zotero-db/storage/9KKSNRLZ/.zotero-ft-cache

Test of Lorentz Invariance in Electrodynamics Using Rotating Cryogenic Sapphire Microwave Oscillators

Paul L. Stanwix1,∗ Michael E. Tobar1,† Peter Wolf2,3, Mohamad Susli1, Clayton
R. Locke1, Eugene N. Ivanov1, John Winterﬂood1, and Frank van Kann1 1University of Western Australia, School of Physics M013, 35 Stirling Hwy., Crawley 6009 WA, Australia
2SYRTE, Observatoire de Paris, 61 Av. de l’Observatoire, 75014 Paris, France 3Bureau International des Poids et Mesures, Pavillon de Breteuil, 92312 S`evres Cedex, France
(Dated: July 20, 2021)

We present the ﬁrst results from a rotating Michelson-Morley experiment that uses two orthogo-

nally orientated cryogenic sapphire resonator-oscillators operating in whispering gallery modes near

10 GHz. The experiment is used to test for violations of Lorentz Invariance in the frame-work of

the photon sector of the Standard Model Extension (SME), as well as the isotropy term of the

Robertson-Mansouri-Sexl (RMS) framework. In the SME we set a new bound on the previously

unmeasured κ˜Ze−Z component of 2.1(5.7) × 10−14, and set more stringent bounds by up to a factor

of 7 on seven other components. In the RMS a more stringent bound of −0.9(2.0) × 10−10 on the

isotropy

parameter,

PM M

=δ−β+

1 2

is

set,

which

is

more

than

a

factor

of

7

improvement.

PACS numbers: 03.30.+p, 06.30.Ft, 12.60.-i, 11.30.Cp, 84.40.-x

arXiv:hep-ph/0506074v3 21 Jul 2005

The Einstein Equivalence Principle (EEP) is a founding principle of relativity [1]. One of the constituent elements of EEP is Local Lorentz Invariance (LLI), which postulates that the outcome of a local experiment is independent of the velocity and orientation of the apparatus. The central importance of this postulate has motivated tremendous work to experimentally test LLI. Also, a number of uniﬁcation theories suggest a violation of LLI at some level. However, to test for violations it is necessary to have an alternative theory to allow interpretation of experiments [1], and many have been developed [2, 3, 4, 5, 6, 7]. The kinematical Roberson-MansouriSexl (RMS) [2, 3] framework postulates a simple parameterization of the Lorentz transformations with experiments setting limits on the deviation of those parameters from their values in special relativity (SR). Because of their simplicity they have been widely used to interpret many experiments [8, 9, 10, 11]. More recently, a general Lorentz violating extension of the standard model of particle physics (SME) has been developed [6] whose Lagrangian includes all parameterized Lorentz violating terms that can be formed from known ﬁelds.
This work presents ﬁrst results of a rotating lab experiment using cryogenic microwave oscillators. Previous non-rotating experiments [10, 12, 13] relied on the earth’s rotation to modulate a Lorentz violating eﬀect. This is not optimal for two reasons. Firstly, the sensitivity is proportional to the noise of the oscillators at the modulation frequency, typically best for periods between 10 and 100 seconds. Secondly, the sensitivity is proportional to the square root of the number of periods of the modulation signal, therefore taking a relatively long time to acquire suﬃcient data. Thus, by rotating the experiment the data integration rate is increased and the relevant signals are translated to the optimal operating regime [14].

Our experiment consists of two cylindrical sapphire resonators of 3 cm diameter and height supported by spindles within superconducting niobium cavities [15], and are oriented with their cylindrical axes orthogonal to each other in the horizontal plane. Whispering gallery modes [16] are excited near 10 GHz, with a diﬀerence frequency of 226 kHz. The frequencies are stabilized using Pound locking, and amplitude variations are suppressed using an additional control circuit. A detailed description of such oscillators can be found in [17, 18]. The resonators are mounted in a common copper block, which provides common mode rejection of temperature ﬂuctuations. The structure is in turn mounted inside two successive stainless steel vacuum cylinders from a copper post, which provides the thermal connection between the cavities and the liquid helium bath. A foil heater and carbon-glass temperature sensor attached to the copper post controls the temperature set point to 6 K with mK stability.
A schematic of the rotation system is shown in Fig.1. The cryogenic dewar along with the room temperature oscillator and control electronics is suspended within a ring bearing. A multiple ”V” shaped suspension made from elastic cord avoids high Q-factor pendulum modes by ensuring that the cord has to stretch and shrink (providing damping) for horizontal and vertical motion. The rotation system is driven by a microprocessor controlled stepper motor. A commercial 18 conductor slip ring connector, with a hollow through bore, transfers power and various signals to and from the rotating experiment. A mercury based rotating coaxial connector transmits the diﬀerence frequency to a stationary frequency counter referenced to an Oscilloquartz oscillator. The data acquisition system logs the diﬀerence frequency as a function of orientation, as well as monitoring systematic eﬀects including the temperature of the resonators, liquid helium

