Atomic Clocks

A literature study

Artwork credit: Brad Baxley and Ye Labs, JILA

Cathelijne Glaser, BScnijn.glaser@gmail.com

supervised by Dr. Jeroen Koelemeij

summer 2015

Abstract

In atomic clocks, the frequency of the oscillator is determined by the energy differencebetween two quantum mechanical states in atoms, ions or molecules. This transitionenergy depends only on fundamental constants and thus provides a stable frequencyreference. Disturbances of the system that may shift the frequency are discussed, as well asbroadening of the peak due to dephasing and the uncertainty principle. This literature study gives an overview of the different types of both microwave and opticalatomic clocks, as well as the different atom species that are suitable as clock atoms. Aqualitative way to determine clock performance is given in terms of the fractionalfrequency uncertainty, and the Allan variance to determine the stability and characterisethe noise of the clock. Various methods and procedures to decrease the uncertainties arediscussed. An outlook on possible future developments is presented, including clocks based onnuclear transitions and techniques to increase the accuracy and stability beyond the curentstate of the art.

Front page: artist's view of an optical lattice clock. [1]

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Table of contents

Abstract 1

1 Introduction 3

2 Atomic transitions 42.1 Frequency shifts 42.2 Line broadening 5

3 The basic clock system 93.1 Active clocks 93.2 Passive clocks 103.3 Measuring and feedback 10

4 Characterizing the performance of clocks 124.1 Accuracy and reproducibility 124.2 Stability 124.3 Comparison and synchronisation 16

5 Active clocks 19

6 Passive clocks 256.1 Microwave regime 256.2 Optical regime 32

7 Future prospects 367.1 (Fundamental) limits 367.2 Suggested further research 38

8 Conclusion 39

References 40

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1 Introduction

Humans have, as far as known, kept track of time, or cyclic events, since the beginning ofdevelopment. To measure time, a reference oscillator is needed. It is impossible to measuretime itself. It is only possible to measure a frequency or duration. In time measurement, it isassumed 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 ofmore subtle recurring events, like solar eclipses and other astronomical phenomena, havebeen found dated at least 4000 years ago. The sundial was the earliest form of a clock thatdivides time into smaller segments than a day. The most sophisticated versions had anerror of 24 seconds for each 0.1 degree of angle measurement. Other historical clocks werebased 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 areexcited by applying an alternating voltage to a crystal. This was the first clock based onmaterial 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 statesof an atom, ion or molecule determines the frequency of the oscillator. Frequency dividersprovide pulses at a desired rate, for example with a frequency of 1 Hz. The development ofthese types of clock boosted the abilities of time measurement. They are so sensitive thatrelativistic effects can be measured. Precise time measurements are for example applied inmetrology, fundamental constant research, the foundations of quantum mechanics, gravity,and geodetics. But it also finds applications in navigation and communication networks. Itmakes a precise measurement of position or length possible, especially since the meter hasbeen defined in terms of the speed of light. [2]This literature study dives into the principle of atomic clocks, and what makes themintrinsically suitable to measure time. After defining a quantitative manner to measure theperformance of a clock, several factors that determine the performance of an atomic clockare described. Although it is impossible to mention every atomic clock ever built, anoverview of the different types is given, in both the microwave domain and the opticaldomain. These are mutually compared and their performance, advantages anddisadvantages are discussed, as well as the different improvements that came with eachnew design. The currently best optical lattice clock is described, evoking a look into thefuture. What is there to be expected of future atomic clocks, and is there a limit to thepotential performance of clocks?

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2 Atomic transitions

According to quantum mechanics, the energy of an atom, molecule, or ion consists ofdiscrete values. Transitions between these levels occurs by emitting or absorbingelectromagnetic radiation of a specific frequency. This resonance frequency ν0 dependsonly on fundamental physical constants and is the same for all atoms of a particularelement, which makes it a reproducible standard. Two other aspects make atomictransitions particularly suitable for timekeeping. The properties of the atoms do not, as faras known, change over time. Additionally, atoms do not wear out, as mechanical clocks alldo. [2][4]

2.1 Frequency shifts

The resonance frequency is an intrinsic property of the unperturbed atom, but severalfactors may cause the actual frequency to be shifted. It is important to accuratelycharacterize the shifts that occur in the clock, to be able to state the correct outputfrequency. Frequency shifts can either be diminished by reducing the environmentalperturbations acting on the clock system, or corrected for when accurate data about theshifts are available. Inhomogeneities and fluctuations in the perturbing fields should alsobe taken into account. Additionally, a shift in some level may influence the energy of anearby level. [5] Depending on the respective design, each type of atomic clock will encounter these andother, minor disturbances, in varying degrees. This is described in more detail in thecorresponding 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 externalmagnetic field. The energy of atomic levels with different magnetic moments depend ontheir orientation with respect to the field. The external magnetic field therefore lifts thedegeneracy of these levels. This is called the Zeeman effect. The energy shifts can be calculated using perturbation theory. The nature of the shiftsdepends on the strength of the external magnetic field relative to the internal magneticfield of the atom, generated by the moving charges. The latter causes spin-orbit coupling,which can also be considered a perturbationThe shift in frequency for a certain transition due to an external magnetic field can bewritten as a Taylor expansion. For small magnetic fields, only the first two terms need to beconsidered. [5] Only constant magnetic fields contribute to the Zeeman effect as long asthe 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 toplay a role as well. [5][6]

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External electric fields

The electrical analogue to the Zeeman effect is the Stark effect. For a static electric field Ɛin the direction of the z-axis, the perturbation operator for N electrons is given by:

H '=e∑i=1

N

Ɛ z i=−Ɛ Dz (2.1)

with e the electron charge, zi the spatial coordinate of electron i, and Dz the z-componentof 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. Otherwisestated, atoms do not have a permanent dipole moment in non-degenerate states. Thelinear Stark effect exists only for hydrogen-like orbitals with n>1. The degeneracy is onlypartly 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, theshifting of the levels depends on the neighbouring levels of opposite parity and thecorresponding parity. [7] The quadratic Stark shift depends on the polarizibility of theatom, which is in turn dependent on the electron configuration. Note that both DC and ACelectric fields may give rise to a quadratic Stark shift. [8]A gradient in the present electric field interacts with the electric quadrupole moment of theatom. 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, theirclock rates differ. The frequency shift depends on the mutual height difference and thelocal value of the gravitational acceleration. [5]

2.2 Line broadening

Atoms will absorb or emit not only electromagnetic radiation with exactly the resonancefrequency, but over a small frequency range surrounding ν0. This range is called theresonance width or linewidth Δν. The ratio of resonance frequency to linewidth is called thequality factor Q:

Q=ν 0

Δν (2.2)

All other parameters equal, the stability of the atomic oscillator is proportional to Q. Animportant 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

