Abstract
We report direct measurements of average vibration energy in a high frequency flexural resonance mode achieved via an-harmonic elastic coupling to a fundamental vibration mode of a nanomechanical resonator. The second order coupling effect produces a frequency shift of the read-out mode as a function of the mean square of the excitation amplitude of the high order mode. We measure frequency shifts at the lowest driving amplitudes, down to the noise floor of the experimental setup. With implementation of existing ultra-sensitive amplifiers, the reported technique will enable direct measurements of quantized energy transitions in low-thermal occupation number nanomechanical resonators.