Loading a Planar RF Paul Trap From a Cold Yb +
Source
by
Brendan John Shields
Submitted to the Department of Physics
in partial fulfillment of the requirements for the degree of
Bachelor of Science in Physics
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
May, 2006
©2006 Brendan Shields
All Rights Reserved
The author hereby grants to MIT permission to reproduce and to distribute publicly paper
and electronic copies of this thesis document in whole or in part.
Signature of Author
Department of Physics
May 18, 2006
Certified by
Vladan Vuletik
Lester Wolfe Associate Professor of Physics
Thesis Supervisor, Department of Physics
Approved by
David E. Pritchard
Cecil and Ida Green Professor of Physics
Senior Thesis Coordinator, Department of Physics
MASSACHUSETtj11WJI
MASSACHU 'NS'n'a'
OF TECHNOLOGY
JUL 0 7 2006
LIBRARIES
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ARCH s8
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2
Loading a Planar RF Paul Trap From a Cold Yb + Source
by
Brendan John Shields
Submitted to the Department of Physics
on May 18, 2006, in partial fulfillment of the
requirements for the degree of
Bachelor of Science in Physics
Abstract
In this thesis, we demonstrate a functioning planar radio frequency, three-rod Paul
Trap, loaded with Yb+ ions that have been photoionized from a source of neutral atoms,
which were cooled in a magneto-optical trap. Planar ion traps have only recently been
successfully loaded[1], and never from a cold ion source. Additionally, ionizing directly
from a MOT allows for isotope selection.
Thesis Supervisor: Vladan Vuleti6
Title: Lester Wolfe Associate Professor of Physics
3
4
Acknowledgements
In my two years working as an undergraduate
in the the Vuleti6 group, I have had
the pleasure of working with a friendly group of extremely talented people on a variety of
projects. In particular, I would like to thank my thesis advisor, Prof. Vladan Vuleti6, for
introducing me to atomic physics and to ion traps, and for giving me the opportunity to
work on this and other projects. He is a wonderful teacher and mentor, and his patience,
inspiration and guidance has been invaluable. A special thanks is due to the graduate
students with whom I worked on this project, Marko Cetina, Andrew Grier, and Jon
Campbell, who gave up much of their time to provide me with help and guidance, and
whose patience and hard work has brought us to where we are. Two undergraduates,
Thaned Pruttivarasin and David Brown, have also made great contributions to the experiment. Finally, I owe thanks to a great many people who I cannot name individually
here, but particularly to my loving parents, who have supported and encouraged me in
all my endeavors.
5
6
Contents
1
Introduction
13
1.1 A Novel Approach to Planar Trap Loading .................
1.2 Outline of Thesis.
14
15
2 Experimental Overview
17
3 Magneto-Optical Traps
21
3.1 The Scattering Force .............................
3.2 A Restoring Force: The Doppler and Zeeman Effects ...........
22
24
4 Ionization and Ion Detection
29
4.1 Ion Detector .
4.1.1 Electric Field Considerations ........
4.1.2
4.1.3
Pulse Counting/Continuous Voltage Mode
Analog/Pulsed Voltage Mode .......
4.2 Photoionization ...................
5 Paul Traps
5.1 Ideal Paul Traps ......
5.1.1 Linear Confinement .
5.1.2 Point Confinement .
5.2 Planar Traps ........
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Results
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6.1 Trap Lifetime.
6.2 Trap Population .......
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7 Discussion
7.1 Future Work .
59
A Ytterbium
65
59
7
8
List of Figures
2.1 Experimental Layout.
2.2
Close-up view of the chamber
18
19
............
3.1 Photon scattering by an atom ............
...........
3.2
3.3
Scattering Force vs. Detuning
Zeeman Trapping.
4.1
4.2
4.3
4.4
Ion Detector Schematic .
Circuit diagram for the ion detector ........
Ionization Apparatus.
Ionization Rate Vs. MOT Fluorescence.
5.1
5.2
5.3
5.4
Electrode Configuration for Classic RF Paul Traps.
Planar RF Paul Trap Electrode Configuration . . .
Chip Layout.
Trap Height vs. Electrode Voltage Ratio ......
6.1
6.2
6.3
6.4
Typical Ion Detector Output .............
Trap Population vs. Load Time ...........
Trap Lifetime .....................
Ion Dispersion vs. Dump Time ...........
A.1 Yb Energy Levels ....................
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23
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List of Tables
4.1 Ionization Rates ................................
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37
12
Chapter
1
Introduction
Ion traps have their origins in the work of F.M. Penning in 1936 [2] and of W. Paul
in the 1950s, for which he shared the nobel prize in 1989 [3]. This thesis describes a
novel kind of radio frequency (RF) Paul trap. The RF Paul trap has found applications in a variety of fields, including mass spectrometry and beam focusing in particle
accelerators. The use of ion traps in atomic physics has for many years focused on precision spectroscopy, especially beginning with the first demostration of laser cooling of
trapped ions in 1978[4]. This opened the door for crystal structures of ions confined in a
trap[5; 6]. In recent years, there has been a renewed interest in ion traps with the advent
of the field of Quantum Computation. Ion traps provide a promising framework for carrying out scalable computations on qubits[7; 8]. Small-scale entanglement and quantum
computing operations have already been carried out on small numbers of qubits in ion
traps[9;
10; 7].
The latest challenge for ion trappers is to load ions into a planar trap configuration.
This differs from previous trapping geometries, in which the trap electrodes exist in a
three dimensional configuration, by flattening out the electrodes into a two dimensional
geometry. A planar trap has great advantages for scalability, since such electrode config13
urations can easily be printed directly onto a chip, allowing for smaller traps as well as
more complex traps or even multiple traps on a single chip[1].
In this thesis, I describe
the successful loading of such a planar trap.
1.1
A Novel Approach to Planar Trap Loading
Our approach to ion trapping is unique, in that we load a planar trap from a source
of cold ions. Typically, the process for obtaining ions is to direct a beam of high energy
electrons at a gas of neutral atoms. The electrons scatter off of the neutral atoms, ionizing
them. Due to the high energy nature of this process, the resulting ions are necessarily
hot, and only the slowest of them may be trapped. In addition to the low efficiency,
it has been suggested [12] that stray electrons scatter in the direction of the trap and
create undesireable "patch" potentials, which interfere with the trapping potential. This
will be even more problematic for planar traps printed on a dielectric material, since the
dielectric easily collects static surface charges.
