Controlled collisions of a single atom and ion guided by movable
  trapping potentials
Controlled collisions of a single atom and ion guided by movable trapping potentials
Zbigniew Idziaszek
CNR-INFM BEC Center, I-38050 Povo (TN), Italy
and Center for Theoretical Physics, Polish Academy of Sciences, 02-668 Warsaw, Poland
Tommaso Calarco
CNR-INFM BEC Center and ECT*, I-38050 Povo (TN), Italy
and ITAMP, Harvard University, Cambridge, MA 02138, USA
Peter Zoller
Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences
and Institute for Theoretical Physics, University of Innsbruck, A-6020 Innsbruck, Austria
We consider a system composed of a trapped atom and a trapped ion. The ion charge induces in
the atom an electric dipole moment, which attracts it with an r−4 dependence at large distances.
In the regime considered here, the characteristic range of the atom-ion interaction is comparable or
larger than the characteristic size of the trapping potential, which excludes the application of the
contact pseudopotential. The short-range part of the interaction is described in the framework of
quantum-defect theory, by introducing some short-range parameters, which can be related to the
s-wave scattering length. When the separation between traps is changed we observe trap-induced
shape resonances between molecular bound states and vibrational states of the external trapping
potential. Our analysis is extended to quasi-one-dimensional geometries, when the scattering exhibit
confinement-induced resonances, similar to the ones studied before for short-range interactions. For
quasi-one-dimensional systems we investigate the effects of coupling between the center of mass and
relative motion, which occurs for different trapping frequencies of atom and ion traps. Finally, we
show how the two types of resonances can be employed for quantum state control and spectroscopy
of atom-ion molecules.
PACS numbers: 32.80.Pj, 34.90.+q
I. INTRODUCTION
Techniques developed in atomic physics during the last
two decades allow the preparation of single atoms and
ions in the laboratory. Single neutral atoms can be stored
and laser cooled in far off-resonance laser traps (FORT)
[1, 2], and arrays of atoms can be prepared in an op-
tical lattice via a Mott insulator phase, where exactly
one atom is stored per lattice site [3]. Similar techniques
are being pursued in the context of atom chips [4]. Fur-
thermore, single ions and arrays of ions can be stored
in Paul and Penning traps, and sideband laser cooling
allows to prepare ions in the vibrational ground state
of these trapping potentials [5]. The controlled prepa-
ration, manipulation and measurement of electronic in-
ternal and motional states of laser driven single atoms
and ions, combined with the possibility of controlled en-
tanglement of atoms and ions provides the basic ingredi-
ents to for investigations of fundamental aspects of quan-
tum mechanics, and applications such as high-precision
measurements and quantum information. In quantum
information processing entanglement of qubits stored in
internal states of single atoms or ions is achieved by con-
trolled interactions between particles, either in the form
of switchable qubit-dependent two-particle interactions
or via an auxiliary collective mode of the systems, which
serves as a quantum data bus. In particular, for neutral
atoms cold controlled collisions of atoms stored in mov-
able spin-(i.e. qubit-)dependent optical lattices has been
proposed as a means to entangle atomic pairs, and has
experimentally implemented to generate N -atom cluster
states [6].
In the present work we study the controlled cold colli-
sion of a single atom and a single ion, where the atom
(ion) is prepared in a given motional state of an atom
(ion) trap, and we move the traps to guide the atom
and ion wave packet to “collide” for a given time (see
Fig. 1). Our focus is the development of a quantum-
defect formalism for trapped atoms and ions, in a form
which is convenient for future applications and exten-
sions, in particular in the context of quantum informa-
tion processing to swap qubits stored in atoms and ions,
and the entanglement of these qubits in a controlled colli-
sion. The atom-ion collision is governed by the potential
V (r) → −αe2/(2r4) (r → ∞), where α is the dipolar
polarizability – to be contrasted to a van der Waals po-
tential V (r) → −C6/(r
6) which represents the collisional
interactions between e.g. alkali atoms in their electronic
ground state. Below we will calculate the dynamics of the
interacting atom-ion system for a given time-dependence
of the motion of the trap. This includes the transition to
excited trap states after the collision, the possible forma-
tion of an excited atom-ion (molecule) complex, and the
description of trap induced resonances.
To study the controlled atom-ion collisions we first
derive an effective Hamiltonian, describing effective net
forces acting on the particles moving in rapidly changing
http://arxiv.org/abs/0704.1037v2
(d) (c)
FIG. 1: (Color online) Schematic drawing of a controlled
collision between a single atom and a single ion, whose center-
of-mass wavepackets are guided by time-dependent atom trap
and ion trap, respectively. Labels distinguish different phases
of the process: a) initially particles prepared in the motional
ground state; b) collision (overlap of the wave packets); c)
excitation of the motional states in the traps; d) particles in
some excited states after the collision.
laser and rf fields. In our approach, we include in the
Hamiltonian only the asymptotic part of the atom-ion
potential, whereas the short-range interactions are taken
into account by imposing appropriate boundary condi-
tions on the wave function at r → 0. In the regime of cold
collisions considered here, the boundary conditions can
be expressed in terms of a single quantum-defect param-
eter, independent of the collisional energy and angular
momentum. The quantum-defect approach is dictated
by the relatively long-range character of the atom-ion in-
teraction, which exceeds the typical size of the trapping
potentials, and, in contrast to the neutral atoms, excludes
the applicability of the pseudopotential.
Our method to describe the dynamics of the controlled
collision is based on the application of the correlation di-
agrams, i.e. energy spectra as a function of the trap
separation. Such correlation diagrams, widely used in
quantum chemistry to characterize reactions of diatomic
molecules, connect in our case the asymptotic vibrational
states with the molecular and vibrational states at zero
trap separations. At the intermediate distances the en-
ergy curves exhibit avoided or diabatic crossings, depend-
ing on the symmetry of eigenstates and of the coupling
term in the Hamiltonian. In the atom-ion system the
avoided crossing can be attributed to the resonances be-
tween molecular and vibrational states, that appear when
the energy of a vibrational level, coincide with the energy
of a molecular state shifted by the external trapping. The
dynamics in the vicinity of such avoided crossings can be
accurately described in the framework of the Landau-
Zener theory. In this paper we determine the level split-
ting at the avoided crossing, at different trap separations,
and for different symmetries of the molecular states. In
this way we can characterize the time scale appropriate
for adiabatic or diabatic traversing of a given avoided
crossing.
In our paper we consider two different geometries of
the trapping potentials: spherically symmetric traps and
very elongated cigar-shape traps (quasi-1D traps). The
former case requires full three-dimensional treatment,
while the latter one can be described using an effec-
tive one-dimensional Hamiltonian. In the latter case,
the tight transverse confinement effectively renormalizes
the one-dimensional quantum-defect parameters. In ad-
dition, very elongated cigar-shape traps exhibit another
type of resonances, that can be observed at zero-trap sep-
arations, but for changing ratio of the transverse confine-
ment to the s-wave scattering length of the atom-ion in-
teraction. Such confinement-induced resonances appear
when the energy of the colliding particles coincide with
energy of a bound state lifted up by the tight transverse
potential.
As already outlined above, the counterpart to the idea
of ultracold controlled trap-guided collisions has been im-
plemented in optical lattices, where application of spin-
dependent potentials has allowed to entangle atoms be-
tween neighboring sites [6]. Moreover, resonance phe-
nomena in trapped systems, similar to the ones consid-
ered here, have been already thoroughly investigated for
interactions between neutral atoms. In quasi-1D systems,
collisions between neutral atoms exhibit confinement-
induced resonances [10], whereas displacement of the
trapping potentials leads to trap-induced resonances [11].
The paper is organized as follows. In section II we
present our model that we use to describe the controlled
atom-ion collisions. In particular, in section II A we in-
troduce an effective Hamiltonian of a single trapped atom
and ion. Section II B discuss the quantum-defect treat-
ment of the short-range part of the interaction potential.
In sections II C and IID we discuss our model for two par-
ticular geometries of the trapping potentials, spherically
symmetric traps of the same frequency and elongated
cigar-shape traps with the same transverse trapping fre-
quency, respectively. We extend the quantum-defect
treatment to quasi-1D systems in section II E. Our ap-
proach to describe the dynamics of the atom-ion system is
discussed in section II F. Section III presents the results
of our calculations. We start from discussing the proper-
ties of the scattering in quasi-1D systems in the presence
of long-range r−4 potential, in particular, the problem
of determining of 1D quantum-defect parameters. Sec-
tions III B and III C present adiabatic energy curves and
adiabatic eigenstates in, respectively, 1D and 3D traps
with the same trapping frequencies for the atom and the
ion. The dynamics in the vicinity of the avoided crossing
is discussed in section III D, where we apply the Landau-
Zener theory and semiclassical methods to address this
problem. Section III E discuss most complicated case of
different trapping frequencies, focusing on the 1D atom-
ion motion. The outlook and final conclusions are pre-
sented in section IV. Finally appendix A presents the
microscopic derivation of the effective Hamiltonian and
appendix B describes the details of the pseudopotential
treatment of the scattering in quasi-1D traps.
