Determination of the equation of state of a polarized Fermi gas at ...
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PHYSICAL REVIEW A 77, 041603共R兲 共2008兲
Determination of the equation of state of a polarized Fermi gas at unitarity Yong-il Shin* Department of Physics, MIT-Harvard Center for Ultracold Atoms, and Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA 共Received 10 January 2008; published 10 April 2008兲 We report on the measurement of the equation of state of a two-component Fermi gas of 6Li atoms with resonant interactions. By analyzing the in situ density distributions of a population-imbalanced Fermi mixture reported in a recent experiment 关Y. Shin et al., Nature 451, 689 共2008兲兴, we determine the energy density of a resonantly interacting Fermi gas as a function of the densities of the two components. We present a method to determine the equation of state directly from the shape of the trapped cloud, where the fully polarized, noninteracting ideal Fermi gas in the outer region provides the absolute calibration of particle density. From the density profiles obtained at the lowest temperature, we estimate the zero-temperature equation of state. DOI: 10.1103/PhysRevA.77.041603
PACS number共s兲: 03.75.Ss, 03.75.Hh, 05.30.Fk, 51.30.⫹i
Interacting fermions are a paradigm of modern physics, encompassing superconductivity and superfluidity. One interesting situation arises when the constituents interact resonantly, i.e., the scattering length for the free fermions diverges. At this so-called unitarity limit, the behavior of the system becomes universal, being independent of the nature of the interactions. Ultracold atomic Fermi gases near Feshbach collisional resonance present a model system for studying strongly interacting fermions 关1兴. Recently, the phase diagram of a two-component Fermi gas with resonant interactions has been experimentally established 关2兴, showing that at zero temperature the system undergoes a firstorder quantum phase transition from a fully paired superfluid to a partially polarized normal gas when the imbalance between the two spin components exceeds a critical value, called the Chandrasekhar-Clogston limit of superfluidity 关3,4兴. The nature of the partially polarized normal phase, however, is still a subject of investigation. The spectral shift observed in the minority rf excitation spectrum has been interpreted as the existence of “pairing” in the normal phase 关5兴, but several theoretical studies for a highly polarized system, e.g. a single minority atom in a majority Fermi sea, suggest that the system is well-described as a normal Fermi liquid, where the minority atoms are associated with weakly interacting quasiparticles 关6–8兴. This picture seems to be supported by the experimental observation that the shape of the minority cloud in the normal phase is similar to a free Fermi gas 关2,9兴. It has been speculated that exotic pairing states might exist in the partially polarized phase 关10兴. In this paper, we determine the equation of state of a polarized Fermi gas at unitarity from the in situ density profiles of a population-imbalanced Fermi mixture confined in a harmonic trap. Since the variation of the external trapping potential across the sample scans the chemical potential, in principle, the density information of a single sample contains the whole equation of state 关11,12兴. We present a method to determine the equation of state directly from the shape of the trapped cloud. Because of its exactly known thermal properties, a fully polarized, noninteracting ideal Fermi gas in the
*[email protected] 1050-2947/2008/77共4兲/041603共4兲
outer region provides the absolute density calibration. The equation of state of a polarized Fermi gas can be parametrized using a normal Fermi liquid description, which includes the binding energy of a single minority atom resonantly interacting with a majority Fermi sea, the effective mass of the quasiparticles, and its correction. This work is a quantitative study of the thermodynamic properties of the polarized normal state with strong interactions, finding reasonable agreement with recent calculations 关6–8,11,12兴. For infinite scattering length, the unitarity limit implies that all interaction energies scale with the Fermi energies of the components Fi = ប2 / 2m共62ni兲2/3 关13兴, where ប is the Planck constant divided by 2, m is the particle mass, ni is the density of component i, and i = 1 , 2. As a result, a simple dimensional scaling argument implies that the energy density E共n1 , n2兲 of a two-component Fermi gas can be parametrized as 3 E共n1,n2兲 = ␣关n1g共x兲兴5/3 , 5
共1兲
introducing a dimensionless universal function g共x兲 关12兴, where ␣ = 共62兲2/3ប2 / 2m and x = n2 / n1 is the density ratio of the two components. Without loss of generality, 0 ⱕ x ⱕ 1 due to the symmetry of the two components. From the chemical potential relation i = E / ni, the universal function can be expressed as g共x兲5/3 =
1 + x2 1 = 共1 + xy兲, F1 ␣n2/3 1
共2兲
where y = 2 / 1 is the chemical potential ratio of the two components. The main result of this paper is the measurement of the universal function g共x兲 for a resonantly interacting Fermi gas. When a Fermi mixture is confined in a harmonic trap, V共r兲 ⬀ r2, the local chemical potential is given as i共r兲 = i0 − V共r兲, where i0 is the global chemical potential with respect to the bottom of the trapping potential. The global chemical potential of the majority component 10 is determined from the radius of the majority cloud R1, i.e., 10 = V共R1兲. Then, the chemical potential ratio y共r兲 = 2共r兲 / 1共r兲 is given as
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©2008 The American Physical Society
