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Atomic Clocks
A literature study
Artwork credit: Brad Baxley and Ye Labs, JILA
Cathelijne Glaser, BSc
[email protected]
supervised by Dr. Jeroen Koelemeij
summer 2015
Abstract
In atomic clocks, the frequency of the oscillator is determined by the energy difference
between two quantum mechanical states in atoms, ions or molecules. This transition
energy depends only on fundamental constants and thus provides a stable frequency
reference. Disturbances of the system that may shift the frequency are discussed, as well as
broadening of the peak due to dephasing and the uncertainty principle.
This literature study gives an overview of the different types of both microwave and optical
atomic clocks, as well as the different atom species that are suitable as clock atoms. A
qualitative way to determine clock performance is given in terms of the fractional
frequency uncertainty, and the Allan variance to determine the stability and characterise
the noise of the clock. Various methods and procedures to decrease the uncertainties are
discussed.
An outlook on possible future developments is presented, including clocks based on
nuclear transitions and techniques to increase the accuracy and stability beyond the curent
state of the art.
Front page: artist's view of an optical lattice clock. [1]
1
Table of contents
Abstract
1
1
Introduction
3
2
Atomic transitions
2.1 Frequency shifts
2.2 Line broadening
4
4
5
3
The basic clock system
3.1 Active clocks
3.2 Passive clocks
3.3 Measuring and feedback
9
9
10
10
4
Characterizing the performance of clocks
4.1 Accuracy and reproducibility
4.2 Stability
4.3 Comparison and synchronisation
12
12
12
16
5
Active clocks
19
6
Passive clocks
6.1 Microwave regime
6.2 Optical regime
25
25
32
7
Future prospects
7.1 (Fundamental) limits
7.2 Suggested further research
36
36
38
8
Conclusion
39
References
40
2
1
Introduction
Humans have, as far as known, kept track of time, or cyclic events, since the beginning of
development. To measure time, a reference oscillator is needed. It is impossible to measure
time itself. It is only possible to measure a frequency or duration. In time measurement, it is
assumed that two identical phenomena acquire the same time to be produced, the socalled reproducibility postulate. [2]
The rhythm of day and night and the four seasons is always present. But descriptions of
more subtle recurring events, like solar eclipses and other astronomical phenomena, have
been found dated at least 4000 years ago. The sundial was the earliest form of a clock that
divides time into smaller segments than a day. The most sophisticated versions had an
error of 24 seconds for each 0.1 degree of angle measurement. Other historical clocks were
based on water or mechanical devices, such as the pendulum clock or the spring-balancewheel clock. The quartz clock is based on the piezoelectric effect, in which vibrations are
excited by applying an alternating voltage to a crystal. This was the first clock based on
material properties instead of astronomical observations or mechanical movement.
However, it still suffers from ageing and is very sensitive to environmental conditions. [3]
In atomic clocks, the electromagnetic transition between two quantum mechanical states
of an atom, ion or molecule determines the frequency of the oscillator. Frequency dividers
provide pulses at a desired rate, for example with a frequency of 1 Hz. The development of
these types of clock boosted the abilities of time measurement. They are so sensitive that
relativistic effects can be measured. Precise time measurements are for example applied in
metrology, fundamental constant research, the foundations of quantum mechanics, gravity,
and geodetics. But it also finds applications in navigation and communication networks. It
makes a precise measurement of position or length possible, especially since the meter has
been defined in terms of the speed of light. [2]
This literature study dives into the principle of atomic clocks, and what makes them
intrinsically suitable to measure time. After defining a quantitative manner to measure the
performance of a clock, several factors that determine the performance of an atomic clock
are described. Although it is impossible to mention every atomic clock ever built, an
overview of the different types is given, in both the microwave domain and the optical
domain. These are mutually compared and their performance, advantages and
disadvantages are discussed, as well as the different improvements that came with each
new design. The currently best optical lattice clock is described, evoking a look into the
future. What is there to be expected of future atomic clocks, and is there a limit to the
potential performance of clocks?
3
2
Atomic transitions
According to quantum mechanics, the energy of an atom, molecule, or ion consists of
discrete values. Transitions between these levels occurs by emitting or absorbing
electromagnetic radiation of a specific frequency. This resonance frequency ν0 depends
only on fundamental physical constants and is the same for all atoms of a particular
element, which makes it a reproducible standard. Two other aspects make atomic
transitions particularly suitable for timekeeping. The properties of the atoms do not, as far
as known, change over time. Additionally, atoms do not wear out, as mechanical clocks all
do. [2][4]
2.1
Frequency shifts
The resonance frequency is an intrinsic property of the unperturbed atom, but several
factors may cause the actual frequency to be shifted. It is important to accurately
characterize the shifts that occur in the clock, to be able to state the correct output
frequency. Frequency shifts can either be diminished by reducing the environmental
perturbations acting on the clock system, or corrected for when accurate data about the
shifts are available. Inhomogeneities and fluctuations in the perturbing fields should also
be taken into account. Additionally, a shift in some level may influence the energy of a
nearby level. [5]
Depending on the respective design, each type of atomic clock will encounter these and
other, minor disturbances, in varying degrees. This is described in more detail in the
corresponding paragraphs in chapter 5 and 6.
External magnetic fields
Magnetic moments are associated with the various angular momenta of atoms: orbital,
spin and nuclear angular momenta. These magnetic moments interact with an external
magnetic field. The energy of atomic levels with different magnetic moments depend on
their orientation with respect to the field. The external magnetic field therefore lifts the
degeneracy of these levels. This is called the Zeeman effect.
The energy shifts can be calculated using perturbation theory. The nature of the shifts
depends on the strength of the external magnetic field relative to the internal magnetic
field of the atom, generated by the moving charges. The latter causes spin-orbit coupling,
which can also be considered a perturbation
The shift in frequency for a certain transition due to an external magnetic field can be
written as a Taylor expansion. For small magnetic fields, only the first two terms need to be
considered. [5] Only constant magnetic fields contribute to the Zeeman effect as long as
the energy shift is linear, because the time average of an oscillating (AC) field is zero.
However, in the intermediate field regime, the quadratic Zeeman effect cause AC fields to
play a role as well. [5][6]
4
External electric fields
The electrical analogue to the Zeeman effect is the Stark effect. For a static electric field Ɛ
in the direction of the z-axis, the perturbation operator for N electrons is given by:
N
H ' =e ∑ Ɛ z i =−Ɛ D z
(2.1)
i=1
with e the electron charge, zi the spatial coordinate of electron i, and Dz the z-component
of the electric dipole moment of the atom. Since this is an odd operator regarding parity,
the first order perturbation energy is zero for wavefunctions of definite parity. Otherwise
stated, atoms do not have a permanent dipole moment in non-degenerate states. The
linear Stark effect exists only for hydrogen-like orbitals with n>1. The degeneracy is only
partly removed.
For weak fields, the Stark splittings are negligible compared to the fine structure effects.
Intermediate field effects can be calculated using full perturbation theory. [7]
The quadratic Stark shift is generally very small. According to perturbation calculations, the
shifting of the levels depends on the neighbouring levels of opposite parity and the
corresponding parity. [7] The quadratic Stark shift depends on the polarizibility of the
atom, which is in turn dependent on the electron configuration. Note that both DC and AC
electric fields may give rise to a quadratic Stark shift. [8]
A gradient in the present electric field interacts with the electric quadrupole moment of the
atom. The resulting energy shift is usually very small and even zero for many energy levels.
[8]
Gravitational red shift
According to relativity, when two clocks experience different gravitational potentials, their
clock rates differ. The frequency shift depends on the mutual height difference and the
local value of the gravitational acceleration. [5]
2.2
Line broadening
Atoms will absorb or emit not only electromagnetic radiation with exactly the resonance
frequency, but over a small frequency range surrounding ν0. This range is called the
resonance width or linewidth Δν. The ratio of resonance frequency to linewidth is called the
quality factor Q:
ν
Q= Δν0
(2.2)
All other parameters equal, the stability of the atomic oscillator is proportional to Q. An
important route to increased stability is thus to narrow the linewidth. [4]
A perfect oscillator displays a pure sine wave. In the frequency domain, this is represented
5
by a Dirac delta function. Line broadening can originate from either homogeneous or
inhomogeneous processes. In the former case, the probability that an atom emits or
absorbs a certain frequency is the same for all atoms in the system. Inhomogeneous
broadening occurs when this probability slightly differs for individual atoms.
Natural lifetime
Spontaneous emission reduces the average amount of time that an atom can be found in
an excited state. Due to the time-energy uncertainty principle, the energy of an excited
state can not be determined with infinite accuracy if its lifetime is finite. Therefore, an
uncertainty in the frequency arises, referred to as the natural linewidth. The frequency
spectrum has a Lorentzian lineshape. The full width at half maximum (FWHM) is inversely
proportional to the natural lifetime of the excited state. If the lower level can undergo
spontaneous emission to a lower level as well, both uncertainties contribute. [9][10]
Transit time
The atom can only interact with the electromagnetic field during a certain finite transit time
tT. The uncertainty principle therefore introduces an uncertainty in the frequency as seen by
the atom. Assuming the interaction to begin and stop abruptly, the intensity profile of the
radiation as seen by the atom is rectangular. This results in a sinc 2 shaped line broadening.
If the atom experiences a Gaussian intensity distribution, either spatially or temporal, the
frequency will be broadened into a Gaussian profile. The FWHM is proportional to the
velocity of the atom perpendicular to the beam, and inversely proportional to the beam
radius.
Another factor that should be mentioned is the contingent diffusion time of the atoms out
of the laser beam, which also decrease the interaction time. This is significant for very long
lifetimes of the excited state.
An additional, inhomogeneous broadening mechanism comes from the curvature R of the
phase surfaces within a focused Gaussian beam. This causes a phase shift depending on
the location of the atoms in the beam. [9][10]
Collisional broadening
Inelastic collisions contribute to depopulation of the excited state. The resulting
broadening is similar to natural broadening and has a Lorentzian shape as well. The FWHM
is proportional to the pressure p because it depends on the number of collisions per unit of
time.
Due to elastic collisions, the lineshape may become asymmetric. The exact broadening
depends on the interaction potential between the particles. Collisions with the wall interact
with a different potential than when collisions occur between particles. This shift is
temperature dependent and is called the wall shift. [3]
Collisional broadening generally also shifts the resonance frequency, because both energy
6
levels involved in the transition may experience a shift due to the interaction with other
particles. Even elastic collisions where the distance between the particles remains relatively
large, such that the broadening effect is weak, can still very efficiently shift the centre
frequency. [9][10]
Doppler effect
Each individual atom will have a certain velocity relative to the beam of radiation. From the
frame of reference of the atom, this means that it will experience the light as if it had a
different frequency, according to the Doppler effect. Alternatively, in the laboratory frame,
the atomic resonance frequency appears to be shifted. In the case of a single atom, the
Doppler effect manifests itself only as a shift in frequency, not as line broadening. This shift
may vary in time if the velocity of the atom changes.
