3.1 Atomic clocks

3.1.1 Atomic spectra and constants

The laboratory constraints on the time variation of fundamental constants are obtained by comparing the long-term behavior of several oscillators and rely on frequency measurements. The atomic transitions have various dependencies in the fundamental constants. For instance, for the hydrogen atom, the gross, fine and hyperfine-structures are roughly given by

2 2 2p − 1s : ν ∝ cR ∞, 2p3∕2 − 2p1∕2 : ν ∝ cR∞ α EM, 1s : ∝ cR∞ α EMgpμ¯,
respectively, where the Rydberg constant set the dimension. gp is the proton gyromagnetic factor and ¯μ = me∕mp. In the non-relativistic approximation, the transitions of all atoms have similar dependencies but two effects have to be taken into account. First, the hyperfine-structures involve a gyromagnetic factor gi (related to the nuclear magnetic moment by μi = giμN, with μN = eℏ ∕2mpc), which are different for each nuclei. Second, relativistic corrections (including the Casimir contribution), which also depend on each atom (but also on the type of the transition) can be included through a multiplicative function Frel(αEM ). It has a strong dependence on the atomic number Z, which can be illustrated on the case of alkali atoms, for which
[ ]−1∕2[ 4 ]−1 11 Frel(αEM ) = 1 − (ZαEM )2 1 − -(Z αEM )2 ≃ 1 + --(Z αEM )2. 3 6
The developments of highly accurate atomic clocks using different transitions in different atoms offer the possibility to test a variation of various combinations of the fundamental constants.

It follows that at the lowest level of description, we can interpret all atomic clocks results in terms of the g-factors of each atoms, gi, the electron to proton mass ration μ and the fine-structure constant αEM. We shall parameterize the hyperfine and fine-structures frequencies as follows.

The hyperfine frequency in a given electronic state of an alkali-like atom, such as 133Cs, 87Rb, 199Hg+, is

νhfs ≃ R ∞c × Ahfs × gi × α2EM × ¯μ × Fhfs(α) (21 )
where gi = μi∕ μN is the nuclear g factor. Ahfs is a numerical factor depending on each particular atom and we have set Frel = Fhfs(α ). Similarly, the frequency of an electronic transition is well-approximated by
νelec ≃ R ∞c × Aelec × Felec(Z, α), (22 )
where, as above, Aelec is a numerical factor depending on each particular atom and Felec is the function accounting for relativistic effects, spin-orbit couplings and many-body effects. Even though an electronic transition should also include a contribution from the hyperfine interaction, it is generally only a small fraction of the transition energy and thus should not carry any significant sensitivity to a variation of the fundamental constants.

The importance of the relativistic corrections was probably first emphasized in [423] and their computation through relativistic N-body calculations was carried out for many transitions in [170, 174Jump To The Next Citation Point, 175Jump To The Next Citation Point, 198Jump To The Next Citation Point]. They can be characterized by introducing the sensitivity of the relativistic factors to a variation of αEM,

κα ≡ --∂ ln-F-. (23 ) ∂ lnαEM
Table 6 summarizes the values of some of them, as computed in [175Jump To The Next Citation Point, 210Jump To The Next Citation Point]. Indeed a reliable knowledge of these coefficients at the 1% to 10% level is required to deduce limits to a possible variation of the constants. The interpretation of the spectra in this context relies, from a theoretical point of view, only on quantum electrodynamics (QED), a theory, which is well tested experimentally [280] so that we can safely obtain constraints on (αEM, μ,gi), still keeping in mind that the computation of the sensitivity factors required numerical N-body simulations.

Table 6: Sensitivity of various transitions on a variation of the fine-structure constant.


Transition sensitivity κ α


1s − 2s 0.00


hf 0.34


2S1∕2(F = 2)− (F = 3) 0.83


2S1∕2 − 2D3∕2 0.9


2S − 2D 1∕2 5∕2 –3.2


1 3 S0 − P0 0.06


1S − 3P 0 0 0.008

From an experimental point of view, various combinations of clocks have been performed. It is important to analyze as many species as possible in order to rule-out species-dependent systematic effects. Most experiments are based on a frequency comparison to caesium clocks. The hyperfine splitting frequency between the F = 3 and F = 4 levels of its 2S1∕2 ground state at 9.192 GHz has been used for the definition of the second since 1967. One limiting effect, that contributes mostly to the systematic uncertainty, is the frequency shift due to cold collisions between the atoms. On this particular point, clocks based on the hyperfine frequency of the ground state of the rubidium at 6.835 GHz, are more favorable.

3.1.2 Experimental constraints

We present the latest results that have been obtained and refer to Section III.B.2 of FCV [500Jump To The Next Citation Point] for earlier studies. They all rely on the developments of new atomic clocks, with the primarily goal to define better frequency standards.

Table 7: Summary of the constraints obtained from the comparisons of atomic clocks. For each constraint on the relative drift of the frequency of the two clocks, we provide the dependence in the various constants, using the numbers of Table 6. From Ref. [379], which can be consulted for other constants.

