††thanks:
†These authors contributed equally to this work.

# Trapped ion-mediated interactions between two distant trapped atoms

Subhra Mudli†, Subhanka Mal†, Anushree Dey and Bimalendu Deb<EMAIL_ADDRESS>School of Physical Sciences, Indian Association for the Cultivation of
Science, Jadavpur, Kolkata 700032, India.

###### Abstract

We theoretically show that when two largely separated trapped atoms interact
with a trapped ion via Rydberg excitation of the atoms, the ion-mediated
interaction between the atoms exceeds the direct atom-atom interaction by
several orders of magnitude. Since the motion of the atoms is much slower than
the motion of the ion, we resort to Born-Oppenheimer approximation to
calculate the ion-mediated adiabatic potential. We also calculate the ion-
mediated phonon modes of the atoms that are separated by more than 10 micron.
For cylindrical geometry of the system and both the atoms being excited to the
same Rydberg state, the stretched and center-of-mass (COM) axial or transverse
phonon modes are found to be almost degenerate, while the phonon modes are
non-degenerate when one atom is in a Rydberg state and the other in the ground
state. We discuss the non-adiabatic effects in the system that give rise to a
Gauge structure and associated geometric phase in the system. This study may
open a new perspective in quantum computing and exploring molecular physics
associated with a conical intersection using an ion-atom hybrid architecture.

##  I Introduction

Manipulating coupling or interactions between ionic or atomic qubits which are
separated on the micrometer scale is an important step towards quantum
information processing in an ion- or atom-based quantum computer. A two-qubit
quantum gate operation Cirac:PRL:1995 ; Sorenson:PRL:1999  in ion traps makes
use of the ion-ion Coulomb interaction that, in combination with the harmonic
motion of the trapped ions, leads to the phononic coupling between the ions.
This is accomplished by addressing two ionic qubits individually with Raman
laser pulses that control the phononic coupling in a deterministic way. Since
the beam waist of a laser is typically of the micrometer scale, the
separations between the qubits in a quantum computing architecture must be
larger than a micrometer in order to be able to address the qubits
individually with lasers. Two neutral atoms in electronic ground or low lying
excited states generally interact with a range of sub-nanometer scale. This
rules out the possibility of any two-qubit gate operation in such neutral
atoms using Raman pulses. However, when the atoms are excited to high Rydberg
states with the principal quantum number $n\sim 100$, they can interact
strongly even at separations on the micrometer scale. Thus Rydberg excitation
provides a promising route to neutral atom-based two- or multi-qubit quantum
gate operation.

In recent times, Rydberg atoms in optical lattices or optical tweezers have
emerged as viable architecture for neutral atom-based quantum computation and
quantum simulation Saffman:RMP:2010 ; Beterov:JPB:2016 ; Browaeys:NatPhys:2020
. Based on Rydberg Blockade which forbids excitation of a second atom when the
first atom is already excited in a Rydberg state provided two atoms are
separated not more than a critical distance known as Rydberg radius, Graham et
al. have experimentally demonstrated a two-qubit quantum gate using neutral
atoms in an optical lattice Graham:PRL:2019 . Several quantum algorithms have
been implemented on a two-dimensional array of hyper-fine-qubit neutral atoms
using Rydberg excitation and individual addressing of single atoms
Graham:Nat:2022 .

With these current advancements, the arrays of neutral atom qubits in tailor-
made optical lattices or optical tweezers have acquired almost all the
essential features to serve as a viable quantum computer or a quantum
simulator. Over the last two decades, a lot of improvements have been made in
design and functionality of the linear and planner ion traps for applications
in quantum computing and simulation Leibfried:RMP:2003 ; Monroe:RMP:2021 .
With current pace of progress in both neutral atom- and ion-trap technologies,
it is expected that in near future a hybrid quantum architecture combining
both trapped ions and trapped atoms will be developed for all or certain tasks
in quantum computation and quantum simulation. In fact, a lot of experimental
and theoretical research are currently being pursued to understand ion-atom
collision physics either in a hybrid trap Tomza:RMP:2019 ; Niranjan:Atoms:2021
; Eberle:JPCS:2015 ; Jyothi:RSI:2019 ; Bahrami:arxiv:2023 ; Cui:arxiv:2023  or
merging an atom-trap containing ultracold atoms with an ion-trap containing
one or a few ions Ewald:PRL:2019 . The purpose of these studies is to
understand the basic elastic and inelastic processes in ion-atom collisions
Rakshit:PRA:2011 ; Tomza:RMP:2019  at different energy scales and eventually
to go to the quantum regime of ion-atom collisions where a few low lying
partial waves become important.

Here we address a different question regarding an ion-atom hybrid quantum
platform. We consider a scheme where single atoms in optical tweezers can be
brought in the vicinity of a trapped ion with relatively large separations to
avoid any direct atom-ion collision, yet there should be significant ion-atom
interaction through Rydberg excitation of the atoms Secker:PRA:2016 ;
Secker:PRL:2017 ; Ewald:PRL:2019 . The trapping frequencies of an ion-trap
differs from that of an atom-trap typically by two to three orders of
magnitude. Consequently, the length scales of these two types of traps differ
by one or two orders of magnitude. Ion traps are extraordinarily stable (hours
or days) while atom traps have limited life time. The coherence time of an ion
qubit is of the order of a second while that of an atomic qubit is typically
of 100 microsecond or millisecond order. The first question that comes to mind
is then how one can symbiotically combine these widely different energy,
length and time scales so that one can form a stable hybrid structure. In
fact, the difficulties and possible routes to integrating a single Rydberg
atom in an optical trap with a single ion in a Paul trap for ion-atom two-
qubit quantum gate operation have been theoretically addressed by Secker et
al. Secker:PRA:2016 . Setting aside the engineering aspect of constructing a
stable hybrid platform combining both ion and atom traps, here we focus on
certain basic physics part of the problem theoretically. Our purpose is to
understand ion-mediated atom-atom interactions and phononic couplings between
the atoms. The long term goal of this study is to use these phononic couplings
for fast two qubit quantum gate operations using atomic qubits while ionic
qubits may be used as quantum memory.

We consider a model system where two neutral atomic qubits confined in two
similar but largely separate optical tweezers simultaneously interact with an
ion in a trap. The center of the ion-trap lies between the two tweezers’
centers as schematically shown in Fig.1. One may assume that both the optical
tweezers operate at a magic wavelength for which one electronic ground-state
and one Rydberg state of the atom can be simultaneously confined in the
tweezers. Otherwise, one of the ground-state hyper fine qubit states of the
atom may be admixed with a Rydberg state so that the atoms interact with the
ion and between themselves primarily through Rydberg excitation. Since the
harmonic trapping frequency of an ion-trap is larger than that of optical
tweezers typically by three orders of magnitude, the motion of the ion is much
faster than that of the atoms. Because of the Rydberg excitation-induced long-
range atom-ion Ewald:PRL:2019  and atom-atom interactions Beguin:PRL:2013  and
much faster center-of-mass (COM) motion of the ion than that of the atoms, an
intriguing interplay between quantum dynamics of atoms and the ion occurs.