2

FIG. 1: Schematic of the cryogenic dewar, mounted in the rotation table. Inside the dewar a schematic of the two orthogonally orientated resonators is shown, along with the Poynting vectors of propagation S1 and S2.
FIG. 2: Square Root Allan Variance fractional frequency instability measurement of the diﬀerence frequency when rotating (crosses) and stationary (circles). The hump at short integration times is due to systematic eﬀects associated with the rotation of the experiment, with a period of 18 seconds. Above 18 seconds the instability is the same as when the experiment is stationary.
bath level, ambient room temperature, oscillator control signals, tilt, and helium return line pressure.
Inside the sapphire crystals standing waves are set up with the dominant electric and magnetic ﬁelds in the axial and radial directions respectively, corresponding to a Poynting vector around the circumference. The experi-

mental observable is the diﬀerence frequency, and to test

for Lorentz violations the perturbation of the observable

with respect to an alternative test theory must be de-

rived. For example, in the photon sector of the SME this

may be calculated to ﬁrst order as the integral over the

non-perturbed ﬁelds (Eq. (34) of [7]), and expressed in

terms of 19 independent variables (discussed in more de-

tail later). The change in orientation of the ﬁelds due

to the lab rotation and Earth’s orbital and sidereal mo-

tion induces a time varying modulation of the diﬀerence

frequency, which is searched for in the experiment. Al-

ternatively, with respect to the RMS framework, we an-

alyze the change in resonator frequency as a function of

the Poynting vector direction with respect to the veloc-

ity of the lab through the cosmic microwave background

(CMB). The RMS parameterizes a possible Lorentz vi-

olation by a deviation of the parameters (α, β, δ) from

their

SR

values

(−

1 2

,

1 2

,

0).

Thus,

a

complete

veriﬁcation

of LLI in the RMS framework [2, 3] requires a test of (i)

the

isotropy

of

the

speed

of

light

(PM M

=

δ

−β

+

1 2

),

a

Michelson-Morley (MM) experiment [19], (ii) the boost

dependence of the speed of light (PKT = β − α − 1), a

Kennedy-Thorndike (KT) experiment [20] and (iii) the

time

dilation

parameter

(PIS

=

α

+

1 2

),

an

Ives-Stillwell

(IS) experiment [21, 22]. Because our experiment com-

pares two cavities it is only sensitive to PMM . Fig.2 shows typical fractional frequency instability of

the 226 kHz diﬀerence with respect to 10 GHz, and com-

pares the instability when rotating and stationary. A

minimum of 1.6 × 10−14 is recorded at 40s. Rotation in-

duced systematic eﬀects degrade the stability up to 18s

due to signals at the rotation frequency of 0.056Hz and

its harmonics. We have determined that tilt variations

dominate the systematic by measuring the magnitude of

the fractional frequency dependence on tilt and the varia-

tion in tilt at twice the rotation frequency, 2ωR(0.11Hz),

as the experiment rotates. We minimize the eﬀect of tilt

by manually setting the rotation bearing until our tilt

sensor reads a minimum at 2ωR. The latter data sets

were up to an order of magnitude reduced in amplitude

as we became more experienced at this process. The re-

maining systematic signal is due to the residual tilt varia-

tions, which could be further annulled with an automatic

tilt control system. It is still possible to be sensitive to

Lorentz violations in the presence of these systematics by

measuring the sidereal, ω⊕ and semi-sidereal, 2ω⊕ side-

bands about 2ωR, as was done in [8]. The amplitude and

phase of a Lorentz violating signal is determined by ﬁt-

ting the parameters of Eq.1 to the data, with the phase

of the ﬁt adjusted according to the test theory used.

∆ν0 = A+Bt+ ν0

i

Cicos(ωit+ϕi)+Sisin(ωit+ϕi) (1)

Here ν0 is the average unperturbed frequency of the two sapphire resonators, and ∆ν0 is the perturbation of the 226 kHz diﬀerence frequency, A and B determine the

