A novel linear ion trap that permits storage of a large number of ions with reduced susceptibility to the second-order Doppler effect caused by the radio frequency (RF) confining fields has been designed and built. This new trap should store about 20 times the number of ions a conventional RF trap stores with no corresponding increase in second-order Doppler shift from the confining field. In addition, the sensitivity of this shift to trapping parameters, i.e., RF voltage, RF frequency, and trap size, is greatly reduced

Dick, G. J.

Maleki, L.

Prestage, J. D.

English

NASA Technical Reports Server

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1189-27875
TDA Progress Report 42-97 January-March 1989
New Ion Trap for Atomic Frequency Standard Applications
J. D. Prestage, G. J. Dick, and L. Maleki
Communications Systems Research Section
A novel linear ion trap that permits storage of a large number of ions with reduced sus-
ceptibility to the second_rder Doppler effect caused by the radio frequency {RF) con-
fining fields has been designed and built. This new trap shouM store about 20 times the
number of ions a conventional RF trap stores with no corresponding increase in second-
order Doppler shift from the confining field. In addition, the sensitiviO, of this shift to
trapping parameters, i.e., RF voltage, RF frequency, and trap size, is greatly reduced.
I. Introduction
There has been much recent activity directed toward the
development of trapped ion frequency standards, in part
because ions confined in an electromagnetic trap are subjected
to very small perturbations of their atomic energy levels. The
inherent immunity to environmental changes that is afforded
by suitably chosen ions suspended in DC or radio frequency
(RF) quadrupole traps has led to the development of fre-
quency standards with very good stability [1], [2]. Indeed,
the trapped 199Hg+ ion clock of [1] is the most stable clock
yet developed for averaging times 1>10 6 seconds [3]. However,
certain applications, such as millisecond pulsar timing [4] and
low frequency gravity wave detection across the solar system
[5], require stabilities beyond that of present day standards.
Although the basic performance of the ion frequency
source depends fundamentally on the number of ions in the
trap, the largest source of frequency offset stems from the
motion of the atoms caused by the trapping fields via the
second-order Doppler or relativistic time dilation effect [6] .I
Moreover, instability in certain trapping parameters (e.g., trap
field strength, temperature, and the actual number of trapped
particles) influences the frequency shift and leads to frequency
instabilities. Since this offset also depends strongly on the
number of ions, a trade-off situation results, where fewer ions
are trapped in order to reduce the (relatively) large frequency
offset that would otherwise result.
A hybrid RF/DC linear ion trap that should allow an
increase in the stored ion number with no corresponding
l RF and Penning traps are both subject to second-order Doppler prob-
lems related to the trapping fields. In this article, attention is confined
specifically to the case of RF traps. It is simply noted that, for the
same number of stored ions, magnetton r_tation of _he ion cloud in a
Penning trap leads to a second-order Doppler shift of magnitude com-
parable to that for an RF trap. A Penning trap-based clock is described
in [61.
58
https://ntrs.nasa.gov/search.jsp?R=19890018504 2020-03-20T01:52:48+00:00Z
increase in second-order Doppler instabilities has been designed
and constructed. The 20 times larger ion storage capacity
should improve clock performance substantially. Alternatively.
the Doppler shift from the trapping fields may be reduced by
a factor of 10 below comparably loaded hyperbolic traps.
II. Second-Order Doppler Shift for Ions
in an RF Trap
Figure 1 shows a conventional RF ion trap along with the
applied voltages. The trapping forces are generated by the
driven motion of the ions (at frequency fZ) in the inhomo-
geneous electric field created by the trap electrodes [7]. Ions
are trapped around the node point of the oscillating electric
