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An easy reading of modern ether-drift experiments

Modern ether-drift experiments look for a preferred reference frame searching for modulations of the beat note of two optical resonators that might be induced by the Earth's rotation. We present a compact formalism to evaluate the signal for most experiments where two arbitrary gaseous media fill the resonating cavities. Our predictions can provide useful hints to optimize the experimental set up and the data taking.


1.
In modern ether-drift experiments with optical resonators, the search for the possible existence of a preferred reference frame is performed by looking for modulations of the signal that might be induced by the Earth's rotation. Descriptions of this important effect are already available in the literature. For instance, within the SME model [1] the relevant formulas are given in the appendix of Ref. [2] and for the RMS test theory [3] one can look at Ref. [4]. However, either due to the great number of free parameters (19 in the SME model) and/or to the restriction to a definite experimental set up, it is not always easy to adapt these papers to the various experimental conditions. For this reason, in this Letter, we will present a set of compact formulas that can be immediately used by the reader to evaluate the signal when two arbitrary gaseous media fill the resonating cavities. The formalism covers most experimental set up including the very recent type of experiment proposed in Ref. [5].
In our presentation one clearly understands that the Earth's rotation enters only through two quantities, v = v(t) and θ 0 = θ 0 (t), respectively the magnitude and the angle associated with the projection of the unknown cosmic Earth's velocity V in the plane of the interferometer. At the same time, our predictions can provide useful hints to optimize the experimental set up and the data taking.

2.
Once the angle θ 0 is conventionally defined when one of the arms of the interferometer is oriented to the North point in the laboratory (counting θ 0 from North through East), we can immediately use the formulas given by Nassau and Morse [6]. These are valid for short-term observations, say 3-4 days, where there are no appreciable changes in the cosmic velocity due to changes in the Earth's orbital velocity around the Sun so that the only time dependence is due to the Earth's rotation.
In this approximation, introducing the magnitude V of the full Earth's velocity with respect to a hypothetic preferred frame Σ, its right ascension α and angular declination γ, we get cos z(t) = sin γ sin φ + cos γ cos φ cos(τ − α) where z = z(t) is the zenithal distance of V, φ is the latitude of the laboratory and τ = ω sid t is the sidereal time of the observation in degrees (ω sid ∼ 2π 23 h 56 ′ ).
Let us now consider two orthogonal cavities oriented for simplicity to North (cavity 1) and East (cavity 2) in the laboratory frame. They are filled with two different gaseous media with refractive indices N i (i=1,2) such that N i = 1 + ǫ i , and 0 ≤ ǫ i ≪ 1. The frequency in each cavity is and the frequency shift is In the above relations we have introduced the parameters k i where m i are integers fixing the cavity modes, L i are the cavity lengths andū ′ i (θ i ) denote the two-way speeds of light, as measured in the Earth's frame, θ i being the angle between V and the axis of the i-th cavity.
Following the point of view of Refs. [7,8,9], that no observable Fresnel's drag has ever been detected in the gaseous regime, we shall assume that the two speeds of light c N i are seen isotropic in the preferred frame Σ. Using Lorentz transformations to connect to the Earth's frame, one then obtains with We emphasize that the structure in Eq. (8), although obtained in connection with Eqs.(9) by using Lorentz transformations, remains also valid under the more general assumptions of the RMS test theory [3]. As such, if A i and B i are considered as free parameters, it provides a physical framework that is equivalent to the RMS model.
Introducing the unit vectorsû i fixing the direction of the two cavities and the projection v of the full V in the interferometer's plane one finds so that (v = |v|) and Therefore, defining the reference frequency ν 0 = ck 1 N 1 and introducing the parameter ξ through one finds the relative frequency shift For a symmetric apparatus where N 1 = N 2 , A 1 = A 2 , B 1 = B 2 = B and ξ = 1, one finds On the other hand for a non-symmetric apparatus of the type considered in Ref. [5] with L 1 = L 2 = L, but where one can conveniently arrange N 1 = 1 (up to negligible terms) so To consider experiments where one or both resonators are placed in a state of active rotation (at a frequency ω rot ≫ ω sid ), it is convenient to modify Eq.(14) by rotating the resonator 1 by an angle δ 1 and the resonator 2 by an angle δ 2 so that the last term in Eq. (14) becomes Therefore, in a fully symmetric apparatus where N 1 = N 2 , A 1 = A 2 , B 1 = B 2 = B and ξ = 1 and both resonators rotate, as in Ref. [10], setting On the other hand, if only one resonator rotates, as in Ref. [11], setting δ 1 = 0 and δ 2 = ω rot t one obtains the alternative result By first filtering the signal at the frequency ω = ω rot ≫ ω sid , the main difference between the two expressions is an overall factor of two.

3.
Let us now return to the general case of a non-rotating set up Eq.(14). Using Eqs. (1)(2)(3)(4) we obtain the simple Fourier expansion where Since the mean signal is most likely affected by systematic effects, one usually concentrates on the daily modulation. In this case, assuming that f 1 , f 2 , f 3 and f 4 can be extracted to good accuracy from the experimental data, one can try to obtain a pair of angular variables through the two independent determinations of α and the relation tan |γ| = |B 1 sin 2 φ + ξB 2 | |2B 1 sin 2φ| Notice that, since the ether-drift is a 2nd-harmonic effect, the pair (α, γ) cannot be distinguished from the pair (α + π, −γ). Notice also that two dynamical models that predict the same anisotropy parameters up to an overall re-scaling B i → λB i would produce the same |γ| from the experimental data.
Finally for a symmetric apparatus, where B 1 = B 2 = B and ξ = 1, one obtains the simpler relation tan |γ| = 1 + sin 2 φ |2 sin 2φ| where any reference to the anisotropy parameters drops out. within the cavities, one should also try to check the trend predicted in Eq.(9), namely On the other hand, for a non-symmetric apparatus of the type proposed in Ref. [5], where one can conveniently fix the cavity oriented to North to have N 1 = 1 (up to negligible terms), by using Eqs.(9) one would predict B 1 ∼ 0 in Eqs. (23) and (24)  Notice that for a complete description of the observations over a one-year period, it is not necessary to modify the simple formulas Eqs.(23) and (24) and introduce explicitly the further modulations associated with the orbital frequency Ω orb ∼ 2π 1 year . Rather, by plotting on the celestial sphere all directions defined by the various (α j , γ j ) pairs obtained in the various short-term observations one can try to reconstruct the Earth's "aberration circle". If this will show up, one can determine the mean magnitude of the cosmic velocity V from the angular opening of the circle and from the known value of the orbital Earth's velocity ∼ 30 km/s. In this way, given the value of K , one will be able to disentangle V from B and get a definitive test of models that predict the absolute magnitude of the anisotropy parameter.