Here we study ion-mediated interactions and quantum correlations between the
atoms. We use Born-Oppenheimer (BO) approximation Born:AdP:1927 well-known in
molecular physics to clearly separate out the fast ionic degrees-of-freedom
(DOF) from the much slower atomic DOF to derive ion-mediated interatomic BO
potentials. We consider that the atomic ground state $|g_{a}\rangle$ is
coupled to a low-lying Rydberg state $|r\rangle=|n,S\rangle$ (where $n$ is the
principal quantum number and $S$ refers to the electronic orbital momentum
$\ell=0$) by a two-photon process via an intermediate $P(\ell=1)$ state. We
have chosen $S$ Rydberg state because we want to get rid of the first order
Stark shift on the Rydberg state due to the electric field produced by the
ion. At long separations, the ion-atom interaction potential is of the form
$-C_{4}/r^{4}$ where $C_{4}$ is a coefficient proportional to polarizability
of the atom due to the ionic field. There will be a small Stark shift of the
$S$ level due to the electric field of the Paul trap electrodes, but we assume
that the tweezers’ centers are placed at positions where electric field due to
the ion-trap electrodes is almost homogeneous over the extent of the atom’s
motion in the tweezers. This is necessary to nullify or minimize any effect of
electric field-induced forces on the atoms. The Stark-shifted $S$ Rydberg
state remains well isolated from all other Stark-shifted atomic levels at a
separation much larger than 0.5 $\mu$m Secker:PRA:2016 . When the atom is in
$|g_{a}\rangle$ there is practically no interaction between the atom and the
ion if the separations between them are large ($>0.5\mu$m). Since the induced
atomic polarizability scales as $n^{7}$ Kamenski:JPB:2014 , even with low-
lying or moderate Rydberg excitation with $20<n<40$, the range of the ion-
mediated potentials between the two trapped Rydberg atoms is greater than the
direct Rydberg-Rydberg interaction and the range can be as large as tens of
microns. In the leading order, the ion-mediated Rydberg-Rydberg interaction
goes as $-1/r^{4}$, and for separations typically larger than 0.5 $\mu$m, it
exceeds the direct interaction between two $S$-state Rydberg atoms by several
orders of magnitude. Remarkably, the ion-mediated long-range interaction
between the two atoms depends on atomic internal state. For simplicity, we
consider the ion and the two atoms are harmonically trapped in their
respective traps. We first diagonalize the Hamiltonian of atom-ion-atom system
considering the atomic COM motion is frozen under BO approximation. We then
calculate axial and transverse phonon modes of the atoms due to ion-induced
small oscillations of the equilibrium positions of the two atoms. The non-
adiabatic coupling in the system leads atom-ion-atom entangled states. The two
atoms may be separated by a distance as large as 10 $\mu$m, still they can be
entangled due to atoms’ internal state-dependent phonon modes. Our study may
open a new perspective for quantum computation and simulation using an atom-
ion-atom hybrid quantum node.

The paper is organized in the following way. In Sec.II we present our model,
formulate the problem and obtain analytical solutions. In Sec.III we
illustrate the numerical results for a typical system considering
experimentally feasible system parameters. Finally we conclude in Sec.IV.

##  II  The model and formulation of the problem

Figure 1: A schematic diagram of the model system: two atoms $A_{1}$ and
$A_{2}$ trapped in two largely separate optical tweezers simultaneously
interact with a single ion $B^{+}$ trapped at the center of a linear Paul
trap. Elimination of much faster ionic degrees-of-freedom under Born-
Oppenheimer approximation leads to an effective ion-mediated long-range atom-
atom interaction and geometric or Berry phase in the system (see text).

A schematic diagram of our proposed model is shown in Fig.1. Two identical
optical tweezers containing identical single atoms $A_{1}$ and $A_{2}$ of mass
$m_{a}$ are placed symmetrically on the two sides of a single ion $B$ of mass
$m_{i}$ trapped in a Paul trap. The origin of the coordinate system is taken
at the electric potential minimum of the ion-trap or the equilibrium position
of the ion. Note that other geometries are also possible, for instance one can
consider a planar ion trap or an ion chip and the optical tweezers may be
placed above the surface of the trap. To begin with, let us assume that the
atoms $A_{1}$ and $A_{2}$ are in the internal (electronic) state
$|\alpha\rangle$ and $|\beta\rangle$ respectively; where $\alpha$ and $\beta$
represent ground $\left(g\right)$ or Rydberg state $\left(r\right)$. Since the
electric field of the Paul trap electrodes primarily influences the internal
electronic states of the atoms, we here do not consider such electric-field-
induced effects. The Hamiltonian of the system is

$\displaystyle\hat{H}^{\alpha\beta}=\hat{H}^{\alpha\beta}_{{\rm
atom}}+\hat{H}^{\alpha\beta}_{\rm{ion-atom}}$ (1) $\displaystyle\hat{H}_{{\rm
atom}}^{\alpha\beta}=\sum_{j=1,2}\left[-\frac{\hbar^{2}}{2m_{a}}\nabla_{j}^{2}+V_{j}^{\rm{trap}}({\bf
r}_{j})\right]+V_{\rm{\alpha\beta}}(|{\bf{r}_{1}}-\bf{r}_{2}|)$ (2)

is the part that depends only on the atomic COM DOF with

$\displaystyle V_{j}^{\rm{trap}}({\bf
r}_{j})=\frac{1}{2}m_{a}\left[\omega_{a\rho}^{2}\left\\{(x_{j}-x_{j0})^{2}+(y_{j}-y_{j0})^{2}\right\\}+\omega_{az}^{2}(z_{j}-z_{j0})^{2}\right]$
(3)

being the harmonic trapping potential of the $j$th atom and
$V_{\rm{\alpha\beta}}(|{\bf{r}_{1}}-\bf{r}_{2}|)$ the interatomic potential.
Here $\omega_{a\rho}$ and $\omega_{az}$ denote the radial and axial trapping
frequencies of the atom, respectively; $(x_{j0},y_{j0},z_{j0})$ is the
coordinate of the $j$the atom’s trap center. The axial and radial length
scales associated with the atoms’ trapping motion are
$a_{z}=\frac{\hbar}{m_{a}\omega_{z}}$ and
$a_{\rho}=\frac{\hbar}{m_{a}\omega_{a\rho}}$. The Hamiltonian $H_{\rm ion-
atom}$ describing the motion of the ion in the presence of the ion-atom
interactions is given by

$\displaystyle\hat{H}_{\rm ion-atom}$ $\displaystyle=$
$\displaystyle-\frac{\hbar^{2}}{2m_{i}}\nabla_{i}^{2}+V_{\rm ion-atom};$ (4)
$\displaystyle V_{\rm ion-atom}$ $\displaystyle=$
$\displaystyle\frac{1}{2}m_{i}\left[\omega_{i\rho}^{2}\rho_{i}^{2}+\omega_{iz}^{2}z_{i}^{2}\right]-\frac{C_{4}^{\alpha}}{|{\bf
r}_{i}-{\bf r}_{1}|^{4}}-\frac{C_{4}^{\beta}}{|{\bf r}_{i}-{\bf r}_{2}|^{4}}$
(5)

where ${\rho}_{i}^{2}=x_{i}^{2}+y_{i}^{2}$ with $(x_{i},y_{i},z_{i})$ being
the position of the ion. Here $\omega_{ir}$ and $\omega_{iz}$ are the radial
and axial trapping frequencies of the ion, respectively, and
$C_{4}^{\alpha(\beta)}$ is the long range coefficient of interaction between
ion and the atom $A_{1}$($A_{2}$).