II. BASIC SETUP AND MODEL
We consider a system consisting of a single atom and
single ion, stored in their respective trapping potentials.
Single atoms can be trapped in experiments with optical
tweezers [1, 2] or optical lattices [3]. Such potentials are
created in far-detuned laser fields due to the AC Stark
effect. In another experimental technique, on atom chips
[4], single atoms can be trapped in microtraps created by
electric and magnetic fields around wires and electrodes.
On the other hand, single ions can be confined in radio-
frequency traps, which use a rapidly oscillating electric
field to create an adiabatic trapping potential for the ion
charge [5]. Typically, the trapping potentials close to
the trap center are with a good approximation harmonic,
and in our approach we consider the simple picture of
the atom and ion confined in the harmonic potentials
with frequencies ωa and ωi, respectively, with d denoting
the displacement between the traps (c.f. Fig. 2.a). We
assume that the atom and ion are prepared initially in a
given trap state, e.g. the ground state of the potential,
and that the atom and ion are brought to collision by
overlapping their center-of-mass (COM) wave packets,
i.e. we change d(t) as a function of time as outlined in
Fig. 1.
In studying this collision with COM wave packets
guided by movable trapping potentials we will study two
cases of (i) three-dimensional (3D) dynamics, where both
traps are assumed to be spherically symmetric, and (ii)
the case of quasi-1D traps, i.e. elongated, cigar-shaped
traps which can be described as an effectively 1D colli-
sion dynamics. This second case is conceptually simpler,
but also provides collisional features specific to 1D sit-
uation (e.g. the appearance of 1D confinement-induced
resonances, reminiscent of resonances in [10]). The val-
ues of trapping frequencies in atom and ion traps are
typically quite different, thus it is natural to consider
ωa 6= ωi. For such condition, however, the COM and
relative motion are coupled, which significantly compli-
cates the theoretical description. Apart from this general
regime, we consider also the special case of ωa = ωi, when
COM and relative motions can be separated. In this case
we perform the full diagonalization for the 3D problem,
which is numerically very demanding in the general case
of ωa 6= ωi.
At large distances, the interaction between atom and
ion is given by a potential V (r) ∼ −αe2/(2r4), where α is
the dipolar polarizability. This potential originates from
the attraction between the ion charge and the electric
dipole that it induces on the atom [24]. At short distances
the interaction is more complicated; however, it turns out
that a detailed knowledge of the form of core region of the
potential is not necessary, and that it can be subsumed in
our model by introducing a set of (energy independent)
quantum-defect parameters.
Let us focus now on characteristic scales that can be
associated with the atom-ion interaction. For the sake
of clarity table I summarizes all the length scales that
a d Ri, Ra
(c)(b)
FIG. 2: (Color online) Panel (a): Schematic drawing of the
considered setup consisting of a trapped atom and ion. Panels
(b) and (c): Long and short-range regimes, respectively, in the
controlled collisions between a trapped atom and ion.
are introduced in the course of the paper. We define
its characteristic length as R∗ ≡
αe2µ/~2, and char-
acteristic energy as E∗ = ~2/
2µ(R∗)2
, where µ is the
reduced mass. Other characteristic lengths and energies
are given by the ground states of the trapping poten-
tials. We associate to the trapping frequencies the har-
monic oscillator lengths for atom (ion): la =
~/(maωa)
(li =
~/(miωi)). For optical and rf traps, the trapping
frequencies are typically ωa = 2π × 10 − 100kHz and
ωi = 2π × 0.1 − 10MHz. In table II we collected some
example values of R∗, li, la, and E
∗ for a few systems
of alkali atoms and alkaline earth ions. One can observe
that the range of the polarization potential is compara-
ble or larger than the size of the trapped ground state.
Therefore, for the trapped particles the polarization in-
teraction cannot be replaced with the standard contact
pseudopotential familiar from s-wave atom-atom scatter-
ing in the context, e.g., of Bose-Einstein condensation.
In the interactions between trapped atom and ion, one
can distinguish two different regimes depending on the
relative distance d between traps: (i) d ≫ Ri, Ra, (ii)
d . Ri ∼ Ra, where Rν =
αe2/mνω
for ν = i, a
is some characteristic distance at which the atom-ion in-
teraction becomes comparable to the trapping potential
of the ion and atom, respectively. Because of the weak
dependence of Rν on ων and mν , the characteristic dis-
tances Ri and Ra are roughly the same. The two regimes
(i) and (ii) correspond to the cases where the motion in
the traps is weakly or strongly affected by the atom-ion
interactions, respectively. The discussed regimes are il-
lustrated schematically in panels (b) and (c) of Fig. 2,
and they correspond to snap shots in the collision pro-
cess when the wavepackets of the particles do not over-
lap / do overlap during the controlled collision. In the
regime (i) of large distances, the atom-ion interaction can
be treated perturbatively as a distortion of the trapping
potential. In this limit the system can be described in
terms of two coupled harmonic oscillators, and all the dy-
Definition Description
αe2µ/~2 characteristic length of the polarization
potential
~/(miωi) harmonic oscillator length of ion trap
~/(maωa) harmonic oscillator length of atom trap
~/(µω) harmonic oscillator length for relative
degrees of freedom (axial direction in
case of quasi 1D traps)
~/(µω⊥) harmonic oscillator length in the trans-
verse direction for relative degrees of
freedom
distance at which the polarization po-
tential becomes equal to the ion trapp-
ping potential
maω2a
distance at which the polarization po-
tential becomes equal to the atom
trappping potential
Rrel =
distance at which the polarization po-
tential becomes equal to the trappping
potential for relative degrees of freedom
distance at which the polarization po-
tential becomes equal to the trapping
potential in the transverse direction for
relative degrees of freedom
R1D = max(R⊥, l⊥) boundary of the quasi-1D regime
R0 size of the core region of the atom-ion
complex
TABLE I: Definitions of the length scales used throughout
the paper.
R∗(a0) li(a0) la(a0) E
∗/h (kHz)
40Ca+ + 87Rb 3989 300 644 4.143
9Be+ + 87Rb 2179 632 644 46.59
40Ca+ + 23Na 2081 300 1252 28.56
TABLE II: Characteristic distance R∗, characteristic energy
E∗, harmonic oscillator length li for ion trap of ωi = 1MHz,
and harmonic oscillator length la for atom trap of ωa =
100kHz, for some combinations of alkali earth ions and al-
kali atoms.
namics can be solved analytically. We discuss this case
in more details in section III F. On the other hand, in
the regime (ii) the description is more difficult, since it
requires inclusion of the short-range part of the interac-
tion, and full treatment of the long-range r−4 part. In
the present paper we focus mainly on the latter case.
A. Effective Hamiltonian
We adopt the following time-dependent Hamiltonian
to describe the system of a single trapped atom and a
single trapped ion (c.f. Fig. 2):
H(t) =
ν=i,a
ν(zν − dν(t))
+ V (|ri − ra|), (1)
Here the label i (a) refers to the ion (atom) respectively,
p and r are the momentum and position operators, dν(t)
denotes the positions of the atom and ion traps, respec-
tively, that can be controlled in the course of dynamics,
and ρ2 = x2 + z2. We assume that the trapping po-
tentials are axially symmetric and displaced along the
axis of symmetry. The trapping frequencies are denoted
by ων and ω⊥ν for the axial and transverse directions,
respectively. Finally, V (r) denotes the interaction po-
tential between the atom and the ion. At large distances
the main contribution to this interaction comes from the
polarization of the atomic cloud, V (r) ∼ −αe2/(2r4).
A microscopic derivation of the Hamiltonian (1) is pre-
sented in appendix A. The basic idea of this derivation
is based on the application of the Born-Oppenheimer ap-
proximation to the motion of the outer shell electrons,
and followed by a time averaging over fast time scale of
the rf and laser frequencies. In this picture, the Hamilto-
nian (1) represents effective net interactions felt by parti-
cles moving in the rapidly changing time-dependent po-
tentials. The interaction potential V (r) can be identi-
fied with the adiabatic Born-Oppenheimer curve of the
electronic ground-state, that depends on the difference
between the atom and ion COM coordinates.
B. Quantum-defect theory
We denote by R0 the characteristic distance at which
V (r) starts to deviate from the asymptotic r−4 law.
Typically, R0 is the size of the core region of the ion-
atom complex, and is much smaller than all the other
length scales in our problem. Therefore, we can describe
the short-range part of interaction in the spirit of the
quantum-defect theory. For R0 ≪ R
∗, the parameters
describing the short-range potential become independent
of energy and angular momentum, since for r ≪ R∗ the
interaction potential is much larger than typical kinetic
energies and heights of the angular-momentum barrier.
This feature is a key ingredient of quantum-defect theory,
where the complicated short-range dynamics can be sum-
marized in terms of few, energy-independent constants
(phase shifts or quantum defects).
