We introduce a coupled-channel method for elastic three-body scattering in systems of identical bosonic alkali-metal atoms. The approach relies on the numerically exact two-body off-the-energy-shell transition matrix, constructed from realistic multichannel molecular interaction potentials that support many bound states. By rigorously accounting for this off-shell structure, the method captures both the short-range physics as well as multichannel couplings characteristic of alkali-metal potentials without resorting to model pseudopotentials. The central output is the complex three-body scattering hypervolume—the three-body analog of the two-body scattering length—which we obtain with controlled and verifiable numerical accuracy. As a realistic benchmark, we apply our framework to spin-polarized potassium-39, performing full coupled-channel three-body scattering calculations and extracting the hypervolume over experimentally relevant conditions. The method is general and transferable to other atomic species and interaction models featuring deep molecular potentials with an arbitrarily large number of bound states.
We introduce a coupled-channel method for elastic three-body scattering in systems of identical bosonic alkali-metal atoms. The approach relies on the numerically exact two-body off-the-energy-shell transition matrix, constructed from realistic multichannel molecular interaction potentials that support many bound states. By rigorously accounting for this off-shell structure, the method captures both the short-range physics as well as multichannel couplings characteristic of alkali-metal potentials without resorting to model pseudopotentials. The central output is the complex three-body scattering hypervolume—the three-body analog of the two-body scattering length—which we obtain with controlled and verifiable numerical accuracy. As a realistic benchmark, we apply our framework to spin-polarized potassium-39, performing full coupled-channel three-body scattering calculations and extracting the hypervolume over experimentally relevant conditions. The method is general and transferable to other atomic species and interaction models featuring deep molecular potentials with an arbitrarily large number of bound states.
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 develop a full multichannel spin model in momentum space to investigate three-body recombination of identical alkali-metal atoms colliding in a magnetic field. The model combines the exact three-atom spin structure and realistic pairwise atom-atom interactions. By neglecting the interaction between two particles when the spectating particle is not in its initial spin state we arrive at an approximate model. With this approximate model we achieve excellent agreement with the recent precise measurement of the ground Efimov resonance position in potassium-39 close to 33.58 G [R. Chapurin et al., Phys. Rev. Lett. 123, 233402 (2019)]. We analyze the limitations of our approximation by comparing to the numerical results for the full system and find that it breaks down for Feshbach resonances at larger magnetic fields in the same spin channel. There the relevant three-body closed channel thresholds are much closer to the open channel threshold, which enhances the corresponding multichannel couplings. Therefore the neglected components of the interaction should be included for those Feshbach resonances.
Abstract We give a detailed and self-contained description of a recently developed theoretical and numerical method for the simulation of three identical bosonic alkali-metal atoms near a Feshbach resonance, where the Efimov effect is induced. The method is based on a direct construction of the off-shell two-body transition matrix from exact eigenfunctions of the embedded two-body Hamiltonians, obtained using realistic parameterizations of the interaction potentials which accurately reproduce the molecular energy levels. The transition matrix is then inserted into the appropriate three-body integral equations, which may be efficiently solved on a computer. We focus especially on the power of our method in including rigorously the effects of multichannel physics on the three-body problem, which are usually accounted for only by various approximations. We demonstrate the method for 7Li, where we recently showed that a correct inclusion of this multichannel physics resolves the long-standing disagreement between theory and experiment regarding the Efimovian three-body parameter. We analyze the Efimovian enhancement of the three-body recombination rate on both sides of the Feshbach resonance, revealing strong sensitivity to the spin structure of the model thus indicating the prevalence of three-body spin-exchange physics. Finally, we discuss an extension of our methodology to the calculation of three-body bound-state energies.
We analyze the collision of three identical spin-polarized fermions at zero collision energy, assuming arbitrary finite-range potentials, and define the corresponding three-body scattering hypervolume DF. The scattering hypervolume D was first defined for identical bosons in 2008. It is the three-body analog of the two-body scattering length. We solve the three-body Schr\"odinger equation asymptotically when the three fermions are far apart or one pair and the third fermion are far apart, deriving two asymptotic expansions of the wave function. Unlike the case of bosons for which D has the dimension of length to the fourth power, here the DF we define has the dimension of length to the eighth power. We then analyze the interaction energy of three such fermions with momenta $\ensuremath{\hbar}{\mathbf{k}}_{1}$, $\ensuremath{\hbar}{\mathbf{k}}_{2}$, and $\ensuremath{\hbar}{\mathbf{k}}_{3}$ in a large periodic cubic box. The energy shift due to DF is proportional to ${D}_{F}/{\mathrm{\ensuremath{\Omega}}}^{2}$, where $\mathrm{\ensuremath{\Omega}}$ is the volume of the box. We also calculate the shifts of energy and pressure of spin-polarized Fermi gases due to a nonzero DF and the three-body recombination rate of spin-polarized ultracold atomic Fermi gases at finite temperatures.
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 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.