Content uploaded by Alan Madej
Author content
All content in this area was uploaded by Alan Madej on Aug 26, 2015
Content may be subject to copyright.
Laser-Cooled Atoms and Ions in Precision Time and Frequency
Standards
John E. Bernard, Louis Marmet, Alan A. Madej, and Pierre Dub´e
Institute for National Measurement Standards
National Research Council, Ottawa, Ontario, Canada K1A 0R6
November 9, 2005
Laser cooling and trapping techniques have led to
dramatic improvements in atomic frequency and time
standards.
Abstract
The science of precision measurement has shown dramatic im-
provements in the past decade due to the introduction of atom
trapping and laser cooling. These techniques have permitted
atoms and ions to be isolated from perturbations and serve as
almost ideal atomic frequency standards. We will review the
various methods used to trap single ions and clouds of atoms
and cool them to temperatures approaching absolute zero. De-
tails on the operation of modern, laser-cooled time and fre-
quency standards will be presented along with a summary of
the current state of the art.
1 Introduction
The highest quality standards of frequency and time that are
available today use atomic oscillators in which electronic tran-
sitions serve as reproducible frequency references. Since 1967,
the base quantity of time in the International System of Units
(SI), the second, has been defined in terms of one such atomic
transition as: “The second is the duration of 9 192 631 770 pe-
riods of the radiation corresponding to the transition between
the two hyperfine levels of the ground state of the cesium-133
atom”[1]. A laboratory anywhere in the world can build a clock
based on this definition and reproduce the same value for the
second to very high accuracy. Besides the cesium atomic clock,
other devices that depend on atomic transitions in the rf re-
gion have been built. Well known examples are rubidium stan-
dards at 6.83 GHz and hydrogen masers at 1.42 GHz. In re-
cent years, benefiting from the introduction of optical frequency
comb technology[2, 3] which enables the counting of optical cy-
clesTO BE REMOVED without error, new standards have ap-
peared which use optical transitions as frequency references.
These rf and optical standards have found numerous applica-
tions ranging from tests of fundamental theories of physics, to
practical applications such as time-keeping, the global position-
ing system (GPS), and the synchronization of telecommunica-
tion systems. The second and its inverse, the hertz, are also of
fundamental importance in the definitions of three other base SI
units of measurement: the meter, the ampere, and the candela.
Atomic standards of frequency and time have experienced
dramatic advances in the past decade largely as a result of the
introduction of trapping and cooling techniques. These ad-
vances have made it possible to isolate the atomic oscillators
from perturbations such as collisions and to reduce or elimi-
nate the effects of first and second order (relativistic time dila-
tion) Doppler shifts. Long observation times are now possible,
permitting unprecedented levels of spectral resolution. Cesium
atomic clocks based on cold atoms are now capable of repro-
ducing the SI second with a fractional uncertainty of less than 1
part in 1015 making it by far the most accurately realized unit
of measurement. Similar levels of precision have recently been
achieved with cold-atom optical standards. In this article, we
will review the methods of trapping and cooling atoms and ions
that have made these advances possible. We will also describe
some of the inner workings of the new standards. These ideas
will be illustrated by referral to two standards which exist in
our laboratories at NRC: a cesium fountain clock and a single
trapped strontium ion.
2 Trapping Atoms and Ions
An ideal trap for an atomic frequency or time standard would
confine atoms for extended periods of time without perturbing
their electronic structure. In this section, we will describe the
types of traps used in cold-atom frequency standards.
2.1 Ion Traps
Because of their charge, ions are somewhat easier to trap and
cool than neutral atoms. However, since coulomb interactions
between ions can perturb the transition frequencies, only single
ions are normally employed in cold frequency standards. These
ions are usually held under ultra-high vacuum conditions in a
form of Paul trap[4, 5]. In its basic form, the Paul trap con-
sists of two end-cap electrodes and a central ring electrode, as
shown in Fig. 1. If a sinusoidally varying electric potential is
applied between the ring electrode and the end-cap electrodes,
the ion will move such that it migrates towards regions of lower
r
zo
o
Ri ng
En dca p
En dca p V co s( t)
oΩ
Figure 1: Cut-away schematic diagram of a Paul trap used for
trapping single ions.
field. Therefore, the ion feels a net force directed towards the
field-free centre of the trap where it can be trapped for peri-
ods of days. The trap’s pseudo-potential is approximately a
three-dimensional harmonic well with a depth of several volts.
If the ion has any thermal energy, it will tend to oscillate in this
potential at well-defined frequencies known as “secular frequen-
cies”, normally in the range of hundreds of kHz. In addition, if
the ion moves away from the centre of the trap, it will feel the
applied rf field and oscillate with “micro-motion” at the fre-
quency of the applied rf. In an ideal standard, the ion is cooled
to the ground vibrational state of the trap potential where both
the secular and micro-motions are minimized.
