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17.1 Introduction

Strongly interacting Fermi systems offer a universal testbed for several subfields of physics such as atomic physics, condensed-matter physics, and nuclear physics. The interatomic interactions can be tuned simply by changing the strength of an external magnetic field. In an ultralow-temperature regime, a two-component Fermi gas of atoms shows superfluidity in various ways. In the positive s-wave scattering-length side of a Feshbach resonance, where interatomic interactions are repulsive, a two-component Fermi gas of atoms behaves as a gas of diatomic molecules and exhibits Bose-Einstein condensation of molecules. In the negative side, where the interactions are attractive, atoms form Cooper pairs and show BCS-type superfluidity. These two different regimes are considered to be the opposite extremes of a single phase and cross over to each other across the Feshbach resonance.

The superfluid phase diagram in the BCS-BEC crossover regime was theoretically predicted by Sa de Melo et al. in 1993 [1]. Back in those days, it was not clear whether a strongly-interacting Fermi gas would show superfluidity at very low temperatures or whether the system would be stable at all. In 2002, the Duke group experimentally demonstrated that a system of fermionic lithium atoms in the unitary regime is stable [2]. They also found a hydrodynamic behavior in the expansion of the gas [3] which is analogous to the elliptic flow in quark-gluon plasmas. In 2004, the Innsbruck group observed a smooth change in the atomic density distribution across the Feshbach resonance, which indicates that the system crosses over from the BEC to BCS regimes without undergoing a phase transition [4]. The Innsbruck group also measured the pairing gap in this crossover region by using RF spectroscopy [5]. In the same year, the JILA group and the MIT group observed the emergence of condensation in ultracold Fermi gases in the BCS-BEC crossover regime [6, 7]. However, the entire superfluid phase diagram in the BCS-BEC crossover regime has not yet been determined, because of the difficulty in determining the thermodynamic quantities including the temperature due to the strong interatomic interaction. Understanding the thermodynamic behavior of a Fermi gas at the unitarity limit is among the most important problems in the strongly interaction Fermi system.

At the unitarity limit, thermodynamics is expected to become universal. In 1999, Bertsch raised the question about the ground state of dilute spin-1/2 particles when the scattering length diverges and the range of interaction is negligible [8]. He envisaged such a situation as an extreme case of the inner crust of neutron stars in which the range of interaction R, neutron density n, and the scattering length a s satisfy R∼1 [fm]<n −1/3<|a s |∼18.5 [fm]. When the scattering length diverges and the zero-range approximation is valid, the interaction parameter drops out of the thermodynamic description and therefore we may expect the same thermodynamic behavior at unitarity, regardless of the atomic species of constituent particles. This is called the universal hypothesis. In this chapter, we present an overview of the universal properties of a unitary Fermi gas.

17.2 Universality in a Unitary Fermi Gas

17.2.1 Universal Thermodynamics

We consider s-wave collisions in a two-component Fermi gas of atoms. The scattering amplitude is described as f(k)=1/(−1/a s +r e k 2/2−ik), where k is the relative wavenumber, a s is the scattering length, and r e is the range of the interactions. At the unitarity and zero-range limits, where k|a s |→∞ and k|r e|→0, the scattering amplitude reaches the maximum f(k)=−1/ik. The characteristic length scale of fermionic atoms is the Fermi wavenumber k F, which is on the order of a typical interatomic distance n −1/3. The condition of the unitarity limit is |r e|≪n −1/3≪|a s |. In the case of fermionic lithium-6 (6Li) near quantum degeneracy, a typical interatomic distance n −1/3 is on the order of a few hundred nanometers and the range of interaction r e is 5 nm. By tuning the s-wave scattering length to infinity using a Feshbach resonance, the unitarity limit can be realized experimentally.

