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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 imaginary part of the current correlation function. For fermionic gases in a strong lattice potential, the same correlation function can be probed via the production rate double occupancy. The phase modulation gives thus direct access to the conductivity of the system, as function of the modulation frequency. We give an example of application in the case of one dimensional bosons at zero temperature and discuss the link between the phase- and amplitude-modulation.