The emission (absorption) spectrum of an ion confined in-
side a Paul trap is simpler than that for a free ion. When an
atom is confined to a volume of space which is small compared
to the wavelength of light, it is said to satisfy the Lamb-Dicke
criterion[6] and the spectrum consists of a sharp, non-Doppler-
broadened central component and a series of regularly spaced
sidebands separated by the frequencies of the secular oscilla-
tions. Any first-order Doppler broadening of the transition
completely disappears including any recoil-associated shifts.
2.2 Atom traps
Neutral atoms, because they are not charged, cannot be held
in a trap which depends on coulomb forces. Instead, atoms
are trapped by position-dependent light forces inside a device
known as a magneto-optical trap (MOT)[7, 8]. A MOT consists
of a pair of anti-Helmholtz coils and three pairs of circularly-
polarized laser beams that are detuned slightly to the red side
of a strong atomic transition possessing Zeeman substructure.
The coils produce a magnetic field, with the ~
B-vector as shown
in Fig. 2a. The magnetic field is zero at the centre of the trap
and increases in magnitude approximately linearly with dis-
tance from the centre. In a weak magnetic field, the ground
and excited states are split into Zeeman sublevels whose shifts
depend linearly on the local magnetic field. For simplicity,
consider only displacements in the x-direction. As shown in
σ
σ
σ
σ
σ
σ
+
+
+
-
-
-
I
I
An t i - H em h o l tz
Co i l s
x
y
z
B-
fie l d
T r ap p i n g
and co o l i n g
las e r b e a m s
a)
B-
fi e l d
x
x
T r an s i t io n
en e r g y o r
fre q u e nc y
σσ
+-
∆∆
m =-1 m =+1
FF
b)
Figure 2: a) Schematic diagram of a magneto-optic trap (MOT)
used for trapping clouds of cold atoms. b) The magnetic field and
the resulting shifts in the energies of the ∆mF= 1 and ∆mF=−1
transitions.
Fig. 2b, the magnetic field is positive ( ~
Bpoints in the positive
x-direction) for x > 0 while for x < 0 the magnetic field is neg-
ative ( ~
Bpoints in the negative x-direction). The ∆mF=−1
transitions are therefore shifted to lower frequencies for x > 0,
and closer to resonance with the laser radiation, and to higher
frequencies, and further from resonance with the laser radia-
tion, for x < 0. The inverse situation applies to the ∆mF= +1
transitions. Therefore, atoms located at x > 0 are more likely
to absorb a photon from the σ−-polarized beam, which prop-
agates in the −xdirection, than from the σ+-polarized beam.
Similarly, atoms located at x < 0 are more likely to absorb
a photon from the σ+-polarized beam. Each time an atom
absorbs a photon it receives a small kick in the direction of
propagation of the photon while the re-emission of photons is
isotropic. The net effect is that atoms, which are displaced
from the centre of the trap, feel a force directed towards the
trap centre. In typical applications, a MOT can capture ap-
proximately 109atoms in a few tenths of a second. Besides
trapping the atoms, it can also serve to cool them, as will be
described in the following section.
3 Laser Cooling of Atoms and Ions
Atoms and ions in frequency standards are cooled, that is,
their kinetic energy is reduced, through the technique of “laser
cooling,”[7, 8, 9] which was first proposed for neutral atoms and
trapped ions in 1975[10, 11]. Cooling occurs either through the
transfer of photon momentum to the atoms or through the di-
pole force[12] in which the laser light polarizes the atoms and
then interacts with the induced dipoles. In this section, we will
review the mechanisms responsible for laser cooling that apply
to cold frequency standards.
3.1 Doppler Cooling
The simplest form of laser cooling through photon momentum is
known as Doppler cooling. In its general form, Doppler cooling
uses three orthogonal pairs of counter-propagating laser beams
that are detuned to the red side of a strong dipole transition.
The atom sees the frequency of the beam, which propagates
in the direction opposite to its motion, as Doppler-shifted to
the blue, closer into resonance with its transition, and the co-
propagating laser beam as red shifted and further from reso-
nance. Therefore, the atom absorbs more photons, each carry-
ing a momentum of hν0/c, from the counter-propagating beam
and receives more kicks in the direction opposite to its mo-
tion. Its speed is reduced by several cm/s with each absorption.
Since, the atom re-emits or scatters the photons in a random
direction, cooling by this process is limited to a minimum “tem-
perature” of approximately 0.5 mK for strongly-allowed transi-
tions, corresponding to a velocity of 40 cm/s. This temperature
is much larger than the limit imposed by the recoil velocity due
to the emission of a single photon of 1 cm/s or ∼1µK. Not
only is the velocity of atoms reduced by Doppler cooling, but
the atoms are also partially confined. Their mean free path
between absorptions is small and so they diffuse very slowly
(about 1 cm/s). This effect is known as “optical molasses”[9].
3.2 Sideband Cooling
Cooling to temperatures approaching the recoil limit is pos-
sible if the atom is tightly confined inside a harmonic well.
