Trapping cold atoms by quantum reflection

We examine the properties of a quantum reflection trap when particle interaction is included. We explore the influence of the particle interaction on the trapping for different regimes: repulsive particle interaction and attractive particle interactions in its stable and unstable limit. With variational techniques, we calculate the phase diagram of the quantum reflection trap and determine the stable and unstable regimes of the system.

Figure 1

Surviving particle densities for $\sigma =20$ and values of $\gamma =0,0.1,0.5,1.0,5.0$ from top to bottom. The presence of the repulsive particle interaction reduces the trapping effect of the quantum reflection by its additional contribution to the total energy of the system.

Figure 2

The total energy of the system ${\mathcal{E}}_{\text{tot}}\left(\tau \right)$ (solid line), the kinetic energy ${\mathcal{E}}_{\text{kin}}\left(\tau \right)$ (dashed line), and the repulsive particle interaction ${\mathcal{E}}_{\text{int}}\left(\tau \right)$ (dotted line) for $\sigma =20$ and $\gamma =5$.

Figure 3

Surviving particle densities for $\sigma =40$ and values of $\gamma =0,-0.1,-0.5,-0.62$ from bottom to top. The presence of the attractive particle interaction increases the trapping effect of the quantum reflection.

Figure 4

The total energy of the system ${\mathcal{E}}_{\text{tot}}\left(\tau \right)$ (solid line), the kinetic energy ${\mathcal{E}}_{\text{kin}}\left(\tau \right)$ (dashed line), and the repulsive particle interaction ${\mathcal{E}}_{\text{int}}\left(\tau \right)$ (dotted line) for $\sigma =40$ and $\gamma =-0.62$. It can clearly be seen how the energetic components of the system balance each other. The dynamical equilibrium between the kinetic energy and the interaction energy stabilizes the system.

Figure 5

Surviving particle densities for $\sigma =40$ and values of $\gamma =-0.63$ (solid line), ${\gamma }_c=-0.627$ (dashed line), and $\gamma =0$ (dot-dashed line) for comparison. The collapse of the wave packet is indicated by the sudden falloff of the surviving particle density.

Figure 6

The kinetic energy ${\mathcal{E}}_{\text{kin}}\left(\tau \right)$ (solid line) and the repulsive particle interaction ${\mathcal{E}}_{\text{int}}\left(\tau \right)$ (dashed line) for $\sigma =40$ and $\gamma =-0.63$. The peak of the kinetic energy is responsible for the breakdown of the trapping mechanism and the strong and sudden loss of particle density (see Fig. 5).

Figure 7

Phase diagram of the quantum reflection trap obtained by a simple Gaussian variational approach. The vertical line $\alpha =0.31\leftrightarrow a=5$ marks our initial data; the horizontal line ${\gamma }_c=-0.627$ marks our numerical result for the critical interaction strength. The region encircled by the ellipse is the critical region of the system where transitions between universal and nonuniversal behavior, as well as transitions between stable and unstable states of the condensate, occur. The region below $\sigma =0$ supports only unstable states. The arrows drawn on the phase-line of $\sigma =10$ indicate exemplarily for any other value of $\sigma$ that a stable state for a certain value of $\sigma$ becomes unstable when its parametric set $\left(\alpha ,\gamma \right)$ is crossing the corresponding phase line from left to right and vice versa.