Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:quant-ph/0604205 · Physical Review A

Analytical solutions for the dynamics of two trapped interacting ultracold atoms

Zbigniew Idziaszek, Tommaso Calarco

Published Aug 8, 2006• 120 citations
arXivHTMLPDF

Abstract

We discuss exact solutions of the Schr\"odinger equation for the system of two ultracold atoms confined in an axially symmetric harmonic potential. We investigate different geometries of the trapping potential, in particular we study the properties of eigenenergies and eigenfunctions for quasi-one-dimensional and quasi-two-dimensional traps. We show that the quasi-one-dimensional and the quasi-two-dimensional regimes for two atoms can be already realized in the traps with moderately large (or small) ratios of the trapping frequencies in the axial and the transverse directions. Finally, we apply our theory to Feshbach resonances for trapped atoms. Introducing in our description an energy-dependent scattering length we calculate analytically the eigenenergies for two trapped atoms in the presence of a Feshbach resonance.

DOI
10.1103/physreva.74.022712

Related papers

Analytical solutions for two atoms in a harmonic trap: p -wave interactions

Zbigniew Idziaszek

We derive analytical solutions for the system of two ultracold spin-polarized fermions interacting in $p$ wave and confined in an axially symmetric harmonic trap. To this end we utilize $p$-wave pseudopotential with an energy-dependent scattering volume. This allows to describe the scattering in tight trapping potentials in the presence of scattering resonances. We verify predictions of the pseudopotential treatment for some model interaction potential obtaining an excellent agreement with exact energy levels. Then we turn to the experimentally relevant case of neutral atom interactions in the vicinity of a $p$-wave Feshbach resonance. In the framework of the multichannel quantum-defect theory we derive relatively simple formula for an energy-dependent scattering volume, and later we apply it to investigate the energy spectrum of trapped atoms close to the $p$-wave Feshbach resonance.

arXiv:0902.2167•Physical Review A•Jun 4, 2009•30 citations

Two atoms in an anisotropic harmonic trap

Zbigniew Idziaszek, Tommaso Calarco

We consider the system of two interacting atoms confined in axially symmetric harmonic trap. Within the pseudopotential approximation, we solve the Schr\"odinger equation exactly, discussing the limits of quasi-one-and quasi-two-dimensional geometries. Finally, we discuss the application of an energy-dependent pseudopotential, which allows us to extend the validity of our results to the case of tight traps and large scattering lengths.

arXiv:quant-ph/0410163•Physical Review A•May 23, 2005•79 citations

Analytical solution for the spectrum of two ultracold atoms in a completely anisotropic confinement

Yue Chen, Da-Wu Xiao, Ren Zhang +1 more

We study the system of two ultracold atoms in a three-dimensional (3D) or two-dimensional (2D) completely anisotropic harmonic trap. We derive the transcendental equation ${J}_{3\mathrm{D}}(E)=1/{a}_{3\mathrm{D}}$ (${J}_{2\mathrm{D}}(E)=ln{a}_{2\mathrm{D}}$) for the eigenenergy $E$ of this system in the 3D (2D) case, with ${a}_{3\mathrm{D}}$ and ${a}_{2\mathrm{D}}$ being the corresponding $s$-wave scattering lengths, and provide the analytical expressions of the functions ${J}_{3\mathrm{D}}(E)$ and ${J}_{2\mathrm{D}}(E)$. In previous research, this type of equation was obtained for spherically or axially symmetric harmonic traps [T. Busch et al., Found. Phys. 28, 549 (1998); Z. Idziaszek and T. Calarco, Phys. Rev. A 74, 022712 (2006)]. However, for our cases with a completely anisotropic trap, only the equation for the ground-state energy of some cases has been derived [J.-J. Liang and C. Zhang, Phys. Scr. 77, 025302 (2008)]. Our results in this work are applicable for arbitrary eigenenergy of this system and can be used for the study of dynamics and thermal dynamics of interacting ultracold atoms in this trap, e.g., the calculation of the second virial coefficient or the evolution of two-body wave functions. In addition, our approach for the derivation of the above equations can also be used for other two-body problems of ultracold atoms.