Since the divergent scattering length and the negligibly small range of interaction do not enter the thermodynamic description, the thermodynamics of a unitary Fermi gas depends only on the Fermi energy ε F and the temperature k B T [9, 10]. According to this universal hypothesis, the internal energy, which is the sum of the kinetic energy and the interaction energy, is described by the following universal thermodynamic function:

$$ E=N E _{\mathrm{F}} f _{E} ( {k _{B} T} / {E _{\mathrm{F}}} ) =N E _{\mathrm{F}} f _{E} ( {T} / {T _{\mathrm{F}}} ). $$
(17.1)

Here, T F is the Fermi temperature and f E is a dimensionless universal function. Other thermodynamic functions can also be described in similar universal forms; for example, the chemical potential μ, Helmholtz free energy F, and entropy S are described as

(17.2)
(17.3)
(17.4)

According to the universal hypothesis, the internal energy of a unitary Fermi gas at T=0 is given by \(E _{0} ^{\mathrm{unitary}} /N= E _{0} ^{\mathrm{ideal}} (1+\beta)\), where \(E _{0} ^{\mathrm{ideal}} = \frac{3}{5} \varepsilon_{\mathrm{F}} N\) is the internal energy of an ideal Fermi gas, and β is a dimensionless universal parameter. The ratio of the ground-state energy of a unitary Fermi gas to that of an ideal Fermi gas \({E _{0} ^{\mathrm{unitary}}} / {E _{0} ^{\mathrm{ideal}}} =1+\beta=\xi\) is called the Bertsch parameter, where β has been measured to be −0.64(15) [11], \(-0.68 ( _{-0.10} ^{+0.15} )\) [12], −0.49(4) [13], \(-0.54 ( _{-0.12} ^{+0.05} )\) [14], −0.54(5) [15], −0.50(7) [16], −0.61(2) [17], −0.565(15) [18]; at the time of writing, the latest experimental result gives ξ=0.376(4) [19]. The fact β<0 shows that the effective interatomic interaction is attractive at unitarity. This can be confirmed by the experimental fact that the size of the cloud of a unitary Fermi gas in a harmonic trap is smaller than that of an ideal Fermi gas with the same number of atoms.

17.2.2 Pressure-Energy Relation and Virial Theorem

A unitary Fermi gas features several key properties that can be derived from the form of the universal thermodynamic functions in (17.1)–(17.4). One is the relation between the pressure P and the energy density E/V [20, 21]. We consider N atoms in a volume V. The pressure of the gas is determined from the relation P=−(∂E/∂V) N,S . Equation (17.4) shows that the constant entropy implies constant T/T F. Therefore, the pressure can be described as

(17.5)

where we used E F∝V −2/3. This relation is the same as the one for a uniform ideal Fermi gas.

Another key property of a unitary Fermi gas is the virial theorem [21]. The virial theorem can be derived from (17.5) and the equation of force balance of a gas. At mechanical equilibrium, an inward force due to a trapping potential and an outward force due to the gas pressure balance everywhere inside the gas:

$$ \boldsymbol{\nabla} P ( \boldsymbol{r} ) +n ( \boldsymbol{r} ) \boldsymbol{ \nabla} U _{\mathrm{trap}} ( \boldsymbol{r} ) =0. $$
(17.6)

Taking the inner product of this equation with r and performing the volume integration, the first term yields

(17.7)

Here E internal is the internal energy per particle. The first term on the right-hand side of the first equality vanishes and (17.5) is used in deriving the last equality. In a harmonic trapping potential which gives r⋅∇ U trap(r)=2U trap(r), the second term in (17.6) is transformed by taking the inner product with r and performing the volume integration to

$$ \int\boldsymbol{r} \cdot n ( \boldsymbol{r} ) \boldsymbol{\nabla} U _{\mathrm{trap}} ( \boldsymbol{r} ) \mathrm{d} \boldsymbol{r} = \int2n ( \boldsymbol{r} ) U _{\mathrm{trap}} ( \boldsymbol{r} ) \mathrm{d} \boldsymbol{r} =2 N E _{\mathrm{pot}}. $$
(17.8)

Thus, (17.7) and (17.8) give

$$ E _{\mathrm{pot}} = E _{\mathrm{internal}}. $$
(17.9)

