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
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, , the electron to proton mass ration
and the
fine-structure constant
. 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
where 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
,
Atom |
Transition | sensitivity ![]() |
1H |
![]() |
0.00 |
87Rb |
hf | 0.34 |
133Cs |
![]() |
0.83 |
171Yb+ |
![]() |
0.9 |
199Hg+ |
![]() |
–3.2 |
87Sr |
![]() |
0.06 |
27Al+ |
![]() |
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 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 |
|
![]() |
|||
87Rb |
133Cs | (0.2 ± 7.0) × 10–16 | ![]() |
[346![]() |
87Rb |
133Cs | (–0.5 ± 5.3) × 10–16 | [58![]() |
|
1H |
133Cs | (–32 ± 63) × 10–16 | ![]() |
[196![]() |
199Hg+ |
133Cs | (0.2 ± 7) × 10–15 | ![]() |
[57![]() |
199Hg+ |
133Cs | (3.7 ± 3.9) × 10–16 | [214![]() |
|
171Yb+ |
133Cs | (–1.2 ± 4.4) × 10–15 | ![]() |
[408![]() |
171Yb+ |
133Cs | (–0.78 ± 1.40) × 10–15 | [407![]() |
|
87Sr |
133Cs | (–1.0 ± 1.8) × 10–15 | ![]() |
[61![]() |
87Dy |
87Dy | (–2.7 ± 2.6) × 10–15 | ![]() |
[100![]() |
27Al+ |
199Hg+ | (–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
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 vibro-rotational
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 vibro-rotation transitions is, in the Born–Oppenheimer approximation, of the form
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]
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 non-gravitational 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 fine-structure constant, , while the vibrational
levels depend mainly on the electron-to-proton 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 hyperfine-structures, for which the
sensitivity to
can reach 600 for instance for 139La32S 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 229Th, are also under consideration. Using a Walecka model for the nuclear potential, it was concluded [199] that the sensitivity of the transition to the fine-structure 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 fine-structure 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/lrr-2011-2 |
Living Rev. Relativity 14, (2011), 2
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