Thermometry of a unitary Fermi gas

Transcription

1 Thermometry of a unitary Fermi gas William Cairncross October 9, 03 Abstract In this brief report, I summarize a method for the thermometry of an ultracold unitary Fermi gas in the specialized case where in situ imaging is infeasible. I use scaling solutions to model the time-of-flight expansion of the gas after release from an optical dipole trap, transform the D density profile (absorption image) into a pressure profile, and fit this using the equation of state of the gas a Fermi-Dirac integral in the case of the polarized (and therefore noninteracting) gas, or an analogous function for the strongly interacting spin mixture, composed of a patchwork of experimental and asymptotic results. I also describe an application of the latter equation of state to determining the change in temperature of a unitary Fermi gas as it evolves from a superposition state to a spin mixture, and present preliminary experimental results for this temperature change. Contents Introduction Universal thermodynamics 3 3 Time-of-flight expansion 5 3. Ballistic expansion Ideal hydrodynamic expansion Viscous hydrodynamic expansion Transformation of absorption images 7 4. Abel inversion Equipotential averaging Temperature rise 0 6 Data fitting in Matlab 7 Conclusion A Sample derivation 3 B Overview of the unitary/ideal Fermi fitter in Matlab 4

2 Acknowledgements First and foremost, I am very grateful to the Natural Sciences and Engineering Research Council of Canada for funding, and to Joseph Thywissen for giving me the opportunity to return to the ultracold atoms lab for a second summer. I would also like to thank Joseph for his guidance in physics both inside and out of the lab. Many thanks as well to my lab-mates Alma, Scott, Chris and Stefan for their advice and good company. I would particularly like to thank Stefan for suggesting this project, which while at first seeming a limited success was in the end very fruitful both in my own education and hopefully in future utility to the lab. Thanks as well to the lattice group for their excellent company, particularly Dylan for his occasional wanders into the chip lab for some late-afternoon laughs. I look forward to seeing you all at DAMOPs in the future! Introduction Low-energy scattering of identical particles, such as that exhibited by ultracold trapped Fermions in a mixture of spin states, is completely characterized by the two-body s-wave scattering length a. A Fano-Feshbach resonance allows tuning of the scattering length over an infinite range, leading to a scattering cross section σ that can be chosen anywhere from zero to a maximum value of 4π/k when the scattering length diverges, where k is the relative momentum between the scattering particles []. The regime for which σ 4π/k is referred to as the unitarity limit, and is hypothesized to be universal: in this limit, the thermodynamic properties of the system are believed independent of the physical origin and the functional form of the inter-particle potential. The universal character of the unitary Fermi gas allows researchers to predict the properties of exotic systems such as neutron stars, or in the case of this report to apply results obtained for unitary 6 Li to 40 K. Temperature is a crucial thermodynamic quantity for the characterization of ultracold gases, and its determination is particularly challenging for Fermions: unlike Bosons forming a Bose- Einstein condensate, Fermions do not undergo a dramatic phase transition at a critical temperature. Instead, below the Fermi temperature T F = ω(6n) /3 /k B (where N is the number of particles and ω is the geometric average trap frequency), effects of quantum degeneracy become important in a continuous way. Particularly at low temperatures and for small numbers of particles when signal-to-noise becomes an issue, the signatures of these effects are subtle enough that accurate thermometry can be difficult. Typically, the general principle of thermometry for an ultracold gas is to fit an absorption image of the cloud i.e. a measurement of its two-dimensional number density distribution to some theoretical model relating the state variables of the gas (an equation of state). In some cases, the image may be transformed before fitting, for example using the inverse Abel transform to reconstruct the three-dimensional density or pressure distribution. In this work, absorption images are manipulated significantly, and the model to which data is fitted is not entirely theoretical, but rather is a patchwork of theoretical predictions and experimental data. Section of this report discusses the equation of state of the unitary Fermi gas, while Section 4 discusses the transformation and fitting of absorption images. Under different experimental schemes, absorption images of atomic clouds are variously taken in situ, where the cloud is trapped up until the time when the image is captured, or in time-of-flight, where the trapping potential is switched off a short time before the image is captured (typically on the order of milliseconds). In situ imaging is viable for systems with large numbers of particles in a

