The essential quantum many-body physics of an ultracold quantum gas relies onthe single-particle Green's functions.\ We demonstrate that it can be extractedby the spectrum of electromagnetically induced transparency (EIT).\ Thesingle-particle Green's function can be reconstructed by the measurements offrequency moments in EIT spectroscopy.\ This optical measurement provides anefficient and nondestructive method to reveal the many-body properties, and wepropose an experimental setup to realize it.\ Finite temperature and finitesize effects are discussed, and we demonstrate the reconstruction steps ofGreen's function for the examples of three-dimensional Mott-insulator phase andone-dimensional Luttinger liquid.
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