Scale Invariance in Expanding Fermi Gases

7/27/2019 Scale Invariance in Expanding Fermi Gases

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arXiv:1308.3162v1

[cond-mat.quant-gas]14Aug2013

Scale Invariance in Expanding Fermi Gases

E. Elliott1,2, J. A. Joseph1, and J. E.Thomas11Department of Physics, North Carolina State University, Raleigh, NC 27695, USA and

2Department of Physics, Duke University, Durham, NC 27708, USA(Dated: August 15, 2013)

We precisely test scale invariance in the hydrodynamic expansion of a Fermi gas of atoms withresonant interactions, using a model-independent method. We observe that the three-dimensionalmean square cloud radius expands ballistically and find that the bulk viscosity is consistent withzero, as predicted for a scale-invariant system. At resonance, the expanding cloud obeys the universalequation of state for a non-relativistic gas, p 2

3E = 0, where p is the pressure and E is the energy

density. Tuning away from resonance, we show that the observed deviation from scale invariant flowcan be explained by the change in the equation of state p 2

3E = 0 (conformal symmetry breaking

parameter) and a finite bulk viscosity, which is compared to predictions.

PACS numbers: 03.75.Ss

The identification and comparison of scale-invariantphysical systems, defined as those without an intrinsiclength scale, has enabled significant advances connect-ing diverse fields of physics. Of recent interest is a deep

connection between scale-invariant strongly interactingsystems and their weakly interacting counterparts. Animportant example is the Anti de Sitter-Conformal FieldTheory correspondence, which connects a broad class ofstrongly interacting quantum fields to weakly interactinggravitational fields in five dimensions, enabling theoreti-cal techniques that span high energy physics, condensedmatter, and ultracold atoms [1]. These techniques havebeen used to predict a universal lower bound for the ra-tio of shear viscosity to entropy density [2], connectingquark-gluon-plasmas [3, 4] to ultra-cold Fermi gases [57].

An ultra-cold Fermi gas is a paradigm for scale-invariant quantum fluids, with the unique trait that in-teractions between spin-up and spin-down atoms are tun-able between two scale-invariant regimes, from an ideal,noninteracting gas to the most strongly interacting, non-relativistic quantum system known [8]. A central con-nection between these two regimes is that in both casesthe pressure p and the energy density E are related byp = 2

3E [9, 10]. However, an obvious distinction be-

tween the ideal and strongly-interacting regimes was firstdemonstrated by observing the aspect ratio of a Fermi gasafter release from the trapping potential [11]. The idealgas was shown to expand ballistically with an isotropic

momentum distribution, whereas the strongly interact-ing gas was found to expand hydrodynamically and toexhibit anisotropic elliptic flow [3, 11]. Despite thisdifference, we demonstrate both theoretically and exper-imentally that scale-invariance requires the mean squarecloud sizes to expand identically.

We report the observation and study of scale invari-ance in an expanding Fermi gas by measuring the meansquare cloud size r2 = x2 + y2 + z2 as a function oftime after release. In the resonantly-interacting, scale in-variant regime, where p 2

3E= 0 and the bulk viscosity

vanishes, we predict and observe that r2 expands bal-

listically and fits a universal curve for all initial energies.In contrast, the aspect ratios exhibit energy-dependentelliptic flow. Tuning away from resonance, where thescattering length a is finite and scale invariance is bro-

ken, we show that the observed deviation of r2 fromscale invariant ballistic expansion can explained by thechange in the equation of state and a finite bulk viscosity,which is estimated and compared to predictions.

In the experiments [12], we employ an optically-trapped cloud of 6Li atoms in a 50-50 mixture of thetwo lowest hyperfine states, which is cooled by evapora-tion [11]. The initial energy per particle E is measuredfrom the trapped cloud profile [12]. To observe the ex-pansion dynamics, the cloud is released from the trapand the cloud profile is measured as a function of timeafter release in all three dimensions using two CCD cam-eras, which simultaneously image different spin states toavoid cross-saturation. As discussed in more detail be-low, the cigar-shaped optical trap has a 2.6:1 transverseaspect ratio, which enables an observation of transverseelliptic flow on a short time scale with very good signalto background ratio, and a precise measurement of thestatic shear viscosity.

To characterize the degree to which the expansionis scale-invariant, we measure the mean-square three-dimensional cloud radius r2 and compare to predic-tions. Using the hydrodynamic equation for the velocityfield v (including pressure and viscous forces), the conser-vation equation for the density n, and conservation of to-tal (kinetic, internal, and potential) energy, we obtain anexactmodel-independent evolution equation for r2 as afunction of time t after release of the cloud. Without anysimplifying assumptions, a single-component fluid com-prising N atoms of mass m obeys [12],

d2

dt2mr2

2= x Uopt0 + 3

N

d3x [(p) (p)0]

3N

d3x B v, (1)

where the subscript (0) denotes the condition at t = 0,
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2

0.0 0.5 1.0 1.5

0.40.6

0.8

1.0

1.2

1.4

1.6

t ms

AspectRatiox

y

FIG. 1: Transverse aspect ratio x/y versus time after re-lease showing elliptic hydrodynamic flow: Top to bottom,resonantly interacting gas, E = 0.66 EF, E = 0.89 EF,E = 1.17 EF, E = 1.46EF, ballistic gas E = 1.78EF. Topfour solid curves: Hydrodynamic theory with the shear vis-cosity as the only fit parameter; Lower solid curve: Ballistictheory with no free parameters. Error bars denote statisticalfluctuations in the aspect ratio.

