  The rare-earth tritellurides ($R$Te$_3$, where $R$ = La, Ce, Pr, Nd, Sm, Gd,
Tb, Dy, Ho, Er, Tm, Y) form a charge density wave state consisting of a single
unidirectional charge density wave for lighter $R$, with a second
unidirectional charge density wave, perpendicular and in addition to the first,
also present at low temperatures for heavier $R$. We present a quantum
oscillation study in magnetic fields up to 65T that compares the single charge
density wave state with the double charge density wave state both above and
below the magnetic breakdown field of the second charge density wave. In the
double charge density wave state it is observed that there remain several
small, light pockets with the largest occupying around 0.5% of the Brillouin
zone. By applying magnetic fields above the independently determined magnetic
breakown field, the quantum oscillation frequencies of the single charge
density wave state are recovered, as expected in a magnetic breakdown scenario.
Measurements of the electronic effective mass do not show any divergence or
significant increase on the pockets of Fermi surface observed here as the
putative quantum phase transition between the single and double charge density
wave states is approached.