To analyze the short-distance behavior of the wave
functions, we omit for the moment the trapping poten-
tials in the Hamiltonian (1), and consider only the part
describing the scattering of atom from the ion in free
space. In this case, the relative and COM degrees of
freedom are decoupled, and the relative motion is gov-
erned by the Hamiltonian H0 = p
2/2µ+V (r). We apply
the partial wave expansion to the relative wave function:
Ψrel(r) =
l Rl(r)Ylm(θ, φ) with l denoting the angular
momentum and Ylm(θ, φ) the spherical harmonics. For
r ≫ R0, we set V (r) = −αe
2/(2r4), which allows to
solve radial Schrödinger equation in terms of the Math-
ieu functions of the imaginary argument [7, 8, 9]. In this
way we obtain the following short-distance behavior of
the radial wave functions
Rl(r, k) ∼ sin (R
∗/r + ϕl(k)) , r ≪
R∗/k, (2)
where ~k is the relative momentum, and ϕl(k) are some
short-range phases. As one can easily verify, the asymp-
totic solution (2) fulfills the radial Schrödinger equa-
tion with the energy ~2k2/(2µ) and centrifugal barrier
2l(l+1)/(2µr2) terms neglected. The short-range phases
constitute our quantum-defect parameters. For R0 ≪ R
we can assume that ϕl(k) are independent of the en-
ergy and angular momentum: ϕl(k) ≡ ϕ, which reduces
description of the short-range interaction to the single
quantum-defect parameter φ. In the calculations we re-
place V (r) by its asymptotic r−4 behavior, effectively
letting R0 → 0 and imposing boundary condition stated
by (2) with the short-range phase φ.
For k = 0 the solution (2) becomes valid at all dis-
tances. Utilizing the fact that zero-energy solution be-
haves asymptotically as Ψ(r) ∼ 1 − as/r (r → ∞), we
can relate the short-range phase to the s-wave scattering
length as
as = −R
∗ cotϕ (3)
In this way the knowledge of the scattering length as
allows us to calculate the quantum defect parameter ϕ.
C. Symmetric 3D trapping potential and identical
trapping frequencies
In this section we consider 3D dynamics assuming,
spherically symmetric trapping potentials ων = ω⊥ν for
ν = i, a and identical trapping frequencies for atom and
ion: ωi = ωa = ω. For such conditions the relative and
CM motions are decoupled.
The COM motion is governed by the Hamiltonian of
a harmonic oscillator with mass M = mi +ma and fre-
quency ω, while the Hamiltonian of the relative motion
is given by
Hrel =
µω2(r− d)2 −
, (4)
where p and r are the relative-motion momentum and
position operators, respectively, and d = da − di. In the
case of relative motion we define
Rrel =
αe2/µω2
= (R∗)1/3l2/3 (5)
0 1 2 3
-5.000
5.000
10.00
15.00
20.00
25.00
30.00
U(r)/
FIG. 3: (Color online) Contour plot of the total potential
U(r) = V (r) + 1
µω2(r − d)2 for the relative motion in the
atom-ion system. The figure shows the case of spherically
symmetric trapping potentials with the same trapping fre-
quencies, separated by d = (0, 0, 2R∗), and for R∗ = 2l
~/(µω)).
as a characteristic length at which the atom-ion inter-
action is comparable to the trapping potential for the
relative degrees of freedom, where l =
~/(µω) denotes
harmonic oscillator length.
Fig. 3 illustrates a typical potential for the atom-ion
relative motion. In addition this figure depicts differ-
ent lengths scales characteristic of our problem. In the
presented case R∗ = 2l, and the distance between trap
centers is d = 2R∗.
D. Quasi-1D trapping potentials
As a second geometry we consider cigar-shape traps
with a transverse trapping frequency much larger than
the axial one: ω⊥ν ≫ ων (quasi-1D traps) for ν = i, a.
For energies smaller than the excitation energy in the
transverse direction, the motion in the transverse direc-
tion is frozen to zero-point oscillations, and the dynamics
takes place along the weakly confined direction. Never-
theless, the transverse motion plays an important role at
short distances, effectively renormalizing the short-range
phase, as we will show in the next section. In this way
quasi-1D traps offer the additional possibility of tuning
the interactions, similarly to the case of neutral atoms
exhibiting confinement-induced resonances [10].
Here, we consider only some particular situation when
ω⊥i = ω⊥a is the same for atom and ion. This simpli-
fies our description of the renormalization effects, since
in this case the transverse COM and relative motions can
be separated. We expect that our results are also qual-
itatively valid in the general case of different transverse
trapping frequencies.
To obtain an effective 1D Hamiltonian, describing the
evolution of the wave packets along the z axis, we de-
compose the total wave function of the atom and ion
into a series over eigenmodes of the transverse part of
the Hamiltonian. For total energies E < 3~ω⊥ only
the transverse ground-state mode contributes to the total
wave function at large distances. In addition, condition
|zi − za| ≫ R⊥ assures that the axial and the transverse
motions are decoupled and the wave function can be writ-
ten as a product of the axial and the transverse compo-
nents. Here, R⊥ =
αe2/µω2⊥
is some characteris-
tic distance at which the atom-ion interaction becomes
comparable to the transverse trapping potential, and the
condition |zi − za| ≫ R⊥ describes the regime where
the transverse oscillation frequency is weakly modified
by the atom-ion interaction. Hence, we have Ψ(ri, ra) →
ψ0(ρi, ρa)Ψ1D(zi, za) (|zi − za| → ∞) for E < 3~ω⊥.
Here, ψ0 is the ground state of the transverse part of
the Hamiltonian: ψ0(ρi, ρa) = e
−ω⊥(miρ
)/2~/π1/2,
and Ψ1D(zi, za) denotes the axial part of the wave func-
tion. Substituting the decomposition over transverse
modes into the Schrödinger equation with the Hamilto-
nian (1), and retaining only the lowest transverse mode
we obtain an effective 1D Hamiltonian, that governs the
dynamics of Ψ1D(zi, za)
H1D =
ν=i,a
ν(zν − dν)
+ V1D(|zi − za|).
Here V1D(|z|) is the 1D interaction potential obtained by
integrating out the transverse degrees of freedom:
V1D(|zi − za|) =
dρidρa|ψ0(ρi, ρa)|
2V (|ri − ra|)
At sufficiently large distances the effective 1D interaction
has similar power dependence as in 3D
V1D(|z|) = −
, |z| ≫ l⊥, (8)
where l⊥ = ~/(µω⊥). Summarizing, in our 1D cal-
culations we apply the Hamiltonian (6) with the ap-
proximation (8), that are valid for |zi − za| ≫ R1D ≡
max(R⊥, l⊥).
Finally in the case of equal longitudinal trapping fre-
quencies: ωi = ωa = ω, the relative and the COM de-
grees of freedom can be separated, and the dynamics is
described by the Hamiltonian of the relative motion
Hrel1D =
µω2(z − d)2 −
, (9)
where z = zi − za and p = pi − pa. In the next Section
we shall solve the problem in the different regimes de-
scribed before, and investigate the peculiar phenomena
that arise from the interplay between the trapping and
the interaction potentials involving the two particles.
E. Quantum-defect theory in 1D
The quantum-defect treatment of the short-range in-
teractions can be extended to the 1D dynamics, described
by the Hamiltonian (6). In 1D the asymptotic behavior of
the relative wave function at short distances is governed
Ψerel(z, k) ∼ |z| sin (R
∗/|z|+ ϕe(k)) , z ≪
R∗/k,
Ψorel(z, k) ∼ z sin (R
∗/|z|+ ϕo(k)) , z ≪
R∗/k,
where labels e and o refer to the even and odd solu-
tions respectively. In our model of 1D dynamics we treat
Eqs. (10)-(11) as boundary conditions for |z| → 0.
In section IIIA we show that the phases ϕe and ϕo are
uniquely determined by l⊥ and ϕ, and that their values
can be calculated by solving the scattering problem in the
quasi-1D geometry. This requires that l⊥ ≫ R0, which
allows to use the quantum-defect description of the short-
range potential. As we discuss later, in contrast to 3D
traps, in quasi-1D traps the values of ϕ are in general
different for scattering waves of different symmetry.
F. Time-dependent problem
The atom-ion collision by moving the trapping po-
tential d(t) is an intrinsically time-dependent problem
which requires the integration of the time dependent
Schrödinger equation for the given initial condition to
predict the transition probabilities for the possible final
states. In our approach to the dynamics we first calcu-
late the correlation diagrams, showing the energy levels
as a function of the trap separations, and on the ba-
sis of these diagrams we predict the possible scenario of
the atom-ion collision. When the motion of the trap is
sufficiently adiabatic, the evolution of the system pro-
ceeds along one of the energy curves, therefore the basis
of adiabatic eigenstates, dependent parametrically on d,
is particularly useful in the analysis of the collision pro-
cess. Of course, the adiabaticity is usually broken in the
vicinity of avoided crossings. In such cases, however, one
can apply e.g. the Landau-Zener theory to calculate the
probability of the adiabatic and diabatic passage through
an avoided crossing.
