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

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.

Figure 1

(a) Geometry of the Bragg beams and level scheme of the $2^3{S}_1-2^3{P}_0$ transition of ${}^4\mathrm{He}$ (at 1083 nm). A Bragg pulse of two $\pi$-polarized laser beams (shown by the two arrows) detuned by $\Delta /2\pi =600\mathrm{MHz}$ produces two counterpropagating BECs that separate along their radial dimension at approximately 45° to the vertical ($z$) axis at relative velocity $2v_0$. (b) Schematic diagram of the collision geometry in the center-of-mass frame in which we denote the collision axis as $Z$. The two disks represent the colliding condensates in momentum space. The sphere represents the halo of scattered atoms. The cigar shaped initial condensate with axial direction $X=x$ is shown in the center. We analyze the experimental data in the $XY$ plane.

Figure 2

(a) Average momentum space density $n(k_X,k_Y)$ (in arb. units, from $\sim 1500$ experimental runs) of the experimentally observed scattering halo on the equatorial plane ($k_X$, $k_Y$); the density is averaged over a disk of thickness [$-0.1k_0$, $+0.1k_0$] along $k_Z$. (b) Same as in (a) but from the positive-$P$ simulation (see text) after $70\mu \mathrm{s}$ collision time, in units of ${10}^{-18}{\mathrm{m}}^3$. (c) Plot of the peak radius of the scattering halo on the equatorial plane versus the azimuthal angle $\varphi$. Black squares are experimental data, while the red circles are from the simulation. The data is binned into 18 angular bins of $\Delta \varphi =20°$, and each data point for the peak radius is derived from a Gaussian fit to the radial distribution $n(k_R,\varphi )\equiv n(k_X=k_R\cos \varphi ,k_Y=k_R\sin \varphi )$ at the respective angle $\varphi$ (the error bars show the statistical uncertainty in the fits; in addition, there is a systematic uncertainty of $\pm 1.5\%$ in the determination of the average radius of the sphere). The smooth line is a sinusoidal fit to the experimental data.

Figure 3

Predictions for the peak radius of the scattering halo as in Fig. 2c, after the end of the collision ($72\mu \mathrm{s}$), with various controlled changes. Red-●: full positive-$P$ calculation, Eq. (2) [same as in Fig. 2c]; Black-□: anisotropic Bogoliubov calculation, Eq. (5); Blue-◃: anisotropic Bogoliubov, but with mean-field potentials $\propto g|{\psi }_0{|}^2$ removed from Eq. (5) and from the GP equation for ${\psi }_0(\mathbf{x},t)$; Green-×: full Bogoliubov, but with spherical BECs and unchanged peak density $\rho (0)$ ($200\mu \mathrm{s}$).