field at the center of the trap for certain trap voltages, ionic
masses, etc. The motion in each of three directions for a single
ion in an RF trap is characterized by two frequencies, the fast
driving frequency _ and a slower secular frequency co. An
exact solution to the equations of motion shows that fre-
quencies k f_ +- co, k = 2, 3 .... are also present. However. in
the limit co/f2 << 1, the co and f2 +- co frequencies dominate,
and the kinetic energy (K.E.) of a particle, averaged over one
cycle of fL separates into the kinetic energy of the secular
motion and the kinetic energy of the driven motion. The
average kinetic energy is transferred from the secular to the
driven motion and back while the sum remains constant,
just as a harmonic oscillator transfers energy from kinetic to
potential and back. The second-order Doppler shift for a small
and/or hot ion cloud, where interactions between ions are
negligible, is
Af _ 1 (v 2 ) _ (totalK.E.)
f) _ C 2
hot _ mc 2
( secular K.E. + driven K.E. )
mc 2
( secular K.E. )
mc 2
3kBT
mc 2
(1)
where () indicates a time average over one cycle of _. An
average over one cycle of _o was obtained to equate the secular
and driven K.E. This is the same as a simple harmonic oscilla-
tor where the average K.E. is equal to the average potential.
Now consider the case in which many ions are contained in
a trap and interactions between ions dominate. In this cold
cloud model [8], [9] of the trapped ions, displacements of
individual ions from the trap center are primarily due to
electrostatic repulsion between the ions, and random thermal
motion associated with temperature can be assumed to be
small compared to driven motion due to the trap fields. Such
clouds have a constant ion density out to the edge of the
plasma where the density falls off in a distance characterized
by the Debye length [9]
k B TcoxD = (2)
tlo q 2
This cold cloud model should be useful provided the ion cloud
size is large compared to the Debye length. For room tempera-
ture Hg ions held in a trap with 50 kHz secular frequency, the
Debye length is about 0.2 mm.
The trap shown in Fig. 1 is described by a pseudo-potential
energy [7]
1 p2 + mcoz z- ) (3)42 : T (mcoa2 2 "
where
2 2q2V°2 2qU°
co - + -- (4)
p m 2 _.-_2e4 me2
and
8q: vo: 4qc),
2 _ (5)
COz m2 _._2 54 r?lc 2
and e describes the trap size. It is assumed that the DC and RF
voltages are adjusted to make the trapping forces spherical so
that the ion cloud is a sphere containing N ions out to radius
Rsp n . The pseudo-potential depicts trap forces as arising from
a uniform "background" charge density computed from
Poisson's equation with the above pseudo-potential:
_o
3eomco 2
= .__
q',
(6)
Trapped positive ions neutralize the negative background of
charge, matching its density out to a radius where the supply
of ions is used up.
The oscillating electric field that generates the trapping
force grows linearly with distance from the trap center. The
corresponding amplitude of any ion's driven oscillation is pro-
portional to the strength of the driving field (i.e., also increas-
59
ing linearly with the distance from the trap center). The aver-
age square velocity of the driven motion for an ion at position
(p, z) is
1 2 2)(v 2) =_ (p2+4z (7)
For a given trapping strength, reflected in force constant
co 2 , the density is fixed by Eq. (6),and the radius of the spheri-
cal cloud is determined once the ion number N has been speci-
fied. The second-order Doppler shift due to the micromotion
is the spatial average of -(l/2)((v 2 )/c 2 ) over the spherical ion
cloud. Using Eq. (7) for the spatial variation of the micro-
motion
_/)_p _ i ( v2 > (8)
_- h 2 c 2
3 c°2Rsph 2
10 c2
(9)
31 [Ncoq2t 2/3
10 c 2 _] (10)
For typical operating conditions [8], N = 2 • 106 and
co = (2n) 50 kHz, Af/f = 2 • 10 -12 . This corresponds to a
second-order Doppler shift that is about 10 times higher than
the shift for free 199Hg ions at room temperature, /xf/f=
3kBT/2mc2 = 2 • 10 -13,
For increased signal.to-noise in the measured atomic
resonance used in frequency standard applications, it is desir-
able to have as many trapped ions as possible. However, as has
just been seen, larger ion clouds have larger second-order