The translational energy of such an atom will then be quan-
tized and it will oscillate with a secular frequency, νsec. If a
weak transition and a narrow-linewidth laser are chosen for
cooling, the unshifted central component at frequency ν0and
the secular-motion sidebands at ν0±mνsec , where mis an in-
teger, are well resolved. Irradiating this atom with radiation
at frequency ν0−νsec then causes it to make a transition to an
excited electronic state having one less quanta of vibrational en-
ergy. See Fig. 3. It can then spontaneously decay back to the
ground state while emitting a photon with an average energy of
hν0. On the average, the atom’s vibrational energy is reduced
by hνsec for each scattering event. This method of cooling is
known as “sideband” cooling. It has been shown capable of
reducing the vibrational energy to the ground state of the trap
potential well[12, 13].
(ν − ν )
hose c ν
ho
Co o l i n g
la s e r
Gr o u n d
el e c t r o n i c
st a t e
Ex c i t e d
el e c t r o n i c
st a t e
T ra p
po t e n t i a l w e ll
Se c u l a r
vi b r a t i o n al
su b - l e v e l s
Sp o n t a n e o u s
em i s s i o n
T i me
n
n- 1
n- 1
Figure 3: The mechanism of sideband cooling of single atoms.
3.3 Sisyphus and Polarization-Gradient Cool-
ing
Although both atoms and ions can be cooled through Doppler
cooling, sideband cooling is normally restricted to single
trapped ions which are confined in deep potential wells. Alter-
native cooling techniques, known as Sisyphus cooling[9, 14] and
polarization-gradient cooling[15] are used to cool alkali atoms
inside a MOT to near the recoil limit.
Sisyphus cooling makes use of the dipole force as well as op-
tical pumping, light (a.c. Stark) shifts, and gradients in laser
polarization. It is illustrated in Fig. 4 for the 1-D case. The
polarization of the wave pattern due to two counter propagat-
ing beams, with perpendicular linear polarization, varies along
the x-axis between linear vertical polarization to σ+to lin-
ear horizontal polarization to σ−, and back to linear vertical-
polarization, repeating the cycle in a distance of λ/2. This spa-
tial variation in the polarization results in a spatial variation
in both the optical pumping rate and light shift. An atom in
Ve rt i ca l Ve rt i ca lHo r iz o n ta l σσ+
σ+
+
σ-
σ-
Ground-state
Energy
x
x
0λ/4 λ/2
g gg
+1 /2 + 1 /2-1 /2
Co o li n g
la s er
Co o li n g
la s er
Op t ic a l p um p in g
vi a e xc i t ed s t a te
y
z
Figure 4: The mechanism of sisyphus cooling of single atoms. The
line thickness in the lower diagram represents the relative population
of atoms in the two ground states.
the ground state g−1/2, located where the field is σ+polarized,
is shifted into resonance with the excited state, resulting in an
increased photon absorption rate with ∆mF= +1. It can then
decay spontaneously either back to the original state, g−1/2,
where it can absorb another σ+photon and repeat the cycle,
or decay to the g+1/2state, where it will remain trapped since
it is not in resonance with an excited state sublevel. (For the
133Cs atom, many Zeeman sublevels are involved, but a similar
process occurs for each of them.) Atoms located in the σ+field
will therefore be optically pumped to the g+1/2ground state,
and similarly, those located in the σ−field will be optically
pumped to the g−1/2ground state. In addition to the optical
pumping, there is a light shift of the atoms’ energy levels, as
shown in Fig. 4, due to the ac Stark effect. Consider what hap-
pens to an atom which has been pumped into the g+1/2state
in the σ+region. As this atoms moves, either left or right, it
feels a dipole force which opposes its motion and it gains po-
tential energy at the expense of its kinetic energy. When it
reaches the σ−region it is likely to be optically pumped into
the g−1/2state, which now has a lower potential energy than
the g+1/2state. The excess potential energy is carried off by
the photon which is emitted as the atom spontaneously decays
from the excited state to the g−1/2state. Therefore, the atom,
after climbing the potential hill, finds itself suddenly back at
the bottom of another hill; the atom loses kinetic energy as it
moves through the standing wave pattern and its temperature
is reduced.
Polarization-gradient cooling uses a setup similar to Sisy-
phus cooling, but circularly polarized, σ+and σ−laser beams
are used to produce a polarization which is always linear but
whose orientation rotates perpendicular to the x-axis. For an
atom moving slowly in the xdirection there is a motion-induced
population difference in its ground-state levels which results in
it scattering more photons from the counter-propagating laser
beam than from the co-propagating beam. It therefore feels a
net force which opposes its motion and is slowed.