arXiv:2002.02168•Physical Review A•May 14, 2020•8 citations

Controlled collisions of two ultracold atoms in separate harmonic traps

Michał Krych, Zbigniew Idziaszek

We consider controlled collisions between two ultracold atoms guided by external harmonic potentials. We derive analytical solutions of the Schr\"odinger equation for this system and investigate the properties of eigenergies and eigenstates for different trap geometries as a function of a trap separation and of the scattering length. When varying the trap separation the energy spectrum exhibits avoided crossings, corresponding to trap-induced shape resonances. Introducing an energy-dependent scattering length we investigate the behavior of the system in the vicinity of a magnetic Feshbach resonance. Finally, we illustrate our analytical results with two examples: the quantum phase gate controlled by the external magnetic field and a scheme for a coherent transport of atoms in optical lattices into higher Bloch bands.

arXiv:0904.4203•Physical Review A•Aug 19, 2009•22 citations

A model study on a pair of trapped particles interacting with an arbitrary effective range

Partha Goswami, Bimalendu Deb

Abstract We study the effects of the effective range of interaction on the eigenvalues and eigenstates of two particles confined in a three-dimensional (3D) isotropic as well as one- or quasi-one dimensional harmonic (1D) traps. For this we employ model potentials which mimic finite-range s -wave interactions over a wide range of s -wave scattering length a s including the unitarity limits a s → ± ∞ . Our results show that when the range is larger than the 3D or 1D harmonic oscillator length scale, the eigenvalues and eigenstates are nearly similar to those of noninteracting two particles in the 3D or 1D trap, respectively. In case of 3D, we find that when the range goes to zero, the results of contact potential as derived by Busch et al (1998 Foundations of Physics 28 549 ) are reproduced. However, in the case of 1D, such reproducibility does not occur as the range goes to zero. We have calculated the eigenvalues and eigenstates in a 1D harmonic trap taking one dimensional finite-range model potential. We have also calculated the bound state properties of two particles confined in a highly anisotropic quasi-1D trap taking three-dimensional finite-range model potential, and examined whether these quasi-1D results approach towards 1D ones as the aspect ratio η of the radial to axial frequency of the trap increases. We find that if the range is very small compared to the axial size of the trap, then one can reach 1D regime for η ≥ 10000 . However, for a large range, one can almost get 1D results for smaller values of η . This study will be important for the exploration of two-body or many body physics of trapped ultracold atoms interacting with narrow Feshbach resonance for which the effective range can be large.

arXiv:1609.04155•Physica Scripta•Jul 18, 2016•7 citations

Quantum States of Interacting Atoms in Quasi-One-Dimensional Ultracold Trapped Gases

Polyanna Guimarães e Miranda, Débora Nazaré de Freitas, André Luiz Mota

We theoretically investigate the quantum states of a Hamiltonian model for quasi-one-dimensional ultracold trapped gases. From the ansatz given by the numerical solution of the Schrödinger equation of the system, we develop a scattering potential functional form and an approximate solution for the analytical approach of the model. We obtain the set of approximate eigenstates and eigenenergies that can be used in future improvements on the study of atomic scattering in low dimensional ultracold gases. We also show that there is a parity inversion of the ground state of the model as the interaction strength increases.