This is the same as the one for an ideal Fermi gas since the derivation starts from (17.5) and (17.6), which hold for both a unitary Fermi gas and an ideal Fermi gas. The experimental verification of (17.9) is reported in [21]. Since E pot is proportional to the mean square size of the cloud 〈x 2〉, (17.9) can be verified by checking the relation 〈x 2(E)〉/〈x 2(E 0)〉=E/E 0, where E 0 denotes the ground-state energy. To measure the energy of the gas, Thomas et al. first prepared the gas in the ground state and then added the energy to the gas by turning off the optical trap for a short time and recapturing the atoms. By recapturing the atoms after the expansion, a controlled amount of energy was added to the gas in the form of a potential energy. Since the unitary Fermi gas is strongly interacting, the expansion of the unitary Fermi gas can be well described by hydrodynamics. By calculating the energy that is added to the gas due to the hydrodynamic expansion, they found the precise energy of the gas after thermalization. Figure 17.1 shows the experimental result of 〈x 2(E)〉/〈x 2(E 0)〉−1 vs. E/E 0−1. The dashed line in Fig. 17.1 shows 〈x 2(E)〉/〈x 2(E 0)〉=1.03(0.02)E/E 0, which is consistent with the virial theorem. Equation (17.9) is quite useful in experiments because the total energy of the gas is derived from the potential energy, which can be measured directly from the density profile.

Fig. 17.1
figure 1

Experimental verification of the virial theorem in a unitary Fermi gas. The linear dependence shows 〈x 2(E)〉/〈x 2(E 0)〉=E/E 0. Reproduced from [21] with permission

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17.2.3 Measurement of Trap-Averaged Thermodynamic Quantities

The first thermodynamic quantity that was measured experimentally was the ground-state energy of a unitary Fermi gas in a harmonic trap. The ground-state energy is described as (1+β)E F, and several measurements of β have been reported from the measurement of the size of the cloud [11–14, 17] and the measurement of the first sound velocity [9] at the low-temperature limit.

Kinast et al. measured the heat capacity of a unitary Fermi gas using the optically trapped 6Li atoms [13]. They measured the trap-averaged energy of the gas as a function of temperature and observed a clear change in the temperature dependence of the heat capacity upon the emergence of superfluidity. To determine the energy, they applied the energy input technique [13]. In this experiment, they determined the temperature by using the calibration curve from an empirical temperature estimated by applying the Thomas-Fermi profile to a unitary Fermi gas. Although the change in the heat capacity due to superfluidity was clearly observed, the accuracy of the estimation of the critical temperature was limited by the empirical temperature calibration. Later, they invented a new scheme to determine the temperature of a unitary Fermi gas more accurately from the measurement of the total energy and entropy [17, 22]. The entropy of the gas can be determined from the density profile at a magnetic field where the scattering length is small enough to guarantee that the entropy versus cloud size curve is to a good approximation given by that for an ideal gas. From the energy and entropy measurements, they determined the relation between energy and temperature by applying the thermodynamic relation 1/T=∂S/∂E to the entropy vs. energy data. Therefore, once we know any one of the energy, entropy or temperature, we can find the other quantities from the relations obtained by the Duke group. This is quite useful because the total energy can easily be obtained from the potential energy which is calculated from the absorption image and the information about the trapping potential.

Stewart et al. measured the potential energy of a unitary Fermi gas of 40K [14]. They investigated the temperature dependence of the potential energy of a gas in a harmonic trap and determined β at the low-temperature limit. Their result is consistent with that obtained from experiments using 6Li [11–13, 15–18], which indicates the universality of the thermodynamics.

Hu et al. analyzed the data of the trap-averaged energy vs. entropy measured at Duke and Rice using 6Li and at JILA using 40K (Fig. 17.2) [23]. All the experimental data are consistent with each other, which indicates that the thermodynamics of a unitary Fermi gas is universal independent of the atomic species and trapping condition over a wide range of temperature.

Fig. 17.2
figure 2

Universal thermodynamics of a unitary Fermi gas. Three experimental results of the trap-averaged energy are plotted as a function of entropy. Blue, red and yellow markers show the experimental result from the Duke, JILA, and Rice groups, respectively

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However, these results are obtained from trap-averaged quantities, and they do not provide direct information on the universal thermodynamic functions. The universal thermodynamic functions for a unitary Fermi gas have k B T/E F as an argument. Since the scattering between atoms occurs so frequently that the thermal equilibrium is always achieved, implying that k B T is constant over the entire atomic cloud. However, since an atomic gas in a harmonic trap has a nonuniform density profile, the Fermi energy E F, which depends on the density n, becomes position-dependent. Therefore k B T/E F is also position-dependent, which makes it difficult to derive the thermodynamic functions from the trap-averaged thermodynamic quantities and to compare experimental results with theoretical ones [24–28]. To determine the detailed shape of the thermodynamic functions, it is necessary to measure both the thermodynamic quantities and k B T/E F at the same position.