3 weak trap, where the atomic cloud is large and the peak density of trapped atoms is not sufficient to cause saturation of absorption images. The issue of high optical density can be remedied by the use of phase contrast imaging or selective imaging of a known fraction of the trapped atoms. In the case of the Thywissen chip experiment, the limited resolution of the imaging system makes in situ imaging infeasible. Replication of the in-trap density distribution requires a model for the expansion of the unitary Fermi gas, which is discussed in Section 3. The final section of this report describes an application of the thermodynamic model of Section to a recent experiment in the Thywissen chip lab. In this experiment, a unitary Fermi gas is prepared in a superposition of two hyperfine states that form an effective spin-/ system. A magnetic field gradient causes the superposition state to decay into a mixture of the two states, during which time the gas transitions from ideal and non-interacting (due to Pauli exclusion) to maximally interacting. Throughout this process, the temperature of the gas increases by an amount that is predicted using the tools of Section, and subsequently measured experimentally. Universal thermodynamics A general approach to the thermodynamics of the unitary Fermi gas was developed in Refs. [,, 3, 4] and others, based on the universality hypothesis: when the scattering length a diverges, it should drop out of all physical quantities. At finite temperatures, the system should be completely described by the chemical potential µ and the temperature T [5]. The relations developed in Refs. [,, 3, 4] make predictions for the gas macroscopic properties, without regard to the microscopic processes giving rise to, e.g., the superfluid phase transition at T 0.7T F [3]. Here, the object of study is an equation of state (EoS) relating the state variables: T, µ, the density n, and the pressure p. Throughout this section, we employ the local density approximation: the local properties of the gas are assumed to be equal to those of the homogeneous gas at a local chemical potential µ(r) = µ 0 V (r), where V is the external trapping potential. The experimentally determined EoS for a homogeneous spin-balanced unitary Fermi gas is given in Refs. [4, 3] in the form n(µ, T )/n 0 (µ, T ) = g n (βµ), () where β = (k B T ), n 0 (µ, T ) = λ 3 Li 3/ ( e βµ ) is the density of a non-interacting Fermi gas, and λ = π /(mk B T ) is the thermal de Broglie wavelength. The functions Li n are polylogarithms (see e.g. Ref. [6]) and g n (βµ) is a universal function of βµ, parameterized as shown in Fig.. This parameterization is composed of three parts: For βµ <.5, we follow Ref. [3] in using the virial expansion nλ 3 = k pλ 3 k B T = k kb k z k () b k z k (3) where z e βµ is the fugacity, and the virial coefficients b k are known up to third order, with b =, b = 3 /8, and b 3 = [7]. For.5 < βµ < 4, we use a spline interpolation of the 3

4 experimental results of Ref. [3]. For βµ > 4, we use the low-temperature results nλ 3 pλ 3 k B T T π ( ) βµ 3/, (4) ξ T (βµ) 5/ π ξ 3/. (5) where ξ is the Bertsch parameter relating the zero-temperature chemical potential of the unitary Fermi gas to the Fermi energy according to µ = ξe F [6]. For the special case of a harmonic trap, many thermodynamic quantities can be written in terms of single integrals over the equation of state. Rewriting Eq. () as where f(x) = g n (x) Li 3/ ( e x ), we obtain the pressure of the gas n(µ, T ) = f(βµ), (6) λ3 p(µ, T ) = k BT λ 3 βµ f(x)dx k BT F (βµ), (7) λ3 using the Gibbs-Duhem relation dp = n dµ + s dt [5]. For a harmonically trapped gas in the local density approximation, Ku et al. [8] define the functions f n (βµ 0 ) = Γ( 3/ + n) βµ0 (βµ 0 x) 5/+n f(x)dx, (8) analogous to the Fermi-Dirac integrals for a non-interacting gas, fn n.i. (βµ 0 ) = Li n ( e βµ 0 ). In terms of these functions, we have ( ) kb T 3 N = f 3 (βµ 0 ), ω (9) E N = 3k BT f 4(βµ 0 ) f 3 (βµ 0 ), (0) T = [3f 3 (βµ 0 )] /3, T F () S = 4f 4(βµ 0 ) Nk B 3f 3 (βµ 0 ) βµ 0, () C V = f 4(βµ 0 ) Nk B f 3 (βµ 0 ) 9f 3(βµ 0 ) f (βµ 0 ), (3) where N is the particle number, E the total energy, S the entropy, and C V the heat capacity at constant volume. Here µ 0 is the local chemical potential at the trap center. For an anharmonic potential, the author has been unable to derive such simple expressions for thermodynamic quantities of interest. We must instead numerically evaluate the triple integrals for these quantities, for example the particle number: N = λ 3 drf[βµ 0 βv (r)], (4) 4

