Efficient computational scheme for ion dynamics in RF-field of Paul trap

We have developed an efficient computational scheme for integration of the classical Hamilton equations describing the ion dynamics confined in the radio-frequency field of the Paul trap. It has permitted a quantitative treatment of cold atom-ion resonant collisions in hybrid atom-ion traps with taking into account unremovable ion micromotion caused by the radio-frequency fields (V.S. Melezhik et. al., Phys. Rev. A100, 063406 (2019)). The important element of the hybrid atom-ion systems is the electromagnetic Paul trap confining the charged ion. The oscillating motion of the confined ion is defined by two frequencies of the Paul trap. It is the frequency of the order of 100 kHz due to the constant electric field and the radio-frequency of about 1–2 MHz defined by the alternating electromagnetic field of the ion trap. The necessity to accurately treat the ion motion in the combined field with two time scales defined by these two very different frequencies has demanded to develop the stable computational scheme for integration of the classical Hamilton equations for the ion motion. Moreover, the scheme must be stable on rather long time-interval of the ion collision with the cold atom ∼ 10 × 2π/ωa defined by the atomic trap frequency ωa ∼ 10 kHz and in the moment of the atom-ion collision when the Hamilton equations are strongly coupled. The developed numerical method takes into account all these features of the problem and makes it possible to integrate the system of coupled quantum-semiclassical equations with the necessary accuracy and quantitatively describes the processes of atomic-ion collisions in hybrid traps, including resonance effects.


Introduction
In the last decade, there has been great interest in ultracold hybrid atomicion systems, which is due to the new opportunities that arise here for quantum simulation of various processes and effects from solid state physics to highenergy physics: electron-phonon coupling in solid state physics, critical phenomena in high-energy physics, quantum information processing etc. [1]. However, a realization of the hot proposals with cold atom and ions is impeded by the unremovable ion micromotion caused by the radio-frequency fields of the Paul traps used for confining ions in the hybrid confined atom-ion systems [1]. In the recent work [2] a quantum-semiclassical computational scheme for treating the collisional atom-ion dynamics in the confined geometry of the hybrid atom-ion traps was suggested where the ion micromotion caused by the radio-frequency fields of the ion trap was taken into account. In this work the following problem was considered: an ion confined in a time-dependent radio-frequency Paul trap with linear geometry, while the atom is constrained to move into a quasi-one-dimensional waveguide within the ion trap. In this approach the atom-ion dynamics was treated semiclassically, namely the atom dynamics is governed by the time-dependent Schrödinger equation, whereas the ion motion is described by the classical Hamilton equations of motion. Both equations were integrated simultaneously.
The quantum-semiclassical computational method [3]- [6] specifically designed for particle collisions such as the problem of ionisation of the helium ion colliding with protons [5] and antiprotons [6] has been employed and extended to the time-dependent domain, as the radio-frequency ionic confinement by the Paul trap requires. It has demanded to develop a new stable computational scheme for integration the classical Hamilton equations for the ion motion. Here, we describe the scheme and demonstrate its efficiency by using as an example of the specific Li/Yb + atom-ion pair, since it is the most promising atomic pair to reach the s-wave regime in Paul traps and it is currently under intense experimental investigations [7]- [9].

