Exact Thermodynamics of High Partial-Wave Normal-Phase Quantum Gas: Equations of State, Contacts and Chemical Reaction Rate

While the thermodynamics for bosonic systems with weak $s$-wave interactions has been known for decades, a general and systematic extension to higher partial-waves has not yet been reported. We provide closed-form expressions for the equations of state for weakly interacting systems with arbitrary partial-waves in the normal phase. All thermodynamics, including contact, loss rate, compressibility, and heat capacity, can be derived over the entire temperature regime. To showcase its power, we calculated the contact as a function of the temperature, which agrees with literature both in the low and high temperatures. Using the virial expansion, we find that, while the contact of weakly-interacting $s$-wave Bose gases in the normal phase is a pure two-body quantity, that of weakly-interacting $p$-wave Fermi gases displays pronounced three-body effects even at temperatures as high as the degeneracy temperature. This effect is shown to arise from many-body dressing, i.e., the emergence of quasi-particles at leading order in the interaction strength. Our results offer an improved thermometer for ultracold atoms and molecules with weak high-partial wave interactions.