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by a Dirac delta function. Line broadening can originate from either homogeneous orinhomogeneous processes. In the former case, the probability that an atom emits orabsorbs a certain frequency is the same for all atoms in the system. Inhomogeneousbroadening 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 inan excited state. Due to the time-energy uncertainty principle, the energy of an excitedstate can not be determined with infinite accuracy if its lifetime is finite. Therefore, anuncertainty in the frequency arises, referred to as the natural linewidth. The frequencyspectrum has a Lorentzian lineshape. The full width at half maximum (FWHM) is inverselyproportional to the natural lifetime of the excited state. If the lower level can undergospontaneous 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 timetT. The uncertainty principle therefore introduces an uncertainty in the frequency as seen bythe atom. Assuming the interaction to begin and stop abruptly, the intensity profile of theradiation 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, thefrequency will be broadened into a Gaussian profile. The FWHM is proportional to thevelocity of the atom perpendicular to the beam, and inversely proportional to the beamradius.Another factor that should be mentioned is the contingent diffusion time of the atoms outof the laser beam, which also decrease the interaction time. This is significant for very longlifetimes of the excited state.An additional, inhomogeneous broadening mechanism comes from the curvature R of thephase surfaces within a focused Gaussian beam. This causes a phase shift depending onthe location of the atoms in the beam. [9][10]

Collisional broadening

Inelastic collisions contribute to depopulation of the excited state. The resultingbroadening is similar to natural broadening and has a Lorentzian shape as well. The FWHMis proportional to the pressure p because it depends on the number of collisions per unit oftime. Due to elastic collisions, the lineshape may become asymmetric. The exact broadeningdepends on the interaction potential between the particles. Collisions with the wall interactwith a different potential than when collisions occur between particles. This shift istemperature dependent and is called the wall shift. [3]Collisional broadening generally also shifts the resonance frequency, because both energy

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levels involved in the transition may experience a shift due to the interaction with otherparticles. Even elastic collisions where the distance between the particles remains relativelylarge, such that the broadening effect is weak, can still very efficiently shift the centrefrequency. [9][10]

Doppler effect

Each individual atom will have a certain velocity relative to the beam of radiation. From theframe of reference of the atom, this means that it will experience the light as if it had adifferent 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, theDoppler effect manifests itself only as a shift in frequency, not as line broadening. This shiftmay 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 tothis inhomogeneous process is Gaussian. It is an accumulation of the different frequencyvalues measured. If the individual atoms also give rise to Lorentzian lineshapes obtainedfrom homogeneous processes, a convolution of the Gaussian broadening results in a Voigtlineshape. 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 themean free path is smaller than the wavelength of the radiative field, this effect causes theDoppler broadening to be decreased, averaging over the sample. This is called Dickenarrowing. For high pressures however, the narrowing effect will be overcompensated bythe collisional broadening. [9][10]The second order Doppler effect is the highest order contribution of time dilation. It isindependent of the direction of the velocity. It instead depends on v2 and is therefore alsocalled the quadratic Doppler effect. Even in confined systems, a quadratic Doppler shiftoccurs 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 Dopplershift. To calculate the second-order Doppler shift, knowledge of the velocity dependenceon temperature is required. [5][10]When an atom absorbs a photon, its momentum changes accordingly. If this is taken intoaccount, a small additional shift appears. This so-called recoil shift can be incorporated inthe Doppler calculations if the velocity is replaced by the arithmetic average, calculatedusing momentum conservation arguments. [11]

Saturation or power broadening

At high intensities, the excitation rate becomes larger than the relaxation rate. Thepopulations of the absorbing levels thus decreases. The system is said to be saturatedwhen the absorption and relaxation processes balance. The saturation parameter is frequency dependent, following a Lorentzian profile. Forsufficiently high radiation intensity, the absorption at each frequency is altered according

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to this distribution, resulting in a Lorentzian broadening. The width depends on the valueof 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 ofRabi 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 electricfield is present, called Stark broadening. The total potential experienced by the electron isaltered due to the electric field, and another minimum will arise, next to the one due to theCoulomb potential of the nucleus. At sufficiently high electric fields, there will be a nonzero probability for the electron to accelerate towards the new minimum and thusescaping its bound state, as pictured in figure 2.1. This tunnel effect decreases the lifetimeof the atomic levels and therefore causes additional broadening of the spectral line. Inpractice, the choice of the clock transition is usually such that this effect plays isinsignificant. [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 withopposite parity occur, because the perturbation H' contains non-diagonal elements.Transitions that are usually forbidden can happen because the state is contaminated withanother state, to or from which the selection rules do allow transitions to occur. Theadditional population decrease is called quenching. Except that this may cause additionalline broadening due to the decreased lifetime of the involved levels, it also means that thetwo-level approximation should be reconsidered because additional transition possibilitiesmay occur when an electric field is present. [7]

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3 The basic clock system

3.1 Active clocks

In active clocks, the atomic system itself is the oscillator and generates radiation with theclock frequency, which is then simply received and converted to an output signal. The firstexample of an active atomic clock is the ammonia maser clock. This principle has sincebeen extended to other atoms. [2]Masers are based on amplification of stimulated emission. To achieve this, populationinversion is required. This means that the population of the upper state is increased suchthat it transcends that of the lower level. According to Boltzmann's distribution, this doesnot occur in thermal equilibrium. It is either accomplished by pumping the atoms into theupper 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 fieldwill cause stimulated emission. In a high quality cavity, the resonant modes will beamplified by subsequent stimulated emission, while other modes, originating fromspontaneous emission, vanish. If the cavity is not tuned correctly to the resonance frequency, the output frequency will beshifted relative to the exact transition frequency. This is called cavity pulling. The amount ofcavity pulling depends on the ratio of the quality factor of the cavity QC to the qualityfactor of the resonance width which was introduced in chapter 2:

δν ∝QC

Q(ν C−ν 0) (3.1)

where νC – ν0 is the amount of mistuning of the cavity. [3] Cavity pulling can be reduced byusing a very high quality cavity or by continually adjusting the cavity properties (e.g. lengthor temperature) according to the output measured. Note that this feedback system differsfrom that of a passive clock, in that not the oscillation itself is adjusted, but the cavity inwhich 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 forminga passive clock. [2]The principles of the active microwave clocks have also been extended to the opticalregime. In general, lasers – the optical equivalents of masers – show long-term frequencydrifts which makes them unsuitable as frequency standards. Several ways to improve thestability of laser systems have been proposed, that will be discussed in more detail inchapter 6. A different way to improve the stability is to lock the laser to a long-term stablereference, and that is the principle of passive atomic clocks. [5]

