Atomic Clocks
A literature studyArtwork credit: Brad Baxley and Ye Labs, JILA
Cathelijne Glaser, BSc nijn.glaser@gmail.com
supervised by Dr. Jeroen Koelemeij
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.
Table of contents Abstract 1 1 Introduction 3 2 Atomic transitions 4 2.1 Frequency shifts 4 2.2 Line broadening 5
3 The basic clock system 9
3.1 Active clocks 9
3.2 Passive clocks 10
3.3 Measuring and feedback 10
4 Characterizing the performance of clocks 12
4.1 Accuracy and reproducibility 12
4.2 Stability 12
4.3 Comparison and synchronisation 16
5 Active clocks 19 6 Passive clocks 25 6.1 Microwave regime 25 6.2 Optical regime 32 7 Future prospects 36 7.1 (Fundamental) limits 36
7.2 Suggested further research 38
8 Conclusion 39
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 so-called 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-balance-wheel 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?
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]
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:
H ' =e
∑
i=1 N
Ɛ zi=−Ɛ Dz (2.1)
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:
Q=ν0
Δν (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]
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 sinc2 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]
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
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]
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: δ ν ∝ QC
Q (νC−ν0) (3.1)
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]
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
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 pre-set 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]
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
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
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)−ν0 ν0 =2 πν1 0 d ϕ (t) 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
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:
σ2y(τ )= 1
2(N−1)
∑
i =1 N−1(
̄yi+1− ̄yi)
2 (4.2)where ̄yi 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:
Sy(f )=h(α )fα (4.3)
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 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]
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. Multi-channel 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
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]
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 H3 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
15NH
3 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 2S
½ 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
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]
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 Physical-Engineering 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]
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
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 spin-exchange 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 room-temperature 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.
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]
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
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
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 3P
1 – 3P0 transition with a frequency of 601 GHz. Efficient pumping,
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 short-term stability. However, long-short-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
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]
[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]
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
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
220Hg 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
