Single Electron Trapping in an Ion Crystal
Weibin Li and Igor Lesanovsky
School of Physics and Astronomy, the University of Nottingham, Nottingham, UK
Introduction
Ion’s local trapping frequency
E
E[GHz]
-1
0
++
2
0
2
!E "
!E"
5
z [10 a0]
2!i
r
l
Rydberg Ion Molecule
FIG. 1. !Color online# Electronic potential if the ions are frozen
Lesanovsky, Mueller & Zoller,
at their equilibrium positions of the configuration A++ + A+. The
PRA 79, 010701(R)(2009)
electronic potential supports bound states which are delocalized between the two nuclei. For too large excitation energies the saddle
point due to the inverted harmonic potential leads to field
ionization.
z
Z0,1
zsymChainZ0,2
Rydberg
Ion
Mueller, Liang, Lesanovsky & Zoller,
NJP 10,
093009(2008)
FIG. 2. !Color online#
Sketch
of the energy-level structure close
to ndel. Here the influence of the inverted parabolic potential can be
approximately neglected and the states can be characterized by their
symmetry property under reflection at zsym. The energies of these
pairs of gerade !ungerade# states are indicated by dashed !solid#
lines. The pairs are almost degenerate with a small energy splitting
+i. Wave functions which are localized in one of the two wells
!"El/r(# can be created by linear superpositions of the gerade and
ungerade states. Examples of two such wave functions are shown.
Electron-ion crystal
either ionic core, the potential is approximately a sum of two
one-dimensional Coulomb potentials V1D = −e2 / !2!"0"z"#
potential
ofpositions.
a radio-frequency
(Paul) trap is
centeredThe
at theelectric
respective ionic
equilibrium
For
small excitation energies the potential wells are well
well !"Er(# can be created by the linear combinations "El(
separated. If the excitation energy surpasses Edel
2
2
2 (' and2 "E ( = !1 2
$ −!2e2# / !!"0D#, the wells merge and hence electronic
= !1 / %2#&"Eg( + "E
/ %2#&"Eg( − "Eu(', respecu
r
states which extend over both ionic cores are possible. Extively. They are not stationary, but evolve under the Hamilpressing Edel in terms of the principal quantum number
tonian Htunnel = #+"El()Er" + H.c., which describes Rabi osciln, one
finds this ismerging
to occur
at ndel of
lations
the electronic
density between
the left and right
where
the driving
frequency
RFoffield,
and electric
gradient
wells at a rate +. As an example we provide two sets of
$ !e / ##%!mD# / !16!"0#. For the employed parameter set one
and
determined
by which
the trap.
gerade and ungerade eigenstates, which were obtained for
finds the
numericalisvalue
ndel = 195.654 &14',
is sigz energies correspond to the printhe above parameters: Their
nificantly
smaller than the
principal
quantum
number wecan
obParticles
with
same
charges
!1#
tained earlier by invoking the argument of overlapping wave
=
195.318,
n
cipal quantum numbers n!1#
g
u = 195.335 and
!2#
beFurther
trapped
while
inverse
charged
functions.
inspection
of the potential
reveals
that the
n!2#
g = 195.599, nu = 195.668. The tunnel coupling in the two
inverted parabolic potential due to the Penning trap eventucases evaluates to +1 = 2! * 29.5 MHz and +2 = 2!
particles
are
repelled
from
the
trap.
ally leads to field ionization, which is shown in Fig. 1. The
* 121.2 MHz. Since these rates are significantly larger than
$ Z0,2# is given
zsd = %Z0,2,
positionSix
of the
saddle
!zsd trapped
the oscillation frequencies of the ionic
cores, it is indeed
Ions
are
in aby linear
y
where % $ 1 is the solution of the equation !% − 1#−2 + !%
justified to use the picture of a frozen ionic motion. More2 2
doubly
2
−1
+ 1 / 2#−2Paul
− &% = 0trap,
and &where
= &!1 / 2#m'four
over, the rates are much larger than the radiative decay rate
z Z0,2'&e / !4!"0"Z0,2"#'
3
2
of Rydberg states, which is usually smaller than 1 MHz &10'.