5 with the crossed dipole potential V (r) = U x ( e (x +y )/w x ) ( ) + U y e (y +z )/wy, (5) where w i are the beam waists, U i = mωi w i /4 are the trap depths, and we have neglected the Rayleigh ranges of the trapping beams. n(µ, T )/n0(µ, T ) p(µ, T )/p0(µ, T ) βµ (a) βµ (b) Figure : Parametrization of the equation of state of a unitary Fermi gas, showing the ratios of (a) density and (b) pressure to those of a non-interacting gas. For βµ <.5 (high temperature), the virial expansion of Ref. [9] is used. Values in the range.5 < βµ < 4 are given by the experimentally determined equation of state of Ref. [3]. The low temperature expressions Eqs. (4) and (5) are used for βµ > 4. 3 Time-of-flight expansion 3. Ballistic expansion Pauli exclusion suppresses collisions between Fermions in the same spin state, so a polarized Fermi gas behaves as an ideal gas in time-of-flight expansion. In the ballistic approximation, we neglect collisions and assume that at the time when the trap is switched off, each particle continues unimpeded with velocity p 0 /m, reaching position r = r 0 + p 0 m t at time t [6]. The integral over momentum space of the semiclassical phase space distribution can be performed analytically, with the result that the density distribution n(r, t) evolves as a scaling solution, n(r, t) = n(r, 0), (6) b x (t)b y (t)b z (t) x i (t) = b i (t)x i0, (7) 5

6 where the scaling parameters b i (t) evolve as b i (t) = + ω i t. (8) In this case, the aspect ratio of the cloud (the ratio of the radial to axis widths) approaches for t ω i. 3. Ideal hydrodynamic expansion A unitary Fermi mixture, where the mean free path is short compared to the cloud size, is collisonally dense, i.e. a particle will collide many times during time-of-flight expansion [6]. At zero temperature, the expansion is well-approximated by the Euler equation of ideal hydrodynamics and the continuity equation, t n + (nv) = 0, (9) m ( t + v ) v = V (r, t) p(r, t)/n. (0) where v is the velocity field and m is the particle mass. There exists a scaling solution to these equations, where n and r(t) take the same form as Eqs. (6) and (7) and v = ḃix i (t)/b i [0, ]. The equations for the scaling parameters in this case are bi b i = 3.3 Viscous hydrodynamic expansion ωi. () b i (b x b y b z ) /3 At nonzero temperatures, the unitary Fermi gas acquires a nonzero shear viscosity η. In Refs. [, 3], an approximation for viscous hydrodynamic expansion at finite temperatures was developed based on the Navier-Stokes equations, continuity, and energy conservation: t (ρv i ) + j Π ij = 0, () t ρ + (ρv) = 0, (3) t E + j ɛ = 0. (4) Here ρ is the mass density, E the energy density, and j ɛ the energy current. Π ij = ρv i v j + pδ ij is the stress tensor, with dissipative part δπ ij = ησ ij, where σ ij = i v j + j v i 3 δ ij k v k. Assuming that the velocity field is linear in the coordinates v i = α i x i and substituting the scaling ansatz n(r, t) = ρ/m = b x (t)b y (t)b z (t) F [ x b x(t) + y b y(t) + z b z(t) in the continuity equation gives α i = ḃi/b i. By taking the gradient of the energy conservation equation [Eq. (4)] and using the Navier Stokes equations ] (5) m ( t + v ) v i = f i + j(ησ ij ), (6) n 6