###  A Born-Oppenheimer approximation

The Born-Oppenheimer approximation we apply here to solve the problem relies
on the smallness of the ratio of the speed of atom’s motion to the speed of
the ion’s motion. Since we consider that the ion-trap as well as the two atom
traps are cylindrical, one may define an average trapping frequency of the ion
$\bar{\omega}_{i}=[\omega_{i\rho}^{2}\omega_{iz}]^{1/3}$, and similarly an
average atomic trapping frequency
$\bar{\omega}_{a}=[\omega_{a\rho}^{2}\omega_{az}]^{1/3}$. Correspondingly, one
can define average time and length scales of the ionic motion by
$T_{i}=2\pi/\bar{\omega}_{i}$ and
$L_{i}=\sqrt{\hbar/(m_{i}\bar{\omega}_{i})}$, respectively; and those of the
atomic motion by $T_{a}=2\pi/\bar{\omega}_{a}$ and
$L_{a}=\sqrt{\hbar/(m_{a}\bar{\omega}_{a})}$. Thus one can define the ratio
$\eta={L_{a}T_{i}}/{L_{i}T_{a}}$ of the average speed of atom to the average
speed of ion by

$\displaystyle\eta=\sqrt{\frac{m_{i}}{m_{a}}}\left(\frac{\bar{\omega}_{a}}{\bar{\omega}_{i}}\right)^{1/2}$
(6)

Since, typically $\bar{\omega}_{a}/\bar{\omega}_{i}\sim 10^{-2}$, for
$m_{i}\leq m_{a}$, $\eta<1$. So, one can safely apply BO approximation to
separate out the atomic and ionic motion as a first approximation to obtain
adiabatic solution of the problem. Later non-adiabatic effects may be included
to improve the solutions. Note that the criterion $\eta<1$ here is different
from the criterion for BO approximation in molecular physics where one can
separate out the much faster motion of electrons from the nuclear motion of a
molecule - in molecular physics the counterpart of $\eta$ is the square root
of the ratio between the electron’s mass to the nuclear mass.

We diagonalize the Hamiltonian $\hat{H}_{\rm{ion-atom}}$ to obtain ion-
mediated adiabatic BO potentials. Let
$\psi_{\mu}^{\alpha\beta}(\bf{r_{i}};\bf{r_{1}},\bf{r_{2}})$ represent an
eigenfunction with corresponding eigenvalue
$V_{\mu}^{{\alpha\beta}}(\bf{r_{1}},\bf{r_{2}})$, where $\mu$ denotes a
collection of quantum numbers that characterize an eigenstate. Both the
eigenfunction and eigenvalue parametrically depend on the atomic position
coordinates $\bf{r_{1}}$ and $\bf{r_{2}}$ which are slowly varying in time.

###  B Adiabatic ionic and atomic states

In order to derive analytical solution, we approximate the potential $V_{\rm
ion-atom}$ of Eq.(6) as

$\displaystyle V_{\rm ion-atom}$ $\displaystyle\approx$
$\displaystyle\frac{1}{2}m_{i}\omega_{i\rho}^{2}\tilde{\rho}_{i}^{2}+\frac{1}{2}m_{i}\omega_{iz}^{2}\tilde{z}_{i}^{2}-\frac{1}{2}m_{i}\omega_{i\rho}^{2}\rho_{0}^{2}$
(7) $\displaystyle-$
$\displaystyle\frac{1}{2}m_{i}\omega_{iz}^{2}\zeta_{0}^{2}-\left(\frac{C_{4}^{\alpha}}{r_{1}^{4}}+\frac{C_{4}^{\beta}}{r_{2}^{4}}\right)-\frac{C^{\alpha\beta}_{6}}{\left(|\bf
r_{1}-\bf r_{2}|\right)^{6}}$

where
$\tilde{\rho}_{i}^{2}=\left(x_{i}-x_{0}\right)^{2}+\left(y_{i}-y_{0}\right)^{2}$,
$\tilde{z}_{i}^{2}=\left(z_{i}-\zeta_{0}\right)^{2}$,
$\rho_{0}^{2}=x_{0}^{2}+y_{0}^{2}$ with
$x_{0}=\frac{4}{m_{i}\omega_{i\rho}^{2}}\left(\frac{C_{4}^{\alpha}x_{1}}{r_{1}^{6}}+\frac{C_{4}^{\beta}x_{2}}{r_{2}^{6}}\right)$,
$y_{0}=\frac{4}{m_{i}\omega_{i\rho}^{2}}\left(\frac{C_{4}^{\alpha}y_{1}}{r_{1}^{6}}+\frac{C_{4}^{\alpha}y_{2}}{r_{2}^{6}}\right)$
and
$\zeta_{0}=\frac{4}{m_{i}\omega_{iz}^{2}}\left(\frac{C_{4}^{\alpha}z_{1}}{r_{1}^{6}}+\frac{C_{4}^{\alpha}z_{2}}{r_{2}^{6}}\right)$.
$C_{6}^{\alpha\beta}$ is the Van der Waals coefficient between two atoms in
$|\alpha\rangle$ and $|\beta\rangle$ internal states. In writing Eq.(7), we
have expanded $V_{\rm ion-atom}$ up to first order in $\frac{r_{i}}{r_{j}}$,
since $\frac{r_{i}}{r_{j}}<<1$. To calculate the ionic state we solve
eigenvalue equation

$\displaystyle\hat{H}_{\rm ion-atom}\psi_{\mu}\left(\bf r_{i},\bf r_{1},\bf
r_{2}\right)=V_{\mu}^{\alpha\beta}\psi_{\mu}\left(\bf r_{i},\bf r_{1},\bf
r_{2}\right)$ (8)

The eigenvalue $V_{\mu}^{\alpha\beta}$ is given by

$\displaystyle V_{\mu}^{\alpha\beta}\left(\bf{r}_{1},\bf{r}_{2}\right)$
$\displaystyle=$
$\displaystyle\left(2n_{\rho_{i}}+\left|m_{i}\right|+1\right)\hbar\omega_{i\rho}+\left(n_{z_{i}}+\frac{1}{2}\right)\hbar\omega_{iz}+\frac{1}{2}m_{i}\omega_{i\rho}^{2}\rho_{0}^{2}\left(r_{1},r_{2}\right)$
(9) $\displaystyle+$
$\displaystyle\frac{1}{2}m_{i}\omega_{iz}^{2}\zeta_{0}^{2}\left(r_{1},r_{2}\right)-\left(\frac{C_{4}^{\alpha}}{r_{1}^{4}}+\frac{C_{4}^{\beta}}{r_{2}^{4}}\right)-\frac{C^{\gamma_{12}}_{6}}{\left(|\bf
r_{1}-\bf r_{2}|\right)^{6}}$

where $\mu\equiv\left(n_{\rho_{i}},m_{i},n_{z_{i}}\right)$ with
$n_{\rho_{i}},m_{i},n_{z_{i}}$ being the radial, azimuthal and axial quantum
numbers respectively, $n_{\rho}=0,1,2...$, $m=0,\pm 1...$ and
$n_{z}=0,1,2...$. The eigen state $\psi_{\mu}$ is given by

