Analytical interaction dependence of the Anderson-orthogonality prefactor

Derive an explicit analytical expression for the dependence of the long-time Ramsey-signal prefactor C on the interaction parameter 1/(k_F a) for an infinitely heavy impurity in a Fermi sea.

Background

For an infinitely heavy impurity, the zero-temperature Ramsey signal has a long-time asymptotic form whose leading term is proportional to C exp(-i ΔE t)/(it/t_F)α. The prefactor C depends on the interaction parameter 1/(k_F a) and determines the prefactor \widetilde{C} in the power-law scaling of the renormalized Rabi splitting.

The paper states that the interaction dependence of C is currently obtained numerically by fitting the long-time tail of the functional-determinant-approach Ramsey signal. An analytical formula would therefore provide a closed-form characterization of the prefactor entering the Anderson-orthogonality scaling law for the driven impurity.

References

While we are not aware of explicit analytical expressions giving the dependence of $C$ in terms of the interaction parameter, it can be accessed numerically by fitting the long time tail of the $|S(t)|$ calculated using FDA with the analytical expression eq:Stail_T0.

— Anderson orthogonality scaling in the Rabi-driven heavy Fermi polaron  (2609.09129 - Rautenberg et al., 8 Sep 2026) in Section AOC-scaling prefactors at T=0, immediately following Eq. (S18) and before Fig. S2