= !1 / 2#Q
4 / 9. This
is solved
by
charged
areequation
confined
in x-y
1 / !Q1 + Q2# =ions
Therefore the described process can be considered coherent.
% = 2.12, and thus the saddle point is for our parameters
charged
ions We are now in position to study the charge-transfer
x prolocated plane
at zsd = 217and
523a0two
. Fromsingly
this we find
that classically
++
+
+
++
+
A
→
A
+
A
. To this end we employ two laser
cess
A
ionization
is
expected
to
occur
beyond
energies
correspondare
sitting
symmetrically
on
z-axis.
fields coupling the ground state of the right ion "Gr( resoing to the principal quantum number nion = 1.456%Z0,2 / a0
2/3%static2/3magnetic
nantly to the state "Er( with a Rabi frequency '2. Due to the
field
= !0.21eA
m# / !#"0 (1/6# $ 466. Hence
there is z
an is
energy
tunnel coupling, the state "Er( evolves into the state "El(,
window ndel ) n ) nion in which delocalized electronic states
applied
in
z-direction
in
order
to
which is then resonantly coupled to the ground state "Gl( of
exist. Here the escape of the electron from the trap is prethe left ion with Rabi frequency '1. During the charge transvented confine
by its attraction
the doubly charged
ion.direction.
For the
theto electron
in x-y
fer, the external positions of the ions are frozen. Hence, the
given parameters the width of this window is about #2!
Assuming one electron is located external state of the final configuration A+ + A++ will be dis* 60 GHz.
placed from its equilibrium. The corresponding energy
We now
focus vicinity
on the electronic
states center,
which lie close to
in
the
of
trap
offset—i.e., the change in potential energy #,—has to be
the top of the Coulomb barrier. Since for these states the
ionsʼ
potential
is due to the anticonfin- accounted for in order to make the transition "El( → "Gl( resoasymmetry
of the
electronic potential
nant. This is shown in Fig. 3 where the potentials of the
ing is small, it makes sense to define gerade and ungerade
ground-state configurations A++ + A+ and A+ + A++ are destates with respect to the approximate
symmetry point zsym
6
picted. The intersections shown are made along the normal
= !Z0,1 + Z0,2# / 2. In Fig. 2 we show a sketch of the energy2
2
2 Z = - Z + - Z at Z = −- Z + - Z set to zero.
coordinate
level structure
of
Hamiltonian
!4#
in
the
vicinity
of
n
.
Gera
− 1
+ 2
b
+ 1
− 2
del
IE
i
i
i
i
ade !"Eg(# and ungerade !"Eu(# states are energetically split
The coefficients are given by -. = %!1 / 2# . 3 / !2%73#. The
by the energy 2#+ = Eu − Ei=1
energy gap between the two configurations evaluates to
g, with + being the tunnel coupling. The splitting increases with increasing degree of exci#, = !e(/2# / !2 * 31/3#, which yields for our parameters
6 in the left !"El(# or right , =62! * 21.4 GHz.
tation. States which are localized
Φ(r, t) = α cos Ωt(x − y ) − β(x + y + z )
Ω
α
β
B
V
=
+
!
a
b
Q [ (x + y ) + z ]
2
2
!!
i=1 j>i
! Qi
Qi Qj
−
010701-3
|Ri − Rj | i=1 |Ri |
where Qi , and Ri are charges and positions of ions and
b=
-6
1
1.2
1.4
r
0.8
1.6
!
2e2 α2
M Ω2
+
2
Bz2
e
4M
× 4eβ
− 2eβ
"
50
40
30
20
0.8
1
1.2
1.4
r
50
45
40
35
0.8
1
1.2
1.4
r
40
30
20
10
0.8
1
1.2
1.4
r
1.4
1.6
(x0 , 0, 0) (0, x0 , 0) (−x0 , 0, 0) (0, −x0 , 0) (0, 0, z1 ) (0, 0, −z1 )
Let z1 = kx0 , equilibrium positions of ions are found by solving
>0
! √
"1/3
4 2−2
2
xs =
+
4a
a(1 + k 2 )3/2
4
3
k
Solve k as a function ofr = a/b .
