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Physical Review A

Resonance states of three self-gravitating bosons and fermions

Sabyasachi Kar, Y. K. Ho

Published Sep 18, 2007• 10 citations
Publisher

Abstract

The resonance states of the systems consisting of three identical bosons and of three fermions interacting with an attractive $1∕r$ potential are investigated using highly correlated exponential basis functions. The method of complex-coordinate rotation is used to extract energies and widths for the resonances lying below the $n=2$ excited state threshold of the respective two-body subsystem. For the system consisting of three identical bosons, we have obtained a total of 12 $S$-wave resonances. It is concluded that nine of them belong to one series of Rydberg states. For the system consisting of three fermions, we have obtained a total of 23 $S$-wave resonances in total spin three-halfs (the $^{4}S$ state) and 17 $S$-wave resonances total spin one-half (the $^{2}S$ states). The resonances for the $^{4}S$ states belong to three different series, while those for the $^{2}S$ states belong to two Rydberg series. The resonance energies and widths of these states for the three-body self-gravitating bosons and fermions are reported.

DOI
10.1103/physreva.76.032711

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The resonance states of the systems consisting of three identical bosons and of three fermions interacting with an attractive 1∕r1∕r1∕r potential are investigated using highly correlated exponential basis functions. The method of complex-coordinate rotation is used to extract energies and widths for the resonances lying below the n=2n=2n=2 excited state threshold of the respective two-body subsystem. For the system consisting of three identical bosons, we have obtained a total of 12 SSS-wave resonances. It is concluded that nine of them belong to one series of Rydberg states. For the system consisting of three fermions, we have obtained a total of 23 SSS-wave resonances in total spin three-halfs (the 4S^{4}S4S state) and 17 SSS-wave resonances total spin one-half (the 2S^{2}S2S states). The resonances for the 4S^{4}S4S states belong to three different series, while those for the 2S^{2}S2S states belong to two Rydberg series. The resonance energies and widths of these states for the three-body self-gravitating bosons and fermions are reported.
The resonance states of systems consisting of three identical bosons and three fermions interacting with an attractive 1/ r potential are investigated using highly correlated exponential basis functions. The method of complex-coordinate rotation is used to extract energies and widths for the resonances lying below the N = 3 excited state threshold of the respective two-body subsystem. We have obtained a total of 24 S-wave resonances for the three-boson system and a total of 19 S-wave resonances in total spin-3/2 (the 4 S state) and a total of 15 S-wave resonances in total spin-1/2 (the 2 S state) below the N = 3 threshold. For the resonances associated with the N = 3 threshold, we have obtained two, three and four series of Rydberg states for the three-boson system, three-fermion system for 2 S states and three-fermion system for 4 S states, respectively. The resonance energies and widths of these states for the three-body self-gravitating bosons and fermions are reported.
We have used the hyperspherical adiabatic representation to describe the system of three identical bosons in a spin stretched state interacting through an attractive 1∕r1∕r1∕r potential. A proposal has been forwarded to enable the experimental realization of such a system in cold trapped atoms using extremely off-resonant laser fields [Phys. Rev. Lett., 84, 5687 (2000)]. We have obtained the effective potentials, channel functions, and nonadiabatic couplings for this gravitylike interaction, allowing us to calculate the ground-state energy with accuracy that substantially improves upon previous results. We have similarly calculated the energies for the first four 0+{0}^{+}0+ excited states. These results show that the simple adiabatic hyperspherical approximation offers an accurate description for such a system. Further, since the effective potentials have a long-range 1∕R1∕R1∕R attraction, the properties of the excited states follow from the well-known results for Rydberg states. We have also analyzed the possible geometries and vibrational modes for this system.
The universal three-body dynamics in ultra-cold binary Fermi and Fermi-Bose mixtures is studied. Two identical fermions of the mass mmm and a particle of the mass m1m_1m1​ with the zero-range two-body interaction in the states of the total angular momentum L=1 are considered. Using the boundary condition model for the s-wave interaction of different particles, both eigenvalue and scattering problems are treated by solving hyper-radial equations, whose terms are derived analytically. The dependencies of the three-body binding energies on the mass ratio m/m1m/m_1m/m1​ for the positive two-body scattering length are calculated; it is shown that the ground and excited states arise at m/m1gelambda1approx8.17260m/m_1 \\ge \\lambda_1 \\approx 8.17260m/m1​gelambda1​approx8.17260 and m/m1gelambda2approx12.91743m/m_1 \\ge \\lambda_2 \\approx 12.91743m/m1​gelambda2​approx12.91743, respectively. For m/m1altlambda1m/m_1 \\alt \\lambda_1m/m1​altlambda1​ and m/m1altlambda2m/m_1 \\alt \\lambda_2m/m1​altlambda2​, the relevant bound states turn to narrow resonances, whose positions and widths are calculated. The 2 + 1 elastic scattering and the three-body recombination near the three-body threshold are studied and it is shown that a two-hump structure in the mass-ratio dependencies of the cross sections is connected with arising of the bound states.
The universal three-body dynamics in ultra-cold binary Fermi and Fermi–Bose mixtures is studied. Two identical fermions of mass m and a particle of mass m 1 with the zero-range two-body interaction in states of total angular momentum L = 1 are considered. Using the boundary condition model for the s -wave interaction of different particles, both eigenvalue and scattering problems are treated by solving hyper-radial equations, whose terms are derived analytically. The dependences of the three-body binding energies on the mass-ratio m / m 1 for the positive two-body scattering length are calculated; it is shown that the ground and excited states arise at m / m 1 ⩾ λ 1 ≈ 8.172 60 and m / m 1 ⩾ λ 2 ≈ 12.917 43, respectively. For m / m 1 ≲ λ 1 and m / m 1 ≲ λ 2 , the relevant bound states turn to narrow resonances, whose positions and widths are calculated. The 2 + 1 elastic scattering and the three-body recombination near the three-body threshold are studied, and it is shown that a two-hump structure in the mass-ratio dependences of the cross sections is connected with the rise of the bound states.
Three quantum particles with on-site repulsion and nearest-neighbour attraction on a one-dimensional lattice are considered. The three-body Schrödinger equation is reduced to a set of single-variable integral equations. Energies of three-particle bound complexes (trions) are found from self-consistency of the approximating matrix equation. In the case of spin- fermions, the ground-state trion energy, the excited-state energies, the trion spectra and stability regions are obtained for total spins S = 1/2 and S = 3/2. In the S = 1/2 sector, a narrow but finite parameter region is identified where the ground state consists of a stable fermion pair and an unbound fermion. Also presented is the reference case of spin-0 bosons.