3

FIG. 3: Spectrum of amplitudes Ci2 + Si2 calculated using WLS, showing systematic leakage about 2ωR for 2 data sets, data set 1 (3.6 days, circles), data set 5 (6.1 days, squares) and the combined data (18 days spanning 3 months, solid triangles). Here ω⊕ is the sidereal frequency (11.6µHz). By comparing a variety of data sets we have seen that leakage is reduced in longer data sets with lower systematics. The insets show the typical amplitude away from the systematic, which have statistical uncetainties of order 10−16.
frequency oﬀset and drift, and Ci and Si are the amplitudes of a cosine and sine at frequency ωi respectively. In the ﬁnal analysis we ﬁt 5 frequencies to the data, ωi = (2ωR, 2ωR±ω⊕, 2ωR±2ω⊕), as well as the frequency oﬀset and drift. The correlation coeﬃcients between the ﬁtted parameters are all between 10−2 to 10−5. Since the residuals exhibit a signiﬁcantly non-white behavior, the optimal regression method is weighted least squares (WLS) [13]. WLS involves pre-multiplying both the experimental data and the model matrix by a whitening matrix determined by the noise type of the residuals of an ordinary least squares analysis.
We have acquired 5 sets of data over a period of 3 months beginning December 2004, totaling 18 days. The length of the sets (in days) and size of the systematic are (3.6, 2.3 × 10−14), (2.4, 2.1 × 10−14), (1.9, 2.6 × 10−14), (4.7, 1.4 × 10−15), and (6.1, 8.8 × 10−15) respectively. We have observed leakage of the systematic into the neighboring side bands due to aliasing when the data set is not long enough or the systematic is too large. Fig.3 shows the total amplitude resulting from a WLS ﬁt to 2 of the data sets over a range of frequencies about 2ωR. It is evident that the systematic of data set 1 at 2ωR is aﬀecting the ﬁtted amplitude of the sidereal sidebands 2ωR ± ω⊕ due to its relatively short length and large systematics. By analyzing all ﬁve data sets simultaneously using WLS the eﬀective length of the data is increased, reducing the width of the systematic suﬃciently as to not contribute signiﬁcantly to the sidereal and semi-sidereal sidebands.
In the photon sector of the SME [7], Lorentz violating terms are parameterized by 19 independent components, which are in general grouped into three traceless and

symmetric 3 × 3 matrices (κ˜e+, κ˜o−, and κ˜e−), one antisymmetric matrix(κ˜o+) and one additional scalar, which all vanish when LLI is satisﬁed. To derive the expected
signal in the SME we use the method of [7, 11] to calculate the frequency of each resonator in the SME and in
the resonator frame. We then transform to the standard
celestial frame used in the SME [7] taking into account the rotation in the laboratory frame in a similar way
to [23]. The resulting relation between the parameters
of the SME and the Ci and Si coeﬃcients are given in Tab.I which, for short data sets, were calculated using
the leading order expansion at the annual phase position of the data. The 10 independent components of κ˜e+ and κ˜o− have been constrained by astronomical measurements to < 2 × 10−32 [7, 25]. Seven components of κ˜e− and κ˜o+ have been constrained in optical and microwave cavity experiments [10, 13] at the 10−15 and 10−11 level
respectively, while the scalar κ˜tr component recently had an upper limit set of < 10−4 [23]. The remaining κ˜Ze−Z component could not be previously constrained in non-
rotating experiments [10, 13]. In contrast, our rotating experiment is sensitive to
κ˜Ze−Z. However, it appears only at 2ωR, which is dominated by systematic eﬀects. From our combined analysis of all data sets, and using the relation to κ˜Ze−Z given in Tab.I, we determine a value for κ˜Ze−Z of 4.1(0.5) × 10−15. However, since we do not know if the systematic has can-
celed a Lorentz violating signal at 2ωR, we cannot reasonably claim this as an upper limit. Since we have ﬁve individual data sets, a limit can be set by treating the
C2ωR coeﬃcient as a statistic. The phase of the systematic depends on the initial experimental conditions, and
is random across the data sets. Thus, we have ﬁve values of C2ωR , ({−4.2, 11.4, 21.4, 1.3, −8.1} in 10−15). If we take the mean of these coeﬃcients, the systematic
signal will cancel if its phase is random, but the possible
Lorentz violating signal (with constant phase) will not. Thus a limit can be set by taking the mean and stan-
dard deviation of the ﬁve coeﬃcient of C2ωR . This gives a more conservative bound of 2.1(5.7) × 10−14, which in-
cludes zero. Our experiment is also sensitive to all other
seven components of κ˜e− and κ˜o+ (see Tab.I) and improves present limits by up to a factor of 7, as shown in
Tab.II.
In the RMS frame-work, a frequency shift due to a putative Lorentz violation is given by Eq.2 [9, 11],

∆ν0 ν0

=

PM M 2πc2

2
v.θˆ1 dϕ1 −

2
v.θˆ2 dϕ2 (2)

Where v is the velocity of the preferred frame wrt the CMB, θˆj is the unit vector in the direction of the azimuthal angle (direction of propagation) of each resonator (labeled by subscripts 1 and 2), and ϕ is the azimuthal variable of integration in the cylindrical coordinates of each resonator. Perturbations due to Lorentz violations occur at the same ﬁve frequencies as the SME,

