Tradeoffs for number squeezing in collisions of Bose-Einstein condensates

We investigate the factors that influence the usefulness of supersonic collisions of Bose-Einstein condensates as a potential source of entangled atomic pairs by analyzing the reduction of the number difference fluctuations between regions of opposite momenta. We show that nonmonochromaticity of the mother clouds is typically the leading limitation on number squeezing, and that the squeezing becomes less robust to this effect as the density of pairs grows. We develop a simple model that explains the relationship between density correlations and the number squeezing, allows one to estimate the squeezing from properties of the correlation peaks, and shows how the multimode nature of the scattering must be taken into account to understand the behavior of the pairing. We analyze the impact of the Bose enhancement on the number squeezing, by introducing a simplified low-gain model. We conclude that as far as squeezing is concerned the preferable configuration occurs when atoms are scattered not uniformly but rather into two well-separated regions.

Figure 1

Outcome of a single realization of a collision of two BECs in the dense $\gamma =1.02$ case [in a simulation of (5,4)]. Shown are cross sections through the density of atoms after the end of the collision ($t=1.6t_{\mathrm{coll}}$), in the $k_z-k_y$ momentum plane (top), and the $k_x-k_y$ plane (bottom). Clearly, a halo of scattered atoms is forming, nevertheless the atoms at the opposite regions of the halo are poorly aligned, which implies the absence of number squeezing in the system. Data scaled to units of $\left[k_0\right]{}^{-3}$. The white areas are saturated at this color scale; the bold (thin) black contours are at $0.1$ ($0.01$) of the peak condensate density.

Figure 2

The number-squeezing parameter ${\eta }^2$ in the dense $\gamma =1.02$ case as a function of time for three different bin volumes given in $k_0^3$ units. The solid lines are from a numerical solution of the full model (5,4). The dashed lines are predictions of the Gaussian correlation model (26). The dash-dotted horizontal line denotes the shot-noise level ${\eta }^2=1$.

Figure 3

The number-squeezing parameter ${\eta }^2$ in the dense $\gamma =1.02$ case as a function of bin volume (log scale) calculated at three different scattering times (in units of $t_{\mathrm{col}}$). The solid lines are from a numerical solution of the full model (5,4). The dashed lines are predictions of the Gaussian correlation model (26). The dash-dotted horizontal line denotes the shot-noise level ${\eta }^2=1$.

Figure 4

The number-squeezing parameter ${\eta }^2$ in the dense $\gamma =1.02$ case as a function of time for three different bin volumes (in $k_0^3$ units), as predicted by the reduced Bogoliubov model (RBM) (15,30). The dash-dotted horizontal line denotes the shot-noise level ${\eta }^2=1$.

Figure 5

The number-squeezing parameter ${\eta }^2$ in the dense $\gamma =1.02$ case as a function of time for three different bin volumes given in $k_0^3$ units. The solid lines are from a numerical solution of the full model (5,4). The dashed lines are predictions of the model without Bose enhancement (8,9). The dash-dotted horizontal line denotes the shot-noise level ${\eta }^2=1$.

Figure 6

The peak height of the back-to-back correlation function $h_{\text{bb}}$ as a function of time for the dense case $\gamma =1.02$. The solid line is from a numerical solution of the full model (5,4), the dashed line from the model without Bose enhancement (8,9), while the dotted line comes from the reduced Bogoliubov model (RBM) (15,30). The dot-dashed line shows the border value of $h_{\mathrm{bb}}=1$ when the back-to-back and collinear peaks are equal, and small-bin number squeezing disappears.

Figure 7

The number-squeezing parameter ${\eta }^2$ in the dilute $\gamma =0.24$ case as a function of time for three different bin volumes given in $k_0^3$ units. The solid lines come from a numerical solution of the full model (5,4). The dashed lines are predictions of the Gaussian correlation model (26). The dash-dotted horizontal line denotes the shot-noise level ${\eta }^2=1$.

Figure 8

The number-squeezing parameter ${\eta }^2$ in the dilute $\gamma =0.24$ case as a function of time for three different bin volumes given in $k_0^3$ units. The solid lines result from a numerical solution of the full model (5,4). The dashed lines are predictions of the model without Bose enhancement (8,9). The dash-dotted horizontal line denotes the shot-noise level ${\eta }^2=1$.

Figure 9

The peak height of the back-to-back correlation function $h_{\text{bb}}$ as a function of time for the dilute case $\gamma =0.24$. The solid line is from a numerical solution of the full model (5,4), and the dashed line is from the model without Bose enhancement (8,9). The dot-dashed line denotes the border value of $h_{\mathrm{bb}}=1$ when the back-to-back and colinear peaks are equal.

Figure 10

The widths of the colinear (left column) and back-to-back (right column) correlation functions for the dense (top row) and dilute (bottom row) cases as a function of time, fitted to the numerical solution of the full model (5,4). The solid black line is the width in the axial direction ${\sigma }_z$, the dotted red line is in the tangential direction to the halo center ${\sigma }_t$, while the dashed blue line is in the radial direction ${\sigma }_r$. The dot-dashed violet line is the Gaussian fitted half-width of the halo density $w_r$, which narrows in time due to energy-time uncertainty principle. All widths are given in $k_0$ units.

Figure 11

The number-squeezing parameter ${\eta }^2$ in the dense $\gamma =1.02$ case as a function of time for several different bin volumes given in $k_0^3$ units. Here, the BECs are replaced with the plane waves. The solid lines come from a numerical solution of the full model (5,4), while the dashed green line comes from a reduced Bogoliubov model (RBM) simulation (15,30). The dash-dotted horizontal lines denote the shot-noise and zero levels.

Figure 12

The peak height of the back-to-back correlation function $h_{\text{bb}}$ as a function of time for the dense case $\gamma =1.02$ and two colliding plane waves. The solid line is from a numerical solution of the full model (5,4), while the dot-dashed line denotes the border value of $h_{\mathrm{bb}}=1$ when the back-to-back and colinear peaks are equal.

Figure 13

The bin widths (black solid line, axial; dotted red line, tangential; dashed blue line, radial) as a function of the total bin volume (dense case $\gamma =1.02$) used in the full model (5,4) simulations.