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arXiv:1908.04482 · Physical Review A

Tight-binding Kondo model and spin-exchange collision rate of alkaline-earth-metal atoms in a mixed-dimensional optical lattice

Ren Zhang, Peng Zhang

Published Jan 30, 2020• 11 citations
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Abstract

We study the two-body problem of ultracold fermionic alkaline-earth-metal-(like) atoms in the electronic $^{1}S_{0}$ state ($g$-state) and $^{3}P_{0}$ state ($e$-state) which are confined in a quasi-one-dimensional (quasi-1D) tube simultaneously, where in the axial direction the $g$-atom experiences a 1D optical lattice and the $e$-atom is localized by a harmonic potential. Due to the nuclear-spin exchange interaction between the $g$- and $e$-atoms, one can use such a quasi-$(1+0)\mathrm{D}$ system to realize the Kondo effect. We suggest two tight-binding models for this system, for the cases that the odd-wave scattering between the $g$- and $e$-atoms is negligible or not, respectively. Moreover, we give a microscopic derivation for the interatomic interaction parameters of these models, by explicitly calculating the quasi-$(1+0)\mathrm{D}$ low-energy scattering amplitude of the $g$- and $e$-atoms in this system and matching this exact result with the ones given by tight-binding models. We illustrate our results for the experimental systems of ultracold $^{173}\mathrm{Yb}$ and $^{171}\mathrm{Yb}$ atoms and show the control effect of the confinement potentials on these model parameters. Furthermore, the validity of the simple ``projection approximation'' is examined. In this approximation, one derives the interaction parameters of the tight-binding models by directly projecting the three-dimensional (3D) Huang-Yang pseudopotential on the ground state of the confinement and the lowest band of the optical lattice. This approximation is supposed to be correct when the 3D interatomic scattering length ${a}_{s}$ is much smaller than the characteristic lengths (CLs) of the confinements. However, we find that for our system this approximation already does not work when ${a}_{s}$ is only of the order of 10% of the confinement CLs. Using the exact two-atom scattering amplitude, we also calculate the spin-exchanging rate (i.e., the cross section of the spin-exchanging collision between the $g$- and $e$-atoms) for the recent experiment [L. Riegger et al., Phys. Rev. Lett. 120, 143601 (2018)] of $^{173}\mathrm{Yb}$ atoms in this quasi-$(1+0)\mathrm{D}$ system and investigate finite-quasimomentum effect of the $g$-atoms in this experiment. Our results show that this effect is very significant.

