Anisotropy in $s$-wave Bose-Einstein condensate collisions and its relationship to superradiance

We report the experimental realization of a single-species atomic four-wave mixing process with Bose-Einstein-condensate collisions for which the angular distribution of scattered atom pairs is not isotropic, despite the collisions being in the $s$-wave regime. Theoretical analysis indicates that this anomalous behavior can be explained by the anisotropic nature of the gain in the medium. There are two competing anisotropic processes: classical trajectory deflections due to the mean-field potential and Bose-enhanced scattering which bears similarity to superradiance. We analyze the relative importance of these processes in the dynamical buildup of the anisotropic density distribution of scattered atoms and compare the Bose enhancement effects to those in optically pumped superradiance.

Figure 1

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 after their mean-field-induced expansion. The sphere represents the halo of scattered atoms. The initial cigar-shaped condensate, whose long axis coincides with $x$, is shown in the center. We analyze the experimental data in the $x\text{−}y$ plane.

Figure 2

Experimental data (averaged over $\sim 1600$ experimental runs). (a) The momentum space density $n(k_x,k_y)$ of the observed scattering halo on the equatorial $k_x\text{−}{k}_y$ plane (in arbitrary units). It has been averaged over a disk of thickness $[-0.1k_0,+0.1k_0]$ along $k_z$, and the momentum is normalized to the collision momentum $k_0=m{v}_0/\hslash$. (b) Atom distribution in the halo as a function of the radial momentum $k_r$. We show the sum of counts over all runs for a single azimuthal sector of width $22.5^{\circ }$. The red line is the fit to Eq. (1).

Figure 3

Halo anisotropy in the experiment. (a) The angular distribution of the number of scattered atoms $N_i$ on the equatorial $k_x\text{−}{k}_y$ plane (calculated from fits to Eq. (1) after counting atoms over all runs). The solid red curve shows a sinusoidal fit to $N_i$, which is anisotropic and maximal along the direction of the long axis of the condensate at $0^{\circ }$ and ${180}^{\circ }$. (b) The fitted radial width $\delta k_i$ of the halo density as a function of the azimuthal angle ${\varphi }_i$. The solid blue curve shows a sinusoidal fit. In both panels, the angular bins (or azimuthal sectors), labeled by $i=1,2,\cdots ,16$, are centered at $0^{\circ },22.5^{\circ },{45}^{\circ },\cdots ,337.5^{\circ }$.

Figure 4

Halo properties in a classical test-particle model: (a) A scatterplot of the distribution of final momenta of classical test particles after their escape from the collision zone. (b) The angular distribution of the number of test particles $N_i$ in azimuthal sectors centered at angles ${\varphi }_i \left(i=1,2,\cdots ,16\right)$. The key observations are that the atom number depends weakly on the angle, but in the opposite manner to that observed in the experiment, and that the halo width exhibits a strong anisotropy that reflects the shape of the source in momentum space.

Figure 5

Direction-dependent properties of the halo from the stochastic Bogoliubov simulation on the $k_x\text{−}{k}_y$ plane after $96 \mu \mathrm{s}$ for $N={10}^5$ atoms. (a) The local peak density ${\mathcal{A}}_i$, (b) mean halo radius $K_i$, and (c) thickness $\delta k_i$ come from a fit of the linear density $n_i\left(k_r\right)$ in each sector (of width ${10}^{\circ }$) to Eq. (1) with no background $\left({\beta }_i={\alpha }_i=0\right)$. The sectors divide up a disk around the plane in the interval $k_z\in [-0.2k_0,+0.2k_0]$. (d) The number of scattered atoms $N_i \left(i=1,2,\cdots ,16\right)$ per azimuthal sector centered at ${\varphi }_i$ from a direct count (histogram) and from the simple estimate $\sqrt{2\pi }{\mathcal{A}}_i\delta k_i$ (purple circles). Statistical fluctuations are visible.

Figure 6

The relative anisotropy of the scattered atom number ${\Gamma }_N$ as a function of time $t$ from the stochastic Bogoliubov simulations. $N$ is the total atom number in the main cloud. Triple lines show the statistical uncertainty. The inset shows the angular variation of the scattered atom count $N_i \left(i=1,2,\cdots ,16\right)$ per azimuthal sector centered at ${\varphi }_i$ for the highest density case at the end of the collision as per Fig. 5.

Figure 7

The number of scattered atoms $N_i \left(i=1,2,\cdots ,16\right)$ per azimuthal sector centered at ${\varphi }_i$ at the earlier time $t=64 \mu \mathrm{s}$, from the stochastic Bogoliubov simulation. Other details are as in Fig. 5.

Figure 8

A schematic comparison of the scattering geometry in (a) atom collisions and (b) superradiance in the rest frame of the upper condensate. The color background shows the long-time atom distribution on the $x\text{−}z$ plane. In (a), it comes from the stochastic Bogoliubov simulation for the $N={10}^5$ case. We use the $k$-space distribution at the end of the collision, scaled out to the far field via $x=(\hslash k/m)t$. The $x\text{−}y$ plane analyzed in the text is shown as a dashed green line. In both panels, arrows show flight paths.

Figure 9

(a) Examples of simulated classical test-particle trajectories. Solid blue and dashed red lines show the paths followed by particles in a pair for 1 ms. The pair is produced at the point where the two lines meet within the collision zone that is shown as a contour plot. (b) magnifies detail near the source zone.

Figure 10

Halo properties in a classical test-particle model as in Fig. 4, but for a static mean-field potential.