Spectroscopy for cold atom gases in periodically phase-modulated optical lattices

The response of cold atom gases to small periodic phase modulation of an optical lattice is discussed. For bosonic gases, the energy absorption rate is given, within linear response theory, by the imaginary part of the current autocorrelation function. For fermionic gases in a strong lattice potential, the same correlation function can be probed via the production rate of double occupancy. The phase modulation gives thus direct access to the conductivity of the system, as a function of the modulation frequency. We give an example of application in the case of bosonic systems at zero temperature and discuss the link between the phase and amplitude modulation.