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3.2 Passive clocks

The most common atomic clocks are of the passive type, where the resonant light is lockedto the atomic transition. The atom then plays the role of frequency discriminator. The localoscillator is a source that produces radiation with more or less the frequency of the desiredtransition. After the system is prepared in a certain quantum state, the response to theradiation is monitored. Maximum response is detected when the frequency equals theresonance 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 apassive clock is shown in figure 3.1.Passive clocks either operate in a continuous mode, or in a cyclic sequence consisting ofpreparation, irradiation, and response measurement and frequency adjustment. Typically,optical clocks operate on a cyclic basis, while for example atomic beam clocks provide acontinuous 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 atomicpopulation in one of the two clock states is measured. Generally this is done using a lightsource resonant with a transition involving the state of interest and measuring the amountof absorption or fluorescence. In clocks operated in cyclic mode, the detection may disturbthe population difference such that state preparation is again necessary when initiating thenext cycle. It may even displace the atoms from their trap or lattice, requiring reloading forthe next cycle. New methods are being investigated that maintain the internal coherencefrom cycle to cycle. Not only does this reduce the time in between interrogations, it alsoenables 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 thecentre frequency of the oscillator. Depending on the retrieved signal, the centre frequencyis adjusted after a certain number of averaging cycles. The latter determines the accuracyof state population measurements, and together with the sensitivity of the adjustment tothe error signal, it determines how responsive the oscillator is. If it takes longer to approach

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the resonance frequency, the clock system will be more dependent on the short-termstability 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 ofstability in the involved electronics is of course required. [3][5]

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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 exactresonance frequency cannot directly be measured, because the atomic system will not beunperturbed and the measurement equipment inherently adds some uncertainties. Boththe 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 todetermine the accuracy of the clock. If the output frequency is denoted νout and the exactresonance frequency of the atomic system ν0, then the ratio νout/ν0 can be determined. Therelative uncertainty of the clock, given in fractional frequency units, then equals the relativeuncertainty of this ratio. [2] It is common practice to divide errors into two categories: systematic errors, and statisticalerrors, 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 orcorrected 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 applyingeventual corrections. For some frequency standards however, the output frequencydepends critically on the value of operational parameters. These standards are calledsecondary and they need to be calibrated against a primary standard. The absoluteaccuracy of secondary standards therefore depends strongly on the quality and validity oftheir calibration. [3]Another important term involving clock performance is its reproducibility. A clock with highreproducibility does not necessarily have to have a precisely known shift from theresonance frequency, as long as it is constant. Reproducibility thus refers to the uncertaintyin the frequency shift. [5]

4.2 Stability

Although the intrinsic frequency of atomic transitions is assumed to be non changing, themeasured frequency will be subject to fluctuations. This usually includes both fluctuationsaround the mean, and drift. In the latter case, the deviation continues to increase (ordecrease) 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, whichdoes not necessarily have to be physically motivated. This way, the frequency drift can becorrected 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

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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 inthis chapter. Note, however, that the amplitude does have a significant implication in thecontext of signal measuring; a signal is only measured after it reaches a certain value. If theamplitude of the oscillation is lower, it takes longer to pass the threshold. In practicehowever, these fluctuations are not significant. [2]

Figure 4.1. Example of the output voltage in the time domain of a frequency measurementwith 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 completelyunderstood. A distinction between noise that modulates the signal itself and additivenoise, independent of the signal, is often useful to make. The short-term stability isdetermined 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 causefluctuations in the signal. Thermal noise can also occur in for example cavity resonancemodes. It is white noise: independent of the frequency. [2] Shot noise, which also gives riseto white noise, occurs because particles are discrete and the signal can therefore not becompletely continuous. This applies to charge carriers in electronic signals, but shot noisealso occurs because the number of atoms in each measurement fluctuates. [3] Anotherimportant phenomenon relevant for atomic clocks is quantum projection noise, whicharises from the discrete nature of state population measurements. If an atom is in asuperposition of both states, the measurement will observe only one of these states withthe corresponding probability. The outcome thus fluctuates depending on the probabilisticcollapse 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 arelarge and these increase with longer averaging times. This is often visible as the graph ofthe Allan deviation (see below) flattening out for longer averaging times, while for shortaveraging times, white noise is dominant, as can be seen in figure 4.2. However, flickernoise can be largely reduced by optimising clock design and operation parameters. It is

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associated with imperfections in the components of the device. [3]For even longer averaging times, the Allan deviation increases with time due to randomwalk noise and frequency drift. In random walk noise, the frequency at some time iscodetermined by the frequency at the previous moment. The increments are Gaussianrandom variables with zero mean. [14] Random walk noise is usually caused byenvironmental perturbations and fluctuations in parameters. [3]

Figure 4.2 Typical Allan deviation for a clock with different types of noise. PM = phasemodulation, FM = frequency modulation, RW = random walk. [12]

The deviation from pure sinusoidal wave can be viewed either in the frequency domain orthe time domain. In the time domain, the time fluctuating fractional frequencies y are thecentral quantities:

y (t )=ν (t )−ν 0

ν 0=

12πν 0

dϕ (t )dt

(4.1)

where ν(t) is the instantaneous frequency and φ(t) the instantaneous phase. It is notpractical to use the standard deviation to specify frequency deviations, because this willget larger with increasing sample number. This is because the deviations are calculatedrelative to the average, which is not stationary for most kinds of noise. There are several

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ways to describe the variance of an oscillator, each of which is particularly useful for certaintypes of analysis. The most widely used variance however, is the Allan variance. Here, thedifferences are determined relative to subsequent measurements, instead of relative tosome average. For N samples, it is calculated as:

σ y2(τ )=

12(N−1)

∑i=1

N−1

( y i+1− y i)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 σy

2(τ) is called the Allandeviation. An example of the difference between the standard deviation and the Allandeviation for increasing N can be seen in figure 4.3. If only white noise is present, thedeviations are equal.

Figure 4.3. Standard deviation (blue curve) and Allan deviation (red curve) as a function ofthe number of samples, for an oscillator with flicker noise.

In the frequency domain, the instability is described by a power spectral density based onFourier analysis. The different types of noise can then be modelled by a law of the form:

S y ( f )=h(α ) f α (4.3)

where Sy(f) is the spectral power density of y, f the Fourier frequency and α an integerbetween -4 and 0 for the most common types of noise. This can also be translated into a τdependence in the Allan variance, as can be seen in figure 4.2. If only white noise ispresent, the Allan deviation show a t-½ dependence, which is often stated empathically inthe σy(τ) value. [12]

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The results of stability analysis can be significantly altered by dead time: the time inbetween measurements, in which no data is acquired. This is specified by the dead timeratio: r = T/τ where T is the time between the (starting point of the) measurements, aspictured 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. Whenthe nature of the present noise is known, it is possible to mathematically correct for deadtime. Most precision measurement techniques however, circumvent this problem bymeasuring 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 referencestandard that is very stable over the time interval under consideration. It is important tokeep in mind that the drift may reverse sign at some time, so measurements at differenttimes and different intervals are needed. [2][3] Quartz oscillators provide good referencesfor short time intervals and hydrogen masers are generally used for time intervals ofseveral 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 isolatedfrom each other. Else, resonance effects will lock both masers to the same frequency andthe difference measured is of course zero. [3]A method to measure a clock's stability without the need for a reference standard has beendeveloped by Camparo et al in 2009. The clock output is interferometically compared witha delayed copy of the signal. It is not particularly accurate, but it may be advantageous incertain applications. An optical variant has been suggested by the authors. [15]Another approach is to monitor various clock parameters that are responsible for drift, butthis is only limitary. It is never possible to reconstruct the complete error analysis of theoutput frequency based on these indicators. [15]

