in Fig. 1A. On the other hand, the turbulent BECs are known to expand in a self-similar manner, without ever inverting its aspect ratio, shown in Fig. 1B. The turbulent cloud presents a self-similar anomalous expansion with a shape and an aspect ratio nearly constant during the TOF (13, 14). Essentially, the preferential vortex line alignment along the radial direction, transverse to the symmetry axis direction, balances the radii expansion rates, keeping the aspect ratio locked during the TOF expansion (13). The rate expansions determined ˙ ˙ in both cases are R˙y = 3.9(1) µm/ms, R˙x = 0.47(1) µm/ms for the standard BEC and R˜y = 5.3(2) µm/ms, R˜x = 1.5(1) µm/ms for the turbulent BECs, clearly showing that the disordered matter wave expands faster than the coherent one. Fig. 1 C and D shows the expansion of a standard (nonturbulent) BEC and a turbulent BEC, respectively, for different TOF intervals. Fig. 1D presents the number density distribution and its characteristic modulations regularly observed in expanding turbulent BECs. These can be seen in detail only on snapshots acquired after sufficiently long TOF intervals (≥ 25 ms), when the optical depth lowers and the probe light passes through the atomic cloud, revealing its typical density distribution pattern. This fact forces a lower limit on the TOF to allow for the experimental study of turbulent density distribution patterns. We mention that, currently, it is not technically possible to perform studies at TOFs longer than 55 ms in our system. To follow up the observations for the propagating matter wave, the propagation of light beams with and without speckle is presented in Fig. 2. A coherent elliptical Gaussian laser beam propagates in space, presenting aspect ratio inversion along the optical axis, similar to the aspect ratio inversion observed in coherent matter waves, such as a free-falling BEC. In that case, the traveling optical wave is characterized by its divergence angle, given by ref. 15: tan θi = λ/πwi , where λ is the light wavelength and wi the beam waist in the i direction. If we now consider the same elliptical beam, but with a speckle pattern on it, the divergence is quite different. The divergence angle for each direction is given by refs. 16 and 17, tan θi = λ/`i , where `i is the speckle correlation length in the i direction. Actually, `i is normally expressed as d, which is the typical particle size found in the rough surface that generated the speckle pattern. For an isotropic speckle field the correlation length is equal in both directions, and thus the divergence is the same in any direction. When dealing with light-wave propagation one has to take into account different regimes, namely the near and the far field. The near-field propagation regime takes place at short distances, compared with the Rayleigh length, and each and every domain expands as λ/`i , which is precisely the case studied here. The overall expansion is given by the sum of individual domains, and therefore the general spatial evolution takes place described based on different expansion rates, for each direction, since there are different numbers in each direction. The propagation does not present the aspect ratio inversion, which tends to unity at large distances (far field). In Fig. 2, the axial evolution of the light beam waists of a coherent Gaussian beam and a speckle beam are shown as a function of the propagation distance from the focus. We point out that both coherent and incoherent optical beams start with identical elliptical cross-sections, but evolve at distinct rates due to the presence or lack of disorder (speckle). For the coherent beam, Fig. 2A, the inversion takes place after a given propagation distance away from the focus. In contrast, for the speckle field, Fig. 2B, the beam propagates in a nearly self-similar shape, and the cross-section along each dimension increases faster compared with the coherent beam. That happens when `i < wi , i.e., for a beam full of speckle grains, and presents a larger −2 −4 divergence angle. From Fig. 2 A and B the determined beam divergences in both cases are w˙ x = 3.964(2) × 10 , w˙ y = 4.8(1) × 10 −2 −1 for the coherent Gaussian beam and w˜˙ x = 9.8(5) × 10 and w˜˙ y = 1.67(6) × 10 for the speckle beam, showing that the disordered light wave expands faster than the coherent one, the same as the turbulent quantum gas, presented in Fig. 1. Fig. 2 C and D shows

Fig. 1. BEC expansion during TOF. (A and B) The Tomas–Fermi radii, Rx and Ry , as a function of the TOF are shown for standard (A) and turbulent (B) BECs, respectively. (C and D) Three different TOF snapshots showing the expansion of standard and turbulent BECs, respectively.

