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arXiv:cond-mat/0507399 · Physical Review Letters

Unitary Quantum Three-Body Problem in a Harmonic Trap

Félix Werner, Yvan Castin

Published Oct 10, 2006• 168 citations
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Abstract

We consider either 3 spinless bosons or 3 equal mass spin-1/2 fermions, interacting via a short-range potential of infinite scattering length and trapped in an isotropic harmonic potential. For a zero-range model, we obtain analytically the exact spectrum and eigenfunctions: for fermions all the states are universal; for bosons there is a coexistence of decoupled universal and efimovian states. All the universal states, even the bosonic ones, have a tiny 3-body loss rate. For a finite range model, we numerically find for bosons a coupling between zero angular momentum universal and efimovian states; the coupling is so weak that, for realistic values of the interaction range, these bosonic universal states remain long-lived and observable.

DOI
10.1103/physrevlett.97.150401

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We consider NNN atoms trapped in an isotropic harmonic potential, with sss-wave interactions of infinite scattering length. In the zero-range limit, we obtain several exact analytical results: mapping between the trapped problem and the free-space zero-energy problem, separability in hyperspherical coordinates, SO(2,1) hidden symmetry, existence of a decoupled bosonic degree of freedom, and relations between the moments of the trapping potential energy and the moments of the total energy.
We derive the asymptotic expansions of the wave function of three particles having equal mass with finite-range interactions and infinite or zero two-dimensional scattering length colliding at zero energy and zero orbital angular momentum, from which a three-body parameter DDD is defined. The dimension of DDD is length squared, and we call DDD three-body scattering area. We find that the ground-state energy per particle of a zero-temperature dilute Bose gas with these interactions is approximately $\frac{{\ensuremath{\hbar}}^{2}D}{6m}{\ensuremath{\rho}}^{2}$, where $\ensuremath{\rho}$ is the number density of the bosons and mmm is the mass of each boson. Such a zero-temperature Bose gas is stable at D>0D>0D>0 in the thermodynamic limit, and metastable at D<0D<0D<0 in the isotropic harmonic trap if the number of bosons is less than ${N}_{\mathrm{cr}}\ensuremath{\approx}3.6413\sqrt{\frac{\ensuremath{\hbar}}{m\ensuremath{\omega}|D|}\phantom{{}^{1}}}$, where $\ensuremath{\omega}$ is the angular frequency of the harmonic trap. If the two-body interaction supports bound states, DDD typically acquires a negative imaginary part, and we find the relation between this imaginary part and the amplitudes of the pair-boson production processes. We derive a formula for the three-body recombination rate constant of the many-boson system in terms of the imaginary part of DDD.
We calculate energy levels of two and three bosons trapped in a harmonic oscillator potential with oscillator length aosc{a}_{\mathrm{o}\mathrm{s}\mathrm{c}}aosc​. The atoms are assumed to interact through a short-range potential with a scattering length aaa, and the short-distance behavior of the three-body wave function is characterized by a parameter $\ensuremath{\theta}$. For large positive a/aosca/{a}_{\mathrm{o}\mathrm{s}\mathrm{c}}a/aosc​, the energies of states that, in the absence of the trap, correspond to three free atoms approach values independent of aaa and $\ensuremath{\theta}$. For other states, the $\ensuremath{\theta}$ dependence of the energy is strong, but the energy is independent of aaa for $|a/{a}_{\mathrm{o}\mathrm{s}\mathrm{c}}|\ensuremath{\gg}1$.