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It is difficult to extract the systematic error of a frequency standard, because no referencestandard is perfect. By comparing three or more copies of the same clocks, it is possible todetermine the noise performance of each of those clocks separately. [3][5]A very sensitive apparatus like an atomic clock cannot readily be transported over longdistances. [16] To link two or more clocks in different laboratories, signals can betransferred via satellites. This applies to both frequency comparisons and time scaledifferences. Either the Global Positioning System (GPS) is used or geostationarytelecommunication satellites in a method called two-way satellite time and frequencytransfer (TWSTFT). [17]GPS satellites contain a reference rubidium or caesium clock, synchronised to each otherand to the International Atomic Time on Earth. They emit data on two differentfrequencies. Disturbances due to the atmosphere or ionospheric refraction can be tracedby comparing the time difference between to two signal frequencies. If only one frequencyis 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 acommon signal from the same satellite. This is only possible if both clocks are in range ofone satellite. To compare time scales, the delays of both receivers is calibrated. [2] Anadvantage of this method is that it reduces the errors that are common for both clocks, likeerrors of the satellite reference itself, and most of the atmospheric errors. [17] The positionsof 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. TheInternational GPS Service (IGS) provides such a time scale The main errors arise fromuncertainties 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 coordinatesof the clocks under consideration, nor do atmospheric delays play a role. In this method, asignal from both clocks is sent to the other via a geostationary satellite. Delays cancelbecause of the simultaneity and equality of the propagation paths. Corrections are madefor the residual movement of the satellite relative to the Earth. For long averaging time, thestability 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 timeinterval, fractional uncertainties of about 10-15 are reached when satellites are used tocompare 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 wasestablished stable enough to compare optical clocks. It is based on an optical carrier wavefrom a continuous wave laser, which provides sufficient resolution and is particularlysuitable for transmission over long distances. The Allan deviation of the fibre link is shownbelow 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. Abeat note is measured between the frequency of each clock with the received signal of theother 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

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treated as the long arm of a Michelson interferometer with a partial reflector at the remoteend. Part of the light is reflected and compared with a reference signal. The frequency sentinto the fibre is adjusted according to the measured phase fluctuations. Because of thelong length, fibre amplifiers have been installed. They are bidirectional to establish equalpath 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 basedon a comparison of two independent frequency locks that are operated alternately. [1]

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5 Active atomic clocks

The ammonia maser

The first prototype atomic clock ever built, actually a molecular clock, was an ammoniamaser. [4] The transition has a particular strong coupling to external fields and is thereforeeasy to exploit. [3] In the ground state, the position of the nitrogen atom can be describedto be a linear combination of the positions on either side of the H3 plane, with theirrespective amplitude oscillating in time. There are two stationary states in which the phasesof the amplitudes have the same frequency: one symmetric and the other antisymmetric. A15NH3 maser is based on a transition between these two stationary states, with a frequencyof 22.8 GHz. [18][19]Several important inventions improved the performance of the ammonia maser, that havebeen applied to modern clocks afterwards. State selection, by means of an electrostaticfield, increased the signal to noise ratio. Line broadening and frequency shift due tocollisions was reduced by using a beam, formed in low pressure channels. Lastly, the firstorder 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 averagingtimes 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 withthe wall, and instabilities in the operation parameters. [3][20] During the sixties, furtherdevelopment ceased quickly, as better clock designs became apparent. [4]

Alkali masers

Masers based on other atoms work similar to the ammonia maser. Alkali vapour frequencystandards were researched in the sixties. Rubidium was preferred because of practicalreasons. [2] The rubidium maser is based on a hyperfine transition of the 2S½ ground stateof 87Rb, with F = 2 and F = 1. The transition frequency is 6.8 GHz. Population inversion isobtained by using an 87Rb discharge lamp with an 85Rb filter to pump atoms from the F = 1level into the P state, followed by relaxation back to the ground state, mainly throughcollisions with a buffer gas. The F = 1 level will be depopulated with respect to the F = 2level due to this absorption-relaxation cycle. [2][3][19]Threshold for oscillation in a rubidium maser is much higher than for ammonia, becausethe transition involves a magnetic interaction instead of the much stronger electric dipoleinteraction in ammonia. A high pumping power and high quality cavity are thereforenecessary. [3]The applied magnetic field to remove the degeneracy of the ground state sublevels causesa second order Zeeman effect. The buffer gas diminishes the first order Dopplerbroadening by decreasing the mean free path of the rubidium atoms, and using a specificmixture of gases, an optimum temperature coefficient can be reached. However, it alsocauses an additional collisional shift. [3][19] Another important shift in the resonancefrequency 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 theoutput signal. Since high powers are needed to achieve sufficient population inversion inrubidium, this problem is severe in rubidium masers. However, the so-called double bulbdesign circumvents light shifts by using separate regions for pumping and interrogation.Another approach is to use pulsed optical pumping, also called the POP technique, wherethe 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. Instabilitiesarise 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. Itsfrequency is somewhat lower with 3.0 GHz. Experimental difficulties made this variantunsuitable 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 isobtained using a multi pole magnet. The atoms are then focused into a storage bulbconnected to a resonant cavity. An automatic tuning device ensures long-term stability ofthe cavity. The bulb is magnetically shielded from environmental disturbances. Because theatoms reside in the bulb for relative long periods, the frequency peak is very narrow. Avacuum is present to decrease collisional broadening. The first order Doppler effect isnegligible because of the Dicke effect, occurring for atoms confined to a small cell. All theabove 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 hydrogenatom is 1.4 GHz. The relative error is 2·10-12. [2][23] The accuracy of active hydrogen maserclocks is moderate because of the uncertainties in the cavity properties and the atomscolliding 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 of1 day. [24] For medium-term averaging times, between 30 and 200 days, Allen deviationsfrom 10-16 to 10-15 can be obtained as measured by the National Institute of Standards andTechnology (NIST). [25]The long-term stability of several hydrogen masers has been measured and reported bythe NIST in 2010. Frequency drifts in the order of 10 -16 per day are common for hydrogenmasers. These drifts are often not linear. At the NIST, a measurement of the fractionalfrequency drift hydrogen maser clocks is made over a period of about 8.5 years. The resultsare shown in figure 5.2. Environmental corrections cause small frequency offsets. The dataare 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 theNIST. 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 thehydrogen masers at NIST. It is calculated that time scale errors due to frequency drift willbe 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 operatingparameters and ageing of parts of the clock system. Stabilising these parameters willincrease the stability of the clock output. However, maintenance itself also infringes theperformance 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 thecavity tuning device [22] or the coating inside the storage bulb [26]. There are, however, norecent developments that significantly improve active hydrogen maser clocks and the basicdesign has been the same since the eighties. [22] In 2014, Boyko and Aleynikov suggestedusing different magnetic sorting systems to improve the Allan deviation by an order ofmagnitude 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 evenbetter stabilities, especially at short-term. These so-called cryogenic masers benefit fromlower 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 wherepredicted by theoretical considerations. However, at low temperatures, fluctuations in theatomic 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 vapourshift, such that at a specific temperature (near 0.5 K), the sum of these is independent oftemperature to first order. Temperature control is therefore crucial. Another importantfactor is the surface coating of the storage bulb. To prevent the hydrogen atoms frombinding to the wall, a layer of inert gases is applied to the wall. This layer has to be thickenough to cover the possible impurities in the wall material, but at the same time it has tobe smooth and uniform in thickness to prevent additional relaxation. This hampers theperformance 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 Allandeviations, obtained at the Smithsonian Institution Astrophysical Observatory inCambridge, 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. Theoptical equivalent is the laser. However, lasers have never been stable or accurate enoughto function as an active frequency standard. [22] Only very recently, lasers that cancompete with microwave frequency standards have been realised. The most stable lasers atthis 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 forexample electro-optical modulators, accousto-optical modulators or piezoelectrictransducers, to alter the frequency as to match the cavity resonance. The cavity itself canbe stabilised by decreasing its sensitivity to temperature variations and environmentalperturbations, as well as mechanical vibration. One strategy is to isolate the cavity, byimmersing it in vacuum, enclosing it in vibrational insensitive and thermally isolatingcontainer, or mounting it on a support capable of damping mechanical motion. [34]Another way is to use materials that have appropriate properties, like low temperaturesensitivity. Both Ultra-Low Expansion (ULE) ceramic glass and monocrystalline silicon havezero first-order thermal expension coefficient at a specific temperature and can be used tofabricate the spacer between the cavity mirrors. The latter has a better intrinsic qualityfactor than the former when used at their respective zero crossing. It is therefore veryinstensive to vibrational noise, and it does not show ageing. It also has a superiour thermalconductivity, which contribute to temperature homogeneity. The thermal noise limit ofcavities 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 ascoating materials, or by using longer spacers. [31]