In contrast to the electron beam technique, we photoionize neutral atoms which
have been cooled to sub-Kelvin temperatures in a magneto-optical trap. Photoionization
as a method for obtaining ions has already been demonstrated in 3D trap geometries[13],
but not in a planar trap, and not from a MOT. The photoionization imparts minimal
heating, leaving the ions cold. This should eliminate the surface charge build-up, and
efficiently produce trappable
ions. The MOT is also isotope-selective, so we can easily
select between the different Yb isotopes for our trap by ionizing from the MOT.
14
1.2
Outline of Thesis
In the remaining sections of this thesis we cover the following areas. An overview
of the experiment and apparatus is given in Chapter 2. The MOT which is our source
of cold neutral atoms is described in Chapter 3. Details of the ionization process and
detection of ions are described in Chapter 4. A brief overview of ion traps in general,
and a description of the planar trap we use are given in Chapter 5. A summary of our
results is presented in Chapter 6. Finally, Chapter 7 offers some concluding remarks.
15
16
Chapter 2
Experimental Overview
A diagram of our experimental setup is shown in Fig.2.1. The main systems involved are the lasers and associated frequency controls, the vacuum chamber which houses
the MOT and the ion trap, and the ion detector. A close-up photograph of the chamber
is shown in Fig.2.2.
The laser setup currently consists of a master laser diode (Nichia model NDHV310ACAE1)
locked to the 'So -+
P1 (398.799 nm) transition of atomic Yb using the DAVLL tech-
nique [14]via an Yb-filled hollow cathode lamp (Hammamatsu model L2783). The master
laser is frequency stabilized via optical feedback from a grating. It outputs
-
2.8mW of
power after an optical isolator, part of which is picked off and injected into a slave laser
which provides the power for the MOT. The slave output of - 12mW is split into three
beams, which are appropriately polarized and aligned so as to create the MOT.
The vacuum chamber has four viewports along two diagonal axes in the vertical
plane, as well as two large horizontal viewports providing optical access along the third
axis. Our Yb source is an oven which is recessed from the main chamber but has a direct
line of sight into the chamber. The ion detector is centered above the trap at the top of
the chamber. An LED used to ionize the neutral atoms in the MOT has optical access
17
Legend
Optical Set-Up
:er
ucer(PZT)
itter(PBS)
Repumper
Slave
Master
Figure 2.1: Experimental layout for the ion trap. The 398.99nm master laser is gratingstabilized and locked to the 398.799nm line of atomic Yb via the DAVLLtechnique. Part
of the 2.8mW of master power is used to inject the slave, whose 12mW output operates
the MOT. Both the slave and master are fed into a Fabry-Perot interferometer to monitor
the slave injection-locking.
To operate optically on Yb+ ions, a 369nm laser is needed,
along with a 935nm repumping laser. (Diagram courtesy Jon Campbell)
18
Figure 2.2: Close-up photo of the chamber. The ion trap is in the center of the chamber,
with the ion. detector directly above. The MOT optics are on the diagonal viewports.
Attached to the left is the Yb oven, and on the right side of the chamber are the electrical
feedthroughs for the trap electrodes. The vacuum pumps are below the chamber. In the
foreground on the left is the LED and its focusing optics.
19
off-axis with the horizontal MOT axis. The LED is a Nichia model NCCU001E, which
outputs 85mW with a spectrum of 385 ± 15nm. A photodiode used for measuring MOT
fluorescence also has optical access off-axis with the horizontal MOT axis.
The magnetic field for the MOT is provided primarily by a main set of water-cooled
copper coils carrying -
30A of current.
Three pairs of offset coils are also available
to manipulate the MOT along each of the three axes. The magnetic field gradient is
- 45G/cm.
The ion trap itself consists of a set of electrodes printed onto a PCB chip. The
excess dielectric is removed to allow optical access for the MOT beams. The trap is
situated 3mm below the center of the chamber.
20
Chapter 3
Magneto-Optical Traps
The novel aspect of our approach to ion trapping is that we are loading the trap
with cold ions, which has never before been done. In addition, we are loading a planar
trap, which has only recently been demostrated for the first time[1]. Conventionally, ion
traps have used a gas of atoms ionized by an electron beam as their source of ions. This
approach has several downsides. First, the electrons are necessarily high energy, and
consequently cause heating, which means only a fraction of the ions are slow enough to
be trapped.
Second, electron beam scattering process is believed to result in a high level
of residual charge adhering to the trap itself, creating "patch" potentials[12]. This added
potential perturbs the trapping potential and is difficult to compensate for. Third, this
ionization process does not provide a way to select for a particular isotope.
We have
eliminated all of these problems by using a magneto-optical trap (MOT) to cool a gas
of neutral atoms. We then photoionize the atoms from the excited state of the MOT
transition. This ionization process requires only a single photon and imparts minimal
energy to the ion, so our trap can be loaded from a cold source of ions. Photoionization
has been used as a source of ions before[13], but never with a planar trap, and never from
a cold source. The MOT also has the advantage of providing a mechanism for isotope
21
selection. The lasers used are sufficiently narrow so as to discriminate against isotope
species.
The MOT is a simple and elegant radiation pressure trap which confines a high
density gas of atoms in a small region of space. Since the first MOT was demonstrated in
1987[15],the MOT has become a staple choice among neutral atom traps. This chapter
describes the operation of the MOT.
3.1
The Scattering Force
Optical traps employ one of two different light forces, either the dipole force which
is due to the interaction of a field gradient with the dipole moment of the atoms, or
radiation pressure from the scattering of photons. A MOT makes use of the scattering
force to cool and spatially confine a gas of atoms.
