Unitary Quantum Three-Body Problem in a Harmonic Trap

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. With a Feshbach resonance, it is now possible to produce a stable quantum gas of fermionic atoms in the unitary limit, i.e. with an interaction of negligible range and scattering length a = ∞ [1]. The properties of this gas, including its superfluidity, are under active experimental investigation [2]. They have the remarkable feature of being universal, as was tested in particular for the zero temperature equation of state of the gas [3]. In contrast, experiments with Bose gases at a Feshbach resonance suffer from high loss rates [4,5,6], and even the existence of a unitary Bose gas phase is a very open subject [7].In this context, fully understanding the few-body unitary problem is a crucial step. In free space, the unitary 3-boson problem has an infinite number of weakly bound states, the so-called Efimov states [8]. In a trap, it has efimovian states [9, 10] but also universal states whose energy depends only on the trapping frequency [9]. Several experimental groups are currently trapping a few particles at a node of an optical lattice [11] and are controlling the interaction strength via a Feshbach resonance. Results have already been obtained for two particles per lattice node [12], a case that was solved analytically [13]. Anticipating experiments with 3 atoms per node, we derive in this Letter exact expressions for all universal and efimovian eigenstates of the 3-body problem for bosons (generalizing [9] to a non-zero angular momentum) and for equal mass fermions in a trap. We also show the long lifetime of the universal states and their observability in a real experiment, extending to universal states the numerical study of [10].If the effective range and the true range of the interaction potential are negligible as compared to the de Broglie wavelength of the 3 particles, the interaction potential can be replaced by the Bethe-Peierls contact conditions on the wavefunction ψ: it exists a function A such thatin the limit r ij ≡ |r i − r j | → 0 taken for fixed positions of the other particle k and of the center of mass R ij of i and j. In the unitary limit considered in this paper, a = ∞. When all the r ij are non zero, the wavefunction ψ obeys the non-interacting Schrödinger equationω is the oscillation frequency and m the mass of an atom.To solve this problem, we exte...

Convergence properties of the effective theory for trapped bosons

“…The eigenvalues D(α) as well as the eigenstates now depend on the cutoff parameter N. Therefore it is not straightforward to derive the leading-order behavior of g (3) analytically. Werner et al have provided a semi-analytic solution for the three-body problem in a harmonic trap in the unitary limit [16]. We use their results for the energy spectrum to benchmark our threebody calculations.…”

“…The corresponding 3-body problem was first solved by Jonsell et al [15]. Werner and Castin calculated the complete 3-body spectrum in the unitary limit of infinite scattering length and provided a semi-analytic solution [16]. More recent work has focused on the universality of the twobody spectrum [17,18], the extension to heteronuclear systems [19] and anisotropic traps [20,21], and the interpretation of the scattering-length dependence of the level structure in terms of the Zel ′ dovich effect [22].…”

Trapped two-component Fermi gases with up to six particles: Energetics, structural properties, and molecular condensate fraction

“…In the strongly-interacting unitary regime, the properties of the system-motivated by analytical treatments that exploit the scale invariance of equal-mass Fermi gases at unitarity [31,32]have been interpreted within the hyperspherical framework [21,22]. In some cases, the excitation spectrum at unitarity has also been investigated [21,22,32,33]. In addition, small two-component Fermi gases have been investigated as a function of the s-wave scattering length a s [22,[34][35][36][37][40][41][42].…”

“…[21][22][23][24][31][32][33][34][35][36][37][38][39][40][41][42]). The ground state of trapped equalmass two-component Fermi gases, e.g., has been investigated numerically by the fixed-node diffusion Monte Carlo approach [21][22][23]39] and the stochastic variational approach [21,22,36,40,42].…”

Comparing and contrasting nuclei and cold atomic gases

“…From The early seminal study of Busch et al (Busch et al 1997) demonstrates that for two trapped fermions in opposite internal states the ground-state energy is 2 ω (see (Zinner 2012) a discussion of this model in both two-and three-dimensional traps and for a review of the relevant theoretical work and experimental support). An important benchmark for the two-component fermionic few-body problem in a harmonic trap is the exact solution for three particles at unitarity where the groundstate energy is 4.27 ω (Werner & Castin 2006a, Werner & Castin 2006b). These results was followed by several numerical studies of three and four trapped fermions (Stetcu et al 2007, Alhassid et al 2008, and for up to 30 fermions (Chang & Bertsch 2007, Blume et al 2007).…”