The low-energy scattering of three bosons or distinguishable particles with short-range interactions is characterized by a fundamental parameter, the three-body scattering hypervolume. Its imaginary part is directly related to the three-body recombination rate in a quantum gas consisting of such particles. We derive an analytical formula of it for weak interactions, and perform its first numerical calculations for bosons with a variable nonzero-range potential. For attractive interactions, we identify several three-body resonances at which the three-body scattering hypervolume becomes divergent or anomalously large.
The low-energy scattering of three bosons or distinguishable particles with short-range interactions is characterized by a fundamental parameter, the three-body scattering hypervolume. Its imaginary part is directly related to the three-body recombination rate in a quantum gas consisting of such particles. We derive an analytical formula of it for weak interactions, and perform its first numerical calculations for bosons with a variable nonzero-range potential. For attractive interactions, we identify several three-body resonances at which the three-body scattering hypervolume becomes divergent or anomalously large.
The low-energy scattering of three bosons or distinguishable particles with short-range interactions is characterized by a fundamental parameter, the three-body scattering hypervolume, which is responsible for the nonuniversal effects in dilute Bose-Einstein condensates. We derive an analytical formula of the three-body scattering hypervolumes for weak interactions. When the interaction supports two-body bound states, the three-body scattering hypervolume gains a negative imaginary part, which is directly related to the rate constant for three-body recombinations. We develop a numerical method to calculate the three-body scattering hypervolumes for some model potentials with variable strengths. For real atoms with van der Waals potential, the three-body scattering hypervolumes are much harder to compute, because the three-body wave function is highly oscillatory at smaller inter-atomic distances. However, they may be extractable from precision data such as the collective-mode frequencies of trapped Bose-Einstein condensates. In many numerical simulations, the system being simulated is put into a large but finite volume, such as a large periodic box of size L. To extract the low-energy scattering properties of two particles in infinite space from such simulations, L\\"uscher's formula must be used. In the second part of this thesis, we generalize L\\"uscher's formula to d spatial dimensions. We obtain its s-wave and p-wave approximations and the systematic expansions of the energies of the low-lying states. At a s-wave resonance in dge4 dimensions, we identify two low-lying states close to the threshold with nearly opposite energies, Esimpm1/Ld/2 when dge5, or Esimpm1/L2sqrtlnL when d=4. These calculations provide important insights to the physics of three particles at a three-body resonance in finite volumes in two or three dimensions. Three-body resonances are important phenomena not yet completely explored in the experiments on cold atoms.
We analyze the zero energy collision of three identical bosons in the same internal state with total orbital angular momentum L=2, assuming short range interactions. By solving the Schrödinger equation asymptotically, we derive two expansions of the wave function when three bosons are far apart or a pair of bosons and the third boson are far apart. The scattering hypervolume D is defined for this collision. Unlike the scattering hypervolume defined by one of us in 2008, whose dimension is length to the fourth power, the dimension of D studied in the present paper is length to the eighth power. We then derive the expression of D when the interaction potentials are weak, using the Born's expansion. We also calculate the energy shift of such three bosons with three different momenta ℏk1, ℏk2 and ℏk3 in a large periodic box. The obtained energy shift depends on D(0)/Ω2 and D/Ω2, where D(0) is the three-body scattering hypervolume defined for the three-body L=0 collision and Ω is the volume of the periodic box. We also calculate the contribution of D to the three-body T-matrix element for low-energy collisions. We then calculate the shift of the energy and the three-body recombination rate due to D(0) and D in the dilute homogeneous Bose gas. The contribution to the three-body recombination rate constant from D is proportional to T2 if the temperature T is much larger than the quantum degeneracy temperature but still much lower than the temperature scale at which the thermal de Broglie wave length becomes comparable to the physical range of interaction.
The elastic scattering properties of three bosons at low energy enter the many-body description of ultracold Bose gases via the three-body scattering hypervolume D. We study this quantity for identical bosons that interact via a pairwise finite-range potential. Our calculations cover the regime from strongly repulsive potentials towards attractive potentials supporting multiple two-body bound states and are consistent with the few existing predictions for D. We present a numerical confirmation of the universal predictions for D in the strongly interacting regime, where Efimov physics dominates, for a local nonzero-range potential. Our findings highlight how D is influenced by three-body quasibound states with strong d-wave or g-wave characteristics in the weakly interacting regime.
We analyze the collision of three particles with arbitrary mass ratio at zero collision energy, assuming arbitrary short-range potentials, and generalize the three-body scattering hypervolume D first defined for identical bosons in 2008. We solve the three-body Schr\"odinger equation asymptotically when the three particles are far apart or one pair and a third particle are far apart, deriving two asymptotic expansions of the wave function, and the parameter D appears at the order 1/B4, where B is the overall size of the triangle formed by the particles. We then analyze the ground state energy of three such particles with vanishing or negligible two-body scattering lengths in a large periodic volume of side length L, where the three-body parameter contributes a term of the order D/L6. From this result we derive some properties of a two-component Bose gas with negligible two-body scattering lengths: its energy density at zero temperature, the corresponding generalized Gross-Pitaevskii equation, the conditions for the stability of the two-component mixture against collapse or phase separation, and the decay rates of particle densities due to three-body recombination.
We study the wave function φ(3) of three identical bosons scattering at zero energy, zero total momentum, and zero orbital angular momentum in two dimensions, interacting via short-range potentials with a finite two-body scattering length a. We derive asymptotic expansions of φ(3) in two regimes: the 111-expansion, where all three pairwise distances are large, and the 21-expansion, where one particle is far from the other two. In the 111-expansion, the leading term grows as ln3(B/a) at large hyperradius B=(s12+s22+s32)/2. At order B−2ln−3(B/a), we identify a three-body parameter D with dimension of length squared, which we term the three-body scattering area. This quantity should be contrasted with the three-body scattering area previously studied for infinite or vanishing two-body scattering length. If the two-body interaction is attractive and supports bound states, D acquires a negative imaginary part, and we derive its relation to the probability amplitudes for the production of two-body bound states in three-body collisions. Under weak modifications of the interaction potentials, we derive the corresponding shift of D in terms of φ(3) and the changes of the two-body and three-body potentials. We also study the effects of D and φ(3) on three-body and many-body physics, including the three-body ground-state energy in a large periodic volume, the many-body energy and the three-body correlation function of the dilute two-dimensional Bose gas, and the three-body recombination rates of two-dimensional ultracold atomic Bose gases.
We study the zero-energy collision of three identical spin-polarized fermions with short-range interactions in one dimension. We derive the asymptotic expansions of the three-body wave function when the three fermions are far apart or one pair and the third fermion are far apart, and the three-body scattering hypervolume DF appears in the coefficients of such expansions. If the two-body interaction is attractive and supports two-body bound states, DF acquires a negative imaginary part related to the amplitudes of the outgoing waves describing the departure of the resultant bound pair and the remaining free fermion. For weak interaction potentials, we derive an approximate formula of the hypervolume by using the Born expansion. For the square-barrier and the square-well potentials and the Gaussian potential, we solve the three-body Schr\"odinger equation to compute DF numerically. We also calculate the shifts of energy and of pressure of spin-polarized one-dimensional Fermi gases due to a nonzero DF and the three-body recombination rate in one dimension.