Stationary point is
2
1
2.5
15
0.8
1
1.2
0.8
1
1.2
0.8
1
1.2
r
1.4
1.6
1.4
1.6
1.4
1.6
30
25
20
15
r
35
30
25
1.6
r
Electronʼs potential near trap center, in a good approximation, is
Ve ≈
with
!
2
mωρ 2
mωz2 2
2
2
ρ +
z + ω L̂z − eα cos Ωt(x − y )
2
2
2 (eBz )
e 1 − 2k
2
ωρ =
+ eβ +
m
8m
4πε k 3 x3s
2
2
3
"
!
2 e 4k − 2
2
ωz =
− 2eβ
3
3
m 4πε k xs
2
3
"
eBz
ω=
2m
the above frequency are in GHz region while RF field is in 100MHz,
indicating that we can solve the time-dependent part adiabatically. The
time-independent Hamiltonian can be solved exactly and time-dependent
part could be solved perturbatively.
Ω =2 π × 20MHz
Bz = 1T
β = 9 × 106 V/m2
α = 1.0 × 109 V/m2
150
At equilibrium, ionʼs coordinates can be found
∂ 2 VIE
∂Ri2
20
Electron trapping
100
50
00
=0
r
25
1.6
0
and
1.2
200
Ions’ Equilibrium
∂VIE
∂Ri
1
30
1.6
55
Energy (GHz)
a=
4πε
e2
4πε
e2
-6
0.8
2!i+1
-2
1
2.5
2
X-axis Trap Frequency (MHz)
-2
2
Y-axis Trap Frequency (MHz)
++
3
gerade
ndel
D
2.25
1.75
delocalized
electronic states
-400
Equilibrium and frequencies of
ions sitting on z-axis
Z-axis Trap Frequency (MHz)
ndel
3
3.5
X-axis Trap Frequency (MHz)
-350
Ω =2 π × 20MHz ，a = 2 × 10 /m and
17
Equilibrium and frequencies of
ions sitting on x-axis
Y-axis Trap Frequency (MHz)
-300
nion
Bz = 1T
Coulomb
potential
ungerade
ionization
saddle
Considering Ca+ , for
Z-axis Trap Frequency (MHz)
-250
U0 ≈ Uc0 +
2
2
M
ω
M
ω
y
z
2
(x − xs ) +
(y − ys )2 +
(z − zs )2
2
2
2
M ωx2
Zs(10 m)
PHYSICAL REVIEW A 79, 010701!R# !2009#
TRAP-ASSISTED CREATION OF GIANT MOLECULES AND…
Expand ionʼs potential as Taylor series to second order, local potential
for ions in the vicinity of equilibrium is
xs(10 m)
When ions are excited to a Rydberg state in ion traps, weakly bound
electrons render the system exotic properties, such as binding repulsive
interacting ions into giant ionic molecules and achieving spin models as
well as quantum gates in one dimensional Rydberg ions.
Traditionally, ion traps are used to confine single ions or crystals of ions
of the same charge. Simultaneous trapping of ions and electrons seems
not possible at first glance because either Paul trap or Penning trap
confine only particles of a certain charge while the oppositely charged
ones are repelled from the trap.
We demonstrate that a single electron can be trapped in the centre of a
small-number ion crystal in a linear Paul trap.
The successful trapping of the electron can be monitored by changes of
RAPID COMMUNICATIONS
ionʼs equilibrium position and oscillating frequency.
5
7.5
r
10
20
40
π
T
ΩT
60
80
2π
Outlook
Electron dynamics has to be studied consistently. When energy splittings
close to driving frequency, perturbation becomes invalid, especially near
degeneracy.
Ion is assumed always in its equilibrium. However, the coupling among
RF field and electron may cause instability, which can be investigated via,
e.g. molecular dynamics.
Electron trapping will be studied in the future. During this process, both
frequency and equilibrium position of ions change dramatically, which
migh bring interesting dynamics.