In 1963 Glauber introduced the modern theory of quantum coherence 1 , which extended the concept of first-order (onebody) correlations, describing phase coherence of classical waves, to include higher-order (n-body) quantum correlations characterizing the interference of multiple particles. Whereas the quantum coherence of photons is a mature cornerstone of quantum optics, the quantum coherence properties of massive particles remain largely unexplored. To investigate these properties, here we use a uniquely correlated 2 source of atoms that allows us to observe n-body correlations up to the sixthorder at the ideal theoretical limit (n!). Our measurements constitute a direct demonstration of the validity of one of the most widely used theorems in quantum many-body theory-Wick's theorem 3 -for a thermal ensemble of massive particles. Measurements involving n-body correlations may play an important role in the understanding of thermalization of isolated quantum systems 4 and the thermodynamics of exotic many-body systems, such as Efimov trimers 5 .Glauber's modern theory of optical coherence and the famous Hanbury Brown-Twiss effect 6 were pivotal in the establishment of the field of quantum optics. Importantly, the definition of a coherent state required coherence to all orders, which for example distinguishes a monochromatic but incoherent thermal source of light from a truly coherent source such as a laser. Higher-order correlation functions therefore provide a more rigorous test of coherence.Higher-order correlations, characterized by an n-body correlation function g (n) , are of general interest and have been investigated in many fields of physics including astronomy 6 , particle physics 7 , quantum optics 8 , and quantum atom optics 9 . In particular they have been a fruitful area of research in the field of quantum optics, where they have been used to investigate the properties of laser light, including heralded single photons 10 , and the statistics of parametric down-conversion sources 11 . State-of-the-art quantum optics experiments have measured photon correlation functions up to sixth order for quasi-thermal sources 8 , allowing the possibility of performing full quantum state tomography 12 .Higher-order correlations experiments with massive particles are currently approaching the same level of maturity as with photons. So far, experiments have directly observed correlations up to fourth order with single-atom-sensitive detection techniques for ultracold atomic bosons 9,13,14 , and second-order correlations for an atomic source of fermions 15 demonstrating the uniquely quantum mechanical property of atom-atom antibunching. Alternative, indirect techniques have also been employed to investigate higher-order correlations, including the measurements of twobody (photoassociation 16 ) and three-body 17 loss rates that are sensitive, respectively, to second-and third-order correlation functions. Interestingly, fermionic atom pairs 18 and fermionic antibunching 19 have also been observed in the atomic shot no...

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The coherence properties of amplified matter waves generated by four-wave mixing (FWM) are studied using the Hanbury-Brown-Twiss method. We examine two limits. In the first case stimulated processes lead to the selective excitation of a pair of spatially separated modes, which we show to be second order coherent, while the second occurs when the FWM process is multimode, due to spontaneous scattering events which leads to incoherent matter waves. Amplified FWM is a promising candidate for fundamental tests of quantum mechanics where correlated modes with large occupations are required.

A fundamental property of a three-dimensional Bose-Einstein condensate is long-range coherence; however, in systems of lower dimensionality, not only is the long-range coherence destroyed but additional states of matter are predicted to exist. One such state is a "transverse condensate," first predicted by van Druten and Ketterle [Phys. Rev. Lett. 79, 549 (1997)], in which the gas condenses in the transverse dimensions of a highly anisotropic trap while remaining thermal in the longitudinal dimension. Here, we detect the transition from a three-dimensional thermal gas to a gas undergoing transverse condensation by probing Hanbury Brown-Twiss correlations.