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The scattering of a few particles with short-range interactions at low energies

Shangguo Zhu

Published Nov 8, 2018• 0 citations
OpenAlex

Abstract

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 $d\\ge4$ dimensions, we identify two low-lying states close to the threshold with nearly opposite energies, $E\\sim \\pm 1/L^{d/2}$ when $d\\ge 5$, or $E \\sim \\pm 1/L^2 \\sqrt{\\ln L}$ 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.

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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 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.
It is shown that the effective interaction strength of three bosons at small collision energies can be extracted from their wave function at zero energy. Asymptotic expansions of this wave function at large interparticle distances are derived, from which is defined a quantity DDD named three-body scattering hypervolume, which is an analog of the two-body scattering length. Given any finite-range interactions, one can thus predict the effective three-body force from a numerical solution of the Schr\"odinger equation. In this way, the constant DDD for hard-sphere bosons is computed, leading to the first complete result for the ground-state energy per particle of a dilute Bose-Einstein condensate (BEC) of hard spheres to order ${\ensuremath{\rho}}^{2}$, where $\ensuremath{\rho}$ is the number density. Effects of DDD are also demonstrated in the three-body energy in a finite box of size LLL, which is expanded to the order ${L}^{\ensuremath{-}7}$, and in the three-body scattering amplitude in vacuum. The three-body scattering amplitude calculated in this paper disagrees with an earlier calculation in the literature, because of the omission of the two-body effective range in that earlier work. Another key prediction is the condensate fractions of dilute BECs, which also disagree with an earlier work in the literature based on the effective field theory (EFT), as a result of short-range physics. An EFT prediction of the BEC ground-state energy, however, is corroborated.
We study the wave function φ(3)φ^{(3)}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 aaa. We derive asymptotic expansions of φ(3)φ^{(3)}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 ln⁡3(B/a)\ln^3(B/a)ln3(B/a) at large hyperradius B=(s12+s22+s32)/2B=\sqrt{(s_1^2+s_2^2+s_3^2)/2}B=(s12​+s22​+s32​)/2​. At order B−2ln⁡−3(B/a)B^{-2}\ln^{-3}(B/a)B−2ln−3(B/a), we identify a three-body parameter DDD 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, DDD 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 DDD in terms of φ(3)φ^{(3)}φ(3) and the changes of the two-body and three-body potentials. We also study the effects of DDD and φ(3)φ^{(3)}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 analyze the zero energy collision of three identical bosons in the same internal state with total orbital angular momentum L=2L=2L=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 DDD 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 DDD studied in the present paper is length to the eighth power. We then derive the expression of DDD 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\hbar \mathbf{k_{1}}ℏk1​, ℏk2\hbar\mathbf{k_{2}}ℏk2​ and ℏk3\hbar\mathbf{k_{3}}ℏk3​ in a large periodic box. The obtained energy shift depends on D(0)/Ω2D^{(0)}/Ω^{2}D(0)/Ω2 and D/Ω2D/Ω^{2}D/Ω2, where D(0)D^{(0)}D(0) is the three-body scattering hypervolume defined for the three-body L=0L=0L=0 collision and ΩΩΩ is the volume of the periodic box. We also calculate the contribution of DDD 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)D^{(0)}D(0) and DDD in the dilute homogeneous Bose gas. The contribution to the three-body recombination rate constant from DDD is proportional to T2T^2T2 if the temperature TTT 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.
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.
We derive the asymptotic expansions of the wave function of three particles having equal mass with finite-range interactions and infinite or zero two-dimensional scattering length colliding at zero energy and zero orbital angular momentum, from which a three-body parameter DDD is defined. The dimension of DDD is length squared, and we call DDD three-body scattering area. We find that the ground-state energy per particle of a zero-temperature dilute Bose gas with these interactions is approximately $\frac{{\ensuremath{\hbar}}^{2}D}{6m}{\ensuremath{\rho}}^{2}$, where $\ensuremath{\rho}$ is the number density of the bosons and mmm is the mass of each boson. Such a zero-temperature Bose gas is stable at D>0D>0D>0 in the thermodynamic limit, and metastable at D<0D<0D<0 in the isotropic harmonic trap if the number of bosons is less than ${N}_{\mathrm{cr}}\ensuremath{\approx}3.6413\sqrt{\frac{\ensuremath{\hbar}}{m\ensuremath{\omega}|D|}\phantom{{}^{1}}}$, where $\ensuremath{\omega}$ is the angular frequency of the harmonic trap. If the two-body interaction supports bound states, DDD typically acquires a negative imaginary part, and we find the relation between this imaginary part and the amplitudes of the pair-boson production processes. We derive a formula for the three-body recombination rate constant of the many-boson system in terms of the imaginary part of DDD.