  Specific heat ($C_V$) measurements in the spin-1/2
Cu$_2$(C$_2$H$_{12}$N$_2$)$_2$Cl$_4$ system under a magnetic field up to
$H=8.25 T$ are reported and compared to the results of numerical calculations
based on the 2-leg antiferromagnetic Heisenberg ladder. While the temperature
dependences of both the susceptibility and the low field specific heat are
accurately reproduced by this model, deviations are observed below the critical
field $H_{C1}$ at which the spin gap closes. In this Quantum High Field phase,
the contribution of the low-energy quantum fluctuations are stronger than in
the Heisenberg ladder model. We argue that this enhancement can be attributed
to dynamical lattice fluctuations. Finally, we show that such a Heisenberg
ladder, for $H>H_{C1}$, is unstable, when coupled to the 3D lattice, against a
lattice distortion. These results provide an alternative explanation for the
observed low temperature ($T_C\sim 0.5K$ -- $0.8K$) phase (previously
interpreted as a 3D magnetic ordering) as a new type of incommensurate gapped
state.