Method
A schematic view of the system under investigation is given in Figure 1. The ion is assumed to be confined in a linear Paul trap, whose electric fields read as [10]: Here, is the ion mass, Ω rf is the radio-frequency (rf), = Ω rf √ /2 is the secular frequency, and are dimensionless geometric parameters (i.e. = 0, = − ≡ , − /2 = = ≡ , and ≪ 2 < 1). We assume that the axis of the waveguide in which is travelling the colliding atom is precisely the -axis of the Paul trap (see Figure 1). The corresponding ion-trap interaction potential is given by ) .
(2) Figure 1. Pictorial illustration of the atom-ion system confined in hybrid trap. The lightand dark-grey electrodes (the big bars in the figure) of the Paul trap generate the time-dependent electric fields needed to confine the ion transversally, whereas longitudinally a static voltage is applied to ensure confinement (not shown). The atom is injected from the right to the left into a waveguide, whose centre hosts the ion. The waveguide is orientated along the longitudinal axis, , of the linear Paul trap. In the transverse directions, , , the confining potential both for the atom and the ion is strong Hence, the classical Hamiltonian describing an ion in a Paul trap is given by When the atom is confined in the optical waveguide within the Paul trap, the ion experiences its presence via the atom-ion interaction (|r − r ( )|), where r defines the atom coordinates. The full classical ion Hamiltonian is therefore given by where is the quantum mechanical average of the atom-ion interaction over the atomic density instantaneous distribution. We see that the ion Hamiltonian has parametric dependence on the atom position r . It leads at the moment of the atom-ion collision to the strong non-separability of the Hamilton equations describing the ion dynamics and, as a consequence, to the requirement of sufficient stability of the computational scheme to this strong perturbation. The set of classical equations (6) together with the Schrödinger equation for the atomic wave function Ψ(r , , r ) form the complete set of dynamical equations for describing the confined atom-ion collision in hybrid traps [2]. In order to integrate simultaneously the equations we need proper initial conditions with physical significance. At the beginning of the collisional process, the atom and the ion are assumed to be far away from each other such that they do not interact ( = 0). In particular, the atom is initially in the ground state of the atomic trap with the longitudinal colliding energy, that is, coll ≪ 2ℏ 0 , whereas the ion performs fast (with respect to atom motion) oscillations in the Paul trap with mean transversal̄⟂ and longitudinal̄∥ energies. Since the atom approaches the region of interaction with the ion very slowly ( coll /ℏ ≪ 0 ≪ , Ω rf ), the initial position of the ion does not influence the scattering process itself, which depends only on̄⟂ and̄∥. Specifically, the classical solution of the ion equations of motion (Mathieu equation) in the Paul trap (without the atom) are well approximated by cos( + )[1 + cos(Ω rf )/2], ∀ = , , [11].
The associated kinetic energy depends on the amplitude , but not on the phase . Therefore, we choose, without loss of generality, the ion position at the initial time = 0 in the trap centre with transversal energy, ⟂ , and longitudinal energy, ‖ . This can be summarised with the following set of These initial conditions set the mean values of the ion transversal and longitudinal energies as̄⟂ = 1.64 ⟂ (calculated numerically for our trap parameters Ω rf = 2 × 2 MHz, = 2 × 63 kHz, = 0.002 and = 0.08) and̄∥ = ∥ /2, which is in qualitative agreement with the estimatē from the first-order solution of the Mathieu equation [11], [12]. For the integration of the Hamilton equations of motion, which involve three considerably different scales of frequencies, namely Ω rf , as well as 0 in the quantum mechanical average ⟨Ψ(r , ; r )| (|r − r ( )|)|Ψ(r , ; r )⟩, we employed the second-order Störmer-Verlet method [13].
Simultaneously to the forward in time propagation → +1 = + Δ of the atom wave-packet Ψ ( , ) → Ψ ( , +1 ) we integrate the Hamilton equations (6) with the initial conditions (7), which describe the dynamics of the ion in the Paul trap. To this end, we have adapted the Störmer-Verlet method [13] to our problem Here, and the same definition for r ( ) .

Numerical Example
The computational scheme (9) was successfully applied for numerical integration of the system of differential equations (6) with the initial conditions (7) for the Li/Yb + atom-ion systems confined in the hybrid traps with three absolutely different time-scales rf = 2 /Ω rf ≪ = 2 / ≪ 0 = 2 / 0 defined by the frequencies of Paul trap (Ω rf = 2 × 2 MHz and = 2 × 63 kHz) and atomic waveguide ( 0 = 2 × 10 kHz). These three time-scales define the demand to the computational scheme. The scheme must be stable in rather long time-interval (time of atom-ion collision) ∼ 10 0 = 10 × 2 / 0 and, from the other side, it must accurately treats the fast oscillations defined by the frequency Ω rf of the rf-field.
In Figure 2 we present the calculated trajectory of the ion in the Paul trap ( variable) when there is no interaction with the atom: = 0. Here, the convergence over the step of integration on time Δ → 0 is demonstrated as well as the stability of the computational scheme over the entire integration interval 0 ⩽ ⩽ 10 0 . The efficiency of the computational scheme was confirmed by the calculation of the scattering parameters in the atom-ion resonant collisions confined in hybrid traps [2] and can be applied for other resonant low-dimensional atomic and atom-ion systems.   (2019)).