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5.3 Cavity experiments

From the Michelson-Morley experiments onward, interferometry has been an excellent method of testing relativity. Modern cavity experiments extend on the ideas of interferometry and provide very precise tests on the bounds of certain photon parameters. The main technique of a cavity experiment is to detect the variation of the resonance frequency of the cavity as its orientation changes with respect to a stationary frequency standard. In this sense, it is similar to a clock comparison experiment. However, since one of the clocks involves photons, cavity experiments constrain the electromagnetic sector of the mSME as well.

The analysis of cavity experiments is much easier if we make a field redefinition of the electromagnetic sector of the mSME [179Jump To The Next Citation Point]. In analogy to the theory of dielectrics, we define two new fields D and H by

( D ) (1 + k k ) (E ) = DE DB . (54) H kHE 1 + kHB B
The k coefficients are related to the mSME coefficients by
(k )jk = - 2(k )0j0k, (55) DE F (k )jk = 1-ejkqekrs(k )pqrs, (56) HB 2 F (kDB)jk = - (kHE)jk = ekpq(kF)0jpq. (57)
With this choice of fields, the modified Maxwell equations from the mSME take the suggestive form
\~/ × H - @0D = 0, \~/ .D = 0, (58) \~/ × E + @0B = 0, \~/ .B = 0.
This redefinition shows that the Lorentz violating background tensor (kF )mnab can be thought of as a dielectric medium with no charge or current density. Hence we can apply much of our intuition about the behavior of fields inside a dielectric to construct tests of Lorentz violation. Note that since H and D depend on the components of (k ) F mnab, the properties of the dielectric are orientation dependent.

Constraints from cavity experiments are not on the k parameters themselves, but rather on the linear combinations

1- ll ~ktr = 3 (kDE) , (59) 1 (~ke+)jk = --(kDE + kHB)jk, (60) 2 (~k )jk = 1-(k - k )jk- djk~k , (61) e- 2 DE HB tr jk 1- jk (~ko+) = 2 (kDB + kHE) , (62) 1 (~ko-)jk = --(kDB - kHE)jk. (63) 2
~ktr, ~ke+, and ~ke- are all parity even, while ~ko+ and ~ko- are parity odd. The usefulness of this parameterization can be seen if we rewrite the Lagrangian in these parameters [179Jump To The Next Citation Point],
1[ ] 1 1 L = --(1 + ~ktr)E2 - (1 - ~ktr)B2 + -E .[k~e+ + ~ke-] .E - -B .[k~e+ - ~ke-] .B + E .[~ko+ + ~ko- ] .B(6.4) 2 2 2
From this expression it is easy to see that ~ktr corresponds to a rotationally invariant shift in the speed of light. It can be shown that ~ke- and ~ko+ also yield a shift in the speed of light, although in a direction dependent manner. The coefficients ~k e+ and ~k o- control birefringence. Cavity experiments yield limits on k~e- and ~ko+, while birefringence (see Section 6.3) bounds ~ke+ and ~ko-.

The most straightforward way to constrain Lorentz violation with cavity resonators is to study the resonant frequency of a cavity. Since we have a cavity filled with an orientation dependent dielectric, the resonant frequency will also vary with orientation. The resonant frequency of a cavity is

mc fr = ----, (65) 2nL
where m is the mode number, c is the speed of light, n is the index of refraction (including Lorentz violation) of any medium in the cavity, and L is the length of the cavity. fr can be sensitive to Lorentz violating effects through c, n, and L. Depending on the construction of the cavity some effects can dominate over others. For example, in sapphire cavities the change in L due to Lorentz violation is negligible compared to the change in c. This allows one to isolate the electromagnetic sector.

In general, all cavities are sensitive to the photon k parameters. In contrast to sapphire, for certain materials the strain induced on the cavity by Lorentz violation is large. This allows sensitivity to the electron parameters cmn at a level equivalent to the photon parameters. Furthermore, by using a cavity with a medium, the dependence of fr on n gives additional electron sensitivity [226Jump To The Next Citation Point].

The complete bounds on the mSME coefficients for cavity experiments are given in [23Jump To The Next Citation Point67227226Jump To The Next Citation Point28020732Jump To The Next Citation Point261Jump To The Next Citation Point]. The strongest bounds are displayed in Table 1. Roughly, the components of ~ke- and cmn are bounded at -15 O(10 ) while ~ko+ is bounded at - 11 O(10 ). The 4 10 difference arises as ~ko+ enters constraints suppressed by the boost factor of the earth relative to the solar “rest” frame where the coefficients are taken to be constant.

Value (- 15 × 10)

e cXY 0.76 ± 0.35
ce YZ 0.21 ± 0.46
ce XZ - 0.16 ± 0.63
cXX - cYY 1.15 ± 0.64
|cXX + cYY - 2cZZ - 0.25~kZZe- | 103

XY ~k e- - 0.63 ± 0.43
YZ ~k e- - 0.45 ± 0.37
~kXZ e- 0.19 ± 0.37
~kXXe- - ~kYe-Y - 1.3 ± 0.9
~kZZe- - 20 ± 2

XY ~k o+ (0.20 ± 4 0.21) × 10
YZ ~k o+ (0.44 ± 4 0.46) × 10
~kXZ o+ (- 0.91 ± 0.46) × 104

Table 1: Cavity limits on cmn, ~ke-, and ~ko+ (taken from [23, 226, 32Jump To The Next Citation Point, 261]). Components are in a sun centered equatorial frame. Error bars are 1s. The non-zero value of ~kZZ e- is argued by the authors to be due to systematics in the experiment [32].

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