Figure 5.5. Allan deviation of several kinds of stabilised lasers, compared to the quantumnoise 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. Thecavity loss rate is then larger than the gain bandwith. Cavity length noise is thensuppressed in return for a stronger cavity pulling effect, of which the latter is much easierto characterise. However, this proposal has not been put to practise yet. [32][33]

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6 Passive atomic clocks

6.1 Microwave regime

Just as in masers, state selection or pumping is needed to be able to passively measuremicrowave transitions. In the microwave range, spontaneous transitions back to the groundstate have a low probability, and since the probabilities of absorption and stimulatedemission are inherently equal, artificial state population difference needs to be created tobe 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 ofatoms. A simplified overview of the first caesium beam clock using magnetic deflection isshown in figure 6.1. Caesium was used because it is relatively heavy, which means it travels slower and has alonger interaction time when moving through an electromagnetic field, and because it hasa higher frequency then other microwave atomic oscillators, which provides betteraccuracy. The transition used in caesium clocks is the hyperfine transition in the groundstate between F = 3 and F = 4 (mF = 0). The resonance frequency is defined to be exactly9,129,631,770 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 selectionis achieved by means of a magnetic field. A quartz-based frequency synthesizer provides a

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microwave field, tuned to match the transition frequency. A second magnetic field directsatoms that changed state toward a detector. Based on the strength of the signal, the servofeedback adjusts the quartz oscillator. [19][4][2]A constant magnetic field is present to separate the hyperfine states. The field needs to behigher than for hydrogen. The second-order dependence of the frequency on the magneticfield is thus much more sensitive to variations. Accordingly, the apparatus needs to beshielded 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 differentplaces along the beam path. The width of the output frequency peak is still determined bythe time the atoms need to cover the whole cavity length, but line broadening mechanismssuch as the first-order Doppler effect are extinguished. Furthermore, the method decreasessensitivity in the output frequency to inhomogeneity effects of the static magnetic fieldand fluctuations in the microwave field. [4][19][35] It does, however, cause an additionaluncertainty, called end-to-end cavity phase bias. This bias arises from the difference inphase of the microwave radiation in the two excitation regions. The value can bedetermined by comparing results with the beam direction reversed, but it still adds asignificant uncertainty. [35] The second-order Doppler effect also plays an important role in the accuracy of thecaesium beam clock. Calculations to determine the shift are based on information aboutthe velocity distribution of the atoms. [19][35] The frequency synthesiser adds anadditional source of uncertainty. The microwave field itself causes a frequency shift, sovariations in the applied frequency transfer to variations in the output frequency. Theamplitude 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. Thisincreases the number of atoms available for transition and thus improves the signal. Anexample of this type is built at the NIST and evaluated in 2001. The minimum accuracy was

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found to be 4·10-15. Its stability is represented in figure 6.2. [35] The short-term stability isdetermined by the quartz oscillator. For sampling periods of a few seconds to a day, thestability mainly depends on the shot noise in the detection. The long-term stability isdetermined by variations in the frequency shifts and ageing. [2]The same design has been considered with other atoms. Thallium-205, providing afrequency of 21.3 GHz, has the advantages that a small magnetic field is sufficient for theseparation of the F = 0 and F = 1 levels (mF = 0) of the 2P½ ground state. The second-orderfrequency dependence on the magnetic field is then of course also small. Difficulties arosebecause magnetic deflection and detection of thallium atoms is difficult, and an oven tocreate a thallium beam has to be operated at very high temperatures. Since the accuracyand stability is subject to the same constraints as the caesium beam clocks, there was noreason 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 asuitable hyperfine transition in the ground state, with frequencies of respectively 1.7 and2.0 GHz. [19]The impossibility to detect hydrogen atoms with sufficient efficiency also hinder the use ofhydrogen in an atomic beam clock. Another disadvantage is that the velocities of theatoms are very large. In a variant of the hydrogen beam clock, the interaction time of thehydrogen atoms with the microwave field is increased by sending the atoms through astorage bulb, in which the atoms reside for some time before continuing through a smallhole on the other side. Adjusting the temperature to diminish the wall shift would be easierand cavity pulling would be negligible. However, the impossibility to detect atomichydrogen 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 ofabsorbance after passing through the microwave field. To increase the signal, the gaugelight crosses the atomic beam several times. Instabilities in the gauge light add to theinstability of this type of clock. This makes rubidium beam clocks subordinate to thecaesium variant. [3][19] However, recent developments using lasers as more stablepumping or detecting source has renewed interest in rubidium beam clocks. Results showa somewhat improved short-term stability, but the frequency drift of the lasers added toinstability at long-term. The rubidium beam clocks have thus far not outperformed thecaesium variant. [36][37]Magnesium beam frequency standards have been built with an accuracy of 10 -12 and ashort-term stability of 10-11τ-½. Although the alkaline earth metals do not show a hyperfinestructure in the ground state, there are some suitable transitions between excited statelevels, 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]

