Atomic clocks: new prospects in metrology andgeodesy

Pacome Delva∗, Jerome Lodewyck

LNE-SYRTE, Observatoire de Paris, CNRS, UPMC ; 61 avenue de l’Observatoire, 75014 Paris, France

September 2, 2013

Abstract

We present the latest developments in thefield of atomic clocks and their applicationsin metrology and fundamental physics. Inthe light of recent advents in the accuracyof optical clocks, we present an introductionto the relativistic modelization of frequencytransfer and a detailed review of chronometricgeodesy.

1 Introduction

Atomic clocks went through tremendous evo-lutions and ameliorations since their inven-tion in the middle of the twentieth cen-tury. The constant amelioration of their ac-curacy (figure 1) and stability permitted nu-merous applications in the field of metrol-ogy and fundamental physics. For a longtime cold atom Caesium fountain clocks re-mained unchallenged in terms of accuracyand stability. However this is no longer

∗Corresponding author. E-mail: Pa-come.Delva@obspm.fr

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1950 1960 1970 1980 1990 2000 2010 2020

EssenandParry

RedefinitionoftheSIsecond

Atomicfountains

Opticalfrequencycombs

Clocksaccuracy

MicrowaveclocksOpticalclocks

H

H

HHg+,Yb+,Ca

Hg+,Yb+Sr

Hg+

Sr

Al+

CaH

Sr+,Yb+

Al+

Figure 1: Accuracy records for microwave andoptical clocks. From the first Cs clock by Es-sen and Parry in the 1950’s, an order of mag-nitude is gained every ten years. The adventof optical frequency combs boosted the per-formances of optical clocks, and they recentlyovercome microwave clocks.

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true with the recent development of opticalclocks. This new generation of atomic clockopens new possibilities for applications, suchas chronometric geodesy, and requires new de-velopments, particularly in the field of fre-quency transfer. The LNE-SYRTE laboratory(CNRS/LNE/Paris Observatory/UPMC) is in-volved in many aspects of the development ofatomic clocks and their applications.

In section 2 we present the latest develop-ments in the field of atomic clocks: microwaveclocks, optical clocks, their relation to inter-national time-scales, means of comparisonsand applications. Section 3 is an introductionto relativistic time transfer, the modelizationof remote frequency comparisons, which liesat the heart of many applications of atomicclocks, such as the realization of internationaltime-scales and chronometric geodesy. Fi-nally, section 4 is a detailed review of the fieldof chronometric geodesy, an old idea whichcould become reality in the near future.

2 Atomic clocks

In 1967, the definition of the SI secondwas changed from astronomical references toatomic references by setting the frequency ofan hyperfine transition in the Cs atom [39].Since then, the accuracy of atomic clocks hasimproved by five orders of magnitude, en-abling better and better time-keeping. Morerecently, a new generation of atomic clocks,based on atomic transitions in the opticaldomain are challenging the well establishedCs standard and thus offer opportunities fornew applications in fundamental physics andgeodesy.

2.1 Microwave clocksIn a microwave atomic frequency standard, amicrowave electro-magnetic radiation excitesan hyperfine electronic transition in the groundstate of an atomic species. Observing the frac-tion of excited atoms p after this interaction (ortransition probability) gives an indicator of thedifference between the frequency ν of the mi-crowave radiation and the frequency ν0 of thehyperfine atomic transition. This frequencydifference ν − ν0 (or error signal) is fed in aservo-loop that keeps the microwave radiationresonant with the atomic transition. Accord-ing to Fourier’s relation, the frequency reso-lution that can be achieved after such an in-terrogation procedure grows as the inverse ofthe interaction time T , and since consecutiveinterrogations are uncorrelated, the frequencyresolution further improves as the square rootof the total integration time τ . Quantitatively,the residual frequency fluctuation of the mi-crowave radiation locked on the atomic reso-nance are (in dimension-less fractional units,that is to say divided by the microwave fre-quency):

σy(τ) =ξ

ν0T√N

√Tcτ, (1)

where Tc is the cycle time (such that τ/Tc isthe number of clock interrogations), and N isthe number of simultaneously (and indepen-dently) interrogated atoms. ξ is a numericalconstant, close to unity, that depends on thephysics of the interaction between the radi-ation and the atoms. This expression is theultimate frequency (in)stability of an atomicclock, also called the Quantum ProjectionNoise (QPN) limit, referring to the quantumnature of the interaction between the radiationand the atoms. It is eventually reached if all

2

other sources of noise in the servo-loop aremade negligible.