In our approach to the atom-ion collisions we consider
that initially the traps are well separated. In this limit,
the asymptotic states of (1) are given by products of
the harmonic oscillator states in the two traps. In the
course of dynamics, the distance d(t) decreases, particle
interact for some definite time, and finally they are again
separated, and the final state evolves into some superpo-
sition of the harmonic oscillator states (see Fig. 1 for a
schematic picture). For such a scheme, we are interested
in predicting the final state for some particular realiza-
tion of d(t). We stress that in the intermediate phase,
the system may evolve into atom-ion molecular complex,
and such possibility is fully accounted for in our model.
We start from the time-dependent Schrödinger equa-
tion with the Hamiltonian (1). We expand the time-
dependent wave function in the basis of the energy-
ordered adiabatic eigenstates Ψn(x1,x2|d) of the Hamil-
tonian (1)
H(d)Ψn(xi,xa|d) = En(d)Ψn(xi,xa|d), (12)
where we explicitly point out its dependence on the dis-
tance d. Substituting the expansion
Ψ(xi,xa, t) =
cn(t) exp
dτ En(d(τ))
×Ψn (xi,xa|d(t)) (13)
into the Schrödinger equation, we obtain a set of coupled
differential equations that govern the dynamics of the
expansion coefficients cn:
ċn = −ḋ
m 6=n
cm(t) exp
dτ (En(d(τ)) − Em(d(τ)))
× 〈Ψn(d)|
|Ψm(d)〉, (14)
In the case of fully adiabatic evolution the coefficients
cn(t) remain constant, and the evolution of the sys-
tem proceeds along the adiabatic energy curves. From
Eqs. (14) one can derive the condition for the adiabaticity
of the transfer process. Adiabaticity requires that ḋ mul-
tiplied by the nonadiabatic coupling 〈Ψn(d)|
|Ψm(d)〉
be much smaller than the frequency of the oscillating
factor in the exponential of (14). In this case the oscillat-
ing factor effectively cancels out the contribution due to
changes of d. The adiabaticity condition can be written
as ~ḋ〈Ψn(d)|
|Ψm(d)〉 ≪ (En−Em)
2∀m,n [13]. At trap
separations where atom and ion can be approximately
described by harmonic oscillator states, one can easily
estimate the nonadiabatic couplings 〈Ψn(d)|
|Ψm(d)〉,
which gives the following constraint on the adiabatic
changes of the trap separation:
(En − Em)
, k = i, a, (15)
where k stands for i (a) when the ion trap (atom trap)
is moved. We stress that the latter condition is valid
for transitions between different vibrational states, that
may occur during the transfer of the atom or of the ion.
In the case of avoided crossings between vibrational and
molecular states, the adiabaticity of the transfer is char-
acterized by the condition that can be determined from
the Landau-Zener theory.
III. RESULTS
We turn now to the analysis of the adiabatic eigenen-
ergies and eigenstates as a function of the trap separa-
tion d, and of the trapping potential geometry. We start
our analysis from the simplest case of quasi-1D traps,
where the dynamics takes place effectively in 1D, while
the assumptions of equal trapping frequencies allows us
to consider COM and relative motions separately. Before
discussing this problem, we first study the dependence of
the 1D short-range phases ϕe and ϕo on l⊥, R
∗ and ϕ.
We argue that in the considered range of parameters they
are practically independent of the kinetic energy of the
scattering particles, which is assumed in our quantum-
defect-theory approach. The reader not interested in de-
tails of the derivation may start at Section III B, where
we analyze 1D relative motion eigenenergies and eigen-
states assuming some particular values of ϕe and ϕo. In
Section III C we switch to 3D geometries, analyzing the
system of two spherically symmetric traps with the same
trapping frequency for atom and ion. Section IIID is de-
voted to the properties of adiabatic energy spectra in the
vicinity of an avoided crossing. Using the semiclassical
theory we calculate the level separation at the avoided
crossing, and then applying Landau-Zener theory we in-
vestigate the conditions for the adiabatic and diabatic
transfer, depending on the distance between the traps.
Finally, in Section III E we address the most complicated
case of different trapping frequencies for atom and ion,
coupling COM and relative degrees of freedom. Because
of the complexity of this problem, we limit our analysis
only to 1D dynamics; we argue, however, that the ob-
served behavior should be qualitatively valid also for the
3D system.
A. Short-range phases in quasi-1D traps
As a preliminary technical step in preparation for the
calculations of adiabatic energy curves that will be re-
ported in the following, the present subsection deals with
calculating the short-range phases ϕe and ϕo in quasi-
one-dimensional geometries.
To find the 1D short-range phases we solve the
Schrödinger equation for the relative part of the Hamil-
tonian (1), assuming the same transverse trapping fre-
quency for atom and ion: ω⊥i = ω⊥a = ω⊥, and ne-
glecting axial trapping frequencies, which is valid for
z ≪ Ri, Ra. The latter condition requires Ri, Ra ≫ R1D,
where R1D = max(R⊥, l⊥) determines the boundaries of
the 1D regime. In this way we obtain the following equa-
tion for the relative wave function Ψ(r)
µω2⊥ρ
2 − E
Ψ(r) = 0, (16)
In the case of interaction potentials at distances R1D
much larger than typical kinetic energies in the trapping
potential: |V (R1D)| ≫ ~ωi, ~ωa, we can neglect the en-
ergy dependence of the short-range phases, and solve (16)
for E = ~ω⊥.
At small distances, r ≪ R1D, the atom-ion interac-
tion dominates over the transverse trapping potential,
and the solution of (16) behaves according to (2). At
large distances, z ≫ R1D, the motion in the transverse
direction is frozen to its zero-point oscillations and in the
-1.0 -0.5 0.0 0.5 1.0
1D regime
1D regime
2l R1D
V(r)/
l  = 0.09 R*
(z)  |z| sin(|z|/R*+
) (r) sin(r/R
3D regime
FIG. 4: (Color online) Potential for the atom-ion relative
motion (upper panel) and the relative wave function (lower
panel) for the quasi-one dimensional system with l⊥ = 0.09R
and without trapping in the axial direction.
asymptotic regime the scattering wave function assumes
the form Ψ(r) → ψ0(ρ)Ψ1D(z) (|z| → ∞), where ψ0(ρ)
is the ground-state wave function of the 2D harmonic os-
cillator: ψ0(ρ) = e
−ρ2/(2l2
)/π1/2, and Ψ1D(z) is a linear
combination of odd and even waves:
Ψ1D(z) = ceΨe(z) + coΨo(z), (17)
with Ψe(z), Ψg(z) given by (10) and (11) respectively
[25].
Fig. 4 shows the potential for the relative motion, and
the axial profile (ρ = 0) of the relative wave function for
l⊥ = 0.09R
∗, which corresponds to a system of 40Ca+
and 87Rb in a trap with ω⊥ = 2π × 1MHz. In addition,
the figure illustrates different length scales present in the
quasi-1D problem, and indicates the 1D (|z| ≫ R1D)
and 3D regimes (r ≪ R1D) in the behavior of the wave
function.
Fig. 5 shows an example of the dependence of ϕe and
ϕo on ϕ, for l⊥ = R
∗. It compares the results of nu-
merical calculations with predictions based on the pseu-
dopotential method, that is dicsussed in appendix B.
We observe that for even waves the agreement is fairly
good, while for odd waves the agreement is poorer, which
is probably due to the fact that the p-wave energy-
dependent pseudopotential does not work already in the
regime of R∗ ∼ l⊥, or contributions of odd partial waves
with l > 1 can be important. The pseudopotentials are
expected to give accurate predictions for R∗ ≪ l⊥.
When R∗ ≫ l⊥, the pseudopotential approach is not
applicable at all, and one has to resort to numerical cal-
culations. As an example, we present in Fig. 6 values
for the short-range phases for l⊥ = 0.1R
∗. We note the
presence of several resonances in the dependence of ϕe
and ϕo. This behavior is related to the contribution of
several partial waves for R∗ ≫ l⊥, leading to resonances
when the energy of a bound state in the combined har-
monic and r−4 potential becomes equal to the energy of
the scattered wave. To analyze this issue more carefully
-0.4 -0.2 0.0 0.2 0.4
-0.4 -0.2 0.0 0.2 0.4
l  = R*
 numerical
 energy-dependent
         pseudopotential
l  = R*
 numerical
 energy-dependent 
        pseudopotential
FIG. 5: (Color online) Even and odd short-range phases ϕe
and ϕo calculated for l⊥ = R
∗. Numerical results (red solid
lines) are compared with predictions of the model replacing
r−4 with the energy dependent pseudopotential.
we have calculated the energies of some particular bound
states in in the combined harmonic and r−4 potential.
Fig. 7 shows the relation between the bound-state energy
and the short-range phase for few bound states that are
responsible for the scattering resonances close to φ = 0
for φe (cf. Fig. 6). For comparison we include the bound
states of pure r−4 potential and we label them by the
angular momentum l. We note that, due to the limited
resolution of our numerical calculations, Fig. 6 does not
allow to resolve the single resonances close to φ = 0,
however, one can observe the rapid changes of φe in this
region.