17.2.4 Tan Relations

In 2008, Tan predicted several remarkable relations between the momentum distribution and thermodynamic quantities [29–31]. These relations, known as Tan relations, hold quite generally for interacting gases in the sense that they can be applied to any system that satisfies

$$ r _{0} \ll a _{s},\ n ^{- {1} / {3}},\ \lambda,\ ( { \hbar} / {m\omega} ) ^{{1} / {2}}, $$
(17.10)

where r 0 is the range of interaction, n −1/3 is the mean interparticle distance, \({\lambda=h} / {\sqrt{2\pi m k _{\mathrm{B}} T}}\) is the thermal de Broglie length, and (ħ/mω)1/2 is the size of the harmonic potential with the energy separation ħω. In typical cold atom experiments near quantum degeneracy, r 0 is much smaller than any of n −1/3, λ, and (ħ/mω)1/2. At the unitarity limit where a s diverges, the Tan relations become exact. When the condition (17.10) is met, it is known that the momentum distribution of a Fermi gas over the range

$$ a _{s} ^{- 1},\ n ^{{1} / {3}},\ \lambda^{-1}, \ ( {\hbar} / {m\omega} ) ^{{-1} / {2}} \ll k\ll r _{0} ^{- 1} $$
(17.11)

can be described as n(k)=C/k 4 [32], and Tan dubbed C as contact. Since the number of atoms with the wavenumber larger than K C is given by N(|k|>K C)=C/2π 2 K C, the contact may be interpreted as a measure of the number of high-momentum atoms [33]. The contact is also related to the density-density correlation of atoms and can be considered as a measure of the number of pairs of atoms in a small volume [33].

The universal properties of interacting fermions have been experimentally confirmed by several groups. Figure 17.3(a) shows the measurement of the momentum distribution of a unitary Fermi gas performed at JILA [34]. The magnetic field was tuned quickly to switch off the atomic interaction when the atoms were released to measure the momentum distribution. The JILA group confirmed that the high-momentum tail of the distribution has the predicted k −4 dependence and obtained the contact from the asymptotic behavior of the rf spectrum. Figure 17.3(b) shows the frequency dependence of the transition rate in the RF spectroscopy [34]. The ν −3/2 dependence of the line shape of the RF spectrum at the high-frequency region is another consequence of the universal behavior. Since the static structure factor is the Fourier transform of the density-density correlation which can be described in terms of the contact [33], the static structure factor S(k) can also be used to test the universal behavior. Figure 17.3(c) shows the measurement of S(k) using the Bragg spectroscopy performed by the Swinburne group [35]. The 1/k dependence of S(k) shows the predicted universal behavior.

Fig. 17.3
figure 3

Experimental confirmation of the universal behavior of a strongly interacting Fermi gas. (a) Momentum distribution of a unitary Fermi gas measured at JILA using 40K. The high-momentum tail shows the k −4 dependence. (b) RF spectrum showing the ν −3/2 dependence at the high-frequency region. (c) Static structure factor measured by the Swinburne group, showing the 1/k dependence. (a) and (b) are taken from [34] and (c) is taken from [35] with permissions

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Although the contact depends on the scattering length, atomic density and temperature, the relations between thermodynamic quantities and the contact are universal. A prime example that has been verified experimentally is the virial theorem [34]. For interacting particles in a harmonic trap, the kinetic energy T, the interaction energy I and the potential energy V satisfy the following equation:

$$ T+I-V=- {C} / {4\pi k _{\mathrm{F}} a}. $$
(17.12)

The virial theorem for the unitary gas mentioned in the previous section is T+I−V=0, which is given as the special case of (k F a)−1=0. Figure 17.4 shows the experimental verification of the virial theorem over a range of the scattering length. The left- and right-hand side of (17.12) are plotted as closed and open circles. Those two quantities agree within the uncertainty, showing the universality of the system. The JILA group also tested the adiabatic sweep theorem [34]. Tan relations show that many thermodynamic quantities are related to one another through the contact, and it will therefore be useful to derive thermodynamic quantities which cannot be measured directly from the experiment.