For a collection of atoms however, each atom gives rise to a slightly different frequency.
The spectrum will therefore be distributed over these frequencies. The lineshape due to
this inhomogeneous process is Gaussian. It is an accumulation of the different frequency
values measured. If the individual atoms also give rise to Lorentzian lineshapes obtained
from homogeneous processes, a convolution of the Gaussian broadening results in a Voigt
lineshape. The FWHM can be determined if the velocity distribution of the atoms is known.
If the atoms frequently collide with other particles, the mean free path will be small. If the
mean free path is smaller than the wavelength of the radiative field, this effect causes the
Doppler broadening to be decreased, averaging over the sample. This is called Dicke
narrowing. For high pressures however, the narrowing effect will be overcompensated by
the collisional broadening. [9][10]
The second order Doppler effect is the highest order contribution of time dilation. It is
independent of the direction of the velocity. It instead depends on v2 and is therefore also
called the quadratic Doppler effect. Even in confined systems, a quadratic Doppler shift
occurs due to the residual motion of the atom around its equilibrium position.
Since the velocity of the atoms depend on temperature, so does the quadratic Doppler
shift. To calculate the second-order Doppler shift, knowledge of the velocity dependence
on temperature is required. [5][10]
When an atom absorbs a photon, its momentum changes accordingly. If this is taken into
account, a small additional shift appears. This so-called recoil shift can be incorporated in
the Doppler calculations if the velocity is replaced by the arithmetic average, calculated
using momentum conservation arguments. [11]
Saturation or power broadening
At high intensities, the excitation rate becomes larger than the relaxation rate. The
populations of the absorbing levels thus decreases. The system is said to be saturated
when the absorption and relaxation processes balance.
The saturation parameter is frequency dependent, following a Lorentzian profile. For
sufficiently high radiation intensity, the absorption at each frequency is altered according
7
to this distribution, resulting in a Lorentzian broadening. The width depends on the value
of the saturation parameter at the resonance frequency. For inhomogeneously (Gaussian)
broadened absorption lines, saturation broadening results in a Voigt shaped profile.
The same results can be derived using the strong field approximation and the influence of
Rabi oscillations on the population of the two levels. [10]
Stark broadening and quenching
Besides an energy shift, another factor that should be considered when an external electric
field is present, called Stark broadening. The total potential experienced by the electron is
altered due to the electric field, and another minimum will arise, next to the one due to the
Coulomb potential of the nucleus. At sufficiently high electric fields, there will be a non
zero probability for the electron to accelerate towards the new minimum and thus
escaping its bound state, as pictured in figure 2.1. This tunnel effect decreases the lifetime
of the atomic levels and therefore causes additional broadening of the spectral line. In
practice, the choice of the clock transition is usually such that this effect plays is
insignificant. [7]
Figure 2.1. Potential of an electron in an external electric field. [7]
Under influence of a static or oscillating electric field, mixing between different states with
opposite parity occur, because the perturbation H' contains non-diagonal elements.
Transitions that are usually forbidden can happen because the state is contaminated with
another state, to or from which the selection rules do allow transitions to occur. The
additional population decrease is called quenching. Except that this may cause additional
line broadening due to the decreased lifetime of the involved levels, it also means that the
two-level approximation should be reconsidered because additional transition possibilities
may occur when an electric field is present. [7]
8
3
The basic clock system
3.1 Active clocks
In active clocks, the atomic system itself is the oscillator and generates radiation with the
clock frequency, which is then simply received and converted to an output signal. The first
example of an active atomic clock is the ammonia maser clock. This principle has since
been extended to other atoms. [2]
Masers are based on amplification of stimulated emission. To achieve this, population
inversion is required. This means that the population of the upper state is increased such
that it transcends that of the lower level. According to Boltzmann's distribution, this does
not occur in thermal equilibrium. It is either accomplished by pumping the atoms into the
upper state or by selectively picking atoms that happen to be in the upper state.
The atoms reside into a cavity which is tuned to resonance. A weak external resonant field
will cause stimulated emission. In a high quality cavity, the resonant modes will be
amplified by subsequent stimulated emission, while other modes, originating from
spontaneous emission, vanish.
If the cavity is not tuned correctly to the resonance frequency, the output frequency will be
shifted relative to the exact transition frequency. This is called cavity pulling. The amount of
cavity pulling depends on the ratio of the quality factor of the cavity QC to the quality
factor of the resonance width which was introduced in chapter 2:
δν ∝
QC
Q
(3.1)
(ν C −ν 0 )
where νC – ν0 is the amount of mistuning of the cavity. [3] Cavity pulling can be reduced by
using a very high quality cavity or by continually adjusting the cavity properties (e.g. length
or temperature) according to the output measured. Note that this feedback system differs
from that of a passive clock, in that not the oscillation itself is adjusted, but the cavity in
which the radiation oscillates.
Although the accuracy of maser clocks is limited because of cavity-related frequency shifts,
they turn out to have excellent stability, which may be invaluable for some applications. [3]
Masers can also be used to lock an external oscillator to the maser frequency, thus forming
a passive clock. [2]
The principles of the active microwave clocks have also been extended to the optical
regime. In general, lasers – the optical equivalents of masers – show long-term frequency
drifts which makes them unsuitable as frequency standards. Several ways to improve the
stability of laser systems have been proposed, that will be discussed in more detail in
chapter 6. A different way to improve the stability is to lock the laser to a long-term stable
reference, and that is the principle of passive atomic clocks. [5]
9
3.2 Passive clocks
The most common atomic clocks are of the passive type, where the resonant light is locked
to the atomic transition. The atom then plays the role of frequency discriminator. The local
oscillator is a source that produces radiation with more or less the frequency of the desired
transition. After the system is prepared in a certain quantum state, the response to the
radiation is monitored. Maximum response is detected when the frequency equals the
resonance frequency. Using feedback loops, the radiative frequency is adjusted. This way,
the oscillator is stabilized with respect to the atomic transition. A schematic diagram of a
passive clock is shown in figure 3.1.
Passive clocks either operate in a continuous mode, or in a cyclic sequence consisting of
preparation, irradiation, and response measurement and frequency adjustment. Typically,
optical clocks operate on a cyclic basis, while for example atomic beam clocks provide a
continuous signal. [2][5]
Figure 3.1. Block diagram of the working principle of a passive atomic clock.
3.3 Measuring and feedback
To determine the response of the atoms to the applied resonance frequency, the atomic
population in one of the two clock states is measured. Generally this is done using a light
source resonant with a transition involving the state of interest and measuring the amount
of absorption or fluorescence. In clocks operated in cyclic mode, the detection may disturb
the population difference such that state preparation is again necessary when initiating the
next cycle. It may even displace the atoms from their trap or lattice, requiring reloading for
the next cycle. New methods are being investigated that maintain the internal coherence
from cycle to cycle. Not only does this reduce the time in between interrogations, it also
enables gaining a signal from the same sample of atoms in each cycle. [5]
A sequence of measurements is performed at frequencies alternately above and below the
centre frequency of the oscillator. Depending on the retrieved signal, the centre frequency
is adjusted after a certain number of averaging cycles. The latter determines the accuracy
of state population measurements, and together with the sensitivity of the adjustment to
the error signal, it determines how responsive the oscillator is. If it takes longer to approach
10
the resonance frequency, the clock system will be more dependent on the short-term
stability of the oscillator.
Frequency drifts in the oscillator are not tolerable if the feedback system cannot keep up.
An automatic drift correction can be built in by adjusting the frequency according to a preset function or based on drift rate measurements made on the fly. A high degree of
stability in the involved electronics is of course required. [3][5]
11
4
Characterizing the performance of clocks
4.1 Accuracy and reproducibility
The accuracy is the capability to measure the exact frequency of the system. The exact
resonance frequency cannot directly be measured, because the atomic system will not be
unperturbed and the measurement equipment inherently adds some uncertainties. Both
the uncertainty in the exact frequency and the uncertainty of the final output frequency,
caused by the measurement and relative to the exact frequency, have to be considered to
determine the accuracy of the clock. If the output frequency is denoted νout and the exact
resonance frequency of the atomic system ν0, then the ratio νout/ν0 can be determined. The
relative uncertainty of the clock, given in fractional frequency units, then equals the relative
uncertainty of this ratio. [2]
It is common practice to divide errors into two categories: systematic errors, and statistical
errors, due to measurement fluctuations which are intrinsic to all physical experiments.
Systematic uncertainties arise from uncertainties in the frequency shift characterisation.
Perturbations causing a shift in the frequency of the transition can either be prevented or
corrected for. An overview of the most common shift origins is given in chapter 2.
Primary standards are expected to depict the true resonance frequency, after applying
eventual corrections. For some frequency standards however, the output frequency
depends critically on the value of operational parameters. These standards are called
secondary and they need to be calibrated against a primary standard. The absolute
accuracy of secondary standards therefore depends strongly on the quality and validity of
their calibration. [3]
Another important term involving clock performance is its reproducibility. A clock with high
reproducibility does not necessarily have to have a precisely known shift from the
resonance frequency, as long as it is constant. Reproducibility thus refers to the uncertainty
in the frequency shift. [5]
4.2 Stability
Although the intrinsic frequency of atomic transitions is assumed to be non changing, the
measured frequency will be subject to fluctuations. This usually includes both fluctuations
around the mean, and drift. In the latter case, the deviation continues to increase (or
decrease) over time, while in the former case, the fluctuations average to zero over time.
[3]
Frequency drifts can arise from many environmental sources, or ageing of the apparatus.
When drift is present, the uncertainty in the measured frequency will increase over time.
The observed trend in frequency change can be fitted to a mathematical model, which
does not necessarily have to be physically motivated. This way, the frequency drift can be
corrected for.
It is assumed that frequency drift is removed or absent when characterising signal noise.