For the numerical calculations we have transformed
(16) into cylindrical coordinates and we were solving
2D elliptic partial differential equation using a finite-
element method. To fix the short-range phase we im-
pose the boundary conditions on the logarithmic deriva-
tive of Ψ(r), ∂rΨ(r)/Ψ(r), at rmin, where rmin = 0.09R
(rmin = 0.022R
∗) for l⊥ = R
∗ (l⊥ = 0.1R
∗) . At large
distances we impose the Dirichlet boundary condition on
the rectangle with boundaries |z| = zmax, ρ = ρmax, as-
suming that the wave function has a Gaussian transverse
profile at |z| = zmax and vanishes at ρ = ρmax. For
l⊥ = R
∗ we took zmax = 6.3R
∗, ρmax = 3R
∗, while for
l⊥ = 0.1R
∗ we have used zmax = R
∗, ρmax = 0.4R
To determine the values of ϕe and ϕo, we fit at large
distances (|z| ≫ R1D) the symmetric and antisymmetric
solutions of (16) for E = ~ω⊥, to the asymptotic formula
(17).
B. Relative motion in a 1D system: adiabatic
eigenenergies and eigenstates
In this section we consider the 1D motion of an atom
and an ion, assuming ωi = ωa. In this case we can focus
only on the relative motion described by the Hamilto-
nian (9), with the boundary conditions at z → 0, stated
by (10) and (11). To find eigenenergies and eigenfunc-
-0.4 -0.2 0.0 0.2 0.4
0.4 (b)
FIG. 6: (Color online) Even short-range phase ϕe (upper
panel) and odd short-range phase ϕo (lower panel) calculated
numerically for l⊥ = 0.1R
∗, versus 3D short-range phase ϕ.
-50 -5 -0.5
0.0 0.5 1.0
FIG. 7: (Color online) 3D short-range phase ϕ versus energy
of bound states in the presence of the transverse confinement
with l⊥ = 0.1R
∗. The bound states in the combined harmonic
and r−4 potentials (solid lines) are compared with the bound
states of pure r−4 interaction (dashed lines).
tions for arbitrary value of d we diagonalize the Hamil-
tonian (9) numerically in the basis of its eigenstates for
d = 0, that are found by numerical integration of the 1D
Schrödinger equation. In the basis we include the low-
est 50 odd and even eigenfunctions, which is sufficient
to perform the diagonalization for d/R∗ . 2.5. Fig. 8
present the adiabatic energy spectrum for some example
parameters: ϕe = −π/4, ϕo = π/4 and R
∗ = 3.48l. The
latter value corresponds to the system of 40Ca+ and 87Rb
in the trap with ω = 2π × 100kHz, while the particular
choice of the short-range phases ϕe and ϕo is explained
later in this section. Points with labels correspond to the
wave functions shown in Fig. 9 presented together with
their eigenenergies and potential energy curves.
The three panels shown in Fig. 9 illustrate three dif-
ferent regimes, where the system exhibits qualitatively
different behavior. In the first regime, represented by
the eigenstate Ψa and realized at large distances between
traps, d≫ Rrel with Rrel defined in (5), the main effect of
the atom-ion interaction is the distortion of the trapping
potential, and in this limit the Hamiltonian can be diago-
nalized analytically in the model of two coupled harmonic
oscillators (see Section III E for more details). The value
of the short-range phase is not important for this model.
At large separations the eigenstate Ψa is typically only
weakly perturbed with respect to the eigenstate of the
harmonic oscillator.
In the second regime, represented by the eigenstates Ψb
and Ψc, and realized at distances d ∼ Rrel, the system
exhibits resonances between vibrational (Ψb) and molec-
ular (Ψc) states, manifesting themselves as avoided cross-
ings in the adiabatic energy spectrum. The resonances
appear when the energies of the two eigenstates become
equal, and the energy splitting at the avoided crossing
is proportional to the tunneling rate through the poten-
tial barrier separating the two regions of the potential.
In section IIID we calculate this splitting in the frame-
work of a semiclassical approximation, and describe the
dynamics in the vicinity of the avoided crossing. The
value of the short-range phase determines the energy of
the molecular states, but it is practically not important
for the atom-ion vibrational states. Since the point of
the avoided crossing depends on the short-range phases,
the controlled collisions can provide some information on
the short-range interaction potential.
Finally, at distances d ≪ Rrel the barrier separating
the external trap from the well given by the atom-ion at-
traction disappears and all eigenstates have some short-
range component behaving at short distances according
to Eqs. (10) and (11). Hence, in this regime all eigen-
states depend on the value of the short-range phases.
The molecular states with energies below the dissociation
threshold (E = 0) are mainly localized in the well of the
atom-ion attractive potential, however for the tight traps
considered here, they can be strongly affected by the ex-
ternal trapping (e.g. state Ψe). The other type of states,
with energies E > 0, are analogs of the vibrational states
and they are localized mainly at distances where the ex-
ternal trapping potential dominates (e.g. state Ψd).
So far we have discussed the properties of the energy
spectrum for some particular choice of ϕe and ϕo. It
turns out that a qualitatively similar behavior can be
observed in all the systems with |ϕe − ϕo| = π/2. In
the general case, however, the adiabatic energy spec-
trum has a slightly more complicated structure, as it
is illustrated in Fig. 10, showing the eigenenergies for
ϕe = ϕo = −π/4. We note that by going from d = 0
0.0 0.5 1.0 1.5 2.0 2.5
 = +  / 4 
 = -  / 4 
FIG. 8: (Color online) Energy spectrum of the relative motion
for an atom and an ion in 1D traps versus the distance d
between traps, calculated for ϕe = −π/4, ϕo = π/4 and R
3.48l (see text for details). The dashed curve shows the line
µω2d2, giving the approximate shift of the bound state in
the trapping potential (see section IIIC for derivation).
to positive d 6= 0 all the energies split into two branches.
To understand the nature of this splitting in Fig. 11 we
present the wave functions for d = 0.1R∗, with the cor-
responding eigenvalues marked with stars in Fig. 10. We
observe that the branches represent the eigenstates local-
ized on the left-hand side and right-hand side of the point
z = 0. The left-localized eigenstates correspond to the
rising branches, because they are mainly localized in the
region of strong atom-ion attraction. We note that this
behavior is similar to the properties of eigenstates in a
double-well potential, where appropriate wave functions
are constructed by taking symmetric and antisymmetric
combination of the states in two wells. In the particular
case of ϕe = ϕo, the symmetric and antisymmetric com-
bination leads to the states localized on the positive and
negative z semi axes, respectively.
C. Relative motion in 3D for spherical ωi = ωa
traps: adiabatic eigenenergies and eigenstates
In this section we extend our analysis to 3D, and con-
sider spherically symmetric traps for the atom and the
ion with the same trapping frequencies: ωi = ωa. In
this case the COM and relative motion can be separated,
and in the following we focus only on the relative motion
governed by the Hamiltonian (4).
We have diagonalized the Hamiltonian (4) taking dif-
ferent values of the short-range phase ϕ, and we have
observed that the adiabatic energy spectra exhibit quali-
tatively the same features as in 1D. In the numerical cal-
culations we first calculated the eigenstates for d = 0, by
solving the radial Schrödinger equation, for angular mo-
menta l ≤ 75 and for energies E ≤ 75~ω. These states
-3 -2 -1 0 1 2 3 4
d = 2.5 R*
 = - /4
 = + /4 
d = 1.2 R*
 = - /4
 = + /4 
d = 0
 = - /4
 = + /4 
FIG. 9: (Color online) Eigenstates of the relative motion for
an atom and an ion in 1D traps at different separations d
between the traps, calculated for ϕe = −π/4, ϕo = π/4
and R∗ = 3.48l. Presented eigenstates correspond to labeled
points in Fig. 8. The horizontal lines present the correspond-
ing eigenenergies, and the thick black line shows the potential
energy.
0.0 0.5 1.0 1.5 2.0 2.5
 = -  / 4 
 = -  / 4 
FIG. 10: (Color online) Energy spectrum of the relative mo-
tion for trapped atom and ion in 1D traps versus distance d
between traps, calculated for ϕe = ϕo = −π/4 and R
∗ = 3.48l
(see text for details).
were used as a basis in the numerical diagonalization of
the Hamiltonian (4).
A sample adiabatic energy spectrum is shown in
Fig. 12, presenting the results for ϕ = −π/4 and R∗ =
3.48l. At d = 0 the angular momentum l is a good
quantum number and the states have definite angular
-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5
d = 0.1 R*
 = - /4
 = - /4
FIG. 11: (Color online) Eigenstates of the relative motion for
trapped atom and ion in 1D at trap separation d = 0.1R∗,
calculated for ϕe = ϕo = −π/4 and R
∗ = 3.48l. The hori-
zontal lines present the corresponding eigenenergies, and the
thick black line shows the potential energy.
