  A unique feature of ultracold atoms is the separation of length scales,
$r_0\ll k_F^{-1}$, where $k_F$ and $r_0$ are the Fermi momentum characterizing
the average particle distance and the range of interaction between atoms
respectively. For $s$-wave scattering, Shina Tan discovered that such
diluteness leads to universal relations, all of which are governed by contact,
among a wide range of thermodynamic quantities. Here, we show that the concept
of contact can be generalized to an arbitrary partial-wave scattering. Contact
of all partial-wave scatterings form a contact spectrum, which establishes
universal thermodynamic relations with notable differences from those in the
presence of $s$-wave scattering alone. Moreover, such a contact spectrum has an
interesting connection with a special bipartite entanglement spectrum of atomic
quantum Hall states, and enables an intrinsic probe of these highly correlated
states using two-body short-ranged correlations.

Quantum Gases