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:0908.2204 · Physical Review A

Two-atom energy spectrum in a harmonic trap near a Feshbach resonance at higher partial waves

Akira Suzuki, Yi Liang, Rajat K. Bhaduri

Published Sep 2, 2009• 24 citations
arXivHTMLPDF

Abstract

Two atoms in an optical lattice may be made to interact strongly at higher partial waves near a Feshbach resonance. These atoms, under appropriate constraints, could be bosonic or fermionic. The universal $l=2$ energy spectrum for such a system, with a caveat, is presented in this paper and checked with the spectrum obtained by direct numerical integration of the Schr\"odinger equation. The results reported here extend those of Yip for $p$-wave resonance [S.-K. Yip, Phys. Rev. A 78, 013612 (2008)], while exploring the limitations of a universal expression for the spectrum for the higher partial waves.

DOI
10.1103/physreva.80.033601

Related papers

Energy spectrum of two atoms interacting via a short-range potential in higher partial waves

Yi Liang

We study a two-atom system interacting via a short-range potential, and trapped by an optical lattice, in higher partial waves. These atoms may be made to interact strongly near a Feshbach resonance. Our main purpose is to find the relation between energy levels of the system and the scattering parameters in higher partial waves. We check our analytical results by numerical calculations and find the conditions under which this method is applicable for the d-wave.

•Aug 1, 2009•0 citations

Universality in s -wave and higher partial-wave Feshbach resonances: An illustration with a single atom near two scattering centers

Shangguo Zhu, Shina Tan

It is well-known that cold atoms near $s$-wave Feshbach resonances have universal properties that are insensitive to the short-range details of the interaction. What is less known is that atoms near higher partial-wave Feshbach resonances also have remarkable universal properties. We illustrate this with a single atom interacting resonantly with two fixed static centers. At a Feshbach resonance point with orbital angular momentum $L\ensuremath{\ge}1$, we find $2L+1$ shallow bound states whose energies behave like $1/{R}^{2L+1}$ when the distance $R$ between the two centers is large. We then compute corrections to the binding energies due to other parameters in the effective range expansions. For completeness we also compute the binding energies near $s$-wave Feshbach resonances, taking into account the corrections. Afterwards we turn to the bound states at large but finite scattering volumes. For $p$-wave and higher partial-wave resonances we derive a simple formula for the energies in terms of a parameter called ``proximity parameter.'' These results are applicable to a free atom interacting resonantly with two atoms that are localized to two lattice sites of an optical lattice, and to one light atom interacting with two heavy ones in free space. Modifications of the low energy physics due to the long range van der Waals potential are also discussed.

arXiv:1304.3791•Physical Review A•Jun 24, 2013•8 citations

Feshbach resonant scattering of three fermions in one-dimensional wells

Tanja Đurić, A. F. Ho, Derek K. K. Lee

We calculate the spectrum for three atoms in a one-dimensional harmonic well near an $s$-wave Feshbach resonance using a two-channel model to describe the Feshbach physics. This is relevant to the study of cold $^{40}\text{K}$ atoms trapped in a one-dimensional optical lattice. Our results indicate that, for this one-dimensional system, one- and two-channel models will differ close to the Feshbach resonance, although the two theories would converge in the limit of strong Feshbach coupling. We also find level crossings in the low-energy spectrum of a single well with three atoms that may lead to quantum phase transition for an optical lattice of many wells. We discuss the stability of the system to a phase with nonuniform density.

arXiv:0904.3477•Physical Review A•Sep 15, 2009•0 citations

Trap-induced resonances in controlled collisions of cesium atoms

R. Stock, Ivan Deutsch

We consider the feasibility of observing a trap-induced resonance [Stock et al., Phys. Rev. Lett. 91, 183201 (2003)] for the case of two $^{133}\mathrm{Cs}$ atoms, trapped in separated wells of a polarization-gradient optical lattice, and interacting through a multichannel scattering process. Due to the anomalously large scattering length of cesium dimers, a strong coupling can occur between vibrational states of the trap and a weakly bound molecular state that is made resonant by the ac Stark shift of the lattice. We calculate the energy spectrum of the two-atom system as a function of the distance between two potential wells by connecting the solutions of the Schr\"odinger equation for the short-range molecular potential to that of the long-range trap in a self-consistent manner. The short-range potential is treated through a multichannel pseudopotential, parametrized by the $K$ matrix, calculated numerically for atoms in free space in a close-coupling approximation. This captures both the bound molecular spectrum as well as the energy-dependent scattering for all partial waves. We establish realistic operating conditions under which the trap-induced resonance could be observed and show that this strong and coherent interaction could be used as a basis for high-fidelity two-qubit quantum logic operations.

arXiv:physics/0509211•Physical Review A•Mar 2, 2006•14 citations

Effective single-band models for strongly interacting fermions in an optical lattice

J. P. Kestner, L.-M. Duan

To test effective Hamiltonians for strongly interacting fermions in an optical lattice, we numerically find the energy spectrum for two fermions interacting across a Feshbach resonance in a double-well potential. From the spectrum, we determine the range of detunings for which the system can be described by an effective lattice model, and how the model parameters are related to the experimental parameters. We find that for a range of strong interactions the system is well described by an effective $t$-$J$ model, and the effective superexchange term, $J$, can be smoothly tuned through zero on either side of unitarity. Right at and around unitarity, an effective one-band general Hubbard model is appropriate, with a finite and small onsite energy, due to a lattice-induced anharmonic coupling between atoms at the scattering threshold and a weakly bound Feshbach molecule in an excited center-of-mass state.

