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arXiv:2105.04497 · Physical Review A

Three-body spin mixing in spin-1 Bose-Einstein condensates

P. M. A. Mestrom, Jing-Lun Li, V. E. Colussi, T. Secker, S. J. J. M. F. Kokkelmans

Published Aug 24, 2021• 9 citations
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Abstract

We study zero-energy collisions between three identical bosons with spin $f=1$ interacting via pairwise potentials. We quantify the corresponding three-body scattering hypervolumes, which parametrize the effective three-body interaction strengths in a many-body description of spin-1 Bose-Einstein condensates. Our results demonstrate universal behavior of the scattering hypervolumes for strong $s$- and $p$-wave two-body interactions. At weak interactions, we find that the real parts of the scattering hypervolumes are predominantly determined by hard-hyperspherelike collisions, which we characterize by a simple formula. With this universal result, we estimate that spin mixing via three-body collisions starts to dominate over two-body spin mixing at a typical particle density of ${10}^{17}\phantom{\rule{4pt}{0ex}}{\mathrm{cm}}^{\ensuremath{-}3}$ for $^{23}\mathrm{Na}$ and $^{41}\mathrm{K}$ spinor condensates. This density can be reduced by tuning the two-body interactions to an $s$- or $p$-wave dimer resonance or to a point where two-body spin mixing effectively vanishes. Another possibility to observe the effects of three-body spin mixing involves the application of weak magnetic fields to cancel out the effective two-body interaction strength in the characteristic timescale describing the spin dynamics.

DOI
10.1103/physreva.104.023321

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We study zero-energy collisions between three identical bosons with spin f=1f=1f=1 interacting via pairwise potentials. We quantify the corresponding three-body scattering hypervolumes, which parametrize the effective three-body interaction strengths in a many-body description of spin-1 Bose-Einstein condensates. Our results demonstrate universal behavior of the scattering hypervolumes for strong sss- and ppp-wave two-body interactions. At weak interactions, we find that the real parts of the scattering hypervolumes are predominantly determined by hard-hyperspherelike collisions, which we characterize by a simple formula. With this universal result, we estimate that spin mixing via three-body collisions starts to dominate over two-body spin mixing at a typical particle density of 1017cm\ensuremath−3{10}^{17}\phantom{\rule{4pt}{0ex}}{\mathrm{cm}}^{\ensuremath{-}3}1017cm\ensuremath−3 for 23Na^{23}\mathrm{Na}23Na and 41K^{41}\mathrm{K}41K spinor condensates. This density can be reduced by tuning the two-body interactions to an sss- or ppp-wave dimer resonance or to a point where two-body spin mixing effectively vanishes. Another possibility to observe the effects of three-body spin mixing involves the application of weak magnetic fields to cancel out the effective two-body interaction strength in the characteristic timescale describing the spin dynamics.
Abstract Optical trapping techniques allow for the formation of bosonic condensates with internal degrees of freedom, so-called spinor condensates. Mean-field models of spinor condensates highlight the sensitivity of the quantum phases of the system to the relative strength of the two-body interaction in the different spin-channels. Such a description captures the regime where these interactions are weak. In the opposite and largely unexplored regime of strongly correlated spinor condensates, three-body interactions can play an important role through the Efimov effect, producing possible novel phases. Here, we study the three-body spinor problem using the hyperspherical adiabatic representation for spin-1, -2 and -3 condensates in the strongly-correlated regime. We characterize the Efimov physics for such systems and the relevant three-body mean-field parameters. We find that the Efimov effect can strongly affect the spin dynamics and three-body mean-field contributions to the possible quantum phases of the condensate through universal contributions to scattering observables.
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 LLL. 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 ddd spatial dimensions. We obtain its sss-wave and ppp-wave approximations and the systematic expansions of the energies of the low-lying states. At a sss-wave resonance in dge4d\\ge4dge4 dimensions, we identify two low-lying states close to the threshold with nearly opposite energies, Esimpm1/Ld/2E\\sim \\pm 1/L^{d/2}Esimpm1/Ld/2 when dge5d\\ge 5dge5, or Esimpm1/L2sqrtlnLE \\sim \\pm 1/L^2 \\sqrt{\\ln L}Esimpm1/L2sqrtlnL when d=4d= 4d=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.
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 DDD. 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 DDD. We present a numerical confirmation of the universal predictions for DDD in the strongly interacting regime, where Efimov physics dominates, for a local nonzero-range potential. Our findings highlight how DDD is influenced by three-body quasibound states with strong ddd-wave or ggg-wave characteristics in the weakly interacting regime.
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{D}_{F}DF​ appears in the coefficients of such expansions. If the two-body interaction is attractive and supports two-body bound states, DF{D}_{F}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{D}_{F}DF​ numerically. We also calculate the shifts of energy and of pressure of spin-polarized one-dimensional Fermi gases due to a nonzero DF{D}_{F}DF​ and the three-body recombination rate in one dimension.
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{D}_{F}DF​. The scattering hypervolume DDD 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 DDD has the dimension of length to the fourth power, here the DF{D}_{F}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{D}_{F}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{D}_{F}DF​ and the three-body recombination rate of spin-polarized ultracold atomic Fermi gases at finite temperatures.
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 DFD_FDF​ appears in the coefficients of such expansions. If the two-body interaction is attractive and supports two-body bound states, DFD_FDF​ 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 Schrödinger equation to compute DFD_FDF​ numerically. We also calculate the shifts of energy and of pressure of spin-polarized one-dimensional Fermi gases due to a nonzero DFD_FDF​ and the three-body recombination rate in one dimension.