DOI
10.1103/physreva.101.013636

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We study the two-body problem of ultracold fermionic alkaline-earth-metal-(like) atoms in the electronic 1S0^{1}S_{0}1S0​ state (ggg-state) and 3P0^{3}P_{0}3P0​ state (eee-state) which are confined in a quasi-one-dimensional (quasi-1D) tube simultaneously, where in the axial direction the ggg-atom experiences a 1D optical lattice and the eee-atom is localized by a harmonic potential. Due to the nuclear-spin exchange interaction between the ggg- and eee-atoms, one can use such a quasi-(1+0)D(1+0)\mathrm{D}(1+0)D system to realize the Kondo effect. We suggest two tight-binding models for this system, for the cases that the odd-wave scattering between the ggg- and eee-atoms is negligible or not, respectively. Moreover, we give a microscopic derivation for the interatomic interaction parameters of these models, by explicitly calculating the quasi-(1+0)D(1+0)\mathrm{D}(1+0)D low-energy scattering amplitude of the ggg- and eee-atoms in this system and matching this exact result with the ones given by tight-binding models. We illustrate our results for the experimental systems of ultracold 173Yb^{173}\mathrm{Yb}173Yb and 171Yb^{171}\mathrm{Yb}171Yb atoms and show the control effect of the confinement potentials on these model parameters. Furthermore, the validity of the simple ``projection approximation'' is examined. In this approximation, one derives the interaction parameters of the tight-binding models by directly projecting the three-dimensional (3D) Huang-Yang pseudopotential on the ground state of the confinement and the lowest band of the optical lattice. This approximation is supposed to be correct when the 3D interatomic scattering length as{a}_{s}as​ is much smaller than the characteristic lengths (CLs) of the confinements. However, we find that for our system this approximation already does not work when as{a}_{s}as​ is only of the order of 10% of the confinement CLs. Using the exact two-atom scattering amplitude, we also calculate the spin-exchanging rate (i.e., the cross section of the spin-exchanging collision between the ggg- and eee-atoms) for the recent experiment [L. Riegger et al., Phys. Rev. Lett. 120, 143601 (2018)] of 173Yb^{173}\mathrm{Yb}173Yb atoms in this quasi-(1+0)D(1+0)\mathrm{D}(1+0)D system and investigate finite-quasimomentum effect of the ggg-atoms in this experiment. Our results show that this effect is very significant.
A nuclear-spin-exchange interaction exists between two ultracold fermionic alkali-earth-metal-like atoms in the electronic 1S0{}^{1}{S}_{0}1S0​ state (ggg state) and 3P0{}^{3}{P}_{0}3P0​ state (eee state) and is an essential ingredient for the quantum simulation of the Kondo effect. We study the control of this spin-exchange interaction for two atoms simultaneously confined in a quasi-one-dimensional (quasi-1D) tube, where the ggg atom is freely moving in the axial direction while the eee atom is further localized by an additional axial trap and behaves as a quasi-0D impurity. In this system, the two atoms experience effective 1D spin-exchange interactions in both even and odd partial-wave channels, whose intensities can be controlled by the characteristic lengths of the confinements via confinement-induced resonances (CIRs). In a previous work, we studied this problem with a simplified pure 1D model [Y. Cheng et al., Phys. Rev. A 96, 063605 (2017)]. In the present work, we go beyond that pure 1D approximation. We model the transverse and axial confinements by harmonic traps with finite characteristic lengths ${a}_{\ensuremath{\perp}}$ and az{a}_{z}az​, respectively, and exactly solve the quasi-1D--quasi-0D scattering problem between these two atoms. Using the solutions, we derive the effective 1D spin-exchange interaction and investigate the locations and widths of the even-odd wave CIRs for our system. It is found that when the ratio ${a}_{z}/{a}_{\ensuremath{\perp}}$ is larger, the CIRs can be induced by weaker confinements, which are easier to be realized experimentally. The comparison between our results and a recent experiment by L. Riegger et al. [Phys. Rev. Lett. 120, 143601 (2018)] shows that the two experimentally observed resonance branches of the spin-exchange effect are due to an even-wave CIR and an odd-wave CIR, respectively. Our results are advantageous for the control and description of either the effective spin-exchange interaction or other types of interactions between ultracold atoms in quasi-(1+0)-dimensional systems.