arXiv:0912.1640•Physical Review A•Apr 19, 2010•10 citations

Two-channel Bose-Hubbard model of atoms at a Feshbach resonance

Philipp‐Immanuel Schneider, Alejandro Sáenz

Based on the analytic model of Feshbach resonances in harmonic traps described in Schneider et al. [Phys. Rev. A 83, 030701(R) (2011)] a Bose-Hubbard model is introduced that provides an accurate description of two atoms in an optical lattice at a Feshbach resonance with only a small number of Bloch bands. The approach circumvents the problem that the eigenenergies in the presence of a delta-like coupling do not converge to the correct energies, if an uncorrelated basis is used. The predictions of the Bose-Hubbard model are compared to nonperturbative calculations for both the stationary states and the time-dependent wave function during an acceleration of the lattice potential. For this purpose, a square-well interaction potential is introduced, which allows for a realistic description of Feshbach resonances within nonperturbative single-channel calculations.

arXiv:1303.4570•Physical Review A•May 28, 2013•5 citations
Two atoms in an optical lattice may be made to interact strongly at higher partial waves near a Feshbach resonance. These atoms, under appropriate constraints, could be bosonic or fermionic. The universal l=2l=2l=2 energy spectrum for such a system, with a caveat, is presented in this paper and checked with the spectrum obtained by direct numerical integration of the Schr"odinger equation. The results reported here extend those of Yip for ppp-wave resonance [S.-K. Yip, Phys. Rev. A 78, 013612 (2008)], while exploring the limitations of a universal expression for the spectrum for the higher partial waves.
It is well-known that cold atoms near sss-wave Feshbach resonances have universal properties that are insensitive to the short-range details of the interaction. What is less known is that atoms near higher partial-wave Feshbach resonances also have remarkable universal properties. We illustrate this with a single atom interacting resonantly with two fixed static centers. At a Feshbach resonance point with orbital angular momentum $L\ensuremath{\ge}1$, we find 2L+12L+12L+1 shallow bound states whose energies behave like 1/R2L+11/{R}^{2L+1}1/R2L+1 when the distance RRR between the two centers is large. We then compute corrections to the binding energies due to other parameters in the effective range expansions. For completeness we also compute the binding energies near sss-wave Feshbach resonances, taking into account the corrections. Afterwards we turn to the bound states at large but finite scattering volumes. For ppp-wave and higher partial-wave resonances we derive a simple formula for the energies in terms of a parameter called ``proximity parameter.'' These results are applicable to a free atom interacting resonantly with two atoms that are localized to two lattice sites of an optical lattice, and to one light atom interacting with two heavy ones in free space. Modifications of the low energy physics due to the long range van der Waals potential are also discussed.
We calculate the spectrum for three atoms in a one-dimensional harmonic well near an sss-wave Feshbach resonance using a two-channel model to describe the Feshbach physics. This is relevant to the study of cold 40K^{40}\text{K}40K atoms trapped in a one-dimensional optical lattice. Our results indicate that, for this one-dimensional system, one- and two-channel models will differ close to the Feshbach resonance, although the two theories would converge in the limit of strong Feshbach coupling. We also find level crossings in the low-energy spectrum of a single well with three atoms that may lead to quantum phase transition for an optical lattice of many wells. We discuss the stability of the system to a phase with nonuniform density.
We consider the feasibility of observing a trap-induced resonance [Stock et al., Phys. Rev. Lett. 91, 183201 (2003)] for the case of two 133Cs^{133}\mathrm{Cs}133Cs atoms, trapped in separated wells of a polarization-gradient optical lattice, and interacting through a multichannel scattering process. Due to the anomalously large scattering length of cesium dimers, a strong coupling can occur between vibrational states of the trap and a weakly bound molecular state that is made resonant by the ac Stark shift of the lattice. We calculate the energy spectrum of the two-atom system as a function of the distance between two potential wells by connecting the solutions of the Schr\"odinger equation for the short-range molecular potential to that of the long-range trap in a self-consistent manner. The short-range potential is treated through a multichannel pseudopotential, parametrized by the KKK matrix, calculated numerically for atoms in free space in a close-coupling approximation. This captures both the bound molecular spectrum as well as the energy-dependent scattering for all partial waves. We establish realistic operating conditions under which the trap-induced resonance could be observed and show that this strong and coherent interaction could be used as a basis for high-fidelity two-qubit quantum logic operations.
To test effective Hamiltonians for strongly interacting fermions in an optical lattice, we numerically find the energy spectrum for two fermions interacting across a Feshbach resonance in a double-well potential. From the spectrum, we determine the range of detunings for which the system can be described by an effective lattice model, and how the model parameters are related to the experimental parameters. We find that for a range of strong interactions the system is well described by an effective ttt-JJJ model, and the effective superexchange term, JJJ, can be smoothly tuned through zero on either side of unitarity. Right at and around unitarity, an effective one-band general Hubbard model is appropriate, with a finite and small onsite energy, due to a lattice-induced anharmonic coupling between atoms at the scattering threshold and a weakly bound Feshbach molecule in an excited center-of-mass state.
Based on the analytic model of Feshbach resonances in harmonic traps described in Schneider et al. [Phys. Rev. A 83, 030701(R) (2011)] a Bose-Hubbard model is introduced that provides an accurate description of two atoms in an optical lattice at a Feshbach resonance with only a small number of Bloch bands. The approach circumvents the problem that the eigenenergies in the presence of a delta-like coupling do not converge to the correct energies, if an uncorrelated basis is used. The predictions of the Bose-Hubbard model are compared to nonperturbative calculations for both the stationary states and the time-dependent wave function during an acceleration of the lattice potential. For this purpose, a square-well interaction potential is introduced, which allows for a realistic description of Feshbach resonances within nonperturbative single-channel calculations.