Noise in the oscillator signal can be viewed either as phase fluctuations or frequency
12
fluctuations, as depicted in figure 4.1, but these two can be converted into each other. [12]
Since the frequency is the quantity of interest, amplitude fluctuations are not considered in
this chapter. Note, however, that the amplitude does have a significant implication in the
context of signal measuring; a signal is only measured after it reaches a certain value. If the
amplitude of the oscillation is lower, it takes longer to pass the threshold. In practice
however, these fluctuations are not significant. [2]
Figure 4.1. Example of the output voltage in the time domain of a frequency measurement
with different kinds of instability. There is no drift present in this case. [13]
Several types of noise can be distinguished, although the origins are not always completely
understood. A distinction between noise that modulates the signal itself and additive
noise, independent of the signal, is often useful to make. The short-term stability is
determined both by external noise processes and the quality factor of the resonance. [3]
In electrical circuits, some noise is always present. Thermal motion of the atoms cause
fluctuations in the signal. Thermal noise can also occur in for example cavity resonance
modes. It is white noise: independent of the frequency. [2] Shot noise, which also gives rise
to white noise, occurs because particles are discrete and the signal can therefore not be
completely continuous. This applies to charge carriers in electronic signals, but shot noise
also occurs because the number of atoms in each measurement fluctuates. [3] Another
important phenomenon relevant for atomic clocks is quantum projection noise, which
arises from the discrete nature of state population measurements. If an atom is in a
superposition of both states, the measurement will observe only one of these states with
the corresponding probability. The outcome thus fluctuates depending on the probabilistic
collapse of the wavefunction. It manifests itself as white noise as well. [5]
Flicker noise is inversely dependent on the frequency. This means that slow fluctuations are
large and these increase with longer averaging times. This is often visible as the graph of
the Allan deviation (see below) flattening out for longer averaging times, while for short
averaging times, white noise is dominant, as can be seen in figure 4.2. However, flicker
noise can be largely reduced by optimising clock design and operation parameters. It is
13
associated with imperfections in the components of the device. [3]
For even longer averaging times, the Allan deviation increases with time due to random
walk noise and frequency drift. In random walk noise, the frequency at some time is
codetermined by the frequency at the previous moment. The increments are Gaussian
random variables with zero mean. [14] Random walk noise is usually caused by
environmental perturbations and fluctuations in parameters. [3]
Figure 4.2 Typical Allan deviation for a clock with different types of noise. PM = phase
modulation, FM = frequency modulation, RW = random walk. [12]
The deviation from pure sinusoidal wave can be viewed either in the frequency domain or
the time domain. In the time domain, the time fluctuating fractional frequencies y are the
central quantities:
y (t )=
ν (t )−ν 0
1 d ϕ (t )
ν0 =
2 πν 0 dt
(4.1)
where ν(t) is the instantaneous frequency and φ(t) the instantaneous phase. It is not
practical to use the standard deviation to specify frequency deviations, because this will
get larger with increasing sample number. This is because the deviations are calculated
relative to the average, which is not stationary for most kinds of noise. There are several
14
ways to describe the variance of an oscillator, each of which is particularly useful for certain
types of analysis. The most widely used variance however, is the Allan variance. Here, the
differences are determined relative to subsequent measurements, instead of relative to
some average. For N samples, it is calculated as:
σ 2y (τ )=
N−1
1
∑ ̄y i+1− ̄y i
2(N −1) i =1
(
2
)
(4.2)
where ̄y i is the ith fractional frequency value, averaged over many equal time intervals τ.
Corresponding to the standard deviation, the square root of σy2(τ) is called the Allan
deviation. An example of the difference between the standard deviation and the Allan
deviation for increasing N can be seen in figure 4.3. If only white noise is present, the
deviations are equal.
Figure 4.3. Standard deviation (blue curve) and Allan deviation (red curve) as a function of
the number of samples, for an oscillator with flicker noise.
In the frequency domain, the instability is described by a power spectral density based on
Fourier analysis. The different types of noise can then be modelled by a law of the form:
S y (f )=h( α ) f
(4.3)
α
where Sy(f) is the spectral power density of y, f the Fourier frequency and α an integer
between -4 and 0 for the most common types of noise. This can also be translated into a τ
dependence in the Allan variance, as can be seen in figure 4.2. If only white noise is
present, the Allan deviation show a t -½ dependence, which is often stated empathically in
the σy(τ) value. [12]
15
The results of stability analysis can be significantly altered by dead time: the time in
between measurements, in which no data is acquired. This is specified by the dead time
ratio: r = T/τ where T is the time between the (starting point of the) measurements, as
pictured in figure 4.4.
Figure 4.4. Dead time in frequency stability measurement. [12]
Dead time introduces a bias in the resulting variance, as can be seen from figure 4.4. When
the nature of the present noise is known, it is possible to mathematically correct for dead
time. Most precision measurement techniques however, circumvent this problem by
measuring with zero dead time. [12]
4.3 Comparison and synchronisation
To measure the frequency drifts of a primary standard, it can be compared to a reference
standard that is very stable over the time interval under consideration. It is important to
keep in mind that the drift may reverse sign at some time, so measurements at different
times and different intervals are needed. [2][3] Quartz oscillators provide good references
for short time intervals and hydrogen masers are generally used for time intervals of
several days. For even longer averaging periods, caesium beam clocks can be used. [12]
When a maser is referenced to another maser, it is important that the devices are isolated
from each other. Else, resonance effects will lock both masers to the same frequency and
the difference measured is of course zero. [3]
A method to measure a clock's stability without the need for a reference standard has been
developed by Camparo et al in 2009. The clock output is interferometically compared with
a delayed copy of the signal. It is not particularly accurate, but it may be advantageous in
certain applications. An optical variant has been suggested by the authors. [15]
Another approach is to monitor various clock parameters that are responsible for drift, but
this is only limitary. It is never possible to reconstruct the complete error analysis of the
output frequency based on these indicators. [15]
16
It is difficult to extract the systematic error of a frequency standard, because no reference
standard is perfect. By comparing three or more copies of the same clocks, it is possible to
determine the noise performance of each of those clocks separately. [3][5]
A very sensitive apparatus like an atomic clock cannot readily be transported over long
distances. [16] To link two or more clocks in different laboratories, signals can be
transferred via satellites. This applies to both frequency comparisons and time scale
differences. Either the Global Positioning System (GPS) is used or geostationary
telecommunication satellites in a method called two-way satellite time and frequency
transfer (TWSTFT). [17]
GPS satellites contain a reference rubidium or caesium clock, synchronised to each other
and to the International Atomic Time on Earth. They emit data on two different
frequencies. Disturbances due to the atmosphere or ionospheric refraction can be traced
by comparing the time difference between to two signal frequencies. If only one frequency
is received, a model may be used to calculate the disturbances. [2]
In the common view method, the two clocks to be compared or synchronised receive a
common signal from the same satellite. This is only possible if both clocks are in range of
one satellite. To compare time scales, the delays of both receivers is calibrated. [2] An
advantage of this method is that it reduces the errors that are common for both clocks, like
errors of the satellite reference itself, and most of the atmospheric errors. [17] The positions
of the receivers must be known in the same coordinate system as the satellite. [2]
In the all-view method, the signals from all satellites within sight are averaged. Multichannel GPS receivers are then necessary, synchronised with a common time scale. The
International GPS Service (IGS) provides such a time scale The main errors arise from
uncertainties in the receiver, both inherent and due to environmental fluctuations.
[17]
When using TWSTFT, it is no longer necessary to have exact knowledge of the coordinates
of the clocks under consideration, nor do atmospheric delays play a role. In this method, a
signal from both clocks is sent to the other via a geostationary satellite. Delays cancel
because of the simultaneity and equality of the propagation paths. Corrections are made
for the residual movement of the satellite relative to the Earth. For long averaging time, the
stability of this method decreases due to noise that is unexplained to date. [17]
For averaging times of one day, provided the stabilities of the clocks allow for such a time
interval, fractional uncertainties of about 10 -15 are reached when satellites are used to
compare clocks. However, for optical clocks, a more precise comparison is recommended.
After regional tests proved promising, in 2012 a long-distance double optical fibre link was
established stable enough to compare optical clocks. It is based on an optical carrier wave
from a continuous wave laser, which provides sufficient resolution and is particularly
suitable for transmission over long distances. The Allan deviation of the fibre link is shown
below in figure 4.5, together with the stability of modern optical clocks.
Signals between two clocks are sent through two independent fibres in both directions. A
beat note is measured between the frequency of each clock with the received signal of the
other clock. It is thus a two-way time and frequency transfer with a direct connection.
Phase noise from environmental influences in the fibre is actively compensated. The fibre is
17
treated as the long arm of a Michelson interferometer with a partial reflector at the remote
end. Part of the light is reflected and compared with a reference signal. The frequency sent
into the fibre is adjusted according to the measured phase fluctuations. Because of the
long length, fibre amplifiers have been installed. They are bidirectional to establish equal
path lengths for both directions. [16]
figure 4.5. Allan deviation of the long-distance optical fibre link established by Predehl et al,
compared with satellite links, together with the typical stability of modern optical clocks. [16]
Self-comparison can be used to characterise the short-term stability of a clock. It is based
on a comparison of two independent frequency locks that are operated alternately. [1]
18
5
Active atomic clocks
The ammonia maser
The first prototype atomic clock ever built, actually a molecular clock, was an ammonia
maser. [4] The transition has a particular strong coupling to external fields and is therefore
easy to exploit. [3] In the ground state, the position of the nitrogen atom can be described
to be a linear combination of the positions on either side of the H 3 plane, with their
respective amplitude oscillating in time. There are two stationary states in which the phases
of the amplitudes have the same frequency: one symmetric and the other antisymmetric. A
15
NH3 maser is based on a transition between these two stationary states, with a frequency
of 22.8 GHz. [18][19]
Several important inventions improved the performance of the ammonia maser, that have
been applied to modern clocks afterwards. State selection, by means of an electrostatic
field, increased the signal to noise ratio. Line broadening and frequency shift due to
collisions was reduced by using a beam, formed in low pressure channels. Lastly, the first
order Doppler effect was eliminated by using a second beam in opposite direction. [19][20]
The short-term stability of the double-beam ammonia maser reached 2·10 -12, for averaging
times of 0.2 seconds, and the long-term stability was estimated to be of the order of 10 -12
as well. Limiting factors are strong cavity pulling, collisions with other molecules and with
the wall, and instabilities in the operation parameters. [3][20] During the sixties, further
development ceased quickly, as better clock designs became apparent. [4]
Alkali masers
Masers based on other atoms work similar to the ammonia maser. Alkali vapour frequency
standards were researched in the sixties. Rubidium was preferred because of practical
reasons. [2] The rubidium maser is based on a hyperfine transition of the 2S½ ground state
of 87Rb, with F = 2 and F = 1. The transition frequency is 6.8 GHz. Population inversion is
obtained by using an 87Rb discharge lamp with an 85Rb filter to pump atoms from the F = 1
level into the P state, followed by relaxation back to the ground state, mainly through
collisions with a buffer gas. The F = 1 level will be depopulated with respect to the F = 2
level due to this absorption-relaxation cycle. [2][3][19]
Threshold for oscillation in a rubidium maser is much higher than for ammonia, because
the transition involves a magnetic interaction instead of the much stronger electric dipole
interaction in ammonia. A high pumping power and high quality cavity are therefore
necessary. [3]
The applied magnetic field to remove the degeneracy of the ground state sublevels causes
a second order Zeeman effect. The buffer gas diminishes the first order Doppler
broadening by decreasing the mean free path of the rubidium atoms, and using a specific
mixture of gases, an optimum temperature coefficient can be reached. However, it also
causes an additional collisional shift. [3][19] Another important shift in the resonance
frequency is the light shift, caused by the presence of the pumping light. Not only does
19
this shift the output frequency, it also transfers the instabilities of the pump source to the
output signal. Since high powers are needed to achieve sufficient population inversion in
rubidium, this problem is severe in rubidium masers. However, the so-called double bulb
design circumvents light shifts by using separate regions for pumping and interrogation.