When a beam of light is shone upon an atom, it exerts an average force due to the
scattering of light by the atom. Suppose we have an atom of mass m in its electronic
ground state moving with speed
x direction (see Fig.3.1).
and a photon of frequency v moving in the positive
The momentum of the atom is mv, while the momentum of
the photon is (hv/c)x. If the photon is of the correct frequency, the atom will absorb
the photon and will be left in an excited state. Since momentum must be conserved,
the momentum of the atom will be my = mfy + (hv/c)x. After some amount of time,
the atom will spontaneously emit a photon into space and be left once again in the
ground state. The process can then be repeated. If instead of a single photon we have
a laser beam, then every time a photon is absorbed, the atom will receive a momentum
kick of (hv/c)x, while the reemitted photons will be directed isotropically into space,
contributing nothing, on average, to the momentum.
The force F which results from the absorption and spontaneous reemission of pho22
I%
-I
Figure 3.1: Photon scattering by an atom. An incident photon is absorbed, imparting
a change of momentum to the atom and leaving it in the excited state. Photons from
relaxation to the ground state are emitted isotropically into space. Thus, over the course
of many scattering events, a beam of photons will exert a net force on the atom in the
direction of the beam.
23
tons is simply F = yp/A, where Afl is the momentum change per scattering event and
yp is the scattering rate. The momentum change, we have just argued, averages to
(AiJ = (hv/c)&. The scattering rate is given by[16]
soy/2
P
= 1 + so + (25/7) 2 '
(31)
where y is the natural linewidth of the atomic transition, 6 is the detuning of the laser
from the resonant frequency of the transition, and so, the on-resonance saturation parameter, is a measure of the laser intensity. So the force exerted on an atom by scattering
photons from a beam of light is
II=
()
(1 + s + (2/)2)
(3.2)
A graph of the 6-dependence of the scattering force for several different saturation
parameters is shown in Fig.3.2. It is maximum for
= 0 and symmetric about
= 0.
The sources of detuning that are most significant for the MOT are the Zeeman effect and
the Doppler effect which will be discussed in §3.2.
3.2
A Restoring Force: The Doppler and Zeeman
Effects
In order to trap something in space, one must create a potential minimum at some
spatial coordinate. At such a minimum, a displacement in any direction will result in a
restoring force. Since a beam of light can only exert a scattering force in the direction
of propagation, it follows that in order to have a restoring force in all directions, an
arrangement of counter-propagating beams is required. Typically, the beams are oriented
24
-U. r,
0.45
0.4
0.35
0.3
> 0.25
ULL
0.2
0.15
0.1
0.05
n
-3
-2
-1
0
1
2
3
8 ()
Figure 3.2: The scattering
force as a function of detuning, 6, for three values of the
saturation parameter, so. The force is maximum for = 0 and is symmetric about 6 = 0.
25
along three orthogonal axes, as is the case in our experiment. Consider first a pair of
counter-propagating
beams in one dimension, propagating along the x-axis.
If the laser is tuned to the atomic resonance, then the force from the beams will
exactly cancel, eliminating any effect. One must therefore cause the effective detuning
from resonance for one beam to be different from that of the other. The two phenomena
that we will consider for this purpose result in exactly the opposite detuning for interaction with one beam as for the other. Since F(6) = F(-6), the first step is therefore
to tune the laser off-resonance by some amount 60, so that the effective detuning for one
beam is o +
a
and for the other o -
. It turns out to make a difference which way
(above or below) one tunes off-resonance, as we shall now see.
If an atom is moving in the positive x direction with speed v, then due to the
Doppler effect, it will see the laser propagating in the same direction as having a frequency
v+ = v(1 - v/c) where v is the frequency the laser is tuned to in the lab frame. On
the other hand, it will see the laser propagating in the negative x direction as having
frequency v_ = v(1 + v/c). We are trying to cool the atom, so we want to push against
its motion. Therefore, we want v'_ to be closer to resonance than v+. This is satisfied if
we tune below resonance.
The Doppler cooling described above will cool the atom, but will not confine it
spatially.
To acheive spatial confinement, we employ the Zeeman effect. If one of the
levels in the transition has total angular momentum J > 0, then the level will split into
distinct energy levels in the presence of a magnetic field. In the weak-field limit, the
splitting is [17]
E = ILBgJBmJ,
where B
(3.3)
eh/2m is the Bohr magneton (m is the electron mass), B is the applied
26
magnetic field, and gj is the Land6 g-factor:
gj
j(j-+1+
1)- 1(1+ 1)+ 3/4
2j(j + 1)
(3.4)
In the case of neutral Yb, the 'P1 (excited) state has J = 1, so the possible values of
mj are mj = 1,0, -1 (see Appendix A for the level scheme of neutral Yb). A transition
from the So (ground) state to mj = -1 requires a- light, while a transition to mj = +1
requires a+ light. Therefore, by polarizing one beam to be a- and the other to be
a+ we can control which beam operates on which transition. We then place the whole
configuration in a magnetic field which is proportional to x. Since the Zeeman splitting
in Eq. 3.3 is proportional to mj, we can make the mj = -1 transition closer to the laser
frequency than the mj = +1 transition for x > 0, and vice versa for x < 0. Thus, the
atoms to the right of x = 0 will preferentially absorb a- light propagating in the negative
x direction, while the atoms to the left of x = 0 will absorb the a+ light propagating in
the positive x direction. The force will thus push towards x = 0 from both directions.
This effect is illustrated
in Fig.3.3.
It is actually possible to create a magnetic field which behaves in the same way
along each of three orthogonal axes, with only two sets of coils carrying opposite currents
[18]. The field varies linearly on the coil axis and in the plane perpendicular to the
coil axis between the two coils. Thus, extending the confinement to three dimensions is
straightforward.
27
Energy
l
I
--
+1
8=0
A
E -E
e
8=-8
g
I
m
m=-1
+
<
I
I
6
f
x -0
Xo
B=0
B
Figure 3.3: The Zeeman effect for spatial confinement in a MOT. The detuning of the
laser is -60, and the energy of the transition is Ee- Eg, with Ee the energy of the
1P state and Eg the energy of the So state. The two diagonal lines represent the
1
energy of the mj = il sublevels of the excited state. Laser light propagating in the
negative x direction is a- polarized, while light propagating in the positive x direction
is a + polarized. The magnetic field increases from negative to positive x, resulting in a
restoring force.
