  The functional determinant approach (FDA) is a simple method to compute
exactly certain observables for ideal quantum systems and has been successfully
applied to the Fermi polaron problem to obtain the dynamical overlap and
spectral function. Unfortunately, its application to Bosonic ultracold gases is
prohibited by the failure of the grand canonical ensemble for these systems. In
this paper, we show how to circumvent this problem and develop a Bosonic FDA.
This yields exact injection and ejection spectra for ideal Bose polarons at
arbitrary temperatures. While coherent features visible at absolute zero
quickly smear out with rising temperature as expected, the line width of the
main peak is, counterintuitively, found to decrease near unitarity.
Furthermore, we provide explicit formulas for the overlap operator, which allow
to compute the necessary determinants for both Bose and Fermi polarons more
efficiently than previously possible.
