Abstract. -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 longrange 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 multi-layer systems.Introduction. -Quantum gases of polar atoms and molecules in their rovibrational ground-state represent a unique opportunity to study the interplay of long-and short-range interactions in the highly controllable trapped gas environment. Early experiments used magnetic dipolar atoms [1-4] which have observable effects in spite of intrinsically weak dipole moments. Recently, heteronuclear molecules with very large electric dipole moments have been realized by a number of groups [5][6][7][8][9]. The goal of a quantum degenerate system of polar molecules with strong 1/r 3 long-range dipole-dipole forces therefore seems close at hand.The attractive force of polar molecules in the headto-tail configuration can lead to collapse of the system [10]. However, as suggested by Wang et al. [11], a onedimensional optical lattice that creates a multilayered stack of pancake systems can stabilize the situation. If we apply a field to polarize the dipoles perpendicular to the layers then the intralayer interaction will be purely repulsive, whereas the interlayer part will be attractive but with the optical lattices separating the dipoles in different layers. This setup is presently being implemented experimentally. As discussed in [11], the dipole-dipole force forms bound chains and the system effectively behaves as a liquid of chains with resemblance to rheological fluids.

We consider an analytic way to make the interacting N -body problem tractable by using harmonic oscillators in place of the relevant two-body interactions. The two-body terms of the N -body Hamiltonian are approximated by considering the energy spectrum and radius of the relevant two-body problem which gives frequency, center position, and zero point energy of the corresponding harmonic oscillator. Adding external harmonic one-body terms, we proceed to solve the full quantum mechanical N -body problem analytically for arbitrary masses. Energy eigenvalues, eigenmodes, and correlation functions like density matrices can then be computed analytically. As a first application of our formalism, we consider the N -boson problem in two-and three dimensions where we fit the two-body interactions to agree with the well-known zerorange model for two particles in a harmonic trap. Subsequently, condensate fractions, spectra, radii, and eigenmodes are discussed as function of dimension, boson number N , and scattering length obtained in the zero-range model. We find that energies, radii, and condensate fraction increase with scattering length as well as boson number, while radii decrease with increasing boson number. Our formalism is completely general and can also be applied to fermions, Bose-Fermi mixtures, and to more exotic geometries.

We consider the N -body problem in a layered geometry containing cold polar molecules with dipole moments that are polarized perpendicular to the layers. A harmonic approximation is used to simplify the Hamiltonian and bound state properties of the two-body inter-layer dipolar potential are used to adjust this effective interaction. To model the intra-layer repulsion of the polar molecules, we introduce a repulsive inter-molecule harmonic potential and discuss how its strength can be related to the real dipolar potential. However, to explore different structures with more than one molecule in each layer, we treat the repulsive harmonic strength as an independent variable in the problem. Single chains containing one molecule in each layer, as well as multi-chain structures in many layers are discussed and their energies and radii determined. We extract the normal modes of the various systems as measures of their volatility and eventually of instability, and compare our findings to the excitations in crystals. We find modes that can be classified as either chains vibrating in phase or as layers vibrating against each other. The former correspond to acoustic and the latter to optical phonons. For the acoustic modes, our model predicts a smaller sound speed than one would naively get from expansion of the dipolar potential to second order around the origin. Instabilities can occur for large intra-layer repulsion and produce diverging amplitudes of molecules in the outer layers, and our model predicts how the breakup takes places. Lastly, we consider experimentally relevant regimes to observe the structures. The harmonic model considerd here predicts that for the multi-layer systems under current study chains with one molecule in each layer are always bound whereas two chains comprised of two molecules in each layer will not be bound. However, since realistic systems have external confinement prevention the molecules from escaping to infinity, we still expect the unstable modes to show up as resonances in the dynamics. PACS. 67.85.-d Ultracold gases, trapped gases -36.20.-r Macromolecules and polymer molecules -03.75.Kk Dynamic properties of condensates arXiv:1106.2102v3 [cond-mat.quant-gas]

We describe a procedure to solve an up to 2N problem where the particles are separated topologically in N groups with at most two particles in each. Arbitrary interactions are allowed between the (two) particles within one group. All other interactions are approximated by harmonic oscillator potentials. The problem is first reduced to an analytically solvable N -body problem and N independent two-body problems. We calculate analytically spectra, wave functions, and normal modes for both the inverse square and delta-function two-body interactions. In particular, we calculate separation energies between two strings of particles. We find that the string separation energy increases with N and interaction strength.

The microscopic properties of few interacting cold fermionic atoms confined in a two-dimensional (2D) harmonic trap are studied by numerical diagonalization. For repulsive interactions, a strong shell structure dominates, with Hund's rule acting at its extreme for the midshell configurations. In the attractive case, odd-even oscillations due to pairing occur simultaneously with deformations in the internal structure of the ground states, as seen from pair correlation functions.