Spin-Energy Correlation in Degenerate Weakly-Interacting Fermi Gases
aa r X i v : . [ c ond - m a t . qu a n t - g a s ] J un Spin-Energy Correlation in Degenerate Weakly-Interacting Fermi Gases
S. Pegahan, J. Kangara, I. Arakelyan and J. E. Thomas Department of Physics, North Carolina State University, Raleigh, NC 27695, USA (Dated: June 21, 2019)Weakly interacting Fermi gases exhibit rich collective dynamics in spin-dependent potentials,arising from correlations between spin degrees of freedom and conserved single atom energies, offer-ing broad prospects for simulating many-body quantum systems by engineering energy-space “lat-tices,” with controlled energy landscapes and site to site interactions. Using quantum degenerateclouds of Li, confined in a spin-dependent harmonic potential, we measure complex, time-dependentspin-density profiles, varying on length scales much smaller than the cloud size. We show that aone-dimensional mean field model, without additional simplifying approximations, quantitativelypredicts the observed fine structure. We measure the magnetic fields where the scattering lengthsvanish for three different hyperfine state mixtures to provide new constraints on the collisional(Feshbach) resonance parameters.
I. INTRODUCTION
Weakly interacting two-component Fermi gases [1],with tunable, nearly vanishing s-wave scattering lengths a , offer a pristine platform for exploring the interplaybetween spin, motion, and statistics in many-body sys-tems [2]. In such gases, the collision rate ∝ | a | is neg-ligible, so that single atom energies are conserved overthe evolution time scale set by the mean field frequency ∝ | a | [3–6]. Since s-wave scattering in Fermi gases is al-lowed only for antisymmetric spin states, two-componentclouds exhibit an effective exchange interaction, enablingsimulations of a variety of spin-lattice models [2], wherethe conserved single atom quantum numbers play therole of the lattice sites [2, 7]. Spin-motion coupling isinduced by spin-dependent trapping potentials, imple-mented using magnetic field gradients [8–10] or magneticfield curvature [1, 5, 6, 11]. Global spreading of quantumcorrelations in real space can occur due to the effectivelong-ranged character of the spin couplings, which is aconsequence of the separation of time scales for the fastharmonic oscillation of atoms and slow macroscopic spindensity evolution [2, 12].The evolution of the spin density in weakly inter-acting Fermi gases has been described by mean fieldmodels employing phase-space representation [3, 4] andenergy representation [5]. The initial implementationof the energy-dependent collective spin-rotation modelof Ref. [5] yielded only semi-quantitative agreementwith the observed spin-density profiles and the time-dependent amplitude, which were measured at high tem-peratures, suggesting that the model was incomplete.Recently, Koller et al., [2] have devised a new descrip-tion in terms of Dicke collective spin states, exploitingconservation of the total spin vector for the exchange in-teraction. This picture suggests that the observed varia-tion of the spin-wave amplitude with time arises from athermal average of Dicke gaps [2], but comparison withthe measured spin density profiles has been only qualita-tive [13].We report measurements of time-dependent spin-density profiles for coherently prepared two-state Fermi gases of Li, confined in a spin-dependent harmonicpotential, providing a precise quantitative test of theunderlying energy-space spin-lattice model and energy-dependent long-range couplings. We employ quantumdegenerate samples to minimize energy shifts of the scat-tering length that become significant at higher temper-atures. This enables precise comparison of predictionswith measured spin-density profiles, which vary fromrelatively smooth to exhibiting complex structure overshort length scales. We find that our collective spin-rotation model [5], extended to degenerate samples, andimplemented without additional simplifying approxima-tions [14], quantitatively predicts the observed spin den-sity profiles. At high temperatures and small scatteringlengths a < Li. Com-paring data at high and low temperatures determines thetemperature shift of the zero crossings. These measure-ments provide new constraints on the Li molecular po-tentials that determine the precise shapes of the Fesh-bach resonances [15, 16], which have been widely usedin studies of strongly interacting Fermi gases [17, 18].At resonance, where the gas is unitary, the thermody-namic and hydrodynamic properties are universal, de-pending only on the density and temperature [19]. Themost precise measurements of the universal thermody-namic properties [20] and of the universal hydrodynamicproperties [21] rely on the the precise location of the Libroad Feshbach resonance near 832.2 G, which is con-strained by the zero crossing [16].
II. EXPERIMENT
Our experiments employ mixtures of the groundZeeman-hyperfine states of Li, which are denoted by | i to | i , in order of increasing energy [1]. We initially pre- FIG. 1. Spin-energy correlation produces spin segregation in a degenerate Fermi gas with an s-wave scattering length of 5.2bohr. The palettes are 50 × µ m. Left to right: n , n , n − n , and n + n in units of ( n + n ) max at t = 0 (upper) and t = 800 ms (lower) after coherent excitation of a | i − | i superposition state. Note that n − n evolves in time while n + n remains constant, due to single particle energy conservation. pare a degenerate sample in state | i [14]. The bias mag-netic field is tuned to B = 527 G, near the zero crossingof the | i − | i scattering length. A 2 ms radio-frequency π/ | i to state | i then creates a | i − | i superposition state.Similarly, | i−| i or | i−| i superposition states are pre-pared close to the corresponding zero crossings near 589G or 569 G [14]. The curvature of the bias magnetic field, B z ( x ), creates a significant spin-dependent harmonic po-tential in the long x-direction of the cigar-shaped cloud,with negligible effect in the narrow transverse directions.The subsequent evolution of the observed spin den-sities, Fig. 1, can be understood using a Bloch vectorpicture [5]. First, the short radio-frequency (rf) pulsecreates a collective spin vector along one axis in the x-y plane. In a frame rotating about the z -axis at theresonant hyperfine frequency, spin vectors for atoms inthe n th axial harmonic oscillator state precess about the z -axis at the detuning frequency, Ω( E ) = − n ( E ) δω x .Here, δω x = ω x − ω x = − π × . × − Hzis the difference in the oscillation frequencies of states | i and | i , arising from magnetic field curvature, and n ( E ) ≃ E/ ¯ h ¯ ω x , with ¯ ω x = ( ω x + ω x ) / π × . E F = 0 . µ K, the detuning forthe average x-energy, ¯ E = E F /
4, is Ω( ¯ E ) ≃ − π × . E ) causes the spin vec-tors for atoms of different energies to fan out in the x − y plane. Second, forward s-wave scattering, which isnot Pauli-blocked in degenerate samples, occurs betweentwo atoms with different energies and corresponding spinvectors, producing a rotation about the total spin vec-tor [6, 22–24]. This creates a mean field rotation of thecollective spin with an energy-dependent z-component,which maps into a spatially varying spin density in theharmonic trap, revealed using absorption imaging of bothhyperfine components, as shown in Fig. 1. The evolutionoccurs on a time scale set by the mean field frequency,Ω MF ≃ π × . | i −| i cloud with a = 3 . a . Fig. 3 shows the differenceof the transversely integrated spin densities n ( x, t ) − n ( x, t ) ≡ S z ( x, t ) at selected times t after excitation,for scattering lengths of larger magnitude, ≃ ± a . Forthe larger scattering lengths, the data are sensitive to the evolution time and exhibit a complex structure, which weexplain using a mean field model, outlined below [14].A thermal average of the Heisenberg equations for thecollective spin vector ˜ S ( E, t ) as a function of axial energy E (in a one dimensional approximation) yields [5, 14], ∂ t ˜ S ( E ) = Ω ( E ) × ˜ S ( E ) + Z dE ′ ˜ g ( E ′ ,E ) ˜ S ( E ′ ) × ˜ S ( E ) , (1)where we suppress t in ˜ S ( E, t ) and ˜ S ( E ′ , t ). In eq. 1, Ω ( E, t ) includes the energy dependent precession rateabout the z axis and a general Rabi vector for radio fre-quency (rf) excitation of the initial superposition state,with R dE ˜ S z ( E, t = 0) = 1, prior to the rf pulse. Theintegral term describes the rotation of the spin vectorfor atoms of energy E arising from collisions with atomsof energy E ′ . Here, the coupling matrix ˜ g ( E ′ , E ) (seeeq. B29) is proportional to the mean field frequency andplays the role of the site to site coupling in a latticemodel.For a degenerate gas, the mean field frequency Ω MF =9¯ h n D a/ (5 m ), where n D is the 3D total atom densityand m is the atom mass. Although it is not necessaryto make a continuum approximation, in eq. 1 we haveassumed that the harmonic oscillator states are closelyspaced compared to the Fermi energy, as is the case forour experiments. Employing a WKB approximation forthe harmonic oscillator wave functions, ˜ g ( E ′ , E ) is pro-portional to 1 / √ E − E ′ , which determines the effectivelong-range character of the spin couplings. Eq. 1 is solvednumerically for ˜ S ( E, t ), from which we obtain the vectorspin density as a function of axial position x , S ( x, t ) = N ω x π Z ∞ dp x ˜ S (cid:18) p x m + m ¯ ω x x , t (cid:19) . (2)For | a | ∼ MF ≃ π × . s − , which is negligible. As N ( E ) + N ( E ) is conserved [14], the total atom spa-tial density, determined by analogy to eq. 2, should beconstant in time, as shown in Fig. 1.For the low temperature, degenerate gas, we find thateq. 1 is in excellent quantitative agreement with the spin-density profiles of Fig. 2 and captures very well the finefeatures of the data shown in Fig. 3, as well as the time FIG. 2. Spin-density profiles for a degenerate (
T /T F = 0 .