just after the optical trap is extinguished. The depar-ture from scale-invariance is determined by the conformalsymmetry breaking parameter p/p, where p p 23 Eis nonzero in general, but vanishes in the scale-invariantresonant regime, p = 23 E. The bulk viscosity B isnonzero in general, but is predicted to vanish in the scale-invariant regime [1316]. Further, the bulk viscosity fre-quency sum rule vanishes when p = 0 [17]. For brevity,we include only the optical trap potential Uopt in Eq. 1,which need not be harmonic. However, for our preci-sion measurements, as described below, it is importantto include also the small potential energy arising fromthe finite curvature of the bias magnetic field [12]. Giventhe time-dependent volume integrals of p and B, Eq. 1can be integrated to determine r2 at time t from themeasured initial value of r20 and the initial condition,dr20/dt = 0. As r2 is a scalar, the contribution ofthe shear viscosity pressure tensor vanishes, since it istraceless. Hence, Eq. 1 is independent of the shear vis-cosity, enabling a very sensitive measurement of the bulkviscosity, where the divergence of the velocity field v iseasily determined as described below. By tuning awayfrom resonance, we find that observed deviations fromscale invariance can be explained by the pressure changep, which is an odd function of 1/a, and a small con-tribution, proportional to 1/a2, which we attribute to anonzero bulk viscosity.

For a scale-invariant hydrodynamically expandingcloud, where both p and B are zero, Eq. 1 yields

r2 = r20 + t2

mx Uopt0, (2)

which corresponds to ballistic expansion of the meansquare cloud size, even though the individual cloud radii

0.6 0.8 1.0 1.2 1.4 1.6

0

1

2

3

EEF

ViscosityCoefficients

FIG. 2: Measurement of bulk and shear viscosity fora scale-invariant Fermi gas: Blue (top): Trap-averagedshear viscosity coefficient

d3x /(Nh) S versus energy

E/EF. Red (bottom): Trap-averaged bulk viscosity coeffi-cient

d3x B/(Nh) B versus energy. The weighted aver-

age value of B = 0.005(0.016) is consistent with zero. (Dot-ted curves added to guide the eye.)

expand hydrodynamically and exhibit elliptic flow, asstudied previously [11] and shown in Fig. 1 for the trans-verse aspect ratio. Indeed, numerical modeling of hy-drodynamic expansion of the cloud in the scale-invariantregime shows that Eq. 2 holds even for large shear viscos-ity, provided that both dissipative forces and heating areproperly included in the hydrodynamic equations [6, 7]to assure that the total energy is conserved [12].

The aspect ratio x/y of the cloud is measured atthe Feshbach resonance (834 G) as a function of timeafter release to establish that the flow is hydrodynamicand to determine the shear viscosity. Fig. 1 shows datafor E/EF = 0.66, 0.89, 1.17, 1.46, where E x U0and EF h

2k2FI

2mis the measured Fermi energy of an

ideal gas at the trap center with kFI the correspondingwavevector [12]. In contrast to the ballistic expansiondata (E/EF = 1.78), where the aspect ratio saturatesto unity, for the resonantly interacting cloud, x/y in-creases to approximately 1.5 over the time range shown,clearly demonstrating that the cloud expands hydrody-namically. The rate at which the aspect ratio increaseswith time substantially decreases as the energy is in-creased, due to the increase in the shear viscosity.

The aspect ratio in the x y plane is very sensitive tothe shear viscosity, which slows the flow in the rapidly ex-panding x-direction and increases the speed in the moreslowly expanding y-direction. To measure the shear vis-cosity at 834 G, we fit the aspect ratio data of Fig. 1 usinga general, energy-conserving hydrodynamic model [6, 7],valid in the scale-invariant regime where p = 0. At theFeshbach resonance, the shear viscosity takes the form = S h n, where n is the total density of atoms andS is a dimensionless function of the local reduced tem-perature. The trap-averaged shear viscosity coefficientS

d3x/(hN) is used as the only free parameter,


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3

0.0 0.5 1.0 1.5 2.00

1

2

3

4

t ms

2tms2

FIG. 3: Scale invariant expansion of a resonantly interactingFermi gas. Experimental values of2(t) m[r2r20]/x U0 versus time t after release, for the same data as inFig. 1 (including ballistic data) collapse onto a single curve,demonstrating universal t2 scaling. Dashed curve 2(t) =t2, as predicted by Eq. 2. Note that the small correctionr2Mag arising from the curvature in the bias magneticfield is subtracted from the r2 data to obtain the 2(t) datathat is shown in the figure [12].

initially neglecting the bulk viscosity, which is expectedto be much smaller. For the shear viscosity in the scale-invariant regime, S is an adiabatic invariant, which istherefore temporally constant in the adiabatic approxi-mation [6, 7]. The fits to the aspect ratio obtained inthis way are shown in Fig. 1 as solid lines. We find thatthe measured shear viscosity coefficients, Fig. 2, are sig-nificantly smaller than those obtained from our previ-

ous measurements in the same energy range, based oncollective mode damping [6, 7]. We note that collectivemode measurements determine the shear viscosity at thecollective mode frequency and may be subject to excessdamping arising near the cloud edges, where the gas isballistic. As the frequency scale for expansion is essen-tially the inverse of the expansion time, the expansionmeasurements correspond to a low frequency, and hencemeasure the static shear viscosity on a time scale wherethe gas is likely to maintain thermal equilibrium and thestress tensor has relaxed to the Navier-Stokes form [18].

Scale invariance of the expanding gas is now directlydemonstrated by measuring 2(t) m[r2 r20]/x Uopt0, which should obey

2

(t) = t2

for a scale-invariant system, according to Eq. 2 [12]. Fig. 3 showsthe experimental values of 2(t) versus t for the samedata as used in Fig. 1. In contrast to the aspect ratioversus time data of Fig. 1, which vary substantially withenergy due to the shear viscosity, the experimental valuesof 2(t) fall on a single t2 curve.