As seen from eq. (1), an efficient way to im-prove the clock stability is to increase the in-teraction time T . The first atomic clocks there-fore comprised a long tube in which a thermalbeam of Cs atoms is travelling while interact-ing with the microwave radiation.

The advance in the physics of cold atomsenabled to prepare atoms with a smaller ve-locity and consequently largely increase theinteraction time, thus the frequency stabilityof atomic standards. In cold atoms clocks,the interrogation time is only limited by thetime after which the atoms, exposed to grav-ity, escape the interrogation zone. In theatomic fountain clock (see fig. 2), a set ofcold atoms are launched vertically in the inter-action zone and are interrogated during theirparabolic flight in the Earth gravitational field.

In micro-gravity, atoms are not subject tothe Earth gravitational field and thus can beinterrogated during a longer time window T .This will be the case for the Pharao space mi-crowave clock [7].

The reproducibility of an atomic clockstems from the fact that all atoms of agiven species are rigorously indistinguishable.For instance, two independent atomic clocksbased on caesium will produce the same fre-quency (9,192,631,770 Hz exactly in the SIunit system), regardless of their manufactureror their location in space-time.

However, in laboratory conditions, atomsare surrounded by an experimental environ-ment that perturbs their electronic state andthus slightly modify their resonance frequencyin a way that depends on experimental condi-tions. For instance, atoms will interact withDC or AC electro-magnetic perturbations, willbe sensitive to collisions between them in a

Inte

ract

ion

Detection

Source

Coolin

g

RamseyFringes

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10-13

1 10 100 1000 10000

Fra

cti

on

al A

llan

de

via

tio

n

Integration time τ (s)

Frequencystability σ (τ)y

Figure 2: Atomic fountain clock. About 106

cold atoms, at a temperature of 1 µK arelaunched upwards by pushing laser beams. Atthe beginning and the end of their trajectory,they interact with a microwave radiation in aresonator and the hyperfine transition prob-ability is measured by an optical detection.When scanning the microwave frequency, thetransition probability follows a fringe (Ram-sey) pattern much like a double slit interfer-ence pattern. During the clock operation, themicrowave frequency is locked on the top ofthe central fringe. The frequency stability,defined by eq. (1), is a few 10−14 after 1 s,and a statistical resolution down to 10−16 isreached after a few days of continuous oper-ation [12, 30].3

way that depends on the atomic density or tothe Doppler effect resulting from their residualmotion. . . Such systematic effects, if not eval-uated and correctd for, limit the universality ofthe atomic standard. Consequently, the accu-racy of a clock quantifies the uncertainty onthese systematic effects. Currently, the bestmicrowave atomic fountain clocks have a rel-ative accuracy of 2 × 10−16, which is pre-sumably their ultimate performances given themany technical obstacles to further improvethis accuracy.

2.2 SI second and time scalesAbout 250 clocks worldwide, connectedby time and frequency transfer techniques(mainly through GNSS signals) realize a com-plete architecture that enable the creation ofatomic time scales. For this, local timescales physically generated by metrology lab-oratories are a posteriori compared and com-mon time-scales are decided upon. the In-ternational Bureau for Weights and Measures(BIPM) in Sevres (France) is responsible forestablishing the International Atomic Time(TAI). First a free atomic time-scale is build,the EAL. However, this time-scale is free-running and the participating clocks do notaim at realizing the SI second. The rate ofEAL is measured by comparison with a fewnumber of caesium atomic fountains whichaim at realizing the SI second, and TAI is thenderived from EAL by applying a rate correc-tion, so that the scale unit of TAI is the SIsecond as realized on the rotating geoid [11].It necessary to take into account the atomicfontain clocks frequency shift due to relativ-ity (see section 3). Recently, a change ofparadigm occured in the definition of TAI. Ac-cording to UAI resolutions [36], TAI is a real-

ization of Terrestrial Time (TT), which is de-fined by applying a constant rate correction toGeocentric Coordinate Time (TCG). This def-inition has been adopted so that the referencesurface of TAI is no longer the geoid, whichis not a stable surface (see section 4). Finally,the Coordinated Universal Time (UTC) differsfrom TAI by an integer number of second inorder to follow the irregularities of the Earthrotation.

