### 5.2 Clock comparison experiments

The classic clock comparison experiments are those of Hughes [150] and Drever [100], and their
basic approach is still used today. Two “clocks”, usually two atomic transition frequencies,
are co-located at some point in space. As the clocks move, they pick out different components
of the Lorentz violating tensors in the mSME, yielding a sidereal drift between the two
clocks. The difference between clock frequencies can be measured over long periods, yielding
extremely high precision limits on the amount of drift and hence the parameters in the
mSME.
Note that this approach is only possible if the clocks are made of different materials or have different
orientations.
The best overall limit is in the neutron sector of the mSME and comes from a maser
system [44, 45]. In this setup, both noble gases are co-located. The gases are placed into a population
inverted state by collisions with a pumped vapor of rubidium. In a magnetic field of , each gas acts
as a maser at frequencies of and for and , respectively. The emission is
used as a magnetometer to stabilize the magnetic field while the emission frequency is tracked over
time, looking for sidereal variation. At lowest order in Lorentz violating couplings, the Lorentz violating
effect for each gas is that of a single valence neutron, so this experiment is sensitive only to
neutron parameters in the mSME. The magnitude of the sidereal variation is given by
where stands for the components of the Lorentz violating tensors in a non-rotating frame that
are orthogonal to the earth’s rotation axis. All parameters are understood to be the ones for the neutron
sector of the mSME. The coefficients , , and are related to the mSME coefficients of Section 4.1.3
by [175]
Here is the neutron mass and is the three-dimensional antisymmetric tensor. Barring conspiratorial
cancellations among the coefficients, the bound on is , which is
the strongest clock comparison limit on mSME parameters. Similarly, one can derive bounds on and
that are two to three orders of magnitude lower. Hence certain components of these coefficients are
bounded at the level of . A continuation of this experiment has recently been able to
directly constrain boost violation at the level of [71] (sidereal variations look at
rotation invariance). Besides the bounds above, other clock comparison experiments [175] are able
to establish the following bounds on other coefficients in the neutron sector of the mSME:
A constraint of the dimension five operators of Equation (40) for neutrons was recently derived in [52]
using limits on the spatial variation of the hyperfine nuclear spin transition in as a function of the
angle between the spin axis and an external magnetic field [64]. Assuming the reference frame of the earth
is not aligned with the four vector , the extra terms in Equation (40) generically introduce a small
orientation dependent potential into the non-relativistic Schrödinger equation for any particle. For ,
the nuclear spin can be thought of as being carried by a single neutron, so this experiment
limits the neutron Lorentz violating coefficients. This extra potential for the neutron leads to
anisotropy of the hyperfine transition frequency, which can be bounded by experiment. The limits
are roughly and if is timelike and coincides with the rest
frame of the CMBR. If is spacelike one has and . If
is lightlike both coefficients are bounded at the level. Note that all these bounds are
approximate, as they depend on the spatial orientation of the experiment with respect to spatial
components of in the lab frame. The authors of [52] have assumed that the orientation is not
special.

The above constraints apply solely to the neutron sector. Other clock comparison experiments have been
performed that yield constraints on the proton sector [84, 243, 196, 46, 241] in the mSME. The best
proton limit, on the parameter, is [241], with corresponding limits
on and of order . Similar bounds have been estimated [175] from
the experiment of Berglund et al. [46] using the Schmidt model [251] for nuclear structure,
where an individual nucleon is assumed to carry the entire nuclear angular momentum. The
experiments of Chupp [84], Prestage [243], and Lamoreaux [196] are insensitive to proton
coefficients in this model, so no proton bounds have yet been established from these experiments.
As noted in [175], proton bounds would be derivable with a more detailed model of nuclear
structure.