28
Chapter 4
Ionization and Ion Detection
Having obtained a cold source of neutral atoms in the MOT, the next step is to
create ions from them. This section describes the ionization process and procedures for
direct measurement of the ions.
4.1 Ion Detector
Ideally, we would observe ions directly via optical means, and this will eventually
become a necessity in order to perform measurements on the ions in the trap. However,
for the time being our laser systems are not fully operational and we have no optical
means of probing the ions. Therefore, we instead detect the presence of ions through
the use of an electron multiplier (Burle Magnum 5901) which I will refer to as the "ion
detector".
The basic idea behind the ion detector's operation is to trigger an electron cascade,
thereby increasing the signal of a single ion by a high gain factor. This is accomplished
by accelerating an incoming ion towards a highly negative potential, where it will collide
with the wall of a glass tube lined with a semiconducting material capable of emitting
29
secondary electrons. The impact causes several electrons to be emitted. The end of the
tube is held at ground, so the electrons are accelerated towards the end, colliding against
the walls of the tube on the way, causing more electrons to be emitted. This results in
an avalanche of up to - 107 electrons for every incident ion. The process is illustrated
in Fig.4.1(a).
In the case of the Magnum 5901, there are actually six tubes twisted around each
other to reduce noise, and a negatively charged ring is positioned at the front end of the
device to attract the ions (see Fig.4.1(b))'1.
4.1.1
Electric Field Considerations
To estimate the electric field generated by the ion detector, we model it as a charged
ring, at a distance of 4cm from the trap, which we take to be at ground for the purposes
of this calculation (the offset electrodes on the trap are close to ground). The actual field
strength may be somewhat less, due to dilution of the field by the grounded walls of the
chamber, but this exercise should give us some idea of the field strength. The electric
field strength, E, along the axis of a charged ring is given by:
E(z) = 27rrA ((2
r 2 )3 / 2
(4.1)
where Ais the linear charge density on the ring, r is the radius of the ring, and z is the
distance along the axis (z = 0 is the center of the ring). Integrating, we find the potential
difference, AV, between z = 0 and z = zo, for some point z on the axis is
v = V(z = )- V(z = zo) = 27r
1
(-
For further details on the Magnum 5901 construction see [19]
30
Z;
).
(4.2)
skeainabccWW"t
HV
SemlconductinI
layer
: aL .
Electroding
ion
Secondary . Gla
electrons
channel
wall
i
' '
Output
electrons
(a) The walls of the ion detector are lined with
a secondary electron emitting semiconductor,
which enables an electron cascade resulting in
a gain of up to _ 107.
(b) The Burle Magnum device consists of six
channels which are twisted helically. The front
end is a charged ring held at negative high voltage, in our case
-2000V. The six-channel
configuration reduces dark counts, which for
the Burle Magnum 5901 occur at a rate of
< .1Hz.
Figure 4.1: Images originally printed in [19].
31
In our case r = 0.5cm and zo = 4cm, so we have 27rA = 1.142AV. Substituting
this into
our formula for the electric field, we find that the field at the trap has a strength of
E(z = 4cm) = 0.0349AVcm-1 .
(4.3)
The detector has a minimum operating voltage of -1200V, for which the electric
field strength is 42 V/cm, and a typical pulse counting mode voltage of -2000V, for
which the field strength is 70 V/cm. This field strength greatly exceeds the depth of
the ion trap by an estimated factor of 50, so it is necessary to switch off the detector
while trapping. Furthermore, for a voltage of -2000V, an ion which is initially at rest
at z = 4cm will have a flight time to the detector of < 4.5ps. Since measurements on
the trap consist of turning the detector on at the end of a trapping cycle, it is therefore
necessary to switch the detector on quickly so that it will be able to produce a signal by
the time the ions arrive from the trap. To satisfy our measurement needs, we operate
the ion detector in two modes, pulse counting mode (continuous voltage), for measuring
ionization rates, and analog mode (pulsed voltage) for measuring trap population.
4.1.2
Pulse Counting/Continuous Voltage Mode
To test the ionization process, we need to measure ionization rates. This requires
integration of the number of ions over an appreciable length of time (seconds). Therefore,
it is simplest to operate the ion detector with a continuous DC voltage source. For this
purpose, we used a Bertan (model 205A-05R) high voltage supply, capable of supplying
up to 5kV continuously, though we typically operate the device at 2kV or less in this
mode.
For these parameters, the device produces a current spike of
20ns duration. We
are therefore able to resolve individual ions for steady-state rates of up to - 5 x 107s-1.
32
The ion detector output is sent through a 50Q resistor and the voltage is amplified by a
factor of 100 with a Minicircuits RF amplifier (model ZFL-500LN) before being sent to a
frequency counter (HP model 5382A). The ionization rate is then read off the frequency
counter. A diagram of the electrical configuration for the detector is illustrated in Fig.4.2.
4.1.3
Analog/Pulsed Voltage Mode
When measuring trap population, the detector is quickly switched on and a pulse
of ions travels from the trap to the detector essentially simultaneously. This introduces
two problems for the pulse counting setup described in §4.1.2. First, the detector must
be switched on quickly due to the short travel time of the ions. Second, the ions arrive
in too short a period of time to be resolved individually, so in order to measure trap
population one must count total charge instead of counting ions.
The problem of fast switching is solved by using a pulsed power supply. For this
purpose we used a solid state Pockels-cell driver (Analog Modules model 829), which is
triggered by a TTL input and provides a pulse of up to 3.5kV for up to 300ps, with a rise
time of < 10ns. Using this device, the ion detector can provide useable output within
2- 3ps of turning on the voltage. The output for pulsed voltage mode is measured on an
oscilloscope (Tektronix model 2430) which is connected to the computer for data storage.
The problem of measuring trap population is a fundamental limitation of the detector. In theory, the detector should be able to output a constant level of total current
over a wide range of gains. In pulse counting mode, the output current can saturate with
a single ion, so if a group of ions arrives at the same time, they will all be counted as
one. Therefore, to measure trap population, we reduce the gain so as not to saturate
the output current. We should then be able to determine the number of ions from signal
strength. In practice, the gain can only be turned down to a level where the output
saturates at - 20 ions. Therefore, we have ben unable to satisfactorily measure the trap
population using this method.