4 Stability of Atom and Ion Stan-
dards
A high quality frequency standard should have low instability,
that is, the fluctuations in its frequency should be small. Nor-
mally, the instability is given in terms of the Allan deviation,
σy(τ), which is a measure of the rms difference between consec-
utive values of the average frequency, each averaged over a time
period, τ, and separated from each other by time, τ[1]. Low in-
stability is important if the standard is to be used to measure
the frequency of a new device without hours of averaging. The
fractional frequency instability of an atomic frequency standard
is determined by the Q (ν/∆ν)of the atomic transition, , and
the effective signal-to noise ratio with which the transition can
be interrogated. At the quantum limit, the S/N is determined
by the transition probability and S/N = 2√N, where Nis the
number of atoms. The instability is given by[16],
σy(τ) = 1
πQrTc
Nτ ,(1)
where Tcis the measurement cycle time. It can be seen from
Eq. 1, that optical standards, because of their high frequency
and line Q, should have superior stability as compared to RF
standards. It is also clear that, at least in terms of stability,
standards which employ millions of atoms should be superior to
single ion standards.
5 Single Ion Standards
In 1982, Hans Dehmelt[17] proposed that a single ion, with a
narrow, weakly-allowed spectral transition and suspended in-
side a Paul trap would exist in “a state of complete rest in free
space” and could serve as the ultimate laser frequency stan-
dard. A single ion, held at the field-free centre of the trap,
can be almost completely isolated from environmental pertur-
bations such as collisions and stray electric and magnetic fields.
Stark and Zeeman shifts disappear. Other broadening effects
such as Doppler shifts and transit time broadening vanish as the
ion’s motion is reduced by laser cooling and localized within a
small volume at the centre of the trap. Dehmelt predicted that
the resolution of single ion standards would eventually reach 1
part in 1018.
A number of different ions standards have been developed[5]
at a handful of laboratories around the world, primarily at na-
tional metrology institutes. These standards have advanced to
the level where reproducibilities between different systems are
now within a few parts in 1016. At NRC, our group has been
studying the single 88Sr+ion. It was chosen because of its sim-
ple energy level structure and the availability of the required
cooling, detection, and probing lasers. This standard will be
discussed in detail in order to illustrate the technologies com-
mon to single ion standards.
5.1 Single Trapped Strontium Ion Standard
Our strontium ion is held inside a miniature Paul trap of di-
mensions z0=r0/√2 = 0.50 mm and employing an applied rf
voltage of V0= 256 V at a frequency of Ω/2π= 12 MHz[18].
Strontium atoms are produced by a small oven located near the
trap and are ionized through collisions with electrons emitted
though field emission from the same oven.
The energy level diagram of the 88 Sr+ion is shown in Fig. 5a.
It has a strong 5s2S1/2−5p2P1/2electric dipole transition with
a natural linewidth of Γ/2π≈22 MHz that is used for Doppler
cooling. A frequency-doubled, 844-nm extended-cavity diode
laser produces the required cooling radiation. When in the
5p2P1/2state, the ion has a 1:13 probability of spontaneously
decaying to the metastable 4d2D3/2state, so an auxiliary laser
at 1092 nm, with rapid polarization switching, is required to
prevent optical pumping into this state. The 1092-radiation
is provided by a ytterbium-doped fibre laser. When these two
lasers are incident on the ion, millions of 422-nm photons are
scattered every second and the ion is rapidly Doppler-cooled to
a kinetic temperature of approximately 10 mK.
The “clock” transition in 88 Sr+is the electric-quadrupole,
5s2S1/2−4d2D5/2transition at 674 nm (445 THz). The nat-
ural linewidth of this transition is only 0.4 Hz, corresponding
P
2
1/2
S
2
1/2
D5/2
2
2D3/2
mj
mj
mj
+5/2
+3/2
+1/2
-1/2
-3/2
-5/2
+3/2
+1/2
-1/2
-3/2
+1/2
-1/2
+1/2
-1/2
422 nm
674 nm
1092 nm
Laser Cooling
and detection
of Quantum
Jumps
Clock Transition
a)
010 15
0
2000
4000
6000
Time [s]
Photomultipler Signal
(arbitrary units)
5
b)
Figure 5: a) Partial energy level diagram of the 88Sr+ion. b)
Photomultiplier output showing quantum jumps in the fluorescence
at 422 nm.
to a Q of 1015. In order to fully exploit this extremely nar-
row transition, the probe laser must possess a similarly nar-
row linewidth and have a very low drift rate. Lasers having
these properties have been developed in several laboratories and
a record linewidth of less than 0.6 Hz has been obtained[19].
They rely on stabilizing the laser’s output frequency by lock-
ing it, through a high speed servo, to a resonance of an ultra-
high finesse (F>100 000) Fabry-Perot interferometer. Our
probe laser system uses a commercial, 674-nm, extended-cavity
diode laser, and a Fabry-Perot cavity utilizing a spacer made
of Ultra-Low Expansion (ULE) glass. The cavity is placed in-
side a temperature-controlled vacuum chamber and mounted
on a vibration-isolating table located inside a concrete bunker.
