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arXiv:1008.2444 · Physical Review A

Two-dimensional scattering and bound states of polar molecules in bilayers

Michael Klawunn, Alexander Pikovski, Luis Santos

Published Oct 20, 2010• 77 citations
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Abstract

Low-energy two-dimensional scattering is particularly sensitive to the existence and the properties of weakly bound states. We show that interaction potentials $V(r)$ with the vanishing zero-momentum Born approximation $\ensuremath{\int}\mathit{dr} rV(r)=0$ leads to an anomalously weak bound state that crucially modifies the two-dimensional scattering properties. This anomalous case is especially relevant in the context of polar molecules in bilayer arrangements.

DOI
10.1103/physreva.82.044701

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Low-energy two-dimensional scattering is particularly sensitive to the existence and the properties of weakly bound states. We show that interaction potentials V(r)V(r)V(r) with the vanishing zero-momentum Born approximation \ensuremath∫drrV(r)=0\ensuremath{\int}\mathit{dr} rV(r)=0\ensuremath∫drrV(r)=0 leads to an anomalously weak bound state that crucially modifies the two-dimensional scattering properties. This anomalous case is especially relevant in the context of polar molecules in bilayer arrangements.
The recent experimental realization of cold polar molecules in the rotational and vibrational ground state opens the door to the study of a wealth of phenomena involving long-range interactions. By applying an optical lattice to a gas of cold polar molecules one can create a layered system of planar traps. Due to the long-range dipole-dipole interaction one expects a rich structure of bound complexes in this geometry. We study the bilayer case and determine the two-body bound-state properties as a function of the interaction strength. The results clearly show that a least one bound state will always be present in the system. In addition, bound states at zero energy show universal behavior and extend to very large radii. These results suggest that non-trivial bound complexes of more than two particles are likely in the bilayer and in more complicated chain structures in multilayer systems.
We propose a framework for calculating scattering and bound-state properties in anisotropic two-dimensional potentials. Using our method, we derive systematic approximations of partial wave phase shifts and binding energies. Moreover, the method is suitable for efficient numerical computations. We calculate the sss-wave phase shift and binding energy of polar molecules in two layers polarized by an external field along an arbitrary direction. We find that scattering depends strongly on their polarization direction and that absolute interlayer binding energies are larger than thermal energies at typical ultracold temperatures.
The exploration of cold polar molecules in different geometries is a rapidly developing experimental and theoretical pursuit. Recently, the implementation of optical lattices has enabled confinement in stacks of planes, the number of which is also controllable. Here we consider the bound state structure of two polar molecules confined in two adjacent planes as function of the polarization angle of the dipole moment of the molecules. We prove analytically and present numerical evidence for the existence of bound states for arbitrary dipole moments and polarization directions in this two-dimensional geometry. The spatial structure of the bound states is dominated by two-dimensional s- and p-waves, where the latter exceeds 40 percent over a large range of polarization angles for intermediate or strong dipole strength. Finally, we consider the influence of the dimer bound states on the potential many-body ground-state of the system.
We investigate the quantum phase transitions of bosonic polar molecules in a two-dimensional double layer system. We show that an interlayer bound state of dipoles (dimers) can be formed when the dipole strength is above a critical value, leading to a zero-energy resonance in the interlayer s-wave scattering channel. In the positive detuning side of the resonance, the strong repulsive interlayer pseudopotential can drive the system into a maximally entangled state, where the wave function is a superposition of two states that have all molecules in one layer and none in the other. We discuss how the zero-energy resonance, dimer states, and the maximally entangled state can be measured in time-of-flight experiments.
We study Fermi-liquid properties of a weakly interacting two-dimensional gas of single-component fermionic polar molecules with dipole moments ddd oriented perpendicularly to the plane of their translational motion. This geometry allows the minimization of inelastic losses due to chemical reactions for reactive molecules and, at the same time, provides a possibility of a clear description of many-body (beyond mean-field) effects. The long-range character of the dipole-dipole repulsive interaction between the molecules, which scales as 1/r31/{r}^{3}1/r3 at large distances rrr, makes the problem drastically different from the well-known problem of the two-species Fermi gas with repulsive contact interspecies interaction. We solve the low-energy scattering problem and develop a many-body perturbation theory beyond the mean field. The theory relies on the presence of a small parameter kFr∗{k}_{F}{r}_{*}kF​r∗​, where kF{k}_{F}kF​ is the Fermi momentum and ${r}_{*}=m{d}^{2}/{\ensuremath{\hbar}}^{2}$ is the dipole-dipole length, with mmm being the molecule mass. We obtain thermodynamic quantities as a series of expansion up to the second order in kFr∗{k}_{F}{r}_{*}kF​r∗​ and argue that many-body corrections to the ground-state energy can be identified in experiments with ultracold molecules, as it has been recently done for ultracold fermionic atoms. Moreover, we show that only many-body effects provide the existence of zero sound and calculate the sound velocity.