$\displaystyle\psi_{\mu}=\frac{1}{\sqrt{2\pi}}\psi_{n_{z_{i}}}\left(\tilde{z}_{i}\right)\psi_{n_{\rho_{i}}}\left(\tilde{\rho}_{i}\right)e^{im_{i}\phi_{i}}$
(10)

where $\psi_{n_{z_{i}}}\left(\tilde{z}_{i}\right)$ represents 1D harmonic
oscillator state in the $z$-axis,
$\psi_{n_{\rho_{i}}}\left(\tilde{\rho}_{i}\right)$ represents the radial wave
function and $\phi$ is the azimuthal angle of the ion. Thus adiabatic BO state
$\phi_{\mu}\left(\bf r_{1},\bf r_{2}\right)$ can be obtained by solving the
eigenvalue equation

$\displaystyle\left[-\sum_{j=1,2}\frac{\hbar^{2}}{2m_{a}}\nabla_{j}^{2}+U_{\mu}^{\alpha\beta}\right]\phi_{\mu}\left(\bf
r_{1},\bf r_{2}\right)=E_{\mu}\phi_{\mu}\left(\bf r_{1},\bf r_{2}\right)$ (11)

where

$\displaystyle U_{\mu}^{\alpha\beta}\left(\bf{r}_{1},\bf{r}_{2}\right)$
$\displaystyle=$
$\displaystyle\frac{1}{2}m_{a}\omega_{az}^{2}\left((z_{1}+z_{0})^{2}+(z_{2}-z_{0})^{2}\right)+\frac{1}{2}m_{a}\omega_{a\rho}^{2}\sum_{j=1}^{2}\rho_{j}^{2}+V_{\mu}^{\alpha\beta}\left({\bf
r_{1},r_{2}}\right)$ (12)

In order to calculate $\phi_{\mu}\left(\bf r_{1},\bf r_{2}\right)$ and
$E_{\mu}$ perturbatively, we make some approximation in
$U_{\mu}^{\alpha\beta}\left(\bf r_{1},\bf r_{2}\right)$. The positions of the
centers of the traps for atom $A_{1}$ and $A_{2}$ are set at
$\left(0,0,z_{0}\right)$ and $\left(0,0,-z_{0}\right)$ respectively. By
introducing the variables $z_{1}^{\prime}=z_{1}-z_{0}$ and
$z_{2}^{\prime}=z_{2}+z_{0}$, Eq.(11) can be approximated as

$\displaystyle U_{\mu}^{\alpha\beta}\left({\bf r_{1},r_{2}}\right)$
$\displaystyle\approx$ $\displaystyle
E_{\mu}^{(0)}+\frac{1}{2}m_{a}\omega_{az}^{2}\left(z_{1}^{\prime
2}+z_{2}^{\prime
2}\right)+\frac{1}{2}m_{a}\sum_{j=1,2}\left(\omega_{a\rho}^{2}-\frac{A_{12}^{\gamma_{j}}}{\left(z_{1}^{\prime}+z_{0}\right)^{12}}-\frac{A_{6}^{\gamma_{j}}}{\left(z_{1}^{\prime}+z_{0}\right)^{6}}\right)\rho_{j}^{2}$
(13) $\displaystyle-$
$\displaystyle\frac{{m_{a}}A_{12}^{\alpha\beta}\rho_{1}\rho_{2}\cos\left(\phi_{1}-\phi_{2}\right)}{\left(z_{1}^{\prime}+z_{0}\right)^{6}\left(z_{2}^{\prime}-z_{0}\right)^{6}}-\frac{m_{a}A_{10}^{\alpha\beta}}{\left(z_{1}^{\prime}+z_{0}\right)^{5}\left(z_{2}^{\prime}+z_{0}\right)^{5}}-\frac{m_{a}A_{10}^{\gamma_{1}}}{2\left(z_{1}^{\prime}+z_{0}\right)^{10}}-\frac{m_{a}A_{4}^{\gamma_{1}}}{2\left(z_{1}^{\prime}+z_{0}\right)^{4}}$
$\displaystyle-$
$\displaystyle\frac{m_{a}A_{10}^{\gamma_{2}}}{2\left(z_{2}^{\prime}-z_{0}\right)^{10}}-\frac{m_{a}A_{4}^{\gamma_{2}}}{2\left(z_{2}^{\prime}-z_{0}\right)^{4}}-\frac{C_{6}^{\alpha\beta}}{\left\\{(\rho_{1}-\rho_{2})^{2}+\left(z^{\prime}_{1}-z^{\prime}_{2}+2z_{0}\right)^{2}\right\\}^{3}}$

where
$E_{\mu}^{(0)}=\left(2n_{\rho_{i}}+\left|m_{i}\right|+1\right)\hbar\omega_{i\rho}+\left(n_{z_{i}}+\frac{1}{2}\right)\hbar\omega_{iz}$,
$A_{12}^{\gamma_{j}}=\frac{16\left(C_{4}^{\gamma_{j}}\right)^{2}}{m_{a}m_{i}\omega_{i\rho}^{2}}$,
$A_{12}^{\alpha\beta}=\frac{16C_{4}^{\alpha}C_{4}^{\beta}}{m_{a}m_{i}\omega_{i\rho}^{2}}$,
$A_{6}^{\gamma_{j}}=\frac{4C_{4}^{\gamma_{j}}}{m_{a}}$,
$A_{10}^{\gamma_{j}}=\frac{16\left(C_{4}^{\gamma_{j}}\right)^{2}}{m_{a}m_{i}\omega_{iz}^{2}}$,
$A_{10}^{\alpha\beta}=\frac{16C_{4}^{\alpha}C_{4}^{\beta}}{m_{a}m_{i}\omega_{iz}^{2}}$,
$A_{4}^{\gamma_{j}}=\frac{2C_{4}^{\gamma_{j}}}{m_{a}}$ with
$\gamma_{1}=\alpha$ and $\gamma_{2}=\beta$. Here $\phi_{1}$ and $\phi_{2}$ are
the azimuthal angles of the atoms $A_{1}$ and $A_{2}$, respectively. As
$2z_{0}>>a_{\rho},a_{z}$, we further approximate the Eq.(12) up to the
quadratic terms in atomic coordinates. The potential takes the form