Another approach is to use pulsed optical pumping, also called the POP technique, where
the pumping and interrogation phases are separated in time instead of space. [2][3]
The stability of a POP rubidium maser at Selex Galileo in Rome, Italy, is in the order of 10 -12,
at averaging times of 1 second, and 10 -15 for averaging times of 105 seconds. Instabilities
arise mainly from thermal noise, the buffer gas, and collisions with the wall. [2][3]
An 85Rb maser is designed comparable to the 87Rb maser, with all isotopes interchanged. Its
frequency is somewhat lower with 3.0 GHz. Experimental difficulties made this variant
unsuitable for further research. [19]
A proposal to build a maser based on 133Cs has apparently never been realised. [19]
Hydrogen masers
Atomic hydrogen is obtained by gas discharge of molecular hydrogen. State selection is
obtained using a multi pole magnet. The atoms are then focused into a storage bulb
connected to a resonant cavity. An automatic tuning device ensures long-term stability of
the cavity. The bulb is magnetically shielded from environmental disturbances. Because the
atoms reside in the bulb for relative long periods, the frequency peak is very narrow. A
vacuum is present to decrease collisional broadening. The first order Doppler effect is
negligible because of the Dicke effect, occurring for atoms confined to a small cell. All the
above factors provide the excellent stability of hydrogen masers. [2][21]
Figure 5.1. Allan deviation for a hydrogen maser at the VNIIFTRI. [22]
20
Active hydrogen maser clocks operate at the hyperfine separation of the F = 0 and F = 1
(mF = 0) levels of the ground state. The transition frequency for an undisturbed hydrogen
atom is 1.4 GHz. The relative error is 2·10-12. [2][23] The accuracy of active hydrogen maser
clocks is moderate because of the uncertainties in the cavity properties and the atoms
colliding with the walls of the storage bulb. [3][2]
An example of the Allan deviation as a function of the averaging time is given in figure 5.1.
It is obtained from a hydrogen maser at the Scientific Research Institute for PhysicalEngineering and Radiotechnical Metrology (VNIIFTRI) in Russia in 2005. [22]
A minimum Allan deviation of σy = 1·10-17 is reported by Ashby et al, for averaging times of
1 day. [24] For medium-term averaging times, between 30 and 200 days, Allen deviations
from 10-16 to 10-15 can be obtained as measured by the National Institute of Standards and
Technology (NIST). [25]
The long-term stability of several hydrogen masers has been measured and reported by
the NIST in 2010. Frequency drifts in the order of 10 -16 per day are common for hydrogen
masers. These drifts are often not linear. At the NIST, a measurement of the fractional
frequency drift hydrogen maser clocks is made over a period of about 8.5 years. The results
are shown in figure 5.2. Environmental corrections cause small frequency offsets. The data
are fit to the NIST-F1 caesium fountain clock using the linear least mean squares method.
[24][25]
Figure 5.2. Long-term fractional frequency of five different hydrogen maser clocks at the
NIST. The linear least mean squares fits are shown as a solid black line. [25]
21
A plot of the typical variation in the drift rate over time is show in figure 5.3 for one of the
hydrogen masers at NIST. It is calculated that time scale errors due to frequency drift will
be in the order of a few nanoseconds, if the masers are monitored every month. [25]
Frequency drifts in hydrogen maser clocks are caused by variations in the operating
parameters and ageing of parts of the clock system. Stabilising these parameters will
increase the stability of the clock output. However, maintenance itself also infringes the
performance of the clock, if not carried out with meticulous care. [23][24][25]
Short-term stabilities can be increased by improving the clock design. For example in the
cavity tuning device [22] or the coating inside the storage bulb [26]. There are, however, no
recent developments that significantly improve active hydrogen maser clocks and the basic
design has been the same since the eighties. [22] In 2014, Boyko and Aleynikov suggested
using different magnetic sorting systems to improve the Allan deviation by an order of
magnitude for short averaging times. This has, however, not been experimentally tested.
[27]
Figure 5.3. Drift rate as a function of time for one of the hydrogen masers at the NIST. [25]
Cryogenic hydrogen masers
Hydrogen masers operating at very low temperatures have been predicted to reach even
better stabilities, especially at short-term. These so-called cryogenic masers benefit from
lower thermal noise, a decrease of collisional broadening, and smaller sensitivity to thermal
22
perturbations Stabilities in the order of σy = 10-18 for averaging times of one hour where
predicted by theoretical considerations. However, at low temperatures, fluctuations in the
atomic density have a complicated non linear influence on the frequency due to spinexchange interactions. This greatly reduces the stability of these clocks. [28]
The increasing wall shift at low temperatures can be balanced by the decreasing vapour
shift, such that at a specific temperature (near 0.5 K), the sum of these is independent of
temperature to first order. Temperature control is therefore crucial. Another important
factor is the surface coating of the storage bulb. To prevent the hydrogen atoms from
binding to the wall, a layer of inert gases is applied to the wall. This layer has to be thick
enough to cover the possible impurities in the wall material, but at the same time it has to
be smooth and uniform in thickness to prevent additional relaxation. This hampers the
performance of cryogenic masers considerably. [29]
Because of these limitations, measured stabilities do not exceed those of the best roomtemperature hydrogen masers. Figure 5.4 shows a comparison of their respective Allan
deviations, obtained at the Smithsonian Institution Astrophysical Observatory in
Cambridge, US-MA. [30]
Figure 5.4. Allan deviations of cryogenic and room-temperature hydrogen masers. [30]
Lasers
Amplified stimulated emission is of course not restricted to the microwave region. The
optical equivalent is the laser. However, lasers have never been stable or accurate enough
to function as an active frequency standard. [22] Only very recently, lasers that can
compete with microwave frequency standards have been realised. The most stable lasers at
this moment reach stabilities of order 10 -16 for averaging times of 1 to 1000 seconds.
However, the long-term stability lags behind. Frequency drifts are at best in the order of
23
10-19 per second. [31][32][32]
The stability can be increased by installing electronic or optical feedback systems using for
example electro-optical modulators, accousto-optical modulators or piezoelectric
transducers, to alter the frequency as to match the cavity resonance. The cavity itself can
be stabilised by decreasing its sensitivity to temperature variations and environmental
perturbations, as well as mechanical vibration. One strategy is to isolate the cavity, by
immersing it in vacuum, enclosing it in vibrational insensitive and thermally isolating
container, or mounting it on a support capable of damping mechanical motion. [34]
Another way is to use materials that have appropriate properties, like low temperature
sensitivity. Both Ultra-Low Expansion (ULE) ceramic glass and monocrystalline silicon have
zero first-order thermal expension coefficient at a specific temperature and can be used to
fabricate the spacer between the cavity mirrors. The latter has a better intrinsic quality
factor than the former when used at their respective zero crossing. It is therefore very
instensive to vibrational noise, and it does not show ageing. It also has a superiour thermal
conductivity, which contribute to temperature homogeneity. The thermal noise limit of
cavities based on this material is limited by the properties of the optical coating.
Suggestions to improve this limit are to use microstructured gratings or III/V materials as
coating materials, or by using longer spacers. [31]
Figure 5.5. Allan deviation of several kinds of stabilised lasers, compared to the quantum
noise limit for a Hg+ frequency standard. CORE =CO2 lasers locked to OsO4. [31]
A completely different method is to operate a laser in the so-called bad cavity regime. The
cavity loss rate is then larger than the gain bandwith. Cavity length noise is then
suppressed in return for a stronger cavity pulling effect, of which the latter is much easier
to characterise. However, this proposal has not been put to practise yet. [32][33]
24
6
Passive atomic clocks
6.1
Microwave regime
Just as in masers, state selection or pumping is needed to be able to passively measure
microwave transitions. In the microwave range, spontaneous transitions back to the ground
state have a low probability, and since the probabilities of absorption and stimulated
emission are inherently equal, artificial state population difference needs to be created to
be able to measure absorption from an external resonance field. [3]
Atomic beam clocks
The first passive clock design somewhat resembled the maser design, based on a beam of
atoms. A simplified overview of the first caesium beam clock using magnetic deflection is
shown in figure 6.1.
Caesium was used because it is relatively heavy, which means it travels slower and has a
longer interaction time when moving through an electromagnetic field, and because it has
a higher frequency then other microwave atomic oscillators, which provides better
accuracy. The transition used in caesium clocks is the hyperfine transition in the ground
state between F = 3 and F = 4 (mF = 0). The resonance frequency is defined to be exactly
9,129,631,770 Hz. [4]
Figure 6.1. Simplified overview of the original caesium beam clock. [4]
In the original design, a beam of caesium atoms emerges from an oven and state selection
is achieved by means of a magnetic field. A quartz-based frequency synthesizer provides a
25
microwave field, tuned to match the transition frequency. A second magnetic field directs
atoms that changed state toward a detector. Based on the strength of the signal, the servo
feedback adjusts the quartz oscillator. [19][4][2]
A constant magnetic field is present to separate the hyperfine states. The field needs to be
higher than for hydrogen. The second-order dependence of the frequency on the magnetic
field is thus much more sensitive to variations. Accordingly, the apparatus needs to be
shielded very carefully. [19]
The so-called separate oscillatory field method or Ramsey's method replaced the single,
long microwave pulse by two short pulses with a fixed mutual phase relation, on different
places along the beam path. The width of the output frequency peak is still determined by
the time the atoms need to cover the whole cavity length, but line broadening mechanisms
such as the first-order Doppler effect are extinguished. Furthermore, the method decreases
sensitivity in the output frequency to inhomogeneity effects of the static magnetic field
and fluctuations in the microwave field. [4][19][35] It does, however, cause an additional
uncertainty, called end-to-end cavity phase bias. This bias arises from the difference in
phase of the microwave radiation in the two excitation regions. The value can be
determined by comparing results with the beam direction reversed, but it still adds a
significant uncertainty. [35]
The second-order Doppler effect also plays an important role in the accuracy of the
caesium beam clock. Calculations to determine the shift are based on information about
the velocity distribution of the atoms. [19][35] The frequency synthesiser adds an
additional source of uncertainty. The microwave field itself causes a frequency shift, so
variations in the applied frequency transfer to variations in the output frequency. The
amplitude should also be very stable, because the light shift is power-dependent. [19]
Figure 6.2. Allan deviation as a function of time for the NIST-7 caesium beam clock.
