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Thermodynamics of Multi-Component Fermi Vapors

We study the thermodynamical properties of Fermi vapors confined in a harmonic external potential. In the case of the ideal Fermi gas, we compare exact density profiles with their semiclassical approximation in the conditions of recent experiments. Then, we consider the phase-separation of a multi-component Fermi vapor. In particular, we analyze the phase-separation as a function of temperature, number of particles and scattering length. Finally, we discuss the effect of rotation on the stability and thermodynamics of the trapped vapors.


I. INTRODUCTION
Recent experiments with dilute vapors of alkali-metal atoms in magnetic or magnetooptical traps at very low temperatures have renewed the theoretical study of Bose and Fermi gases. For atomic gases, the Bose-Einstein condensation has been achieved in 1995 [1][2][3] and the Fermi quantum degeneracy in 1999 [4,5].
The s-wave scattering between Fermions in the same hyperfine state is inhibited due the Pauli principle. It follows that at low temperature the dilute Fermi gas, in a fixed hyperfine state, is practically ideal. Nevertheless, the effect of interaction could be very effective for a Fermi vapor with two or more hyperfine states (components). In the recent experiment with dilute 40 K Fermions [4,5], to favor the evaporative cooling, a 40 K Fermi vapor in two hyperfine states (|9/2, 9/2 and |9/2, 7/2 ) is used. When the system is below the Fermi temperature, one hyperfine component is removed and it remains a trapped quasi-ideal degenerate Fermi gas.
In this paper we analyze the thermodynamical properties of a many-component interacting Fermi vapor by using a harmonic potential that models the trap of recent experiment with 40 K [4,5]. These trapped Fermi gases are quite interesting because the quantum degeneracy shows up not only in momentum space, as in uniform systems, but also in coordinate space. First, we investigate the accuracy of the semiclassical approximation for the study of the properties of the Fermi gas in the conditions of 40 K experiment. For the ideal Fermi gas we discuss the finite-temperature spatial and momentum distribution and derive, in the large impulse approximation, the dynamic structure factor. Then, we consider Fermi vapors with many components. Due to the nonzero interaction between different hyperfine states, one can obtain the phase-separation of Fermi components by varying temperature, number of particles and scattering length. Finally, we discuss the effect of an imposed rotation around a fixed axis on stability and thermodynamics of the trapped Fermi vapors.