4

TABLE I: Coeﬃcients Ci and Si in (1) for the ﬁve frequencies of interest and their relation to the components of the SME parameters κ˜e− and κ˜o+, derived using a short data set approximation including terms up to ﬁrst order in orbital velocity, where Φ0 is the phase of the orbit since the vernal equinox (see [24] for details of the calculation). Note that for short data sets the upper and lower sidereal sidebands are redundant, which reduces the number of independent measurements to 5. To lift the redundancy, more than a year of data is required so annual oﬀsets may be de-correlated from the twice rotational and sidereal sidebands listed.

ωi 2ωR 2ωR + ω⊕
2ωR + 2ω⊕
2ωR − ω⊕ 2ωR − 2ω⊕

Ci
0.21κ˜Ze−Z 2.5 × 10−5 sin Φ0κ˜X o+Y − 1.0 × 10−5 cos Φ0κ˜Yo+Z
−0.27 κ˜X e−Z −2.1 × 10−5 cos Φ0κ˜X o+Z +2.3 × 10−5 sin Φ0κ˜Yo+Z − 0.11(κ˜X e−X − κ˜Ye−Y )
−0.31C2ωR +ω⊕ 9.4 × 10−2C2ωR+2ω⊕

Si
− cos Φ0 2.3 × 10−5κ˜X o+Y − 1.0 × 10−5κ˜X o+Z
−0.27κ˜Ye−Z −2.3 × 10−5 sin Φ0κ˜X o+Z −2.1 × 10−5 cos Φ0κ˜Yo+Z − 0.23 κ˜X e−Y
0.31S2ωR +ω⊕ −9.4 × 10−2S2ωR+2ω⊕

TABLE II: Results for the SME Lorentz violation parameters,
assuming no cancelation between the isotropy terms κ˜e− (in 10−15) and ﬁrst order boost terms κ˜o+ (in 10−11) [12].

κ˜X e−Y

κ˜X e−Z

κ˜Ye−Z

(κ˜X e−X − κ˜Ye−Y )

this work -0.63(0.43) 0.19(0.37) -0.45(0.37) -1.3(0.9)

from [13] -5.7(2.3) -3.2(1.3) -0.5(1.3) -3.2(4.6)

this work from [13]

κ˜Ze−Z 21(57)
−

κ˜X o+Y

κ˜X o+Z

0.20(0.21) -0.91(0.46)

-1.8(1.5) -1.4(2.3)

κ˜Yo+Z 0.44(0.46)
2.7(2.2)

improve these results, tilt and environmental controls will be implemented to reduce systematic eﬀects. To remove the assumption that κ˜o+ and κ˜e− do not cancel each other, data integration will continue for more than a year. Note added: Two other concurrent experiments have also set some similar limits [26, 27].
This work was funded by the Australian Research Council.

TABLE III: Dominant coeﬃcients in the RMS, using a short
data set approximation calculated from Eq.2. The measured values of PMM (in 10−10) are shown together with the sta-
tistical uncertainties in the bracket. From this data the mea-
sured and statistical uncertainty of PMM is determined to be −0.9(2.0) × 10−10, which represents more than a factor of 7 improvement over previous results 2.2(1.5) × 10−9[10].

ωi

C ui

PM M

2ωR + ω⊕ [−1.13 × 10−7 − 3.01 × 10−8 cos Φ0 −2.1(7.2)

+8.83 × 10−9 sin Φ0]PMM

2ωR − ω⊕ [3.51 × 10−8 + 9.31 × 10−9 cos Φ0 62.4(23.3)

−2.73 × 10−9 sin Φ0]PMM

2ωR + 2ω⊕ [4.56 × 10−7 − 1.39 × 10−8 cos Φ0 −1.3(2.1)

−7.08 × 10−8 sin Φ0]PMM

2ωR − 2ω⊕ [4.37 × 10−8 − 1.34 × 10−9 cos Φ0 −7.5(22.1)

−6.78 × 10−9 sin Φ0]PMM

but for the RMS analysis we do not consider the 2ωR frequency due to the large systematic, as we only need to put a limit on one parameter. The dominant coeﬃcients are due to only the cosine terms with respect to the CMB right ascension, Cui, which are shown in Tab.III.
In conclusion, we set bounds on 7 components of the SME photon sector (Tab.II) and PMM (Tab.III) of the RMS framework, which are up to a factor of 7 more stringent than those obtained from previous experiments. We have also set an upper limit [2.1(5.7) × 10−14] on the previously unmeasured SME component κ˜Ze−Z . To further

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