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Communications in Theoretical Physics

Path of Momentum Integral in the Skorniakov-Ter-Martirosian Equation*

Chao Gao, Peng Zhang

Published Dec 1, 2018• 0 citations
Publisher

Abstract

Abstract The Skorniakov-Ter-Martirosian (STM) integral equation is widely used for the quantum three-body problems of low-energy particles (e.g., ultracold atom gases). With this equation these three-body problems can be efficiently solved in the momentum space. In this approach the boundary condition for the case that all the three particles are gathered together is described by the upper limit of the momentum integral, i.e., the momentum cutoff. On the other hand, in realistic systems, the three-body recombination (TBR) process can occur when all these three particles are close to each other. In this process two particles form a deep dimer and the other particle can gain high kinetic energy and then escape from the low-energy system. In the presence of the TBR process, the momentum-cutoff in the STM equation would include a non-zero imaginary part. As a result, the momentum integral in the STM equation should be done in the complex-momentum plane. In this case the result of the integral depends on the choice of the integral path. Obviously, only one integral path can lead to the correct result. In this paper we consider how to correctly choose the integral path for the STM equation. We take the atom-dimer scattering problem in a specific ultracold atom gas as an example, and show the results given by different integral paths. Based on the result for this case we explore the reasonable integral paths for general case .

