  We give a detailed account of a stationary impurity in an ideal Bose-Einstein
condensate, which we call the ideal Bose polaron, at both zero and non-zero
temperatures and arbitrary strength of the impurity-boson coupling. The time
evolution is solved exactly and it is found that, surprisingly, many of the
features that have been predicted for the real BEC are already present in this
simpler setting and can be understood analytically therein. We obtain explicit
formulae for the time evolution of the condensate wave function at $T=0$ and of
the one-particle density matrix at $T>0$. For negative scattering length, the
system is found to thermalize even though the dynamics are perfectly coherent.
The time evolution and thermal values of the Tan contact are derived and
compared to a recent experiment. We find that contrary to the Fermi polaron,
the contact is not bounded at unitarity as long as a condensate exists. An
explicit formula for the dynamical overlap at $T=0$ allows us to compute the rf
spectrum which can be understood in detail by relating it to the two-body
problem of one boson and the impurity.