Doppler shifts. This frequency offset must be stabilized to a
high degree in order to prevent degradation of long-term
performance. To reduce this susceptibility to second-order
Doppler shift, a hybrid RF/DC ion trap has been designed and
constructed that replaces the single-field node of the hyper-
bolic trap with a line of nodes. The RF electrode structure
producing this line of nodes of the RF field is shown in Fig. 2.
The ions are trapped in the radial direction by the same RF
trapping forces used in a conventional RF trap, and a secular
motion in that direction of frequency co is assumed. To
prevent ions from escaping along the axis of the trap, DC
biased "endcap" needle electrodes are mounted on each end.
Unlike conventional RF or Paul traps, this linear trap holds
positive or negative ions but not both simultaneously.
Near the central axis of the trap, one assumes a quadrupolar
RF electric field
V(x 2 _ y2) cos (_t)
2R 2
11)
which gives the corresponding pseudo-potential energy
q2V2
(I} - (x 2 +y2) (12)
4R 4m _ 2
2
6O p2
- rn T (13)
Poisson's equation leads to a limiting charge density
2 fro IH 40 2
no (14)
q2
for ions held in this linear trap, where
60 2 _ q2 V2 (15}
2m2_2R4
The motion induced by the RF trapping field is purely trans-
verse and is given by
( V2 ) = CO2p 2 (16)
As before this quantity is averaged over the ion cloud to find
the second-order Doppler shift
AJ'(7_) 1< v2_____2)= c°2R 2
= - _ c2 - " (17)l/n _ 4{,2
A cylindrical ion cloud of radius R¢ and length L is assumed
for simplicity. Equation (17) can be written in terms of total
ion number, N, and trap length, L,
8n% mc 2 L
08)
In contrast to the spherical case, this expression contains no
dependence on trap parameters except for the linear ion
density N/L. This is also true for the relative Debye length:
6o
/,f
2 1 (f lhot
(19)
which must be small to ensure the validity of the "cold cloud"
model. From this it is seen that the transverse dimension R of
the trap may be reduced without penalty of performance, pro-
viding that operational parameters are appropriately scaled.
This requires co and _2 to vary as R-a, and the applied voltage
Vo to be held constant.
Assuming the traps hold the same number of ions, the
second-order Doppler shift for the two traps can be compared
Af
(_-)lin - 5 Rwh(Lf-)wh3L (20)
As more ions are added to the linear trap, their average second-
order Doppler shift increases. It equals that of the spherical
ion cloud in the hyperbolic trap when
N. m _ 3 L (21)5 R o,.h
sph
A linear trap can thus store (3/5)(L/Rsph) times the ion
number a conventional RF trap stores with no increase in
average second-order Doppler shift. For the JPL-built trap,
L = 75 ram. Taking Rsp a = 2.5 mm for 2 • 10 6 199Hg+ ions
[8], one sees that this linear trap should hold about 18 times
the number of ions as that of [8].
III. Construction of a Linear Ion Trap
The linear trap built consists of four molybdenum rods
equally spaced on an approximately 1-cm radius. Oxygen-
free high-conductivity copper pins with DC bias are located
at each end to confine ions in the axial direction and are about
75 mm apart.
The input optical system that performs state selection and
also determines which hyperfine state the ions are in has been
modified from the previous system [2], The present system
illuminates about 1/3 of the 75-mm long cylindrical ion cloud.
An ion's room temperature thermal motion along the axis of
the trap gives an average round-trip time of 1.4 msec, a value
that is much smaller than any optical pumping, interrogation,
or microwave resonance time. Thus, all the ions are illumi-
nated, but with a lower average intensity.
In order to operate within the Lambe-Dicke regime [10],
the 40.5-GHz microwave resonance radiation is propagated
perpendicularly to the line of ions. The ions should then all
feel phase variations of this radiation that is less than n, so
that the first-order Doppler absorption in sidebands induced
by an ion's motion does not degrade the 40.5-GHz funda-
mental.
IV. Conclusions
Trapped ion frequency standards eliminate containing walls
and their associated perturbations of the atomic transition fre-
