  From its discovery, the WASP-18 system with its massive transiting planet on
a tight orbit was identified as a unique laboratory for studies on tidal
planet-star interactions. In an analysis of Doppler data, which include five
new measurements obtained with the HIRES/Keck-I instrument between 2012 and
2018, we show that the radial velocity signal of the photosphere following the
planetary tidal potential can be distilled for the host star. Its amplitude is
in agreement with both theoretical predictions of the equilibrium tide
approximation and an ellipsoidal modulation observed in an orbital phase curve.
Assuming a circular orbit, we refine system parameters using photometric time
series from TESS. With a new ground-based photometric observation, we extend
the span of transit timing observations to 28 years in order to probe the rate
of the orbital period shortening. Since we found no departure from a
constant-period model, we conclude that the modified tidal quality parameter of
the host star must be greater than 3.9x10^6 with 95% confidence. This result is
in line with conclusions drawn from studies of the population of hot Jupiters,
predicting that the efficiency of tidal dissipation is 1 or 2 orders of
magnitude weaker. As the WASP-18 system is one of the prime candidates for
detection of orbital decay, further timing observations are expected to push
the boundaries of our knowledge on stellar interiors.