7 the authors of Ref. [] obtain ( t + v + 3 ) v f i + ( i v j )f j 5 3 ( i j v j ) p n = i q 3 n, (7) where f i = ( i p)/n are the forces, and q = η( i,j σ ij )/ is the heating rate. In a harmonic trap in the local density approximation, the initial pressure is given by the hydrostatic equilibrium solution p 0 = n 0 µ 0 = n 0 V 0, so that the initial forces f i (t = 0) = i V 0 = mωi x i are linear in the coordinates. Following Refs. [, 3], we assume that the forces remain linear in the coordinates throughout expansion, so that f i = a i (t)x i. Finally, we assume that the shear viscosity is proportional to the density η = α n n βn/3. Making these substitutions into Eqs. (6) and (7), we obtain a system of nonlinear ordinary differential equations for the parameters a i and b i, bi b i = a i βω i ȧ i = 3 a i b i ( ( ḃi b i ḃj b j ḃk 4ḃi b i + ḃj b j + ḃk b k ) b k ) + 4βω i 9b i, (8) ijk S 3 ( ḃi b i ḃj b j ḃk b k ), (9) with the initial conditions b i (0) =, ḃi(0) = 0, a i (0) = ωi, ȧ i(0) = 0. Here S 3 is the group of symmetric permutations of the indices ijk. Figure shows the time-of-flight evolution of the cloud aspect ratio AR(t) (the ratio of its radial to axial rms width) for all three of the above cases. In the ballistic expansion, the aspect ratio quickly approaches unity, while in both ideal and viscous hydrodynamic expansion, the asymptotic value is approached more slowly. In ideal hydrodynamics, the asymptotic value of the aspect ratio is approximately.ω x /(πω z ) [6]. 4 Transformation of absorption images 4. Abel inversion This section summarizes the derivation of a modified inverse Abel transform technique developed by Félix Werner, and its application to the case of a Gaussian trap by Martin Zwierlein [4, 5]. The three-dimensional density distribution of a trapped atomic gas is related to an absorption image by the inverse Abel transform n 3D (ρ, x) = π ρ n D (y, x) y dy y ρ, (30) where ρ y + z. The pressure p(ρ, x) is related to the density by the Gibbs-Duhem relation p = µ which in the local density approximation can be written p(ρ, x) = ρ n(µ)dµ, (3) n 3D (ρ, x) V (ρ, x) ρ dρ. (3) 7

8 .5 Aspect ratio AR(t) ω t Figure : Aspect ratio of expanding clouds, showing ballistic (blue), inviscid hydrodynamic (red), and viscous hydrodynamic expansion (green). Substituting Eq. (30), exchanging the order of integration, and integrating by parts to shift the derivative from the experimental data n D to the potential, we obtain where By defining p(ρ, x) = π I(y) J(y, y ) y ρ y taking J (y, y ( ) = lim y y y + y ), and substituting the identity Werner obtains the formula p(ρ, x) = π ρ n D (y, x) y y = y y ρ + ρ ρ n D (y, x) I(y) y, (33) V (ρ, x) dρ ρ. (34) y ρ V (ρ, x) dρ ρ, (35) y ρ ρ ρ dρ (y ρ, (36) ) 3/ [ [ ρ V (ρ, x)] [ y ρ=y + dρ ρ ρ V (ρ, x) ] ] ρ =y y ρ V (ρ, x) y ρ ρ (y ρ ) 3/. (37) To apply Eq. (37) to the Gaussian potential in Eq. (5), we must neglect the y-trapping contribution of the beam propagating along z (called the x beam because it traps primarily in the x-direction). We can partially compensate for this approximation by scaling the final result by a 8

