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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.13 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 3He/129Xe maser system [4445]. 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 1.5 G, each gas acts as a maser at frequencies of 4.9 kHz and 1.7 kHz for He and Xe, respectively. The Xe emission is used as a magnetometer to stabilize the magnetic field while the He 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 DfJ is given by
|| ( )|| 2p|DfJ |= |-3.5~bJ + 0.012 ~dJ - ~gD,J |, (51)
where J stands for the X, Y 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 ~b, d~, and ~g are related to the mSME coefficients of Section 4.1.3 by [175Jump To The Next Citation Point]
1 1 ~bJ = bJ- mdJ0 + -meJKLgKL0 - --eJKLHKL, 2 2 ~ 1- dJ = m(d0J + dJ0) - 4 (2mdJ0 + eJKLHKL), (52) ( ) ~gD,J = meJKL gK0L + 1gKL0 - bJ. 2
Here m is the neutron mass and eIJK is the three-dimensional antisymmetric tensor. Barring conspiratorial cancellations among the coefficients, the bound on V~ -------- ~b _L = ~b2X + ~b2Y is 6.4 ± 5.4 × 10-32 GeV, which is the strongest clock comparison limit on mSME parameters. Similarly, one can derive bounds on ~ d_ L and ~gD, _L that are two to three orders of magnitude lower. Hence certain components of these coefficients are bounded at the level of 10-28 GeV. A continuation of this experiment has recently been able to directly constrain boost violation at the level of 10- 27 GeV [71] (sidereal variations look at rotation invariance). Besides the bounds above, other clock comparison experiments [175Jump To The Next Citation Point] are able to establish the following bounds on other coefficients in the neutron sector of the mSME:
-25 |~cQ,J|= |m(cJZ + cZJ)| < 10 GeV, -27 |~c- |= |m(cXX - cYY)|< 10 GeV, (53) |~cXY |= |m(cXY + cY X)|< 10-27 GeV.

A constraint of the dimension five operators of Equation (40View Equation) for neutrons was recently derived in [52Jump To The Next Citation Point] using limits on the spatial variation of the hyperfine nuclear spin transition in Be+ 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 ua, the extra terms in Equation (40View Equation) generically introduce a small orientation dependent potential into the non-relativistic Schrödinger equation for any particle. For Be+, 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 |j1|< 6× 10 -3 and |j2|< 3 if ua is timelike and coincides with the rest frame of the CMBR. If a u is spacelike one has - 8 |j1|< 2 × 10 and - 8 |j2|< 10. If a u is lightlike both coefficients are bounded at the -8 10 level. Note that all these bounds are approximate, as they depend on the spatial orientation of the experiment with respect to spatial components of ua 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 [84Jump To The Next Citation Point243Jump To The Next Citation Point196Jump To The Next Citation Point46Jump To The Next Citation Point241Jump To The Next Citation Point] in the mSME. The best proton limit, on the ~b _L parameter, is |~b _L | < 2 .10-27 GeV [241], with corresponding limits on d~_ L and ~gD, _L of order 10- 25 GeV. Similar bounds have been estimated [175Jump To The Next Citation Point] 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.


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