Ground-State Thermodynamic Quantities of Homogeneous Spin-$1/2$ Fermions from the BCS Region to the Unitarity Limit

The understanding of physical properties of fermions in the unitary regime, where the $s$-wave scattering length in the collisional channel of particles is longer than both the interparticle distance and the size of the interaction potential, is a crucial issue for electron systems of high-temperature superconductivity, dilute nucleons in nuclei, and neutron stars. We experimentally determine various thermodynamic quantities of interacting two-component fermions at the zero-temperature limit from the BCS region to the unitarity limit. The obtained results are very accurate in the sense that the systematic error is within 4% in the unitary regime. Using this advantage, we can compare our data with various many-body theories. We find that an extended $T$-matrix approximation, which is a strong-coupling theory involving fluctuations in the Cooper channel, well reproduces our experimental results. We also find that the superfluid order parameter $\mathrm{\Delta }$ calculated by solving the ordinary BCS gap equation with the chemical potential of interacting fermions is close to the binding energy of the paired fermions directly observed in a spectroscopic experiment and that obtained using a quantum Monte Carlo method.

Popular Summary

When fermions—subatomic particles such as electrons—get together at temperatures approaching absolute zero, they can form a bizarre state of matter known as a superfluid, which is a fluid that flows with zero viscosity. Understanding such Fermi superfluids can provide insight into such diverse materials as superconductors and neutron stars. Currently, the relationship among thermodynamic quantities (such as temperature and pressure) in a Fermi superfluid and other parameters that describe interactions among particles is not fully understood. We report the first experiment to comprehensively determine the thermodynamic properties of fermions in a particular type of superfluid state and the first attempt to use those properties to calculate other parameters that describe the system.

Our experimental setup creates a Fermi superfluid from pairs of trapped, ultracold lithium atoms. By tuning the strength of the interactions from weakly interacting to strongly interacting and probing the superfluid with a laser, we are able to extract precise thermodynamic quantities of the fermions. Using these quantities, we calculate the superfluid gap, an important parameter because of its influence on the superfluid transition temperature and the ground-state energy. Surprisingly, the gap is close to the binding energy of the paired fermions observed in experiments and derived from computer simulations. Our thermodynamic measurements are accurate enough to compare several theories for how particles in the superfluid interact with one another.

Since understanding the strong-coupling properties of superfluid fermions is important for developing high-temperature superconductors as well as understanding both dilute nuclear matter and neutron stars, we expect our results to be useful in the further development of these fields.

Figure 1

Schematic diagram of the experimental apparatus. The $\widehat{z}$ axis is defined as the axial direction of the optical trap.

Figure 2

Typical experimental data at 882.86 G. (a) Absorption image. (b) Local pressure $P(U_{\text{trap}})$. (c) Local number density $n(U_{\text{trap}})$. The red curves in (b) and (c) show the data region used in the analysis, where the number density is larger than half of the peak density.

Figure 3

Experimental data of isothermal compressibility from the BCS region to the unitarity limit. The red circles are experimental data. The dotted curve and the dashed curve show their asymptotic behavior at the BCS limit up to the first and second order of $1/x$, respectively. The vertical dash-dotted line indicates interaction parameter $x^{*}$, where superfluid transition occurs. The fermions are in the superfluid state in the region of $x>x^{*}$.

Figure 4

Construction of dimensionless function $\mathcal{G}(\chi )$. The different colors indicate the contributions of data taken at each magnetic field for the Feshbach resonance. (a) Around the BCS limit. The dash-dotted red curve and the dashed red curve indicate the lower and upper limits of $\mathcal{G}(\chi )$. The gray areas correspond to the forbidden area for $\mathcal{G}(\chi )$. (i) Unconnected data with ${\mu }_0^B=0$ for all data. (ii) Connected data located in the forbidden area. (iii) Connected data for which the offset value is tuned appropriately. (b) Around the unitarity limit. (i) Unconnected data with ${\mu }_0^B=0$ for all data. (ii) Connected data for which the offset value is tuned appropriately.

Figure 5

Dimensionless thermodynamic functions derived from $\mathcal{G}(\chi )$. (a) Pressure. (b) Number density. (c) Internal energy density. (d) Chemical potential. (e) Contact density. (f) Canonical interaction parameter. The red colors indicate ranges of statistical errors caused by the uncertainty of the offset value of $\mathcal{G}(\chi )$. The blue colors indicate additional statistical errors resulting from the uncertainty of the absorption cross section. The dotted curves and the dashed curves show their asymptotic behavior at the BCS limit up to the first and second order of $1/x$, respectively. The vertical dash-dotted lines indicate interaction parameters $X^{*}$ and $x^{*}$ where superfluid transition occurs. The Fermi system is in the superfluid state in the region of $X>X^{*}$ or $x>x^{*}$.

Figure 6

Experimental and theoretical dimensionless isothermal compressibility. The vertical axis follows a logarithmic scale. The red circles indicate the experimental data of the present study. The green diamonds, blue triangles, and black square indicate experimental values obtained based on the speed of sound [21], density fluctuations [20], and thermodynamic measurements at the unitarity limit [19], respectively. The light blue curve indicates the theoretical calculation of ETMA [31, 32]. The vertical dash-dotted line indicates the superfluid transition point $x^{*}$ obtained in the present study.

Figure 7

Experimental and theoretical dimensionless pressure. The vertical axis follows a logarithmic scale. The red curve indicates the experimental data of the present study. The width indicates the total systematic error. The green circles and the black square indicate the experimental values obtained using spin imbalanced Fermi gases [18] and spin balanced unitary Fermi gas [19], respectively. The light blue curve indicates the theoretical calculation of the ETMA [31, 32]. The vertical dash-dotted line indicates the superfluid transition point $X^{*}$ of the present study.

Figure 8

Experimental and theoretical dimensionless internal energy density. The red curve indicates the experimental data of the present study in the superfluid state ($x>x^{*}$). The width indicates the total systematic error. The black square indicates the experimental data obtained using spin-balanced unitary Fermi gas [19], and the dotted green curve is an approximated curve based on pressure measurements of Ref. [18]. The light blue curve indicates the ETMA [31, 32]. The open brown squares indicate data obtained by the QMC method [33]. The long dashed green curve was obtained by the FNDMC method [34]. The dash-dotted purple curve was obtained by the NSR theory [35]. The dashed black curve was obtained by the LW approach [36].

Figure 9

Experimental and theoretical dimensionless contact density. The vertical axis follows a logarithmic scale. The red curve indicates the experimental data of the present study. The width indicates the total systematic errors. The blue triangles and the green inverse triangles indicate experimental values determined by radio frequency spectroscopy (rf line shape) and photon emission spectroscopy [17]. The light blue curve is the ETMA [31, 32]. The long dashed green curve indicates the results obtained by the FNDMC method [34]. The dash-dotted purple curve indicates the results obtained by the NSR theory [37]. The short dashed black curve indicates the LW approach [38]. The brown diamonds indicate the results obtained by TMA [39].

Figure 10

Dimensionless superfluid gap. The red curve indicates the gap calculated by substituting our $f_{\mu }(x)$ into the gap equation in Eq. (31). The dashed black curve was obtained by BCS-MF calculation [42]. The green triangle and the green inverse triangles indicate the theoretical values obtained by the QMC method [43, 44]. The blue circles are experimental values obtained by quasiparticle spectroscopy [14].

Figure 11

Condensate fraction of paired fermions evaluated at various magnetic fields for the Feshbach resonance. The circles with error bars indicate experimental data. The dashed curve shows the fitting result. The red square corresponds to the point at which the condensate fraction disappears.