0.0 0.5 1.0 1.5 2.0
 l = 0
 l = 1
 l = 2
 l = 3
 l = 4
 = - /4
FIG. 12: (Color online) Energy spectrum of the relative mo-
tion for an atom and an ion confined in spherically symmet-
ric traps versus the distance d between traps, calculated for
ϕ = −π/4, and R∗ = 3.48lr (see text for details). The blue
dashed line shows the shift of the bound states with d, given
approximately by 1
µω2d2.
symmetry, which is depicted by the appropriate symbols
in Fig. 12. For nonzero d, angular momentum is not
conserved, hence in the dynamics the state can change
its angular symmetry. We note that the energies of the
bound states shift to a good approximation according to
µω2d2/2. This behavior can be explained by noting that
bound states |Ψmol〉 are concentrated around r = 0, and
〈Ψmol(d)|Hrel(d)|Ψmol(d)〉 ≈ Eb +
µω2d2, where Eb is
the binding energy at d = 0.
D. Avoided crossings: semiclassical analysis
In the considered setup, level anticrossings reflect res-
onances between molecular and trap states. A simple
picture of such an avoided crossing is shown in Fig. 13.
If one passes an avoided crossing from the direction of
the asymptotic trap state, then for adiabatic evolution
the system at small distances evolves into a molecular
state. On the other hand, for diabatic passage, the
particles remain in their traps, and the state basically
does not change, apart from the modification due to the
smaller trap separation. In addition, for fast changes of
the trap positions, the particles can be excited to higher
motional states. To describe quantitatively the dynamics
in the vicinity of the avoided crossing one can apply the
Landau-Zener theory. Assuming that close to the avoided
crossing the eigenenergies are linear in d, and that d(t)
varies linearly in time, the probability that the crossing
is traversed diabatically is given by [14, 15]
p|1〉→|1〉′ = exp
|〈Ψ1|H |Ψ2〉|
~|ḋ||∂E12/∂d|
, (18)
where the labels |1〉, |2〉 (|1〉′, |2〉′) respectively refer to
the vibrational and molecular states before (after) pass-
ing the avoided crossing (cf. Fig. 13), and E12(d) =
E1(d) − E2(d). For |〈Ψ1|H |Ψ2〉|
2 ≪ ~|ḋ||∂E12/∂d|,
the probability (18) is close to unity, and the avoided
crossing is passed diabatically. In the opposite case:
|〈Ψ1|H |Ψ2〉|
2 ≫ ~|ḋ||∂E12/∂d|, p is small, and the
avoided crossing is traversed adiabatically. The matrix
element 〈Ψ1|H |Ψ2〉 can be related to the energy gap at
the avoided crossing, and in this way we obtain the fol-
lowing constraint on the adiabaticity of the transfer close
to the avoided crossing:
≪ (E1(d) − E2(d))
2 (19)
The position of the avoided crossings is directly related
to the energies of the bound states; thus, a measurement
of the final state after controlled collisions, provides tech-
nique for spectroscopy of the trapped atom-ion complex.
Since the avoided crossings become weaker as the separa-
tion between traps increases, this scheme allows to probe
only excited molecular states, having sufficiently small
binding energies, comparable to the energy scales of the
trapping potentials.
In the case when the tunneling barrier is sufficiently
large, the energy splitting at the avoided crossings can be
estimated using a semiclassical approximation. We first
focus on the relative motion in the 1D system, assuming
ωi = ωa = ω. Using the standard WKB method, one can
derive the following result for the energy splitting ∆E
∆E = EΨmol(x1)Ψvib(x2)
v(x1)v(x2)Te
−W/~. (20)
Here x1, x2 are some arbitrary points located in the
classically forbidden region close to the classical turning
FIG. 13: (Color online) Schematic drawing of the avoided
crossing between vibrational and molecular states in the sys-
tem of a trapped atom and an ion. An adiabatic change of
the distance d between traps induces a transition from the
vibrational to the molecular states, while a diabatic process
leaves the particles in the vibrational states.
points, W (x1, x2, E) =
2m(V (x) − E) is the ac-
tion along the classical trajectory from x1 to x2, T =
is the tunneling time, Ψmol(x1), Ψvib(x2) denote, respec-
tively, the molecular and vibrational states (cf. Fig. 13)
with the same eigenvalues E (the resonance case), and
v(x) =
2(V (x)− E)/m is the velocity of a particle with
energy −E in the inverted potential −V (x). In princi-
ple the choice of x1 and x2 is arbitrary, but in our cal-
culations we take x1 and x2 at fixed distance from the
location of the molecular (x = 0) and vibrational states
(x = d), close to the classical turning points. In this
way at sufficiently large separations Ψmol(x1), Ψvib(x2)
becomes independent of d.
We turn now to the case of relative motion in 3D traps.
Similar to the analysis for 1D systems, the energy split-
ting at the avoided crossings can be calculated semiclassi-
cally by applying the instanton technique. In our calcula-
tion we adopt the formulation based on the path decom-
position expansion developed by Auerbach and Kivelson
[16]. We obtain the following formula describing the level
splitting:
∆E = EAΨmol(x1)Ψvib(x2)
v(x1)v(x2)Te
−W/~ (21)
The meaning of all the quantities is the same as in the 1D
case, the only difference is the prefactorA which accounts
for the fluctuations around the instanton path (see [16]
for details). In the case ωi = ωa the instanton path that
minimizes the classical action is simply a straight line
connecting the trap centers.
Fig. 14 compares the semiclassical formulas Eqs. (20)
and (21) with the exact energy splitting determined from
the numerical energy spectra, as those presented in Fig. 8.
The presented results correspond to the avoided crossings
between the vibrational ground state molecular states. In
1D the energy splitting at the avoided crossing does not
0.8 1.0 1.2 1.4 1.6 1.8
 WKB 1D
 instanton 3D (s-wave)
 1D numerical
 3D numerical (s-wave)
R* = 3.48 l
FIG. 14: (Color online) Energy splitting at the avoided cross-
ing between vibrational ground-state and the molecular state
in 1D and 3D systems. Semiclassical results (solid lines) are
compared to the numerical values extracted from the adia-
batic energy spectra for different combinations of ϕe and ϕo.
3D results present avoided crossings widths for s-wave bound
states.
depend on the symmetry of the molecular state, while
in 3D it does depend on its angular momentum l, and
we present here only the case l = 0. In the first ap-
proximation, the splitting depends on the short-range
phases only through the critical trap separation at which
the resonance occurs. The numerical data are obtained
for different combinations of the short-range phases, for
which the adiabatic spectra exhibit avoided crossings at
different values of d. In our approach, instead of calculat-
ing Ψmol(x1) and Ψvib(x2), we fix the overall amplitude
in (20) by fitting to the single point at the largest trap
separation, where we expect the WKB approximation to
be most accurate. The semiclassical curves stop at the
distances where the potential barrier disappears. We ob-
serve that for the same ratio ofR∗ to l the splittings in 1D
are larger than in 3D. Fig. 14 shows the prediction of the
semiclassical formula Eq. (20), and the energy splittings
calculated numerically for molecular states with different
l. We note that separations at the avoided crossing are
largest for the spherically symmetric molecular states,
with l = 0.
The knowledge of the energy splitting ∆E can be used
to calculate the probabilities of an adiabatic and dia-
batic passage of avoided crossings. Assuming that the
avoided crossing is traversed at constant rate, we apply
the formula (18) with ∂E1(d)/∂d ≈ 0 and ∂E2(d)/∂d =
〈Ψmol(d)|∂Hrel(d)/∂d|Ψmol(d)〉 ≈ µω
2d (c.f. Figs. 8 and
12), which leads to
p|1〉→|1〉′ = exp
(∆E)2
~|ḋ|µω2d
, (22)
where we use the same notation as in Fig. 13. Analyzing
Figs. 14 and 15 we can now estimate the rates ḋ required
1.0 1.2 1.4 1.6
 s-wave semiclassical
 s-wave numerical
 p-wave semiclassical
 p-wave numerical
 d-wave semiclassical
 d-wave numerical
R* = 3.48 l
FIG. 15: (Color online) Energy splitting at the avoided cross-
ing between vibrational ground state and molecular states
with different angular momenta, for a 3D spherically sym-
metric trap. Semiclassical results obtained by means of the
instanton technique (solid lines) are compared to the numeri-
cal values extracted from the adiabatic energy spectra. Semi-
classical calculations stop at the distance d = 1.1R∗ where
the potential barrier disappears.
for adiabatic and diabatic transitions. For instance in
3D, for the parameters of Fig. 14, the diabatic transfer
of particles across the avoided crossings up to distances
d ≈ R∗ (∆E . 0.1~ω) can be realized by keeping ḋ/R∗ ≫
0.001ω .
In summary, the analysis carried out here provides the
basis for a description of the dynamics of various pro-
cesses of interaction between an atom and an ion manip-
ulated through external trapping potentials, and it gives
a way to estimate quantitatively with simple analytical
means the outcome of controlled interaction experiments
in the different regimes described in Sect. II.
E. Center of mass coupled to the relative motion:
1D analysis for ωi 6= ωa
In this section we consider the effects of coupling be-
tween COM and relative motions in 1D, that appear
when the trapping frequencies for atom and ion are not
equal.