The publication of such time-scales en-ables world-wide comparison of Cs fountainclocks [30, 31] and the realization of the SIsecond.

2.3 Optical clocksA new generation of atomic clocks have ap-peared in the last 15 years. These clocks con-sist in locking an electromagnetic radiation inthe optical domain (ν = 300 to 800 THz) toa narrow electronic transition. As seen fromeq. (1), increasing the frequency ν0 of theclock frequency by several orders of magni-tude drastically improves the ultimate clockstability, even though the number of interro-gated atoms N is usually smaller in opticalclocks. Increasing the clock frequency alsoimproves the clock accuracy since most sys-tematic effects (sensitivity to DC electromag-netic fields, cold collisions. . . ) are of thesame order of magnitude in frequency units,and thus decrease in relative units. However,two notable exceptions remain, and both havetriggered recent research in optical clocks.First, the sensitivity of the clock transition fre-quency on the ambient black-body radiationis mostly rejected in microwave clocks butnot for optical transitions. Thus, the uncer-tainty on this effect usually only marginallyimproves when going to optical clocks (with

4

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Ultra-stablecavity

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00.10.20.30.40.50.60.70.80.9

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Figure 3: Sr optical lattice clock. The clocklaser, prestabilized on an ultra-stable opti-cal cavity, probes an optical transition of 104

atoms confined in an optical lattice. The ex-citation fraction (or transition probability) isdetected by a fluorescence imaging and a nu-merical integrator acts on a frequency shifter(FS) to keep the laser on resonance with theatomic transition. The width of the resonanceis Fourier-limited at 3 Hz, which, given theclock frequency ν0 = 429 THz, yields a reso-nance quality factorQ = 1.4×1014, comparedto 1010 for microwave clocks.

the notable exception of a few atomic speciessuch as Al+ for which the sensitivity is acci-dentally small). Therefore, extra care has to betaken to control the temperature of the atomsenvironment. Second, the Doppler frequencyshift δν due to the residual velocity v of theatoms scales as the clock frequency ν0, suchthat the relative frequency shift remains con-stant:

δν

ν0=v

c(2)

For this reason, the fountain architecture, forwhich the Doppler effect is one of the limita-tions, cannot be applicable to optical clocks,and the atoms have to be tightly confined ina trapping potential to cancel their velocity v.To achieve this goal, two different technolo-gies have been developed. First, a single ionis trapped in a RF electric field. These ion op-tical clocks [34, 8, 16, 25] achieve record ac-curacies at 9 × 10−18. However, since only asingle ion is trapped (N = 1, because of theelectrostatic repulsion of ions), the stability oftheses clocks is limited at the QPN level of 2×10−15 at 1 s. Second, a more recent technol-ogy involves trapping of a few thousands neu-tral atoms in a powerful laser standing wave(or optical lattice) by the dipolar force. Due toits power, this trapping potential is highly per-turbative, but for a given ”magic” wavelengthof the trapping light [38], the perturbation isequal for the two clock levels, hence cancelledfor the clock transition frequency. These opti-cal lattice clocks [38, 21, 29, 23, 26] have al-ready reached an accuracy of 1×10−16 and arerapidly catching up with ion clocks. Further-more, the large number of interrogated atomsallowed the demonstration of unprecedentedstabilities (a few 1 × 10−16 at 1 s), headingtoward their QPN below 1× 10−17 at 1 s.

5

2.4 Remote comparison of opticalfrequencies

The recent breakthrough of performances ofoptical clocks was permitted by the develop-ment of frequency combs [44], which real-izes a ”ruler” in the frequency domain. Theselasers enable the local comparison of differentoptical frequencies and comparisons betweenoptical and microwave frequencies. However,although optical clocks now largely surpassmicrowave clocks, a complete architecture hasto be established to enable remote compari-son of optical frequencies, in order to validatethe accuracy of optical clocks, build ”optical”time-scales, or enable applications of opticalclocks such as geodesic measurements Indeed,the conventional remote frequency compar-isons techniques, mainly through GPS links,cannot reach the level of stability and accuracyrealized by optical clocks.

