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 ($T_Csim 0.5K$ -- $0.8K$) phase (previously interpreted as a 3D magnetic ordering) as a new type of incommensurate gapped state.
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