12692 | www.pnas.org/cgi/doi/10.1073/pnas.1713693114 Tavares et al. 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PHYSICS Fig. 3. Normalized correlation function, Ce(2)(∆r). (A) For a regular BEC (red) and a turbulent cloud (green) as a function of the thermal de Broglie wavelength, λdB = 0.45 µm. (B) For the numerical simulations of a Gaussian (red) and a speckle (green) beam of light as a function of ∆r, in units of λ, at a propagation distance where the Gaussian beam aspect ratio matches with the regular BEC of A. The Ce(2)(∆r) for a regular BEC (Gaussian beam) is zero except for very small ∆r, whereas for a turbulent cloud (speckle beam) it has an exponential decay with a correlation length of `Turb = 8.5(2) µm (`Speckle = 13.8(8)λ). The typical dimensions are Rx = 63(3) µm and Ry = 198(6) µm for the BEC and wx = 1876λ and wy = 2528λ for the beam waists, showing that the correlation length is much smaller than the BEC/beam sizes. The colored regions represent the confidence bands of the exponential decay fittings.

speckle light beam and compared them with the expanding BEC and nonequilibrium BEC. In Fig. 4B, we present the normalized correlation function Ce (2)(∆r) of a Gaussian and a speckle beam at a propagation distance where the Gaussian beam aspect ratio matches that of a regular BEC in Fig. 4A. Essentially, the same qualitative decay behavior of the correlation function is observed on the light field and on the expanding quantum gas. In Fig. 4, we present the atomic density absorption profile of an expanding turbulent cloud together with the cross-section intensity profile of a propagating speckle field, showing the similarities in the distribution of the spatial fluctuation amplitudes. An important point concerning our measurements is that they are obtained from absorption images. This corresponds to a projection of the condensate’s atomic (number) density distribution onto an image plane. Despite the fact that this procedure smooths out the fluctuation amplitudes (as shown in Fig. 4), the correlation length measured after projection remains the same, as in three dimensions. This is supported by 2D simulations of an optical speckle beam and its unidimensional projection. It was observed that the effect on the correlation length, extracted from Ce (2)(∆r), remains unaffected. In fact, the observation of a similar behavior found in the spatial evolution of such different physical systems, starting from completely different initial conditions, is truly remarkable. The observation and study of disorder existing in matter waves freely moving in space, which can be accomplished by driving BECs to turbulence, may open up a broad research field with opportunities for understanding the inner nature of the disorder phenomena.

Fig. 4. Direct comparison between a turbulent BEC and an elliptical speckle beam. (A) Typical atom number density map of a turbulent condensate shown in units of optical depth (OD). (B) Light intensity map from a cross-section of a speckle beam. The integration in one of the 3Ds of the BEC results in slightly smaller amplitude for the density fluctuations.