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1 − r2/R21 where y 0 = 20 / 10. With population imbalance, i.e. y 0 ⬍ 1, x共r兲 and y共r兲 vary over the sample. From Eq. 共2兲, the spatial correlation of the local particle densities and the local chemical potentials in a trapped sample determines g共x兲 under the local density approximation. We estimate the zero-temperature equation of state by analyzing the density profiles obtained at the lowest temperature 共T / TF ⬇ 0.03 where TF is the Fermi temperature of the majority component兲 in Ref. 关2兴. A spin mixture of the two lowest hyperfine states of 6Li atoms was prepared in a threedimensional 共3D兲 harmonic trap on a broad Feshbach resonance, located at a magnetic field of 834 G 关14兴, resonantly enhancing the interactions between the two spin states. The detailed description of the experimental procedure for the sample preparation and the signal processing has been provided in Ref. 关2兴. Figure 1 shows the spatial structure of a resonantly interacting Fermi mixture in a harmonic trap. According to the zero-temperature phase diagram 关2兴, three distinctive spatial regions can be identified in the inhomogeneous sample with population imbalance 共y 0 ⬍ 1兲. When the chemical potential ratio at center y 0 is larger than a critical value y c for the superfluid-to-normal phase transition, the sample has 共I兲 a fully paired superfluid core 共x = 1, y ⬎ y c兲 surrounded by 共II兲 a partially polarized normal gas 共0 ⬍ x ⱕ xc , y ⬍ y c兲, showing a discontinuity in the density ratio x at the phase boundary. In the outer region where the minority component is completely depleted, 共III兲 a fully polarized noninteracting Fermi gas forms 共x = 0, y ⬍ y m ⬍ y c兲. The radii, Rc and R2 are the I-II and II-III transition points, defining y共Rc兲 = y c and y共R2兲 = y m, respectively. The noninteracting ideal Fermi gas in the outer 共r ⬎ R2兲 region provides a reliable method to measure the local chemical potential 1共r兲 in the strongly interacting, inner 共r ⬍ R2兲 region. Since F1 = 1 in the outer region, the extension of the noninteracting ideal Fermi distribution n0共r兲, fit to the outer region, into the inner region gives the local chemiconsequently 1 / F1 cal potential as 1共r兲 = ␣n2/3 0 , = 共n0 / n1兲2/3. This method allows us to measure the equation of state directly from the shape of the cloud without any absolute calibration for particle density. Furthermore, when a sample has a superfluid core, i.e. the whole range of the density ratio 0 ⱕ x ⱕ 1, a single shot image of the sample can provide all information for the determination of the equation of state. In the experiments, the total population imbalance was controlled to be ␦ = 共N1 − N2兲 / 共N1 + N2兲 = 44共4兲% less than the critical imbalance ␦c共⬇75% 兲 关15,16兴 to have a superfluid core, i.e. y 0 ⬎ y c, where Ni is the total atom number of component i. The phase boundary Rc, located by the kink and peak position in the column density difference profile, was measured to be Rc / R1 = 0.430共3兲 and the critical density ratio was measured to be xc = 0.53共5兲. The reference density n0共r兲 and the radius R1共R2兲 were determined from the fit of the outer region, r ⬎ R2共r ⬎ Rc兲 of the majority共minority兲 column density profile to a zero-temperature Thomas-Fermi 共TF兲 distribution.
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FIG. 1. 共Color online兲 Spatial profiles of a populationimbalanced Fermi mixture confined in a harmonic trap at unitarity. 共a兲 The averaged column density profiles and 共b兲 the reconstructed 3D profiles at the lowest temperature from Ref. 关2兴 关green 共gray兲: majority; blue 共dark gray兲: minority; black: difference兴. R1 and R2 are the radii of the majority 共spin 1兲 and the minority 共spin 2兲 cloud, respectively. The core radius Rc was determined as the kink and peak position in the column density difference. The black dotted line in 共a兲 is a zero-temperature Thomas-Fermi 共TF兲 distribution fit to the outer region 共r ⬎ R2兲 of the majority column density profile and the black dotted line in 共b兲 is the corresponding 3D distribution, n0. The sample has three regions: 共I兲 The superfluid core region 共0 ⱕ r ⬍ Rc兲, 共II兲 the partially polarized, intermediate region 共Rc ⬍ r ⬍ R2兲, and 共III兲 a fully polarized noninteracting outer region 共R2 ⬍ r ⬍ R1兲. 共c兲 The density ratio of the two components, x共r兲 = n2 / n1. 共d兲 The majority density normalized by the reference density, n0.
The well-known zero-temperature thermodynamics of a balanced superfluid and a noninteracting Fermi gas provide physical constraints on the form of the universal function. Since the chemical potential of a fully paired 共x = 1兲 superfluid is proportional to the Fermi energy, i.e. s = 共1 + 2兲 / 2 = F1, we have 1 / F1 = 2 / 共1 + y兲 and g共1兲 = 共2兲3/5. On the other hand, a fully polarized 共x = 0兲 noninteracting Fermi gas has 1 = F1 so that g共0兲 = 1. The universal parameter can be independently determined from the
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PHYSICAL REVIEW A 77, 041603共R兲 共2008兲 (a)
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Density ratio (x=n2/n1) FIG. 2. 共Color online兲 Thermodynamic potential at unitarity. The universal function g共x兲 for energy density was constructed as a function of the density ratio x of the two components, from five independent measurements of the density profiles 共Fig. 1兲. The critical chemical potential ratio and the critical density ratio were measured to be y c = 0.03共2兲 and xc = 0.53共5兲, respectively 共see text for determination method兲. The red 共dark gray兲 solid line is obtained by fitting the model equation Eq. 共4兲 with 兵 , y c , y m , xc其 = 兵0.42, 0.03, −0.58, 0.53其 to the normal region 共x ⬍ xc兲 of g共x兲. A red 共dark gray兲 dashed line in the phase separation region 共xc ⬍ x ⬍ 1兲 connects the two points 关x = 1 , g共1兲 = 共2兲3/5兴 and 关x = xc , g共xc兲兴.
