Verification of Universal Relations in a Strongly Interacting Fermi Gas

Many-body fermion systems are important in many branches of physics, including condensed matter, nuclear, and now cold atom physics. In many cases, the interactions between fermions can be approximated by a contact interaction. A recent theoretical advance in the study of these systems is the derivation of a number of exact universal relations that are predicted to be valid for all interaction strengths, temperatures, and spin compositions. These equations, referred to as the Tan relations, relate a microscopic quantity, namely, the amplitude of the high-momentum tail of the fermion momentum distribution, to the thermodynamics of the many-body system. In this work, we provide experimental verification of the Tan relations in a strongly interacting gas of fermionic atoms by measuring both the microscopic and macroscopic quantities in the same system.

Figure 1

Extracting the contact from the momentum distribution and rf line shape. (a) Measured momentum distribution for a Fermi gas at $\frac{1}{k_{F}a}=-0.08\pm 0.04$. Here, the wave number $k$ is given in units of $k_F$, and we plot the normalized $n(k)$ multiplied by $k^4$. The dashed line corresponds to $2.2$, which is the average of $k^{4}n(k)$ for $k>1.85$. (Inset) The measured value for $C$ depends on the rate of the magnetic-field sweep that turns off the interactions before time-of-flight expansion. (b) rf line shape measured for a Fermi gas at $\frac{1}{k_{F}a}=-0.03\pm 0.04$. Here, $\nu$ is the rf detuning from the single-particle Zeeman resonance, given in units of $E_F/h$. We plot the normalized rf line shape multiplied by $2^{3/2}{\pi }^2{\nu }^{3/2}$, which is predicted to asymptote to $C$ for large $\nu$. Here, the dashed line corresponds to $2.1$, from an average of the data for $\nu >5$.

Figure 2

The contact. We measure the contact, $C$, as a function of $(k_{F}a{)}^{-1}$ using three different methods. Filled circles correspond to direct measurements of the fermion momentum distribution $n(k)$ using a ballistic expansion, in which a fast magnetic-field sweep projects the many-body state onto a noninteracting state. Open circles correspond to $n(k)$ obtained using atom photoemission spectroscopy measurements. Stars correspond to the contact obtained from rf spectroscopy. The values obtained with these different methods show good agreement. The contact is nearly zero for a weakly interacting Fermi gas with attractive interactions (left hand side of plot) and then increases as the interaction strength increases to the unitarity regime where $(k_{F}a{)}^{-1}=0$. The line is a theory curve obtained from Ref. 5.

Figure 3

Testing the adiabatic sweep theorem. (Inset) The measured potential energy, $V$, and release energy, $T+I$, per particle in units of $E_F$ are shown as a function of $1/k_{F}a$. (Main) Taking a discrete derivative of the inset data, we find that $2\pi \frac{dE}{d[-1/(k_{F}a)]}$ ($•$) agrees well with the average value of $C$ obtained from the measurements shown in Fig. 2 ($\circ$).

Figure 4

Testing the generalized virial theorem. The difference between the measured release energy and potential energy per particle $T+I-V$ is shown as filled circles. This corresponds to the left-hand side of Eq. (3). Open circles show the right-hand side of Eq. (3) obtained from the average values of the contact shown in Fig. 2. The two quantities are equal to within the measurement uncertainty.