Correlations in Amplified Four-Wave Mixing of Matter Waves Wu RuGway, S. S. Hodgman, R. G. Dall, M. T. Johnsson, and A. G. Truscott * ARC Centre of Excellence for Quantum-Atom Optics and Research School of Physics and Engineering, The Australian National University, Canberra, ACT 0200, Australia (Received 3 January 2011; revised manuscript received 29 June 2011; published 9 August 2011) 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. DOI: 10.1103/PhysRevLett.107.075301 PACS numbers: 67.85.Fg, 03.75.Gg, 03.75.Pp The concept of second order coherence was first consid- ered by Hanbury-Brown and Twiss (HBT), who measured intensity correlations between separate thermal light sources [1]. Although initially controversial, as it required correlations between independent photons, the concept was placed on a sound theoretical footing by Glauber, who extended the idea of coherence to arbitrary orders [2]. Second order coherence is one of the simplest mea- surements that shows the difference between classical and quantum particles [3]; when applied to atoms it distin- guishes between Bose-Einstein condensation and thermal or chaotic sources and is a more stringent test of coherence. It has also been used to distinguish between bosonic [4,5] and fermionic sources [6,7]. These recent efforts to make correlation measurements on atoms are part of a trend to extend the well-tested techniques of quantum optics to quantum-atom optics. For example, the creation of nonclassical states of light exhibiting squeezing and entanglement is now routine, while recent advances in the field of quantum-atom optics are starting to allow the creation of nonclassical states of matter [8]. Fundamental tests of quantum mechanics that have until now only been possible with photons, such as nonlocality and the EPR paradox, are now close to being realized using massive particles [9,10]. A major goal of quantum-atom optics is to produce entangled (or at least correlated) pairs of atoms. One of the ways this can be accomplished is through atomic four- wave mixing (FWM) [11,12]. FWM in the atomic regime is achieved through the intrinsic nonlinearities in atomic interactions, in particular, atom pair collisions. The FWM process can be spontaneous or stimulated, resulting in atoms in numerous modes with low occupation or highly occupied amplified modes that result from bosonic en- hancement. Bosonic stimulation is a process whereby tran- sition rates into a particular mode are enhanced by other identical bosons already occupying that mode. Such an effect has also been used to demonstrate various processes including superradiance [13], the exponential growth of a Bose-Einstein condensate (BEC) [14], stimulated FWM [15], and the pumping of an atom laser [16]. Another difference between optical and atomic FWM is that while photons do not interact with each other, atoms emphatically do, leading to many sources of decoherence that are not present in their optical analogue. This is potentially a problem, as the coherence properties of mat- ter waves generated via bosonic stimulation are critically important for active atom optical devices, and it is not clear that methods of producing such atomic fields result in coherence to all orders. The coherence properties of matter waves produced in the FWM of atoms, or indeed any method involving bo- sonic stimulation, have only been tested to first order [13,17]. In this Letter we present the first higher order test of the coherence of amplified matter waves. We are able to access both the spontaneous regime, where we observe atom bunching due to the multimode nature of the process, and the transition to stimulated behavior where the correlation function is unity, indicating that the output modes are coherent to second order. The concept of coherence in the sense of classical optics refers to first order or phase coherence, meaning the tendency for two field values at separated points in space or time to acquire correlated values. This is evident when the fields are superimposed and show interference fringes. The work by Glauber [2] extends the notion of coherence by defining higher order correlation functions. The nth order correlation function expresses the correla- tion of field values at n points in time and space, and a wave is coherent to nth order if the condition that g ðnÞ ¼ 1 holds for all particle separations. Higher order coherence of matter waves can be tested using single atom detection to discern the arrival time and position of each individual atom. Using such methods the well- known HBT effect [1] has been demonstrated for both bosonic [4] and fermionic [7] atoms as well as third order coherence for a BEC [18]. The HBT effect deals with correlations in intensity fluctuations, using the normalized PRL 107, 075301 (2011) PHYSICAL REVIEW LETTERS week ending 12 AUGUST 2011 0031-9007= 11=107(7)=075301(4) 075301-1 Ó 2011 American Physical Society