majority profile by comparing the curvature of the Fermi energy distribution F1共r兲 ⬀ n1共r兲2/3 in the core region and in the outer region, as = 共d2F1 / dr2兲r⬎R2 / 共d2F1 / dr2兲r⬍Rc. This determination is, however, technically limited due to the low signal-to-noise ratio. In the following analysis we use the theoretically predicted value th = 0.42共1兲关17,18兴, confirmed in previous measurements 关19–26兴. Figure 2 displays the universal function g共x兲 constructed from the density profiles at the lowest temperature. The critical value y c was determined to be y c = 0.03共2兲 such that the average value of g共x兲 for x ⬎ 0.9 gives 共2th兲3/5 = 0.90. This critical value y c has been discussed in Ref. 关2兴 to demonstrate the stability of a fully paired superfluid state at zero temperature. At zero temperature, g共x兲 is not defined for a homogeneous system with xc ⬍ x ⬍ 1. The sparse population of the data points in the region of xc ⬍ x ⬍ 1 indicates the phase separation in the sample, associated with the first-order phase transition. The other critical value y m represents the binding energy Eb of a single minority resonantly interacting with a majority Fermi sea as Eb = limx→0+2 = y mF1. y = g⬘共x兲 / 关g共x兲 − xg⬘共x兲兴 and g共0兲 = 1 give y m = g⬘共0兲. By fitting 1 + y mx to g共x兲 for x ⬍ 0.1, we estimated y m = −0.58共5兲, which is in good agreement with the recent theoretical results of −0.6 关11兴, −0.58共1兲 关6兴, −0.6066 关7兴, −0.54共4兲 关12兴, and −0.618 关8兴. From the definitions, y c = y共Rc兲 and y m = y共R2兲, the estimated critical values 兵y c , y m其 = 兵0.03共2兲 , −0.58共5兲其 suggest R2 = 0.707共20兲R1 for Rc = 0.43R1. However, the minority radius was measured to be R2 / R1 = 0.73共1兲, suggesting y m = −0.69共8兲. We attribute this discrepancy to the fact that the shape of the minority density profile in the intermediate region cannot be completely captured with a zero-temperature TF distribution 关27兴. Following a normal Fermi liquid description 关6–8兴, we
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FIG. 3. 共Color online兲 Comparison between the experimental data and the model. 共a兲 The 3D density profiles and 共b兲 the column density profiles for ␦ = 44% 共dotted-dashed: majority; dashed: minority; dotted: difference兲 generated from the model equation of state 关the red 共dark gray兲 line in Fig. 2兴. Experimental data in Fig. 1 are plotted together for comparison 共same color usage兲.
consider a model for the equation of state of a partially polarized Fermi gas in the following form: 5 1 + cx ␥ g共x兲5/3 = 1 + y mx + x 3 m*
for x ⬍ xc ,
共4兲
satisfying the boundary conditions g共0兲 = 1 and g⬘共0兲 = y m at x = 0. The second term corresponds to the momentumindependent binding energy for the minority atoms and the third term describes the deviation from the free particle behavior, regarding m* as the effective mass and c as its correction 关for the noninteracting case, g0共x兲5/3 = 1 + x5/3兴. The equilibrium condition for the coexistence of two spatially separate phases requires that the two phases have the same chemical potential and pressure at the critical point 关28兴, imposing the boundary conditions at x = xc such as g共xc兲 = 关共1 + xcy c兲 / 共1 + y c兲兴g共1兲 and g⬘共xc兲 = 关y c / 共1 + y c兲兴g共1兲. Then, for given values 兵 , y c , y m , xc其, this model relies on only one free parameter. With 兵 , y c , y m , xc其 = 兵0.42, 0.03, −0.58, 0.53其, the fit of Eq. 共4兲 to the intermediate region 共x ⬍ xc兲 gives ␥ = 1.60共13兲, having m* = 1.06 and c = −0.019. Quantum Monte Carlo calculations for small x predicts m* = 1.04共3兲 with ␥ = 5 / 3 关6兴, which is very close to the observed behavior. Figure 3 displays the density profiles generated from the model equation of state, together with the experimental data. Our observation of ␥ ⬇ 5 / 3 and the small change in the effective mass suggests that a polarized Fermi gas with resonant interactions can be described as a normal Fermi liquid with weakly interacting quasiparticles. However, it is an open question whether the Fermi liquid description is still valid for high minority concentrations, where the Pauli blocking effect of the minority Fermi sea might play an important role. The possibility of the exotic ground state of a partially polarized system has been suggested by the recent observation of the temperature-dependent spectral shift in the minority rf excitation spectrum 关5兴. We note that it is not clear how to distinguish possible exotic states 关10兴 via the equation of state. More experimental studies for microscopic properties of the system, e.g. majority rf spectroscopy, are necessary to clarify the issue. In conclusion, we measure the equation of state of a two-
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component Fermi gas with resonant interactions by analyzing the in situ density distributions of the trapped sample. In a similar way, the density profiles at finite temperature may reveal the excitation spectrum of the system 关29兴.
