Analytic Harmonic Approach to the N-body problem

Abstract

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 zero-range 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.

30 pages, 12 figures, updated references

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• Rényi entropies of the highly-excited states of multidimensional harmonic oscillators by use of strong Laguerre asymptotics
• Impenetrable Mass-Imbalanced Particles in One-Dimensional Harmonic Traps
• Number-parity effect for confined fermions in one dimension
• Bound states of Dipolar Bosons in One-dimensional Systems
• Analytic solutions of topologically disjoint systems
• Theoretical description of two ultracold atoms in finite 3D optical lattices using realistic interatomic interaction potentials
• Layers of Cold Dipolar Molecules in the Harmonic Approximation
• Virial expansion coefficients in the harmonic approximation
• Pinning of fermionic occupation numbers: Higher spatial dimensions and spin
• Variance of an anisotropic Bose-Einstein condensate
• Universal Two-Body Spectra of Ultracold Harmonically Trapped Atoms in Two and Three Dimensions
• Analysis of a trapped Bose-Einstein condensate in terms of position, momentum, and angular-momentum variance
• Quantum Statistics and Thermodynamics in the Harmonic Approximation
• Heisenberg and entropic uncertainty measures for large-dimensional harmonic systems
• Solvable Model of a Generic Trapped Mixture of Interacting Bosons: Reduced Density Matrices and Proof of Bose-Einstein Condensation
• Formation of classical crystals of dipolar particles in a helical geometry
• Quantum few-body bound states of dipolar particles in a helical geometry
• Solvable model of a generic driven mixture of trapped Bose-Einstein condensates and properties of a many-boson Floquet state at the limit of an infinite number of particles
• Many-particle Systems in One Dimension in the Harmonic Approximation
• Bound Chains of Tilted Dipoles in Layered Systems
• Thermodynamics of Dipolar Chain Systems
• Effects Beyond Center-of-Mass Separability in a Trapped Bosonic Mixture: Exact Results
• Properties of a trapped multiple-species bosonic mixture at the infinite-particle-number limit: A solvable model
• Entanglement and correlations in an exactly-solvable model of a Bose-Einstein condensate in a cavity
• Fragmentation of a trapped bosonic mixture
• Morphology of an interacting three-dimensional trapped Bose-Einstein condensate from many-particle variance anisotropy
• Fragmentation of a trapped multiple-species bosonic mixture
• A Solvable Model for Decoupling of Interacting Clusters
• Correlation Effects in a Trapped Bose-Fermi Mixture: Exact Results
• Spatial correlations and entanglement in a hybrid system of N fermion pairs with harmonic interaction
• Clusters in Separated Tubes of Tilted Dipoles
• Effective theory for strongly attractive one-dimensional fermions
• Fragmentation of identical and distinguishable bosons' pairs and natural geminals of a trapped bosonic mixture