35) Fermi gas at t = 800 ms relative to coherent excitation. Data(blue dots) versus prediction (red curves) showing quantitative agreement. Left to right: n , n , n − n , n + n in units of thepeak total density. Each solid curve is the mean field model with a fixed scattering length of a = 3 .
04 bohr ( B = 528 .
147 G)and a fitted cloud size σ F x ≡ σ = 329 µ m, obtained by fitting the total density n + n to a 1D Thomas-Fermi profile, eq. B38.FIG. 3. Spin-density profiles in a degenerate sample T /T F = 0 .
35 at selected times relative to coherent excitation. ∆ n (0) = n (0 , t ) − n (0 , t ) is given in units of n (0)+ n (0). Solid curves: Mean field model with the same scattering length for each timeand a fitted cloud size within a few percent of the measured average value, σ = 322 . . µ m. Top three panels: B = 528 . a = 5 . a . Bottom three panels: B = 525 .
478 G, a = − . a . Note that the spin density inverts when the scatteringlength changes sign. dependence of the spin-density profiles shown in Fig. 13for a fixed scattering length [14].We fit the mean field model to the data of Fig. 3 inthe following way. First, we plot the dimensionless spindensity ( n − n ) / ( n + n ) at the center ( x = 0) as afunction of time, Fig. 4, for each value of the magneticfield. Second, we fit the model to the data of Fig. 4 to findthe scattering length that gives the best fits (red curves).The fits to the spatial density profiles of are then ob-tained by fixing the scattering length at each field to thevalue obtained from Fig. 4 and adjusting the Thomas-Fermi radius by a few per cent to fit the measured profileat each time. The mean of the measured radii is found tobe 322.0(1.5) µ m. Magnetic field stability is better than5 mG, limited by measurement precision. The absolutevalue of the field is calibrated using radio frequency spec-troscopy of the hyperfine transitions. Increasing the scattering length to a = − . a , wemeasure the amplitude of the spin density at the cloudcenter for a degenerate sample as a function of time rela-tive to coherent excitation, Fig. 5. Although the collisionrate ≃ . s − is still negligible, we observe a decay ofthe amplitude that is not predicted. We believe thatthe decay arises from the variation of the atom densityover several runs, which are averaged to determine eachdata point. The average of the predictions (red curve) ofFig. 5 yields the observed decay, because the sensitivityto the mean field frequency, and hence to the atom den-sity variation, increases with increasing time, resultingin a decreasing amplitude for the average. The corre-sponding spatial profiles are shown in Fig. 6, where pre-dicted curves are obtained for a fixed scattering lengthof − . a and fitting the Fermi width, within a fewpercent of the mean. FIG. 4. Central spin density versus evolution time for various magnetic fields near the zero crossing of the | i − | i scatteringlength. ∆ n (0) = n (0 , t ) − n (0 , t ) is given in units of n (0) + n (0). Solid curves show the mean-field model with the scatteringlength a as a fit parameter. The fitted values of a are plotted in Fig. 7.FIG. 5. Decay of the amplitude of the central spin density versus time for a = − . a . The dashed curve shows the predictedamplitude for the average density. The red curve shows the the average of the predictions based on the measured atom numbersand cloud widths for each shot. III. SCATTERING LENGTH PARAMETERS
The small a region, where the mean field model pre-cisely fits the data, enables measurement of the tuningrate a ′ (in bohr per gauss) of the scattering length nearthe zero crossing field B , where a ( B ) = a ′ ( B − B ) . (3)Here, we assume that the energy shift is negligible forthe degenerate sample, in contrast to the hot sample dis-cussed below. Using the data in Fig. 4, the fitted | i− | i scattering length for each magnetic field is plotted inFig. 7. The corresponding plot for | i − | i scatteringis discussed in Appendix A. The slopes of the linear fitsto the data yield the tuning rates a ′ , Table I.Next, we measure the magnetic field B at which thescattering length vanishes by using the spin evolutionas a sensitive probe: The profiles of the individual spincomponents remain unchanged at the zero crossing inthe degenerate regime. Fig. 8 shows the change in sizefor each spin profile between t = 0 and t = 800 ms, asa function of magnetic field. In addition, we show thedifference between the sizes of the state 1 and state 2profiles at t = 800 ms. Each method gives a field value B for the zero crossing. We report the mean in Table I. The corresponding uncertainties are estimated as onehalf of the difference between the maximum and the min-imum of B . The zero crossing for a , 527.18(2) G, issmaller than the value 527.5(2) G obtained by the samemethod at high temperature [1], and is consistent withthe calculated value 527.32(25), based on the most re-cent Li molecular potentials determined from 1D dimerspectra [16]. The zero crossings for a , 567.98(01) G andfor a , 588.68(01), listed Table I, are in very good agree-ment with the values 568.07 G and 588.80 G estimatedfrom the Feshbach resonance data of Ref. [16], which dif-fer only slightly from Ref. [15].Table I compares that the tuning rates a ′ = 3 . a / Gand a ′ = 4 . a / G, which we obtain from the fittedscattering length versus magnetic field in the presentwork, to estimates based on the Feshbach resonance pro-files a [ B ], which are obtained from the molecular po-tentials reported in Ref. [15] and in Ref. [16, 26]. Us-ing the profiles of Ref. [15], we find a ′ = 4 . a / Gand a ′ = 6 . a / G. These slopes are 50% larger thanthose estimated in the present work, but the ratios,4 . / .
14 = 1 .
44 and 6 . / .
12 = 1 .