$\displaystyle
U_{\mu}^{\alpha\beta}\left(\bf{r}_{1},\bf{r}_{2}\right)=T_{1}+T_{2}+\bar{E}_{\mu}^{(0)}$
(14)

where

$\displaystyle T_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{2}m_{a}\sum_{j=1,2}\left[\omega_{a\rho}^{2}-\frac{A_{6}^{\gamma_{j}}}{z_{0}^{6}}-\frac{A_{12}^{\gamma_{j}}}{z_{0}^{12}}+\frac{3C_{6}^{\alpha\beta}}{128m_{a}z_{0}^{8}}\right]\rho_{j}^{2}$
$\displaystyle+$
$\displaystyle\frac{1}{2}m_{a}\sum_{j=1,2}\left[\omega_{az}^{2}-\frac{10A^{\gamma_{j}}_{4}}{z_{0}^{6}}-\frac{55A^{\gamma_{j}}_{10}}{z_{0}^{12}}-\frac{15A_{10}^{\alpha\beta}}{z_{0}^{12}}-\frac{21C_{6}^{\alpha\beta}}{128m_{a}z_{0}^{8}}\right]z_{j}^{\prime
2}$ $\displaystyle T_{2}$ $\displaystyle=$ $\displaystyle-
m_{a}\left(\frac{A_{12}^{\alpha\beta}}{z_{0}^{12}}+\frac{3C_{6}^{\alpha\beta}}{128m_{a}z_{0}^{8}}\right)\cos\left(\phi_{1}-\phi_{2}\right)\rho_{1}\rho_{2}+m_{a}\left(\frac{25A_{10}^{\alpha\beta}}{z_{0}^{12}}+\frac{21C_{6}^{\alpha\beta}}{128m_{a}z_{0}^{8}}\right)z^{\prime}_{1}z^{\prime}_{2}$
$\displaystyle+$ $\displaystyle
m_{a}{\Bigg{[}}{\Bigg{(}}\frac{5A_{10}^{\alpha\beta}}{z_{0}^{11}}+\frac{5A_{10}^{\gamma_{1}}}{z_{0}^{11}}+\frac{2A_{4}^{\gamma_{1}}}{z_{0}^{5}}-\frac{3C_{6}^{\alpha\beta}}{64m_{a}z_{0}^{7}}{\Bigg{)}}z^{\prime}_{1}-{\Bigg{(}}\frac{5A_{10}^{\alpha\beta}}{z_{0}^{11}}+\frac{5A_{10}^{\gamma_{2}}}{z_{0}^{11}}+\frac{2A_{4}^{\gamma_{2}}}{z_{0}^{5}}-\frac{3C_{6}^{\alpha\beta}}{64m_{a}z_{0}^{7}}{\Bigg{)}}z^{\prime}_{2}{\Bigg{]}}$

and
$\bar{E}_{\mu}^{(0)}=E_{\mu}^{(0)}-m_{a}\left(\frac{A_{4}^{\gamma_{1}}}{2z_{0}^{4}}+\frac{A_{4}^{\gamma_{2}}}{2z_{0}^{4}}+\frac{A_{10}^{\gamma_{1}}}{2z_{0}^{10}}+\frac{A_{10}^{\gamma_{2}}}{2z_{0}^{10}}+\frac{A_{10}^{\alpha\beta}}{2z_{0}^{10}}\right)-\frac{C_{6}^{\alpha\beta}}{64z_{0}^{6}}$.
The first term $T_{1}$ describes uncoupled harmonic motion of the two atoms
with modified radial and axial frequencies

$\displaystyle\bar{\omega}_{a\rho}^{j}$ $\displaystyle=$
$\displaystyle\sqrt{\omega_{a\rho}^{2}-\frac{A_{6}^{\gamma_{j}}}{z_{0}^{6}}-\frac{A_{12}^{\gamma_{j}}}{z_{0}^{12}}+\frac{3C_{6}^{\alpha\beta}}{128m_{a}z_{0}^{8}}}$
(15) $\displaystyle\bar{\omega}_{az}^{j}$ $\displaystyle=$
$\displaystyle\sqrt{\omega_{az}^{2}-\frac{10A^{\gamma_{j}}_{4}}{z_{0}^{6}}-\frac{55A^{\gamma_{j}}_{10}}{z_{0}^{12}}-\frac{15A_{10}^{\alpha\beta}}{z_{0}^{12}}-\frac{21C_{6}^{\alpha\beta}}{128m_{a}z_{0}^{8}}}$
(16)

The term $T_{2}$ will couple even radial harmonic function with odd ones and
vice versa of each atom while the even azimuthal angular function will be
coupled to odd ones and vice versa. This means the third term will lead to
transitions
$|n_{\rho_{1}},n_{\rho_{2}},m_{1},m_{2}\rangle\rightarrow|n_{\rho_{1}}\pm
1,n_{\rho_{2}}\pm 1,m_{1}\pm 1,m_{2}\pm 1\rangle$. For the stability of the
system, we must have $\bar{\omega}_{a\rho}^{j}>0$ and
$\bar{\omega}_{az}^{j}>0$, implying that there exists a minimum separation
$2z_{0}$ below which the system is unstable due to direct collision between
the ion and the atoms.

###  C Phonon modes

To calculate the phonon modes of the atoms, we rewrite the potential of
Eq.(14) in terms of scaled relative and centre-of-mass (COM) coordinates ${\bf
r}=({\bf r_{1}-r_{2}})/\sqrt{2}$ and ${\bf R}=({\bf r_{1}+\bf
r_{2}})/\sqrt{2}$, respectively. In cartesian coordinates Eq.(14) can be
written as

$\displaystyle U_{\mu}^{\alpha\beta}$ $\displaystyle=$
$\displaystyle\frac{1}{2}m_{a}\left[\left\\{\left(\omega_{a\rho}^{\prime}\right)^{2}+\left(\omega_{xy}\right)^{2}\right\\}\left(x^{2}+y^{2}\right)+\left\\{\left(\omega_{az}^{\prime}\right)^{2}-\left(\omega_{zz}\right)^{2}\right\\}z^{2}\right]$
(17) $\displaystyle+$
$\displaystyle\frac{1}{2}m_{a}\left[\left\\{\left(\omega_{a\rho}^{\prime}\right)^{2}-\left(\omega_{xy}\right)^{2}\right\\}\left(X^{2}+Y^{2}\right)+\left\\{\left(\omega_{az}^{\prime}\right)^{2}+\left(\omega_{zz}\right)^{2}\right\\}Z^{2}\right]$
$\displaystyle+$
$\displaystyle\frac{1}{2}m_{a}\left\\{\left(\bar{\omega}_{a\rho}^{\alpha}\right)^{2}-\left(\bar{\omega}_{a\rho}^{\beta}\right)^{2}\right\\}\left(xX+yY\right)+\frac{1}{2}m_{a}\left\\{\left(\bar{\omega}_{az}^{\alpha}\right)^{2}-\left(\bar{\omega}_{az}^{\beta}\right)^{2}\right\\}zZ$
$\displaystyle+$ $\displaystyle
m_{a}z_{0}\frac{\left(\Omega_{1}^{2}+\Omega_{2}^{2}\right)}{\sqrt{2}}z+m_{a}z_{0}\frac{\left(\Omega_{1}^{2}-\Omega_{2}^{2}\right)}{\sqrt{2}}Z$
$\displaystyle+$
$\displaystyle\left(n_{x_{i}}+\frac{1}{2}\right)\hbar\omega_{ix}+\left(n_{y_{i}}+\frac{1}{2}\right)\hbar\omega_{iy}+\left(n_{z_{i}}+\frac{1}{2}\right)\hbar\omega_{iz}$