II. IDEAL FERMI GAS
For an ideal Fermi gas at thermal equilibrium, the average number of particles in the single-particle state |α with energy ǫ α is given by where µ is the chemical potential and β = 1/(kT ) with k the Boltzmann constant and T the temperature. The average number N = α n α of particles of the system fixes the chemical potential [6]. In the case of a harmonic potential U(r) = (m/2)(ω 2 1 x 2 1 + ω 2 2 x 2 2 + ω 2 3 x 2 3 ) one finds the exact density profile by using the Eq. (1) and Hermite polynomials, from which one obtains the eigenfunctions φ n 1 n 2 n 3 (r) of the harmonic oscillator The exact density profile can be compared with the semiclassical one. The semiclassical approximation means that, instead of ǫ α , one uses the classical single-particle phase-space energy ǫ(r, p) = p 2 2m + U(r). In this way one obtains the single-particle phase-space distribution n(r, p) = 1 e β(ǫ(r,p)−µ) + 1 .
At zero temperature, i.e., in the limit β → ∞ where µ = E F (the Fermi energy), one has n(r, p) = Θ (E F − ǫ(r, p)), where Θ(x) is the Heaviside step function. At temperature well below the Fermi temperature T F = E F /k, the Fermions begin to fill the lowest available single-particle states in accordance with the Pauli exclusion Principle (Fermi quantum degeneracy) [6].
Because of the Heisenberg principle, the quantum elementary volume of the singleparticle phase-space is given by (2πh) 3 , whereh is the Planck constant. It follows that the average number N of particles can be written as where n(r) is the spatial distribution andñ(p) is the momentum distribution. The finite temperature spatial distribution is given by where λ = (2πh 2 β/m) 1/2 is the thermal length and with Γ the factorial function. In the limit of zero temperature, with µ = E F the Fermi energy, the spatial distribution gives the Thomas-Fermi approximation n(r) = where Θ is the Heaviside step function [7][8].
The Eq. (5) is the generalization of well-known formula for an ideal homogenous Fermi gas in a box that is exact in the thermodynamical limit [6]. Note that for |z| < 1 one has Moreover, by using g n (z) = −f n (−z) instead of f n (z), one finds the spatial distribution of the ideal Bose gas in external potential.
A comparison between exact and semiclassical results has been recently performed at zero [9] and finite temperature [10]. It has been shown that the semiclassical approximation is good for (kT /hω H ) >> 1, where ω H = (ω 1 ω 2 ω 3 ) 1/3 , or, at a fixed temperature, for a large number N of trapped particles. Our calculations confirm this prediction. In fact, we find that when (kT /hω H ) > 1 there are no appreciable deviations between exact and semiclassical results. Instead, when (kT /hω H ) < 1 some differences are observable, in particular for "magic" numbers of particles (N = 1, 4, 10, 20, 35, 56, 84, ...) that correspond to a complete shell occupation of single-particle energy levels (see also [10]). The differences are reduced by increasing N showing that semiclassical approximation provides an excellent representation of Fermi distribution for a wide range of parameters. In Figure 1 we plot the exact density profile, obtained by a direct calculation of Eq. (2), and that obtained with the semiclassical function (5) for (kT /hω H ) = 10 −3 and three "magic" numbers. In correspondence of the "magic numbers", the spatial density profile shows local maxima, which suggest a spatial shell structure. The magic numbers are particularly stable; in fact, for small variations of 4 the chemical potential µ the magic number N remains unchanged. The shell structure in the density profile is washed out by increasing the number of particles and is completely absent in the semiclassical approximation.
As shown in the experiment [4], for the quasi-ideal one-component Fermi gas of  Table 1 we show some relevant quantities of the Fermi gas at different temperatures: the energy E = 3/(2βλ 3 ) d 3 rf 5/2 e β(µ−U (r)) + d 3 rU(r)n(r), the density at the origin n(0) and the widths ρ 2 1/2 and z 2 1/2 of the Fermi cloud in the radial and axial directions, respectively.
In addition to the spatial distribution n(r), also the momentum distributionñ(p) gives an useful signature of the quantum degeneracy of the Fermi gas. In the semiclassical approximation, the finite temperature momentum distribution of an ideal Fermi gas in a harmonic trap is calculated from Eq. (4) and reads At zero temperature, one recovers the formula found by Butts and Rokhsar [7], namelỹ Despite the spatial anisotropy of the trap, the momentum distribution of the Fermi gas is isotropic when the semiclassical approximation is used. Moreover, it is not difficult to show that, both in the exact and semiclassical cases, the momentum distributionñ can be found from the spatial one n by the simple relationñ (p 1 , p 2 , where p = (p 1 , p 2 , p 3 ). In general, the exact momentum distribution is not isotropic and it presents the same shell structures of the spatial one when N is close to a magic number.
Finally, the Eq. (7) can be used to calculate the dynamic structure factor S(q, E) of the Fermi gas. In the large impulse approximation [11] one finds the remarkable result where q and E are the momentum and energy transferred by the probe to the sample.
Recently, the dynamic structure factor of a trapped Bose-Einstein condensate has been measured using two-photon optical Bragg spectroscopy with a time of flight technique by which the number of optically excited atoms can be counted [12].

III. MULTI-COMPONENT FERMI VAPOR
The problem of a dilute Fermi vapor with M hyperfine states (components) can be studied by using the s-wave scattering approximation, the mean-field approach and semiclassical formulas. The spatial density profile n i (r) of the i-th component of a Fermi vapor can be written as where i = 1, 2, ..., M, µ i is the chemical potential of the i-th component, and g ij = 4πh 2 a ij /m, with a ij the s-wave scattering length between i-th and j-th component (a ii = 0).
Thus, the effect of the other M − 1 Fermi components on the i-th component is the appearance of a mean-field effective potential. Note that in the limit of zero temperature, one finds the equations used by Amoruso et al [13]. The total energy of a Fermi vapor with M components is given by where the first term is the kinetic energy E kin , the second term is the external potential energy E ext and the third term is the interaction energy E int . At zero temperature the kinetic energy assumes the familiar Thomas-Fermi form We numerically solve the set of equations (9) with a self-consistent iterative procedure.
Recently, we have used a similar iterative procedure to study 7 Li Bose gas [14]. If the components of the Fermi vapor are non-interacting then they can occupy the same spatial region. Instead, if the interaction is strong enough (repulsive interaction) or for N very large one finds a phase-separation, i.e., the Fermi components stay in different spatial regions. This effect, that has been first discussed at T = 0 by Amoruso et al [13], is shown in Figure 2, where we plot the density profiles of two components of the 40 K Fermi vapor for different values of the scattering length a (a=a 12 ) and N = 0.5 · 10 7 atoms in each component.
When two components have the same number of particles, the onset of phase-separation is also an example of spontaneous symmetry breaking. In fact, if the chemical potentials µ i of the two components are equal, Eq. (9) always admits a symmetric solution n 1 (r) = n 2 (r).
However, for particle number N larger than a threshold N c the solution bifurcates and a pair of symmetry breaking solutions appears. Just beyond threshold the asymptotic solutions begin to differ from the symmetric one in a neighborhood of the origin r = 0, i.e. at the point of higher density. Therefore, an analytic formula for the critical chemical potential µ c can be obtained by standard bifurcation analysis of Eq. (9) at the origin, which gives gβ λ 3 e β(µ−gn(0)) f ′ 3/2 (e β(µ−gn(0)) ) = 1 .
At T = 0, by the use of the first term of the large z expansion of f 3/2 (z) [6], one finds analytical expressions for the critical density n c (0) and the critical chemical potential µ c : These remarkably simple formulas can be very useful to determine the onset of phaseseparation in future experiments. Moreover, by knowing the critical chemical potential one numerically finds the number of particles via Eq. (9).
At finite temperature, we numerically solve the Eq. (11). In Figure 3 we plot the critical number of Fermions as a function of the temperature for different values of the scattering length. Figure 3 shows that, by using a realistic scattering length [4,5], one needs 7 about N = 10 12 atoms to get the spontaneous symmetry breaking. One sees that at higher temperatures the phase-separation appears with a larger number of particles or a larger scattering length. In the lower part of Figure 3 we plot the same critical line as a function of temperature for different values of the scattering length. The conclusion is that by increasing the interaction between the two components one can use lower number of particles to obtain the phase-separation. Such a behavior can be extracted form Eq. (12) in the case of zero temperature. Note that at finite temperature one can use the first two terms of the large z expansion of f 3/2 (z) [6]. Under the conditions kT <<hω H and µ >> gn(0) one finds the following equation for the onset of phase-separation Phase-separation also appears in a Fermi vapor with three or more components. In Figure   5 we plot the density profiles of the 40 K Fermi vapor with three components. In this case we numerically solve Eq. for N = 10 7 particles. In conclusion, to obtain the phase-separation with several millions of Fermions, the only chance is to strongly enhance the scattering length, for example by using the Feshbach resonances. The Feshbach resonances, already seen in Bose condensates [15], should allow to vary the interaction strength between atoms. One could use them to go smoothly from a quasi-ideal to a strongly interacting Fermi system.