A laser linewidth below 50 Hz and a drift rate of less than
0.04 Hz/s have been obtained in the NRC experiment.
The absorption or emission of a single photon by a single
ion would be almost impossible to detect though direct mea-
surements of the 674-nm radiation . However, by exploiting
the technique of electron shelving[17], also referred to as the
quantum jump technique, it is possible to detect transitions in
single ions with almost 100% effeciency. When the 88Sr+ion
absorbs a photon at 674 nm and makes the transition to the
meta-stable 4d2D5/2state, the scattering of photons on the
5s2S1/2−5p2P1/2cooling transition stops and reappears only
when the ion spontaneously decays back to the 5s2S1/2ground
level. An example of the fluorescence signal showing quantum
jumps, as detected by a photomultiplier, is shown in Fig. 5b.
These quantum jumps can be counted by a computer.
Figure 6 shows the experimental arrangement of our
strontium-ion standard. The radiation from the 674-nm probe
Sr Ion
Trap
+
Frequency Doubled
Diode Laser System
( =422 nm)
λ
Diode Laser
Locked to Ultra-
Stable Cavity
( =674 nm)
Chopper
Chopper
AOM
Photo-
multiplier
To Frequency
Comb
λ
Diode Pumped
Nd Fiber Laser
( =1092 nm)
λ
3+
"Clock"
Laser
Optical Pumping
Laser
Cooling
Laser
Computer
Probed
Frequencies
Centre
Frequency
Figure 6: Diagram of the setup used at NRC to lock the frequency
of an ultra-stable laser to a narrow transition in the strontium ion
at 674 nm.
laser is frequency shifted by an acousto-optic modulator (AOM)
in order to permit the measurement of spectra. For frequency
standards work, the centre frequency of the S-D Zeeman spec-
trum is determined. A small magnetic field is applied to the
ion to split the Zeeman components and prevent the overlap
of several components which would broaden the spectrum. A
computer, which controls the AOM, then counts the number of
quantum jumps that occur in a period of 10-20 s at each of four
frequency positions, one on each side of each line in a symmetric
pair of Zeeman lines. An algorithm then determines the offset
of the unshifted laser from line centre and keeps the shifted
laser locked to the S-D transition. Simultaneous measurements
of the absolute frequency of the unshifted laser are performed
with respect to the NRC ensemble of atomic clocks, through a
femto-second optical frequency comb[2, 18]. Such combs have
been shown to transfer the frequency accuracy of a microwave
standard to the optical region with an uncertainty approach-
ing 1×10−19[3]. By combining the comb measurements with
the offset values, we have determined the absolute frequency of
the S-D transition with an uncertainty of 15 Hz[20], limited by
second-order Doppler shifts and ac Stark shifts.
5.2 Other Single Ion Standards and Future
Prospects
Several other single ion systems are currently being studied.
These include several transitions in barium, calcium, indium,
mercury, strontium and ytterbium ions. Except for the 674-
nm transition in 88 Sr+, which is studied at both NRC and the
National Physical Laboratory in the United Kingdom, each of
the other transitions is studied in only one laboratory. Re-
cently, the transition frequencies of 88Sr+[21], 199 Hg+[22], and
171Yb+[23] have been measured with respect to the SI second
with inaccuracies of less than 2 Hz using cesium fountain clocks
and similar systems have shown reproducibilities of better than
1 Hz[23]. Instabilities below 1×10−15 for averaging times of
several hundred seconds have been reported[22, 23].
In order to achieve a precision of 1 part in 1018, perturbations
which can shift the clock frequency must be understood and ac-
counted for[18]. The effect of collisions is practically negligible
for a single ion located in an ultra-high vacuum vessel. Back-
ground magnetic fields, which can result in time-varying linear
and quadratic Zeeman shifts, can be reduced through magnetic
shielding. For some ions, however, the quadratic Zeeman shift
amounts to hundreds of hertz and must be calculated from the
measured linear shifts. Significant shifts can occur if the ion is
not cooled sufficiently or if it is displaced from the centre of the
trap. The latter can occur due to stray electric fields caused
by the presence of foreign material deposited onto the trap’s
electrodes. If the ion is displaced from the trap centre, it feels
the applied rf field and experiences driven micromotion. The ac
field produces an ac Stark shift, while the associated micromo-
tion causes a second-order Doppler (relativistic time dilation)
shift. Both of these shifts can be of the order of several hertz
but can be reduced through sideband cooling and auxiliary elec-
trodes that control the ion position within the trap. However,
stray electric field gradients can remain, which can couple to the
ion’s quadrupole moment, leading to a frequency shift. We have
recently developed a method to cancel this quadrupole shift[20].
So far, only technical limits on the precision of single-ion stan-
dards have been encountered. As work continues, fundamental
limits due to our ability to calculate the blackbody, Stark, and
Zeeman shifts will become important along with stability issues
associated with the quantum nature of the interogation of single
ions (Eq. 1). Work is progressing rapidly and it can be expected
that in the next few years, systems with irreproducibilities in the
10−17, and even 10−18 range, should be achievable.