Here we introduce some frequency scales as

$\displaystyle\omega_{a(\rho,z)}^{\prime 2}$ $\displaystyle=$
$\displaystyle\frac{\left(\bar{\omega}_{a(\rho,z)}^{\alpha}\right)^{2}+\left(\bar{\omega}_{a(\rho,z)}^{\beta}\right)^{2}}{2},\omega_{xy}^{2}=\left(\frac{A_{12}^{\gamma_{12}}}{z_{0}^{12}}+\frac{3C_{6}^{\alpha\beta}}{128m_{a}z_{0}^{8}}\right),$
$\displaystyle\omega_{zz}^{2}$ $\displaystyle=$
$\displaystyle\left(\frac{25A_{10}^{\gamma_{12}}}{z_{0}^{12}}+\frac{21C_{6}^{\alpha\beta}}{128m_{a}z_{0}^{8}}\right),\Omega_{j}^{2}=\frac{5A_{10}^{\alpha\beta}}{z_{0}^{12}}+\frac{5A_{10}^{\gamma_{j}}}{z_{0}^{12}}+\frac{2A_{4}^{\gamma_{j}}}{z_{0}^{6}}-\frac{3C_{6}^{\alpha\beta}}{64m_{a}z_{0}^{8}}$

and
$\\{x,y,z\\}=\\{\frac{x_{1}-x_{2}}{\sqrt{2}},\frac{y_{1}-y_{2}}{\sqrt{2}},\frac{z_{1}-z_{2}}{\sqrt{2}}\\}$
and
$\\{X,Y,Z\\}=\\{\frac{x_{1}+x_{2}}{\sqrt{2}},\frac{y_{1}+y_{2}}{\sqrt{2}},\frac{z_{1}+z_{2}}{\sqrt{2}}\\}$
are relative and COM certesian coordinates, respectively. The transverse and
axial frequency of the harmonic oscillators are modified. From Eq.(17) it is
evident that the transverse and longitudinal phonon modes are separable. Also
if the two atoms are prepared in the same internal state, i.e.,
$\alpha=\beta$, then the relative or stretched and COM modes are separable.
Phonon modes are calculated by diagonalizing the Heissian matrix

$\displaystyle\mathcal{A}^{\alpha\beta}_{\mu}=\frac{1}{m_{a}}\begin{pmatrix}\frac{\partial^{2}{U_{\mu}^{\alpha\beta}}}{\partial{u^{2}}}&\frac{\partial^{2}{U_{\mu}^{\alpha\beta}}}{\partial{u\partial{v}}}\\\
\frac{\partial^{2}{U_{\mu}^{\alpha\beta}}}{\partial{v\partial{u}}}&\frac{\partial^{2}{U_{\mu}^{\alpha\beta}}}{\partial{v^{2}}}\end{pmatrix}$

where for transverse modes $\\{u,v\\}\equiv\\{x,X\\}$ or $\\{y,Y\\}$ and for
axial modes $\\{u,v\\}\equiv\\{z,Z\\}$.

###  D Nonadiabatic effects

By incorporating non-adiabatic effects, a general wave function
$\Psi(\bf{r}_{i},\bf{r}_{1}(t),\bf{r}_{2}(t),t)$ of the entire system may be
expanded in terms of the ionic eigenfunctions
$\psi_{\mu}(\bf{r_{i}};\bf{r_{1}}(t),\bf{r_{2}}(t))$

$\displaystyle\Psi(\bf{r}_{i},\bf{r}_{1}(t),\bf{r}_{2}(t),t)=\sum_{\mu}\tilde{\phi}_{\mu}(\bf{r}_{1}(t),\bf{r}_{2}(t),t)\psi_{\mu}(\bf{r_{i}};\bf{r_{1}}(t),\bf{r_{2}}(t))$
(18)

where $\tilde{\phi}_{\mu}(\bf{r}_{1}(t),\bf{r}_{2}(t),t)$ denotes two-atom
wave function with non-adiabatic effects. Substituting Eq.(18) in the
Schrödinger equation $\hat{H}\Psi=i\hbar\frac{\partial\Psi}{\partial t}$, we
have

$\displaystyle\left[-\frac{\hbar^{2}}{2m_{a}}\sum_{j=1,2}\nabla_{j}^{2}+U_{\nu}^{\alpha\beta}\left(\bf{r}_{1},\bf{r}_{2}\right)\right]\tilde{\phi}_{\nu}+\frac{i\hbar}{m_{a}}\sum_{\mu}\sum_{j}\bf{A}_{\nu\mu}^{j}\cdot\nabla_{r}\tilde{\phi}_{\mu}$
(19)
$\displaystyle+\frac{1}{2}\sum_{\mu}\sum_{j}\left(\bf{A}_{\nu\mu}^{j}\right)^{2}\tilde{\phi}_{\mu}$
$\displaystyle=$ $\displaystyle
i\hbar\frac{\partial\tilde{\phi}_{\nu}}{\partial t}$ $\displaystyle{\rm
where}\hskip 36.135pt{\bf A}_{\mu\nu}^{j}({\bf
r_{1},r_{2}})=i\hbar\int\psi_{\nu}({\bf r_{i};r_{1},r_{2}})\nabla_{\bf
r_{j}}\psi_{\mu}({\bf r_{i};r_{1},r_{2}})d^{3}r_{i}$ (20)

is a vector potential or a Gauge field. If one considers non-adiabatic effects
under BO approximation, then $\nu=\mu$. Equation(19) resembles to the
Hamiltonian of a charged particle in a Gauge field. One can make the
transformation

$\displaystyle\tilde{\phi}_{\mu}({\bf
r_{1}(t),r_{2}(t)},t)=e^{i\gamma_{\mu}(C)}\phi_{\mu}({\bf
r_{1}(t),r_{2}(t)},t)$ (21)

where

$\displaystyle\gamma_{\mu}(C)=\oint_{C}\underline{\underline{\bf{A}}}_{\mu}\cdot
d\underline{\underline{\bf{r}}}$ (22)

is the well-known geometric or Berry phase. Here $C$ refers to a closed path
in $\underline{\underline{\bf{r}}}\equiv({\bf r_{1},r_{2}})$ space and
$\underline{\underline{\bf{A}}}_{\mu}=\left(\bf{A}_{\mu}^{1},\bf{A}_{\mu}^{2}\right)$.
Thus $\phi_{\mu}({\bf r_{1}(t),r_{2}(t)},t)$ satisfies the equation

$\displaystyle\left[-\frac{\hbar^{2}}{2m_{a}}\sum_{j=1,2}\nabla_{j}^{2}+U_{\nu}^{\alpha\beta}\left(\bf{r}_{1},\bf{r}_{2}\right)\right]\phi_{\mu}({\bf
r_{1}(t),r_{2}(t)},t)=i\hbar\frac{\partial\phi_{\mu}({\bf
r_{1}(t),r_{2}(t)},t)}{\partial t}$ (23)

The geometric phase $\gamma^{j}_{\mu}(C)$ will be non zero provided the closed
path encloses a singularity or conical intersection Larson:Springer:2020 .