Fig. 17.4
figure 4

The left- and right-hand sides of (17.12) are plotted as closed and open circles, respectively. Those two quantities agree within the experimental uncertainty. Figure is taken from [34] with permission

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17.3 Experimental Determination of Universal Thermodynamics

17.3.1 Determination of Universal Energy Function E(T)

One way to determine the universal thermodynamic functions is to measure the local energy density [36]. As mentioned in Sect. 17.2.2, the local pressure of a trapped gas is related to the internal energy through (17.5) at the unitarity limit. Therefore, we can determine the local energy by measuring the local pressure in a gas. At thermal equilibrium, (17.6) can be used to determine the local pressure P(r) from the information of the trapping potential U trap(r) and the atomic density profile n(r). By relating f E (r)=3P(r)/2n(r)E F(r) to the reduced temperature T/T F at the same position, we determine the universal energy function f E (T/T F). In fact, we can determine f E (T/T F) over a wide range of temperature from a single density profile because it contains the information of the universal functions ranging from \(f _{E} ( {T} / {T _{\mathrm{F}} ^{0}} )\) at the cloud center to f E (∞) at the edge of the cloud, where \(T _{\mathrm{F}} ^{0}\) is the Fermi temperature at the center of the cloud.

The degenerate Fermi gas of 6Li are prepared in the two lowest spin states in an optical dipole trap, and the magnetic field of 834 gauss is applied to realize the unitary gas condition. The temperature of the gas is controlled by the final trap depth of evaporative cooling, and the gas is held until the system reaches thermal equilibrium. The atomic density distribution n(r) is determined from the absorption image taken perpendicular to the axial direction after a 3 ms free expansion at the same magnetic field. From the image, we construct an in situ three-dimensional atomic density distribution under the assumption of local density approximation (LDA) and hydrodynamic expansion. The temperature T is determined by using thermometry for the trapped unitary Fermi gas, which allows us to estimate T/T F,trap from E total/E F,trap [17, 37]. Here, \(E _{\mathrm{F},\mathrm {trap}} = k _{\mathrm{B}} T _{\mathrm{F},\mathrm{trap}} = \hbar\,\overline{\omega} ( 3 N ) ^{1/3}\) is the Fermi energy in the trap and \(E _{\mathrm{total}} =3 m \omega_{z} ^{2} \langle z ^{2} \rangle\) is the total energy per particle. The absolute temperature T is obtained by multiplying the given T/T F,trap by T F,trap.

Since the thermodynamic function of an ideal Fermi gas also has the universal form and obeys (17.5) and (17.6), we first applied the measurement scheme mentioned above to an ideal Fermi gas to check the validity of the scheme. 50 profiles taken at 526 gauss, where the scattering length is zero, are analyzed according to the above procedure. The green open circles in Fig. 17.5 show the experimentally obtained thermodynamic function of the internal energy for an ideal Fermi gas \(f _{E} ^{\mathrm{ideal}} ( {T} / {T _{\mathrm{F}}} )\). The experimental data show excellent agreement with the theoretical curve (green curve), which indicates that we have successfully determined the thermodynamic function for an ideal Fermi gas.

Fig. 17.5
figure 5

Universal functions of the internal energy (f E (T/T F)=E/NE F) for an ideal Fermi gas (green open circles) and for a unitary Fermi gas (red dots). The green curve shows the theoretical universal function for an ideal Fermi gas

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Next, the same scheme is applied to the unitary Fermi gas. By analyzing 800 measured profiles, we determined f E (T/T F) for various trap geometries and temperatures, and confirmed that f E (T/T F) determined by this method is independent of the trap geometry. The red dots in Fig. 17.5 show the experimentally determined f E (T/T F) for the unitary Fermi gas. Because of the effective attractive interaction at the unitarity limit, f E (T/T F) for the unitary Fermi gas is smaller than \(f _{E} ^{\mathrm{ideal}} ( {T} / {T _{\mathrm{F}}} )\).