New caesium beam clocks use optical pumping instead of magnetic state selection. This
increases the number of atoms available for transition and thus improves the signal. An
example of this type is built at the NIST and evaluated in 2001. The minimum accuracy was
26
found to be 4·10-15. Its stability is represented in figure 6.2. [35] The short-term stability is
determined by the quartz oscillator. For sampling periods of a few seconds to a day, the
stability mainly depends on the shot noise in the detection. The long-term stability is
determined by variations in the frequency shifts and ageing. [2]
The same design has been considered with other atoms. Thallium-205, providing a
frequency of 21.3 GHz, has the advantages that a small magnetic field is sufficient for the
separation of the F = 0 and F = 1 levels (mF = 0) of the 2P½ ground state. The second-order
frequency dependence on the magnetic field is then of course also small. Difficulties arose
because magnetic deflection and detection of thallium atoms is difficult, and an oven to
create a thallium beam has to be operated at very high temperatures. Since the accuracy
and stability is subject to the same constraints as the caesium beam clocks, there was no
reason to develop a similar, but technically more challenging clock with thallium. [19]
Silver atoms show similar problems for application in an atomic beam clock. Additionally,
the low resonance frequency provide a poor quality factor. Both 107Ag and 109Ag have a
suitable hyperfine transition in the ground state, with frequencies of respectively 1.7 and
2.0 GHz. [19]
The impossibility to detect hydrogen atoms with sufficient efficiency also hinder the use of
hydrogen in an atomic beam clock. Another disadvantage is that the velocities of the
atoms are very large. In a variant of the hydrogen beam clock, the interaction time of the
hydrogen atoms with the microwave field is increased by sending the atoms through a
storage bulb, in which the atoms reside for some time before continuing through a small
hole on the other side. Adjusting the temperature to diminish the wall shift would be easier
and cavity pulling would be negligible. However, the impossibility to detect atomic
hydrogen with sufficient efficiency hampered the realisation of a hydrogen beam clock.
[19]
Rubidium beam clocks not only use optical pumping to excite the atoms to the upper level,
but the detection is also carried out using a vapour lamp, measuring the amount of
absorbance after passing through the microwave field. To increase the signal, the gauge
light crosses the atomic beam several times. Instabilities in the gauge light add to the
instability of this type of clock. This makes rubidium beam clocks subordinate to the
caesium variant. [3][19] However, recent developments using lasers as more stable
pumping or detecting source has renewed interest in rubidium beam clocks. Results show
a somewhat improved short-term stability, but the frequency drift of the lasers added to
instability at long-term. The rubidium beam clocks have thus far not outperformed the
caesium variant. [36][37]
Magnesium beam frequency standards have been built with an accuracy of 10 -12 and a
short-term stability of 10-11τ-½. Although the alkaline earth metals do not show a hyperfine
structure in the ground state, there are some suitable transitions between excited state
levels, such as the 3P1 – 3P0 transition with a frequency of 601 GHz. Efficient pumping,
however, is complicated, and the potential of this type of clock is limited. [2][38]
27
Gas cell clocks
The basic design of rubidium gas cell clocks is almost the same as for rubidium masers.
Optically pumped rubidium atoms enter a cavity tuned to resonance with the transition
frequency, but without reaching threshold for masing. A photodetector measures the
amount of absorption of resonant microwave light passing through the cavity, comparable
to the interrogation in rubidium beam clocks. The signal is used to tune the frequency
synthesiser.
Compared to the rubidium maser, the constraints to the cavity are less severe and there is
no need for a particular high pumping power. The light shift is therefore smaller.
Additionally, there is more freedom to use a specific mixture of buffer gases to obtain an
optimal temperature coefficient and pressure shift. [2][19] Pulsed optical pumping (POP)
techniques can also be applied to rubidium gas cell clocks, reducing the light shift to
negligible levels. [39] The dependency on the operation parameters make this type of
clocks secondary standards.
The development of lasers as both pumping and interrogation source has improved shortterm stability. However, long-term frequency shifts are present due to the light shift and
ageing effects, comparable to those encountered in maser clocks. Added thereto is the
frequency drift of the lasers. [2]
The small size of rubidium cell clocks make these type of clocks very useful. Size reduction,
however, further impairs long-term stability because of the increased collisional shift. Effort
has been made to develop a coating for the inside surface of the cell to prevent rubidium
atoms from reacting with it and to remove the need for a buffer gas, but haven't been
applied thus far. The advantage to discard the buffer gas is assumed to be small, because
collisional frequency shifts still will still be present. [3][40]
Recent rubidium cell clocks have shown a short-term stability of order 10 -13τ-½, with a
minimum Allan deviation of 10-14 for averaging times of about 2 minutes. Long-term
frequency drifts are below 10 -15 per day. Suggestions that have been put forward to
improve the stability are for example to operate under vacuum to decrease environmental
effects, optimisation of cell and cavity sizes to decrease geometric (inhomogeneity) effects,
and cavities of different materials to increase thermal en mechanical stabilities. [39][41]
In general, the same limits on accuracy and stability apply to gas cell clocks based on
atoms other than rubidium. For caesium, it was initially quite difficult to obtain a sufficient
population difference. In the first versions, a caesium lamp with an interference filter and
circular polariser was used. [19] Nowadays, optical pumping with lasers is available. Double
resonance is again obtained by adding a buffer gas. This leads to a reduced linewidth, but
it also introduces a frequency shift, which renders it slightly inferior to the caesium beam
clock. The performance of a gas cell clock is comparable whether caesium or rubidium is
used. For caesium, short-term stabilities of the order of 10−13τ−½ with a minimum Allan
deviation of order 10-14 at averaging times of 100 to 1000 seconds are reported. [42][43]
28
Atomic fountain clocks
Caesium fountain clocks where invented to increase the interaction time by building an
atomic beam clock in vertical direction. The atoms are fired in upward direction, slow down
and reverse under the influence of gravity. Only one microwave field is necessary, through
which the atoms pass twice. This removes the end-to-end phase bias, although a spatial
variation in the trajectories of the atoms leave a small residual first-order Doppler effect.
However, in this initial design, the atoms where scattered out of the beam by mutual
collisions with atoms of a different velocity and no signal was detected. Laser cooling
provided the solution. Caesium fountain clocks generally make use of atomic molasses, but
several laser cooling techniques exist. By tuning the lasers, a ball of atoms can be launched
at a specific velocity. The low temperature ensures small velocity fluctuations.
The fountain clock works in cyclic mode. After optically pumping the atoms, the pumping
and cooling lasers are screened off to avoid a light shift. Detection after crossing the cavity
is done using a light source tuned to the F = 4 to F = 5 transition. A schematic overview of
an atomic fountain clock is shown in figure 6.3.
The low velocities of the atoms render the second-order Doppler effect very small. The
uncertainty in this shift is also very small, because the velocities are known very precisely. A
density shift remains due to the collisions of caesium atoms with each other. The so-called
multiple ball toss scheme, or juggling method, reduces this by launching several batches of
atoms in quick succession, with different initial velocities such that they arrive at the
detection point at the same time. [2][4]
In cold atom fountain clocks, device limitations are so small that fundamental limits are
significant. A blackbody shift is caused by the relatively warm outer surface of the vacuum
cavity emitting radiation. This can be corrected for quite accurately, but cryogenic vacuum
systems have also been developed to decrease this shift. [44] A gravitational red shift
occurs because the atoms move up and down in Earth's gravitational potential. Accurate
corrections for this are possible. [2][4]
The accuracy of modern caesium fountain clocks is of order 10 -16. The main sources of
uncertainty arise from the collisional shift and microwave amplitude fluctuations. [45][46]
[44]
Slow atoms show a high resonance quality factor, implying that high stabilities can be
obtained. The instability of the oscillator is a limiting factor, but this influence is reduced by
replacing the original quartz oscillator by a cooled sapphire ring, that provides enhanced
spectral purity. [2] The cyclic mode of operation introduces dead time in the frequency
measurement. Techniques to decrease the cycle duration, such as a faster loading time,
thus increase stability. [45] The short-term stability of modern caesium fountain clocks is in
the order of 10-13τ−½ [45][46][44] The Allan deviation for a modern caesium fountain clock
at the NIST is shown in figure 6.4. [44]
When using rubidium instead of caesium in an atomic fountain clock, the collisional shift is
much lower. Although in turn the quadratic Zeeman shift is larger, rubidium fountain clocks
that have been built show an accuracy and stability comparable to the caesium variants,
sometimes even more accurate due to the smaller uncertainty in the collisional shift. [48]
29
[48] A dual fountain clock has even been built, in which both caesium and rubidium atoms
are used simultaneously It preserves the accuracy and stability of the clocks containing
only one species. It can be used to compare both species within exactly the same
environment. [49]
Figure 6.3. An atomic fountain clock. The lasers designated with M provide the optical
molasses, those with D are the detecting lasers. [2]
30
Figure 6.4. Allan deviation of a second generation caesium fountain clock at the NIST. [44]
Trapped ion clocks
Ions can be confined to a limited volume of space using electromagnetic fields. In a Paul
trap, a high frequency alternating electric field is used. Other kinds of traps are not
practically applicable in atomic clocks, because of the presence of a disturbing magnetic
field. Using radiation to form a trap removes the wall shift that exists in systems where the
atoms are confined in a cell or cavity, while the shift from the presence of the field itself
can be accurately calculated. Trapped ion clocks use either a single ion or a confined cloud.