##  III Results and discussions

Figure 2: The lowest ($\mu\equiv\left(0,0,0\right)$) Born-Oppenheimer
potentials in quasi-1D case are plotted as a function of separation $z$ along
the $z$-axis for the 87Rb+40Ca++87Rb system. (a) $V_{\mu}^{rr}$ (in kHz): both
atoms are in the same Rydberg state ($30S-30S$), (b) $V_{\mu}^{rg}$ (in kHz):
one atom in the Rydberg state and other one is in the ground state ($30S-5S$)
and (c) $V_{\mu}^{gg}$ (in Hz): both atoms are in the ground state ($5S-5S$).
Insets of (a) and (b) show BO potential (in MHz) in the range 10-20 $\mu$m.
Figure 3: Axial probability densities of the pair of atoms as a function of
individual atomic coordinates $z_{1}$ and $z_{2}$ for three different values
of separation $2z_{0}$ between the atom-trap centers for the quasi-1D system
as mentioned in Fig.2.

In the preceding section, we have derived some analytical results on the ion-
mediated interaction between two Rydberg atoms, atomic phonon modes, and a
synthetic Gauge structure and associated geometric phase. Here we present
numerical results to further corroborate our analytical findings about phonon
modes. We discuss briefly the consequences of the synthetic Gauge field and
possible realization of a conical intersection in the system. For simplicity
of calculations, we assume that the atom traps are quasi-one dimensional with
$\omega_{a\rho}>>\omega_{az}$. Assuming that atoms are cooled to their radial
ground state and the temperature is much smaller than $\hbar\omega_{a\rho}$,
we use an ansatz for the atomic wave function as

$\displaystyle\phi_{\mu}\left(\bf
r_{1},r_{2}\right)=\chi_{00}^{(1)}\left(\rho_{1}\right)\chi_{00}^{(2)}\left(\rho_{2}\right)\phi_{\mu}\left(z_{1},z_{2}\right)$
(24)

where $\chi_{00}^{(n)}\left(\rho_{n}\right)$ refers to the radial motional
ground state of $n$th atom. By integrating over the radial ground states of
the two atoms, we obtain an effective BO potential.

Figure 4: Variation of the square of axial (a) and transverse (b) phonon mode
frequencies (in (kHz)2) as a function of $2z_{0}$ for quasi-1D case of Fig.2
when both atoms are in the state $30S-30S$ (solid line) and $25S-25S$ (dashed
line). Note that here st and COM modes are degenerate.

Figure 5: Same as in Fig. 4 but when one atom is in the Rydberg $30S$ state
and other is in the ground $5S$ state.

For numerical illustrations, we consider 87Rb+40Ca++87Rb system where 40Ca+
ion is trapped at the center of a linear Paul trap and the two 87Rb atoms
reside in two separated, identical optical tweezers. We assume that the
tweezers beams are coaxial along the $z$-axis of the Paul trap. Taking the
ion-trap’s center as the origin of the coordinate frame, the two tweezers’s
trap centers are located at $x_{10}=y_{10}=0,z_{10}=z_{0}$ and
$x_{20}=y_{20}=0,z_{20}=-z_{0}$. We set the radial and axial harmonic trapping
frequencies of the ion at $\omega_{i\rho}=2\pi\times 1$ MHz and
$\omega_{iz}=2\pi\times 0.2$ MHz, respectively; which are realistic values
(Idziaszek:PRA:2007, ; Suchorowski:PRA:2022, ). The harmonic trapping
frequencies of the two optical tweezers in radial and axial directions are
taken to be $\omega_{a\rho}=2\pi\times 100$ kHz and $\omega_{az}=2\pi\times 9$
kHz, respectively. We assume that the atomic qubit is composed of the ground
state $|g\rangle=|5S,F=1\rangle$ and the Rydberg state
$|r\rangle=|30S,F=0\rangle$, where $F$ represents hyperfine quantum number.
These two atomic states may be coupled by a two-photon process via a
$P$-excited state. These two long-lived atomic states may be co-trapped in the
same optical tweezers using an optical beam with magic wave-length
Zhang:PRA:2011 ; Lundblad:PRA:2010 . The coefficient of the long-range
interaction between the Ca+ ion and the Rb atom in the Rydberg state is
$C_{4}=3.94\times 10^{7}C_{4}^{0}$ (Kamenski:JPB:2014, ), where $C_{4}^{0}$ is
the coefficient when 87Rb is in ground state, $C_{4}^{0}=5.46\times 10^{-57}$
J-m4 (Tomza:RMP:2019, ). The value for Van der Waals coefficient of 87Rb is
$-h\times 26.61$ MHz-$\mu$m6 (Low:JPB:2012, ). The axial length scale of the
tweezers trap is $a_{z}=\sqrt{\frac{\hbar}{m_{a}\omega_{az}}}=0.11$ $\mu$m.
The length scales that characterize atom-ion and atom-atom potentials are
$R_{ia}^{*}=\sqrt{\frac{2C_{4}{\mu_{ia}}}{{\hbar}^{2}}}=1333$ $\mu$m and
$R_{aa}^{*}=\left(\frac{2{C_{6}}\mu_{a}}{{\hbar}^{2}}\right)^{\frac{1}{4}}=21.92$
$\mu$m, where $\mu_{ia}=\frac{m_{a}m_{i}}{m_{a}+m_{i}}$ and
$\mu_{a}=\frac{m_{a}}{2}$ are ion-atom and atom-atom reduced mass,
respectively. The value of $\eta$, which is the ratio between the average
speeds of atom and ion, is $0.19$ for our system.

Figure 2 shows the 1D BO potential $V_{\mu}^{\alpha\beta}\left(z\right)$ for
$\mu=(0,0,0)$ for three cases, $\left(a\right)$ when both the atoms are in
Rydberg state $\left(\alpha\beta=rr\right)$, $\left(b\right)$ when one of the
atoms is in ground state and other is in Rydberg state
$\left(\alpha\beta=rg\right)$ and $\left(c\right)$ both are in ground state
$\left(\alpha\beta=gg\right)$. Here $\mu=(0,0,0)$ implies that the ion is in
the ground state of the 3D harmonic potential. We notice that, even at
separations larger than $20$ $\mu$m, both $V_{\mu}^{rr}$ and $V_{\mu}^{rg}$
are of $10$ kHz order while $|V_{\mu}^{gg}|$ is quite small ($<1$ Hz). As
expected, $|V_{\mu}^{rr}|>|V_{\mu}^{rg}|$, and $|V_{\mu}^{rr}|$ exceeds
$|V_{\mu}^{rg}|$ almost by a factor of 2. The Van der Waals potential between
the two Rydberg atoms at $z=10$ $\mu$m is $\sim 26$ Hz for $30S-30S$ Rydberg
pair.