The author thanks C. H. Schunck and A. Schirotzek for experimental assistance, and W. Ketterle, A. Bulgac, M. M. Forbes, C. Lobo, and S. Reddy for discussions. This work was supported by NSF, ONR, MURI, and DARPA.
关1兴 S. Giorgini, L. P. Pitaevskii, and S. Stringari, e-print arXiv:0706.3360 and references therein. 关2兴 Y. Shin, C. H. Schunck, A. Schirotzek, and W. Ketterle, Nature 共London兲 451, 689 共2008兲. 关3兴 B. S. Chandrasekhar, Appl. Phys. Lett. 1, 7 共1962兲. 关4兴 A. M. Clogston, Phys. Rev. Lett. 9, 266 共1962兲. 关5兴 C. H. Schunck, Y. Shin, A. Schirotzek, M. W. Zwierlein, and W. Ketterle, Science 316, 867 共2007兲. 关6兴 C. Lobo, A. Recati, S. Giorgini, and S. Stringari, Phys. Rev. Lett. 97, 200403 共2006兲. 关7兴 R. Combescot, A. Recati, C. Lobo, and F. Chevy, Phys. Rev. Lett. 98, 180402 共2007兲. 关8兴 N. Prokof’ev and B. Svistunov, Phys. Rev. B 77, 020408共R兲 共2008兲. 关9兴 M. W. Zwierlein, C. H. Schunck, A. Schirotzek, and W. Ketterle, Nature 共London兲 442, 54 共2006兲. 关10兴 A. Bulgac, M. McNeil Forbes, and A. Schwenk, Phys. Rev. Lett. 97, 020402 共2006兲. 关11兴 F. Chevy, Phys. Rev. A 74, 063628 共2006兲. 关12兴 A. Bulgac and M. McNeil Forbes, Phys. Rev. A 75, 031605共R兲 共2007兲. 关13兴 T.-L. Ho, Phys. Rev. Lett. 92, 090402 共2004兲. 关14兴 M. Bartenstein, A. Altmeyer, S. Riedl, R. Geursen, S. Jochim, C. Chin, J. H. Denschlag, R. Grimm, A. Simoni, E. Tiesinga et al., Phys. Rev. Lett. 94, 103201 共2005兲. 关15兴 Y. Shin, M. W. Zwierlein, C. H. Schunck, A. Schirotzek, and W. Ketterle, Phys. Rev. Lett. 97, 030401 共2006兲. 关16兴 M. W. Zwierlein, A. Schirotzek, C. H. Schunck, and W. Ketterle, Science 311, 492 共2006兲. 关17兴 G. E. Astrakharchik, J. Boronat, J. Casulleras, and S. Giorgini, Phys. Rev. Lett. 93, 200404 共2004兲.
关18兴 J. Carlson and S. Reddy, Phys. Rev. Lett. 95, 060401 共2005兲. 关19兴 K. M. O’Hara, S. L. Hemmer, M. E. Gehm, S. R. Granade, and J. E. Thomas, Science 298, 2179 共2002兲. 关20兴 M. E. Gehm, S. L. Hemmer, S. R. Granade, K. M. O’Hara, and J. E. Thomas, Phys. Rev. A 68, 011401共R兲 共2003兲. 关21兴 M. Bartenstein, A. Altmeyer, S. Riedl, S. Jochim, C. Chin, J. H. Denschlag, and R. Grimm, Phys. Rev. Lett. 92, 120401 共2004兲. 关22兴 T. Bourdel, L. Khaykovich, J. Cubizolles, J. Zhang, F. Chevy, M. Teichmann, L. Tarruell, S. J. J. M. F. Kokkelmans, and C. Salomon, Phys. Rev. Lett. 93, 050401 共2004兲. 关23兴 J. Kinast, A. Turlapov, J. E. Thomas, Q. Chen, J. Stajic, and K. Levin, Science 307, 1296 共2005兲. 关24兴 G. B. Partridge, W. Li, R. I. Karmar, Y. Liao, and R. G. Hulet, Science 311, 503 共2006兲. 关25兴 J. T. Stewart, J. P. Gaebler, C. A. Regal, and D. S. Jin, Phys. Rev. Lett. 97, 220406 共2006兲. 关26兴 Y. Shin, C. H. Schunck, A. Schirotzek, and W. Ketterle, Phys. Rev. Lett. 99, 090403 共2007兲. 关27兴 The critical values y c and y m are related with the radii R1, R2, and Rc as ⬅ 共1 − y c兲 / 共1 − y m兲 = 共R21 − R22兲 / 共R21 − R2c 兲. Our measured critical values 兵y c , y m其 = 兵0.03, −0.58其 suggest ⬇ 0.61. The experimental data of the radii reported in Ref. 关9兴 show ⬇ 0.70, where the expansion dynamics was involved. 关28兴 The pressures of a fully paired superfluid is given as PS 4 = 5 ␣−3/2−3/2s5/2. The pressure of a partially polarized normal 2 gas can be expressed as PN = 5 ␣−3/2关1 / 共g − xg⬘兲兴5/2 关12兴. PS = PN at x = xc gives g⬘共xc兲 = 关g共1兲 − g共xc兲兴 / 共1 − xc兲 using y c = g⬘共xc兲 / 关g共xc兲 − xcg⬘共xc兲兴 and s = 共1 + 2兲 / 2. 关29兴 J. Carlson and S. Reddy, e-print arXiv:0711.0414v1.