A single trapped ion is free from interactions with other atoms, but offers a lower stability
than a clock based on a cloud of ions. [50]
There is a limit to the density of the ion cloud due to mutual repulsions, which
automatically limits the number of collisions of the ions. A lower number of ions, however,
also decreases the signal-to-noise ratio. The spread in kinetic energy of the ions is lower in
case of a lower density, and therefore the second-order Doppler shift is also lower. For
higher densities, the so-called collisional cooling method introduces a low pressure helium
gas to reduce the kinetic energies without introducing a significant collisional shift. In a
linear Paul trap, the ions spread out of a cylindrical region, and the spread in kinetic energy
is thus smaller. The linear trap also shows less sensitivity to fluctuations in the applied
electric fields. [2][3]
Several ions have suitable microwave transitions, but the heavy mercury ion provides a
small second-order Doppler effect and is thus preferred. Ionisation is achieved by treating
the atoms with an electron beam or ionizing radiation. When the outermost electron is
removed, 199Hg has an electronic structure very similar to the alkali metals. The transition
that is used in atomic clocks is the F = 0 to F = 1 (mF = 0) transition in the ground state
with a frequency of 40.5 GHz. A population difference is obtained using either a laser
31
source or a 202Hg+ discharge lamp with a similar process as that used to pump rubidium
atoms. One of the emission lines of 202Hg+ lies very close to the transition wavelength from
the ground state F = 1 level to the P state. From the P state, spontaneous decay occurs to
both ground state levels, but the F = 1 level atoms will be re-excited such as to create an
accumulation of atoms in the F = 0 level. Although laser pumping is more efficient and
thus provides a better signal-to-noise ratio, 202Hg+ lamps are still used when the setup is
preferred to be compact. [3]
Trapped ion clocks operate in cyclic mode. First, the trap is reloaded to make sure every
cycle starts with the same number of atoms. After optical pumping, the microwave field is
applied in two successive pulses, thus applying Ramsey's method in the time domain. The
amount of ions that have undergone the transition is probed using the pumping laser or
220
Hg lamp and measuring the fluorescence response. If the microwave field is tuned to
resonance, a maximum number of ions have been excited into the F = 1 level and can thus
respond to the pumping laser, after which they relax back to the ground state emitting a
signal. This method of measuring the population of a state indirectly is described as the
double resonance method, or electron shelving if there is only a single ion present. [2]
The main frequency shifts arise from the quadratic Zeeman effect, the second-order
Doppler effect and collisions. By laser cooling the ions, the second-order Doppler effect
can be reduced to negligible levels. Cryogenic techniques reduce the collisional shift. [2]
Electric quadrupole interactions from a residual electric field gradient can be eliminated to
first order by averaging frequencies measured over any three mutually orthogonal. It is
also possible to average over different Zeeman levels and extrapolate to find the shift. [8]
Instabilities like those arising from fluctuations in the number of ions, the pressure, or the
applied electromagnetic fields can be adequately controlled. [2] These types of clocks show
therefore a very good long-term stability, with frequency drifts of order 10 -17 per day. [51]
Short-term stabilities show an Allan deviation of order 10 -14τ–½. [52] The accuracy of the
trapped mercury ion cloud clock is about 10-15. [53]
Trapped ion clocks have also been investigated using 9Be+, 113Cd+, 137Ba+ and 171Yb+. All of
these species exploit a hyperfine transition in the ground state, except for beryllium, which
has a clock transition between two Zeeman levels of the F = 1 sublevel in the groundstate.
Ytterbium has achieved extra attention because the pumping light of 369.5 nm is easily
obtained using relatively cheap and compact available lasers. Otherwise, the same limits
apply for these elements and the best results have been achieved using mercury. [50]
In the case of a single trapped ion, a technique called electron shelving can be applied.
6.2
Optical regime
A higher frequency provides a more accurate atomic clock. However, until recently there
were no reliable frequency counters that could handle these high frequencies. The
invention of the optical frequency comb turned out to solve this problem. The output of a
mode-locked laser consists of equally spaced laser modes. If these span more than an
octave, it is possible to accurately determine the frequencies of all these modes. An optical
clock frequency can then be measured by comparing it with the nearest mode of the
32
frequency comb by taking a beat note. [54] All optical clocks that have been realised to
date are passive clocks with a cyclic mode of operation. They operate with frequencies of
400 to 1200 THz. [5] The development of the increasing accuracy of optical clocks is
compared with the development of microwave clocks in figure 6.5. These results may lead
to a new definition of the second on short notice. [55]
Figure 6.5. Development of the increasing accuracy of both microwave (blue squares) and
optical (red dots) clocks with trend lines. The green dots represent optical clocks with
estimated accuracy, because their frequency can only be determined with a less accurate
microwave reference. [55]
The first examples of lasers stabilised to an atomic transition where iodine and methane
secondary cell standards. These could use standard laser wavelengths, as tunable
continuous wave lasers where not available in the rising era of optical clocks. Iodine has
several suitable transitions and has a narrow bandwith because it is quite heavy. Methane
provided a quite stable clock with a maximum stability of 10-15τ-½ for averaging times of
100 s. Other molecules used for conventional laser wavelengths are N 2, CO and Te2. In
newer generation optical clocks, research is limited to atoms because there is no efficient
method to cool neutral molecules. [56]
Optical trapped ion clocks and quantum logic clocks
Ions that have been used in optical trapped ion clocks are 40Ca+, 88Sr+, 171Yb+, 199Hg+, 137Ba+,
138
Ba+, and 115In+. Ytterbium has shown the longest storage times, while indium is most
suitable for laser cooling. Both aluminium and indium show very low sensitivity to
frequency shifts induced by external fields. The used clock transition is the quadrupole
transition from the 2S½ ground state to one of the 2D excited states. For ytterbium, an
octopole transition to the 2F7/2 state is also suitable. The mentioned ions all provide
possibilities for cyclic pumping and optical cooling as described earlier in the paragraph on
trapped ion clocks. The excited states for these ions do have a quadrupole moment and a
quadrupole shift thus arises. For mercury-199 and ytterbium-171, the first-order Zeeman
33
effect can be suppressed by restricting the transitions to those with mF = 0 → mF' = 0.
Other shifts and restrictions apply as described in the former paragraph. [5]
Transitions that have a vanishing angular momentum do not show quadrupole shifts and
have usually small Zeeman shifts. However, these transitions have frequencies that are too
high for current laser technologies, or the transitions are too narrow for laser cooling. [5]
In ion traps, atoms without a suitable laser cooling transition can be cooled by interaction
with a different species in a process called sympathetic cooling. The other species is laser
cooled and the ion of interest is cooled along with it by mutual interaction. Detection is
based on a quantum logic technique, in which information about the clock ion is obtained
by interrogating the logic ion based on a common motional state. The constraints on the
ions suitable for this type of optical clock are loosened because the logic ion plays the
important role of cooling and interrogation. [5]
The choice of logic ion codetermines the systematic frequency shifts of a quantum logic
clock. The logic ion can be laser cooled during interrogation, lowering the second-order
Doppler shift but at the same time causing an additional Stark shift on the clock transition.
The cooling linewidth and the mass ratio between clock and logic ion determine which
ions are suitable as logic ion. For Al+, good results are achieved with Be+, Mg+, and Ca+.
The largest source of uncertainty for current Al + quantum logic clocks is in the relativistic
shift due to residual motion of the ions. Compensation is limited by the measurement and
controlling of the electric fields in the trap. Frequency shifts caused by external fields are
measured and corrected for very precisely in these clocks. Collisions with background gas
are rare in ion traps, but can be detected in the laser cooling fluorescence signal. Such
events can then be discarded. [5]
The most accurate optical ion trap clocks are based either on the octopole transition in a
single trapped Yb+ ion, or on an Al+ logic clock, both with fractional uncertainties in the
order of 10-18. [55]
Free-space optical standards
Frequency standards based on free calcium, magnesium and strontium have been
researched. However, Doppler shifts were very significant. The best results have been
achieved with laser cooled, ballistically expanding calcium. [5] These clocks are based on
clouds of cold neutral atoms that are allowed to expand freely under influence of gravity. It
reached a fractional uncertainty of order 10 -15. The greatest improvement arose from more
stable probe lasers, and from a reduction in the uncertainty of the laser cooling, which
allowed for a reduced Doppler effect uncertainty. Residual uncertainty is caused mainly by
collisions between the cold cloud and atoms impinging from the thermal beam atom
source, vibration in the apparatus and motion of the atoms. [57]
Optical lattice clocks
Using a magneto-optical trap or a standing-wave optical dipole trap, neutral atoms can
also be confined in space. The resonance is then almost free from Doppler and recoil
34
effects. Trapped neutral atoms have smaller interactions than ions, which makes it possible
to include more atoms in the trap, thus increasing the signal-to-noise ratio.
Trapping many neutral atoms in an optical lattice is achieved through the spatially
dependent Stark shift induced by a standing-wave electric field created by interfering
lasers. Near anti-nodes of the standing waves, the Stark shift acts as a harmonic potential,
confining the atoms to a subspace to prevent collisions. The radiative field used to create
the optical lattice induce a light shift, so this technique is only available for atoms which
obey light-shift cancelling condition that the polarizibility is equal for both upper and
lower state. This will happen for some wavelength, called the magical wavelength. The light
shift can thus be controlled with high precision by tuning the wavelength of the lattice
laser. At first, a collisional shift was present because several atoms where traps in each
potential well. [58]
The most accurate clock to date is an optical lattice clock with 87Sr atoms at the Joint
Institute for Laboratory Astrophysics (JILA) in Boulder, US-CO. Its fractional frequency
uncertainty is 2·10-18. The most significant contribution to its uncertainty are in the Stark
shifts induced by the blackbody radiation and by the lattice field. The uncertainty in the
black body radiation shift is reduced significantly by careful measurements of the
temperature inside the vacuum chamber and inhomogeneities thereof, and by measuring
additional decay from the upper state due to background radiation. [1]
The stability of optical lattice clocks is better than other types of clocks because of the high
optical frequency, long interaction time and large atom number. [58] The stability is limited
by frequency noise in the interrogation laser at short averaging times. Long-term drift can
be controlled to within order of 10 -18 after thousands of seconds averaging time. The
stability of the 87Sr optical lattice clock at the JILA is estimated to follow a 2.2·10-16τ-½ line,
as can be seen in figure 6.6.
Figure 6.6. Allan deviation of the 87Sr optical lattice clock at JILA. The black circles are
measurements, with 1σ error bars, the red line is a linear fit and the blue dashed line
represents the result of the formerly most stable clock. [1]
35
7
Future prospects
Progress in the design and implementation of atomic clocks has reached the point that
fundamental limits are in sight. It is to be expected that uncertainties of technical nature
will continue to decrease, but the Heisenberg uncertainty principle and the gravitational
red shift impose a limit on the performance of clocks. [5][1][59]
7.1 Fundamental limits
Improving the resonance Q-factor
In highly technologically advanced clocks, quantum projection noise is the dominant noise
limit. The number of atoms in the measurement in can be maintained constant by trapping
them, while statistical fluctuations in the detection are nearly diminished by the electron
shelving technique. [60] The quantum projection noise limited Allan deviation is given by:
σ y (τ )=C Δν
ν
√
tT
(7.1)
Nτ
where C is a constant of order unity, tT the transit time, and N the number of atoms. Note
that the inverse of the resonance quality factor is in this equation. [59]
To increase performance, higher frequencies can be used as already proven in optical clock
development. Nuclear transitions would provide frequencies in the X-ray or γ-ray regime.
