  We discuss the long-time relaxation of a qubit linearly coupled to a finite
bath of $N$ spins (two-level systems, TLSs), with the interaction Hamiltonian
in rotating wave approximation. We focus on the regime $N\gg 1$, assuming that
the qubit-bath coupling is weak, that the range of spin frequencies is
sufficiently broad, and that all the spins are initialized in the ground state.
Despite the model being perfectly integrable, we make two interesting
observations about the effective system relaxation. First, as one would expect,
the qubit relaxes exponentially towards its zero-temperature state at a well
characterized rate. Second, the bath spins, even when mutually coupled, do not
relax towards a thermal distribution, but rather form a Lorentzian distribution
peaked at the frequency of the initially excited qubit. This behavior is
captured by an analytical approximation that makes use of the property $N\gg 1$
to treat the TLS frequencies as a continuum and is confirmed by our numerical
simulations.