We estimate the bulk viscosity at resonance by assum-ing that p = p 23 E= 0 in Eq. 1, which follows from theuniversal hypothesis [9, 10]. This equation of state for aresonantly interacting Fermi gas has been verified experi-mentally to high precision [19]. We therefore assume that

0.0 0.5 1.0 1.5 2.00

1

2

3

4

t ms

2tms2

FIG. 4: Breaking of scale invariance in the expansion for aFermi gas near a Feshbach resonance. The data are the ex-perimental values of 2(t) m[r2 r20]/x U0 forE/EF 1.0, versus time t after release. Solid curves arethe predictions using Eq. 1 with B = 0, where the pressurechange p is approximated using the second virial coefficientwithout any free parameters [12]. Top: 1/(kFIa) = 0.59;Center: 1/(kFIa) = 0; Bottom: 1/(kFIa) = +0.61.

at resonance the bulk viscosity term in Eq. 1 producesthe only deviation from scale-invariance in the evolutionof r2. Analogous to the shear viscosity, we take thebulk viscosity to be of the form B = B h n, where Bis dimensionless, and consider first a large finite scatter-ing length. Since the bulk viscosity must be positive, theleading contribution in powers of the inverse scatteringlength must be of the form B = fB() h n/(kF a)2, wherekF = (3

2n)1/3 is the local Fermi wavevector. HerefB() is a dimensionless function of the reduced temper-

ature , which is an adiabatic invariant, and hence time-independent in the adiabatic approximation that we usefor the small bulk viscosity contribution. As the cloudexpands, the density decreases as n 1/ in the scalingapproximation, where the fits to the aspect ratio datain all three dimensions accurately determine the volumescale factor (t) [12]. Since 1/(kFa)2 2/3, the trap-averaged bulk viscosity coefficient, B

d3x B/(hN)

is time-dependent and scales as

B(t) = B(0)2/3(t). (3)

With the scaling approximation v = /,the bulk viscosity term then takes the simple form3 h B(0) /1/3. We determine B(0) with high pre-cision, by using a least-squares fit of Eq. 1 to the mea-sured r2 data [12]. In contrast to the shear viscositycoefficient, which increases with increasing energy, thebulk viscosity coefficient at resonance remains nearly zeroover the entire energy range, Fig. 2. We find that theweighted-average B(0) = 0.005(0.016), which is consis-tent with zero, as predicted for a scale-invariant cloud[1316].

We investigate the departure from scale invariance andthe thermodynamics of the expanding gas at finite scat-


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0 0.5 1 1.5 21.51

0.5

0

0.5

1

1.5

p scaling

B

scaling

1. 1.2 1.4 1.6 1.80.8

0.9

1.0

1.1

1.2

EEF

c1

c0

c1

c0

834

FIG. 5: Pressure change p and bulk viscosity B contribu-tions to conformal symmetry breaking as a function of en-ergy. The expansion data are fit with r2 = c0 + c1 t

2.The ratio (c1/c0)/(c1/c0)834 is shown for the resonantly in-teracting gas 1/(kFIa) = 0 (black dots, black line-theory), for

1/(kFIa) = 0.59 (top, red dots) and for 1/(kFIa) = +0.61(bottom, blue dots). The ratios are compared to predictionsusing two scaling parameters, B for B of Ref. [16] and p forp based on the second virial coefficient approximation. Thecontour plot shows 2 as a function ofB and p. Solid curvestop and bottom show the best fit, where p = 1.08(0.32) andB = 0.16(0.61). The dashed (dotted) curves show the pre-dictions for p = 1 and B = 0 (B = 1), to illustrate theeffect of the bulk viscosity.

tering length by tuning the bias magnetic field above andbelow the Feshbach resonance. We measure r2 r20for E 1.0 EF, as shown in Fig. 4. Compared to the res-onant case, we see qualitatively that the cloud expandsmore rapidly when the scattering length is negative andmore slowly when the scattering length is positive. Thisbehavior is a signature of the [(p)(p)0] term in Eq. 1,where |p(t)| |p(0)| for any time t after release andp has the same sign as the scattering length.

To obtain a model-independent quantitative compari-son with predictions for finite scattering length, we pa-rameterize the data using a 2 fit with r2 = c0 + c1 t2,where c1 is the effective coefficient oft

2 [12]. This methodavoids utilizing model-dependent expansion factors for

the cloud radii in the data analysis. We determine c0and c1 parameters for both the on-resonance and off-resonance data at several different energies. We correctall of the c1/c0 ratios for the effective potential arisingfrom magnetic field curvature [12], and finally determinethe ratio (c1/c0)/(c1/c0)834, Fig. 5. If p and B werezero at all magnetic fields, then this ratio would be unityeverywhere, corresponding to the black line (black dots)

in the figure. The systematic deviation from unity arisesfrom finite p and B , where the red dots (top) showdata at 986 G where a < 0 and the blue dots (bottom)show data for 760 G, where a > 0.

We fit the data shown in Fig. 5, by modeling the c1/c0ratio as a function of energy using two parameters, ascale factor p for p calculated to second order in fu-gacity (high temperature limit) [12, 20] and a scale fac-tor B for the high temperature bulk viscosity predictedin Ref. [16]. We assume that all contributions to pthat require three-body and higher order interactions tomaintain equilibrium are frozen over the expansion timescale, and therefore do not affect (p)

(p)0 in Eq. 1.

In particular, the molecular contribution to the secondorder virial coefficient can be neglected. Retaining onlythe translational degrees of freedom in the two-body scat-tering contribution, p is evaluated using an adiabaticapproximation for the translational temperature in thissmall correction, so that p is then a known function oftime and is odd in 1/a [12].