These transitions are highly insensitive to external electromagnetic fields and a large
number of atoms can be used in the solid state. The 2 PHz transition between the ground
state of 229Th and the 229mTh isomer has been studied in this context. A single ion in a trap,
or a crystal lightly doped with thorium have been mentioned as the best options for a
frequency standard. [61]
Traditional interferometric techniques and crystal diffraction can be used to measure X-ray
frequencies, and dual X-ray-optical interferometry has been suggested as the best method
to measure X-ray and gamma frequencies. Gamma ray experiments like the electronpositron annihilation process turned out to be quite inaccurate because of centre-of-mass
motion. In general, the necessary fast and efficient techniques are not sufficiently
developed yet to create an X-ray or gamma frequency standard. [56] The high frequency
radiation add extra difficulties to the experimental setup because of its reactivity with air,
and optical instruments in this regime are not available or expensive. [61]
To narrow the linewidth, broadening factors should be decreased as much as possible. This
is mainly a technological issue, as described in detail in the previous chapters. Coherent
population control might be used to alter the upper state lifetime, hereby even decreasing
the natural linewidth broadening. [56]
There is, however, a fundamental limit to the attainable resonance Q-factor. Even if
technological abilities increase without limit, there is a fundamental limit to the
temperature that can be reached with laser cooling. [10] And even if a clock atom would be
36
cooled to absolute zero, quantum zero point motion still remains. An uncertainty in the
position of a clock ion, σx, in the gravitational potential imposes a fractional uncertainty in
the clock frequency by time dilation. The uncertainty in the momentum, σp, can be viewed
as random motion, causing a second order Doppler effect. The Heisenberg uncertainty
relation between position and momentum implies a fundamental uncertainty for frequency
standards. By altering the harmonic potential of the trap, the consequential values of σx
and σp can be varied. There is a minimum fractional frequency uncertainty as a function of
σx and σp. An approximate result for the maximum achievable Q-factor is given by equation
7.2:
( )
mc 3
Q≤
√3 ħ g
2
2
3
(7.2)
where m is the mass of the clock atom, c is the speed of light, ħ Planck's constant divided
by 2π and g the local gravitational acceleration. [59] The limit prevails even for increasing
number of clock atoms and averaging their frequencies, because the limit scales faster with
atom number. The order of magnitude of this limit to the quality factor is 10 21 to 1022,
depending on the species of atom used. [59]
It is possible to increase the stability of an atomic clock despite this limit, by increasing the
transit time or the number of atoms. Both approaches are, however, restricted by practical
drawbacks as described earlier in chapter 5 and 6.
Quantum entanglement
The ultimate quantum mechanical limit to the uncertainty in systems with uncorrelated
atoms is given by the Heisenberg relation in the limit of high atom number. Quantum
projection noise can be reduced by implementing squeezed states. So far, squeezed states
have been demonstrated for two ions in the radio frequency regime, but not for optical
frequencies. Another way to entangle the atoms is by creating so-called GHZ states, which
are superpositions of all atoms being in the upper state and all atoms being in the lower
state. Up to 14 entangled atoms have been demonstrated using this method, of which one
experiment focused on the optical Ca+ ion. Scaling experiments up to hundreds of ions
have been proposed. Systems with GHZ states are, however, more sensitive to laser phase
noise, decreasing the stability.
Entanglement could in theory also act as an alternative for averaging over different
transitions to create insensitivity to external fields, less sensitive for fluctuations and
inhomogeneities of the fields, but there are no schemes developed yet to produces these
designer atoms. [5] Technologically, this is quite a challenge. These methods seem only
promising for situations in which scaling of the atom number or interrogation time is
impossible. [5][55]
37
7.2 Suggested research
Progress has been made in developing techniques to defy the quantum limit to clock
performance, as described in the previous paragraph. However, the quantum limit is not in
sight yet. The best clock in the world now has a fractional frequency uncertainty of 10 -18,
while the quantum limit is a factor 10 3 to 104 lower. [59] On the other hand, comparing this
leap to the development of clock performance since the first atomic clock was built in
1948, it is not inconceivable that uncertainties of order 10 -21 can be reached in a few
decades, especially when looking at the trend of the previous few decades (see also figure
6.5). The realisation of nuclear clocks, [56][61] superradiant lasers [5] or other,
unforeseeable great inventions might speed up the development rate, just like the uprising
of optical clocks have done after invention of the frequency comb. [54]
Nuclear frequency standards could be a valuable tool for specific research of the constancy
of physical constants, and therewith contribute to the development of a grand unification
theory. It is predicted that nuclear transitions are far more sensitive to variations in the fine
structure constant, compared to atomic experiments. This is still quite controversial though,
and measurements are proposed to elucidate the issue. [61] However, as time-measuring
tools, nuclear clocks are not expected to become available in the near future. A frequency
standard based on a nuclear transition has not been realised yet, although research
towards ultrahigh-resolution X-ray spectroscopy is being conducted in e.g. the group of
Eikema at the VU University Amsterdam. [J. Koelemeij, personal communication, 21
October 2015] Techniques to efficiently create X-ray or γ-ray frequencies with sufficient
stability and tunability are not available yet, and specific necessary techniques for this type
of clock are not developed, nor is it completely clear what the challenges in building this
type of clock will be. [56] That has to become clear in the process of trial and error,
comparable to the development of the current clocks, although much of what has been
learned from building these clocks can be used in clever designs for new, nuclear clocks.
Improvements in the technical effectuation of the current state-of-the-art clocks is much
easier to achieve. The recent breakthroughs, after the first optical lattice clock in 2005 [58],
where not based on new approaches, but on refinement of existing procedures. Although
inherently new methods have been published, for example to measure the local
temperature inhomogeneities without disturbing the system [1], they were all based on the
familiar underlying concepts to decrease the uncertainties. This has, however, led to great
results, so further optimisation of current clocks may lead to significantly better performing
clocks on relatively short notice. This applies for example to further decreasing fluctuations
and inhomogeneities in temperature or external fields, decreasing background radiation,
stabilising the laser systems, and improving the optical lattices. [1]
Ways that lead to a considerable acceleration in clock performance development may
come from unexpected areas of research: not so much the engineering part, but a more
fundamental, quantum mechanical point of view. The question how to decrease
uncertainties in the existing systems, may be replaced by the question what systems
intrinsically have less uncertainty. Coherent techniques for example, have a lot of
unexplored potential. [56]
38
8
Conclusion
The frequency of atomic clocks is determined by an atomic transition. This provides a
stable and independent reference for time measurement. External influences, like a nonzero temperature or the presence of electromagnetic fields, can disturb this process by
either shifting or broadening the measured frequency. Fluctuations in parameters and
fundamental noise processes add to the uncertainty and instability of a clock.
Since the first prototype, a microwave ammonia maser, enormous progress in increasing
the accuracy and stability of atomic clocks has been made. In the microwave regime, the
active masers evolved into gas cell clocks, atomic beam clocks and fountain clocks, in
which an external oscillator is stabilised on the atomic transition with a servo feedback
system. Each new design resolved several important issues concerning uncertainties and
instabilities. For example, the first-order Doppler effect, light shift, thermal noise, sensitivity
to external fields and collisional shift where greatly reduced using various techniques.
Additionally, the transit time was significantly increased, leading to a better signal-to-noise
ratio. Each type has its own advantages and disadvantages, just like each atom or
molecule. The hydrogen maser still proves useful because of its great stability, while its
small size make rubidium cell clock invaluable. The caesium fountain clock provides the
definition of the second, although it may be replaced by the advanced accuracies of optical
clocks.
The optical counterpart of the maser has never been established as a frequency standard
because long-term frequency drift turned out to be a problem. However, passive optical
clocks have outperformed microwave clocks since the invention of the frequency comb,
that made it possible to measure the high frequencies of optical transitions.
Trapped ion clocks have been realised in the microwave regime, but became the regular
implementation of optical clocks. The optical lattice made it possible to confine many
neutral atoms without disturbing mutual interaction. A few free-space and gas cell optical
clocks have been built, but the absolute record in both accuracy and stability is held by a
87
Sr optical lattice clock. Its fractional frequency uncertainty is 2·10-18.
Eventually, quantum projection noise provides the fundamental limit to the resonance Qfactor, and thus to clock performance. These limits are estimated to be of order 10 21 to 1022.
Quantum entanglement may further stretch this limit. Possibilities in increasing clock
performance is by further refining current technologies and procedures to reduce
uncertainties, increasing the number of atoms and increasing the transit time. Higher
frequency clocks based on nuclear transitions have been proposed, which theoretically
could perform better. However, no nuclear frequency standard has been realised yet.
Although it might be very interesting to develop nuclear clocks, especially in the context of
specific applications in fundamental research, realisation of these clocks is still far away
because of technological challenges.
Clock performance will likely continue to increase steadily during the next few decades,
until quantum limits come in sight. Fundamentally different approaches will be needed to
either accelerate the rate of improvement, or ultimately stretch quantum limits.
39
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
[18]
[19]
Nicholson, T.L., Campbell, S.L., Hutson, R.B., et al, “Systematic evaluation of an atomic
clock at 2×10-18 total uncertainty.” Nature Communications, 2015, Vol. 6, 6896.
Audoin, C. and Guinot, B., The Measurement of Time. Time, Frequency and the Atomic
Clock; Cambridge University Press: Cambridge UK, 2001.
Major, F. G., The Quantum Beat. Principles and Applications of Atomic Clocks. 2nd ed.;
Springer: New York, 2007.
Lombardi, M. A., Heavner, T. P., Jefferts, S. R., “NIST Primary Frequency Standards and
the Realization of the SI Second.” NCSLI Measure J. Meas. Sci., 2007, Vol. 2, No. 4, 7389.
Ludlow, A.D., Boyd, M.M., and Ye, J., “Optical atomic clocks.” Rev. Mod. Phys., 2015, Vol.
87, No. 2, 637-701.
Griffiths, D., Introduction to Quantum Mechanics. 2nd ed.; Pearson Prentice Hall: Upper
Saddle River NJ, 2005.
Bransden, B. H. and Joachain, C. J., Physics of Atoms and Molecules. Longman Scientific
& Technical: New York, 1983.
Itano, W.M., “External-field shifts of the 199Hg+ optical frequency standard.” J. Res. Natl.
Inst. Stand. Technol., 2000, Vol. 105, No. 6, 829–837.
Bernath, P. F., Spectra of Atoms and Molecules. 2nd ed.; Oxford University Press: Oxford
UK, 2005.
Demtröder, W., Laser Spectroscopy. Basic Concepts and Instrumentation. 3rd ed,;
Springer: New York, 2003.
Barnet, S.M., “On the recoil and Doppler shifts.” Journal of Modern Optics., 2010, Vol.
57, No. 14–15, 1445–1447.
Riley, W. J., Handbook of Frequency Stability Analysis. NIST: Gaithersburg US-MD.,
2008.
Ferre-Pikal, E. S., Vig, J. R., et al, IEEE Standard Definitions of Physical Quantities for
Fundamental Frequency and Time Metrology – Random Instabilities. IEEE: New York
US-NY, 2009.