4.2
Photoionization
With the MOT and the ion detector in place, we are prepared to test the pho-
toionization process. To ensure that we ionize cold atoms from the MOT, we ionize from
the excited state of the MOT transition in a single-photon process. The relevant energy
levels for neutral Yb are illustrated in Fig.A (Appendix A). The ionization energy of
Yb in its ground state is 6.2542eV[20].The '1P state has an energy 3.1081eV above the
ground state, therefore to ionize from the
1P1
state an energy of 3.1461eV is needed.
This corresponds to a wavelength of 394.09nm. (Note: since this is a shorter wavelenth
than the 398.799nm MOT transition, we do not expect ions from the MOT alone via
photoionization. This is necessary for our ability to switch the ion source on and off
without switching the MOT. It is possible that atom collisions within the MOT could
cause ionization, though unlikely.) We use an LED whose frequency distribution is centered on 385nm, which gives sufficient energy to ionize. The LED/MOT
setup is shown
in Fig.4.3. We would like to verify that this process is producing ions.
Since photons from the LED have a higher energy than the MOT transition, it
is conceivable that an atom might be excited to the 'P1 state by the LED, and then
ionized by a second photon from the LED. This is unlikely, however, since the intensity
of the LED at the
So --
1P, transition is extremely small. Nevertheless, such a two-
photon process is undesireable, because it could ionize hot atoms from outside the MOT.
Therefore we would like to verify that the ionization process is indeed single-photon as
expected.
These suspicions are easily tested with the use of the ion detector to measure
ionization rates. With the ion detector operating in pulse counting mode, we can measure
34
Optional -HV Input from
-c_ 1
11A1
I£-+
TTL Trigger
) for
supply
Analog Modules Model 829
Pockels-Cell Driver
I
F
Figure 4.2: Circuit diagram for the ion detector. In continuous voltage mode, the Bertan
high voltage supply is switched on, while the Pockels-cell driver is switched off, and vice
versa for pulsed voltage mode. A 2000:1 voltage divider is used to monitor the Pockels-cell
voltage on an oscilloscope. The Pockels-cell driver takes a TTL trigger with duty cycle of
up to 50Hz, to trigger pulses of up to 300jpsin duration and up to 3.5kV in amplitude,
with a rise time of < 10ns. The current output of the ion detector is sent through a
500 resistor and amplified by a factor of 100 before being sent to the oscilloscope and
frequency counter.
35
v~·
ci
siO
v
Aoc
MOT
Main
Bias
Lasers
Coils
Coils
/
J/
/
./
/
/
/
/
/
LED
Rate (kHz)
/
/
-
v/
Count
3.1
6.3
3.9
118
4.9
227
Table 4.1: Ionization rates for various configurations. The MOT requires that both the
lasers and magnetic field be on. When the lasers are on without a magnetic field, a higher
population of excited state atoms will exist than with the lasers off, however they will
not be densely confined. The ionization rate increases dramatically when the magnetic
field is optimized for MOT population and the LED is on. All measurements were with
a supply voltage of 2kV and a gate time of s on the frequency counter.
the steady state ion production rate with the frequency counter. Table 4.1 shows the
ionization rates for various combinations of MOT conditions with and without the LED
on. The numbers are as expected. Without the magnetic field, there is no MOT, but
some atoms are still excited by the lasers even though there is no spatial confinement.
Thus, the LED is able to ionize more atoms with the lasers on than off. However, a
dramatic increase in ionization comes when the magnetic field is introduced, creating a
MOT. This shows that the ionization process is ionizing primarily atoms from the MOT.
These results also give a good indication that the ionization is a single-photon
process, since the ionization rate is quite low without the lasers on (i.e. with most
atoms in the ground state). As further evidence, we measured ionization rates against
MOT fluorescence, by scanning the MOT laser frequencies through resonance to vary
the excited state population in the MOT, with the fluorescence being measured on a
photodiode preceeded by a 400nm interference filter. The results of this measurement
are shown in Fig.4.4. The ionization rate appears to vary roughly linearly with the excited
state population, which is a good indication that the process is indeed photoionization
37
by the LED from the excited state of the MOT transition.
38
Ion Count Rate Vs. MOT Fluorescence
%A
4'4U
·
·
·
200
300
400
__
220
N
I(1
OC
0o
C
o
TO
200
180
160
140
120
100
80
-·,,
Uv
0
100
500
MOT Fluorescence (mV)
Figure 4.4: Ionization rate vs. MOT fluorescence. By scanning the MOT laser frequency
through resonance with the MOT transition, we were able to vary the excited state
population of the MOT as measured by MOT fluorescence. The ionization rate appears
to vary linearly with excited state population, indicating that the ionization process is
photoionization by the LED from the excited state of the MOT transition.
39
40
Chapter 5
Paul Traps
Ion traps are classified into two types, Paul traps, which use radio frequency electric
fields, and Penning traps, which a static magnetic field and a static electric field. Our
experiment uses a Paul trap, the operation of which is the subject of this chapter. The
Paul trap was originally developed by W. Paul in the 1950sfor use in particle accelerators
and mass spectrometry. For his work in developing the trap and its applications, Paul
shared the 1989 Nobel prize[3].
The original ion traps consisted of three dimensional configurations of electrodes.
The two classic cases of linear and point confinement are discussed in §5.1. Our experiment utilizes a planar trap, which is discussed in §5.2.
5.1 Ideal Paul Traps
The mathematical details concerning ion motion in a Paul trap are treated extensively in several books and reviews[21; 22; 3; 23]. We present a brief description here,
which closely follows Ghosh[21].
The principle behind the Paul trap is to confine an ion in a quadratic electric
41
potential. Quite simply, the desired potential, O(r) is of the form
(
=
2(ax2
+ by2 + cz2),
(5.1)
where 00 is some constant potential, ro is a length depending on the electrode geometry,
and a, b, and c are constants chosen to satisfy the Poisson equation. The electric field,
E, arising from such a potential is
E = -V(0
which for positive
= -- (ax + byO+ czi),
(5.2)
o0is a restoring force in x if a > O, in y if b > 0 and in z if c > 0.