The bulk viscosity at finite scattering length [16] makesthe only 1/a2 correction in Eq. 1, since the 1/a2 part ofp is generally time-independent and does not contributeto (p) (p)0 [12]. Dusling and Schafer [16] point outthat the leading order bulk viscosity must be second or-der in the fugacity z

n3T/2 in the high-temperature

limit. They obtain the form B =1

242

2T

a2h3T

z2, where

T h2mkBT is the thermal wavelength. Assumingthat the temperature evolves adiabatically in this smallbulk viscosity term, the integral over the trap volume [12]

takes the same form as Eq. 3, with B(0) = cBEFE

4and cB =

932

1(kFI a)2

, which is symmetric in 1/a in con-

trast to p, which changes sign with 1/a.Our measurements provide a test of the degree to

which equilibrium thermodynamics holds in the expan-sion of an ultra-cold cloud. In Fig. 5, the 2 contourplot as a function of p and B shows a maximum for

p = 1.08(0.32) and B = 0.16(0.61). To show the rel-ative scale of the bulk viscosity and the p corrections,the predictions for p = 1 and B = 0 are shown asdashed curves, while the dotted curves show the predic-tions for p = 1 and B = 1. The contribution of the bulkviscosity appears smaller than predicted. As p 1, theobserved breaking of scale invariance is primarily due thedirect change in the pressure p = p 2

3E, and our p

model adequately describes the data, showing that thepressure in the expanding cloud is not far from thermalequilibrium in the translational degrees of freedom.

This research is supported by the Physics Division of


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the National Science Foundation (Quantum transport instrongly interacting Fermi gases) and by the Division ofMaterials Science and Engineering, the Office of BasicEnergy Sciences, Office of Science, U.S. Department ofEnergy (Thermodynamics in strongly correlated Fermi

gases). Additional support has been provided by thePhysics Divisions of the Army Research Office and theAir Force Office of Scientific Research. The authors arepleased to acknowledge K. Dusling and T. Schafer, NorthCarolina State University, for stimulating conversations.

[1] A. Adams, L. D. Carr, T. Sch afer, P. Steinberg, , andJ. E. Thomas, New J. Phys. 14, 115009 (2012).

[2] P. K. Kovtun, D. T. Son, and A. O. Starinets, Phys. Rev.Lett. 94, 111601 (2005).

[3] P. F. Kolb and U. Heinz, Quark Gluon Plasma 3 (WorldScientific, 2003), p. 634.

[4] E. Shuryak, Prog. Part. Nucl. Phys. 53, 273 (2004).[5] T. Schafer, Phys. Rev. A 76, 063618 (2007).[6] C. Cao, E. Elliott, J. Joseph, H. Wu, J. Petricka,

T. Schafer, and J. E. Thomas, Science 331, 58 (2011).[7] C. Cao, E. Elliott, H. Wu, and J. E. Thomas, New J.

Phys. 13, 075007 (2011).[8] G. Rupak and T. Schafer, Phys. Rev. A 76, 053607

(2007).[9] T.-L. Ho, Phys. Rev. Lett. 92, 090402 (2004).[10] J. E. Thomas, J. Kinast, and A. Turlapov, Phys. Rev.

Lett. 95, 120402 (2005).[11] K. M. OHara, S. L. Hemmer, M. E. Gehm, S. R.

Granade, and J. E. Thomas, Science 298, 2179 (2002).[12] See Appendix: Supplemental Material.[13] D. T. Son, Phys. Rev. Lett. 98, 020604 (2007).[14] M. A. Escobedo, M. Mannarelli, and C. Manuel, Phys.

Rev. A 79, 063623 (2009).[15] Y.-H. Hou, L. P. Pitaevskii, and S. Stringari (2013),

arXiv:1302.2258v1 [cond-mat.quant-gas].[16] K. Dusling and T. Schafer (2013), arXiv:13054688v1

[cond-mat.quant-gas].

[17] E. Taylor and M. Randeria, Phys. Rev. A 81, 053610(2010).[18] This idea was suggested to us by Thomas Schafer, North

Carolina State University, private communication.[19] M. Ku, A. T. Sommer, L. W. Cheuk, and M. W. Zwier-

lein, Science (2012).[20] T.-L. Ho and E. Mueller, Phys. Rev. Lett. 92, 160404

(2004).[21] L. Luo and J. E. Thomas, J. Low Temp. Phys. 154, 1

(2009).[22] F. Werner, Phys. Rev. A 78, 025601 (2008).

Appendix A: Supplemental Material

1. Experimental Methods

In the experiments, we employ an optically-trappedcloud of 6Li atoms in a 50-50 mixture of the two low-est hyperfine states, which is tuned to a broad collisional(Feshbach) resonance in a bias magnetic field of 834 G,and cooled by evaporation. The initial energy per par-ticle E is measured from the trapped cloud profile, asdiscussed below. A focused CO2 laser beam forms thecigar-shaped optical trap with a transverse aspect ratio

of 2.6:1, which enables an observation of transverse el-liptic flow and a precise measurement of the static shearviscosity even at low temperature, where the shear viscos-ity is small. The transverse aspect ratio is controlled byusing two sets of cylindrical ZnSe lenses, one set just afterthe acousto-optic modulator that controls the laser inten-sity and one just before an expansion telescope, which isused to increase the beam radii before focusing into ahigh vacuum chamber, where the optical trap is loadedfrom a standard magneto-optical trap. The first set ofcylindrical lenses controls the transverse aspect ratio offocused beam, while the second set enables matching of

the beam curvatures to achieve a common focal plane.To observe the expansion dynamics, the cloud is re-

leased from the trap and the cloud profile is measuredas a function of time after release in all three dimen-sions using two identical CCD cameras, Fig. 6 which si-multaneously image different spin states to avoid cross-saturation. The magnifications of the imaging systemsare measured by translating the trap focus. The mea-sured magnifications yield average axial dimensions zthat are consistent within 1%. To obtain the most pre-cise measurements of the cloud profile, we adjust theeffective magnification of one camera so that the aver-age axial dimensions precisely agree. In this way, the

cloud radii i in all three dimensions are consistentlymeasured, to determine the aspect ratios x/y , x/z,and y/z , as well as the mean square cloud radius, r2.Two-dimensional density distributions are fit to the cloudprofiles to extract the cloud radii. For fast data handling,gaussian profiles are assumed for most of the data anda zero-temperature Thomas-Fermi profile is assumed forthe lowest energies. Both types of fit profiles are com-pared to full finite-temperature Thomas-Fermi profilesto estimate multiplicative corrections to the cloud radii,which are needed to correct for the small error arisingfrom the form of the fit functions.

