Observation of Shock Waves in a Strongly Interacting ... - Physics

PRL 106, 150401 (2011) PHYSICAL REVIEW LETTERSweek ending15 APRIL 2011Observation of Shock Waves in a Strongly Interacting Fermi GasJ. A. Joseph and J. E. ThomasDepartment of Physics, Duke University, Durham, North Carolina 27708, USAM. Kulkarni 1,2 and A. G. Abanov 11 Department of Physics and Astronomy, Stony Brook University, Stony Brook, New York 11794-3800, USA2 Department of Condensed Matter Physics and Material Science, Brookhaven National Laboratory, Upton, New York 11973, USA(Received 13 December 2010; published 11 April 2011)We study collisions between two strongly interacting atomic Fermi gas clouds. We observe exoticnonlinear hydrodynamic behavior, distinguished by the formation of a very sharp and stable density peakas the clouds collide and subsequent evolution into a boxlike shape. We model the nonlinear dynamics ofthese collisions by using quasi-1D hydrodynamic equations. Our simulations of the time-dependentdensity profiles agree very well with the data and provide clear evidence of shock wave formation in thisuniversal quantum hydrodynamic system.DOI: 10.1103/PhysRevLett.106.150401 PACS numbers: 05.30.Fk, 03.75.Ss, 67.85. dFermi gases with magnetically tunable interactionsprovide a new universal medium for studies of nonlinearhydrodynamics in quantum matter. Near a collisional(Feshbach) resonance, where the s-wave scattering lengthdiverges, a bias magnetic field can continuously tune thecloud from a weakly interacting molecular Bose-Einsteincondensate to a weakly interacting Fermi gas, i.e., the socalledBEC-BCS crossover. At resonance, a Fermi gas isthe most strongly interacting nonrelativistic system known[1]. It exhibits anisotropic hydrodynamic expansion [2] or‘‘elliptic flow,’’ in common with a quark-gluon plasma[3,4], a state of matter that existed microseconds after thebig bang and was recently recreated in gold ion collisions.Further, both systems have an extremely low (quantum)viscosity [5] and nearly the same ratio of shear viscosity toentropy density, just a few times the conjectured lowerbound for a perfect fluid [6].In this Letter, we report the observation of shock wavesin a strongly interacting Fermi gas. Shock waves are ofrecent interest in nonequilibrium electron Fermi gases[7,8] and occur generally in hydrodynamic systems whenregions of high density move with a faster local velocitythan regions of low density, resulting in increasingly largedensity gradients. In the absence of dissipative or dispersiveforces, this process leads to a ‘‘gradient catastrophe,’’where the density develops infinite gradients. We find thata gradient catastrophe is avoided in a strongly interactingFermi gas by dissipative forces, which we model as arisingfrom the local shear viscosity. The data are well fit with akinetic viscosity 10@=m, where m is the atom mass [9].In contrast, previous experiments on nonlinear hydrodynamicsin quantum matter have focused on weaklyinteracting Bose-Einstein condensates (BECs) [10].There, the mean field and quantum pressure (i.e., theGross-Pitaevski equation) lead to dispersive shock waves,characterized by density oscillations [11–13]. For BECs,dispersive shock waves produce soliton trains [14], whichalso have been observed and modeled for rapidly rotatingBECs [15] and for merging and splitting BECs [16].Recently, quantum turbulence has been observed inBECs, by using oscillating potentials [17,18].Our experiments employ a 50:50 mixture of the twolowest hyperfine states of 6 Li, confined in a cigar-shapedCO 2 laser trap and bisected by a blue-detuned beam at532 nm, which produces a repulsive potential. The gas isthen cooled via forced evaporation near a broad Feshbachresonance at 834 G [19]. After evaporation, the trap isadiabatically recompressed to 0.5% of the initial trapdepth. This procedure produces two spatially separatedatomic clouds, containing a total of ’ 10 5 atoms per spinstate. In the absence of the blue-detuned beam, the trappingpotential is cylindrically symmetric with a radial trapfrequency of ! x ¼ ! y ¼ ! ? ¼ 2 437 Hz and an axialtrap frequency of ! z ¼ ! 