Journal of Modern Physics•Jan 1, 2018•0 citations
We discuss exact solutions of the Schr"odinger equation for the system of two ultracold atoms confined in an axially symmetric harmonic potential. We investigate different geometries of the trapping potential, in particular we study the properties of eigenenergies and eigenfunctions for quasi-one-dimensional and quasi-two-dimensional traps. We show that the quasi-one-dimensional and the quasi-two-dimensional regimes for two atoms can be already realized in the traps with moderately large (or small) ratios of the trapping frequencies in the axial and the transverse directions. Finally, we apply our theory to Feshbach resonances for trapped atoms. Introducing in our description an energy-dependent scattering length we calculate analytically the eigenenergies for two trapped atoms in the presence of a Feshbach resonance.
We derive analytical solutions for the system of two ultracold spin-polarized fermions interacting in ppp wave and confined in an axially symmetric harmonic trap. To this end we utilize ppp-wave pseudopotential with an energy-dependent scattering volume. This allows to describe the scattering in tight trapping potentials in the presence of scattering resonances. We verify predictions of the pseudopotential treatment for some model interaction potential obtaining an excellent agreement with exact energy levels. Then we turn to the experimentally relevant case of neutral atom interactions in the vicinity of a ppp-wave Feshbach resonance. In the framework of the multichannel quantum-defect theory we derive relatively simple formula for an energy-dependent scattering volume, and later we apply it to investigate the energy spectrum of trapped atoms close to the ppp-wave Feshbach resonance.
We study the system of two ultracold atoms in a three-dimensional (3D) or two-dimensional (2D) completely anisotropic harmonic trap. We derive the transcendental equation J3D(E)=1/a3D{J}_{3\mathrm{D}}(E)=1/{a}_{3\mathrm{D}}J3D​(E)=1/a3D​ (J2D(E)=lna2D{J}_{2\mathrm{D}}(E)=ln{a}_{2\mathrm{D}}J2D​(E)=lna2D​) for the eigenenergy EEE of this system in the 3D (2D) case, with a3D{a}_{3\mathrm{D}}a3D​ and a2D{a}_{2\mathrm{D}}a2D​ being the corresponding sss-wave scattering lengths, and provide the analytical expressions of the functions J3D(E){J}_{3\mathrm{D}}(E)J3D​(E) and J2D(E){J}_{2\mathrm{D}}(E)J2D​(E). In previous research, this type of equation was obtained for spherically or axially symmetric harmonic traps [T. Busch et al., Found. Phys. 28, 549 (1998); Z. Idziaszek and T. Calarco, Phys. Rev. A 74, 022712 (2006)]. However, for our cases with a completely anisotropic trap, only the equation for the ground-state energy of some cases has been derived [J.-J. Liang and C. Zhang, Phys. Scr. 77, 025302 (2008)]. Our results in this work are applicable for arbitrary eigenenergy of this system and can be used for the study of dynamics and thermal dynamics of interacting ultracold atoms in this trap, e.g., the calculation of the second virial coefficient or the evolution of two-body wave functions. In addition, our approach for the derivation of the above equations can also be used for other two-body problems of ultracold atoms.
We consider controlled collisions between two ultracold atoms guided by external harmonic potentials. We derive analytical solutions of the Schr\"odinger equation for this system and investigate the properties of eigenergies and eigenstates for different trap geometries as a function of a trap separation and of the scattering length. When varying the trap separation the energy spectrum exhibits avoided crossings, corresponding to trap-induced shape resonances. Introducing an energy-dependent scattering length we investigate the behavior of the system in the vicinity of a magnetic Feshbach resonance. Finally, we illustrate our analytical results with two examples: the quantum phase gate controlled by the external magnetic field and a scheme for a coherent transport of atoms in optical lattices into higher Bloch bands.
Abstract We study the effects of the effective range of interaction on the eigenvalues and eigenstates of two particles confined in a three-dimensional (3D) isotropic as well as one- or quasi-one dimensional harmonic (1D) traps. For this we employ model potentials which mimic finite-range s -wave interactions over a wide range of s -wave scattering length a s including the unitarity limits a s → ± ∞ . Our results show that when the range is larger than the 3D or 1D harmonic oscillator length scale, the eigenvalues and eigenstates are nearly similar to those of noninteracting two particles in the 3D or 1D trap, respectively. In case of 3D, we find that when the range goes to zero, the results of contact potential as derived by Busch et al (1998 Foundations of Physics 28 549 ) are reproduced. However, in the case of 1D, such reproducibility does not occur as the range goes to zero. We have calculated the eigenvalues and eigenstates in a 1D harmonic trap taking one dimensional finite-range model potential. We have also calculated the bound state properties of two particles confined in a highly anisotropic quasi-1D trap taking three-dimensional finite-range model potential, and examined whether these quasi-1D results approach towards 1D ones as the aspect ratio η of the radial to axial frequency of the trap increases. We find that if the range is very small compared to the axial size of the trap, then one can reach 1D regime for η ≥ 10000 . However, for a large range, one can almost get 1D results for smaller values of η . This study will be important for the exploration of two-body or many body physics of trapped ultracold atoms interacting with narrow Feshbach resonance for which the effective range can be large.
We theoretically investigate the quantum states of a Hamiltonian model for quasi-one-dimensional ultracold trapped gases. From the ansatz given by the numerical solution of the Schrödinger equation of the system, we develop a scattering potential functional form and an approximate solution for the analytical approach of the model. We obtain the set of approximate eigenstates and eigenenergies that can be used in future improvements on the study of atomic scattering in low dimensional ultracold gases. We also show that there is a parity inversion of the ground state of the model as the interaction strength increases.