We derive an exact, model-independent evolutionequation for r2 based on hydrodynamics and energyconservation in A 2. This enables precise characteri-zation of scale-invariance and thermodynamics in an ex-panding cloud. The primary result, Eq. A17, is indepen-dent of the shear viscosity and includes the correctionsto the flow arising from the bulk viscosity and the devia-tion p p 2

3Eof the pressure from the scale invariant

equation of state, p = 23E. We also include the potential

energy arising from the finite bias magnetic field curva-ture, as required for our precision measurements.

The pressure change p is determined for the high


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6

FIG. 6: Imaging the expanding cloud in three dimensions.Two CCD cameras are used to measure the density profile ofthe cloud. The cloud is released from an asymmetric opticaltrap with a 2.6 : 1 transverse aspect ratio, enabling observa-tion of elliptic flow in the x y plane.

temperature limit in A 3. Then the method of estimat-ing the bulk viscosity is described in A 4. Finally, wediscuss the method of fitting the mean square size r2data in the resonantly and interacting regimes A 5.

2. Expansion of the Mean-Square Cloud Radius

We employ a hydrodynamic description for a singlecomponent fluid [6, 7], where the velocity field v(x, t)is determined by the scalar pressure and the viscositypressure tensor,

n m (t + v ) vi = ip +j

j( ij + B

ij)

n iUtotal, (A1)

where p is the scalar pressure and m is the atom mass.Utotal is the total trapping potential energy arising fromthe optical trap and the bias magnetic field curvature.The second term on the right describes the friction forcesarising from both shear and bulk B viscosities, whereij = vi/xj + vj/xi 2ij v/3 and v.Current conservation for the density n(x, t) requires

nt

+ (nv) = 0. (A2)

Finally, conservation of total energy yields

d

dt

d3x

n

1

2mv2 + E+ n Utotal

= 0. (A3)

Here, the first term is the kinetic energy arising from thevelocity field and E is the internal energy density of thegas. As shown below, Eq. A3 will play an importantrole in determining a general evolution equation for the

volume integral of the pressure in both the scale-invariantregime and away from scale-invariance.

To explore scale invariance for an expanding cloud,we measure the mean-square cloud radius, r2, whichis a scalar quantity. In this section, we derive gener-ally the equation of motion for r2, with no simplify-ing assumptions, except that of a single component fluid,which is appropriate in the normal fluid regime above

the superfluid transition temperature as well as in thesuperfluid regime when the normal and superfluid com-ponents move together [15]. We show that in the scale-invariant regime at a Feshbach resonance, where p = 23Eand B = 0, conservation of total energy leads to bal-listic expansion of r2 for a hydrodynamic gas. Awayfrom resonance, the departure from scale-invariance isdetermined by the change in the equation of state, char-acterized by p p 2

3E and finite B, permitting a

measurement of the bulk viscosity, given p.We begin by noting that for each direction i = x,y,z,

the mean square size x2i 1N

d3xn(x, t) x2i obeys

dx2i dt

= 1N

d3x nt

x2i =1

N

d3x [ (nv)]x2i

=1

N

d3xnv x2i = 2xi vi, (A4)

where N is the total number of atoms. We have usedintegration by parts and n = 0 for xi to obtainthe second line. Similarly,

dxividt

=1

N

d3xn xi

vit

+1

N

d3x

n

txivi

=1

N d3xn xi

vi

t

+1

N d3xnv

(xivi)

= xi(t + v )vi + v2i . (A5)

Combining Eq. A4 and Eq. A5, we obtain,

d2

dt2x2i

2= xi(t + v )vi + v2i . (A6)

To proceed, we use Eq. A1, which yieldsd3xn xi(t + v )vi = 1

m

d3xxi(ip n iUtotal)

+1

m j d3xxij( ij + B

ij)

Integrating by parts on the right hand side, assumingthat the surface terms vanish, we obtain

xi(t + v )vi = 1N m

d3xp 1

mxiiUtotal

1N m

d3x ( ii + B

)(A7)

with v. Defining the viscosity coefficients Sand B by S h n and B B h n, respectively, we


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can write,

xi(t + v )vi = 1N m

d3xp 1

mxiiUtotal

hm

Sii + B , (A8)

where

Sii + B 1Nd3xn (Sii + B ). (A9)

Using Eq. A8 in Eq. A6, we then obtain for one directionxi,

d2

dt2x2i

2=

1

N m

d3xp + v2i

1

mxiiUtotal

hm

Sii + B . (A10)

Eq. A10 determines the evolution of the mean squarecloud radii along each axis, x2i , which depends on theconservative forces arising from the scalar pressure andthe trap potential, as well as the viscous forces arisingfrom the shear and bulk viscosities.

Summing Eq. A10 over all three directions, the shearviscosity term vanishes, since ij is traceless, yielding

d2

dt2r2

2=

3

N m

d3xp + v2 1

mx Utotal

3hm

B v. (A11)

At t = 0, before release from the trap, v = 0, Eq. A11shows that the volume integral of the pressure is,

3Nd3xp0 = x Utotal0, (A12)

where the subscript ( )0 denotes the initial condition.Here Utotal = UOpt+UMag is the total trapping potential,comprising an optical component from the laser trap anda magnetic component arising from the curvature of thebias magnetic field used in the experiments, as describedfurther below.