Fibre links

To overcome the limitations of remote clockcomparisons using GNSS signals, comparisontechniques using optical fibres are being de-veloped. For this, a stable and accurate fre-quency signal produced by an optical clock issent through a fibre optics that links metrol-ogy institutes, directly encoded in the phase ofthe optical carrier. Because of vibrations andtemperature fluctuations, the fibre adds a sig-nificant phase noise to the signal. This addednoise is measured by comparing the signal af-ter a round trip in the fibre to the original sig-nal, and subsequently cancelled. Such a sig-nal can be transported through a dedicated fi-bre (dark fibre) when available [33], or, morepractically along with the internet communi-cation (dark channel) [24]. These compari-

son techniques are applicable at a continentalscale ; such a network will presumably be inoperation throughout Europe in the near fu-ture. In particular, the REFIMEVE+ projectwill provide a shared stable optical oscilla-tor between a large number of laboratories inFrance, with a number of applications besidemetrology. This network will be connectedto international fibre links (NEAT-FT project1)that will enable long distance comparisons be-tween optical frequency standards located invarious national metrology laboratories in Eu-rope. The ITOC project2 aims at collectingthese comparisons result to demonstrate highaccuracy frequency ratios measurements andgeophysical applications.

Space-based links

When considering inter-continental time andfrequency comparisons, only satellite link areconceivable. In this aspect, The two-waysatellite time and frequency transfer (TW-STFT) involves a satellite that actively relays afrequency signal in a round-trip configuration.Also, the space mission Pharao-ACES [7] thatinvolves an ensemble of clocks on board theInternational Space Station will comprise anumber of ground receiver able to remotelycompare optical clocks.

2.5 Towards a new definition ofthe SI second based on opticalclocks

Unlike microwave clocks, for which Cs hasbeen an unquestionable choice over half a cen-

1http://www.ptb.de/emrp/neatft_home.html

2http://projects.npl.co.uk/itoc/

6

tury, a large number of atomic species seem tobe equally matched as a new frequency stan-dard based on an optical transition. Ion clockswith Hg+, Al+, Yb+, Sr+, Ca+ have beendemonstrated, as well as optical lattice clockswith Sr, Yb and more prospectively Hg andMg. As an illustration, four optical transitionshave already been approved by the CIPM (In-ternational Committee for Weights and Mea-sures) as secondary representation of the SIsecond. Therefore, although the SI secondwould already gain in precision with opticalclock, a clear consensus has yet to emerge be-fore the current microwave defined SI secondcan be replaced.

2.6 Applications of optical clocksThe level of accuracy reached by opti-cal clocks opens a new range of applica-tions, through the very precise frequency ra-tios measurements they enable. In funda-mental physics, they enable the tracking ofdimension-less fundamental constants such asthe fine structure constant α, the electron toproton mass ratio µ = me/mp, or the quantumchromodynamics mass scale mq/ΛQCD. Be-cause each clock transition frequency have adifferent dependence on these constants, theirvariations imply drifts in clock frequency ra-tios that can be detected by repeated mea-surements. Currently, combined optical-to-optical and optical-to-microwave clock com-parisons put an upper bound on the rela-tive variation of fundamental constants in the10−17/yr range [34, 13, 21, 22]. Other testsof fundamental physics are also possible withatomic clocks. The local position invariancecan be tested by comparing frequency ratiosin the course of the earth rotation around theearth [13, 21], and the gravitational red-shift

will be tested with clocks during the Pharao-ACES mission.

The TAI time scale is created from Cs stan-dards, but recently, the Rb microwave tran-sition started to contribute. In the near fu-ture, secondary representations of the SI sec-ond based on optical transitions could con-tribute to TAI, which would thus benefit fromtheir much improved stability.

Clock frequencies, when compared to a co-ordinate time-scale, are sensitive to the gravi-tational potential of the Earth (see section 3).Therefore it is necessary to take into accounttheir relative height difference when build-ing atomic time-scales such as TAI. However,the most accurate optical clocks can resolvea height difference below 10 cm, which is ascale at which the global gravitational poten-tial is unknown. Because of this, optical clockcould become a tool to precisely measure thispotential. They come as a decisive additionto relative and absolute gravimeters which aresensitive to the gravitational field, and to satel-lite based measurement which lack spatial res-olution (see section 4).