48, are in good agree-ment. This suggests that the discrepancy may be ex-plained by an overall scale factor in our estimate of thetransverse averaged 3D density n D (see eq. B20), which FIG. 6. Spin density profiles versus time for a = − . a versus predictions (red curves) with the same scattering length foreach time and a fitted cloud size within a few percent of the measured average value, σ = 330 . µ m.TABLE I. Zero crossings B (G) and tuning rates a ′ ( a / G) for the scattering lengths of the broad Feshbach resonances in Li.States T( µ K) B (G) [This work] B (G) [15] B (G) [16] a ′ ( a /G ) [This work] a ′ ( a /G )[15] a ′ ( a /G ) [16]1-2 0.2 527.18(2) 534.15 527.32(25) 3.14(8) 4.12 3.491-2 45.7 527.42(1) - - - - -2-3 0.2 588.68(1) 588.92 588.75 4.52 (23) 6.11 5.821-3 0.2 567.98(1) 568.13 568.02 - 13.87 13.29FIG. 7. Fitted scattering length a versus measured magneticfield for a | i − | i mixture ( a = 1 bohr). Error bars denoteone standard deviation, obtained for each χ fit of Fig. 4. determines the scattering lengths from the mean field fre-quencies Ω MF ∝ n D a used to fit Fig. 4. However, usingthe Feshbach resonance profiles of Ref. [16, 26], we esti-mate the tuning rate a ′ = 3 . a / G, which only 11%larger than tuning rate obtained from our experiments,and a ′ = 5 . a / G, which is 29% larger.
IV. ENERGY SHIFT
We also observe the energy dependent shift in the zerocrossing, by preparing a | i − | i superposition at a hightemperature of T = 45 . µ K. There, we measure a shift of0 .
22 G relative to the degenerate sample. This yields anenergy tuning rate of 4 . µ K, confirming that the
FIG. 8. Measurement of the zero crossing field for a degener-ate Li | i − | i mixture. The plots show the change in cloudsize between t = 0 and t = 800 ms for state 1 (squares), state2 (diamonds), and the difference in the cloud sizes of the twospin states at t = 800 ms (circles). Solid lines are correspond-ing linear fits, crossing zero (dashed line) when a = 0. Errorbars denote the standard deviation of the mean of five runs. energy dependent shift is negligible for the degeneratesamples, compared to the precision of the magnetic fieldmeasurement.To directly illustrate the energy dependence, we mea-sure the spin density at 45 . µ K for B = 527 .
466 G,Fig. 9. We see that the high temperature spin densityprofile crosses the zero axis four times, in contrast to thelow temperature data of Fig. 3, which only crosses twice.The modification of the spin-density profile at hightemperature is not likely to arise from the | i − | i p-wave resonance in Li, which is located near 186.2(6) G
FIG. 9. High temperature spin density profile of a | i − | i mixture for t = 400 ms. T = 45 . µ K and B = 527 .
466 G,where the zero-energy s-wave scattering length is 0 .
90 bohr.Here σ G = 323 µ m is the gaussian 1 /e radius of the totaldensity profile. and has a width of 0 .
5G [27]. To understand this pro-file and the energy shift, we include the energy depen-dence of the scattering length and of the average mag-netic field, by replacing a in ˜ g ( E ′ , E ) of eq. B31 with a ( E ′ , E ) = a ′ [ B eff ( E ′ , E ) − B ] [14], with B eff ( E ′ , E ) theeffective magnetic field. Then, for small positive B − B ,atoms with small energies E, E ′ have B eff − B > E, E ′ , have B eff − B < .
28 G above the zero cross-ing to 0 .
08 G above, where a = 0 . a for atoms with E = E ′ = 0.In summary, we have shown that a mean field collectivespin rotation model, including the full energy-dependentcoupling matrix, quantitatively describes the spin den-sity evolution in the collisionless regime, precisely test-ing the underlying energy-space spin-lattice model. Themeasurements provide an essential benchmark for futurework on collective spin evolution with designer energylandscapes in the weakly interacting regime, and pave theway studies of beyond mean field physics in weakly in-teracting gases, measurement of spatially correlated spinfluctuations [2] and measurement of correlated spin cur-rents [28].Primary support for this research is provided bythe Physics Divisions of the Army Research Office(W911NF-14-1-0628) and the Division of MaterialsScience and Engineering, the Office of Basic EnergySciences, Office of Science, U.S. Department of Energy(de-sc0008646). 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A cloud comprising a 50-50 mixture of the two low-est hyperfine states, denoted | i and | i , is evaporativelycooled to degeneracy near the | i − | i Feshbach reso-nance at 832 . | i spin component is eliminated by means of a resonant op-tical pulse. To create a | i − | i superposition state, themagnetic field is ramped to 527 G, near the zero crossingof the scattering length. The atoms in spin state | i arethen excited by a 2 ms radio-frequency π/ | i . Similarly, a | i−| i superposition state is prepared by employing an rf transi-tion from state | i to state | i close to the correspondingzero crossing around 589 G. For the | i − | i superpo-sition state, we prepare a single | i spin component at1200 G. The magnetic field is then ramped down to thevalue of interest around 568 G, near the zero crossing ofthe | i − | i scattering length. The atoms are excitedby a 2 ms radio-frequency π/ | i to state | i , creating abalanced | i−| i superposition state. Then a 4 ms radio-frequency π pulse is applied, which is resonant with thetransition from state | i to state | i , to create a balanced | i − | i superposition state. The trap parameters forour experiments are: ω = (2 π × . B(G) / ω x = 2 π ×
23 Hz, ω ⊥ = 2 π ×
625 Hz, for the degenerategas, and ¯ ω x = 2 π ×
174 Hz, ω ⊥ = 2 π × .