12694 | www.pnas.org/cgi/doi/10.1073/pnas.1713693114 Tavares et al. Downloaded by guest on September 26, 2021 Downloaded by guest on September 26, 2021 0 il ,e l 20)Drc bevto fAdro oaiaino atrwvsi otolddisorder. condensate. controlled Bose-Einstein a non-interacting in a waves of matter localization of Anderson localization (2008) potentials. Anderson al. random of et in observation G, condensates Direct Roati Bose-Einstein (2008) expanding light. al. 21. of of et function localization beams J, correlation Anderson coherent Billy density (2007) two the al. in 20. of et photons measurement L, between direct Sanchez-Palencia Correlation A (1956) light: RQ 19. Twiss scattered R, the Hanbury-Brown of field 18. near (1985) the JW in Goodman correlations intensity 17. Space (2000) condensate. A Bose-Einstein Vailati model. turbulent resonators. M, hydrodynamic a and Carpineti generalized in beams M, A profile Laser Giglio condensate: density (1966) Bose-Einstein the T 16. of turbulent Li expansion a H, Self-similar Kogelnik of (2012) expansion al. 15. Self-similar et (2013) M, VS Caracanhas condensate. Bagnato Bose-Einstein SR, 14. trapped Muniz turbulent condensates. G, a Bose–Einstein Baym of atomic distribution AL, trapped momentum Fetter in the waves. M, turbulence in matter Caracanhas Quantum behavior guided (2016) law in al. 13. power correlations et Brown-Twiss of MC, Hanbury Evidence Tsatsos and (2014) speckle al. 12. atomic et of KJ, Observation Thompson (2011) al. 11. et RG, Dall 10. aae tal. et Tavares upr.W prcaedsusoswt .Cutil n h ehia sitnepoie yE enadG ot.MSH loakolde Senior a acknowledges also M.S.H. Roati. G. and Henn E. by provided assistance technical the and Cient CAPES. Courteille Desenvolvimento by P. de granted with Nacional Professorship Visiting discussions Conselho and appreciate (CAPES), We Superior dimensions. an support. Nivel three as of de in emerge Pessoal properties traveling may de fields statistical speckle equilibrium, the of of investigate details ACKNOWLEDGMENTS. to out the possibility driven and the patterns fluids, disordered open quantum explore well free-expanding to may system in finding candidate of observed This ideal established, behavior modulation field. is speckle-like speckle via phase matter–wave The (e.g., and a matter–waves. gases amplitude like work. quantum in much this into disorder space in (19–21). regime once disorder in with localization turbulent achieved nucleating the wave Anderson be of via matter may on only means the results it results different explored similar of interesting be have that interaction we very also believe the although produced might do that already We there note interactions). has that We microscopic fields out speckle. point light matter–waves we speckle of Finally, by propagation created spatial potentials the random mention we particular, In .Pto S eot P lit S ea D(06 pcl atrswt tmcadmlclrd rgi waves. Broglie de molecular and atomic with patterns Speckle (2006) SD Kevan GS, condensate. Elliott Bose-Einstein DP, all-optical Deponte an FS, from Patton laser atom 9. guided (2007) quasi-monomode JW laser. A Goodman atom (2008) nonideal al. a 8. et of A, quality Couvert Beam (2006) 7. al. et JF, Riou divergence. 6. laser K T, Atom Busch (2001) propagation. al. atom-laser-beam 5. et Y, for Coq tools Le Theoretical (2008) 4. al. et JF, Riou traps. dependent time 3. in Bord F, condensates Impens Bose-Einstein (1996) 2. R Dum Y, Castin 1. ncnlso,w aeivsiae h xaso foto-qiiru truet Esadfudsrn vdneta tevolves it that evidence strong found and BECs (turbulent) out-of-equilibrium of expansion the investigated have we conclusion, In Lett Phys 85:1416–1419. 170:133–142. h ,EsigrT ømrK(02 rnvremd fa tmlaser. atom an of mode Transverse (2002) K Mølmer T, Esslinger M, ohl ¨ J(09 Generalized (2009) CJ e ´ ttsia Optics Statistical pcl hnmn nOtc:Ter n Applications and Theory Optics: in Phenomena Speckle etakFnaa eApr eqiad saod a al FPS)(rn 030261,Codncod Aperfeicoamento de Coordenacao 2013/07276-1), (Grant (FAPESP) Paulo Sao de Estado do Pesquisa a Amparo de Fundacao thank We Wly e York). New (Wiley, abcd rpgto o neatn tmcclouds. atomic interacting for propagation hsRvLett Rev Phys rcIEEE Proc hsRvLett Rev Phys 87:170403. 54:1312–1329. hsRvA Rev Phys 96:070404. hsRvLett Rev Phys Rbrs&CmayPbihr,GenodVlae CO). Village, Greenwood Publishers, Company & (Roberts hsRvA Rev Phys 77:033630. hsRvA Rev Phys 77:5315–5319. Nature 65:043615. hsRep Phys Nature 453:895–898. 79:043613. fioeTecnol e ´ ıfico 622:1–52. Nature 177:27–29. uohsLett Europhys PNAS 453:891–894. o epPhys Temp Low J hsRvLett Rev Phys gc CP)(rn V 02821-)frfinancial for 400228/2014-9) PVE (Grant (CNPq) ogico ´ a Commun Nat | hsRvLett Rev Phys oebr2,2017 28, November 83:50001. 97:013202. 2:291. 166:49–58. 98:210401. ae hsLett Phys Laser | o.114 vol. 11:015501. | o 48 no. g(r o Temp Low J ). | hsRev Phys 12695

PHYSICS