Potential harmonics expansion method for trapped interacting bosons: Inclusion of two-body correlation

We study a system of A identical interacting bosons trapped by an external field by solving ab initio the many-body Schroedinger equation. A complete solution by using, for example, the traditional hyperspherical harmonics (HH) basis develops serious practical problems due to the large degeneracy of HH basis. Symmetrization of the wave function, calculation of the matrix elements, etc., become an immensely formidable task as A increases. Instead of the HH basis, here we use a new basis, called 'potential harmonics' (PH) basis, which is a subset of HH basis. We assume that the contribution to the orbital and grand orbital [in 3(A-1)-dimensional space of the reduced motion] quantum numbers comes only from the interacting pair. This implies inclusion of two-body correlations only and disregard of all higher-body correlations. Such an assumption is ideally suited for the Bose-Einstein condensate (BEC), which is required, for experimental realization of BEC, to be extremely dilute. Hence three and higher-body collisions are almost totally absent. Unlike the (3A-4) hyperspherical variables in HH basis, the PH basis involves only three active variables, corresponding to three quantum numbers--the orbital l, azimuthal m, and the grand orbital 2K+l quantum numbers for any arbitrary A. It drastically reduces themore » number of coupled equations and calculation of the potential matrix becomes tremendously simplified, as it involves integrals over only three variables for any A. One can easily incorporate realistic atom-atom interactions in a straightforward manner. We study the ground and excited state properties of the condensate for both attractive and repulsive interactions for various particle number. The ground state properties are compared with those calculated from the Gross-Pitaevskii equation. We notice that our many-body results converge towards the mean field results as the particle number increases.« less

[1]  A. Jensen,et al.  Two-body correlations in N-body boson systems , 2002, cond-mat/0210095.

[2]  A. Jensen,et al.  Correlated trapped bosons and the many-body Efimov effect. , 2002, Physical review letters.

[3]  A. Jensen,et al.  Two-body correlations in Bose-Einstein condensates , 2001, cond-mat/0110069.

[4]  C. Pethick,et al.  Bose–Einstein Condensation in Dilute Gases: Appendix. Fundamental constants and conversion factors , 2008 .

[5]  A. Leggett,et al.  Bose-Einstein condensation in the alkali gases: Some fundamental concepts , 2001 .

[6]  T. K. Das,et al.  STUDY OF THE EXCITED STATE OF DOUBLE-Λ HYPERNUCLEI BY HYPERSPHERICAL SUPERSYMMETRIC APPROACH , 2001 .

[7]  H. Smith,et al.  Bose–Einstein Condensation in Dilute Gases: Contents , 2001 .

[8]  C. Greene,et al.  Validity of the shape-independent approximation for Bose-Einstein condensates , 1999 .

[9]  F. Dalfovo,et al.  Theory of Bose-Einstein condensation in trapped gases , 1998, cond-mat/9806038.

[10]  C. Greene,et al.  Effective potentials for dilute Bose-Einstein condensates , 1998 .

[11]  T. Haugset,et al.  Exact diagonalization of the Hamiltonian for trapped interacting bosons in lower dimensions , 1997, cond-mat/9707107.

[12]  T. K. Das,et al.  ADIABATIC APPROXIMATION IN ATOMIC THREE-BODY SYSTEMS , 1997 .

[13]  B. Esry,et al.  Hartree-Fock theory for Bose-Einstein condensates and the inclusion of correlation effects , 1997 .

[14]  K. B. Davis,et al.  Bose-Einstein Condensation in a Gas of Sodium Atoms , 1995, EQEC'96. 1996 European Quantum Electronic Conference.

[15]  G. Baym,et al.  Ground-state properties of magnetically trapped Bose-condensed rubidium gas. , 1995, Physical review letters.

[16]  Bradley,et al.  Evidence of Bose-Einstein Condensation in an Atomic Gas with Attractive Interactions. , 1995, Physical review letters.

[17]  C. Wieman,et al.  Observation of Bose-Einstein Condensation in a Dilute Atomic Vapor , 1995, Science.

[18]  Das,et al.  Convergence of the hyperspherical-harmonic expansion for two-electron atoms. , 1994, Physical review. A, Atomic, molecular, and optical physics.

[19]  T. K. Das,et al.  Solution of integro-differential equations by projection , 1991 .

[20]  Das,et al.  Adiabatic approximation in few-body molecular and nuclear systems. , 1989, Physical review. A, General physics.

[21]  M. D. L. Ripelle.  A bridge between hyperspherical and integro-differential approaches to the many-body bound states , 1986 .

[22]  I. Elminyawi,et al.  A model calculation for the ground state of 4He tetramer , 1985 .

[23]  M. D. L. Ripelle,et al.  The potential harmonic expansion method , 1983 .

[24]  M. D. L. Ripelle,et al.  Coupled adiabatic approximation in the three-body problem , 1982 .

[25]  M. D. L. Ripelle,et al.  Contribution of three-body force to the trinucleon problem by an essentially exact calculation , 1982 .

[26]  M. Fabre de la Ripelle,et al.  Uncoupled adiabatic approximation for the hyperspherical harmonic approach , 1982 .

[27]  T. K. Das,et al.  Trinucleon Coulomb energy with inclusion of a three-nucleon force , 1982 .

[28]  M. D. L. Ripelle,et al.  Application of the Hyperspherical Formalism to the Trinucleon Bound State Problems , 1980 .

[29]  B. R. Johnson.  The renormalized Numerov method applied to calculating bound states of the coupled‐channel Schroedinger equation , 1978 .

[30]  M. Beiner,et al.  Convergence of the hyperspherical formalism applied to the trinucleons , 1971 .

[31]  Bose.  Plancks Gesetz und Lichtquantenhypothese , 1924 .