3 Relativistic frequencytransfer

In distant comparisons of frequency standards,we are face with the problem of curvature ofspace and relative motion of the clocks. Thesetwo effects change locally the flow of propertime with respect to a global coordinate time.In this section we describe how to comparedistant clock frequencies by means of an elec-tromagnetic signal, and how the comparison isaffected by these effects.

7

3.1 The Einstein EquivalencePrinciple

a) b)

Figure 4: A photon of frequency νA is emitted atpoint A toward point B, where the measured frequencyis νB . a) A and B are two points at rest in an acceler-ated frame, with acceleration ~a in the same direction asthe emitted photon. b) A and B are at rest in a non ac-celerated (locally inertial) frame in presence of a gravi-tational field such that ~g = −~a.

Before we treat the general case, let’s tryto understand in simple terms what is the fre-quency shift effect. Indeed, it can be seen asa direct consequence of the Einstein Equiv-alence Principle (EEP), one of the pillars ofmodern physics [27, 41]. Let’s consider a pho-ton emitted at a point A in an accelerated ref-erence system, toward a point B which lies inthe direction of the acceleration (see fig.4). Weassume that both point are separated by a dis-tance h0, as measured in the accelerated frame.The photon time of flight is δt = h0/c, and theframe velocity during this time increases byδv = aδt = ah0/c, where a is the magnitudeof the frame acceleration ~a. The frequency atpoint B (reception) is then shifted because ofDoppler effect, compared to the frequency at

point A (emission), by an amount:

νBνA

= 1− δv

c= 1− ah0

c2(3)

Now, the EEP postulates that a gravitationalfield ~g is locally equivalent to an accelerationfield ~a = −~g. We deduce that in a non accel-erated (locally inertial) frame in presence of agravitational field ~g:

νBνA

= 1− gh0c2

(4)

where g = |~g|, νA is the photon frequencyat emission (strong gravitational potential) andνB is the photon frequency at reception (weakgravitational field). As νB < νA, it is usual tosay that the frequency at the point of receptionis “red-shifted”. One can consider it in termsof conservation of energy. Intuitively, the pho-ton that goes from A to B has to “work” tobe able to escape the gravitational field, then itlooses energy and its frequency decreases byvirtue of E = hν, with h the Planck constant.

Figure 5: Two clocks A and B are measuring propertime along their trajectory. One signal with phase S isemitted by A at proper time τA, and another one withphase S + dS at time τA + dτA. They are received byclock B respectively at time τB and τ + dτB .

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3.2 General case

The principle of frequency comparison is tomeasure the frequency of an electromagneticsignal with the help of the emitting clock, A,and then with the receiving clock, B. Weobtain respectively two measurements νA andνB

3. Let S(xα) be the phase of the electro-magnetic signal emitted by clock A. It canbe shown that light rays are contained in hy-persurfaces of constant phase. The frequencymeasured by A/B is:

νA/B =1

2π

dSdτA/B

(5)

where τA/B is the proper time along the world-line of clock A/B (see fig.5). We introducethe wave vector kA/Bα = (∂αS)A/B to obtain:

νA/B =1

2πkA/Bα uαA/B (6)

where uαA/B = dxαA/B/dτ is the four-velocityof clock A/B. Finally, we obtain a fundamen-tal relation for frequency transfer:

νAνB

=kAαu

αA

kBα uαB

(7)

This formula does not depend on a particulartheory, and then can be used to perform testsof general relativity. Introducing vi = dxi/dtand ki = ki/k0, it is usually written as:

νAνB

=u0Au0B

kA0kB0

1 +kAi v

iA

c

1 +kBi v

iB

c

(8)

3However, in general one measures the time of flightof the electromagnetic signal between emission and re-ception. Then the ratio νA/νB can be obtain by derivingthe time of flight measurements.

