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We propose a detection method for ultracold atoms which allows reconstructionof the full one-particle and two-particle correlation functions from themeasurements. The method is based on Fourier sampling of the time-of-flightimages through two consecutive impulsive Raman pulses. For applications of thismethod, we discuss a few examples, including detection of phase separationbetween superfluid and Mott insulators, various types of spin or superfluidorders, entanglement, exotic or fluctuating orders.Comment: 4 pages, 2 figure

Detecting Correlation Functions of Ultracold Atoms through Fourier Sampling of Time-of-Flight Images