Hartree-Fock-Bogoliubov model and simulation of attractive and repulsive Bose-Einstein condensates

We describe a model of dynamic Bose-Einstein condensates near a Feshbach resonance that is computationally feasible under assumptions of spherical or cylindrical symmetry. Simulations in spherical symmetry approximate the experimentally measured time to collapse of an unstably attractive condensate only when the molecular binding energy in the model is correct, demonstrating that quantum fluctuations and coupling between atoms and (un)bound pairs of atoms included in the model are the dominant mechanisms during collapse. Simulations of condensates with repulsive interactions find some quantitative disagreement, suggesting that pairing and quantum fluctuations are not the only significant factors for condensate loss or burst formation. Inclusion of three-body recombination was found to be inconsequential in all of our simulations, although we do not consider recent experiments [Phys. Rev. A 84, 033632 (2011)] conducted at over an order of magnitude higher density.

Figure 1

Coordinate axes for spherical symmetry. Since all fields are independent of the orientation of $\mathbf{R}$, we are free to rotate the axes (keeping the origin fixed), and so we align the $z$ axis with $\mathbf{k}$. Then the dependence of the correlation functions' values on the relative orientations of $\mathbf{k}$ and $\mathbf{R}$ (bold arrows) can be expressed in spherical coordinates.

Figure 2

Coordinate axes for cylindrical symmetry. Since all fields are independent of the azimuthal angle of $\mathbf{R}$, we are free to rotate the axes so long as the $z$ axis and origin remain fixed. We align the $x$ axis with the component of $\mathbf{k}$ lying in the $x$-$y$ plane. Then the dependence of the correlation functions' values on the relative orientations of $\mathbf{k}$ and $\mathbf{R}$ (bold arrows) can be expressed in cylindrical coordinates.

Figure 3

Comparison of measured [18] (circles) and analytical (solid line) three-body loss rate of Eq. (33) as a function of magnetic field. Equation (33) has been multiplied by a factor of 6 here, because the measurements were done in a thermal gas. The dashed vertical line marks $B=154.9$ Gauss, where the effective scattering length diverges.

Figure 4

Number of condensed atoms in the collapse simulation. The dashed vertical line marks $t=2.04$ ms, the last sample point for which the simulation is stable.

Figure 5

Condensed atom density in the collapse simulation. In the simulation, $R$ extends to 10 $\mu$m.

Figure 6

Number of condensed atoms during a brief period in the middle of the collapse simulation. The plotted points are $0.5$ nanoseconds apart, so the visible oscillations of period $1.1$ microseconds are not aliased.

Figure 7

Difference over the average of the number of condensed atoms between collapse simulations with and without three-body effects. The dashed vertical line marks $t=2.04$ ms, the last sample point for which the simulation is stable. At no time does the average difference exceed $0.2\%$, although it is consistently increasing.

Figure 8

Density of atoms in the atomic condensate as predicted by the Gross-Pitaevskii equation (solid line) during a simulated collapse and by various configurations of our model. For readability, the abscissa here is truncated before the GPE simulation ends; its density at the origin climbs until instability terminated the simulation at $7.35$ milliseconds. Our model with $g_c=1.816$ (long dashed line) collapses much earlier than the GPE for a given magnetic field, but also collapses for $B=B_{\mathrm{res}}+\Delta B$ (short dashed line). For $g_c=2$, our model predicts a rise in density very similar to the GPE (a plot would overlap the GPE result shown here), and no effective change in density when $B=B_{\mathrm{res}}+\Delta B$ (dotted line). The “C” and “S” labels in the legend are reminders of which series are for nominally collapsing and stationary values of $a_{\mathrm{eff}}$, respectively.

Figure 9

Atomic condensate density at the origin for simulations with effective scattering lengths of 0 (solid line), $-12a_0$ (long dashes), $-26.5a_0$ (short dashes), and $-54.5a_0$ (dotted line). These simulations become unstable after $2.31$, $2.04$, $1.88$, and $1.6$ milliseconds, respectively. All simulations use $g_c=1.816$.

Figure 10

Number of condensed atoms in the two-pulse simulation. The oscillations in the middle of the simulation occur during the free precession time after the first pulse and before the second, when the magnetic field is held constant.

Figure 11

Time dependence of the magnetic field for simulations with a single pulse that induces repulsive interatomic interactions. The lowest value of the magnetic field is $B_{\mathrm{pulse}}$, and the time to descend from the initial value $B_{\mathrm{ini}}\approx 165.68$ Gauss, where $a_{\mathrm{eff}}=7a_0$, to $B_{\mathrm{pulse}}$ is called $t_{\mathrm{ramp}}$. Our simulations had the magnetic field near resonance for 1 microsecond, although experiments were conducted [2] with many other values.

Figure 12

Numbers of condensed atoms remaining at the ends of simulations (glyphs connected by dashed lines) and experiments [2, 45] (glyphs connected by solid lines) as a function of $t_{\mathrm{ramp}}$. Curves are provided as a guide to the eye; actual data are represented by glyphs.

Figure 13

Difference over the average of the number of condensed atoms between simulations with and without three-body effects. The scenario was a single pulse to $B=156$ Gauss. At no time does the average difference exceed $2.7\%$, although it increases most dramatically when the magnetic field is near resonance.

Figure 14

Density of atoms in the atomic condensate as predicted by our model (solid line) and the Gross-Pitaevskii equation (dashed line) during a simulation of a pulse to $B_{\mathrm{pulse}}=156$ Gauss.

Figure 15

Velocities of molecules during a portion of a simulation involving a pulse to 156 G.

Figure 16

Condensed atom density during a pulse to $B_{\mathrm{res}}$. The inset magnifies the region where the oscillations in density are most pronounced.

Figure 17

Molecule density during a pulse to $B_{\mathrm{res}}$. The inset magnifies the same region as the inset in Fig. 16.

Figure 18

Noncondensed atom density during a pulse to $B_{\mathrm{res}}$. The inset magnifies the same region as the inset in Fig. 16.