6 Cold Atom Standards
As shown in Eq. 1, the main advantage of cold atom stan-
dards over those based on single trapped ions is that millions
of atoms are interrogated, which can result in very low insta-
bilities. However, unlike single trapped ions, cold atoms are
perturbed by the MOT fields and, therefore, these fields must
be turned off before the atoms can be interrogated. Once the
trapping fields are removed, the atoms move ballistically due
to their random thermal velocities and the effect of gravity.
This leads to Doppler broadening and shifts of the transition
frequencies as well as limits on the observation time.
Both optical and microwave cold-atom standards have been
developed during the past decade. As a result of their high Q
values, optical standards typically have very low instabilities.
The most advanced of these optical standards is based on the
intercombination transition, 1S0(mj= 0)−3
P1(mj= 0) of 40 Ca
at 657 nm with a natural linewidth of 370 Hz[24, 25]. Atom
clouds as cold as 6 µK have been achieved[24]. A pulsed exci-
tation scheme is used to probe the clock transition. Atoms can
be trapped and interrogated many times every second resulting
in an impressive instability of only 4 ×10−15 for 1 s averaging.
Unfortunately, first-order Doppler shifts, collisions, and Stark
shifts due to black-body radiation, currently limit the accuracy
of this standard to a few parts in 1014.
6.1 Cesium Fountains
To overcome the limit on the observation time imposed by the
ballistic motion of the atoms, it was proposed in the 1950’s by
Jerrold Zacharias to launch atoms upward and observe them as
they fell under the force of gravity. It was reasoned that there
would be some slow moving atoms in the Maxwell-Boltzmann
distribution. Unfortunately, collisions with the faster atoms
prevented this scheme from working and it was only after laser
cooling methods became available that an atomic fountain, us-
ing sodium atoms, was built[26]. The advantages of a clock
based on a cold atom fountain over atomic beam clocks, both
in terms of stability and accuracy, became apparent and the
first cesium-fountain clock was constructed in the mid-1990’s
at BNM-SYRTE in France[27]. Today, these clocks, located
in various national metrology institutes, are the dominant con-
tributers to the accuracy of international atomic time (TAI). In
this section, we will describe how a cesium fountain clock works
by describing the NRC fountain which is in the final stages of
construction.
An energy level diagram of 133Cs, showing the ground and
the first excited states, is given in Fig. 7. TO BE REMOVED
F=5
F=4
F=3
F=2
F=4
F=3
6 S
21/ 2
6 P
23/ 2
mF
+5
0
-5
+4
0
-4
+4
0
-4
+3
0
-3
+3
0
-3
+2
0
-2
Gro u n d
State
Excite d
State
Zeeman
leve ls
H yp e r fi n e
leve ls
9 19 2 6 3 1 7 70 H z C lo c k
’
’
’
’
’
85 2 n m
(C o o l i ng , t r a p pi n g a n d
st a t e p r ep a r a t i on )
Figure 7: Partial energy level diagram for cesium-133 showing the
hyperfine and Zeeman spitting of the ground and first excited states.
The cooling and clock transitions are indicated.
The cesium-133 atom has a nuclear spin of I= 7/2while the
outer electron has a spin of J= 1/2in the ground state, yield-
ing a total angular momentum for the ground state of F= 3 or
F= 4.Cesium clocks are based on the weakly-allowed hyper-
fine transition between the F= 3(mF= 0) and F= 4(mF= 0)
ground states. The NRC fountain uses a MOT employing six
laser beams at 852 nm and arranged as shown in Fig. 8 to trap
and cool up to 109atoms in a 2-mm-diameter volume over a pe-
riod of approximately 500 ms. The laser beams are tuned to the
red side of the 6 2S1/2(F= 4) −62P3/2(F′= 5) dipole tran-
sition and cool the atoms through Doppler and polarization-
gradient cooling to a temperature of approximately 2 µK. Se-
lection rules and a repumper laser tuned to the 6 2S1/2(F=
3) −62P3/2(F′= 4) transition guarantee that all the atoms
eventually get pumped to the F= 4 ground state. After 500 ms
have elapsed, the magnetic field is turned off and the frequen-
cies of the laser beams which point upward and downward along
the diagonal to the vertical are tuned to launch the atoms up-
ward at a velocity of 4 m/s by means of a “moving optical
molasses”. The atoms, which are all in various mFstates of
the F= 4 ground state then pass through a microwave cavity,
where those in the F= 4, mF= 0 state receive a πpulse and
make a transition into the F= 3, mF= 0 state. The remaining
atoms in the F= 4 states are pushed away by a broadband laser
tuned to the 6 2S1/2,(F= 4) −62P3/2(F′= 5) transition, leav-
ing only the atoms in the F= 3, mF= 0 state. Approximately
107atoms are launched by this process.
MO T
St a t e-
pr e p ar a ti o n
ca v i ty
De t e ct i o n of
F= 4
De t e ct i o n of
F= 3
Ra m s ey
int e rr o g at i on
ca v i ty
Mic r o wa v es
Mic r o wa v es
Fo u n ta i n
Figure 8: Schematic diagram of the NRC cesium fountain.
