The reactions systems are studied according to the Boundary Conditions Method. The scattering of a low-energy particle on two scatterers is described by a system of two coupled equations; the corresponding single channel interactions are described by one-channel scattering lengths. The problem of two-channel scattering could be described also in terms of one-channel scattering by means of Reduced Scattering Length. The Reduced Scattering Length takes into account the effect of eliminated channel via its scattering length as well as by multiple scattering of incident particle on two scatterers. One demonstrates that a virtual two-body state becomes bound in a three-body system, provided all scattering lengths are negative.
The reactions systems are studied according to the Boundary Conditions Method. The scattering of a low-energy particle on two scatterers is described by a system of two coupled equations; the corresponding single channel interactions are described by one-channel scattering lengths. The problem of two-channel scattering could be described also in terms of one-channel scattering by means of Reduced Scattering Length. The Reduced Scattering Length takes into account the effect of eliminated channel via its scattering length as well as by multiple scattering of incident particle on two scatterers. One demonstrates that a virtual two-body state becomes bound in a three-body system, provided all scattering lengths are negative.
So far, we have dealt with one- and two-body systems in a variety of settings. These are quite simple, at least a priori, since the equations needed to solve one- and twoparticle scattering problems, as well as bound states, are usually easy enough to handle. The situation becomes much more intricate in the three-body problem, as we shall see in this chapter. Firstly, while two particles in free space can either scatter or bind together, three bodies have extra channels (for attractive interactions) corresponding to two bound particles scattering off a third. Moreover, the energies of these extra channels can be degenerate with those of three unbound particles, and the total momentum of a pair within the triplet is not conserved. Fortunately, we will see that getting our hands dirty with the three-body problem pays off, since there are a number of exotic phenomena associated with the quantum three-body problem, for both bound and scattering states. In this chapter, we introduce the formalism necessary to investigate the three-body problem and study in detail some of the most striking phenomenology known to date associated with, at least, the simplest interactions.
The universal three-body dynamics in ultra-cold binary Fermi and Fermi-Bose mixtures is studied. Two identical fermions of the mass m and a particle of the mass m1 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/m1 for the positive two-body scattering length are calculated; it is shown that the ground and excited states arise at m/m1gelambda1approx8.17260 and m/m1gelambda2approx12.91743, respectively. For m/m1altlambda1 and m/m1altlambda2, 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.
1. Highlight “short-ranged,” “dilute,” and “low-energy” as three main features of interactions between ultracold atoms. 2. Introduce the important concept of the phase shift. 3. Introduce the s-wave scattering length as a universal parameter describing the low-energy interaction between ultracold atoms. 4. Discuss the relation between divergent scattering length, low-energy bound state, and jump of phase shift. 5. Discuss the relation between the scattering length and the scattering amplitude. 6. Discuss under what condition a positive scattering length describes repulsive interaction. 7. Discuss the condition when an algebraically decayed potential can be treated as a finite range one. 8. Introduce two types of zero-range single-channel potentials to capture the universal low-energy s-wave interaction between ultracold atoms. 9. Introduce the concept of renormalization condition and renormalizable theory. 10. Discuss how the spin rotational symmetry imposes constraints on interaction form for both alkali-metal and alkaline-earth-metal atoms. 11. Introduce Feshbach resonance as an important tool to tune scattering length. 12. Compare the two-channel Feshbach resonance with the single-channel shape resonance and compare the wide and the narrow resonances. 13. Introduce a zero-range two-channel model. 14. Introduce the confinement-induced resonance to tune interaction strength by an external potential. 15. Summarize three key conditions for a Feshbach resonance, and unify the optical Feshbach resonance, the orbital Feshbach resonance, and the confinement-induced resonance all in terms of these three conditions. 16. Introduce the Efimov effect as an important three-body effect at the vicinity of the two-body scattering resonance. 17. Highlight the symmetry aspect of the Efimov effect. 18. Discuss various connections between few-body and many-body physics. 19. Illustrate that few-body calculation can be used to determine properties of many-body systems by using high-temperature expansion as an example.
Universal low-energy properties are studied for three identical bosons confined in two dimensions. The short-range pairwise interaction in the low-energy limit is described by means of the boundary condition model. The wave function is expanded in a set of eigenfunctions on the hypersphere, and the system of hyperradial equations is used to obtain analytical and numerical results. Within the framework of this method, exact analytical expressions are derived for the eigenpotentials and the coupling terms of hyperradial equations. The derivation of the coupling terms is generally applicable to a variety of three-body problems provided the interaction is described by the boundary condition model. The asymptotic form of the total wave function at a small and a large hyperradius $\ensuremath{\rho}$ is studied, and the universal logarithmic dependence $\ensuremath{\sim}{\mathrm{ln}}^{3}\phantom{\rule{0.2em}{0ex}}\ensuremath{\rho}$ in the vicinity of the triple-collision point is derived. Precise three-body binding energies and the (2+1)-scattering length are calculated.
We construct an eigenvalue problem by confining many-body system to a bounded domain with the boundary condition that the wave function vanishes. By changing the boundary, however, the eigenvalues of the energy can be varied continuously. The D-matrix is defined for a series of bounded problems with the same value for the ground state energy. The D-matrix is related to the S-matrix, enabling us to calculate the S-matrix at a given energy. The Schrodinger equation for the system is transformed to a diffusion equation by regarding time as imaginary. Initial ensemble, representing an approximate wave function, is evolved, through Monte Carlo simulation of random walks and branching, to the ground state ensemble. The limitations of investigation are: 1. Ingoing and outgoing channels have two fragments. 2. The interaction between the fragments is negligible outside the boundary mentioned above. 3. The particles are bosons or we know the zeros of the wave function. First we consider the scattering of a particle by a potential, which is equivalent to the two-body problem, in one dimension. Here we use the Poschl-Teller potential for which the exact solution is known. We use this case to investigate a new sampling method and study of various parameters. Next we consider three particles in one dimension. Here we take interaction to be a potential well, where at least one of the interactions is attractive so that a two-body bound state is possible.
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.