It will be convenient to rewrite Eq. A11 using

p =2

3E+

p 2

3E

, (A13)

where the first term defines the equation of state forthe pressure in the scale-invariant regime, and the sec-ond term describes the departure from scale invariance.Then,

d2

dt2r2

2=

2

N m

d3xE+ v2 + 3

N m

d3x

p 2

3E

1m

x Utotal 3hm

B v. (A14)

Just after release from the trap, t 0+, from theoptical trap, the trapping potential changes abruptly,

Utotal UMag . To determine the evolution of r2 af-ter release, we use total energy conservation to eliminatev2 from Eq. A14. From Eq. A3, the final total energyper particle is equal to the initial total energy. Then,

1

N

d3xE+ m

2v2+ UMag = 1

N

d3x E0 + UMag 0.

(A15)

To determine the initial internal energy, 1N d3xE0,we use E0 = 32p0 32(p 23E)0. With Eq. A12, this yields

1

N

d3xE0 = 1

2xUtotal0 3

2N

d3x

p 2

3E0

.

(A16)We have x Utotal0 = x UOpt0 + x UMag 0in Eq. A16. Multiplying Eq. A16 by 2/m determines thefirst two terms in Eq. A14. Then, with x Utotal x UMag for t 0+ in Eq. A14, we obtain finally ourcentral result for studying scale invariance,

d2

dt2

m

r2

2 = x UOpt0 +3

N d3x [(p) (p)0]3 h B v + UMag , (A17)

where

p p 23

E (A18)

describes the departure of the pressure from the scale-invariant regime and

UMag 2UMag 0+xUMag 02UMag xUMag (A19)

corrects for the small potential energy arising from thebias magnetic field curvature. As the bias coils are ori-ented along the x direction, the effective potential is re-pulsive along the x axis and twice the magnitude of theattractive potential along y and z,

UMag (x) =1

2m 2Mag

y2 + z2 2x2 , (A20)

where Mag = 2 21.5(0.25) Hz is measured by fromthe oscillation frequency of the cloud in the y z plane.

The first three terms in Eq. A17 reproduce the Eq. 1of the main text, where the small magnetic contributionwas neglected for brevity. The potential energy arising

from the magnetic field curvature depends on the meansquare cloud radii, x2i , which are determined as a func-tion of time by fitting the aspect ratio data using a scalingapproximation for the density profile. x UOpt0 is de-termined by parametric resonance measurements of thetrap frequencies and measurements of the cloud radii,as discussed in more detail below. The cloud radii fort = 0+ are dominated by z20, the longest direction ofthe cigar-shaped cloud in the trap. This is consistentlymeasured both by in-situ imaging and by measurementsafter expansion, using the calculated expansion factor,which is close to unity.


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Eq. A17 is easily integrated using the initial conditionsr20 and tr20 = 0. To clearly demonstrate scale-invariant expansion after the optical trap is extinguished,we integrate Eq. A17 in two steps. First, we integrate themagnetic contribution, r2Mag determined from

d2

dt2mr2Mag

2=

1

2m 2Mag

y20 b2y(t) + z20 b2z(t) 2x20 b2x(t) . (A21)We employ homogeneous initial conditions for the mag-netic contribution, r20Mag = 0 (which does notchange r20) and tr20Mag = 0. The time depen-dent expansion factors bi(t) are determined by fittingthe measured cloud radii as a function of time after re-lease, using the hydrodynamic equations, Eq. A10 in ascaling approximation [6, 7]. As the magnetic contribu-tion to r2 arising from Eq. A21 is only a few percent,the expansion factors are readily determined with suffi-cient precision. After integration, the quantity r2Magis subtracted from the measured

r2

data to determine

the effective r2, which then expands according to theremaining terms in Eq. A17,

d2

dt2mr2

2= x UOpt0 + 3

N

d3x [(p) (p)0]

3 h B v. (A22)

The first term is the optical trap contribution, which isdominant and leads to a t2 scaling for the mean squarecloud radius of a resonantly interacting gas or for a bal-listic gas, with the initial condition r20. The remainingsmall pressure change and bulk viscosity terms can be in-

tegrated separately, using the scale factors bi(t) and thesame homogeneous initial conditions as for the magneticcontribution. These contributions are described in moredetail below.

3. High Temperature Approximation to p

We study the regime away from the Feshbach reso-nance by using in Eq. A17 a nonzero correction to thepressure p. As a consequence of energy conservation,only the difference between the pressure at time t andat time t = 0, i.e., (p) (p)0, appears in Eq. A17.Hence, any static contribution to p has no effect.

In the high-temperature limit, we can evaluate p us-ing a virial expansion [20]. To determine p, we makethe assumption that three-body and higher order contri-butions are frozen at their equilibrium values over thetime scale of the expansion, and do not contribute to(p) (p)0. In particular, the molecular contributionto the second virial coefficient [20] requires three-bodycollisions to populate and depopulate the molecular stateas the gas expands and cools in the translational degreesof freedom. Therefore, the molecular contribution is ex-pected to be negligible.

With the frozen equilibrium hypothesis, we neglectthree-body and higher order processes, and evaluate pin the high temperature limit to second order in the fu-gacity [20] , where p = p 2

3Eis given by,

p = p 23

E=

2

3n kBT

T

b2T

(n3T). (A23)

Here T h/2mkBT is the thermal wavelength andb2 is the part of the second virial coefficient that describestwo-body interactions. Ignoring the molecular contribu-tion, which is frozen on the short time of the expansionas discussed above, we take

b2(x) = sgn[a]2

ex2

erfc(x), (A24)

where erfc(x) = 1 2

x0

dx ex2

and x = T|a|2 , witha the s-wave scattering length.

