Yue Chen,Da-Wu Xiao and Peng Zhang,2

1 Department of Physics,Renmin University of China,Beijing 100872,China

2 Beijing Computational Science Research Center,Beijing 100193,China

3 Department of Physics,Centre for Quantum Coherence,and The Hong Kong Institute of Quantum Information Science and Technology,The Chinese University of Hong Kong,Shatin,New Territories,Hong Kong,China

Abstract We calculate the energy spectrum of three identical fermionic ultracold atoms in two different internal states confined in a two-dimensional anisotropic harmonic trap.Using the solutions of the corresponding two-body problems obtained in our previous work(Chen et al 2020 Phys.Rev.A 101,053624),we derive the explicit transcendental equation for the eigen-energies,from which the energy spectrum is derived.Our results can be used for the calculation of the 3rd Virial coefficients or the studies of few-body dynamics.

Keywords:ultracold atom,three-body problem,two-dimensional system

1.Introduction

The study of few-body problems is an important direction of ultracold atom physics.The few-atom scattering amplitudes and energy spectra can be used to derive physical quantities which are important for the study of many-body physics,e.g.the effective inter-atomic interaction intensity and the Viral coefficients[1–4].One can also obtain basic understandings for the effects induced by inter-atomic interaction via the properties of few-body systems[5–12],or develop new techniques for the control of effective inter-atomic interaction via studying scattering problems of two ultracold atoms in various confinements[13–28],or under laser and magnetic fields[29–36].On the other hand,the ultracold atom system is an excellent platform for the study of important few-body problems,e.g.the Efimov effect[37,38].

Due to the above reasons,theoretical calculations for few-body problems of ultracold atoms have attracted much attention.For confined atoms,so far people have obtained analytical solutions of two-body energy spectra for the systems in various harmonic confinements with arbitrary dimensions and anisotropicity[39–43].The three-body energy spectra have also been derived for the ultracold atoms in a two-dimensional(2D)[44]or 3D[45]isotropic harmonic confinement,or a 3D axial symmetric anisotropic harmonic confinement[2].Nevertheless,the energy spectra of three ultracold atoms in a 2D anisotropic or a 3D completely anisotropic harmonic confinement have not been obtained.

In this work,we calculate the energy spectrum of three ultracold fermionic atoms in an anisotropic 2D confinement.Explicitly,we consider three identical fermionic atoms 1,2,and 3.As shown in figure 1,atoms 1 and 2 are in the same pseudo-spin state(↑),and atom 3 is in another pseudo-spin state(↓).We assume that there is ans-wave short-range interaction between any two atoms in different internal states,i.e.the atoms 1–3 and 2–3,which is described by the 2D scattering lengtha2D.The eigen-energies of these three atoms are calculated as functions ofa2Dand the trapping frequencies in the two spatial directions.Our approach is based on the analytical solution of the corresponding two-body problem,which was obtained in our previous work[43],as well as the method of[2]for the construction of the transcendental equation for the three-body eigen-energy with the solution of the two-body problem.Our results can be used for the study of many-body,or few-body physics of 2D ultracold gases,e.g.the calculation of the 3rd Viral coefficients for twocomponent Fermi gases or the research for the three-body dynamics.In addition,since the confined three-body systems of ultracold systems have been realized in experiments of recent years,the energy spectrum we obtained can be experimentally examined.

Figure 1.The three-atom system and the definitions of the Jacobi coordinates(ρ,R)(a)and(ρ*,R*)(b).

The remainder of this paper is organized as follows.In section 2,we show our calculation approach and derive the equation of the three-body eigen-energy.The results of our calculation are shown in section 3.In section 4 there is a summary and some discussions.

2.Equation of energy spectrum

2.1.System and model

2.2.Schr?dinger equation and boundary conditions

2.3.Equation for eigen-energies

Figure 2.The energy spectra of the relative motion of three fermions in a 2D harmonic trap,for the cases with aspect ratio η=ωx/ωy=1((a1)–(a4)),η=3((b1)–(b4))and η=6((c1)–(c4)).For each given value of η,we show the results for the four parity combinations(ee),(eo),(oe)and(oo)defined in section 2.4.In the horizontal labels,the length scale d is defined as .

2.4.Confinement parity

3.Results

In figure 2 we illustrate the energy spectrum given by the numerical solutions of equation(35),for the systems with the aspect ratio

For each fixed η,the energy spectrum four parity combinations(ee),(eo),(oe)and(oo)defined in section 2.4 are displayed separately in the subfigures of figure 2.

4.Summary

In this work,we calculate the energy spectrum of three identical fermionic ultracold atoms in two different internal states in a 2D anisotropic harmonic confinement.We derive the explicit transcendental equation for the eigen-energy of the relative motion of these three atoms,i.e.equation(35).By numerically solving this equation,the eigen-energies can be calculated as functions of the 2D scattering lengtha2Dbetween two atoms in different internal states.Our results can be used for the study of few-body dynamics or many-body properties,e.g.the calculation of the second 3rd Virial coefficients.Our method can be generalized to the three-body problems in 2D confinements of Bosonic or distinguishable atoms.

Acknowledgments

We thank Mr Ze-Qing Wang for very helpful discussions.This work is supported in part by the National Key Research and Development Program of China Grant No.2018YFA0306502,NSAF(Grant No.U1930201),and also supported by the Fundamental Research Funds for the Central Universities and the Research Funds of Renmin University of China under Grant No.21XNH088.