From eq. (5) we deduce that:

νAνB

=dτBdτA

=

(dtdτ

)A

dtBdtA

(dτdt

)B

(9)

We suppose that space-time is stationary, ie.∂0gαβ = 0. Then it can be shown that k0is constant along the light ray, meaning thatkA0 = kB0 . We introduce the time transfer func-tion:

T (xiA, xiB) = tB − tA (10)

Deriving the time transfer function with re-spect to tA one obtains:

dtBdtA

=1 + ∂T

∂xiAviA

1− ∂T∂xiB

viB(11)

Inserting eq. (11) in eq. (9), and comparingwith eq. (8), we deduce:

kAi = c∂T∂xiA

, kBi = −c ∂T∂xiB

(12)

General formula for non-stationary space-times can be found in [14, 20].

As an exemple, let’s take the simple timetransfer function:

T (xiA, xiB) =

RAB

c+O

(1

c3

)(13)

where RAB = |RiAB| and Ri

AB = xiB − xiA.Then we obtain:

dtBdtA

=1 +

~NAB ·~vAc

+O(

1c3

)1 +

~NAB ·~vBc

+O(

1c3

) (14)

where N iAB = Ri

AB/RAB. Up to the secondorder, this term does not depend on the gravi-tational field but on the relative motion of thetwo clocks. It is simply the first order Dopplereffect.

9

The two other terms in eq. (9) depend onthe relation between proper time and coordi-nate time. In a metric theory one has c2dτ =√gαβdxαdxβ . We deduce that:

u0Au0B

=

[g00 + 2g0i

vi

c+ gij

vivj

c2

]1/2B[

g00 + 2g0ivi

c+ gij

vivj

c2

]1/2A

(15)

3.3 Application to a static, spheri-cally symmetric body

As an example, we apply this formalism forthe case of a parametrized post-Minkowskianapproximation of metric theories. This met-ric is a good approximation of the space-timemetric around the Earth, for a class of theorieswhich is larger than general relativity. Choos-ing spatial isotropic coordinates, we assumethat the metric components can be written as:

g00 = −1 + 2(α + 1)GM

rc2+O

(1

c4

)g0i = 0 (16)

gij = δij

[1 + 2γ

GM

rc2+O

(1

c4

)]where α and γ are two parameters of the the-ory (for general relativity α = 0 and γ = 1).Then:

dτdt

= 1−(α+1)GM

rc2− v2

2c2+O

(1

c4

)(17)

where v =√δijvivj . The time transfer func-

tion of such a metric is given in eq. (108)of [20]. Then we calculate:

dtBdtA

=qA +O

(1c5

)qB +O

(1c5

) (18)

where

qA =1−~NAB · ~vA

c− 2(γ + 1)GM

c3× (19)

RAB~NA · ~vA + (rA + rB) ~NAB · ~vA

(rA + rB)2 −R2AB

qB =1−~NAB · ~vB

c− 2(γ + 1)GM

c3× (20)

RAB~NA · ~vB − (rA + rB) ~NAB · ~vB

(rA + rB)2 −R2AB

Let assume that both clocks are at rest withrespect to the chosen coordinate system, ie.~vA = ~0 = ~vB and that rA = r0 and rB =r0 + δr, where δr � r0. Then we find:

νA − νBνB

= (α + 1)GM

r20c2δr +O

(1

c4

)(21)

The same effect can be calculated with a dif-ferent, not necessarily symmetric gravitationalpotential w(t, xi). The results yields:

νA − νBνB

=wA − wB

c2+O

(1

c4

)(22)

This is the classic formula given in textbooksfor the “gravitational red-shift”. However oneshould bear in mind that this is valid for clocksat rest with respect to the coordinate sys-tem implicitly defined by the space-time met-ric (one uses usually the Geocentric CelestialReference System), which is (almost) neverthe case. Moreover, the separation betweena gravitational red-shift and a Doppler effectis specific to the approximation scheme usedhere. One can read the book by Synge [37] fora different interpretation in terms of relativevelocity and Doppler effect only.

We note that the lowest order gravitationalterm in eq. (21) is a test of the Newtonian limit

10

of metric theories. Indeed, if one wants torecover the Newtonian law of gravitation forGM/rc2 � 1 and v � 1 then it is necessarythat α = 0. Then this test is more fundamentalthan a test of general relativity, and can be in-terpreted as a test of Local Position Invariance(which is a part of the Einstein EquivalencePrinciple). See [41, 42] for more details onthis interpretation, and a review of the experi-ments that tested the parameter α.