Now that the universal function of the internal energy is determined, other thermodynamic functions for the unitary gas can also be determined from the energy function. The universal function of the Helmholtz free energy, chemical potential, and entropy are derived from the standard thermodynamic relations as \(f _{E} ( {T} / {T _{\mathrm{F}}} ) = f _{F} ( {T} / {T _{\mathrm {F}}} ) - {T} / {T _{\mathrm{F}}} f _{F} ' ( {T} / {T _{\mathrm{F}}} )\), \(f _{\mu } ( {T} / {T _{\mathrm{F}}} ) = \{ 5 f _{F} ( {T} / {T _{\mathrm{F}}} ) - 2 {T} / {T _{\mathrm{F}}} f _{F} ' ( {T} / {T _{\mathrm{F}}} ) \} /3\), and \(f _{S} ( {T} / {T _{\mathrm{F}}} ) = - f _{F} ' ( {T} / {T _{\mathrm{F}}} )\), respectively, and plotted in Fig. 17.6.

Fig. 17.6
figure 6

Universal functions for (a) the Helmholtz free energy, (b) chemical potential, and (c) entropy. The red dots show the measured data for a unitary Fermi gas, and the green solid curves show the corresponding functions for an ideal Fermi gas

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17.3.2 Determination of the equation of state P(μ,T)

Nascimbène et al. determined the equation of state P(μ,T) for a unitary Fermi gas of 6Li [38]. They determined the local pressure of the gas from the in situ absorption images, following the proposal by Ho and Zhou [39] who showed that the local pressure is related to the doubly-integrated density profiles:

$$ P \bigl( 0,0,z;\mu( 0,0,z ),T \bigr) = \frac{m \omega_{x} \omega _{y}}{2\pi} \overline{n} ( z ), $$
(17.13)

where P(0,0,z;μ,T) is the local pressure in the gas on the z-axis, m is the atomic mass, ω x and ω y are the angular frequencies of the trap in the x and y directions, and \(\overline{n} ( z ) = \int n ( x,y,z ) \mathrm{d} x \mathrm{d} y\) is the doubly-integrated density with n(x,y,z) being the atomic density. They trapped a small number of 7Li atoms together with 6Li, made both atomic species thermally equilibrated with each other, and determined the temperature of 6Li from that of 7Li which was measured by the time-of-flight measurement. To determine the chemical potential, they started with the data at a high-temperature region, where the second-order virial expansion can be used as a reference. Once the equation of state for the high-temperature cloud is determined, it can be used as a new reference for a colder gas in determining the chemical potential. By iterating this procedure, they were able to determine the equation of state over a wide range of temperature.

The pressure of the gas can be described as

$$ P ( \mu_{1}, \mu_{2},T ) = P _{\mathrm{ideal}} ( \mu _{1},T ) h \biggl( \eta= \frac{\mu_{1}}{\mu_{2}},\zeta= \exp\biggl ( - \frac{\mu_{1}}{k _{\mathrm{B}} T} \biggr) \biggr), $$
(17.14)

where μ 1 and μ 2 are the chemical potentials for two different species and P ideal(μ 1,T) is the pressure of an ideal Fermi gas. The equation of state for a spin-balanced gas h(1,ζ) is plotted as black dots in Fig. 17.7. From this data, they derived the third and fourth virial coefficients and compared them with theoretical predictions. They analyzed the data to focus on the temperature dependence of the pressure by plotting P(μ,T)/P ideal(μ,0) as a function of (k B T/μ)2. The data show the T 2 dependence of the pressure from the high-temperature side down to (k B T/μ)2∼0.1. The T 2 dependence of the pressure above the critical temperature is consistent with the Fermi liquid behavior. By fitting the data with Fermi liquid theory, they determined the Fermi liquid parameters. They observed the deviation of the data from the T 2 dependence at (k B T/μ)2∼0.1, and interpreted it as a signature of the superfluid phase transition.

Fig. 17.7
figure 7

Equation of state for a unitary Fermi gas h(1,ζ) measured by the ENS group [38]. Black dots show the experimental data, and the red dashed and solid curves show the result of the second- and fourth-order virial expansion

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Now that there are two independent sets of experimental results on the thermodynamics, we can compare them and check the consistency between them. The result discussed in Sect. 17.3.1 is the canonical equation of state E(n,T) and the result mentioned in this section is the grand-canonical equation of state P(μ,T). By converting P(μ,T) obtained by the ENS group to E(n,T) [40], we can directly compare these two results. Figure 17.8 shows the thermodynamic function f E [T/T F]=E/NE F obtained by the Tokyo group (red dots) and the ENS group (blue squares). The two data sets are consistent with each other.

