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Within the Schwinger-Keldysh formalism we derive a Ginzburg-Landau theory forthe Bose-Hubbard model which describes the real-time dynamics of the complexorder parameter field. Analyzing the excitations in the vicinity of the quantumphase transition it turns out that particle/hole dispersions in the Mott phasemap continuously onto corresponding amplitude/phase excitations in thesuperfluid phase, which have been detected recently by Bragg spectroscopymeasurements.

Excitation Spectra of Bosons in Optical Lattices from Schwinger-Keldysh Calculation