A more realistic case of the space-time met-ric is treated in article [3], in the context ofgeneral relativity: all terms from the Earthgravitational potential are considered up to anaccuracy of 5.10−17, specifically in previsionof the ACES mission [7].

4 Chronometric geodesy

Instead of using our knowledge of the Earthgravitational field to predict frequency shiftsbetween distant clocks, one can revert theproblem and ask if the measurement of fre-quency shifts between distant clocks can im-prove our knowledge of the gravitational field.To do simple orders of magnitude estimates itis good to have in mind some correspondencescalculated thanks to eqs. (21) and (22):

1 meter ↔ ∆ν

ν∼ 10−16

↔ ∆w ∼ 10 m2.s−2 (23)

From this correspondence, we can alreadyimagine two direct applications of clocks ingeodesy: if we are capable to compare clocksto 10−16 accuracy, we can determine heightdifferences between clocks with one meter ac-curacy (levelling), or determine geopotentialdifferences with 10 m2.s−2 accuracy.

4.1 A review of chronometricgeodesy

The first article to explore seriously this pos-sibility was written in 1983 by Martin Ver-meer [40]. The article is named “chronomet-ric levelling”. The term “chronometric” seemswell suited for qualifying the method of us-ing clocks to determine directly gravitationalpotential differences, as “chronometry” is thescience of the measurement of time. How-ever the term “levelling” seems too restrictivewith respect to all the applications one couldthink of using the results of clock compar-isons. Therefore we will use the term “chrono-metric geodesy” to name the scientific disci-pline that deals with the measurement and rep-resentation of the Earth, including its gravita-tional field, with the help of atomic clocks. Itis sometimes named “clock-based geodesy”,or “relativistic geodesy”. However this lastdesignation is improper as relativistic geodesyaims at describing all possible techniques (in-cluding e.g. gravimetry and gradiometry) in arelativistic framework [18, 28, 35].

The natural arena of chronometric geodesyis the four-dimensional space-time. At thelowest order, there is proportionality betweenrelative frequency shift measurements – cor-rected from the first order Doppler effect –and (Newtonian) geopotential differences (seeeq.(22)). To calculate this relation we haveseen that we do not need the theory of gen-eral relativity, but only to postulate Local Posi-tion Invariance. Therefore, it is perfectly pos-sible to use clock comparison measurements –corrected from the first order Doppler effect –as a direct measurement of geopotential differ-ences in the framework of classical geodesy, ifthe measurement accuracy does not reach themagnitude of the higher order terms.

11

Comparisons between two clocks on theground generally use a third clock in space.For the comparison between a clock on theground and one in space, the terms of orderc−3 in eqs. (19)-(20) reach a magnitude of∼ 10−15 and ∼ 3 × 10−14 for respectively theground and the space clock, if they are sep-arated radially by 1000 km. Terms of orderc−4 omitted in eqs. (21)-(22) can reach ∼ 5×10−19 in relative frequency shift, which corre-sponds to a height difference of∼ 5 mm and ageopotential difference of ∼ 5× 10−2 m2.s−2.Clocks are far from reaching this accuracy to-day, but it cannot be excluded for the future.

In his article, Martin Vermeer explores the“possibilities for technical realisation of a sys-tem for measuring potential differences overintercontinental distances” using clock com-parisons [40]. The two main ingredients are ofcourse accurate clocks and a mean to comparethem. He considers hydrogen maser clocks.For the links he considers a 2-way satellitelink over a geostationary satellite, or GPS re-ceivers in interferometric mode. He has alsoto consider a mean to compare the properfrequencies of the different hydrogen maserclocks. However today this can be overcomeby comparing Primary Frequency Standards(PFS), which have a well defined proper fre-quency based on a transition of Caesium 133,used for the definition of the second. How-ever, this problem will rise again if one usesSecondary Frequency Standards which are notbased on Caesium atoms. Then the proper fre-quency ratio between two different kinds ofatomic clocks has to be determined locally.This is one of the purpose of the Europeanproject “International timescales with opticalclocks”4, where optical clocks based on dif-

4http://projects.npl.co.uk/itoc/

ferent atoms will be compared one each otherlocally, and to the PFS. It is planned also to doa proof-of-principle experiment of chronomet-ric geodesy, by comparing two optical clocksseparated by a height difference of around1 km using an optical fibre link.