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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 transitionfrequency, but without reaching threshold for masing. A photodetector measures theamount of absorption of resonant microwave light passing through the cavity, comparableto the interrogation in rubidium beam clocks. The signal is used to tune the frequencysynthesiser. Compared to the rubidium maser, the constraints to the cavity are less severe and there isno 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 anoptimal 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 tonegligible levels. [39] The dependency on the operation parameters make this type ofclocks secondary standards.The development of lasers as both pumping and interrogation source has improved short-term stability. However, long-term frequency shifts are present due to the light shift andageing effects, comparable to those encountered in maser clocks. Added thereto is thefrequency 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. Efforthas been made to develop a coating for the inside surface of the cell to prevent rubidiumatoms from reacting with it and to remove the need for a buffer gas, but haven't beenapplied thus far. The advantage to discard the buffer gas is assumed to be small, becausecollisional 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 aminimum Allan deviation of 10-14 for averaging times of about 2 minutes. Long-termfrequency drifts are below 10-15 per day. Suggestions that have been put forward toimprove the stability are for example to operate under vacuum to decrease environmentaleffects, 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 onatoms other than rubidium. For caesium, it was initially quite difficult to obtain a sufficientpopulation difference. In the first versions, a caesium lamp with an interference filter andcircular polariser was used. [19] Nowadays, optical pumping with lasers is available. Doubleresonance is again obtained by adding a buffer gas. This leads to a reduced linewidth, butit also introduces a frequency shift, which renders it slightly inferior to the caesium beamclock. The performance of a gas cell clock is comparable whether caesium or rubidium isused. For caesium, short-term stabilities of the order of 10−13τ−½ with a minimum Allandeviation of order 10-14 at averaging times of 100 to 1000 seconds are reported. [42][43]

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Atomic fountain clocks

Caesium fountain clocks where invented to increase the interaction time by building anatomic beam clock in vertical direction. The atoms are fired in upward direction, slow downand reverse under the influence of gravity. Only one microwave field is necessary, throughwhich the atoms pass twice. This removes the end-to-end phase bias, although a spatialvariation 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 mutualcollisions with atoms of a different velocity and no signal was detected. Laser coolingprovided the solution. Caesium fountain clocks generally make use of atomic molasses, butseveral laser cooling techniques exist. By tuning the lasers, a ball of atoms can be launchedat a specific velocity. The low temperature ensures small velocity fluctuations. The fountain clock works in cyclic mode. After optically pumping the atoms, the pumpingand cooling lasers are screened off to avoid a light shift. Detection after crossing the cavityis done using a light source tuned to the F = 4 to F = 5 transition. A schematic overview ofan atomic fountain clock is shown in figure 6.3.The low velocities of the atoms render the second-order Doppler effect very small. Theuncertainty in this shift is also very small, because the velocities are known very precisely. Adensity shift remains due to the collisions of caesium atoms with each other. The so-calledmultiple ball toss scheme, or juggling method, reduces this by launching several batches ofatoms in quick succession, with different initial velocities such that they arrive at thedetection point at the same time. [2][4]In cold atom fountain clocks, device limitations are so small that fundamental limits aresignificant. A blackbody shift is caused by the relatively warm outer surface of the vacuumcavity emitting radiation. This can be corrected for quite accurately, but cryogenic vacuumsystems have also been developed to decrease this shift. [44] A gravitational red shiftoccurs because the atoms move up and down in Earth's gravitational potential. Accuratecorrections for this are possible. [2][4]The accuracy of modern caesium fountain clocks is of order 10-16. The main sources ofuncertainty 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 beobtained. The instability of the oscillator is a limiting factor, but this influence is reduced byreplacing the original quartz oscillator by a cooled sapphire ring, that provides enhancedspectral purity. [2] The cyclic mode of operation introduces dead time in the frequencymeasurement. 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 inthe order of 10-13τ−½ [45][46][44] The Allan deviation for a modern caesium fountain clockat the NIST is shown in figure 6.4. [44] When using rubidium instead of caesium in an atomic fountain clock, the collisional shift ismuch lower. Although in turn the quadratic Zeeman shift is larger, rubidium fountain clocksthat 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 atomsare used simultaneously It preserves the accuracy and stability of the clocks containingonly one species. It can be used to compare both species within exactly the sameenvironment. [49]

Figure 6.3. An atomic fountain clock. The lasers designated with M provide the opticalmolasses, 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 Paultrap, a high frequency alternating electric field is used. Other kinds of traps are notpractically applicable in atomic clocks, because of the presence of a disturbing magneticfield. Using radiation to form a trap removes the wall shift that exists in systems where theatoms are confined in a cell or cavity, while the shift from the presence of the field itselfcan 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 stabilitythan a clock based on a cloud of ions. [50]There is a limit to the density of the ion cloud due to mutual repulsions, whichautomatically 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 incase of a lower density, and therefore the second-order Doppler shift is also lower. Forhigher densities, the so-called collisional cooling method introduces a low pressure heliumgas to reduce the kinetic energies without introducing a significant collisional shift. In alinear Paul trap, the ions spread out of a cylindrical region, and the spread in kinetic energyis thus smaller. The linear trap also shows less sensitivity to fluctuations in the appliedelectric fields. [2][3]Several ions have suitable microwave transitions, but the heavy mercury ion provides asmall second-order Doppler effect and is thus preferred. Ionisation is achieved by treatingthe atoms with an electron beam or ionizing radiation. When the outermost electron isremoved, 199Hg has an electronic structure very similar to the alkali metals. The transitionthat is used in atomic clocks is the F = 0 to F = 1 (mF = 0) transition in the ground statewith 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 rubidiumatoms. One of the emission lines of 202Hg+ lies very close to the transition wavelength fromthe ground state F = 1 level to the P state. From the P state, spontaneous decay occurs toboth ground state levels, but the F = 1 level atoms will be re-excited such as to create anaccumulation of atoms in the F = 0 level. Although laser pumping is more efficient andthus provides a better signal-to-noise ratio, 202Hg+ lamps are still used when the setup ispreferred to be compact. [3]Trapped ion clocks operate in cyclic mode. First, the trap is reloaded to make sure everycycle starts with the same number of atoms. After optical pumping, the microwave field isapplied in two successive pulses, thus applying Ramsey's method in the time domain. Theamount of ions that have undergone the transition is probed using the pumping laser or220Hg lamp and measuring the fluorescence response. If the microwave field is tuned toresonance, a maximum number of ions have been excited into the F = 1 level and can thusrespond to the pumping laser, after which they relax back to the ground state emitting asignal. This method of measuring the population of a state indirectly is described as thedouble 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-orderDoppler effect and collisions. By laser cooling the ions, the second-order Doppler effectcan be reduced to negligible levels. Cryogenic techniques reduce the collisional shift. [2]Electric quadrupole interactions from a residual electric field gradient can be eliminated tofirst order by averaging frequencies measured over any three mutually orthogonal. It isalso 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 theapplied electromagnetic fields can be adequately controlled. [2] These types of clocks showtherefore 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 thetrapped mercury ion cloud clock is about 10-15. [53]Trapped ion clocks have also been investigated using 9Be+, 113Cd+, 137Ba+ and 171Yb+. All ofthese species exploit a hyperfine transition in the ground state, except for beryllium, whichhas 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 easilyobtained using relatively cheap and compact available lasers. Otherwise, the same limitsapply 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 therewere no reliable frequency counters that could handle these high frequencies. Theinvention of the optical frequency comb turned out to solve this problem. The output of amode-locked laser consists of equally spaced laser modes. If these span more than anoctave, it is possible to accurately determine the frequencies of all these modes. An opticalclock frequency can then be measured by comparing it with the nearest mode of the

