Two-dimensional system

There are different possible geometries of the experiment. One can create a two-dimensional perturbation in the three-dimensional condensate. Such a two-dimensional impurity can be created, analogously to the MIT experiment [RKO+99,ORV+00], by a thin laser beam. Such a beam creates a cylindrical hole in the condensate, which is stirred by moving the position of the laser beam. Another possibility is to fix the position of the laser beam along the long axis of an elongated condensate, so that the dissipation can be studied by shaking the trap and exciting the breathing modes. The problem is to create a beam with a diameter which is small with compared to the correlation length. The theory can be easily generalized for beams of finite diameter. The intensity of the beam can be tuned to satisfy the condition of a weak perturbation.

The more interesting possibility is the investigation of true two-dimensional condensates, which can be created in plane optical traps, produced by a standing light wave. If the light intensity is large enough, tunneling between planes is small and the condensates behave as independent two dimensional systems. The impurity can again be created by a laser beam perpendicular to the condensate plane. Another possibility is to use impurity atoms, which can be drive by a laser beam, with a frequency close to the atomic resonance of the impurity.

We expand the two dimensional differential $d^2k$ by its representation in the polar coordinates $d^2k=kdkd\vartheta=-\frac{k}{\sqrt{1-\cos^2\vartheta}}dk\,d\!\cos\vartheta$. The 3D integrate rule (7.24) should be substituted by

$$\int_{-\infty}^\infty dk_x\int_{-\infty}^\infty dk_y f(k_x,k_y) =... ...\cos\vartheta \left(\frac{kf(k,\cos\vartheta)}{\sqrt{1-\cos^2\vartheta}}\right)$$ (7.31)