The essential quantum many-body physics of an ultracold quantum gas relies on the single-particle Green's functions. We demonstrate that it can be extracted by the spectrum of electromagnetically induced transparency (EIT). The single-particle Green's function can be reconstructed by the measurements of frequency moments in EIT spectroscopy. This optical measurement provides an efficient and nondestructive method to reveal the many-body properties, and we propose an experimental setup to realize it. Finite temperature and finite size effects are discussed, and we demonstrate the reconstruction steps of Green's function for the examples of three-dimensional Mott-insulator phase and one-dimensional Luttinger liquid. (C) 2014 Optical Society of America
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