Fig. 16 shows the adiabatic levels as a function of trap
separation for some example parameters: ωi = 5.5ωa and
la = 0.9R
∗. This choice corresponds to the interaction
of 40Ca+ and 87Rb in the traps with ωa = 2π × 10kHz
and ωi = 2π × 55kHz. To obtain the adiabatic spectrum
presented in Fig. 16, we performed the diagonalization
of the Hamiltonian (6) in the product basis of the COM
and relative motion eigenstates evaluated at d = 0. In
the calculations we consider all the states with a total
energy E ≤ 460~ωa, which leads to about 7100 states in
the basis.
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5
 = 5.5 
 = -  / 4
 = +  / 4
FIG. 16: Energy spectrum for atom and ion confined in har-
monic traps with ωi = 5.5ωa, as a function of the distance
d. Calculations are performed for ϕe = −π/4, ϕo = π/4 and
la = 0.9R
∗ (see text for details).
The arrows on the right-hand side of Fig. 16 indi-
cate the asymptotic states for large separations, which
can be labeled by the number of phonons for the atom
and ion trap, respectively. In comparison to the case
where relative and COM motions are decoupled, we ob-
serve the following new features: (i) the molecular spec-
trum contains states with different numbers of excita-
tions in the COM degree of freedom. This can be ob-
served at d = 0, when the molecular levels are with a
good approximation equally separated by ~ωCM where
ω2CM = (maω
a +miω
i )/M . [26]; (ii) the avoided cross-
ings between molecular and vibrational states are weaker
for the states involving vibrational excitations of the ion.
This can be observed by comparing the avoided cross-
ings for the state |0〉a|1〉i with the avoided crossings for
the neighboring states. This behavior can be understood
when we notice that for weaker atom trap (for ωa < ωi),
the particles have larger probability to tunnel when the
atom is excited.
F. Large distances between traps: two coupled
oscillators approximation
Finally we turn to the limit of large distances: d ≫
Ri, Ra. Expansion of the interaction up to second-order
terms in the distance r leads to
H1D ≃
ν=i,a
ν(zν − z̄ν)
(zi − z̄i)(za − z̄a), (23)
where z̄ν denotes the equilibrium position of the parti-
cles and we neglect the second-order terms modifying the
trapping frequencies. The Hamiltonian (23) describes the
system of two coupled harmonic oscillators, which can be
written in the form
H1D = ~ωaa
†a+ ~ωib
†b+ ~ωc(a+ a
†)(b+ b†), (24)
where we have introduced the usual annihilation and cre-
ation operators
zi + i
za + i
. (26)
Here ωc denotes the coupling frequency: ~ωc =
10E∗(R∗)4lali/d
6. The validity of the considered model
is limited by the assumption that a stable equilibrium
position exists, which is fulfilled for Ri, Ra . 0.57d, as
it can be easily verified. Typical, maximal values of ωc
for atoms in optical potentials and ions in rf traps are
of the order of 10kHz. In practice the model ceases to
be valid already at weaker conditions, when the terms
higher than the second order cannot be neglected. In
any case, this model is relevant for an important class of
processes involving coherent transfer of quanta between
the atom and the ion, of direct application in a quantum
information processing context.
IV. CONCLUSIONS AND OUTLOOK
In this paper we analyzed in detail the interaction be-
tween a single atom and a single ion guided by external
trapping potentials. This kind of work is motivated by re-
cently opened experimental possibilities within combined
systems – currently being built in several groups world-
wide – where magneto-optical traps or optical lattices
for neutral atoms coexist with electromagnetic traps for
ions.
Tight confinement of single particles, associated with
independent control of the respective confining poten-
tials, allows for exploring different regimes of the two-
body dynamics involving one atom and one ion at a time.
At large distances, the interaction is dominated by the
inverse quartic term arising from the attraction between
the ion’s charge and the electric dipole induced by it on
the atomic electron wave function. A pseudopotential
approximation, similar to that familiar from ultracold-
atom collision theory, is not applicable in our case, as the
characteristic range of the atom-ion molecular potential
often exceeds the size of the tight traps experimentally
available. To describe the interaction at short distances,
smaller than that range, we employ quantum-defect the-
ory, which allows to deal with different geometries from
isotropic three-dimensional traps to very elongated quasi-
one-dimensional ones.
A good description of adiabatic dynamics, involving
processes where the traps are moved toward or across
each other at a rate much slower than the trapping fre-
quencies, can be obtained from quasistatic eigenenergy
curves, which we calculate for various trapping configu-
rations based on the methods outlined above. A remark-
able feature displayed by the system is the presence of
resonances between molecular-ion bound states and mo-
tional excitations within the trap. These trap-induced
resonances are similar in nature to Feshbach resonances
driven by external fields, and they could as well be spec-
troscopically probed in simple experiments where the in-
teraction is controlled via the external guiding potentials.
In addition to the above aspects, an important moti-
vation for the interest in systems of trapped atoms and
ions and their trap-induced resonances resides in possible
applications to quantum information processing. In this
context, one can utilize controlled atom-ion interactions
to effect coherent transfer of qubits, thereby creating in-
terfaces between atoms and ions. By storing quantum in-
formation in internal atomic states, and performing gate
operations with ions, one would combine the advantages
of both: (i) long decoherence times for neutral atoms
and (ii) short gate-operation times for charged particles
due to the relatively strong interactions. In the present
work we have focused on the motional degrees of free-
dom, which can serve as auxiliary degrees of freedom for
quantum gates involving internal-state qubits, provided
an appropriate coupling mechanism between internal and
external degrees of freedom is employed – e.g., sideband
excitation via a laser.
Another potential application of our results is cooling
of the atomic motion. In typical neutral-atom quantum
computation schemes with qubits stored in internal states
of atoms trapped in optical-lattice sites, motional excita-
tions constitute a serious source of errors, and the atomic
motion needs to be cooled in a state-insensitive manner
between computational steps in order to avoid qubit de-
coherence. Since the long-range part of the atom-ion in-
teraction is not sensitive to the internal state [27], our
setup can be applied for sympathetic cooling of atoms,
through the exchange of energy with laser cooled ions
[12].
Beside these applications in the context of quantum
information processing, our work opens broader perspec-
tives for the study of new interesting collisional physics
in a physical situation never explored before. In prin-
ciple, our scheme allows for production of ultracold
trapped atom-ion molecules (molecular complexes), when
the trapping potentials are lowered adiabatically at the
stage when the particles remain close to each other. In
this way our method can be regarded as a way to per-
form cold chemical reactions, where the final state of
the molecule can be well controlled. Beyond sufficiently
large separation between traps, however, the survival
probability of such molecules in the final state is neg-
ligible. Indeed, in the calculations presented here, we
have only regarded processes leading to a final state in
which the atom and the ion are still separated, and no
molecule has been formed as an outcome of the inter-
action. In other words, the Hamiltonian describing our
single-channel model does not include the possibility of
transitions to other Born-Oppenheimer curves. While
this is a justified assumption for the dynamics considered
here, it is in a sense a limitation of our current approach.
Investigation of molecular formation in traps beyond this
approximation is certainly among the interesting devel-
opments that can arise from our work. Its long-term
motivation is really to open, beyond the present exam-
ples, a new paradigm for cold collision physics, which can
be described as the mechanical control of single-particle
chemical reactions.
Acknowledgments
The authors thank J. Denschlag, P. Schmidt, R. Stock,
and A. Simoni for helpful discussions. This work was
supported by the Austrian Science Foundation (FWF),
the European Union projects OLAQUI (FP6-013501-
OLAQUI), CONQUEST (MRTN-CT-2003-505089), the
SCALA network (IST-15714), the Institute for Quantum
Information GmbH, the EU Marie Curie Outgoing In-
ternational Fellowship QOQIP, and the National Science
Foundation through a grant for the Institute for Theoret-
ical Atomic, Molecular and Optical Physics at Harvard
University and Smithsonian Astrophysical Observatory.
APPENDIX A: DERIVATION OF THE
EFFECTIVE HAMILTONIAN (1)
While the Hamiltonian (1) is intuitively obvious, we
find it nonetheless worthwhile to summarize the micro-
scopic derivation in an adiabatic approximation, and to
discuss its validity. For simplicity we consider an atom
with a single outer-shell electron (alkali atom) and an
ion with single positive charge +e. In addition we do
not consider the internal structure of the ion. The total
Hamiltonian can be written as
+He +Hlas +Hrf . (A1)
Here the labels 1 and 2 correspond, respectively, to
atomic nucleus and ion, and He is the Hamiltonian of
the electron, which includes the Coulomb interactions,
|xe − x1|
|xe − x2|
|x2 − x1|
. (A2)
Here, for simplicity, we omit the contributions of the
core regions for both the atom and ion. In this way our
model refers in fact to H+2 molecule. For alkali atoms and
alkaline-earth ions one should treat the complete struc-
ture of core regions, however, our approach can be readily
generalized to this more complicated case.