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Determination of the equation of state of a polarized Fermi gas at unitarity Yong-il Shin* Department of Physics, MIT-Harvard Center for Ultracold Atoms, and Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA 共Received 10 January 2008; published 10 April 2008兲 We report on the measurement of the equation of state of a two-component Fermi gas of 6Li atoms with resonant interactions. By analyzing the in situ density distributions of a population-imbalanced Fermi mixture reported in a recent experiment 关Y. Shin et al., Nature 451, 689 共2008兲兴, we determine the energy density of a resonantly interacting Fermi gas as a function of the densities of the two components. We present a method to determine the equation of state directly from the shape of the trapped cloud, where the fully polarized, noninteracting ideal Fermi gas in the outer region provides the absolute calibration of particle density. From the density profiles obtained at the lowest temperature, we estimate the zero-temperature equation of state. DOI: 10.1103/PhysRevA.77.041603
PACS number共s兲: 03.75.Ss, 03.75.Hh, 05.30.Fk, 51.30.⫹i
Interacting fermions are a paradigm of modern physics, encompassing superconductivity and superfluidity. One interesting situation arises when the constituents interact resonantly, i.e., the scattering length for the free fermions diverges. At this so-called unitarity limit, the behavior of the system becomes universal, being independent of the nature of the interactions. Ultracold atomic Fermi gases near Feshbach collisional resonance present a model system for studying strongly interacting fermions 关1兴. Recently, the phase diagram of a two-component Fermi gas with resonant interactions has been experimentally established 关2兴, showing that at zero temperature the system undergoes a firstorder quantum phase transition from a fully paired superfluid to a partially polarized normal gas when the imbalance between the two spin components exceeds a critical value, called the Chandrasekhar-Clogston limit of superfluidity 关3,4兴. The nature of the partially polarized normal phase, however, is still a subject of investigation. The spectral shift observed in the minority rf excitation spectrum has been interpreted as the existence of “pairing” in the normal phase 关5兴, but several theoretical studies for a highly polarized system, e.g. a single minority atom in a majority Fermi sea, suggest that the system is well-described as a normal Fermi liquid, where the minority atoms are associated with weakly interacting quasiparticles 关6–8兴. This picture seems to be supported by the experimental observation that the shape of the minority cloud in the normal phase is similar to a free Fermi gas 关2,9兴. It has been speculated that exotic pairing states might exist in the partially polarized phase 关10兴. In this paper, we determine the equation of state of a polarized Fermi gas at unitarity from the in situ density profiles of a population-imbalanced Fermi mixture confined in a harmonic trap. Since the variation of the external trapping potential across the sample scans the chemical potential, in principle, the density information of a single sample contains the whole equation of state 关11,12兴. We present a method to determine the equation of state directly from the shape of the trapped cloud. Because of its exactly known thermal properties, a fully polarized, noninteracting ideal Fermi gas in the
*[email protected] 1050-2947/2008/77共4兲/041603共4兲
outer region provides the absolute density calibration. The equation of state of a polarized Fermi gas can be parametrized using a normal Fermi liquid description, which includes the binding energy of a single minority atom resonantly interacting with a majority Fermi sea, the effective mass of the quasiparticles, and its correction. This work is a quantitative study of the thermodynamic properties of the polarized normal state with strong interactions, finding reasonable agreement with recent calculations 关6–8,11,12兴. For infinite scattering length, the unitarity limit implies that all interaction energies scale with the Fermi energies of the components Fi = ប2 / 2m共62ni兲2/3 关13兴, where ប is the Planck constant divided by 2, m is the particle mass, ni is the density of component i, and i = 1 , 2. As a result, a simple dimensional scaling argument implies that the energy density E共n1 , n2兲 of a two-component Fermi gas can be parametrized as 3 E共n1,n2兲 = ␣关n1g共x兲兴5/3 , 5
共1兲
introducing a dimensionless universal function g共x兲 关12兴, where ␣ = 共62兲2/3ប2 / 2m and x = n2 / n1 is the density ratio of the two components. Without loss of generality, 0 ⱕ x ⱕ 1 due to the symmetry of the two components. From the chemical potential relation i = E / ni, the universal function can be expressed as g共x兲5/3 =
1 + x2 1 = 共1 + xy兲, F1 ␣n2/3 1
共2兲
where y = 2 / 1 is the chemical potential ratio of the two components. The main result of this paper is the measurement of the universal function g共x兲 for a resonantly interacting Fermi gas. When a Fermi mixture is confined in a harmonic trap, V共r兲 ⬀ r2, the local chemical potential is given as i共r兲 = i0 − V共r兲, where i0 is the global chemical potential with respect to the bottom of the trapping potential. The global chemical potential of the majority component 10 is determined from the radius of the majority cloud R1, i.e., 10 = V共R1兲. Then, the chemical potential ratio y共r兲 = 2共r兲 / 1共r兲 is given as