As p causes only a small perturbation to the flow, wemake an adiabatic approximation for the temperature,

T = T0 2/3, where T0 is the initial temperature of thetrapped cloud and = bxbybz is the volume scale factor,i.e., the density n of the expanding gas scales as 1/.We determine by fitting the aspect ratio data with ascaling approximation to the hydrodynamics [6, 7], usingthe shear viscosity as the only free parameter. Then, x =

x0 1/3, where x0 =

T0|a|2 . Using the high temperature

approximation for the energy per particle, E = 3 kB T0

and EF =h2k2

FI

2m= (3N)1/3h, the Fermi energy of an

ideal gas at the cloud center, we have

x = x0 1/3

x0 = 6kFI|a|

EFE1/2 , (A25)

where kFI =

2m EF/h2 is the Fermi wavevector of an

ideal gas at the trap center. Note that EF is measuredusing the total atom number and the oscillation frequen-cies in the trap, (xyz)1/3, where the i are givenin A 5.

Now, Tb2T = xb2/2, where b2(x) sgn[a] f2(x) with

f2(x)

1

xex2erfc(x)

. (A26)

Integrating over the trap volume, and using the adiabaticapproximation for the temperature and a scaling approx-imation for the density, we obtain

1

N

d3x

p 2

3E

=

2

3

kBT02/3

z [x b2(x)], (A27)

where the trap-averaged fugacity z is an adiabatic invari-ant. For a gaussian density profile,

z 1N

d3xn

n3T0

2

=

9

4

2

EFE

3. (A28)


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In Eq. A25 and Eq. A28, we have made the harmonic ap-proximation 2xx20 = 2yy20 = 2zx20 and we haveused the high temperature approximation E = 3 kB T0,We determine the energy E and temperature T0, using

E = 3 kBT0 = 3

z

Utotalz

0

= x Utotal0, (A29)

which follows from force balance in the trapping potential

and p = n(0)kBT0 in the high temperature limit. Thisapproximation is adequate for evaluating p in the sec-ond virial approximation. We discuss the measurementof zzUtotal0 using the axial cloud profile in A 5.

To use Eq. A17 to determine r2 as a function of time,the volume integral Eq. A27 is written as,

1

N

d3xp =

x UTrap0

6

4

EFE

7/21/3

kFIaf2(x),

(A30)where the time dependence ofx is determined by Eq. A25and f2(x) is given by Eq. A26. We note that the totaldependence of p on the Fermi energy is E

7/2F /kFI

E3F

which is proportional to N, the total atom number, as itshould be. We have used sgn[a]/|a| = 1/a to explicitlyshow that the volume integral of p changes sign withthe scattering length as the bias magnetic field is tunedacross the Feshbach resonance.

As discussed above, the net pressure correction inEq. A17 is (p)(p)0. Hence, we also evaluate Eq. A30in the limit t = 0, where 1 and x x0. As|(p)| |(p)0| for all t, the net pressure correction ispositive (negative) when p is negative (positive). Then,compared to the resonant case, where 1/(kFIa) = 0 ,the cloud is expected to expand more rapidly when thescattering length 1/(kFIa) < 0 and more slowly when1/(kFIa) > 0, as observed in the experiments (see themain text).

4. Estimating the Bulk Viscosity

By tuning the bias magnetic field away from the Fesh-bach resonance, we explore the departure from the scale-invariant regime, where the correction to the pressurep p 2

3E is nonzero, as described above. In this

case, the bulk viscosity can be nonzero. We estimate thebulk viscosity by measuring r2 and fitting the data us-ing Eq. A17. This measurement exploits the fact thatthe bulk viscosity provides the onlycontribution propor-tional to 1

a2, while the 1

a2contribution to the volume

integral of (p) (p)0 generally vanishes, as we nowshow.

The bulk viscosity is positive for finite a and mustvanish in the scale-invariant regime, where |a| .Hence, to leading order in 1/a, the bulk viscosity must bequadratic in 1kFa . Using dimensional analysis, the bulkviscosity then takes the general form

B = h nfB()

(kFa)2, (A31)

where h n is the natural unit of viscosity, kF n1/3 isthe local Fermi wavevector and fB() is a dimensionlessfunction of the reduced temperature T /TF(n), wherekB TF(n) F(n) n2/3 is the local Fermi energy. Asdiscussed in the main text, by using an adiabatic approxi-mation for , one obtains the trap-averaged bulk viscositycoefficient, which takes the form

B(t) = B(0)2/3(t), (A32)

where the 2/3 factor arises from the 1/k2F scaling.Again using dimensional analysis, the most general

1/(kFa)2 contribution to p, which we define as p2must be of the form

p2 = n F(n)fp()

(kFa)2,

where F(n) is the local Fermi energy and fp() is a di-mensionless function of the reduced temperature. AsF(n)(kFa)2

= h2

2ma2, 1

N

d3xp = h

2

2ma2fp() is time-

independent, since the number of atoms in a vol-ume element is conserved during the expansion and1N

d3xn fp() is constant in the adiabatic approxima-

tion. Hence, the 1a2 part of (p) (p)0 in Eq. A17vanishes and the bulk viscosity provides the only time-dependent 1

a2contribution, which increases as 2/3, ac-

cording to Eq. A32.As described in the main text, the estimation of the

bulk viscosity is accomplished using a 2 fit to the datafor the mean square cloud radii as a function of time tafter release of the cloud with the fit function

r2 = c0 + c1 t2. (A33)

Eq. A33 is exactly valid for a resonantly interacting gasand for a ballistic gas. For the off-resonance experiments,the c1 determined from the fit function is an averagevalue, which is perturbed by p and B(0).