In the foreseen applications of chronomet-ric geodesy, Martin Vermeer mentions brieflyintercontinental levelling, and the measure-ment of the true geoid (disentangled from geo-physical sea surface effects) [40]. Few au-thors have seriously considered chronometricgeodesy. Following Vermeer idea, Brumbergand Groten [5] demonstrated the possibility ofusing GPS observations to solve the problemof the determination of geoid heights, howeverleaving aside the practical feasibility of such atechnique. Bondarescu et al. [4] discuss thevalue and future applicability of chronometricgeodesy, including direct geoid mapping oncontinents and joint gravity-geopotential sur-veying to invert for subsurface density anoma-lies. They find that a geoid perturbationcaused by a 1.5 km radius sphere with 20 percent density anomaly buried at 2 km depthin the Earth’s crust is already detectable byatomic clocks of achievable accuracy. FinallyChou et al. demonstrated the potentiality ofthe new generation of atomic clocks, based onoptical transitions, to measure heights with aresolution of around 30 cm [9].

4.2 The chronometric geoidArne Bjerhammar in 1985 gives a precise def-inition of the “relativistic geoid” [1, 2]:

“The relativistic geoid is the surfacewhere precise clocks run with thesame speed and the surface is near-est to mean sea level”

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This is an operational definition. Soffel etal. [35] in 1988 translated this definition inthe context of post-Newtonian theory. Theyalso introduce a different operational defini-tion of the relativistic geoid, based on gravi-metric measurements: a surface orthogonaleverywhere to the direction of the plumb-lineand closest to mean sea level. He calls the twosurfaces obtained with clocks and gravimetricmeasurements respectively the “u-geoid” andthe “a-geoid”. He proves that these two sur-faces coincide in the case of a stationary met-ric. In order to distinguish the operational defi-nition of the geoid from its theoretical descrip-tion, it is less ambiguous to give a name basedon the particular technique to measure it. Rel-ativistic geoid is too vague as Soffel et al. havedefined two different ones. The names chosenby Soffel et al. are not particularly explicit,so instead of “u-geoid” and “a-geoid” one cancall them chronometric and gravimetric geoidrespectively. There can be no confusion withthe geoid derived from satellite measurements,as this is a quasi-geoid that do not coincidewith the geoid on the continents [15]. Otherconsiderations on the chronometric geoid canbe found in [18, 19, 28].

Let two clocks be at rest with respect tothe chosen coordinate system (vi = 0) in anarbitrary space-time. From formula (9), (11)and (15) we deduce that:

νAνB

=dτBdτA

=[g00]

1/2B

[g00]1/2A

(24)

In this case the chronometric geoid is definedby the condition g00 = cst.

We notice that the problem of defining a ref-erence surface is closely related to the prob-lem of realizing Terrestrial Time (TT). TT isdefined with respect to Geocentric Coordinate

Time (TCG) by the relation [36, 32]:

dTTdTCG

= 1− LG (25)

where LG is a defining constant. This constanthas been chosen so that TT coincides with thetime given by a clock located on the classi-cal geoid. It could be taken as a formal def-inition of the chronometric geoid [43]. If so,the chronometric geoid will differ in the fu-ture from the classical geoid: a level surfaceof the geopotential closest to the topographicmean sea level. Indeed, the value of the poten-tial on the geoid, W0, depends on the globalocean level which changes with time [6]. Witha value of dW0/dt ∼ 10−3 m2.s−2.y−1, thedifference in relative frequency between theclassical and the chronometric geoid wouldbe around 10−18 after 100 years (the cor-respondence is made with the help of rela-tions (23)). However, the rate of change ofthe global ocean level could change during thenext decades. Predictions are highly model de-pendant [17].

5 ConclusionWe presented recent developments in the fieldof atomic clocks, as well as an introductionto relativistic frequency transfer and a detailedreview of chronometric geodesy. If the con-trol of systematic effects in optical clocks keeptheir promises, they could become very sensi-ble to the gravitational field, which will ulti-mately degrade their stability at the surface ofthe Earth. One solution will be to send verystable and accurate clocks in space, which willbecome the reference against which the Earthclocks would be compared. Moreover, bysending at least four of these clocks in space,

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it will be possible to realize a very stable andaccurate four-dimensional reference system inspace [10].

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