The Kondo effect describes the spin-exchange interaction between localized impurities and itinerant fermions. The ultracold alkaline-earth atomic gas provides a natural platform for quantum simulation of the Kondo model, utilizing its long-lived clock state and the nuclear-spin exchange interaction between clock state and ground state. One of the key issue now is whether the Kondo temperature can be high enough to be reached in current experiments, for which we have proposed to use transverse confinement to confine atoms into a one-dimensional tube and to use the confinement-induced resonance to enhance Kondo coupling. In this work, we further consider the (1+0)(1+0)(1+0)-dimensional scattering problem when the clock state is further confined by an axial harmonic confinement. We show that this axial confinement for the clock-state atoms not only plays a role for localizing them, but can also act as an additional control knob to reach the confinement-induced resonance. We show that, in the presence of both the transverse and the axial confinements, the confinement-induced resonance can be reached in the practical conditions and the Kondo effect can be attainable in this system.
Ultracold gases of fermionic alkaline-earth(-like) atoms are hopeful candidates for the quantum simulation of many-body physics induced by magnetic impurities (e.g., the Kondo physics), because there are spin-exchange interactions (SEIs) between two atoms in the electronic ground (1S0^{1}\mathrm{S}_{0}1S0​) and metastable (3P^{3}\mathrm{P}3P) state, respectively. Nevertheless, this SEI cannot be tuned via magnetic Feshbach resonance. In this paper, we propose three methods to control the SEI between one atom in the 1S0^{1}\mathrm{S}_{0}1S0​ state and another atom in the 3P2^{3}\mathrm{P}_{2}3P2​ states or $^{3}\mathrm{P}_{2}\text{\ensuremath{-}}^{3}\mathrm{P}_{0}$ dressed states, with one or two laser beams. These methods are based on the spin-dependent ac-Stark shifts of the 3P2^{3}\mathrm{P}_{2}3P2​ states or the $^{3}\mathrm{P}_{2}\text{\ensuremath{-}}^{3}\mathrm{P}_{0}$ Raman coupling. We show that due to the structure of alkaline-earth (like) atoms, the heating effects induced by the laser beams of our methods are very weak. For instance, for ultracold Yb atoms, ac-Stark-shift difference of variant spin states of the 3P2(F=3/2)^{3}\mathrm{P}_{2}(F=3/2)3P2​(F=3/2) level, or the strength of the $^{3}\mathrm{P}_{2}\text{\ensuremath{-}}^{3}\mathrm{P}_{0}$ Raman coupling, could be of the order of $(2\ensuremath{\pi})\phantom{\rule{0.28em}{0ex}}\mathrm{MHz}$, while the heating rate (photon scattering rate) is only of the order of Hz. As a result, the Feshbach resonances, with which one can efficiently control the SEI by changing the laser intensity, may be induced by the laser beams with a low-enough heating rate, even if the scattering lengths of the bare interatomic interaction are so small that they are comparable with the length scale associated with the van der Waals interaction.
We calculate the entanglement of the ground state of alkaline-earth-metal fermionic atoms confined in one-dimensional optical lattices. This system can be described using the Kondo lattice model plus a harmonic confining potential, and we adopt the density-matrix renormalization group to study its ground state. We find that the local von Neumann entropy is constant in the insulating domains, and a one-to-one correspondence with the variance of the local density is observed. We show that the average entropy and its derivative are useful tools for identifying quantum transitions in impurity systems.
Alkaline-earth-metal atoms have a long-lived electronic excited state, and when atoms in this excited state are localized in the Fermi sea of ground-state atoms by an external potential, they serve as magnetic impurities, due to the spin-exchange interaction between the excited- and the ground-state atoms. This can give rise to the Kondo effect. However, in order to achieve this effect in current atomic gas experiments, it requires the Kondo temperature to be increased to a sizable portion of the Fermi temperature. In this paper we calculate the confinement-induced resonance (CIR) for the spin-exchanging interaction between the ground and the excited states of the alkaline-earth-metal atoms and propose that the spin-exchange interaction can be strongly enhanced by utilizing the CIR. We analyze this system by the renormalization-group approach and show that near a CIR, the Kondo temperature can be significantly enhanced.
We studied the scattering problem of two distinguishable atoms with unequal mass, where one atom (atom $\ensuremath{\alpha}$) is trapped in a quasi-one-dimensional (quasi-1D) tube and the other one (atom $\ensuremath{\beta}$) is localized by a 3D harmonic trap. We show that in such a system if atom $\ensuremath{\alpha}$ is much heavier than $\ensuremath{\beta}$, confinement-induced resonance (CIR) can appear when the 3D sss-wave scattering length as{a}_{s}as​ of these two atoms is much smaller than the characteristic lengths (CLs) of the confinements, for either as>0{a}_{s}>0as​>0 or as<0{a}_{s}<0as​<0. This is quite different from the usual CIRs which occur only when as{a}_{s}as​ is comparable with the CL of confinement. Moreover, the CIRs we find are broad enough to serve as a tool for the control of effective interatomic interaction. We further show the mechanism of these CIRs via the Born-Oppenheimer approximation. Our results can be used for the realization of strongly interacting systems with ultracold atoms with weak 3D background interaction (i.e., small as{a}_{s}as​), e.g., the realization of ultracold gases with strong spin-dependent interaction at zero magnetic fields.