We compare the fitted B(0) to the recent predictionof Dusling and Schafer, Ref. [16] for the bulk viscosity inthe high temperature limit, which is second order in thefugacity z,

B = cB

Ta

2h

3Tz2. (A34)

Here we can approximate z = n 3T/2 and Dusling and

Schafer give cB = 1242 .Integrating over the trap volume, we obtain B(t) =

B(0)2/3(t) as in Eq. A32 with

B(0) = cB 272

2

1

(kFIa)2

EFE

4 cB

EFE

4.

(A35)Here, the value of cB based on the prediction is

cB =9

32

1

(kFIa)2, (A36)

which is measurable in the experiments.


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5. Fitting r2 = c0 + c1 t2 for Resonant and Ballistic

Gases.

We begin by noting that the energy scale E in theexperiments is twice the total potential energy in theharmonic oscillator approximation. In terms of the mea-sured mean square cloud size along the axial z-direction,we have

E = 3 m (2z opt + 2Mag )z20, (A37)where z opt is the axial oscillation frequency for atomsin the optical trap, in the harmonic approximation, andMag is the corresponding oscillation frequency arisingfrom the bias magnetic field curvature. For the reso-nantly interacting gas, where the virial theorem holds,E is precisely the energy of the cloud for a harmonictrapping potential. For an anharmonic trap, the virialtheorem gives the total energy for the resonantly inter-acting gas in terms of the trapping potential [21], butat finite scattering length, the relation between the totalenergy and the trapping potential energy is scattering

length dependent [22]. By using E to characterize theenergy, we avoid the scattering length dependence.

As discussed in A 3, for evaluating p and B in thehigh temperature limit, we approximate the energy perparticle as E = 3 kB T0 and take

E = x Utotal0. (A38)

In the experiments, we determine the harmonic oscil-lation frequencies, i by parametric resonance methods.We subtract off the contribution from the magnetic po-

tential and extrapolate to the harmonic values of theoptical frequencies by correcting for trap anharmonicity.We obtain x = 2 2210(4) Hz, y = 2 830(2),and z opt = 2 60.7(0.1). The corresponding trapdepth is U0 = 60.3(0.2) K. The Fermi energy of anideal gas at the trap center is EF = (3N)1

/3h, where (xyz)1/3. With a typical total number of atomsN 2.5 105, EF kB 2.0 K.

The quantity c1 = x Uoptical(x)0/m is crucial fordetermining the time evolution of

r20

. Here, we recallthat the total trapping potential takes the form

Utotal(x) = Uopt(x) + UMag (x), (A39)

where Uopt arises from the optical trap and UMag arisesfrom the magnetic field curvature, Eq. A20.

For precise measurements, it is necessary to determineboth the harmonic value and the anharmonic correctionsto x Utotal(x)0. To accomplish this, we note thatfor a scalar pressure p, force balance for the trappedcloud requires

d3xp = xxUtotal0 = yyUtotal0 =

zzUtotal0. Then,

x Utotal(x)0 = 3

zUtotal

z

0

. (A40)

After subtraction of the magnetic bowl correction, asdescribed above, the atoms effectively expand accordingto the initial optical potential, where

x Uopt(x)0 = 3

zUtotal

z

0

x UMag (x)0.(A41)

Since

x2

0 and

y2

0 are small compared to

z2

0, the

last term is just m 2Mag z20. Using Eq. A39 in Eq. A41,we then have

xUopt(x)0 = 3

zUopt

z

0

+2 m 2Mag z20. (A42)

The harmonic oscillator frequency for atoms in the op-tical trap is generally energy dependent, because the trapis less confining at higher energy, causing the frequencyto decrease. We model this by writing

zUopt

z

0

= m 2z opt z20 hA(z20), (A43)

where m 2z opt z20 arises from the harmonic trappingpotential. Here hA is the anharmonic correction factor,

hA(z20) 1 + 1 z20, (A44)Note that at low energy, where the cloud size is small

compared to the width of the trapping potential, we takehA 1. Finally, we write the optical trap contributionas

c1 x Uopt(x)0m

= 3 2z optz2

0 hA(

z2

0) + 2

2Mag

z2

0.(A45)

For p = 0 and B = 0, Eq. A17 shows that the effec-tive mean square cloud radius (with the magnetic contri-bution subtracted as described above), expands ballisti-cally according to

r2 = c0 + c1 t2. (A46)where

c0 = r20 = z20

1 +2z2x

+2z2y

, (A47)

and 2

z 2

z opt + 2

Mag .We fit Eq. A46 to the mean square radius of the res-onantly interacting clouds at a bias magnetic field ofB = 834 G to determine c1 and z20. For differentinitial cloud sizes, we determine the quantity

hA[B, z20] =c1

z20 2 2Mag3 2z opt

= 1 + 1 z20, (A48)

where Mag B. We find that hA[834, z20] decreaseswith increasing z20 (1 < 0), as expected for a correc-tion arising from trap anharmonicity, where the quartic


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terms in the optical trapping potential decrease the av-erage oscillation frequency.

The optical frequency z opt in Eq. A48 is most pre-cisely determined by demanding hA = 1 for energies closeto the ground state, where the anharmonic correction issmall and the resonantly interacting gas is nearly a puresuperfluid, with a negligible bulk viscosity.

The slope 1 is determined by measuring the ballistic

expansion of the noninteracting Fermi gas at 528 G as afunction of initial cloud size, using the same method asfor the resonantly interacting gas. In the experiments,we find that the 1 obtained from ballistic expansion at

528 G is within 10% of that obtained from the hA datafor the highest energies of the resonantly interacting gas.

We use the linear anharmonic correction hA andthe z opt determined in this manner to predict x Uopt(x)0 for the resonantly interacting gas at all initialenergies and find that the corresponding bulk viscosity isthen very small, consistent with zero, as described in the

main text.For the off-resonant studies, we measure the quantity

c1/z2022Mag at a bias field B and divide by the fittedresonant value, 3 2z opt hA[834, z20].

Scale Invariance in Expanding Fermi Gases

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