9 factor ω z /ω y, as discussed in the supplementary information to Ref. [3]. The approximate potential is then V (r) = U x ( e x /wx ) + Uy ( e ρ /wy) with ρ y + z, for which Zwierlein obtains p(ρ, x) = ω z 4U y ( e ρ /wy du n D u π ω + ρ, x) [ u ( )] u D, (38) y w y w y w y 0 where D(x) e x x 0 eu du is the Dawson function [5]. Figure 3 shows sample 3D density and pressure distributions obtained from a simulated absorption image with normally distributed noise. The Abel inversion involves a derivative of the density distribution with respect to the radial coordinate, which reduces signal-to-noise. In contrast, the formula in Eq. (37) improves the signal-to-noise due to the integration over the potential. It should be noted that Eq. (37) is completely equivalent to simply performing the Abel inversion followed by the integration in Eq. (3). Equation (37) simply provides a way to perform the two steps simultaneously, and the results in Fig. 3 demonstrate that fitting the pressure is likely to generate the most accurate results, due to the improvement in signal-to-noise. (a) (b) (c) Figure 3: Comparison of Abel inversion methods: (b) shows a 3D density distribution obtained by the standard Abel inversion of the simulated absorption image shown in (a), while (c) shows the 3D pressure distribution obtained by integration of (a) over the known trapping potential. The pressure distribution (c) is obtained in one step using Eq. (37). 4. Equipotential averaging Equipotential averaging after performing an Abel inversion gives further improvements of signalto-noise. Figure 4 shows a pressure profile with equipotential lines overlaid. The close agreement between equidensity and equipotential lines in the figure demonstrate that the expansion was modeled accurately in this case, by an ideal ballistic expansion. Conversely, errors can be exacerbated if the center of the trap is not properly located, e.g. by fitting a D Gaussian to the raw image. It is not clear whether the possible gain in signal-to-noise justifies equipotential averaging in the event that Gaussian fitting fails. 9

10 y [µm] 6 4 p [kbµk/µm 3 ] x [µm] (a) V [µk] (b) Figure 4: (a) Quadrant averaged pressure slice p(x, y, z = 0), showing equipotential lines from Eq. (5). This image is of a polarized (non-interacting) gas. The close agreement between equidensity and equipotential lines indicates that the expansion was close to the ideal ballistic case. (b) Equipotential-averaged pressure distribution p(v ) obtained from (a) (red) fitted with the equation of state for a non-interacting Fermi gas (blue). 5 Temperature rise In a recent experiment in the Thywissen chip lab, unitary Fermions initially prepared in a superposition of spin states were allowed to dephase in a magnetic field gradient, and the dynamics of the growth of interactions investigated. The transition from a superposition of spin states to a balanced spin mixture is synonymous with the appearance of two Fermi surfaces, and therefore two equal trap-center chemical potentials µ 0f that are different from the initial value µ 0i. Since energy must be conserved, there must be an increase in temperature during the dephasing process. Using the thermodynamics developed in Section, we can predict this temperature rise. With the assumptions that the particle number N and energy E are conserved in the dephasing process from a spin superposition to a spin-balanced mixture, we can determine the final temperature of the cloud using Eqs. (9)-(3) and their equivalents for the non-interacting gas. With an accurate model for losses, we could do away with the first assumption. Equation (9) for the non-interacting gas defines µ 0i implicitly in terms of β i and N, N = f n.i. 3 (β i µ 0i ) (β i ω) 3, (39) where i denotes the initial (polarized) state. We equate the energies per particle for the initial and final state, f4 n.i. (β i µ 0i ) β i f3 n.i. (β i µ 0i ) = f 4(β f µ 0f ) β f f 3 (β f µ 0f ), (40) 0