However, the Poisson equation requires that
2(r=
(a + b + c) =
a + b+ c=
(5.3)
Therefore, it is not possible to create a potential which is simultaneously harmonic in
all directions. Amazingly, it is nevertheless possible to create a time varying potential
which is effectively harmonic in all directions. The two cases which we will discuss here
are a = -b, c = 0 (linear trap) and 2a = 2b = -c (point trap).
5.1.1
Linear Confinement
For a linear trap, the potential becomes
O() =
2X2 _ y 2).
(5.4)
This is the equation for a hyperbola, and it is possible to achieve such a potential with a
configuration of four electrodes having hyperbolic cross-sections, as shown in Fig.5.1(a).
42
(In practice, many traps now use cylindrical rods as electrodes, which closely approximate
the hyperbolic shape[24].) We apply a potential b0 = (U-V cos Qt)/2 to the x electrodes,
and the negative of that potential to the y electrodes (
is a free parameter).
The
potential is then
(x, y, t) = (U -V cosmt)
22
) ·
(5.5)
For this geometry, ro is the shortest distance from the center of the trap to the electrodes.
We now take the gradient to determine the equations of motion for a particle of charge
e and mass m in the trap:
x= -
=
( q) 0(x, Y,,) = -(U - Vcos2t) (ex)
(5.6)
(q)
(5.7)
a(zX Y t) = (U - V os t)
ey( )
0.
(5.8)
The x and y equations are generally rewritten in the form of the Mathieu equation
[25]:
d2 u
d2
i
(a - 2q cos2)u = 0,
(5.9)
where
4eU
a =mr2f
,
q
2eV
'
mro2Q2
it
=2'Qt
(5.10)
and u = [x, y] with a positive sign for u = x and a negative sign for u = y.
We will now solve the Mathieu equation for x by making the simplifying assump-
tions that the x motion is the sum of a low-frequency term, X, and a high-frequency
term,
:
x = X +6,
(5.11)
with I61< IXI and dS3/dtl > IdX/dtl. We will also consider the case where U < V, or
43
hyperbolic
ele trodes
I
FEI~~Er
idcaps
r
Ring
Electrode
Vcos(Omega t) )
U
(a) Linear RF Paul Trap.
Lr
T
(b) Ring RF Paul Trap for point confinement.
Figure 5.1: Electrode configuration for RF Paul traps. For the ideal case, electrodes have
hyperbolic surfaces. For the linear trap, it is frequently more convenient and provides
greater optical access to use cylindrical rods instead of hyperbolic electrodes.
44
in other words a < q. With these simplifications, (5.9) becomes
d2 6
d - 2q cos(2C)X = 0,
(5.12)
which we integrate1 to obtain
6 = -q cos(24)X.
(5.13)
We now substitute X + 6 in the original Mathieu equation, using (5.13) for 6:
d42 = -aX
+
d(2
2
cos 24 + 2qX cos 24 - q 2 X cos 2 2C.
(5.14)
We can average this equation over one period of the RF potential to obtain the following2:
d2X
4 d2X
d4---~
- f2 dt 2
a+
2
)2X.
(5.16)
If there is no DC component, a = 0, and we are left with a harmonic oscillator of
frequency w = q/2v2.
The frequency w is known as the "secular" frequency.
The
motion of frequency Q associated with 6 is known as the "micromotion".
If we repeat this exercise for y, the sign in (5.12) becomes positive, so the sign in
(5.13) also becomes positive. Since the sign in (5.16) is the product of these two signs,
the sign reversals cancel and we have a harmonic oscillator equation for y as well:
d 2Y
(q 2
)Y
(5.17)
'Since X is slowly varying relative to 6, we can treat it as constant for this integration.
2
The expression for d2 6/d(2 is
_d2_
d~2
q d2 X2
2
d
dX
cos 2 -
d, sin 2 - 4Xcos 2
Our assumption that X is slowly varying allows us to treat dX/d( and d2 X/d(
well. Therefore, the righthand side of (5.15) averages to zero over an RF cycle.
45
(5.15)
2
as slowly varying as
Thus, we have a "pseudopotential" which is harmonic in both directions. Its depth, Do,
can be calculated by integrating the restoring force md2 X/dt 2 = - mQ 2 q2 X from the
trap center to the edge at ro. Substituting
for q from (5.10) we have:
e2V 2
Do= 4mQ2
4m2r
2
ro
5.1.2
(5.18)
Point Confinement
In the case of the point trap, 2a = 2b = -c, the potential takes on the form:
O(x,
, z) =
2(X 2 + Y2 _ 2z2).
(5.19)
The symmetry between x and y prompts us to use a cylindrical coordinate system:
(r, ) =
2r2
(r2- 2 2),
(5.20)
where r2 = x2 + y2 . This potential is generated by a ring shaped electrode with hyperbolic cross section and two hyperbolic "endcap" electrodes, as illustrated in Fig.5.1(b).
Applying the same voltages, +(U - V cosQ t)/2 to the ring electrode and the endcaps,
we get the following equations of motion:
r=
-(U-Vcos t)(mro2),
= (U-VcosQt)
,
(5.21)
(5.22)
which are seen to be of the same form (5.9), with the same parameters a, q and ( for the
r equation as in (5.10), while a and q have twice the value for the z equation. Therefore
46
Compensation Electrodes
I
I
_
I~~
I
~~~~~~~,
I
\
I
~Outer
electrodes
p/
llllsI__
, ,
I
I'
If
=IIIIIer
Electrode
Figure 5.2: Our planar trap is essentially a distorted linear trap. Many electrode configurations are possible, but we use a three electrode configuration, with two outer RF
electrodes (dark grey) and one inner RF electrode (light grey). The outermost offset
electrodes are typically held at ground but can be biased to offset DC electric fields in
the trapping region.
the secular frequencies are now different for the r coordinate and the z coordinate:
Wr
eV
/
Vmro
=
2eV
(5.23)
The depth of the trap is the same as in the linear case.