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1 − r2/R21 where y 0 = 20 / 10. With population imbalance, i.e. y 0 ⬍ 1, x共r兲 and y共r兲 vary over the sample. From Eq. 共2兲, the spatial correlation of the local particle densities and the local chemical potentials in a trapped sample determines g共x兲 under the local density approximation. We estimate the zero-temperature equation of state by analyzing the density profiles obtained at the lowest temperature 共T / TF ⬇ 0.03 where TF is the Fermi temperature of the majority component兲 in Ref. 关2兴. A spin mixture of the two lowest hyperfine states of 6Li atoms was prepared in a threedimensional 共3D兲 harmonic trap on a broad Feshbach resonance, located at a magnetic field of 834 G 关14兴, resonantly enhancing the interactions between the two spin states. The detailed description of the experimental procedure for the sample preparation and the signal processing has been provided in Ref. 关2兴. Figure 1 shows the spatial structure of a resonantly interacting Fermi mixture in a harmonic trap. According to the zero-temperature phase diagram 关2兴, three distinctive spatial regions can be identified in the inhomogeneous sample with population imbalance 共y 0 ⬍ 1兲. When the chemical potential ratio at center y 0 is larger than a critical value y c for the superfluid-to-normal phase transition, the sample has 共I兲 a fully paired superfluid core 共x = 1, y ⬎ y c兲 surrounded by 共II兲 a partially polarized normal gas 共0 ⬍ x ⱕ xc , y ⬍ y c兲, showing a discontinuity in the density ratio x at the phase boundary. In the outer region where the minority component is completely depleted, 共III兲 a fully polarized noninteracting Fermi gas forms 共x = 0, y ⬍ y m ⬍ y c兲. The radii, Rc and R2 are the I-II and II-III transition points, defining y共Rc兲 = y c and y共R2兲 = y m, respectively. The noninteracting ideal Fermi gas in the outer 共r ⬎ R2兲 region provides a reliable method to measure the local chemical potential 1共r兲 in the strongly interacting, inner 共r ⬍ R2兲 region. Since F1 = 1 in the outer region, the extension of the noninteracting ideal Fermi distribution n0共r兲, fit to the outer region, into the inner region gives the local chemiconsequently 1 / F1 cal potential as 1共r兲 = ␣n2/3 0 , = 共n0 / n1兲2/3. This method allows us to measure the equation of state directly from the shape of the cloud without any absolute calibration for particle density. Furthermore, when a sample has a superfluid core, i.e. the whole range of the density ratio 0 ⱕ x ⱕ 1, a single shot image of the sample can provide all information for the determination of the equation of state. In the experiments, the total population imbalance was controlled to be ␦ = 共N1 − N2兲 / 共N1 + N2兲 = 44共4兲% less than the critical imbalance ␦c共⬇75% 兲 关15,16兴 to have a superfluid core, i.e. y 0 ⬎ y c, where Ni is the total atom number of component i. The phase boundary Rc, located by the kink and peak position in the column density difference profile, was measured to be Rc / R1 = 0.430共3兲 and the critical density ratio was measured to be xc = 0.53共5兲. The reference density n0共r兲 and the radius R1共R2兲 were determined from the fit of the outer region, r ⬎ R2共r ⬎ Rc兲 of the majority共minority兲 column density profile to a zero-temperature Thomas-Fermi 共TF兲 distribution.
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FIG. 1. 共Color online兲 Spatial profiles of a populationimbalanced Fermi mixture confined in a harmonic trap at unitarity. 共a兲 The averaged column density profiles and 共b兲 the reconstructed 3D profiles at the lowest temperature from Ref. 关2兴 关green 共gray兲: majority; blue 共dark gray兲: minority; black: difference兴. R1 and R2 are the radii of the majority 共spin 1兲 and the minority 共spin 2兲 cloud, respectively. The core radius Rc was determined as the kink and peak position in the column density difference. The black dotted line in 共a兲 is a zero-temperature Thomas-Fermi 共TF兲 distribution fit to the outer region 共r ⬎ R2兲 of the majority column density profile and the black dotted line in 共b兲 is the corresponding 3D distribution, n0. The sample has three regions: 共I兲 The superfluid core region 共0 ⱕ r ⬍ Rc兲, 共II兲 the partially polarized, intermediate region 共Rc ⬍ r ⬍ R2兲, and 共III兲 a fully polarized noninteracting outer region 共R2 ⬍ r ⬍ R1兲. 共c兲 The density ratio of the two components, x共r兲 = n2 / n1. 共d兲 The majority density normalized by the reference density, n0.
The well-known zero-temperature thermodynamics of a balanced superfluid and a noninteracting Fermi gas provide physical constraints on the form of the universal function. Since the chemical potential of a fully paired 共x = 1兲 superfluid is proportional to the Fermi energy, i.e. s = 共1 + 2兲 / 2 = F1, we have 1 / F1 = 2 / 共1 + y兲 and g共1兲 = 共2兲3/5. On the other hand, a fully polarized 共x = 0兲 noninteracting Fermi gas has 1 = F1 so that g共0兲 = 1. The universal parameter can be independently determined from the
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Density ratio (x=n2/n1) FIG. 2. 共Color online兲 Thermodynamic potential at unitarity. The universal function g共x兲 for energy density was constructed as a function of the density ratio x of the two components, from five independent measurements of the density profiles 共Fig. 1兲. The critical chemical potential ratio and the critical density ratio were measured to be y c = 0.03共2兲 and xc = 0.53共5兲, respectively 共see text for determination method兲. The red 共dark gray兲 solid line is obtained by fitting the model equation Eq. 共4兲 with 兵 , y c , y m , xc其 = 兵0.42, 0.03, −0.58, 0.53其 to the normal region 共x ⬍ xc兲 of g共x兲. A red 共dark gray兲 dashed line in the phase separation region 共xc ⬍ x ⬍ 1兲 connects the two points 关x = 1 , g共1兲 = 共2兲3/5兴 and 关x = xc , g共xc兲兴.