11 and require that the particle number stay constant: N = f 3(β f µ 0f ) (β f ω) 3, (4) where the factor of accounts for the presence of two spin states in the final mixture. Given N and β i, equations (39)-(4) have a unique solution for µ 0i, µ 0f and β f. Results of a numerical calculation and experimental data are shown in Fig. 5. Very good agreement between experiment and theory is seen for T 0.6T F, however for higher initial temperatures significant disagreement is apparent, with two points entering the unphysical region of a temperature decrease with growth of interactions. The data at high T i /T F are slightly suspect, however, for two reasons: First, the unphysical points were taken at very low particle numbers of N 0 4. A small, warm cloud makes thermometry difficult by providing worse signal-to-noise than a large, cold and therefore optically dense cloud, and by the fact that its physical dimensions approach the finite resolution of our imaging system. The point with T i 0.68T F however was obtained with a large number of atoms, and still exhibits a negative deviation from the prediction. While it would be excellent to have a physical explanation for the observed deviation, such an explanation evades the author. Nonetheless, the excellent agreement at low T i /T F is interpreted as a (limited) confirmation of the theoretical prediction.. Tf/TF f T i /T F i Figure 5: Increase in temperature of the unitary Fermi gas with growth of interactions. The bottom (black) line represents zero change in absolute temperature, but has a slope of /3 due to the change in definition of the chemical potential with the appearance of two spin states. The dashed (red) line is a calculation of the temperature rise for a non-interacting gas, and the top (blue) line is the prediction calculated from Eqs. (39)-(4).

12 6 Data fitting in Matlab The unitary/ideal Fermi fitter in Matlab is based on the structure of Boris Braverman s 0 Fermi fit monitor. The fitter implements the viscous hydrodynamic expansion of Section 3.3, the Abel inversion of Section 4, and the thermodynamics of Section to obtain the temperature, particle number, degeneracy, and energy per particle of an ideal or unitary Fermi gas from absorption images, and could easily be expanded to calculate both the entropy S and heat capacity C V. Figure 6 shows typical results for atom number, absolute temperature, energy per particle, and degeneracy obtained using the unitary Fermi fitter. The data set contains times-of-flight of 4.0, 4.75 and 5.5 ms, sorted from left to right and with ten data points for each. The data were taken in a random order to reduce the effects of drifts in the atom number. Still, we observe a negative trend in the atom number at longer time-of-flight (higher image number), implying that loss continues to occur after release from the optical trap. The high rate of loss could suggest that these data were taken slightly away from Feshbach resonance. It is also worth noting that the absolute temperature does not exhibit any appreciable trend during time-of-flight, which is reassuring for the robustness of the fitter, and for the accuracy of the viscous hydrodynamic expansion model. Figure 7 shows the rms percent error in the degeneracy T/T F for a set of simulated images with added Gaussian noise with a standard deviation equal to 0, 0.04, 0.08, and 0. times the maximum density in the image, for particle numbers ranging from 0 4 to 0 5 and T/T F between 0. and 0.9. The fitter becomes inaccurate for T/T F below approximately 0. near the superfluid transition. The decreased accuracy for large particle numbers and high T/T F is not yet well understood, in particular since it seems to disappear when more Gaussian noise is added. Further investigation into this effect is warranted. 7 Conclusion The thermodynamics of unitary Fermi gases is a complex topic comprising the work of many authors. It is my hope that this report will be useful in summarizing the elements that went into a new fitting routine, but it is by no means a comprehensive overview of the subject. There is also much room for improvement in the accompanying Fermi fitting code, particularly in the area of ease of use. A rigorous assessment of uncertainties in measured parameters is also warranted, particularly the effects of the assumption of trap harmonicity in the viscous hydrodynamic expansion. References [] Ho, T.-L. Physical Review Letters 9(9), March (004). [] Nascimbène, S., Navon, N., Jiang, K. J., Chevy, F., and Salomon, C. Nature 463(784), February (00). [3] Ku, M. J. H., Sommer, A. T., Cheuk, L. W., and Zwierlein, M. W. Science 335(6068), February (0). An overview of the author s fitter is given in Appendix B.

