The laboratory constraints on the time variation of fundamental constants are obtained by comparing the longterm 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 hyperfinestructures are roughly given by
It follows that at the lowest level of description, we can interpret all atomic clocks results in terms of the gfactors of each atoms, , the electron to proton mass ration and the finestructure constant . We shall parameterize the hyperfine and finestructures frequencies as follows.
The hyperfine frequency in a given electronic state of an alkalilike atom, such as ^{133}Cs, ^{87}Rb, ^{199}Hg^{+}, is
where is the nuclear factor. is a numerical factor depending on each particular atom and we have set . Similarly, the frequency of an electronic transition is wellapproximated by where, as above, is a numerical factor depending on each particular atom and is the function accounting for relativistic effects, spinorbit couplings and manybody 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 body calculations was carried out for many transitions in [170, 174, 175, 198]. They can be characterized by introducing the sensitivity of the relativistic factors to a variation of ,
Table 6 summarizes the values of some of them, as computed in [175, 210]. 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 , still keeping in mind that the computation of the sensitivity factors required numerical body simulations.
Atom 
Transition  sensitivity 
^{1}H 
0.00  
^{87}Rb 
hf  0.34 
^{133}Cs 
0.83  
^{171}Yb^{+} 
0.9  
^{199}Hg^{+} 
–3.2  
^{87}Sr 
0.06  
^{27}Al^{+} 
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 ruleout speciesdependent systematic effects. Most experiments are based on a frequency comparison to caesium clocks. The hyperfine splitting frequency between the and levels of its 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.
We present the latest results that have been obtained and refer to Section III.B.2 of FCV [500] for earlier studies. They all rely on the developments of new atomic clocks, with the primarily goal to define better frequency standards.
Clock 1 
Clock 2  Constraint (yr^{–1})  Constants dependence  Reference 


^{87}Rb 
^{133}Cs  (0.2 ± 7.0) × 10^{–16}  [346]  
^{87}Rb 
^{133}Cs  (–0.5 ± 5.3) × 10^{–16}  [58]  
^{1}H 
^{133}Cs  (–32 ± 63) × 10^{–16}  [196]  
^{199}Hg^{+} 
^{133}Cs  (0.2 ± 7) × 10^{–15}  [57]  
^{199}Hg^{+} 
^{133}Cs  (3.7 ± 3.9) × 10^{–16}  [214]  
^{171}Yb^{+} 
^{133}Cs  (–1.2 ± 4.4) × 10^{–15}  [408]  
^{171}Yb^{+} 
^{133}Cs  (–0.78 ± 1.40) × 10^{–15}  [407]  
^{87}Sr 
^{133}Cs  (–1.0 ± 1.8) × 10^{–15}  [61]  
^{87}Dy 
^{87}Dy  (–2.7 ± 2.6) × 10^{–15}  [100]  
^{27}Al^{+} 
^{199}Hg^{+}  (–5.3 ± 7.9) × 10^{–17}  [440]  
While the constraint (33) 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 (33) on , we can then extract the two following bounds
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 . All these constraints are summarized in Table 7 and Figure 2.A solution is to consider diatomic molecules since, as first pointed out by Thomson [488], molecular lines can provide a test of the variation of . The energy difference between two adjacent rotational levels in a diatomic molecule is inversely proportional to , being the bond length and the reduced mass, and the vibrational transition of the same molecule has, in first approximation, a dependence. For molecular hydrogen so that the comparison of an observed vibrorotational spectrum with a laboratory spectrum gives an information on the variation of and . Comparing pure rotational transitions with electronic transitions gives a measurement of . It follows that the frequency of vibrorotation transitions is, in the Born–Oppenheimer approximation, of the form
where , and are some numerical coefficients.The comparison of the vibrorotational transition in the molecule SF6 was compared to a caesium clock over a twoyear period, leading to the constraint [464]
where the second error takes into account uncontrolled systematics. Now, using again Table 6, we deduce thatThe 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 factors in terms of the quark mass and the QCD scale [198]. Under the assumption of a unification of the three nongravitational interaction (see Section 6.3), the dependence of the magnetic moments on the quark masses was investigated in [210]. The magnetic moments, or equivalently the factors, are first related to the ones of the proton and a neutron to derive a relation of the form
To simplify, we may assume that , which is motivated by the Higgs mechanism of mass generation, so that the dependence in the quark masses reduces to . For instance, we have
Further progresses in a near future are expected mainly through three types of developments:
Concerning diatomic molecules, it was shown that this sensitivity can be enhanced in transitions between narrow close levels of different nature [13, 15]. In such transitions, the fine structure mainly depends on the finestructure constant, , while the vibrational levels depend mainly on the electrontoproton mass ratio and the reduced mass of the molecule, . There could be a cancellation between the two frequencies when with a positive integer. It follows that will be proportional to so that the sensitivity to and can be enhanced for these particular transitions. A similar effect between transistions with hyperfinestructures, for which the sensitivity to can reach 600 for instance for ^{139}La^{32}S or silicon monobrid [42] that allows one to constrain .
Nuclear transitions, such as an optical clock based on a very narrow ultraviolet nuclear transition between the ground and first excited states in the ^{229}Th, are also under consideration. Using a Walecka model for the nuclear potential, it was concluded [199] that the sensitivity of the transition to the finestructure constant and quark mass was typically
The SAGAS (Search for anomalous gravitation using atomic sensor) project aims at flying highly sensitive optical atomic clocks and cold atom accelerometers on a solar system trajectory on a time scale of 10 years. It could test the constancy of the finestructure constant along the satellite worldline, which, in particular, can set a constraint on its spatial variation of the order of 10^{–9} [433, 547].
http://www.livingreviews.org/lrr20112 
Living Rev. Relativity 14, (2011), 2
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