Layers of cold dipolar molecules in the harmonic approximation

Abstract

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]

Correlation effects and collective excitations in bosonic bilayers: Role of quantum statistics, superfluidity, and the dimerization transition

“…The spectrum of collective density excitations will be strongly modified by the inter-chain dynamics and intra-chain modes, being in strong dependence with variation of interlayer distance and a strength of interlayer coupling. 21 Similar to the predictions for a single 2D layer, 22-24 dipolar bilayer and multilayers are expected to undergo a crystallization transition at high densities. 25 All these possibilities, existing in multilayers, form an interesting topic for future analyses.…”

“…The long wavelength limit of (21) has been analyzed by Golden and Kalman 54 and can be expressed via the average interaction energy per volume…”

Analytic solutions of topologically disjoint systems

“…The inter-layer problem is reduced to pure oscillator properties which has been discussed in detail before [42]. If there is no one-body potential, the center-of-mass degree of freedom solved through eqs.…”

“…The prime on the frequency, ω ′ k , indicates that the k solutions may be complicated functions of the parameters in the initial Hamiltonian [42]. These energies then correspond to the normal modes for the relative coordinates.…”

Dipoles on a two-leg ladder

“…These problems can be overcome in low-dimensional geometries where the dipolar particles are confined to either twodimensional (2D) planes or one-dimensional (1D) tubes with and without the presence of lattice potentials. A number of interesting predictions have been made for the phases of system with dipolar interactions, including exotic superfluids [15][16][17][18][19] , Luttinger liquids [20][21][22][23][24][25] , Mott insulators 26,27 , interlayer pairing [28][29][30][31] , non-trivial quantum critical points 32,33 , modified confinement-induced resonances [34][35][36][37] , roton modes and stripe instabilities [38][39][40][41][42][43][44][45] , and crystallization [46][47][48][49][50][51][52][53][54] , as well as formation of chain complexes [55][56][57][58][59][60]…”