In Fig.3 we show the motional ground state probability density for the atom-
pairs along the axial direction for three different trap separations. For
large trap separation, the BO potential is small and the atoms are in the
ground state of the harmonic trap. As the separation decreases the motional
states of the two atoms become coupled so that the ground state becomes
admixed with the excited states. Further decrease in separation makes the pair
of atoms more strongly coupled and the ground state becomes more distorted due
to the coupling with the excited states as shown in Fig.3(a).

We plot the square of the phonon modes as a function of the trap separation
$2z_{0}$ in Fig.4 and 5. As observed in Eq.(17), the axial and transverse
phonon modes in relative DOF are $\sqrt{\omega_{az}^{\prime
2}-\omega_{zz}^{2}}$ and $\sqrt{\omega_{a\rho}^{\prime 2}+\omega_{xy}^{2}}$,
and those in COM DOF are $\sqrt{\omega_{az}^{\prime 2}+\omega_{zz}^{2}}$ and
$\sqrt{\omega_{a\rho}^{\prime 2}-\omega_{xy}^{2}}$, respectively for both
atoms in same internal sate ($\alpha=\beta$). The terms $\omega_{xy}$ and
$\omega_{zz}$ are 2 to 3 orders of magnitude smaller than both
$\omega^{\prime}_{a\rho}$ and $\omega^{\prime}_{az}$. As a result, the
stretched and COM phonon modes are almost degenerate. For the system
87Rb$-^{40}$Ca${}^{+}-^{87}$Rb with two atoms in $30S$ Rydberg state, the
square of the axial phonon frequency $(\omega_{ax}/2\pi)^{2}$ is shifted more
than 40 kHz2 which is quite significant. However, as the interaction is weaker
for the atoms in $25S$ Rydberg state, the shift is relatively small ($<20$
kHz2). Figure 4(b) shows the variation of the square of the transverse phonon
frequency $(\omega_{tr}/2\pi)^{2}$. The shift in the transverse modes from the
asymptotic value of $10^{4}$ kHz2 ($=\omega_{a\rho}^{2}/4\pi^{2}$) is much
smaller as compared to the shift in $(\omega_{ax}/2\pi)^{2}$ owing to the
choice of the quasi 1D trap geometry of our system.

For the case where one of the atoms is in ground state and the other in
Rydberg state, the stretched and COM phonon frequencies are far from
degenerate since ion-Rydberg atom interaction strength is larger than the ion-
ground state atom by several orders of magnitude. As a result the COM mode is
almost unchanged in both axial and transverse directions, i.e.,
$\omega_{ax}^{COM}\approx\omega_{az}$ and
$\omega_{tr}^{COM}\approx\omega_{a\rho}$. However, as shown in Fig.5, both
axial and transverse stretched modes are largely shifted from the asymptotic
value as $2z_{0}$ decreases. It is worth mentioning that there exists a
critical value of $2z_{0}$ below which the system becomes unstable and the
corresponding phonon frequencies become imaginary. For the
87Rb$-^{40}$Ca${}^{+}-^{87}$Rb system, the critical value of trap separation
is $2z_{0}=9.20$ $\mu$m for $30S-30S$ pair, $2z_{0}=9.19$ $\mu$m for $30S-5S$
pair and $2z_{0}=7.58$ $\mu$m for $25S-25S$ pair.

From the foregoing analysis of the phonon modes, we have noticed that while
the axial phonon modes depend on the internal electronic states of the two
atoms, the transverse phonon are very little sensitive to the internal atomic
states. This behavior of the phonon is reflective of the particular geometry
that we have chosen for our numerical works. We have assumed a geometric
scheme with $z$-axes of the linear ion trap and the two quasi-1D cylindrical
atom traps being co-linear. But it is possible to consider other geometry,
such as, atom traps’ $z$-axes are not co-linear with but parallel to the
$z$-axis of the ion-trap and atom traps are either quasi-2D or 3D. In that
case the transverse phonon will also be sensitive to the internal states of
the atoms.

Before ending this section, we wish to discuss briefly the consequences of the
gauge field or vector potential that arises due to non-adiabatic effects in
the system. This is akin to non-adiabatic effects in molecular physics. In the
present scenario, the ion and the atoms play the role of electron and nuclei
of a molecule, respectively. This analogy indicates that an ion-atom hybrid
system may be useful to explore some aspects of molecular physics. It may be
possible to create a conical intersection by coupling two adiabatic BO
potentials by driving the ionic qubit with Raman pulses. Then by changing the
transverse phonon states of the atoms over a cyclic path by Raman pulses, it
may be possible to create geometric phase in the system. Recently, conical
intersection has been experimentally studied using trapped ions by optically
manipulating spin-phonon coupling Whitlow:NatChem:2023 . Although, there are
observations of conical intersections in molecular physics, the detection of
the associated geometric phase in a molecule remains elusive. In this context,
our work opens the possibility to simulate the molecular conical intersection
and the geometric phase using an ion-atom hybrid system.

##  IV Conclusions

In conclusion, we have proposed a model for creating ion-mediated interaction
and phononic coupling between two distant atoms which are otherwise almost
non-interacting. This interaction results from the Rydberg excitation of the
atoms. We have used Born-Oppenheimer approximation to separate out the much
faster ionic degrees-of-freedom from the the much slower atomic ones, allowing
us to calculate the atom-atom BO potentials. We have analyzed in some detail
the characteristics of the axial and transverse phonon modes of the atoms. We
have also discussed non-adiabatic effects which lead to a gauge structure and
geometric phase in the system. Since this geometric phase arises within the
degenerate sub-space of an adiabatic potential, it will entangle the two atoms
with the ion. This will result in an intriguing quantum dynamics in the
system. Since the phonon states can be optically manipulated for a two-qubit
gate operation, our study may be important for quantum computing in such
hybrid quantum platform. Furthermore, the ionic qubit may be used as quantum
memory to store the outcome of a quantum gate operation. Our system my be
scaled up by adding more trapped single atoms arranged in different geometric
order around the trapped ion. Such cluster of atoms around an ion may be used
as a node for a quantum network or distributed quantum computing using
photonic quantum channels to interconnect the nodes. The biggest challenge in
realizing such system is to design an electric potential landscape in an ion
trap (not necessarily the Paul trap, but other chip-based trap architecture
may also be considered) with some positions where the electric field produced
by the ion-trap’s electrodes is small and homogeneous. Then the optical
tweezers can be placed at or near those positions for the stability of the
structure.

##  V Acknowledgements

One of us (Subhra Mudli) is grateful to the Department of Science &
Technology, Govt. of India, for DST INSPIRE fellowship. BD is thankful to Dr.
Rajibul Islam, University of Waterloo, and to Dr. Saikat Ghosh, IIT Kanpur,
for helpful discussions.

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