Fig. 17.8
figure 8

Comparison of f E [T/T F]. Red dots show f E [T/T F] measured by the Tokyo group [36] and blue squares show f E [T/T F] measured by the ENS group [38, 40]

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17.4 Signatures of the Superfluid Phase Transition

17.4.1 Detection of the Phase Superfluid Transition

To detect an emergence of a condensate, we release an atomic cloud from a trap to measure its momentum distribution and observe the bimodal distribution which is composed of a zero-momentum component at the center and a thermal component surrounding it. This technique can be used to probe the condensation of Bose gases but cannot be used for a fermion pair condensate except in the deep BEC regime. This is because the fermion pairs are formed due to the many-body effect and therefore pairs are broken as the density of atoms decreases during the expansion. Therefore, to detect a condensation in a Fermi gas, it is necessary to measure the center-of-mass momentum distribution of the pairs. In 2004, Regal et al. demonstrated a new scheme to detect the condensation of fermions using 40K [6]. When they release the Fermi gas, they sweep the magnetic field from the BCS-BEC crossover region to the deep BEC regime to convert correlated fermion pairs into tightly-bound molecules. When the magnetic field sweep is slow enough to satisfy the adiabatic condition for the atom pairs to follow the two-body bound state, and fast enough to ensure that collisions between atoms can be neglected within the sweep time, atom pairs are efficiently converted into tightly-bound dimers (Fig. 17.9(a)). By applying the magnetic field sweep just before expansion of the gas, we are able to measure the center-of-mass momentum distribution of atom pairs. Figure 17.9(b) shows a typical center-of-mass momentum distribution of atom pairs obtained in the actual experiment. An emergence of the condensate can clearly be seen when the temperature is lower than the critical temperature. The onset of superfluidity can be identified by observing a sudden change in the thermodynamic properties. The changes in the frequency and damping rate of collective excitations were extensively studied to observe a superfluid hydrodynamic behavior. However, since the atoms near a Feshbach resonance are strongly interacting, they show collisional hydrodynamic behavior even at temperature higher than T c. It was difficult to distinguish between superfluid and collisional hydrodynamics. In 2005, the Duke group determined the superfluid transition point as a change in the heat capacity of a unitary Fermi gas [13]. More recently, the ENS group measured the pressure as a function of (k B T/μ)2 and observed the deviation from a Fermi liquid behavior at T c. Very recently, the MIT group measured the compressibility and the specific heat as a function of temperature for a homogeneous gas and observed the superfluid lambda transition at T c [19].

Fig. 17.9
figure 9

Schematic illustration of converting pairs of fermions into tightly-bound dimers. (a) To measure the center-of-mass momentum of correlated atom pairs at the Feshbach resonance B 0, the magnetic field is swept to the BEC side of the resonance as indicated by the green arrow. (b) Time-of-flight images taken after the magnetic field sweep. Top image shows the center-of-mass momentum of atom pairs at T c, and the lower ones were taken at lower temperatures, where the central peak shows the condensate

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17.4.2 Measurements of Critical Parameters

Figure 17.10 shows the condensate fraction as a function of the ratio of the total energy to the Fermi energy E F,trap in a harmonic trap [36]. The condensate fraction was determined by fitting the momentum distribution of molecules measured after the magnetic-field sweep with a bimodal distribution. In this measurement, the sweep time required to leave the strongly interacting regime is 2 μs. This time scale is much shorter than the typical relaxation time of 500 μs. Therefore, the condensate fraction is not likely to change noticeably during the sweep.

Fig. 17.10
figure 10

Condensate fraction plotted against the ratio of the total energy to the Fermi energy in a harmonic trap. The critical point is identified as the point where the condensate fraction starts to increase from zero

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In our experiment [36], the critical total energy is determined to be E c/E F,trap=0.76(1) and the reduced temperature is determined to be T c/T F=0.17(1) from the peak atomic density and the cloud temperature. The universal functions have the critical values of f E [T c/T F]=0.32(2), f F [T c/T F]=0.21(1), f μ [T c/T F]=0.42(2), and f S [T c/T F]=0.7(2). The experimental and theoretical results of the critical temperature, energy, chemical potential and entropy obtained by several groups are listed in Table 17.1.