majority profile by comparing the curvature of the Fermi energy distribution F1共r兲 ⬀ n1共r兲2/3 in the core region and in the outer region, as = 共d2F1 / dr2兲r⬎R2 / 共d2F1 / dr2兲r⬍Rc. This determination is, however, technically limited due to the low signal-to-noise ratio. In the following analysis we use the theoretically predicted value th = 0.42共1兲关17,18兴, confirmed in previous measurements 关19–26兴. Figure 2 displays the universal function g共x兲 constructed from the density profiles at the lowest temperature. The critical value y c was determined to be y c = 0.03共2兲 such that the average value of g共x兲 for x ⬎ 0.9 gives 共2th兲3/5 = 0.90. This critical value y c has been discussed in Ref. 关2兴 to demonstrate the stability of a fully paired superfluid state at zero temperature. At zero temperature, g共x兲 is not defined for a homogeneous system with xc ⬍ x ⬍ 1. The sparse population of the data points in the region of xc ⬍ x ⬍ 1 indicates the phase separation in the sample, associated with the first-order phase transition. The other critical value y m represents the binding energy Eb of a single minority resonantly interacting with a majority Fermi sea as Eb = limx→0+2 = y mF1. y = g⬘共x兲 / 关g共x兲 − xg⬘共x兲兴 and g共0兲 = 1 give y m = g⬘共0兲. By fitting 1 + y mx to g共x兲 for x ⬍ 0.1, we estimated y m = −0.58共5兲, which is in good agreement with the recent theoretical results of −0.6 关11兴, −0.58共1兲 关6兴, −0.6066 关7兴, −0.54共4兲 关12兴, and −0.618 关8兴. From the definitions, y c = y共Rc兲 and y m = y共R2兲, the estimated critical values 兵y c , y m其 = 兵0.03共2兲 , −0.58共5兲其 suggest R2 = 0.707共20兲R1 for Rc = 0.43R1. However, the minority radius was measured to be R2 / R1 = 0.73共1兲, suggesting y m = −0.69共8兲. We attribute this discrepancy to the fact that the shape of the minority density profile in the intermediate region cannot be completely captured with a zero-temperature TF distribution 关27兴. Following a normal Fermi liquid description 关6–8兴, we
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FIG. 3. 共Color online兲 Comparison between the experimental data and the model. 共a兲 The 3D density profiles and 共b兲 the column density profiles for ␦ = 44% 共dotted-dashed: majority; dashed: minority; dotted: difference兲 generated from the model equation of state 关the red 共dark gray兲 line in Fig. 2兴. Experimental data in Fig. 1 are plotted together for comparison 共same color usage兲.
consider a model for the equation of state of a partially polarized Fermi gas in the following form: 5 1 + cx ␥ g共x兲5/3 = 1 + y mx + x 3 m*
for x ⬍ xc ,
共4兲
satisfying the boundary conditions g共0兲 = 1 and g⬘共0兲 = y m at x = 0. The second term corresponds to the momentumindependent binding energy for the minority atoms and the third term describes the deviation from the free particle behavior, regarding m* as the effective mass and c as its correction 关for the noninteracting case, g0共x兲5/3 = 1 + x5/3兴. The equilibrium condition for the coexistence of two spatially separate phases requires that the two phases have the same chemical potential and pressure at the critical point 关28兴, imposing the boundary conditions at x = xc such as g共xc兲 = 关共1 + xcy c兲 / 共1 + y c兲兴g共1兲 and g⬘共xc兲 = 关y c / 共1 + y c兲兴g共1兲. Then, for given values 兵 , y c , y m , xc其, this model relies on only one free parameter. With 兵 , y c , y m , xc其 = 兵0.42, 0.03, −0.58, 0.53其, the fit of Eq. 共4兲 to the intermediate region 共x ⬍ xc兲 gives ␥ = 1.60共13兲, having m* = 1.06 and c = −0.019. Quantum Monte Carlo calculations for small x predicts m* = 1.04共3兲 with ␥ = 5 / 3 关6兴, which is very close to the observed behavior. Figure 3 displays the density profiles generated from the model equation of state, together with the experimental data. Our observation of ␥ ⬇ 5 / 3 and the small change in the effective mass suggests that a polarized Fermi gas with resonant interactions can be described as a normal Fermi liquid with weakly interacting quasiparticles. However, it is an open question whether the Fermi liquid description is still valid for high minority concentrations, where the Pauli blocking effect of the minority Fermi sea might play an important role. The possibility of the exotic ground state of a partially polarized system has been suggested by the recent observation of the temperature-dependent spectral shift in the minority rf excitation spectrum 关5兴. We note that it is not clear how to distinguish possible exotic states 关10兴 via the equation of state. More experimental studies for microscopic properties of the system, e.g. majority rf spectroscopy, are necessary to clarify the issue. In conclusion, we measure the equation of state of a two-
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component Fermi gas with resonant interactions by analyzing the in situ density distributions of the trapped sample. In a similar way, the density profiles at finite temperature may reveal the excitation spectrum of the system 关29兴.