DOI
10.1088/0253-6102/70/6/753

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Abstract The Skorniakov-Ter-Martirosian (STM) integral equation is widely used for the quantum three-body problems of low-energy particles (e.g., ultracold atom gases). With this equation these three-body problems can be efficiently solved in the momentum space. In this approach the boundary condition for the case that all the three particles are gathered together is described by the upper limit of the momentum integral, i.e., the momentum cutoff. On the other hand, in realistic systems, the three-body recombination (TBR) process can occur when all these three particles are close to each other. In this process two particles form a deep dimer and the other particle can gain high kinetic energy and then escape from the low-energy system. In the presence of the TBR process, the momentum-cutoff in the STM equation would include a non-zero imaginary part. As a result, the momentum integral in the STM equation should be done in the complex-momentum plane. In this case the result of the integral depends on the choice of the integral path. Obviously, only one integral path can lead to the correct result. In this paper we consider how to correctly choose the integral path for the STM equation. We take the atom-dimer scattering problem in a specific ultracold atom gas as an example, and show the results given by different integral paths. Based on the result for this case we explore the reasonable integral paths for general case .
We consider fermion-dimer scattering in the presence of a large positive scattering length in the frame of functional renormalization group equations. A flow equation for the momentum dependent fermion-dimer scattering amplitude is derived from first principles in a systematic vertex expansion of the exact flow equation for the effective action. The resummation obtained from the nonperturbative flow is shown to be equivalent to the one performed by the integral equation by Skorniakov and Ter-Martirosian (STM). The flow equation approach allows to integrate out fermions and bosons simultaneously, in line with the fact that the bosons are not fundamental but build up gradually as fluctuation induced bound states of fermions. In particular, the STM result for atom-dimer scattering is obtained by choosing the relative cutoff scales of fermions and bosons such that the fermion fluctuations are integrated out already at the initial stage of the RG evolution.
We consider the problem of obtaining the scattering length for a fermion colliding with a dimer, formed from a fermion identical to the incident one and another different fermion. This is done in the universal regime where the range of interactions is short enough that the scattering length aaa for nonidentical fermions is the only relevant quantity. This is the generalization to fermions with different masses of the problem solved long ago by Skorniakov and Ter-Martirosian for particles with equal masses. We solve this problem analytically in the two limiting cases where the mass of the solitary fermion is very large or very small compared to the mass of the two other identical fermions. This is done for both the value of the scattering length and the function entering the Skorniakov--Ter-Martirosian integral equation, for which simple explicit expressions are obtained.
Abstract The characterization of few-body interactions is fundamental for understanding the properties and collisional stability of ultracold quantum gases in reduced dimensions. In this work, we theoretically investigate the atom-dimer scattering and three-body recombination processes in a two-dimensional quantum gas. By generalizing the Skorniakov–Ter-Martirosian equations to two dimensions, we obtain the exact atom-dimer scattering amplitude and systematically evaluate the validity of common theoretical approximations, revealing a distinct feature of two-dimensional (2D) collisions: while mean-field theory fails rapidly with increasing interaction strength, the single-pole approximation remains remarkably accurate across a broad interaction regime. Furthermore, we extend our exact framework to non-zero incident momenta to calculate the three-body recombination rate. Our results demonstrate that the lifetime of the 2D quantum gas exhibits a power-law dependence on the 2D gas parameter n a 2D 2 with an exponent of approximately − 2.45 , showing a significant reduction in stability as interactions are enhanced. This study bridges the gap between exact few-body theory and practical many-body approximations, offering a robust benchmark for predicting the stability of cold atomic mixtures and guiding future experimental explorations of universal few-body physics in highly controllable 2D environments.
Generally, the three-body recombination (TBR) of the ultracold helium atoms in the zero-collision-energy limit could be described by the original zero-range (Efimov) universal function [E. Braaten, D. Kang, and L. Platter, Phys. Rev. A 75, 052714 (2007)] using the scattering length aaa of the He-He interaction and a three-body parameter a∗{a}_{*}a∗​ as input, where a∗{a}_{*}a∗​ is associated with the positive scattering length at which an interference minimum in the TBR rate occurs at the threshold. This universal property has been reinvestigated by considering various post-Born-Oppenheimer (post-BO) effects on the relevant scattering process. It is found that a linear correction to 1/a∗1/{a}_{*}1/a∗​ versus 1/a1/a1/a should be introduced to give a better description of these considered TBR cases because of the finite-range effects. Particularly, such a correction may be ascribed not only to the general scattering properties associated with the Efimov channel at the short hyperradial distance, but also to the tunneling probability in the lowest incident channel. In addition, a successful fit of the modified universal function that includes correctly the finite-range effects [E. Garrido, M. Gattobigio, and A. Kievsky, Phys. Rev. A 88, 032701 (2013)] or the zero-range universal function with a linearly variational 1/a∗1/{a}_{*}1/a∗​ versus 1/a1/a1/a to numerically accurate data also indicates a clear manifestation of the van der Waals universality. Finally, the post-BO effects on the TBR process for the partial wave with the total angular momentum J>0J>0J>0 are also discussed.
For identical bosons with a large scattering length, the dependence of the three-body recombination rate on the collision energy is determined in the zero-range limit by universal functions of a single scaling variable. There are six scaling functions for angular momentum zero and one scaling function for each higher partial wave. We calculate these universal functions by solving the Skorniakov-Ter-Martirosian equation. The results for the three-body recombination as a function of the collision energy are in good agreement with previous results from solving the three-body Schr\"odinger equation for 4He^{4}\mathrm{He}4He atoms. The universal scaling functions can be used to calculate the three-body recombination rate at nonzero temperature. We obtain an excellent fit to the data from Grimm and co-workers for 133Cs^{133}\mathrm{Cs}133Cs atoms with a large positive scattering length.
The increase of the number of the two-body recombination channels strongly challenges the numerical calculation of the accurate rates for the three-body recombination (TBR) process and its reverse process, collision-induced dissociation (CID), at ultracold temperatures. By taking the 4He–4He–20Ne collision system as an example, we have obtained the rates for its TBR and CID processes involving all four recombination channels, including the two-body states 4He2 (l = 0) and 4He20Ne (l = 0, 1, 2) with l the rotational quantum number. By using the adiabatic hyperspherical method, we have considered not only total angular momentum J = 0 but also J > 0 in the ultracold collision energies (E = 0.01 − 100 mK × kB). It is found that 4He2 (l = 0) is the major product after the TBR process in the ultracold limit (E ≤ 0.1 mK × kB). The TBR rate into 4He2 (l = 0) is nearly one order of magnitude larger than the sum of the other three products, 4He20Ne (l = 0, 1, 2). Moreover, the CID rates for the three 4He20Ne (l = 0, 1, 2) + 4He initial states are close to each other and are smaller than that for the 4He2 (l = 0) + 20Ne initial state. Additionally, we have, for the first time, performed the channel-resolved scattering calculation that can explain the above-mentioned findings quantitatively.