The atoms are interrogated by a technique called the “Ram-
sey method of separated fields” as they pass twice through a
microwave cavity, once on their way upward and again as they
fall downward[1, 28]. REMOVE The Ramsey method is a di-
rect application of the quantum superposition of states. During
their first pass through the microwave cavity, the atoms receive
aπ/2 pulse and are excited into an equal superposition of the
F= 3 and F= 4 ground states. They then travel through the
field-free region where their magnetic moments precess billions
of times at a frequency (EF=4 −EF=3)/h=9 192 631 770 Hz.
If, when they enter the microwave cavity for the second time,
the phase of the microwaves is in phase with this atomic res-
onance, the atoms will receive a second π/2 pulse and their
excitation to the F= 4 state will be completed. If, on the
other hand, the radiation is out of phase with the precession of
the magnetic moments by an angle of 180◦, then the atom will
be returned to the F= 3 state. Secondary maxima will occur if
the frequency of the radiation is tuned such that some, but not
all, of the atoms will have precessed by the amount required
to be in phase with the radiation. However, it is only at the
atomic resonance frequency that all the atoms, regardless of
their velocity, will have precessed so as to be in phase with the
radiation in the second cavity. For the NRC fountain, the drift
period is approximately 500 ms resulting in resonance peaks
with widths of the order of 1 Hz. This should be compared to
resonance widths of tens of hertz for cesium beam clocks which
use cesium atoms at a temperature of 300 K and two microwave
cavities spaced by 1 m.
The fraction of atoms that have made the transition to the
F= 4 state is measured in the region below the microwave
cavity. A standing wave, tuned to the F= 4(mF= 0) −
F′= 5(m′
F= 0) transition, is applied and the fluorescence
is measured to determine the number of atoms in the F=
4(mF= 0) state. A single beam, tuned to the same transition,
is then used to push the F= 4(mF= 0) atoms out of the falling
cloud. The number of atoms in the F= 3(mF= 0) state is then
measured by first driving the F= 3(mF= 0)−F′= 4(m′
F= 0)
transition to optically pump the atoms into the F= 4(mF= 0)
state and then measuring the fluorescence from these atoms. In
clock operation, the frequency or phase of the microwave source
is varied in a square-wave manner to permit the determination
of the transition centre frequency. A complete cycle of trapping,
launching, and interrogation requires over 1.5 s. To increase the
data rate, the NRC fountain uses mechanical shutters which
will allow the launch cycles to be overlapped so that a cloud of
atoms is present in the drift region at all times.
Because their relatively low line Q value of ∼1010 is only
partly offset by the large number of atoms interrogated, the sta-
bility of cesium-fountain standards is inferior to that of single-
ion and atom-cloud optical standards. To date, the best insta-
bility obtained for a fountain clock is 1.6×10−14 τ−1/2(τis the
averaging time), within 20% of the quantum limit, and reach-
ing 1.6×10−16 at 50 000 s[16]. This performance was obtained
with a cryogenic-sapphire-based microwave synthesizer and is
approximately one order of magnitude better than that ob-
tained with the best ultra-stable quartz oscillators. The ability
of cesium-fountain clocks to realize the SI second is impressive.
Current inaccuracy levels are below 1 ×10−15 and are expected
to eventually reach 1 ×10−16. Systematic uncertainties arise
primarily due to collisions, which can result in spin-exchange
interactions in the atom cloud, uncertainty in the blackbody
shift (Stark shift) resulting from poor knowledge of the Stark
shifts and the local temperature, and uncertainty in the local
gravitational red shift[16, 29, 30].
7 Future Standards
For single ion standards, the choice of which ions to study has
been largely determined by practical constraints, particularly
concerning the required laser sources. A recent development
may lessen those constraints. Using methods developed in re-
search on quantum computing, it has become possible to trap,
cool, and interrogate, by means of a second ion species, ions
which possess clock transitions that are relatively insensitive
to external perturbations. A recent experiment has employed
27Al+and 9Be+ions[31]. Both ions are held in a linear Paul
trap, which couples their motional quantum state. The 27Al+
ion has a transition at 267 nm with a natural linewidth of
5 mHz, a small blackbody shift, and no quadrupole shift. How-
ever, the transition for cooling and quantum-jump detection
is at 167 nm and, therefore, experimentally difficult to realize.
Instead of direct cooling, the beryllium ion is laser cooled and
it sympathetically cools the aluminum ion through Coulomb
coupling. The quantum state of the aluminum ion, after prob-
ing at 267 nm, can also be read by interrogating the beryllium
ion. This technique is showing great promise and it should be
possible to apply it to other pairs of ions.
The stability of single-ion optical standards is limited by the
time required to count quantum jumps and determine the line
centre of the clock transition. Recently, a new kind of standard,
with potentially much better stability, called an “optical-lattice
clock” has been proposed in which millions of neutral atoms are
trapped and probed simultaneously[32]. Atoms are trapped by
the dipole force at the antinodes of a standing optical wave pat-
tern and form a lattice with only one atom at each lattice site.
