Amplitude and phase mode in a Bose Einstein condensate
  We show that starting from the Heisenberg equations of motion for Bose annihilation and creation operators and using an appropriate transformation, we can split the Bogoliubov mode into a free particle mode and the amplitude mode. We show this for both the free Bose gas as well as the Bose gas in an optical lattice. 
Aranya B. Bhattacherjee
cond-mat.quant-gas