The term Hlas describes the interaction of the atom
with a laser beam creating an optical potential, written
in the electric dipole representation
Hlas = −dE⊥(x1, t), (A3)
where d = e(x1 − xe) is the dipole moment of the atom,
and for simplicity we neglected the influence of the laser
on the ion, which has typically a different electronic
structure that the atom. In addition we have applied
the long-wavelength approximation, neglecting changes
of the electric field on the scale of the atom. In the case
of the optical lattice the transverse part of the electric
field E⊥ can be assumed to have the form of a standing
E⊥(x, t) = E0 cos(ωLt) [cos(kLx+ φx) + (x→ y, z)] ,
where for simplicity we have assumed the same amplitude
E0, the same wave vector kL, and the same frequency of
the laser light ωL for all three laser beams creating the
optical lattice potential. The abbreviation (x → y, z)
denotes sum of the terms with x replaced by y and z,
and ϕk for k = x, y, z is the phase factor characterizing
the standing wave. Finally, Hrf is the electric potential
creating the rf-trap
Hrf = eΦ(x1, t) + eΦ(x2, t)− eΦ(xe, t), (A5)
where Φ(x1, t) is the time-dependent electric field of the
rf trap
Φ(x, t) =
2 + uyy
2 + uzz
2 + vyy
2 + vzz
2) cosωrft. (A6)
Here, ωrf is the frequency of the time-dependent part of
the electric potential, and uk, vk (k = x, y, z) are am-
plitudes depending on the geometry of the trap [5]. The
electric field at every instant of time has to fulfill the
Laplace equation: ∆Φ = 0, hence, the coefficients uk, vk
are subject to the following conditions: ux+uy+uz = 0,
vx + vy + vz = 0.
Below we indicate the basic steps of the derivation.
Expansion in the basis of Born-Oppenheimer wave
functions for electron motion. We start from generating
a complete set of electronic wave functions, parametrized
by the positions of the atomic core and of the ion
HeΦn(xe|x1,x2) = En(|x2 − x1|)Φn(xe|x1,x2) (A7)
In this way the total wave function can be expanded in
the basis of Born-Oppenheimer electronic wave functions
Ψ(x1,x2,xe, t) =
cn(x1,x2, t)Φn(xe|x1,x2). (A8)
Since the basis is complete, the expansion of the wave
function does not involve any approximations.
Retaining in the expansion only the modes coupled by
the laser. In the expansion (A8) we keep only two modes
coupled by the laser light creating the optical lattice: the
electronic ground state Φg(xe|x1,x2) and the electronic
excited state Φe(xe|x1,x2). In this way we utilize the
Born-Oppenheimer approximation, treating the electron
motion in the adiabatic approximation. This assumes
that the time scale of the electron dynamics is much
faster than the dynamics of the atomic nucleus and of the
ion, which is typically fulfilled since the electron is much
lighter than the other two particles. The approximation
of the two coupled channels can be easily generalized to
more, or even infinite number of channels, since the other
channels are weakly populated and we treat them within
the perturbation theory.
Adiabatic elimination of the excited electronic state
coupled through the laser: derivation of the optical trap
potential. The expansion coefficients cg(x1,x2, t) and
ce(x1,x2, t) fulfill the following set of coupled equations:
∂cg(x1,x2, t)
+ Eg(|x2 − x1|) + eΦ(x2, t)
cg(x1,x2, t)− degE⊥(x1, t)ce(x1,x2, t) (A9)
∂ce(x1,x2, t)
+ Ee(|x2 − x1|) + eΦ(x2, t)
ce(x1,x2, t)− degE⊥(x1, t)cg(x1,x2, t), (A10)
where deg(x1,x2) = 〈Φe|d|Φg〉 is the dipole matrix ele-
ment between the ground and excited electronic states,
which in general depends on the position of atom and
ion. In the derivation of (A9)-(A10) we have neglected
the action of the rf field on the atom core and the elec-
tron, putting
〈Φk|Hrf |Φk〉 ≈ eΦ(x2, t), k = e, g (A11)
〈Φe|Hrf |Φg〉 ≈ 0. (A12)
We focus on the regime of far-detuned laser: ∆ ≫ ΩL,
where ∆ denotes the detuning ∆(|x2 − x1|) = Ee(|x2 −
x1|)−Eg(|x2−x1|)−ωL, and ΩL = −degE0/~ is the the
Rabi frequency. For such conditions, the excited state
is only weakly populated and can be adiabatically elimi-
nated. Additional simplification comes from the fact that
the transitions between states g and e, due to the laser
light, occur on a time scale much shorter than the mo-
tion of atom and ion, and the dynamics of the atom and
ion centers of masses can be decoupled from the internal
dynamics, in full analogy to the Born-Oppenheimer ap-
proximation. Eliminating the excited state, in basically
the same manner as in the standard derivation of the AC
Stark shift, we obtain the following equation that governs
the dynamics of the atom and ion center of masses:
∂cg(x1,x2, t)
+ Eg(|x2 − x1|) + Vopt(x1)
+ eΦ(x2, t)
cg(x1,x2, t), (A13)
where
Vopt(x) = −
cos(kLx+ φx)
2 + (x→ y, z)
(A14)
is the effective potential due to the laser field. In the for-
mula (A14) we have neglected the position dependence
of the detuning ∆(|x2 −x1|) and of the dipole matrix el-
ement deg(x1,x2), assuming that they are modified only
at very short distances between the particles. This can
be justified, since in the atomic collisions, the probabil-
ity of finding the particles at short distances, comparable
to the range of chemical binding forces, is typically very
small.
Time averaging over fast oscillations of the rf field:
derivation of the Paul trapping potential. We replace the
time-dependent rf field by an effective, adiabatic poten-
tial. In this way we neglect the fast micromotion of the
ion on the time scale of ωrf [5]. Finally we obtain the fol-
lowing equation that describes the dynamics of the wave
function cg(x1,x2, t) dependent on the atom and ion po-
sitions
+ Eg(|x2 − x1|) + Vopt(x1)
Vrf(x2)
cg(x1,x2, t). (A15)
Here Vrf denotes the effective potential
Vrf(x) =
2 + ωyy
2 + ωzz
, (A16)
where ωk = (ak + q
1/2ωrf/2, ak = 4euk/(m2ω
and qk = 2evk/(m2ω
rf) for k = x, y, z. The interac-
tion between atom and ion is given by Eg(|x2 − x1|) –
the ground-state energy of the electron motion, which at
large distances behaves as Eg(r) ∼ −αe
2/(2r4).
APPENDIX B: DETERMINING OF ϕe AND ϕo
FROM PSEUDOPOTENTIALS
In principle, the replacement of the r−4 interaction
by the pseudopotential is strictly valid when R∗ ≪ l⊥.
Nevertheless, it is possible to try to determine ϕe and ϕo
forR∗ ∼ l∗ using the energy-dependent pseudopotentials.
To this end we replace the r−4 interaction with [17, 18]
Vs(r) =
2π~2a(k)
r (B1)
for even scattering, and [19]
Vp(r) =
π~2ap(k)
∇ δ(r)
r2, (B2)
for odd scattering. Here, a(k) = − tan δ0(k)/k is the
energy-dependent s-wave scattering length, ap is the p-
wave scattering length: ap(k)
3 = − tan δ1(k)/k
3, the
symbol
∇) denotes the gradient operator that acts to
the left (right) of the pseudopotential, and δ0(k), δ1(k)
are the s- and p-wave phase shifts, respectively.
In 1D the even (ψe) and odd (ψo) scattering waves have
the following asymptotic behavior: ψe(z, k) ∼ sin(k|z|+
δe(k)), and ψo(z, k) ∼ z/|z| sin(k|z| + δo(k)) (|z| → ∞),
where δe(k) and δo(k) denote even and odd scattering
phases, respectively. In analogy to the three-dimensional
scattering theory, we define 1D even (ae1D) and 1D odd
(ao1D) scattering lengths: a
1D = limk→0 −(tan δe,o(k))/k.
From this definition it follows that for k = 0 the even and
odd scattering waves behaves as: ψe(z, k = 0) ∼ |z|−a
and ψo(z, k = 0) ∼ z − a
1Dz/|z| (|z| → ∞). The latter
result applied to Eqs. (10) and (11) leads to
ae1D/R
∗ = − cotϕe (B3)
ao1D/R
∗ = − cotϕo (B4)
In the pseudopotential approximation one can solve the
quasi-1D scattering problem exactly, and calculate values
of the 1D scattering lengths for even [20, 21] and odd [22]
waves
ae1D(k) = −
2as(E)
, (B5)
ao1D(k) =
12ap(E)3
where E = ~ω⊥+~
2k2/(2µ) and ζ(s, a) denotes the Hur-
witz Zeta function: ζ(s, a) =
k=0(k + a)
−s [23]. Fi-
nally, to relate the energy-dependent scattering lenghts
as(E) and ap(E) to ϕ, we apply the exact solutions of
the Schrödinger equation for r−4 potential, given by the
Mathieu functions [7, 8, 9].
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