Spontaneous Four-Wave Mixing of de Broglie Waves: Beyond Optics

Abstract

We investigate the atom-optical analog of degenerate four-wave mixing by colliding two Bose-Einstein condensates of metastable helium. The momentum distribution of the scattered atoms is measured in three dimensions. A simple analogy with photon phase matching conditions suggests a spherical final distribution. We find, however, that it is an ellipsoid with radii smaller than the initial collision momenta. Numerical and analytical calculations agree with this and reveal the interplay between many-body effects, mean-field interaction, and the anisotropy of the source condensate. The field of atom optics has developed to the point that one can now speak of the beginning of ''quantum-atom optics'' [1] in which atoms are manipulated in ways similar to photons and in which quantum fluctuations and entanglement play an important role. The demonstration of atom pair production [2,3], either from the dissociation of ultracold molecules, a process analogous to parametric downconversion [4-6], or from collisions of Bose-Einstein condensates (BECs) [7][8][9][10], analogous to four-wave mixing (FWM) [11][12][13][14][15][16][17][18][19][20][21], holds considerable promise for generating atomic squeezed states and demonstrating nonlocal Einstein-Podolsky-Rosen (EPR) correlations [4,5,22,23]. In both these systems, atom-atom interactions play the role of the nonlinear medium that allows conversion processes. Atoms are not, however, exactly like photons, and in spite of their formal similarity, the processes of pair production of photons and of atoms exhibit some interesting and even surprising differences that must be understood in order for the quantum-atom optics field to advance. In this work, we discuss one such effect.In optical FWM or parametric down-conversion [24], energy conservation requires that the sum of the energies of the outgoing photons be fixed by the energy of the input photon(s). Phase matching requirements impose constraints on the directions and values of the individual photon momenta. A simple case is degenerate, spontaneous FWM (i.e., two input photons of equal energy) in an isotropic medium, for which energy conservation and phase matching require that the momenta of the output photons lie on a spherical shell whose radius is that of the momenta of the input photons.We have performed the atom-optical analog of degenerate FWM in colliding BECs while paying careful attention to the momenta of the outgoing atoms. We find that unlike the optical case, the output momenta do not lie on a sphere, but rather on an ellipsoid with short radius smaller than the input momentum. This behavior is due to a subtle combination of atom-atom interactions, which impose an energy cost for pair production, and the anisotropy of the condensates, which affects the scattered atoms as they leave the interaction region.Although an analogous effect could exist in optics, optical nonlinearities are typically so small that the effect is negligible. However, in the process of high-harmonic generation in intense laser fields, a simil...

Matter Waves and Quantum Correlations

“…The figure also shows the results of a quantum-mechanical calculation of C using a stochastic Bogoliubov approach (green thick solid curve) [20,21,28]. The calculation is for the initial total number of atoms N ¼ 85 000 and is in good agreement with the observations.…”

“…K. V. K. is supported by the ARC FT100100285 grant, P. D. [21] ) calculated using the experimental parameters and a stochastic Bogoliubov approach [20,28].…”

“…Atoms interact via binary, momentum conserving s-wave collisions and scatter onto a nearly spherical halo [see Fig. 1(b)] whose radius in velocity space is about the recoil velocity [11,20]. The scattered atoms fall onto a detector that records the arrival times and positions of individual atoms [13] with a quantum efficiency of $10%.…”

Anisotropy ins-wave Bose-Einstein condensate collisions and its relationship to superradiance

“…This feature is counterintuitive when the process is viewed as four-wave mixing of matter waves, while a discussion in terms of particles and their kinetic and potential energies leads to a simple explanation of the observations [22][23][24]. In the present work, we discuss a different aspect of the same data, the anisotropy in the angular distribution of the number of scattered atoms in the halo.…”

“…In a previous paper, we reported a surprising variation of the radius of the collision halo with the scattering angle when elongated condensates collide [22]. This feature is counterintuitive when the process is viewed as four-wave mixing of matter waves, while a discussion in terms of particles and their kinetic and potential energies leads to a simple explanation of the observations [22][23][24].…”