Low-Dimensional Few-Body Processes in Confined Geometry of Atomic and Hybrid Atom-Ion Traps

TL;DR

This paper presents an efficient computational approach for analyzing low-dimensional few-body quantum processes in atomic and hybrid atom-ion traps, enabling better understanding of ultracold atomic interactions in confined geometries.

Contribution

The authors introduce a split-operator method in 2D DVR for solving the time-dependent Schrödinger equation in low-dimensional systems, advancing computational techniques in this field.

Findings

Application to resonant ultracold atomic processes demonstrated

Results on hybrid atomic-ion systems discussed

Method shows promise for other low-dimensional few-particle problems

Abstract

We have developed an efficient approach for treating low-dimensional few-body processes in confined geometry of atomic and hybrid atom-ion traps. It based on the split-operator method in 2D discrete-variable representation (DVR) suggested by V. Melezhik for integration of the few-dimensional time-dependent Schr\"odinger equation. We give a brief review of the application to resonant ultracold atomic processes and discuss our latest results on hybrid atomic-ion systems. Prospects for the application of the method in other hot problems of the physics of low-dimensional few-particle systems are also discussed.

Equations

$$U({\rho }_A,{\rho }_I)=\frac{1}{2}(m_A{\omega }_A^2{\rho }_A^2+m_I{\omega }_I^2{\rho }_I^2)$$

$$H({\rho }_R,\mathbf{r})=H_{CM}({\rho }_R)+H_{rel}(\mathbf{r})+W({\rho }_R,\mathbf{r}).$$

$$H_{CM}=-\frac{1}{2M}(\frac{{\partial }^2}{\partial {\rho }_R^2}+\frac{1}{{\rho }_R^2}\frac{{\partial }^2}{\partial {\varphi }^2}+\frac{1}{4{\rho }_R^2})+\frac{1}{2}(m_A{\omega }_A^2+m_I{\omega }_I^2){\rho }_R^2$$

$$H_{rel}=-\frac{1}{2\mu }\frac{{\partial }^2}{\partial r^2}+\frac{L^2(\theta ,\varphi )}{2\mu r^2}+\frac{{\mu }^2}{2}(\frac{{\omega }_A^2}{m_A}+\frac{{\omega }_I^2}{m_I}){\rho }^2+V_{AI}(r)$$

$$W({\rho }_R,\mathbf{r})=\mu ({\omega }_A^2-{\omega }_I^2)r{\rho }_R\sin \theta \cos \varphi$$

Full text

I Introduction

Impressive progress of the physics of ultracold quantum gases has stimulated the necessity of detailed and comprehensive investigations of collisional processes in the confined geometry of atomic and ionic traps. The traditional free-space scattering theory is no longer valid here and the development of the low-dimensional few-body theory including the influence of the confinement is needed. In our works we have developed quantitative models [1-4] for pair collisions in tight atomic waveguides and have found several novel effects in its application: the confinement-induced resonances (CIRs) in multimode regimes including effects of transverse excitations and deexcitations [2], the so-called dual CIR yielding a complete suppression of quantum scattering [1], and resonant molecule formation with a transferred energy to center-of-mass excitation while forming molecules [5]. Last effect was confirmed experimentally in [6]. Our calculations have also been used for planning and interpretation of the Innsbruck experiment where CIRs in ultracold Cs gas were observed [7]. Mention also the calculation of the Feshbach resonance shifts and widths induced by atomic waveguides [8]. In the frame of our approach we have predicted dipolar CIRs [9] which may pave the way for the experimental realization of, e.g., Tonks-Girardeau-like or super-Tonks-Girardeau-like phases in effective one-dimensional dipolar gases.

Our latest results on hybrid atomic-ion systems and prospects are discussed in this report.

II Atom-Ion Collisions in Hybrid Atom-Ion Traps

Recently, we have predicted the atom-ion CIRs [10] which are important for a hot problem of control of the confined hybrid atom-ion systems having many promising applications [11]. The condition of appearance of CIR in a atom-ion collision confined in a harmonic waveguide-like trap was found in [10] in ”static” ion approximation. This approach, when one neglects by the ion motion, is well defined for for the Li-Yb+ collision considered in [10]. However, in real experiments an actual problem is controlling of the unremovable effect of ion micromotion in the ion Paul traps [11].

To evaluate the effect of the ion motion on the CIR we performed full quantum calculation for the 6Li-atom scattering by 171Yb+ for a special case of harmonic transversal traps with $\omega_{A}\ll\omega_{I}$ frequencies for atom (A) and ion (I)

[TABLE]

(where $\rho_{i}=r_{i}sin\theta_{i}$). For that, we have integrated the 4D time-dependent Schrödinger equation with the Hamiltonian

[TABLE]

by using computational scheme developed earlier for confined distinguishable atom collisions [1,3,5]. Here ($\hbar=1$)

[TABLE]

and

[TABLE]

describe the CM and relative (rel) atom-ion motions. The potential $V_{AI}(r)$ describes the atom-ion interaction, $\rho_{R}$ and $\bf{r}=\bf{r}_{A}-\bf{r}_{I}\mapsto(r,\theta,\phi)\mapsto(\rho,\phi,z)$ are the polar radial CM and the relative coordinates and $M=m_{A}+m_{I}$, $\mu=m_{A}m_{I}/M$. The term $\frac{L^{2}(\theta,\phi)}{2\mu r^{2}}$ represents the angular part of the kinetic energy operator of the relative atom-ion motion. The term

[TABLE]

leads to a coupling of the CM and relative motion, i.e. to the nonseparability of the quantum two-body problem in confined geometry of the harmonic trap.

In Fig.1 we present the calculated time-evolution of the probability density distribution of 6Li and 171Yb+ near the CIR in the harmonic waveguides. This quantum calculation confirms the surviving of the CIR in the case of ion-motion and demonstrates the molecule ion LiYb+ formation during this collision.

III Conclusion

The efficiency of the splitting-up method based on the 2D DVR for the time-dependent Schrödinger equation makes the method promising in application to actual problems of low-dimensional few-body physics in atomic and atom-ion traps. One can mention the problem of ultracold atomic collisions in anharmonic and asymmetric waveguides and in quasi-2D confining traps. Of great interest in connection with possible important applications is the two-center ( and N-center) problem in a confining trap 12 ; 13 . Note also a collisional three-body problem in tight traps, and non-linear time-dependent Schrödinger equation with a few spatial variables arising in physics of Bose-Einstein condensates.

This work was supported by the Russian Foundation for Basic Research, Grant No. 18-02-00673 and the “RUDN University Program 5-100”.