quencies by using electromagnetic fields alone to confine the
particles. For any given trap, however, there exists a trade-off
between the number of ions in the trap and a frequency shift
due to second-order Doppler effects. This trade-off directly
affects performance of the standard since the frequency shift
is typically very much larger than the ultimate stability required
and since the statistical limit to performance is directly related
to ion number.
This performance trade-off was calculated for an RF trap
with cylindrical geometry, a case not previously considered
for a trapped ion frequency source. By replacing the single
node in the RF trapping field for a spherical trap with a line
of nodes, a cylindrical trap increases effective volume with-
out increasing overall size. Furthermore, this performance is
found to be independent of its transverse dimensions, as
long as the driving frequency is scaled appropriately, with the
driving voltage unchanged. More specifically, for the same fre-
quency shift, a linear trap with length L can hold as many
ions as a spherical trap with ion cloud diameter 6L/5. In
addition to the practical advantage of greatly reduced overall
volume, a fundamental advantage is also allowed since opera-
tion within the Lambe-Dicke regime places a limit on the size
of the ion cloud, a requirement that may be met for a cylindri-
cal trap by irradiating the microwave atomic transition in a
direction perpendicular to the trap's longitudinal axis.
A trapped ion frequency source has been designed in which
a cylindrical trap is implemented with a combination of RF
and DC electric fields. With similar overall size and improved
optical performance, this trap has 15 to 20 times the ion stor-
age volume as conventional RF traps with no increase in
second-order Doppler shift.
61
Acknowledgments
We wish to thank Dave Seidel for assisting in the design of the linear trap described
here and G. R. Janik for helpful comments.
References
[1] L. S. Cutler, R. P. Giffard, P. J. Wheeler, and G. M. R. Winkler, "Initial Opera-
tional Experience with a Mercury Ion Storage Frequency Standard," Proc. 41st
Ann. Symp. Freq. Control, pp. 12-19, 1987.
[2] J.D. Prestage, G. J. Dick, and L. Maleki, "The JPL Trapped Mercury Ion Frequency
Standard," TDA Progress Report 42-92, vol. October-December 1987, Jet Propul-
sion Laboratory, Pasadena, California, pp. 13-19, February 1988.
[3] D. W. Allan, "A Study of Long-Term Stability of Atomic Clocks," Proc. 19th Ann.
PTTI App. Plan. Meeting, pp. 375-412, 1987.
[4] L. A. Rawley, J. H. Taylor, M. M. Davis, and D. W. Allan, "Millisecond Pulsar PSR
1937+21 : A Highly Stable Clock," Science, vol. 238, pp. 761-765, 1987.
[5] J. W. Armstrong, F. B. Estabrook, and H. D. Wahlquist, "A Search for Sinusoidal
Gravitational Radiation in the Period Range 30-2000 Seconds," Astrophys. J.,
vot. 318, pp. 536-541, 1987.
[6] J.J. Bollinger, J. D. Prestage, W.M. Itano, and D. J. Wineland, "Laser-Cooled-
Atomic Frequency Standard," Phys. Rev. Lett., vol. 54, pp. 1000-1003, 1985.
[7] H. G. Dehmelt, "Radio-Frequency Spectroscopy of Stored Ions I: Storage," Adv.
At. Mol. Phys., vol. 3, pp. 53-72, 1967.
[8] L. S. Cutler, R.P. Giffard, and M. D. McGuire, "Thermalization of 199Hg Ion
Macromotion by a Light Background Gas in an RF Quadrupote Trap," Appl.
Phys. B, vol. 36, pp. 137-142, 1985.
[9] S. A. Prasad and T. M. O'Neil, "Finite Length Thermal Equilibria of a Pure Electron
Plasma Column," Phys. FluMs, vol. 22(2), pp. 278-281, 1979.
[10] R. H. Dicke, "The Effect of Collisions upon the Doppler Width of Spectral Lines,"
Phys. Rev., vol. 89, pp. 472-473, 1953.
62
x,¥
Fig. 1. A conventional hyperbolic RF ion trap. A node of the RF and
DC fields is produced at the origin of the coordinate system shown.
Fig. 2. The RF electrodes for a linear ion trap. Not shown are the DC
endcap needle electrodes used to prevent the ions from escaping
along the longitudinal axis. lons are trapped around the line of
nodes of the RF fleid wlth reduced susceptibility to second-order
Doppler frequency shift.
63


New ion trap for atomic frequency standard applications

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