  Within the Schwinger-Keldysh formalism we derive a Ginzburg-Landau theory for
the Bose-Hubbard model which describes the real-time dynamics of the complex
order parameter field. Analyzing the excitations in the vicinity of the quantum
phase transition it turns out that particle/hole dispersions in the Mott phase
map continuously onto corresponding amplitude/phase excitations in the
superfluid phase, which have been detected recently by Bragg spectroscopy
measurements.

Quantum Gases