Galleani, L., A, “Tutorial on the two-state model of the atomic clock noise.” Metrologia,
2008, Vol. 45, No. 6, S175–S182.
Camparo, J., Chan, Y., Johnson, W., et al, “Autonomously measuring an atomic clock’s
Allan variance.” Proceedings from the 2009 Frequency Control Symposium, Joint with
the 22nd European Frequency and Time forum. IEEE: New York US-NY, 2009.
Predehl, K., Grosche, G., Raupach, S. M. F., et al, “A 920-kilometer optical fiber link for
frequency metrology at the 19th decimal place.” Science, 2012, Vol. 336, No. 6080, 441444.
Piester, D. and Schnatz, H., “Novel techniques for remote time and frequency
comparisons.” Special Issue / PTB-Mitteilungen, 2009, Vol. 199, No. 2, 33-44.
Feynman, R.P., Leighton, R.B., and Sands, M.L., The Feynman Lectures on Physics. Vol. 3:
Quantum Mechanics. Perseus Basis Books: New York US-NY, 2010.
Audoin, C., and Vanier, J., “Atomic frequency standards and clocks.” Journal of Physics
40
[20]
[21]
[22]
[23]
[24]
[25]
[26]
[27]
[28]
[29]
[30]
[31]
[32]
[33]
[34]
[35]
[36]
[37]
E: Scientific Instruments, 1976, Vol. 9, No. 9, 697-720.
De Prins, J., “NH3 double-beam maser as a frequency standard.” IRE Transactions on
Instrumentations, 1962, Vol. I-11, No. 3, 200-203.
Ramsey, N.F., “The atomic hydrogen maser.” American Scientist, 1968, Vol. 56, No. 4,
420-438.
Domnin, Y., Gaigerov, B., et al, “Fifty years of atomic time-keeping at VNIIFTRI.”
Metrologia, 2005, Vol. 42, No. 3, S55-S63.
Vasil'ev, V.I., “A physical model of the systematic variation of the frequency of the
output signal of hydrogen frequency and time standards.” Measurement Techniques,
2014, Vol. 57, No. 2, 160-165.
Ashby, N., Heavner, T.P., et al, “Testing local position invariance with four cesiumfountain primary frequency standards and four NIST hydrogen masers.” Phys. Rev.
Lett., 2007, Vol. 98, No. 7, 070802.
Parker, T.E., Jefferts, S.R., and Heavner, T.P., Medium-Term Frequency Stability of
Hydrogen Masers as Measured by a Cesium Fountain. NIST: Gaithersburg US-MD.,
2010.
Bernier, L.G., Busca, G., and Schweda, H., On the Line Q Degradation in Hydrogen
Masers. Observatiore Cantonel de Neuchâtel: Neuchâtel Switzerland, 1990.
Boyko, A.I., and Aleynikov, M.S., “An active hydrogen maser with enhanced short-term
stability.” Measurement Techniques, 2014, Vol. 56, No. 10, 1140-1145.
Hayden, M.E., Hürlimann, M.D., and Hardy, W.N., “Measurement of atomic-hydrogen
spin-exchange paramteers at 0.5 K using a cryogenic hydrogen maser. “ Phys. Rev. A:
At., Mol. Opt. Phys., 1996, Vol. 53, No. 3, 1589-1604.
McAllaster, D.R., Krupczak, J.J., et al, “Cryogenic hydrogen maser at 10 Kelvin.”
Proceedings from the 48th IEEE International Frequency Control Symposium. IEEE: New
York US-NY, 1994.
Walsworth, R.L., Investigation of Atomic Physics and Frequency Stability with a
Cryogenic Hydrogen Maser. Defense Technical Information Center: Fort Belvoir, US-VA,
1997.
Kessler, T., Hagemann, C., Grebing, C., et al, “A sub-40 mHz linewidth laser based on a
silicon single-crystal optical cavity.” Nature Photonics, 2012, Vol. 6, No. 10, 687-692.
Kuppens, S. J. M., Van Exter, M. P., and Woerdman, J. P., “Quantum-limited linewidth of
a bad-cavity laser.” Phys. Rev. Lett. 1994, Vol 72, No. 24, 3815-3818.
Chen, J., “Active Opical Clock.” Chinese Science Bulletin Vol. 2009, 54, No. 3, 348-352.
Young, B. C., Cruz, F. C., Itano, W.M., and Bergquist, J.C., "Visible lasers with subhertz
linewidths." Physical Review Letters, 1999, Vol. 82, No. 19, 3799.
Shirley, J.H., Lee, W.D., and Drullinger, R.E., “Accuracy evaluation of the primary
frequency standard NIST-7.” Metrologia, 2001, Vol. 38, No. 5, 427-458.
Wang, Y.H., Huang, J.Q., et al, “Improved rubidium atomic beam clock based on lamppumping and fluorescence-detection scheme.” J. Eur. Opt. Soc. - Rapid., 2011, Vol. 6,
No. 11005, 1-4.
Besedina, A., Gevorkyan, A., et al, “Preliminary results of investigation of the highstable rubidium atomic beam frequency standard with laser pumping/detection for
41
[38]
[39]
[40]
[41]
[42]
[43]
[45]
[46]
[44]
[48]
[48]
[49]
[50]
[51]
[52]
space application.” Proceedings from the 20th European Frequency and Time Forum.
EFTF: Besançon France, 2006.
Godone, A., and Novero, C., “The magnesium frequency standard.” Metrologia, 1993,
Vol. 30, No. 3, 163-181.
Micalizio, S., Godone, A., et al, “Medium-long term frequency stability of pulsed vapor
cell clocks.” IEEE Trans. Ultrason., Ferroelect., Freq. Control, 2010, Vol. 57, No. 7, 15241534.
Seltzer S. J. and Romalis, M.V., “High-temperature alkali vapor cells with antirelaxation
surface coatings.” J. Appl. Phys. 2009, Vol. 106, No. 11, 114905.
Bandi, T., Affolderbach, C., et al, “Compact high-performance continuous-wave
double-resonance rubidium standard with 1.4×10−13τ−1/2 stability.” IEEE Trans.
Ultrason., Ferroelect., Freq. Control, 2014, Vol. 61, No. 11, 1769-1778.
Hirayama, T., Yoshida, M., et al, “Mode-locked laser-type optical atomic clock with an
optically pumped Cs gas cell.” Opt. Lett., 2007, Vol. 32, No. 10, 1241-1243.
Ohuchi, Y., Suga, H., Fujita, M., Suzuki, T., A high-stability laser-pumped Cs Gas Cell
Frequency Standard.” Proceedings from the 2000 IEEE International Frequency Control
Symposium and PDA Exhibition. IEEE: New York US-NY, 2000.
Gerginov, V., Nemitz, N., Griebsch, D., et al, “Recent improvements and current
uncertainty budget of PTB fountain clock CSF2.” Proceedings from the 24th European
Frequency and Time Forum. EFTF: Besançon France, 2010.
Weyers, S., Gerginov, V., Nemitz, N., et al, “Distributed cavity phase frequency shifts of
the caesium fountain PTB-CSF2.” Metrologia, 2012, Vol. 49, No. 1, 82-87.
Heavner, T.P., Donley, E.A., Levi, F., et al, “First accuracy evaluation of NIST-F2.”
Metrologia, 2014, Vol. 51, No. 3, 174-182.
Sortais, Y., Bize, S., Nicolas, C., et al, “ 87Rb versus 133Cs in cold atom fountains: a
comparison.” IEEE Trans. Ultrason., Ferroelect., Freq. Control, 2000, Vol. 47, No. 5, 10931097.
Pereira Dos Santos, F., Marion, H., Abgrall, M., et al, “ 87Rb and 133Cs laser cooled clocks:
testing the stability of fundamental constants.” Proceedings from the 2003 IEEE
International Frequency Control Symposium and PDA Exhibition Jointly with the 17th
European Frequency and Time Forum. IEEE: New York US-NY, 2003.
Guéna, J., Rosenbusch, P., Laurent, P., et al, “Demonstration of a dual alkali Rb/Cs
fountain clock.” IEEE Trans. Ultrason., Ferroelect., Freq. Control, 2010, Vol. 57, No. 3,
647-653.
Fisk, P.T.H., “Trapped-ion and trapped-atom microwave frequency standards.” Rep.
Prog. Phys. 1997, Vol. 60, No. 8, 761–817.
Burt, E.A., Taghavi-Larigani, S., and Tjoelker, R.L., “A new trapped ion atomic clock
based on 201Hg+.” IEEE Trans. Ultrason., Ferroelect., Freq. Control, 2010, Vol. 57, No. 3,
629-635.
Chung, S.K., Prestage, J.D., Tjoelker, R.L., and Maleki, L., “Buffer gas experiments in
mercury (Hg+) ion clock.” Proceedings from the 2004 IEEE International Ultrasonics,
Ferroelectrics and Frequency Control Joint 50th Anniversary Conference. IEEE: New York
US-NY, 2004.
42
[53] Levi, B. G., “An ion clock reaches the accuracy of the best atomic fountain.” Physics
Today, Feb, 1998, Vol.51, No. 2, 21-23.
[54] Udem, T., Holzwarth, R., and Hänsch, T.W., “Optical frequency metrology.” Nature,
2002, Vol. 416, No. 6877, 233-237.
[55] Riehle, F. “Towards a redefinition of the second based on optical atomic clocks. “ C. R.
Physique, 2015, Vol. 16, No 5, 506–515.
[56] Conroy, R.S., “Frequency standards, metrology and fundamental constants.” Contemp.
Phys., 2003, Vol. 44, No. 2, 99-135.
[57] Wilpers, G., Oates, C.W., and Hollberg, L., “Improved uncertainty budget for optical
frequency measurements with microkelvin neutral atoms: results for a high-stability
40
Ca optical frequency standard.” Appl. Phys. B, 2006, Vol. 85, No. 1, 31–44.
[58] Takamoto, M., Hong, F-L., Higasashi, R., and Katori, H., “An optical lattice clock.”
Nature, 2005, Vol. 435, No. 7040, 321-324.
[59] Sinha, S., and Samuel, J., “Quantum limit on time measurement in a gravitational field.”
Class. Quantum Grav., 2015, Vol. 32, No. 1, 015018.
[60] Itano, W.M., Bergquist, J.C., Bollinger, J.J., et al, “Quantum projection noise: population
fluctuations in two-level systems.” Phys. Rev. A., 1993, Vol. 47, No. 5, 3554-3570.
[61] Peik, E., and Okhapkin, M., “Nuclear clocks based on resonant excitation of γtransitions.” C. R. Physique, 2015, Vol. 16, No. 5, 516-523.
43