Table 17.1 Experimental and theoretical superfluid critical parameters of a unitary Fermi gas
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17.4.3 Fermi Liquid vs. Non-Fermi Liquid

In high-T c materials, a gap structure in the single-particle density of states has been observed above the critical temperature. This phenomenon is known as a pseudo-gap, the origin of which has long been debated. Among the possible scenarios for explaining the pseudo-gap phenomenon is pre-formed pairing which implies that formation of pairs and emergence of the superfluidity occur at different temperatures. Identification of the microscopic mechanism of pairing and characterization of the pseudo-gap phase are important for understanding the superfluidity in a strongly-interacting Fermi gas.

The JILA group conducted a photoemission spectroscopy of 40K atoms in a strongly-interacting regime [42, 43]. By resolving momentum and energy at the same time, which is analogous to the angle-resolved photoemission spectroscopy, they observed a change in the dispersion relation of a strongly-interacting Fermi gas from above to below the superfluid critical temperature. An RF field was applied to drive one of the two components of the Fermi gas to a third state that has negligible interactions with the original two states. The dispersion relation of atoms in the third state is then the same as the one for a free particle, and therefore it is possible to measure the energy of a unitary Fermi gas from the resonant frequency of the applied RF field. Furthermore, they released the atoms at the same time and measure the momentum of atoms in the third state by reconstructing the three-dimensional momentum distribution from the time-of-flight images. This technique cannot be applied to 6Li atoms because there is no useful third state with negligible interactions with the initial states. The JILA group measured the temperature dependence of the dispersion relation by the photoemission spectroscopy and observed the back-bending of the dispersion around k∼k F, which is a characteristic behavior of the BCS-type dispersion relation even above the critical temperature. They concluded that the pseudo-gap theory based on the existence of a finite excitation gap due to preformed pairs describes the properties of a strongly-interacting Fermi gas.

On the other hand, Nascimbène et al. observed the T 2 dependence of the pressure, which is a characteristic feature of the Fermi liquid above the critical temperature [38]. The MIT group also measured the temperature dependence of the magnetic susceptibility of a unitary Fermi gas and showed that their result is consistent with no excitation gap above the critical temperature, which is consistent with Fermi liquid theory [44]. It seems that the fundamental understanding of the unitary Fermi gas above T c is still elusive, as in high-T c superconductivity.

17.5 Summary and Outlook

In this chapter, we have reviewed the recent developments on the thermodynamics of a unitary Fermi gas. Experimental studies on the thermodynamics using ultracold atoms had been conducted through trap-averaged thermodynamic quantities. Recently, it has become possible to determine local thermodynamic quantities of harmonically trapped gases. Universal thermodynamic functions and the equation of state of a unitary Fermi gas have thus been quantitatively determined. Such experimental advances have enables us to study thermodynamics of a homogeneous gas and to directly compare experimental results with theories.

A future challenge is to improve the accuracy of the measurement of the thermodynamics. In this direction, Ku et al. recently performed the measurement of the equation of state of a unitary Fermi gas without any external data input [19]. In their work, clear thermodynamic signatures of the superfluid phase transition have been identified via high precision measurement of the compressibility and specific heat. Another challenge will be to extend the measurement of the thermodynamics to a Fermi gas away from unitarity. ENS reported the experimental determination of the thermodynamics of a Fermi gas in the BCS-BEC crossover regime at the low-temperature limit. Measurements of thermodynamic properties over a broad temperature range in the crossover regime will be the next challenge. New schemes that can be applied at finite temperature need to be developed.

As for thermodynamic measurements on a Bose gas near unitarity, Papp et al. measured the excitation spectra of a strongly-interacting Bose gas of 85Rb using Bragg spectroscopy, and observed the beyond mean-field effect in the spectra [45]. More recently, the ENS group observed the equation of state of a Bose gas of 7Li with the scattering length of 2000a 0 [46]. The same group also determined the upper bound of the universal parameter for a Bose gas in a non-equilibrium condition. However, the thermodynamics of bosons at the unitarity limit is not yet fully understood because of the strong inelastic loss near Feshbach resonances.