77 kHz, for thehigh temperature gas.After preparation, we obtain degenerate samples witha typical total atom number of N = N ↑ + N ↓ ≃ . × and an ideal gas Fermi temperature of k B T F =¯ h (3 N ¯ ω x ω ⊥ ) / = k B × . µ K for our trap frequen-cies. To determine the temperature T , the measuredone dimensional total density versus x is fit with a fi-nite temperature Thomas-Fermi profile for a noninter-acting gas, which is appropriate for our weakly inter-acting gas. Using the calculated Thomas-Fermi radius σ T F = p k B T F / ( m ¯ ω x ) = 270 µ m, we find T = 0 . T F .In the main text, we reported measurements of the zerocrossing field of the scattering length for Li | i − | i , | i − | i , and | i − | i mixtures and the tuning rate ofthe scattering length for | i − | i and | i − | i mixtures.Fig. 10 shows the data that was used to obtain the tuningrate for the | i − | i mixture. FIG. 10. Tuning rate of the scattering length a of a | i − | i mixture versus measured magnetic field ( a = 1 bohr) Errorbars denote one standard deviation, obtained for each χ fitto the time dependent central amplitude for the given B . Figs. 11 and 12 show the data that was used to obtainthe zero crossings fields for the | i−| i , and | i−| i mix-tures. In Fig. 11 and Fig. 12, and for Fig. 5 of the mainpaper, we take into account cloud size variations aris-ing from small changes in the atom number. Each datapoint represents an average of 10 experimental runs. For FIG. 11. Measurement of the zero crossing field for a degener-ate Li | i − | i mixture. The plots show the change in cloudsize between t = 0 and t = 800 ms for state 3 (squares), state2 (diamonds), and the difference in the cloud sizes of the twospin states at t = 800 ms (circles). Solid lines are correspond-ing linear fits, crossing zero (dashed line) when a = 0. Errorbars denote the standard deviation of the mean of five runs.FIG. 12. Measurement of the zero crossing field for a degen-erate Li | i − | i mixture. The plots show the change incloud size between t = 0 and t = 800 ms for state 3 (squares),state 1 (diamonds). Solid lines are corresponding linear fits,crossing zero (dashed line) when a = 0. Error bars denote thestandard deviation of the mean of five runs. each run i , we extract the atom number N i and the axialcloud size σ i for each spin component. The cloud sizesscale as N / i for zero temperature Thomas-Fermi pro-files. Therefore, to correct for the varying atom number,we calculate the reduced size σ i /N / i for each run anduse D σ i /N / i E D N / i E as the effective mean cloud sizefor each field. Appendix B: Mean-Field Model
We employ a mean field model in energy representa-tion to describe the spin-density profiles observed in our experiments. The bias magnetic field tunes the s-wavescattering length near the zero crossing, where the gas isvery weakly interacting and the energy changing collisionrate is negligible. For this reason, we begin with the sin-gle particle Hamiltonian for a noninteracting Fermi gaswith two spin states, a lower hyperfine state denoted ↑ and an upper hyperfine state denoted ↓ . For an atom atrest, these states differ in energy by ¯ hω HF , where ω HF isthe hyperfine resonance frequency. A spin-independentcigar-shaped optical trap confines the atom cloud weaklyalong the cigar axis, denoted x , and tightly in the per-pendicular ρ direction, so that ρ << | x | . Curvature inthe bias magnetic field produces a significant harmonicconfining potential along the x -axis, while for the ρ direc-tion, the magnetic contribution to the confining potentialis negligible compared to that of the optical trap. Thenet optical and magnetic trapping potential along x isthen spin-dependent, with harmonic oscillation frequen-cies ω x ↑ and ω x ↓ . The Hamiltonian for the motion alongthe x -axis (without the hyperfine energies) is H = X n | n ih n | h ( n + 1 /
2) ¯ hω x ↑ | ↑ih↑ | +( n + 1 /
2) ¯ hω x ↓ | ↓ih↓ | i . (B1)For later use, we define the dimensionless single particlespin operators, s z = | ↑ih↑ | − | ↓ih↓ | s x = | ↑ih↓ | + | ↓ih↑ | s y = | ↑ih↓ | − | ↓ih↑ | i , (B2)where [ s x , s y ] = s x s y − s y s z = is z and cylic permutations.A radio-frequency transition does not change the har-monic oscillator quantum number n . Hence, the reso-nance frequency for a transition from the lower ↑ to theupper ↓ hyperfine state of an oscillating atom in state | n i is ω res = ω HF + ( n + ) δω x with δω x ≡ ω x ↓ − ω x ↑ .Working in a frame rotating at the hyperfine resonancefrequency ω HF and defining the energy E = ( n + ) ¯ h ¯ ω x ,where the mean oscillation frequency, ¯ ω x ≡ ( ω x ↑ + ω x ↓ ) / H = X E | E ih E | " E ( | ↑ih↑ | + | ↓ih↓ | )+ ¯ h Ω( E ) | ↑ih↑ | − | ↓ih↓ | , (B3)where h E ′ | E i = δ E ′ ,E and the last term is proportionalto s z , with Ω( E ) ≡ − δω x E ¯ h ¯ ω x . (B4)To treat the many-body problem for a very weakly in-teracting gas, where the single particle energies do notchange during the evolution time, we define the field op-erator in energy representation,ˆ ψ ≡ X E,σ = ↑ , ↓ ˆ a σ ( E ) | E i| σ i . (B5)With the anticommutation relations { ˆ a σ ( E ) , ˆ a † σ ′ ( E ′ ) } = δ σ,σ ′ δ E,E ′ , (B6)we have { ˆ ψ, ˆ ψ † } = ˆ1, the product of the energy and spinidentity operators. The many-body Hamiltonian for thenoninteracting atoms is then defined by ˆ H = ( ˆ ψ † H ˆ ψ ),where the parenthesis ( ... ) denotes inner products for the single particle energy and spin states. Then,ˆ H = X E ′ E ′ [ ˆ N ↑ ( E ′ ) + ˆ N ↓ ( E ′ )]+ X E ′ ¯ h Ω( E ′ ) ˆ S z ( E ′ ) . (B7)Here, the number operators are ˆ N ↑ ( E ) = a †↑ ( E ) a ↑ ( E )and ˆ N ↓ ( E ) = a †↓ ( E ) a ↓ ( E ) and the dimensionless many-body spin operators are given (in the Schr¨odinger pic-ture) byˆ S z ( E ) = ( ˆ ψ † s z ˆ ψ ) = ˆ N ↑ ( E ) − ˆ N ↓ ( E )2 (B8)ˆ S x ( E ) = ( ˆ ψ † s x ˆ ψ ) = ˆ a †↑ ( E ) ˆ a ↓ ( E ) + ˆ a †↓ ( E ) ˆ a ↑ ( E )2ˆ S y ( E ) = ( ˆ ψ † s y ˆ ψ ) = ˆ a †↑ ( E ) ˆ a ↓ ( E ) − ˆ a †↓ ( E ) ˆ a ↑ ( E )2 i . The corresponding field operators in position represen-tation are ˆ ψ ( x ) = ( h x | ˆ ψ ) = X E,σ ˆ a σ ( E ) φ E ( x ) | σ i≡ X σ ˆ ψ σ ( x ) | σ i . (B9)The Schr¨odinger picture operator of the z component ofthe spin density is thenˆ S z ( x ) = ( ˆ ψ † ( x ) s z ˆ ψ ( x )) (B10)= 12 X E,E ′ φ ∗ E ′ ( x ) φ E ( x ) h ˆ a †↑ ( E ′ ) ˆ a ↑ ( E ) − ˆ a †↓ ( E ′ ) ˆ a ↓ ( E ) i . Note that the orthonormality of the φ E ( x ) yields R dx ˆ S z ( x ) = P E ˆ S z ( E ) = ˆ S z , the total z -component ofthe spin operator.For our mean-field treatment, we assume initially thatthere is no coherence between E ′ = E for a thermal av-erage, i.e., h ˆ a †↑ ( E ′ )ˆ a ↑ ( E ) i = h ˆ N ↑ ( E ) i δ E ′ ,E . Then the z -component of the c-number spin density is given by S z ( x ) ≡ h ˆ S z ( x ) i = X E | φ E ( x ) | h ˆ S z ( E ) i . (B11) Hence, we need only to determine S z ( E, t ) to predict themeasured S z ( x, t ).Using the anticommutation relations of Eq. B6, it iseasy to evaluate the elementary commutators, h ˆ a † σ ′ ( E ′ ) ˆ a σ ( E ) , ˆ a † σ ′ ( E ) i = ˆ a † σ ′ ( E ′ ) δ E ,E δ σ ,σ ′ (B12) h ˆ a † σ ′ ( E ′ ) ˆ a σ ( E ) , ˆ a σ ( E ) i = − ˆ a σ ( E ) δ E ′ ,E δ σ ′ ,σ , which are formally identical to the results obtained forbosons. With eq. B12, it is straightforward to show thatthe spin operators of eq. B8 satisfy the usual cyclic com-mutation relations,[ ˆ S i ( E ′ ) , ˆ S j ( E )] = i ǫ ijk ˆ S k ( E ) δ E ′ ,E . (B13)With eq. B7, the Heisenberg operator equations for thecollisionless spin evolution are then ∂ ˆ S ( E, t ) ∂t = i ¯ h h ˆ H , ˆ S ( E, t ) i = Ω ( E, t ) × ˆ S ( E, t ) , (B14)where Ω ( E ) = ˆ e z Ω( E ) (B15)and Ω( E ) is given by eq. B4. For sample preparation us-ing radio frequency excitation, eq. B15 is readily general-ized to include a time dependent Rabi frequency rotationrate Ω R ( t ) ˆ e y and an additional time dependent detuningterm ∆( t ) ˆ e z , with ∆ = ω ( t ) − ω HF in the rotating frame.Next, we consider collisional interactions, assuming s-wave scattering between atoms of different spin, whichis dominant at low temperature. Short range scatteringis modeled by a contact interaction between spin-up andspin-down atoms with an s-wave scattering length a S , H ′ ( x − x ) = 4 π ¯ h a S m δ ( x − x ) ≡ g δ ( x − x ) , (B16)For the many-body system,ˆ H ′ = Z d x d x (cid:16) ˆ ψ † ( x ) ˆ ψ † ( x ) H ′ ( x − x ) ˆ ψ ( x ) ˆ ψ ( x ) (cid:17) = g Z d x ˆ ψ †↑ ( x ) ˆ ψ †↓ ( x ) ˆ ψ ↓ ( x ) ˆ ψ ↑ ( x ) , (B17)where the factor 1 / ψ ↑ , ↓ ( x ) = 0. For simplicity, we initially neglect the de-pendence of a S on the relative kinetic energy of the col-liding pair, which will be included later.For our experiments, where atoms are confined in acigar-shaped cloud, the x dimension is large compared tothe radial dimension ρ , so that the bias field curvature isnegligible along the ρ direction, as noted above. There-fore, we treat the problem as one-dimensional by takingthe field operators to be of the form,ˆ ψ σ ( x ) = φ ( ρ ) ˆ ψ σ ( x ) , (B18)0Carrying out the ρ integration in eq. B17, we determinethe effective one-dimensional interaction Hamiltonian,ˆ H ′ = ˜ g Z dx ˆ ψ †↑ ( x ) ˆ ψ †↓ ( x ) H ′ ˆ ψ ↓ ( x ) ˆ ψ ↑ ( x ) . (B19)where ˜ g ≡ g ¯ n ⊥ and¯ n ⊥ = Z πρdρ [ n ⊥ ( ρ )] . (B20)Here, we have let | φ ( ρ ) | → n ⊥ ( ρ ), where R πρdρ n ⊥ ( ρ ) = 1. Eq. B20 determines an effec-tive mean transverse density, ¯ n ⊥ , as a fraction per unittransverse area. Using eq. B9, eq. B19 takes the formˆ H ′ = ˜ g X E ,E ,E ′ ,E ′ Z dx φ ∗ E ′ ( x ) φ ∗ E ′ ( x ) φ E ( x ) φ E ( x ) × ˆ a †↑ ( E ′ )ˆ a †↓ ( E ′ )ˆ a ↓ ( E )ˆ a ↑ ( E ) . (B21)With the anticommutation relations, eq. B6, we canrewrite the operator product of eq. B21 asˆ O ′ ≡ ˆ a †↑ ( E ′ )ˆ a ↑ ( E )ˆ a †↓ ( E ′ )ˆ a ↓ ( E ) . (B22)We simplify the interaction Hamiltonian by using amean field approximation to evaluate eq. B22. To firstorder, we obtainˆ O ′ ≃ h ˆ a †↑ ( E ′ ) ˆ a ↑ ( E ) i ˆ a †↓ ( E ′ ) ˆ a ↓ ( E )+ h ˆ a †↓ ( E ′ ) ˆ a ↓ ( E ) i ˆ a †↑ ( E ′ ) ˆ a ↑ ( E ) −h ˆ a †↑ ( E ′ ) ˆ a ↓ ( E ) i ˆ a †↓ ( E ′ ) ˆ a ↑ ( E ) −h ˆ a †↓ ( E ′ ) ˆ a ↑ ( E ) i ˆ a †↑ ( E ′ ) ˆ a ↓ ( E ) , (B23)where h ... i denotes a thermal average, which vanishes un-less the energy arguments are the same. Further, we willrequire a thermal average of the Heisenberg equationsof motion, i.e., h [ ˆ O ′ , ˆ S i ( E )] i . This will vanish unless theenergy arguments in the operator factors are the same.Hence, Eq. B21 can be rewritten asˆ H ′ = ˜ g X ˜ E,E ′ Z dx | φ E ′ ( x ) | | φ ˜ E ( x ) | × n h ˆ a †↑ ( E ′ ) ˆ a ↑ ( E ′ ) i ˆ a †↓ ( ˜ E ) ˆ a ↓ ( ˜ E )+ h ˆ a †↓ ( E ′ ) ˆ a ↓ ( E ′ ) i ˆ a †↑ ( ˜ E ) ˆ a ↑ ( ˜ E ) −h ˆ a †↑ ( E ′ ) ˆ a ↓ ( E ′ ) i ˆ a †↓ ( ˜ E ) ˆ a ↑ ( ˜ E ) −h ˆ a †↓ ( E ′ ) ˆ a ↑ ( E ′ ) i ˆ a †↑ ( ˜ E ) ˆ a ↓ ( ˜ E ) o . (B24)With the collective spin operators, eq. B8, we rewriteeq. B24 asˆ H ′ = 2 ˜ g X ˜ E,E ′ Z dx | φ E ′ ( x ) | | φ ˜ E ( x ) | × n N ( E ′ ) ˆ N ( ˜ E ) − S ( E ′ ) · ˆ S ( ˜ E ) o , (B25) where ˆ N ( ˜ E ) = ˆ N ↑ ( ˜ E ) + ˆ N ↓ ( ˜ E ) is the total number oper-ator and ˆ S ( ˜ E ) is the total spin vector operator for atomsof energy ˜ E . N ( E ′ ) is a c-number scalar and S ( E ′ ) is a c-number vector, i.e., the corresponding thermal averagedHeisenberg operators for energy E ′ .To evaluate of the collisional contribution to theHeisenberg equations of motion, we require [ ˆ H ′ , ˆ S ( E )].Here, [ ˆ N ( ˜ E ) , ˆ S ( E )] = 0, and using eq. B13, [ S ( E ′ ) · ˆ S ( ˜ E ) , ˆ S ( E )] = − i S ( E ′ ) × ˆ S ( ˜ E ) δ ˜ E,E . With eq. B14, theHeisenberg equation ˙ˆ S ( E, t ) = i ¯ h h ˆ H + ˆ H ′ , ˆ S ( E, t ) i forthe spin vector operator of energy E takes the simpleform, ∂ ˆ S ( E, t ) ∂t = Ω ( E, t ) × ˆ S ( E, t )+ X E ′ g ( E ′ , E ) S ( E ′ , t ) × ˆ S ( E, t ) . (B26)In eq. B26, g ( E ′ , E ) = − g ¯ n ⊥ ¯ h I ( E ′ , E ) , (B27)where I ( E ′ , E ) ≡ R dx | φ E ′ ( x ) | | φ E ( x ) | , ¯ n ⊥ is given byeq. B20, and g = 4 π ¯ h a S /m .In our experiments, where the energy E >> ¯ h ¯ ω x , | φ E ( x ) | can be evaluated in a WKB approximation, | φ E ( x ) | ≃ Θ[ a ( E ) − | x | ] π p a ( E ) − x , (B28)where a ( E ) = p E/ ( m ¯ ω x ) is the classical turning pointand Θ is a Heaviside function. Then, the x -integral ineq. B27 takes the form I ( E ′ , E ) = 1 π a min Z − du rh EE min − u i h E ′ E min − u i , where we have taken x = u a min . Here, a min = p E min / ( m ¯ ω x ) determines the overlap region, with E min the minimum of E, E ′ . Using u = sin θ , and byconsidering separately the cases E min = E < E ′ and E min = E ′ < E , we obtain I ( E ′ , E ) = 1 π s m ¯ ω x | E − E ′ | Z π/ − π/ dθ q E min | E − E ′ | cos θ . The integral is readily evaluated, yielding g ( E ′ , E ) = − g ¯ n ⊥ π ¯ h s m ¯ ω x | E − E ′ |× EllipticK (cid:20) − E min | E − E ′ | (cid:21) , (B29)where E ′ = E , since the sum in the last term of eq. B26vanishes for E ′ = E , i.e., we can take g ( E ′ = E, E ) = 0in eq. B26.1Taking the thermal average of the evolution equations,we replace the vector operators by the c-number vec-tors S ( E, t ) ≡ h ˆ S ( E, t ) i . Since E >> ¯ h ¯ ω x , we evaluateeq. B26 in the continuum limit. We replace the sum P ′ E ≡ P ′ n by R dE ′ ¯ h ¯ ω x and define S ( E, t )¯ h ¯ ω x ≡ N S ( E, t ) , (B30)where N = N ↑ + N ↓ is the total number of atoms. Then, ∂ ˜ S ( E, t ) ∂t = Ω ( E, t ) × ˜ S ( E, t ) (B31)+ Z dE ′ ˜ g ( E ′ , E ) ˜ S ( E ′ , t ) × ˜ S ( E, t ) , where ˜ g ( E ′ , E ) ≡ N g ( E ′ , E ) has a dimension of s − .Note that the factor N/ ↑ hyperfine state, the total spin inthe z-direction is N/ R dE ˜ S ( E, t ), since ˜ g ( E, E ′ ) is symmetric under E ′ ↔ E and the cross product is antisymmetric. In con-trast, Ω ( E ) is an energy dependent rotation rate thatdoes not conserve the total spin ˜ S ( E, t ). However, with-out radio frequency excitation, Ω ( E ) is along the z -axisand the z-component of the total spin R dE ˜ S z ( E ) is con-served. Finally, since eq. B31 describes a rotation of˜ S ( E, t ), | ˜ S ( E, t ) | ≡ S ( E ) is conserved for each E .We integrate eq. B31 subject to the initial conditionthat all atoms are in the lower hyperfine (spin-up) state.A radio frequency pulse is then used to prepare a col-lective spin vector with components in the x − y plane.With eq. B30, the thermal averaged z-component of theinitial collective spin operator, eq. B8, requires˜ S z ( E, t = 0) = S ( E ) = P ( E ) , (B32)where P ( E, T ) is the fraction of atoms with axial energy E at temperature T and R ∞ dE P ( E ) = 1 in the contin-uum limit. In the high temperature limit, P ( E ) = 1 Z e − EkBT , (B33)with the partition function Z = R ∞ dEe − EkBT = k B T . Inthe low temperature limit, T →
0, we use the occupationnumber for a Fermi distribution in three dimensions andsum over the energies in the two perpendicular directionsto obtain the normalized axial ( x ) energy distribution, P ( E ) = 3 E F (cid:18) − EE F (cid:19) Θ (cid:18) − EE F (cid:19) , (B34)where for N ↑ = N , E F = (6 N ) / ¯ h ¯ ω , with ¯ ω ≡ ( ω ⊥ ¯ ω x ) / .The measured axial spin density profiles are given bythe continuum limit of eq. B11, S ( x, t ) = N Z dE | φ E ( x ) | ˜ S ( E, t ) , (B35) where we neglect coherence between states of differentenergy and R dx S ( x, t ) = N R dE ˜ S ( E, t ). Evaluation ofeq. B35 is simplified by rewriting the WKB wave func-tions of eq. B28 in the form | φ E ( x ) | = ¯ ω x π Z ∞ dp x δ (cid:18) E − p x m − m ¯ ω x x (cid:19) (B36)so that the spin density is S ( x, t ) = N ω x π Z ∞ dp x ˜ S (cid:18) p x m + m ¯ ω x x , t (cid:19) . (B37)The initial spatial densities for the spin components aresimilarly determined. For the degenerate gas, we approx-imate the energy distribution by the zero temperaturelimit, eq. B34, as discussed above. The correspondingspatial density for each spin component, just after prepa-ration, is then a normalized zero temperature Thomas-Fermi profile. Analogous to eq. B35, using eq. B28 (oreq. B36), it is easy to show that the initial density profilesfor each state are of the one dimensional Thomas-Fermiform, n ↑ , ↓ ( x,
0) = N ↑ , ↓ Z dE | φ E ( x ) | P ( E ) (B38)= N ↑ , ↓ π σ F x (cid:18) − x σ F x (cid:19) / Θ (cid:18) − x σ F x (cid:19) , where σ F x = p E F / ( m ¯ ω x ) is the Fermi radius and N ↑ = N ↓ = N/ n ( x ) is time indepen-dent, i.e., n ↑ ( x, t ) + n ↓ ( x, t ) = n ↑ ( x,
0) + n ↓ ( x,
0) = n ( x ),as shown in Fig. 1 of the main paper. For the non-degenerate gas, the Maxwell-Boltzmann energy distribu-tion of eq. B33 yields the corresponding gaussian spatialprofile.
1. Small Angle Approximation
We can make contact with the first order, large Dickegap approximation of Koller et al [2], by consider-ing the evolution equations for small amplitude spinwaves, expressed in terms of angles. As the magni-tude of | ˜ S ( E, t ) | ≡ S ( E ) is conserved for each E , where R dE S ( E ) = 1, we can write the spin components interms of two angles, a polar angle θ E and an azimuthalangle, ϕ E , ˜ S x ( E, t ) = S ( E ) sin θ E ( t ) cos ϕ E ( t )˜ S y ( E, t ) = S ( E ) sin θ E ( t ) sin ϕ E ( t )˜ S z ( E, t ) = S ( E ) cos θ E ( t ) . (B39)Using eq. B31, it is straightforward to obtain the evolu-tion equations for the angles. For times after the radio-frequency preparation pulse,˙ θ E = Z dE ′ g ( E, E ′ ) S ( E ′ ) sin θ E ′ sin( ϕ E ′ − ϕ E ) (B40)2˙ ϕ E = γ E + Z dE ′ g ( E, E ′ ) S ( E ′ ) (B41) × [ cos θ E ′ − cot θ E sin θ E ′ cos( ϕ E − ϕ E ′ )] , where the energy-dependent rotation rate about the z-axis, eq. B4, is Ω( E ) = − δω x / (¯ h ¯ ω x ) E ≡ γ E . Here, wetake the initial conditions to be ˜ S x ( E, t = 0) = S ( E ) and˜ S z ( E, t = 0) = ˜ S y ( E, t = 0) = 0, just after the radio fre-quency pulse. From eq. B40, we see that R dE ˙˜ S z ( E, t ) = − R dE S ( E ) sin θ E ˙ θ E = 0, since sin( ϕ E ′ − ϕ E ) is odd in E ′ , E and R dE ˜ S z ( E, t ) is conserved as it should be.The angle equations take a simple approximate formfor small amplitude spin waves, where θ E = π/ δθ E with δθ E <<
1. Then,˜ S z ( E, t ) ≃ − S ( E ) δθ E ( t ) (B42)and the spatial profile, eq. B35, is given by S z ( x, t ) = − N Z dE | φ E ( x ) | S ( E ) δθ E ( t ) , (B43)where | φ E ( x ) | is easily evaluated using the WKB ap-proximation.For γ E >> g ( E, E ′ ), with sin θ E ′ ≃ ϕ E ′ − ϕ E ≃ γ ( E ′ − E ) t , eq. B40 immediately yields δθ E ( t ) ≃ Z dE ′ g ( E, E ′ ) S ( E ′ ) × − cos[ γ ( E ′ − E ) t ] γ ( E ′ − E ) . (B44)To make contact with the first order, large Dicke gapapproximation of Koller et al [2], we consider the oppositelimit, g ( E, E ′ ) >> γ E . Here, we make the simplifyingassumption that g ( E, E ′ ) ≃ ¯Ω g is energy independent.Then we can approximate ϕ E − ϕ E ′ <<
1, over the rele-vant time scale t ≃ / ¯Ω g and eqs. B40 and B41 take thesimple forms, δ ˙ θ E = ¯Ω g Z dE ′ S ( E ′ ) ( ϕ E ′ − ϕ E ) (B45) δ ˙ ϕ E = γ E + ¯Ω g Z dE ′ S ( E ′ ) ( δθ E − δθ E ′ ) . Differentiating the first equation with respect to t yields, δ ¨ θ E = ¯Ω g Z dE ′ S ( E ′ ) ( ˙ ϕ E ′ − ˙ ϕ E ) . (B46)From the second equation,˙ ϕ E ′ − ˙ ϕ E = γ ( E ′ − E ) + ¯Ω g ( δθ E ′ − δθ E ) , (B47)where we have used R dE ′ S ( E ′ ) = 1. After substitutingeq. B47 into eq. B46, we take R dE ′ S ( E ′ ) δθ E ′ = 0. Here,we assume for simplicity that the initial spin is in the x-yplane, so that the conserved total ˜ S z vanishes. Then, δ ¨ θ E + ¯Ω g δθ E = ¯Ω g γ ( ¯ E − E ) , (B48) where ¯ E ≡ R dE ′ S ( E ′ ) E ′ . For the initial conditions, δθ E (0) = 0 and δ ˙ θ E (0) = 0, δθ E ( t ) = γ ( ¯ E − E )¯Ω g [1 − cos( ¯Ω g t )] . (B49)With eq. B43, we see that eq. B49 is equivalent to eq.3 of Koller et al [2], which was obtained by first orderperturbation theory in the Dicke spin state basis.