Clock 1

Clock 2 Constraint (yr–1) Constants dependence Reference


( ) -dln νclock1- dt νclock2      


133Cs (0.2 ± 7.0) × 10–16 ggCRsbα0E.M49 [346Jump To The Next Citation Point]


133Cs (–0.5 ± 5.3) × 10–16 [58Jump To The Next Citation Point]


133Cs (–32 ± 63) × 10–16 gCs¯μα2E.M83 [196Jump To The Next Citation Point]


133Cs (0.2 ± 7) × 10–15 gCs¯μα6E.M05 [57Jump To The Next Citation Point]


133Cs (3.7 ± 3.9) × 10–16 [214Jump To The Next Citation Point]


133Cs (–1.2 ± 4.4) × 10–15 gCs¯μα1E.M93 [408Jump To The Next Citation Point]


133Cs (–0.78 ± 1.40) × 10–15 [407Jump To The Next Citation Point]


133Cs (–1.0 ± 1.8) × 10–15 gCs¯μα2E.M77 [61Jump To The Next Citation Point]


87Dy (–2.7 ± 2.6) × 10–15 αEM [100Jump To The Next Citation Point]


199Hg+ (–5.3 ± 7.9) × 10–17 α−EM3.208 [440Jump To The Next Citation Point]

View Image

Figure 2: Evolution of the comparison of different atomic clocks summarized in Table 7.

While the constraint (33View Equation) was obtained directly from the clock comparison, the other studies need to be combined to disentangle the contributions of the various constants. As an example, we first use the bound (33View Equation) on αEM, we can then extract the two following bounds

( ) d-- gCs- − 16 −1 d-- −16 − 1 dt ln g = (0.48 ± 6.68) × 10 yr , dt ln (gCs¯μ) = (4.67 ± 5.29 ) × 10 yr , (34 ) Rb
on a time scale of a year. We cannot lift the degeneracies further with this clock comparison, since that would require a constraint on the time variation of μ. All these constraints are summarized in Table 7 and Figure 2View Image.

A solution is to consider diatomic molecules since, as first pointed out by Thomson [488Jump To The Next Citation Point], molecular lines can provide a test of the variation of μ. The energy difference between two adjacent rotational levels in a diatomic molecule is inversely proportional to M r−2, r being the bond length and M the reduced mass, and the vibrational transition of the same molecule has, in first approximation, a √ --- M dependence. For molecular hydrogen M = mp ∕2 so that the comparison of an observed vibro-rotational spectrum with a laboratory spectrum gives an information on the variation of mp and mn. Comparing pure rotational transitions with electronic transitions gives a measurement of μ. It follows that the frequency of vibro-rotation transitions is, in the Born–Oppenheimer approximation, of the form

( √ -- ) ν ≃ EI celec + cvib ¯μ + crot¯μ (35 )
where celec, cvib and crot are some numerical coefficients.

The comparison of the vibro-rotational transition in the molecule SF6 was compared to a caesium clock over a two-year period, leading to the constraint [464]

( ) d-ln νSF6 = (1.9 ± 0.12 ± 2.7) × 10 −14 yr− 1, (36 ) dt νCs
where the second error takes into account uncontrolled systematics. Now, using again Table 6, we deduce that
νSF6 1∕2 − 2.83 −1 ν ∝ ¯μ αEM (gCs¯μ) . Cs
It can be combined with the constraint (26View Equation), which enjoys the same dependence to cesium to infer that
˙μ-= (− 3.8 ± 5.6) × 10 −14 yr− 1. (37 ) μ
Combined with Equation (34View Equation), we can obtain independent constraints on the time variation of gCs, gRb and μ.

3.1.3 Physical interpretation

The theoretical description must be pushed further if ones wants to extract constraints on constant more fundamental than the nuclear magnetic moments. This requires one to use quantum chromodynamics. In particular, it was argued than within this theoretical framework, one can relate the nucleon g-factors in terms of the quark mass and the QCD scale [198Jump To The Next Citation Point]. Under the assumption of a unification of the three non-gravitational interaction (see Section 6.3), the dependence of the magnetic moments on the quark masses was investigated in [210Jump To The Next Citation Point]. The magnetic moments, or equivalently the g-factors, are first related to the ones of the proton and a neutron to derive a relation of the form

ap an g ∝ g p g n .
[198, 210Jump To The Next Citation Point] argued that these g-factors mainly depend on the light quark mass m = 1(m + m ) q 2 u d and ms, respectively for the up, down and strange quarks, that is in terms of Xq = mq ∕ΛQCD and Xs = ms ∕ΛQCD. Using a chiral perturbation theory, it was deduced, assuming ΛQCD constant, that
g ∝ X −0.087X −0.013, g ∝ X − 0.118X0.0013, p q s n q s
so that for a hyperfine transition
2+ κα κq κs νhfs ∝ α EM X q X sμ¯.
Both coefficients can be computed, leading to the possibility to draw constraints on the independent time variation of Xq, Xs and Xe.

To simplify, we may assume that Xq ∝ Xs, which is motivated by the Higgs mechanism of mass generation, so that the dependence in the quark masses reduces to κ = 1(κ + κ ) 2 q s. For instance, we have

κCs = 0.009, κRb = − 0.016, κH = − 0.10.
For hyperfine transition, one further needs to take into account the dependence in μ that can be described [204Jump To The Next Citation Point] by
0.037 0.011 mp ∼ 3ΛQCDX q Xs ,
so that the hyperfine frequencies behaves as
2+κα κ−0.048 νhfs ∝ α EM X q Xe,
in the approximation Xq ∝ Xs and where Xe ≡ me ∕ΛQCD. This allows one to get independent constraints on the independent time variation of Xe, Xq and αEM. Indeed, these constraints are model-dependent and, as an example, Table III of [210Jump To The Next Citation Point] compares the values of the sensitivity κ when different nuclear effects are considered. For instance, it can vary from 0.127, 0.044 to 0.009 for the cesium according to whether one includes only valence nucleon, non-valence non-nucleon or effect of the quark mass on the spin-spin interaction. Thus, it is a very promising framework, which still needs to be developed and the accuracy of which must be quantified in detail.

3.1.4 Future evolutions

Further progresses in a near future are expected mainly through three types of developments:

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