The author thanks C. H. Schunck and A. Schirotzek for experimental assistance, and W. Ketterle, A. Bulgac, M. M. Forbes, C. Lobo, and S. Reddy for discussions. This work was supported by NSF, ONR, MURI, and DARPA.
关1兴 S. Giorgini, L. P. Pitaevskii, and S. Stringari, e-print arXiv:0706.3360 and references therein. 关2兴 Y. Shin, C. H. Schunck, A. Schirotzek, and W. Ketterle, Nature 共London兲 451, 689 共2008兲. 关3兴 B. S. Chandrasekhar, Appl. Phys. Lett. 1, 7 共1962兲. 关4兴 A. M. Clogston, Phys. Rev. Lett. 9, 266 共1962兲. 关5兴 C. H. Schunck, Y. Shin, A. Schirotzek, M. W. Zwierlein, and W. Ketterle, Science 316, 867 共2007兲. 关6兴 C. Lobo, A. Recati, S. Giorgini, and S. Stringari, Phys. Rev. Lett. 97, 200403 共2006兲. 关7兴 R. Combescot, A. Recati, C. Lobo, and F. Chevy, Phys. Rev. Lett. 98, 180402 共2007兲. 关8兴 N. Prokof’ev and B. Svistunov, Phys. Rev. B 77, 020408共R兲 共2008兲. 关9兴 M. W. Zwierlein, C. H. Schunck, A. Schirotzek, and W. Ketterle, Nature 共London兲 442, 54 共2006兲. 关10兴 A. Bulgac, M. McNeil Forbes, and A. Schwenk, Phys. Rev. Lett. 97, 020402 共2006兲. 关11兴 F. Chevy, Phys. Rev. A 74, 063628 共2006兲. 关12兴 A. Bulgac and M. McNeil Forbes, Phys. Rev. A 75, 031605共R兲 共2007兲. 关13兴 T.-L. Ho, Phys. Rev. Lett. 92, 090402 共2004兲. 关14兴 M. Bartenstein, A. Altmeyer, S. Riedl, R. Geursen, S. Jochim, C. Chin, J. H. Denschlag, R. Grimm, A. Simoni, E. Tiesinga et al., Phys. Rev. Lett. 94, 103201 共2005兲. 关15兴 Y. Shin, M. W. Zwierlein, C. H. Schunck, A. Schirotzek, and W. Ketterle, Phys. Rev. Lett. 97, 030401 共2006兲. 关16兴 M. W. Zwierlein, A. Schirotzek, C. H. Schunck, and W. Ketterle, Science 311, 492 共2006兲. 关17兴 G. E. Astrakharchik, J. Boronat, J. Casulleras, and S. Giorgini, Phys. Rev. Lett. 93, 200404 共2004兲.
关18兴 J. Carlson and S. Reddy, Phys. Rev. Lett. 95, 060401 共2005兲. 关19兴 K. M. O’Hara, S. L. Hemmer, M. E. Gehm, S. R. Granade, and J. E. Thomas, Science 298, 2179 共2002兲. 关20兴 M. E. Gehm, S. L. Hemmer, S. R. Granade, K. M. O’Hara, and J. E. Thomas, Phys. Rev. A 68, 011401共R兲 共2003兲. 关21兴 M. Bartenstein, A. Altmeyer, S. Riedl, S. Jochim, C. Chin, J. H. Denschlag, and R. Grimm, Phys. Rev. Lett. 92, 120401 共2004兲. 关22兴 T. Bourdel, L. Khaykovich, J. Cubizolles, J. Zhang, F. Chevy, M. Teichmann, L. Tarruell, S. J. J. M. F. Kokkelmans, and C. Salomon, Phys. Rev. Lett. 93, 050401 共2004兲. 关23兴 J. Kinast, A. Turlapov, J. E. Thomas, Q. Chen, J. Stajic, and K. Levin, Science 307, 1296 共2005兲. 关24兴 G. B. Partridge, W. Li, R. I. Karmar, Y. Liao, and R. G. Hulet, Science 311, 503 共2006兲. 关25兴 J. T. Stewart, J. P. Gaebler, C. A. Regal, and D. S. Jin, Phys. Rev. Lett. 97, 220406 共2006兲. 关26兴 Y. Shin, C. H. Schunck, A. Schirotzek, and W. Ketterle, Phys. Rev. Lett. 99, 090403 共2007兲. 关27兴 The critical values y c and y m are related with the radii R1, R2, and Rc as ⬅ 共1 − y c兲 / 共1 − y m兲 = 共R21 − R22兲 / 共R21 − R2c 兲. Our measured critical values 兵y c , y m其 = 兵0.03, −0.58其 suggest ⬇ 0.61. The experimental data of the radii reported in Ref. 关9兴 show ⬇ 0.70, where the expansion dynamics was involved. 关28兴 The pressures of a fully paired superfluid is given as PS 4 = 5 ␣−3/2−3/2s5/2. The pressure of a partially polarized normal 2 gas can be expressed as PN = 5 ␣−3/2关1 / 共g − xg⬘兲兴5/2 关12兴. PS = PN at x = xc gives g⬘共xc兲 = 关g共1兲 − g共xc兲兴 / 共1 − xc兲 using y c = g⬘共xc兲 / 关g共xc兲 − xcg⬘共xc兲兴 and s = 共1 + 2兲 / 2. 关29兴 J. Carlson and S. Reddy, e-print arXiv:0711.0414v1.
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