2. Numerical Implementation
To determine ˜ S ( E, t ) from eq. B31, we divide theenergy range into discrete intervals ∆ E , taking E =( n − E , with n an integer, 1 ≤ n ≤ n max . Typ-ically, n max = 500. This method determines thespin components i = x, y, z as column vectors in dis-crete energy space, ˜ S discr i ( n, t ), where n labels the row(rather than the harmonic oscillator state). We take˜ S ( E, t ) = ˜ S discr ( n, t ) / ∆ E in eq. B31. With the replace-ment R dE ′ / ∆ E = R dn ′ → P n ′ , the discrete energyevolution equations are ∂ ˜ S discr ( n, t ) ∂t = Ω ( n, t ) × ˜ S discr ( n, t ) (B50)+ X n ′ ˜ g ( n ′ , n ) ˜ S discr ( n ′ , t ) × ˜ S discr ( n, t ) . where˜ g ( n ′ , n ) = ˜Ω p | n − n ′ | EllipticK (cid:20) − n min − | n − n ′ | (cid:21) . (B51)Here, n min is the minimum of n and n ′ and˜Ω = − N g ¯ n ⊥ π ¯ h r m ¯ ω x E , (B52)with g = 4 π ¯ h a S /m .We define ∆ E differently for the high and low tempera-ture limits. In the low temperature limit, we take ∆ E = s E F . Since 0 ≤ E ≤ E F , we have s = 1 / ( n max − E = s k B T and take s so that exp[ − s ( n max − E = s m ¯ ω x σ x .Then, for T = 0, σ x = p E F / ( m ¯ ω x ) ≡ σ F x is theFermi radius, which is measured in the experiments. Forthe high temperature limit, σ x = p k B T / ( m ¯ ω x ) is themeasured gaussian (Boltzmann factor) 1 /e radius. With∆ E = s m ¯ ω x σ x , eq. B52 yields˜Ω = − √ s hπ a S m ¯ n ⊥ Nσ x ≡ − √ s Ω MF , (B53)where we have defined the mean field frequency Ω MF , h = 2 π ¯ h , and ¯ n ⊥ given by eq. B20. In the low tem-perature limit, with n ⊥ ( ρ ) = 3(1 − ρ /σ F ⊥ ) / ( πσ F ⊥ ), weobtain ¯ n ⊥ = πσ F ⊥ . In the high temperature limit, with3 n ⊥ ( ρ ) = exp[ − ρ /σ ⊥ ] / ( πσ ⊥ ), we obtain ¯ n ⊥ = πσ ⊥ .Then, Ω MF = 920 π h a S m n F T = 0Ω MF = 1 π / h a S m n High
T . (B54)Here n F = 8 N/ ( π σ F ⊥ σ F x ) is the 3D central den-sity for a T = 0 Thomas-Fermi profile with σ F ⊥ = p E F / ( mω ⊥ ) and n = N/ ( π / σ ⊥ σ x ) is the 3Dcentral density in the Boltzmann limit, where σ ⊥ = p k B T / ( mω ⊥ ) .With our choices of ∆ E , the initial conditions are anal-ogous to eq. B32,˜ S discr z ( n, t = 0) = P ( n ) , (B55)where for the high temperature limit, P ( n ) = exp[ − s ( n − /Z , and for the T = 0 limit, P ( n ) = 3 s [1 − s ( n − /Z , with Z = P n max n =1 P ( n ).Now we evaluate the first term on the right sideof eq. B50, which is the energy-dependent frequency Ω ( n, t ) = ˆ e z Ω z ( n )+ Ω Rabi ( t ). As discussed above, Ω z ( n )arises from the bias magnetic field curvature. For a gen-eral radio-frequency excitation with a time-dependentdetuning ∆( t ) and Rabi frequency Ω R ( t ), Ω Rabi ( t ) =ˆ e z ∆( t ) + ˆ e y Ω R ( t ). Using E = ( n − s E F for the T = 0limit and E = ( n − s k B T in the high temperature limit,we have Ω z ( n ) ≡ Ω z ( n − . (B56)where Ω z = − δω x s E F / (¯ h ¯ ω x ) at T = 0 and Ω z = − δω x s k B T / (¯ h ¯ ω x ) in the high temperature limit.Next, we evaluate the resonance frequency difference, δω x = ω x ↓ − ω x ↑ , which arises from the curvature ofthe bias magnetic field in the axial x direction, ∆ B z = x B ′′ z (0) /
2. The harmonic oscillation frequencies for theupper hyperfine state ( ↓ ) and lower hyperfine state ( ↑ )are determined by the sum of optical and magnetic springconstants, ω x ↓ , ↑ = ω + ω ↓ , ↑ = ω + 1 m ∂ B z ∂ x ∂E ↓ , ↑ ∂B , (B57)where ω opt arises from the optical trap and ω mag fromthe bias field curvature.For our experiments in Li, the hyperfine energies E ↓ , ↑ are dominated by the Zeeman shift of the (spin down)electron for each of the lowest three hyperfine states,while the much smaller difference E ↓ − E ↑ arises from thedifference between the nuclear parts of the magnetic mo-ment and the difference in the hyperfine mixing. Then,with ω ≡ ( ω ↓ + ω ↑ ) / ω x ≡ ω + ω ,we have ω x ↓ , ↑ = s ω + ω ± ω ↓ − ω ↑ ≃ ¯ ω x ± ω ↓ − ω ↑ ω x ! (B58) and δω x ¯ ω x = ω x ↓ − ω x ↑ ¯ ω x = ω ¯ ω x ω ↓ − ω ↑ ω . Then, δω x = ω ¯ ω x ∂E ↓ ∂B − ∂E ↑ ∂B∂E ↓ ∂B + ∂E ↑ ∂B ! ≃ ω ¯ ω x ¯ hω ′↓↑ g J µ B , (B59)where ω ′↓↑ is the tuning rate of the transition, with ↓ the upper hyperfine state. Here, we have assumed that thedenominator of eq. B59 is approximately twice the Zee-man tuning rate of a spin-down electron, 2 × g J µ B / − π × . Li. For our experiments, ω = (2 π × . B(G) / ω x = 2 π ×
23 Hz, ω ⊥ = 2 π ×
625 Hz; for the hightemperature gas, ¯ ω x = 2 π ×
174 Hz, ω ⊥ = 2 π × .
77 kHz.For a mixture of two hyperfine states, as noted above, ↓ denotes the upper hyperfine state, and ↑ denotes thelower hyperfine state. The hyperfine energies for thethree lowest states of Li, denoted 1 , , ω ′ [527G] = 2 π × . ω ′ [589G] = − π × . ω ′ [568G] = − π × . ω x = 2 π ×
23 Hz,we obtain δω x = − π × . − δω x = +2 π × . − δω x = +2 π × . − { n − , ˜ S discr i ( n, t ) } for 1 ≤ n ≤ n max . Note that n − E = ( n − E = 0for n = 1. The energy-dependent ˜ S discr ( n, t ) is thenconverted to an interpolator function of ( n −
1) = E/ ∆ E and eq. B37 used to find the spin density S ( x, t ).