5.2 Planar Traps
The trap geometries discussed in §5.1 are the two ideal cases for linear and point
confinement. To create a planar trap, the linear trap is essentially flattened, as illustrated
in Fig.5.2 with the two outer electrodes connected at the ends to form endcaps, as
illustrated in Fig.5.3. In our trap, we apply a variable voltage to the inner electrode and
middle electrode, with the outer electrodes being left available for DC offset voltages.
By varying the voltage ratio of the inner to the outer electrodes, we can manipulate the
position of the trap center above the plane.
This geometry results in a non-ideal, more complicated potential. Such potentials
47
Printed Circuit Board (PCB) Trap
Cut-away for
Imaging Lens
0
A,
1
R--RF
Electrode
Ground
/RF
I
Three Groups of 4 x DC
Compensation Electrodes
--
Laser Clearance Slots
(All independently controllable)
Milled Away
~Dielectric
Mounting Holes
L
!
m
I0
0
Dielectric:Rogers 4350
Electrical Connection Pads Underneath
Figure 5.3: Electrode placement for our planar trap. The middle three elctrodes are the
RF electrodes, with the two outer electrodes also serving as endcaps of a sort. The twelve
outermost electrodes are compensation electrodes, allowing for DC field compensation.
Typically we have found that the trap operates best with the compensation electrodes
shorted to ground and the RF electrodes uniformly biased to -7V. Excess dielectric is
milled away to reduce surface charge and increase optical access. (Diagram courtesy Jon
Campbell)
48
have been modeled [11]and do not destroy the important features of the trap, namely the
pseudopotential minimum. We have modeled the trapping potential using the Boundary
Element Method, which fits an electrostatic Green's function to the specified electrode
potentials to determine the electrode charge distribution and potential throughout the
region. Using this model, we were able to plot the trap height as a function of the ratio
of the inner electrode voltage to the outer electrode voltage. This relationship is shown
in Fig.5.4. We also used the model to estimate trap depth at 170meV. The characteristic
length scale of the trap is on the order of 1mm.
49
I0
4.5
4
*
*
*
,
3.5
*
E
*
3
*
-
.
I
a)
2.5
1.5
,
I
,-
-0.6
-0.4
-0.2
--
I, --
,r- --
I
1
0.5
-1
-0.8
0
0.2
0.4
0.6
0.8
RF Electrode Voltage Ratio
Figure 5.4: Trap height in mm vs. ratio of inner electrode voltage to outer electrode
voltage, as modeled with the Boundary Element Method. Negative values for the voltage
ratio indicate that the voltages are 180 degrees out of phase with each other. Positive
values are in phase.
Our trapping region lies
courtesy of Andrew Grier.)
50
3mm above the trap surface.
(Plot
Chapter 6
Results
Having produced ions, we are now prepared to test the operation of the trap. Since
we do not currently have optical means of verifying that the trap is indeed trapping ions,
we instead use the ion detector, in the following way. We first ionize from the MOT
while the trap is on, for a period of loading time, T1 , which is in the millisecond to second
range. During this time, the ion detector is off. We next turn off the ionizing LED and
allow the trap to store the ions for a period of storage time, T8, which is also in the
millisecond to second range. We next turn off the trap for an optional period of dumping
time Td, which is on the order of tens of microseconds, during which the trapped ions can
drift out of the trapping region. Finally, we switch on the ion detector which attracts
all ions in the chamber within roughly 3s of turning on. During the storage time, any
ions which are not trapped will be quickly drawn to the grounded walls of the chamber
by stray electric fields. Thus, by the time we switch on the ion detector, we expect that
the only ions in the chamber are those that were trapped. By varying the loading and
storage times, we are able to verify that the trap works and to determine its lifetime.
For these measurements, we operated the ion detector in pulsed voltage mode as
described in §4.1.3. The graphs presented here are averages of ten oscilloscope traces,
51
triggered off of the TTL signal to the Pockels-celldriver. The first peak is from switching
on the ion detector. The second peak at insert time here is the signal from the ion
trap, as illustrated in Fig.6.1.
The RF electrodes on the ion trap have a voltage of up to 410V (amplitude) with
roughly 2% of the power going into the quadrupole field of the trap. For these measurements, the offset electrodes were held at +18V. Subsequently, we have found that the
trap functions well without any offset voltage applied to the offset electrodes, but with
-7V applied uniformly to the RF electrodes.
The first test was to turn on the ion trap and see if the detector saw anything.
After some tweaking of the offset voltages on the trap, we saw a signal, so we next varied
the trap loading time to try to maximize the signal. We held T8 constant at 100ms to
ensure than any stray ions which had not been trapped would have drifted to the walls,
and varied Tl from 20ms to s. The results of these measurements are shown in Fig.6.2.
The ion detector signal is clearly larger for larger trap times.
We next examined the effect of varying storage time to estimate the trap lifetime.
We took measurements
for values of T8 from 100ms to 900ms with T1 = 500ms. The
results are shown in Fig.6.3. Clearly, the signal is strongest for short storage times.
Finally, we examined the effect of a "dump" time, during which the trap is off and
the ion detector has not yet been turned on. This results in a spread in the arrival time
of the ions, since they become dispersed without the trap to hold them together. The
results of dump times of Ops to 50ps are shown in Fig.6.4
6.1
Trap Lifetime
From the data we took, we were able to get some idea of the trap lifetime. Unfor-
tunately, it seems the ion detector saturates at a fairly low signal level, so to determine
52
IonCounts(mV)
IonCount (V)
tirn (s)
-'''
.
(a) Ion detector background. When the trapping potential is incorrect, no counts are seen
in the signal. For this measurement, the offset
on the RF electrodes was set to zero, destroying
the trapping potential.
(b) Typical ion detector signal from the trap.
The pusl of ions arrives - 3.5/ts after the ion
detector is turned on.
Figure 6.1: Oscilloscope traces from the ion detector. When the trapping potential is
destroyed, no signal is seen. When the trap is working, a pulse of ions arrives at the
detector approximately 3.5/is after the detector is turned on. The noise and peaks to the
left of the pulse of ions are artifacts of the fast turn on for the ion detector.