2 Oz þ !2 Mz ¼ 2 27:7 Hz,qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiwhere the axial frequency of the optical trap is ! Oz ¼2 18:7 Hzand ! Mz ¼ 2 20:4 Hzarises from curvaturein the bias magnetic field. When the repulsivepotential is abruptly turned off, the two clouds acceleratetoward each other and collide in the CO 2 laser trap. After achosen hold time, the CO 2 laser is turned off, allowingthe atomic cloud to expand for 1.5 ms, after which it isdestructively imaged with a 5 s pulse of resonant light.Figure 1 shows false color absorption images for acollision of the atomic clouds at different times after theblue-detuned beam is extinguished. Two distinctive featuresare clearly seen in these data: (i) the formation of acentral peak, which is well-pronounced and robust and(ii) the evolution of this peak into a boxlike shape withvery sharp boundaries, which propagates outward. Theobserved large density gradients provide strong evidenceof shock wave formation in this system, where the sharpboundaries of the ‘‘box’’ are identified as shock wave0031-9007=11=106(15)=150401(4) 150401-1 Ó 2011 American Physical Society

PRL 106, 150401 (2011) PHYSICAL REVIEW LETTERSweek ending15 APRIL 2011FIG. 1 (color online). Collision between two strongly interacting Fermi gas clouds in a cigar-shaped optical trap. The clouds areinitially separated by a repulsive 532 nm optical beam. After the 532 nm beam is extinguished (0 ms), the clouds approach each other.False color absorption images show the spatial profiles versus time. Initially, a sharp rise in density occurs in the center of the collisionzone. At later times the region of high density evolves from a ‘‘peaklike’’ shape into a ‘‘boxlike’’ shape as the shock front propagatesoutward. The well defined edges of the central zone in the last four images provide evidence of shock wave formation in the stronglyinteracting Fermi gas.fronts. Numerical modeling of the hydrodynamic theoryfor one-dimensional motion is used to predict the evolutionof the atomic density, yielding profiles in good agreementwith the data.For simplicity, we assume that the cloud is a stronglyinteracting Fermi gas at zero temperature; i.e., we modelthe cloud as a single fluid, consistent with our measurementsof the sound velocity [20]. In this case, the localchemical potential has the universal form ðn 3D Þ¼ð1 þÞ F ðn 3D Þ, where F ðn 3D Þ¼ @22m ð32 n 3D Þ 2=3 is the idealgas local Fermi energy corresponding to the threedimensionaldensity n 3D . Here, ¼ 0:61 is a universalscale factor [2,21,22].Neglecting viscous forces, the dynamics for the densityn 3D ðr;tÞ and the velocity field vðr;tÞ are described by thecontinuity equationand the Euler equation@ t n 3D þrðn 3D vÞ¼0 (1)m@ t v þr½ðn 3D ÞþU trap ðr; zÞþ 1 2 mv2 Š¼0; (2)where we assume irrotational flow. Here U trap ðrÞ ¼12 m!2 ? r2 þ 1 2 m!2 zz 2 is the confining harmonic potentialof the cigar-shaped trap.To determine the initial density profile for the separatedclouds, we consider the equilibrium 3D density of theFermi gas in the trap, including a knife-shaped repulsivepotential V rep ðzÞ. A blue-detuned laser beam is shaped by acylindrical lens telescope; i.e., the spot size is small comparedto the long dimension of the cigar-shaped cloud andlarge compared to the transverse dimension. Therefore, therepulsive potential varies only in the z (axial) direction:V rep ðzÞ ¼V 0 exp½ ðz z 0 Þ 2 = 2 zŠ. We measure the width z ¼ 21:2 m. The offset z 0 ¼ 5 m of the focus fromthe center in the long direction of the optical trap isdetermined by a fit to the first density profile at 0 ms.Using the beam intensity and the ground state static polarizabilityof 6 Li at 532 nm, we find V 0 ¼ 12:7 K. Theinitial density profile is thenn 3D ðr; zÞ ¼~n 1r 2R 2 ?z 2R 2 zV rep ðzÞ G 3=2;(3)where ~n ¼½ð2m G =@ 2 Þ=ð1 þ ÞŠ 3=2 =ð3 2 Þ. In Eq. (3),qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR z;? ¼ 2 G =ðm! 2 z;? Þ and G is the global chemicalpotential, which is determined by normalizing the integralof the 3D density to the total number N of atoms in bothspin states. For N ¼ 2 10 5 , we find G ¼ 0:53 K,R z ¼ 220 m, and R ? ¼ 14 m.We note that G =ð@! ? Þ¼27, which means that thetypical number of filled energy levels of transverse quantizationis large. Therefore, in this Letter, we use 3Dhydrodynamics [Eqs. (1) and (2)] and neglect effects oftransverse quantization even though they are more pronouncedin regions with lower density.We model the dynamics for the one-dimensional motionin the long direction of the cigar-shaped trap. Just after theblue-detuned beam is extinguished, the initial 1D densityprofile is determined by integrating n 3D of Eq. (3) over thetransverse dimension r:n 1D ðzÞ ¼ 2 5 R2 ? ~n 1z 2R 2 zV rep ðzÞ G 5=2:(4)In the following, we assume that during the evolution ther dependence of Eq. (3) is preserved with the effective sizeof the cloud being a slow function of z and t. We alsoassume that the hydrodynamic velocity is along the z axis150401-2

PRL 106, 150401 (2011) PHYSICAL REVIEW LETTERSweek ending15 APRIL 2011and does not depend on r. Then the subsequent timeevolution of the density follows the quasi-1D nonlinearhydrodynamic equations:@ t n ¼ @ z ðnvÞ; (5)@ t v ¼ @ z v22 þ Cn2=5 þ 1 2 !2 zz 2 þ @ zðn@ z vÞ; (6)nwhere C ¼ 1 2 !2 ? l2 ? ð15 2 l ?Þ 2=5 ð1 þ Þ 3=5 and l ? ¼pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi@=ðm! ? Þ is the oscillator length. For brevity, we haveomitted the subscript 1D in Eqs. (5) and (6). The last‘‘viscosity’’ term in Eq. (6) is added phenomenologicallyto describe dissipative effects. For the strongly interacting1D fluid, is the effective kinematic viscosity, which hasa natural scale @=m. It is the only fitting parameter in thetheory [9].For our previous sound wave experiments [20], weobserved nonlinear (amplitude-dependent) propagationwithout shock waves. By reducing the density perturbation,we observed linear propagation. In this regime, one canexpand the differential equations (5) and (6) around anequilibrium density configuration n 0 ðzÞ in a harmonictrap. By defining nðz; tÞ n 0 ðzÞþnðz; tÞ, the linearizedevolution equation for nðz; tÞ (neglecting viscosity) is 2C@ 2 t n ¼ @ z n 0 @ z5m n ð3=5Þ0n : (7)With a flat background density, i.e., constant n 0 , with G ¼ Cn 2=50 , Eq. (7) reduces to @ 2 t n ¼ c 2 @ 2 zn with thepsound velocity c ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 G =5m, in agreement with predictions[23,24].To compare the numerical solutions of Eqs. (5) and (6)with experiment, we note that the images are taken after anadditional free expansion for 1.5 ms, during which n 1Dcontinues to slowly evolve in the axial potential of the biasmagnetic field, i.e., ! z ! ! Mz ¼ 2 20:4 Hz. We assumethat, during this expansion, the transverse densityprofiles keep the same form, but the radius increases withtime. Then n 3D ðr; zÞ !n 3D ðr=b ? ;zÞ=b 2 ? , where b ?ðtÞ is atransverse scale factor, which obeys €b ? ¼ ! 2 ? b 7=3? , withb ? ð0Þ ¼1 and b _ ? ð0Þ ¼0 [2,25,26]. Since the 3D pressurescales as n 5=34=33D, the 1D pressure scales as b? . This leads toa simple modification of Eq. (6): C ! CðtÞ ¼C=b 4=3? ðtÞ.We numerically integrate Eqs. (5) and (6) by using themeasured values of the trap frequencies, the atom number,and the offset, depth, and width of the repulsive potential.In the numerical simulation, we create and load a densityarray as well as a velocity array with grid spacing z . Theinitial velocity is set to zero. The simulation then updatesthe density and velocity field in discrete time steps taccording to Eqs. (5) and (6). The 1D density profiles arecalculated as a function of time after the repulsive potentialis extinguished. Figure 2 shows the predictions and theFIG. 2 (color online). 