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frequency comb by taking a beat note. [54] All optical clocks that have been realised todate are passive clocks with a cyclic mode of operation. They operate with frequencies of400 to 1200 THz. [5] The development of the increasing accuracy of optical clocks iscompared with the development of microwave clocks in figure 6.5. These results may leadto a new definition of the second on short notice. [55]

Figure 6.5. Development of the increasing accuracy of both microwave (blue squares) andoptical (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 accuratemicrowave reference. [55]

The first examples of lasers stabilised to an atomic transition where iodine and methanesecondary cell standards. These could use standard laser wavelengths, as tunablecontinuous wave lasers where not available in the rising era of optical clocks. Iodine hasseveral suitable transitions and has a narrow bandwith because it is quite heavy. Methaneprovided a quite stable clock with a maximum stability of 10-15τ-½ for averaging times of100 s. Other molecules used for conventional laser wavelengths are N2, CO and Te2. Innewer generation optical clocks, research is limited to atoms because there is no efficientmethod 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+,138Ba+, and 115In+. Ytterbium has shown the longest storage times, while indium is mostsuitable for laser cooling. Both aluminium and indium show very low sensitivity tofrequency shifts induced by external fields. The used clock transition is the quadrupoletransition from the 2S½ ground state to one of the 2D excited states. For ytterbium, anoctopole transition to the 2F7/2 state is also suitable. The mentioned ions all providepossibilities for cyclic pumping and optical cooling as described earlier in the paragraph ontrapped ion clocks. The excited states for these ions do have a quadrupole moment and aquadrupole 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 andhave usually small Zeeman shifts. However, these transitions have frequencies that are toohigh 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 interactionwith a different species in a process called sympathetic cooling. The other species is lasercooled and the ion of interest is cooled along with it by mutual interaction. Detection isbased on a quantum logic technique, in which information about the clock ion is obtainedby interrogating the logic ion based on a common motional state. The constraints on theions suitable for this type of optical clock are loosened because the logic ion plays theimportant role of cooling and interrogation. [5]The choice of logic ion codetermines the systematic frequency shifts of a quantum logicclock. The logic ion can be laser cooled during interrogation, lowering the second-orderDoppler 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 whichions 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 relativisticshift due to residual motion of the ions. Compensation is limited by the measurement andcontrolling of the electric fields in the trap. Frequency shifts caused by external fields aremeasured and corrected for very precisely in these clocks. Collisions with background gasare rare in ion traps, but can be detected in the laser cooling fluorescence signal. Suchevents can then be discarded. [5]The most accurate optical ion trap clocks are based either on the octopole transition in asingle trapped Yb+ ion, or on an Al+ logic clock, both with fractional uncertainties in theorder of 10-18. [55]

Free-space optical standards

Frequency standards based on free calcium, magnesium and strontium have beenresearched. However, Doppler shifts were very significant. The best results have beenachieved with laser cooled, ballistically expanding calcium. [5] These clocks are based onclouds of cold neutral atoms that are allowed to expand freely under influence of gravity. Itreached a fractional uncertainty of order 10-15. The greatest improvement arose from morestable probe lasers, and from a reduction in the uncertainty of the laser cooling, whichallowed for a reduced Doppler effect uncertainty. Residual uncertainty is caused mainly bycollisions between the cold cloud and atoms impinging from the thermal beam atomsource, 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 canalso 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 possibleto 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 spatiallydependent Stark shift induced by a standing-wave electric field created by interferinglasers. 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 createthe optical lattice induce a light shift, so this technique is only available for atoms whichobey light-shift cancelling condition that the polarizibility is equal for both upper andlower state. This will happen for some wavelength, called the magical wavelength. The lightshift can thus be controlled with high precision by tuning the wavelength of the latticelaser. At first, a collisional shift was present because several atoms where traps in eachpotential well. [58]The most accurate clock to date is an optical lattice clock with 87Sr atoms at the JointInstitute for Laboratory Astrophysics (JILA) in Boulder, US-CO. Its fractional frequencyuncertainty is 2·10-18. The most significant contribution to its uncertainty are in the Starkshifts induced by the blackbody radiation and by the lattice field. The uncertainty in theblack body radiation shift is reduced significantly by careful measurements of thetemperature inside the vacuum chamber and inhomogeneities thereof, and by measuringadditional 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 highoptical frequency, long interaction time and large atom number. [58] The stability is limitedby frequency noise in the interrogation laser at short averaging times. Long-term drift canbe controlled to within order of 10-18 after thousands of seconds averaging time. Thestability 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 aremeasurements, 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]

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7 Future prospects

Progress in the design and implementation of atomic clocks has reached the point thatfundamental limits are in sight. It is to be expected that uncertainties of technical naturewill continue to decrease, but the Heisenberg uncertainty principle and the gravitationalred 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 noiselimit. The number of atoms in the measurement in can be maintained constant by trappingthem, while statistical fluctuations in the detection are nearly diminished by the electronshelving technique. [60] The quantum projection noise limited Allan deviation is given by:

σ y (τ )=C Δνν √ tT

Nτ (7.1)

where C is a constant of order unity, tT the transit time, and N the number of atoms. Notethat 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 clockdevelopment. Nuclear transitions would provide frequencies in the X-ray or γ-ray regime.These transitions are highly insensitive to external electromagnetic fields and a largenumber of atoms can be used in the solid state. The 2 PHz transition between the groundstate 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 afrequency standard. [61]Traditional interferometric techniques and crystal diffraction can be used to measure X-rayfrequencies, and dual X-ray-optical interferometry has been suggested as the best methodto measure X-ray and gamma frequencies. Gamma ray experiments like the electron-positron annihilation process turned out to be quite inaccurate because of centre-of-massmotion. In general, the necessary fast and efficient techniques are not sufficientlydeveloped yet to create an X-ray or gamma frequency standard. [56] The high frequencyradiation 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. Thisis mainly a technological issue, as described in detail in the previous chapters. Coherentpopulation control might be used to alter the upper state lifetime, hereby even decreasingthe natural linewidth broadening. [56] There is, however, a fundamental limit to the attainable resonance Q-factor. Even iftechnological abilities increase without limit, there is a fundamental limit to thetemperature that can be reached with laser cooling. [10] And even if a clock atom would be

