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Self-Bound Dilute Quantum Systems

Aurel Bulgac

Published Feb 10, 1999• 0 citations
OpenAlex

Abstract

In the limit when the two-body scattering length $a$ is negative and much larger than the effective two-body interaction radius the contribution to the ground state energy due to the three-body correlations is given by the Efimov effect. For particular values of the diluteness parameter $\rho |a|^3$ the three-body contribution can become the dominant term of the energy density functional. Under these conditions both Bose-Einstein and Fermi-Dirac systems could become self-bound and either boson droplets or fermion ``designer nuclei'' of various sizes and densities could be manufactured.

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In the limit when the two-body scattering length aaa is negative and much larger than the effective two-body interaction radius the contribution to the ground state energy due to the three-body correlations is given by the Efimov effect. For particular values of the diluteness parameter ρ∣a∣3\rho |a|^3ρ∣a∣3 the three-body contribution can become the dominant term of the energy density functional. Under these conditions both Bose-Einstein and Fermi-Dirac systems could become self-bound and either boson droplets or fermion ``designer nuclei'' of various sizes and densities could be manufactured.
In the limit when the two-body scattering length a is negative and much larger than the effective two-body interaction radius the contribution to the ground state energy due to the three-body correlations is given by the Efimov effect. For particular values of the diluteness parameter rho/a/(3) the three-body contribution can become the dominant term of the energy density functional. Under these conditions both Bose-Einstein and Fermi-Dirac systems could become self-bound and either boson droplets or fermion "designer nuclei" of various sizes and densities could be manufactured.
Small weakly bound droplets determine a number of properties of ultracold Bose and Fermi gases. For example, Efimov trimers near the atom-atom-atom and atom-dimer thresholds lead to enhanced losses from bosonic clouds. Generalizations to four- and higher-body systems have also been considered. Moreover, Efimov trimers have been predicted to play a role in the Bose polaron with large boson-impurity scattering length. Motivated by these considerations, the present work provides a detailed theoretical analysis of weakly bound NNN-body clusters consisting of $N\ensuremath{-}1$ identical bosons (denoted by ``B'') of mass mmm that interact with a single distinguishable impurity particle (denoted by ``X'') of mass MMM. The system properties are analyzed as a function of the mass ratio $\ensuremath{\kappa}$ (values from $\ensuremath{\kappa}=1$ to 50 are considered), where $\ensuremath{\kappa}$ is equal to m/Mm/Mm/M, and the two-body sss-wave scattering length aBX{a}_{\text{BX}}aBX​ between the bosons and the impurity. To reach the universal Efimov regime in which the size of the BBX trimer as well as those of larger clusters is much larger than the length scales of the underlying interaction model, three different approaches are considered: resonance states are determined in the absence of BB and BBX interactions, bound states are determined in the presence of repulsive three-body boson-boson-impurity interactions, and bound states are determined in the presence of repulsive two-body boson-boson interactions. The universal regime, in which the details of the underlying interaction model become irrelevant, is identified.
A two-body correlated basis set is used to develop a many-body theory which is valid for any number of bosons in the trap. The formalism incorporates the van der Waals interaction and two-body correlations in an exact way. The theory has successfully been applied to Bose-Einstein condensates---dilute weakly interacting and also dilute but having a large scattering length. Even in the extreme dilute condition, we observe the breakdown of the shape-independent approximation and the interatomic correlation plays an important role in the large particle-number limit. This correlated many-body calculation can handle, within the two-body correlation approximation, the entire range of atom number of experimentally achieved condensates. Next we successfully push the basis function for large scattering lengths where the mean-field results are manifestly bad. The sharp increase in correlation energy clearly shows the beyond-mean-field effect. We also calculate one-particle densities for various scattering lengths and particle numbers. Our many-body calculation exhibits the finite-size effect in the one-body density.