IV. ROTATING FERMI VAPOR
The semiclassical approximation can be easily applied to a rotating gas. For an ideal gas, that is rotating with angular velocity Ω around the x 3 -axis, the classical single-particle energy is ǫ(r, p) = p 2 2m + U(r) − Ω(x 1 p 2 − x 2 p 1 ). One immediately finds that the spatial distribution of an ideal Fermi gas is given by where ρ = (x 2 1 + x 2 2 ) 1/2 . Note that the effective potential of the rotating (Ω > 0) Fermi gas is U(r) − (m/2)Ωρ 2 . If U(r) ∼ ρ s with s < 2 for ρ → ∞ then the effective potential goes to −∞ for ρ → ∞ and the spatial integral of the density profile is divergent. It follows that for s < 2 the system is unstable against rotation. The same result applies for ideal Bose and Boltzmann gases. Obviously, the Fermi gas can remain confined near the center of the trap for a very long time depending on the tunneling probability.
In the case of the harmonic potential, s = 2 and the gas is stable against rotation until Ω is smaller than the smaller of the harmonic frequencies ω 1 and ω 2 . The effect of the rotation is simply a shift in two trap frequencies: ω 1 → (ω 2 1 − Ω 2 ) 1/2 and ω 2 → (ω 2 2 − Ω 2 ) 1/2 . In this way one gets the Fermi energy of the rotating ideal gas A similar formula has been found in [16] for the Bose-Einstein transition temperature of a Bose gas in axial-symmetric harmonic potential. Also the density profile of the rotating quantum gas is modified according to the frequency shift discussed above.
The discussion can be extended to a Fermi vapor with two or more components. In particular, one can use Eq. (9) and all the other equations by simply shifting the trap frequencies. Such a shift determines an enhancement of the characteristic length of the harmonic trap: a H → a H / (1 − Ω 2 /ω 2 1 ) 1/12 (1 − Ω 2 /ω 2 2 ) 1/12 . It means that the rotating gas is more dilute and consequently the interatomic interaction is less effective. As shown in Eq.
(12) and Eq. (13), the rotation increases the critical number of Fermions that are necessary to get phase-separation.

V. CONCLUSIONS
First of all we have analyzed the accuracy of the semiclassical approximation. We have found that the semiclassical approach is very accurate for kT >hω H or, at a fixed temperature, for a large number of particles. In particular, the semiclassical formulas are extremely reliable to study recent experiments with a trapped quasi-ideal 40 K Fermi gas where kT /(hω H ) ≃ 10 2 -10 3 and N ≃ 10 6 . By using the semiclassical approach, we have analyzed the spatial and momentum distributions of an ideal Fermi gas in harmonic potential. In par-ticular, we have derived, in the large impulse approximation, the dynamic structure factor of the system.
Then, we have studied the finite-temperature density profiles of a Fermi vapor with many hyperfine states. In the two-component case, by using the actual scattering length, we have found that the two components occupy the same spatial region. We have shown that the onset of phase-separation appears by increasing the scattering length or, for a fixed scattering length, by increasing the number of particles. By raising the temperature, a larger scattering length or a larger number of particles is needed to obtain the phase-separation.