13 [4] Van Houcke, K., Werner, F., Kozik, E., Prokof ev, N., Svistunov, B., Ku, M. J. H., Sommer, A. T., Cheuk, L. W., Schirotzek, A., and Zwierlein, M. W. Nature Physics 8(5), March (0). [5] Schirotzek, A. Radio-Frequency Spectroscopy of Ultracold Atomic Fermi Gases. PhD thesis, Massachusetts Institute of Technology, February (00). [6] Ketterle, W. and Zwierlein, M. W. Nuovo Cimento Rivista Serie 3, 47 4 May (008). [7] Liu, X.-J., Hu, H., and Drummond, P. D. Physical Review Letters 0(6), 6040 April (009). [8] Ku, M. J. H., Sommer, A. T., Cheuk, L. W., Schirotzek, A., Zwierlein, M. W., Van Houcke, K., Werner, F., Kozik, E., Prokofev, N., and Svistunov, B. July (0). [9] Liu, X.-J. and Hu, H. Physical Review A 8(4), October (00). [0] Menotti, C., Pedri, P., and Stringari, S. Physical Review Letters 89(5), 5040 December (00). [] Guéry-Odelin, D. Physical Review A 66(3), September (00). [] Cao, C., Elliott, E., Joseph, J., Wu, H., Petricka, J., Schäfer, T., and Thomas, J. E. Science 33(603), 58 6 January (0). [3] Schäfer, T. Physical Review A 8(6), December (00). [4] Werner, F. Technical report, January (00). [5] Zwierlein, M. W. Technical report, July (03). [6] Luo, L. and Thomas, J. E. Journal of Low Temperature Physics 54(-), 9 November (008). Appendix A Sample derivation I show here the expression for the total particle number N, which can be simply expressed for a harmonically trapped gas in the local density approximation. N = dr n(r) = λ 3 drf [βµ 0 βv (r)] = ( m ) 3/ π β dx dy dzf [ βµ 0 βm ( ω x x + ω yy + ω zz )] (4) 3

14 Substituting x i = ω ix i, ( ) m 3/ dx dy dz N = π f β ω x ω y ω z ( ) m 3/ = π β ω r dr f Finally, substituting u = βµ 0 βmr /, we have 0 [ βµ 0 βm ( x + y + z )] (βµ 0 βm r ). (43) N = ( ) kb T 3 βµ0 du βµ 0 u f(u) π ω ( ) kb T 3 = f 3 (βµ 0 ). (44) ω This expression naturally holds for any density distribution n(r) = λ 3 f[β(µ 0 V )]. The other f n are similarly straightforward to derive. The energy expression Eq. (0) requires the virial theorem E = V + r V/, which is valid for non-interacting and unitary Fermi gases [6]. B Overview of the unitary/ideal Fermi fitter in Matlab The following is a brief description of each of the functions in the directory Fitter as of October 9, 03. Some functions contain sub-functions that are hopefully self-explanatory, but in rare cases may be illegible, uncommented, obtained from the Matlab file exchange and therefore opaque in their methods, or simply ugly. Control FermiFitMonitor6 Will(folder,bgfolder,refit,to refit,ni or u,pol) Main fitter function. Arguments folder and bgfolder specify directories containing atom images and light-only images, respectively; refit is a flag for whether or not to refit all images; to refit is an array containing specific image numbers to refit; in or u is a flag for either a non-interacting or unitary gas; pol is the numerical value of the polarization. The value of pol is necessarily 0.5 when ni or u = 0 (a unitary mixture), but can take values or 0.5 for a non-interacting gas. As of this writing, certain aspects of the fitting program are not perfect: in many cases, the time of flight vector must be defined along with the experimental constants expv. There is also not currently an easy way to exclude shots where a shutter malfunctioned. fitparams = FermiPhysicalFitter6 Will(img,bg,expv,ni or u,pol) Called by FermiFitMonitor6 Will, this function deals with the top-level physical aspects of temperature fitting. It calls sub-functions to manipulate images and calculate experimental results, then returns these results in a struct fitparams. imageaverage(folder) This function averages images in directory folder and saves them in a directory folder avg. The directory folder should have in it a text file folder.txt that contains some array that distinguishes 4