3. Energy Dependent Scattering Length
For experiments in the non-degenerate regime athigher temperatures, we find that the energy dependenceof the scattering length cannot be neglected. This en-ergy dependence strongly modifies the spin-density pro-files for small positive scattering lengths, as shown inFig. 6 of the main paper, and produces a shift of the zerocrossing field. We include this dependence in g ( E ′ , E )of eq. B27 by replacing the energy-independent s-wavescattering length a S with an energy dependent scatter-ing length a ( E ′ , E ). The s-wave scattering length is givenby the energy-dependent scattering amplitude f ( k ), a [ B, k ] = f (cid:18) − µ B B + ¯ h k µ (cid:19) , (B60)where ¯ h k is the relative momentum and µ = m/ B z ≡ B − µ B B , with µ B the Bohr magne-ton. For our experiments in the degenerate regime, wherethe relative kinetic energy term in eq. B60 is negligible,we assume that the scattering length varies linearly withapplied magnetic field near the zero crossing field B , a ( B ) = a ′ ( B − B ) , (B61)where the tuning rate of the scattering length a ′ is givenin the main text in units of a / G, where a is the Bohrradius.Including the relative kinetic energy K rel in eq. B60 isequivalent to replacing the magnetic field B by an effec-tive magnetic field, B eff = h B z i − h K rel i µ B . (B62)Here, we include an additional average of the spatiallyvarying bias field B z over the position of the center ofmass (CM) of a colliding atom pair.We begin by evaluating h B z i . The bias field is cylin-drically symmetric about the z axis, and oriented per-pendicular to the long x -axis of the trapped cloud, sothat B z = B z [1 + b ( z − ( x + y ) / B z isthe bias field at the cloud center and b B z is the fieldcurvature. For the cigar-shaped clouds utilized in theexperiments, the variation of B z in the z and y di-rections is negligible compared to that in the x direc-tion, so that B z ( x ) = B z [1 − b x / b B z from the measured spring constant of the result-ing harmonic confining potential, − µ B B z ( x ), where for Li, the magnetic moment, + µ B , of the three lowesthyperfine states at high B field is dominated by theelectron spin down contribution, m s = − /
2. With µ B b B z ≡ mω , where ω mag is given in § B 2, the biasfield, averaged over the center of mass position, is then h B z i = B z − mω h X i / (2 µ B ). Using the virial the-orem for a harmonic trap, which holds for weakly inter-acting atoms, we obtain 2 m ¯ ω x h X i = h E x CM i , where2 m is the total mass. Hence, B eff = B z − ω ¯ ω x h E x CM i µ B − h K x rel i µ B − h K ⊥ rel i µ B . (B63)Here, we have separated the relative kinetic energy termof eq. B62 into axial and transverse parts.Next, we evaluate the relative kinetic energy contribu-tions. For the axial x -direction, we select the energy ofthe two colliding atoms E and E ′ in g ( E, E ′ ), eq. B27.Hence, the total energy is E + E ′ = E x CM + E x rel . Forharmonic confinement, the kinetic and potential energiesare quadratic degrees of freedom, which requires E x CM =( E + E ′ ) / φ E ( x ) φ E ′ ( x ). We alsohave E x rel = ( E + E ′ ) /
2, where E x rel = K x rel + µ ¯ ω x x / h K x rel i , we note thatfor a collision to occur, the relative position x rel of the two atoms must vanish for a contact interaction. Hence, K x rel = E x rel = ( E + E ′ ) /
2. For the transverse directions,we have defined a mean fractional spatial density ¯ n ⊥ ,by eq. B20. Assuming that the corresponding relativemomentum average for the two transverse directions isdetermined by a Boltzmann distribution, h K ⊥ rel i ≃ k B T .Using these results in eq. B63, we obtain finally, B eff = B z − h K ⊥ rel i µ B − ω ω x ! E + E ′ µ B , (B64)where we leave h K ⊥ rel i as an adjustable parameter, of or-der k B T . Replacing a S with a ( E ′ , E ) = a ′ ( B eff − B )in g ( E ′ , E ) of eq. B27 and in the results for ˜ g ( n ′ , n ) thatfollow from it, we obtain a reasonable fit to the hightemperature spin density profile of Fig. 6 in the mainpaper with h K ⊥ rel i = 0 . k B T . For T = 45 . µ K, thiscorresponds to a shift of − . B eff , consistent withthe upward shift of the applied field for which a = 0,as reported in Table I of the main paper. For the lowtemperature data, where the energy scale is < µ K, thecorresponding energy shift is negligible.
4. Measured Spatial Profiles versus Predictions
To compare the data for degenerate samples to thezero temperature theoretical model discussed above, weassume that the measured initial densities n ↑ ( x ), n ↓ ( x )and the conserved total density are zero temperatureThomas-Fermi profiles (see eq. B38), with an effectivezero temperature Fermi radius σ , which we use as a fitparameter. From the profile of the total density, we find σ = 329 µ m, corresponding to an effective Fermi tem-perature of m ¯ ω x σ / . µ K and a transverse radius( ¯ ω x /ω ⊥ ) σ . For the high temperature sample, the totalatom number is ∼ . × , and the measured gaus-sian 1 /e radius is σ x = p k B T / ( m ¯ ω x ) = 325 µ m, whichdetermines T = 45 . µ K.Fig. 2 of the main text demonstrates the excellentquantitative agreement between the predicted and mea-sured density profiles of each hyperfine state for a degen-erate sample, in units of the conserved average centraldensity ( n + n ), for a scattering length of a = 3 . a .Fig. 3 of the main text shows the transversely inte-grated spin densities n ( x, t ) − n ( x, t ) ≡ S z ( x, t ) with a = 5 . a and a = − . a at selected times t afterexcitation. The data are quite sensitive to the evolutiontime and exhibit a complex structure, which are verywell fit by the collective spin rotation model. Fig. 13shows additional measurements and predictions for thetime evolution of ( n − n ) between t = 0 and 800 ms, rel-ative to coherent excitation, for a fixed scattering lengthof a = 5 . a at B = 528 .
844 G, which corresponds tothe evolution of the central spin-density shown in Fig. 4.Here, ( n − n ) is given in units of the total central den-sity n (0) + n (0).5 FIG. 13. Spin density profiles (blue dots) for a degenerate sample
T /T F = 0 .
35 versus evolution time relative to coherentexcitation. Each data profile is the average of 5 runs, taking in random time order. Each solid red curve is the mean field modelwith a fixed scattering length of a = 5 .
23 bohr ( B = 528 .
844 G) and a fitted cloud size within a few percent of the averagevalue σ = 329 µµ