1D density profiles divided by the totalnumber of atoms versus time for two colliding strongly interactingFermi gas clouds. The normalized density is in units of10 2 =m per particle. Red dots show the measured 1D densityprofiles. Black curves show the simulation, which uses themeasured trap parameters and the number of atoms, with thekinetic viscosity as the only fitting parameter.data, which are in very good agreement. For the simulationcurves shown in the figure, we use a grid of 150 points.To check for numerical consistency, we also employ asmoothed-particle-hydrodynamics [27] approach, wherethe fluid is described by discrete pseudoparticles. Theresults obtained indeed coincide with the discretized-gridapproach described above.As shown in Fig. 2, we observe a dramatic evolution forthe density of the gas. During the collision, a distinct andstable density peak forms at the point of collision in thecenter of the trap [28]. The density gradient at the side ofthe central peak increases from its onset until 3ms,at which point the gradient reaches its maximum value.A large gradient at the edge of the collision zone is maintainedthroughout the rest of the experiment. For mostof the data, we find relatively small deviations from thesimulation. The largest deviation occurs at 4 ms, where themaximum density of the observed central peak exceedsthat of the simulation by ’ 20%.The steep density gradients observed in Fig. 1 suggestshock wave formation. A deeper analysis of the simulation150401-3

PRL 106, 150401 (2011) PHYSICAL REVIEW LETTERSweek ending15 APRIL 2011curves provides additional evidence for shock waves.Without any dissipation, the numerical integration of thequasi-1D theory breaks down due to a gradient catastrophe.We find that the dissipative force in Eq. (6), which isdescribed by the kinematic viscosity coefficient , isrequired to attenuate the large density gradients and avoidgradient catastrophe. For the data shown in Fig. 2, we findthat the best fits are obtained with the viscosity parameter ¼ 10@=m. For smaller values of , the simulation producesqualitatively similar results to those shown in thefigure, only with steeper density gradients at the edges ofthe collision zone. The dissipative term / has a relativelysmall effect on the density profiles, unless we are in a shockwave regime, where the density gradients are large. Hence,the numerical model suggests that the large density gradientobserved at the edge of the collision zone is theleading edge of a dissipative shock wave.Our one-dimensional data for a strongly interactingFermi gas are very well described by a model based ondissipative nonlinear quantum hydrodynamics. The modelemploys an effective chemical potential 1D ¼ Cn 2=51D ,assuming a single fluid near the ground state. However,we expect that at higher temperatures, even in the normalfluid regime, rapid collisional equilibrium in the stronglyinteracting gas will produce nearly adiabatic evolutionwith a three-dimensional pressure / n 5=3 and, hence, anidentical power-law dependence for the effective onedimensionalchemical potential. The radial density variationsobserved in the two-dimensional image are notcaptured in the one-dimensional profiles but may bestudied by expanding the numerical analysis to threedimensions.In conclusion, we have observed shock waves in astrongly interacting Fermi gas, which provides an entirelynew regime for studies of nonlinear wave propagation incold quantum gases. Studies of nonlinear hydrodynamicscan now be done over a wide range of temperatures, in boththe superfluid and normal fluid regimes, and magnetic fieldcontrol of the interaction strength enables continuous tuningfrom a dispersive BEC to a dissipative Fermi gas. Infuture work, it will be interesting to study the origin of theeffective viscosity, the effects of transverse quantization,and the higher derivative dispersive terms in the stresstensor [7,8,11,29], which become important for large densitygradients.The work of the Duke group is supported by the PhysicsDivisions of the National Science Foundation, the ArmyResearch Office, the Air Force Office of SponsoredResearch, and the Division of Materials Science andEngineering, the Office of Basic Energy Sciences, Officeof Science, U.S. Department of Energy. The work ofA. G. A. was supported by the NSF under GrantNo. DMR-0906866. We are grateful to P. Wiegmann, E.Shuryak, D. Schneble, and F. Franchini for usefuldiscussions.[1] G. Rupak and T. Schäfer, Phys. Rev. A 76, 053607 (2007).[2] K. M. O’Hara, S. L. Hemmer, M. E. Gehm, S. R. Granade,and J. E. Thomas, Science 298, 2179 (2002).[3] J. E. Thomas, Phys. Today 63, No. 5, 34 (2010).[4] P. F. Kolb and U. 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