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cooled to absolute zero, quantum zero point motion still remains. An uncertainty in theposition of a clock ion, σx, in the gravitational potential imposes a fractional uncertainty inthe clock frequency by time dilation. The uncertainty in the momentum, σp, can be viewedas random motion, causing a second order Doppler effect. The Heisenberg uncertaintyrelation between position and momentum implies a fundamental uncertainty for frequencystandards. 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 ofσx and σp. An approximate result for the maximum achievable Q-factor is given by equation7.2:

Q≤2√3(

mc3

ħg )23 (7.2)

where m is the mass of the clock atom, c is the speed of light, ħ Planck's constant dividedby 2π and g the local gravitational acceleration. [59] The limit prevails even for increasingnumber of clock atoms and averaging their frequencies, because the limit scales faster withatom number. The order of magnitude of this limit to the quality factor is 1021 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 thetransit time or the number of atoms. Both approaches are, however, restricted by practicaldrawbacks as described earlier in chapter 5 and 6.

Quantum entanglement

The ultimate quantum mechanical limit to the uncertainty in systems with uncorrelatedatoms is given by the Heisenberg relation in the limit of high atom number. Quantumprojection noise can be reduced by implementing squeezed states. So far, squeezed stateshave been demonstrated for two ions in the radio frequency regime, but not for opticalfrequencies. Another way to entangle the atoms is by creating so-called GHZ states, whichare superpositions of all atoms being in the upper state and all atoms being in the lowerstate. Up to 14 entangled atoms have been demonstrated using this method, of which oneexperiment focused on the optical Ca+ ion. Scaling experiments up to hundreds of ionshave been proposed. Systems with GHZ states are, however, more sensitive to laser phasenoise, decreasing the stability.Entanglement could in theory also act as an alternative for averaging over differenttransitions to create insensitivity to external fields, less sensitive for fluctuations andinhomogeneities of the fields, but there are no schemes developed yet to produces thesedesigner atoms. [5] Technologically, this is quite a challenge. These methods seem onlypromising for situations in which scaling of the atom number or interrogation time isimpossible. [5][55]

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7.2 Suggested research

Progress has been made in developing techniques to defy the quantum limit to clockperformance, as described in the previous paragraph. However, the quantum limit is not insight yet. The best clock in the world now has a fractional frequency uncertainty of 10-18,while the quantum limit is a factor 103 to 104 lower. [59] On the other hand, comparing thisleap to the development of clock performance since the first atomic clock was built in1948, it is not inconceivable that uncertainties of order 10-21 can be reached in a fewdecades, especially when looking at the trend of the previous few decades (see also figure6.5). The realisation of nuclear clocks, [56][61] superradiant lasers [5] or other,unforeseeable great inventions might speed up the development rate, just like the uprisingof optical clocks have done after invention of the frequency comb. [54] Nuclear frequency standards could be a valuable tool for specific research of the constancyof physical constants, and therewith contribute to the development of a grand unificationtheory. It is predicted that nuclear transitions are far more sensitive to variations in the finestructure constant, compared to atomic experiments. This is still quite controversial though,and measurements are proposed to elucidate the issue. [61] However, as time-measuringtools, nuclear clocks are not expected to become available in the near future. A frequencystandard based on a nuclear transition has not been realised yet, although researchtowards ultrahigh-resolution X-ray spectroscopy is being conducted in e.g. the group ofEikema at the VU University Amsterdam. [J. Koelemeij, personal communication, 21October 2015] Techniques to efficiently create X-ray or γ-ray frequencies with sufficientstability and tunability are not available yet, and specific necessary techniques for this typeof clock are not developed, nor is it completely clear what the challenges in building thistype 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 beenlearned 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 mucheasier 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. Althoughinherently new methods have been published, for example to measure the localtemperature inhomogeneities without disturbing the system [1], they were all based on thefamiliar underlying concepts to decrease the uncertainties. This has, however, led to greatresults, so further optimisation of current clocks may lead to significantly better performingclocks on relatively short notice. This applies for example to further decreasing fluctuationsand 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 maycome from unexpected areas of research: not so much the engineering part, but a morefundamental, quantum mechanical point of view. The question how to decreaseuncertainties in the existing systems, may be replaced by the question what systemsintrinsically have less uncertainty. Coherent techniques for example, have a lot ofunexplored potential. [56]

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8 Conclusion

The frequency of atomic clocks is determined by an atomic transition. This provides astable and independent reference for time measurement. External influences, like a non-zero temperature or the presence of electromagnetic fields, can disturb this process byeither shifting or broadening the measured frequency. Fluctuations in parameters andfundamental noise processes add to the uncertainty and instability of a clock. Since the first prototype, a microwave ammonia maser, enormous progress in increasingthe accuracy and stability of atomic clocks has been made. In the microwave regime, theactive masers evolved into gas cell clocks, atomic beam clocks and fountain clocks, inwhich an external oscillator is stabilised on the atomic transition with a servo feedbacksystem. Each new design resolved several important issues concerning uncertainties andinstabilities. For example, the first-order Doppler effect, light shift, thermal noise, sensitivityto external fields and collisional shift where greatly reduced using various techniques.Additionally, the transit time was significantly increased, leading to a better signal-to-noiseratio. Each type has its own advantages and disadvantages, just like each atom ormolecule. The hydrogen maser still proves useful because of its great stability, while itssmall size make rubidium cell clock invaluable. The caesium fountain clock provides thedefinition of the second, although it may be replaced by the advanced accuracies of opticalclocks.The optical counterpart of the maser has never been established as a frequency standardbecause long-term frequency drift turned out to be a problem. However, passive opticalclocks 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 regularimplementation of optical clocks. The optical lattice made it possible to confine manyneutral atoms without disturbing mutual interaction. A few free-space and gas cell opticalclocks have been built, but the absolute record in both accuracy and stability is held by a87Sr optical lattice clock. Its fractional frequency uncertainty is 2·10-18.Eventually, quantum projection noise provides the fundamental limit to the resonance Q-factor, 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 clockperformance is by further refining current technologies and procedures to reduceuncertainties, increasing the number of atoms and increasing the transit time. Higherfrequency clocks based on nuclear transitions have been proposed, which theoreticallycould 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 ofspecific applications in fundamental research, realisation of these clocks is still far awaybecause 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 toeither accelerate the rate of improvement, or ultimately stretch quantum limits.

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