The atoms can be sufficiently confined to permit Doppler-free
probing in the Lamb-Dicke regime and sideband cooling to mi-
crokelvin temperatures. Normally one would expect a strong
ac Stark shift in the atomic energy levels due to the trapping
laser fields as there is in a MOT. However, for a particular
choice of trapping laser frequency and polarization, the shifts
for the ground and excited levels are identical and, therefore,
the ac Stark shift in the transition frequency is zero. The 87 Sr
atom which has a 1S0−3P0clock transition at 698 nm and
a “magic” trapping wavelength of 813.4 nm has been investi-
gated with a 1-D standing wave. Other systems are proposed
and work is progressing rapidly. Provided systematic shifts can
be controlled, it is expected that the optical lattice clock will
eventually have a precision equal to that of single ion standards
and an instability several orders of magnitude better.
8 Conclusions
The methods of trapping and laser-cooling of ions and atoms
have resulted in new standards of frequency and time possess-
ing unprecedented levels of stability, precision, and accuracy.
Cesium fountain clocks are currently able to realize the SI sec-
ond with an accuracy at least one order of magnitude better
than the best cesium beam clocks. Optical frequency stan-
dards, based on single trapped ions, have advanced rapidly and
already surpass cesium fountains in terms of stability and pre-
cision. New optical standards, based on sympathetic cooling or
optical lattices, show great promise. It is likely that some day
an optical standard may be used in a new definition of the sec-
ond. However, at the present moment, no atom or ion system
has won the hearts of the research community and it may be
some time before a new definition is adopted.
References
[1] J. Vanier and C. Audoin, “The Quantum Physics of
Atomic Frequency Standards”, Vol. 1 and 2 (Adam Hilger)
1989.
[2] S.T. Cundiff et al., Rev. Sci. Instr. 72, 3749, 2001.
[3] L.-S. Ma et al. Science 303, 1843 (2004).
[4] D.J. Wineland et al., Adv. Atom. Mol. Phys. 19, 135, 1983.
[5] A.A. Madej and J.E. Bernard, Topics Appl. Phys. 79, 153,
2001.
[6] R.H. Dicke, Phys. Rev. 89, 472, 1953.
[7] H. Metcalf and P. van der Straten Phys. Rep 244, 203,
1994.
[8] W.D. Phillips, Rev. Mod. Phys. 70, 721 (1998).
[9] C.N. Cohen-Tannoudji and W.D. Phillips, Physics Today,
33, Oct. 1990.
[10] T. H¨ansch and A. Schawlow, Opt. Comm. 13, 68, 1975.
[11] D. Wineland and H. Dehmelt, Bull. Am. Phys. Soc. 20,
637, 1975.
[12] W.M. Itano et al., Science 237, 612, 1987.
[13] F. Diedrich et al., Phys. Rev. Lett. 62, 403, 1989.
[14] D.J. Wineland et al., J. Opt. Soc. Am. B 9, 32, 1992.
[15] J. Dalibard and C. Cohen-Tannoudji, J. Opt. Soc. Am. B
6, 2023, 1989.
[16] S. Biz, et al., C. R. Physique 5, 829, 2004.
[17] H.G. Dehmelt, IEEE Trans. Instr. Meas. 31, 83, 1982.
[18] A.A. Madej et al., Phys. Rev. A 70, 012507, 2004.
[19] B. C. Young et al., Phys. Rev. Lett. 82, 3799, 1999.
[20] P. Dub´e et al., Phys. Rev. Lett. 95, 033001, 2005.
[21] H.S. Margolis et al., Science 306, 1355, 2004.
[22] W.H. Oskay et al., Phys. Rev. Lett.,94, 163001, 2005 (and
private communication).
[23] T. Schneider et al., Phys. Rev. Lett.,94, 230801, 2005 (and
private communication).
[24] F. Riehle et al., Lect. Notes Phys. 648, 229, 2004.
[25] E.A. Curtis et al., Proc. 6th Symp. on Frequency Standards
and Metrology ed. P. Gill, 331 (World Scientific, Singapore,
2002).
[26] M.A. Kasevich et al., Phys. Rev. Lett. 63, 612, 1989.
[27] A. Clairon et a., Proc. 5th Symposium on Frequency Stan-
dards and Metrology, (World Scientific, London) 49, 1995.
[28] S. Chu, Nature 416, 206 (2002).
[29] S.R. Jefferts et al., Proc. of the 2003 IEEE Int. Freq. Cntrl.
Symp. and 17th European Frequency and Time Forum,
1084, 2003.
[30] T.P. Heavner et al., IEEE Trans. Inst. Meas. 54, 842, 2005.
[31] P.O. Schmidt et al., Science 309, 749, 2005.
[32] M. Takamoto et al., Science 435, 321, 2005.