15 paramsout = temprise(totf i,n) Predicts the increase in absolute temperature and degeneracy with the growth of interactions. Assumes a harmonic trapping potential, conservation of energy and conservation of particle number. A version of this function for a Gaussian trap is unused due to negligible differences in results. The Gaussian version uses the functions numberintegral and energyintegral to calculate the triple integrals to obtain N and E/N, and thus is much slower than this function. Hydrodynamic expansion [bx,by,bz] = expansionparams(wx,wy,wz,t,beta) Scaling parameters for viscous hydrodynamic expansion, according to Eqs. (8) and (9). Takes as arguments the trapping frequencies wx, wy, and wz, time of flight t, and viscosity parameter beta. beta = expansionparamsfit(tof,ar,expv) Fit the time-of-flight evolution of the aspect ratio of a hydrodynamically expanding cloud, with beta as the fit parameter. Experimental constants (namely trapping frequencies) are provided in the struct expv. Image manipulation [V,p] = densdtopresd(x,y,n,expv) Converts absorption images (D density profiles) to D pressure profiles p(v ). Calls pressureabellinversion to obtain the 3D pressure p(r), then averages along equipotential lines. Equipotential averaging is performed by sorting the matrix containing V, sorting p according to the same order, then calling BinMean to bin the data into approximately 000 points. This final step is not strictly necessary, since the results of the fitting routine should be independent of noise as long as it does not dominate the profile, however binning data significantly improves computation times. [x,r,n3] = AbelInversion(x,y,n) Currently unused. Standard inverse Abel transform to convert n D (x, y) to a cylindrically symmetric n 3D (x, y, z) = n 3D (ρ, x). [x,r,p] = pressureabelinversion(x,y,n,expv) Implements Eq. (38) to transform absorption images to slices of the 3D pressure distribution. Uses the open-source package chebfun to compute the Dawson function in the integrand. Certain sub-functions in this file were simply obtained by the author from the Matlab file exchange without a deep understanding of their workings, and should at some point be replaced. Fitting [paramsout,fitfun] = presdfit(v,p d,expv,ni or u) Fits a D pressure distribution p(v ) obtained by equipotential averaging of p(r). Uses lsqnonlin for its speed, which could be replaced by fminsearch for reduced dependence on the initial guesses, which are arbitrary. 5

16 [paramsout,fitfun] = GaussianRotdFit Will(x,y,z) Fit a skewed D Gaussian to image z. Arguments x and y are vectors. Uses lsqnonlin. [paramsout,fitfun] = GaussiandFit Will(x,y,z) Fit a D Gaussian to image z. Arguments x and y are vectors. Uses lsqnonlin. [paramsout,fitfun] = GaussianFit Will(x,y) Fit a D Gaussian using lsqnonlin. Integrals for physical quantities N = numberintegral(mu0,t,expv,ni or u) Computes the total particle number in a crossed optical dipole trap by a direct triple integration using the Matlab R03 function integral3. This builtin may not exist in earlier versions, and may need to be replaced by the harmonic approximation in Eq. (9). E = energyintegral(mu0,t,expv,ni or u) Similar to numberintegral. Computes the energy per particle in a crossed optical dipole trap by a direct triple integration using the Matlab R03 function integral3. Fermi-Dirac integrals out = FermiFunc(x,n,option) Computes Fermi-Dirac integrals fn n.i. if ni or u =, or the equivalent functions f n from Eq. (8) for the unitary Fermi gas. Important special cases are n = 3/ for calculating the 3D density, 5/ for calculating the 3D pressure, 3 for calculating the total particle number, and 4 for calculating the total energy Low-level functions [Q,err] = quadgr(fun,a,b,tol,trace,varargin) From the Matlab file exchange. Equivalent to the builtin quadgk from Matlab version 7.4. Computes definite or indefinite integrals in one dimension. a = boxselector(img) Written by Boris Braverman. Used to select a region of interest in an absorption image. [x,y] = BinMean Will(x,y,n) Modified from Boris Braverman s original. Bins vectors of unequal spacing and unspecified length x and y to equally-spaced vectors x and y of length n. hh = herrorbar(x, y, l, u, symbol) Similar to errorbar. Generates plots with horizontal error bars. 6

17 N 6 4 T [nk] T/TF E/(NEF ) image number image number Figure 6: Results of a typical experimental run, showing (clockwise from top left) atom number, absolute temperature, energy per particle, and degeneracy. 7

18 N N T/TF rms % error T/T F T/T F Figure 7: Contour plots of the rms percent error in fitted T/T F for different values of T/T F and N, using simulated images with added Gaussian noise of 0, 0.04, 0.08, and 0. times the maximum optical density. 8