Sub-Poissonian Number Differences in Four-Wave Mixing of Matter Waves

We demonstrate sub-Poissonian number differences in four-wave mixing of Bose-Einstein condensates of metastable helium. The collision between two Bose-Einstein condensates produces a scattering halo populated by pairs of atoms of opposing velocities, which we divide into several symmetric zones. We show that the atom number difference for opposing zones has sub-Poissonian noise fluctuations, whereas that of nonopposing zones is well described by shot noise. The atom pairs produced in a dual number state are well adapted to sub–shot-noise interferometry and studies of Einstein-Podolsky-Rosen–type nonlocality tests.

Figure 1

View of the halo after the collision of two BECs and a subsequent ballistic expansion. (a) The experimental data plotted in momentum space, with each dot corresponding to a detected atom. Atoms on the collision halo are black, while the colliding, pancake-shaped BECs at the top and the bottom of the halo are orange/yellow. The collision axis $v_z$ and the optical trap axis are both almost vertical. (b) Schematic view of the analyzed part ($|v_z|<0.5v_{\mathrm{rec}}$) of the collision halo. Here we use $N_Z=8$ zones that are separated from each other for better visualization. An example of two correlated zones is shown (red arrows). The number difference between these two zones shows sub–shot-noise fluctuations.

Figure 2

Variance of all possible pairs of zones for the halo cut into 16 zones and summing $N_s=3600$ shots. The normalized variance is $V_{i,j}$ and the error bars reflect its standard deviation $\delta V_{i,j}$ with $\delta V_{i,j}^2=\frac{1}{N_s}\frac{⟨(N_i-N_j{)}^4⟩-⟨(N_i-N_j{)}^2{⟩}^2}{⟨N_i+N_j⟩}$. Circles correspond to the eight correlated zones and crosses to the 112 uncorrelated ones. The two horizontal lines correspond to the mean of each data set with a thickness given by twice the standard deviation of the mean, considering each pair of zones as independent.

Figure 3

Observed variance, as a function of the number of zones into which we cut the halo. Red circles: average over all correlated zones; blue squares: average over all uncorrelated zones. Error bars show the standard deviation of the mean of the variances for a given $N_Z$. The solid curve is the prediction of the simple model discussed in the text. The dashed curve results from the stochastic Bogoliubov simulation. Both models assume a 12% quantum efficiency.