  We investigate the statistics of the fluctuations of the energy transfer
between an overdamped Brownian particle, whose motion is confined by a
stationary harmonic potential, and a surrounding viscoelastic fluid at constant
temperature. We derive an analytical expression for the probability density
function of the energy exchanged with the fluid over a finite time interval,
which implicitly involves the friction memory kernel that encodes the coupling
with such a non-Markovian environment, and reduces to the well known expression
for the heat distribution in a viscous fluid. We show that, while the odd
moments of this distribution are zero, the even moments can be explicitly
expressed in terms of the auto-correlation function of the particle position,
which generally exhibits a non-mono-exponential decay when the fluid bath is
viscoelastic. Our results are verified by experimental measurements for an
optically-trapped colloidal bead in semidilute micellar and polymer solutions,
finding and excellent agreement for all time intervals over which the energy
exchange takes place.