53
0
O .1
o.e
-a.r
-u
-
.
,
.
s
(b) Tl = 200ms, T = lOOms,Td = Ops
(a) T1 = 20ms, T = lOOms,Td = 0/is
0
- Wi
..........
=
wI
A
0
I
n
0-1
i
~
i
dbeo-;*
i
I
1
-11.
-11
.
(d) T = 10OOms,T, = lOOms,Td = 0s
(c) Tl = 500ms, T, = lOOms,Td = Ous
Figure 6.2: Various load times for 100ms storage time and no dump time. The trap
population is seen to increase with increased loading time.
54
II
(a) T1 = 500ms, T. = 1OOms,
Td = OS
(b) T = 500ms, T, = 200ms,
Td = Os
(d) T = 500ms, T, = 400ms,
Td = Os
(e) T = 500ms, T, = 500ms,
Td = Ots
(c) Tl = 500ms, Ts = 300ms,
Td = Ots
(f) Tl = 500ms, T = 600ms,
Td = Ots
L
A
II vu
(g) T = 500ms, T, = 700ms,
(h) T = 500ms, T. = 800ms,
Td = Opts
Td = Ops
(i) T = 500ms, T. = 900ms,
Td = Ops
Figure 6.3: Trap lifetime. Storage time is varied with constant load time of 500ms.
Trap population is seen to decrease with increasing storage time. At the time of these
measurements, trap lifetime was estimated to be 150ms. Currently, we have been able
to store ions in the trap in excess of 2 seconds.
55
_
A
s
r
0.05
-v
..
(b) T = 200ms, T = 100ms, Td = 5 s
(a) T = 200ms, T = 100ms, Td = Os
A
A
e
q
_
0.05
. A.
.
I
A
A -A
,2
-o005
(c) TL= 200ms, Ts = lOOms,Td = 20zs
(d) T, = 200ms, Ts = lOOms,Td = 50/ts
Figure 6.4: Various dump times for 200ms load time and 100ms storage time. The ions
begin to disperse with as little as 5s of dump time, and by Td = 50Iusthe ions have
been attracted to the chamber walls and the signal is gone.
56
the trap lifetime, we increased storage time, T until the signal was below saturation,
and then looked for a decay time. The result was a trap lifetime of r
150ms, for the
results shown here. Subsequently we have been able to improve the trap's lifetime and
other characteristics by adjusting offset voltages. We have been able to store ions in the
trap for periods in excess of 2 seconds.
6.2
Trap Population
Despite our best efforts, the ion detector continues to reach saturation. Therefore,
we are unable to make a determination of the trap's maximum population. In fact,
we have not yet been able to satisfactorily calibrate the detector, so we really cannot
say anything absolute about trap population. Only that the population increases with
increased loading time and decreases with increased storage time, as expected. When
the laser systems for the Yb+ wavelengths are ready, we expect to be able to make better
population measurements.
57
58
Chapter 7
Discussion
We have shown that it is possible to trap ions from a cold source in a planar RF
Paul trap. The trap lifetime is estimated to be - 150ms, and the trap population is
shown to increase with increased load time and to decrease with increased storage time.
7.1
Future Work
The crude measurements which we were able to take with the ion detector should be
greatly improved upon with the addition of a 369nm laser which will be able to optically
probe the ions in the trap. The energy levels of singly-ionized Yb are somewhat more
complicated than for the neutral atom, which means we will need a repumping laser at
935nm in order to avoid pumping the ions into a dark state. Once these two lasers are in
place, we should not only be able to probe the trap properties, but also to optically cool
the ions in the trap. This is a necessary step before quantum information operations will
be viable, and also opens up interesting physics in the realm of ion crystals, along the
lines of[5; 6].
Besides the improved imaging techniques and the introduction of laser cooling to
59
the trap, much can be done to understand the trap's dependence on offset voltages and
electrode voltage ratios. We expect that the trap lifetime of 150ms can be improved upon
simply by tweaking these parameters.
Clearly, there is much work to be done in the future, but these early results offer
exciting promise for the physics of planar ion traps!
60
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[24] G. R. Janik, J. D. Prestage, and L. Maleki. Simple analytic potentials for linear ion
traps. Journal of Applied Physics, 67(10):6050-6055, 1990.
[25] M. Abramowitz and I.A. Stegun, editors. Handbook of Mathematical Functions With
Formulas, Graphs and Mathematical Tables. National Bureau of Standards, ninth
edition, 1970.
63
64
Appendix A
Ytterbium
The choice of Yb as the element for our trap rested on two main reasons. First,
neutral Yb has two valence electrons, so singly ionized Yb has only one valence elec-
tron, making its electronic structure compartitively simple, requiring only one repumping laser for the dark 2 D3/2 state. Also, its neutral structure is perfect for operating a
MOT, with the P1 state splitting into three Zeeman sublevels, and no repumper needed.
Furthermore, diode lasers are becoming available in the necessary wavelengths for these
transitions.
65
3
1
F=2
7
T =42ns
V+
I1U,.
r
--
D[3/2
A
.M
r /20
k
zF
f
o
F=3
-'D['D5/21s
,, 2.5 GHI
2
6s6p
r /2
I
394nm
'
P2
F=1
=2 2 MHz=
T =5.5ns
2.1GHz
A F=
//
/
41
.3% '
i
I
- F=2
/ '-.
r
-
I
Cii
I
lpi
-
r /2
If
860Mz
. -- = 1
/I
=
3
/
I
22MHz
II
as.
?O
:4
/
I =398.799nr
,. R
/
=3.109eV
-
Yb
L-
6s
-
2
2S1
sa
12.6 GHz
Neutral Yb
Singly Ionized Yb
F=0
Figure A.1: Energy levels for neutral and singly ionized Yb. The ' P state of neutral Yb
splits into three Zeeman sublevels. For singly ionized Yb + , the cooling transition to 2P1/2
decays to the 2 D3/ 2 state .3% of the time, requiring a repumper at 935.2nm. (Diagram
courtesy Jon Campbell)
66