zotero/storage/9TB2AZZU/.zotero-ft-cache

VOLUME 91, NUMBER 2

PHYSICAL REVIEW LETTERS

week ending 11 JULY 2003

Modern Michelson-Morley Experiment using Cryogenic Optical Resonators

Holger Mu¨ller,1,2,* Sven Herrmann,1,2 Claus Braxmaier,2 Stephan Schiller,3 and Achim Peters1,†,‡
1Institut fu¨ r Physik, Humboldt-Universita¨t zu Berlin, Hausvogteiplatz 5-7, 10117 Berlin, Germany 2Fachbereich Physik, Universita¨t Konstanz, 78457 Konstanz, Germany
3Institut fu¨ r Experimentalphysik, Heinrich-Heine-Universita¨t Du¨sseldorf, 40225 Du¨sseldorf, Germany
(Received 27 January 2003; published 10 July 2003)

We report on a new test of Lorentz invariance performed by comparing the resonance frequencies of

two orthogonal cryogenic optical resonators subject to Earth’s rotation over 1 yr. For a possible

anisotropy of the speed of light c, we obtain 
c=c0 ˆ …2:6  1:7†  10ÿ15. Within the Robertson-

Mansouri-Sexl (RMS) test 1:5†  10ÿ9, about 3 times

theory, lower

this than

implies a the best

n isotropy violation parameter  ÿ previous result. Within the general



ÿ

1 2

ˆ

…ÿ2:2

extension of t

 he

standard model of particle physics, we extract limits on seven parameters at accuracies down to 10ÿ15,

improving the best previous result by about 2 orders of magnitude.

DOI: 10.1103/PhysRevLett.91.020401

PACS numbers: 03.30.+p, 12.60.–i, 06.30.Ft, 11.30.Cp

Special relativity (SR) underlies all accepted theories of nature at the fundamental level. Therefore, it has been and must be tested with ever increasing precision to provide a firm basis for its future application. Such tests are also motivated by the efforts to unify gravity with the other forces of nature, one of the outstanding open challenges in modern science. In fact, many currently discussed models of quantum gravity do violate the principles of SR. In string theory, for example, violation of Lorentz invariance might be caused by spontaneous symmetry breaking [1]; in loop gravity, effective low-energy equations (e.g., modified Maxwell equations [2]) that violate Lorentz invariance have been derived. Since the natural energy scale for quantum gravity is the Planck scale, Ep  1019 GeV, a direct experimental investigation of quantum gravity effects is not feasible. It is, however, possible to search for residual effects at an attainable energy scale in experiments of outstanding precision. High-precision bounds on Lorentz violation might give valuable hints for or against particular models of quantum gravity.
A sensitive probe for Lorentz violation (in electrodynamics) is the Michelson-Morley (MM) experiment [3], that even predated the formulation of SR. It tests the isotropy of the speed of light, one of SR’s foundations. In the classic setup, one compares the speed of light c in two orthogonal interferometer arms by observing the interference fringes. If c depends on the direction of propagation, the fringes move if the setup is rotated (using, e.g., Earth’s rotation or a turntable). In 1979, Brillet and Hall [4] introduced the modern technique of measuring a laser frequency stabilized to a resonance of an optical reference cavity. The frequency of such a resonance is given by cav ˆ mc=…2L†, where L denotes the cavity length and m ˆ 1; 2; . . . the mode number. Thus, a violation of the isotropy of c can be detected by rotating the cavity and looking for a resulting variation of cav by comparing the frequency of the cavity-stabilized laser against a suitable reference. In our experiment

(Fig. 1), we compare the frequencies x and y of two similar cavities oriented in orthogonal directions, in analogy with the classical interferometer tests. Compared to the single-cavity setup, this arrangement doubles the hypothetical signal amplitude and provides some common-mode rejection of systematic effects.
Our results are obtained making use of the high dimensional stability of cryogenic optical resonators (COREs). Constructed from crystalline sapphire, COREs show a low thermal expansion coefficient (10ÿ10=K at 4.2 K) and a remarkable absence of creep (i.e., intrinsic length changes due to material relaxation); upper limits for the CORE frequency drift are <2 kHz=6 month and
FIG. 1 (color online). Setup: Inside a 4 K cryostat, two COREs are located in a copper block to provide commonmode rejection of thermal effects. LN2 is refilled automatically every 3 h, LHe manually every 2 days. Laser beams are coupled to the COREs via windows, with polarizers P and lock detectors PD inside the cryostat. For active beam positioning, beams pass through galvanometer (G) mounted glass plates. The horizontal and vertical displacements are adjusted to maximize coupling into the cavities, as measured using the 2fm signal from the detector in reflection. Not shown is the setup for measuring the frequency difference x ÿ y of the lasers, in which the beams are overlapped on a high-speed photodetector and the beat frequency is measured against a quartz oscillator stabilized to the Global Positioning System.

020401-1

0031-9007=03=91(2)=020401(4)$20.00  2003 The American Physical Society

020401-1

VOLUME 91, NUMBER 2

PHYSICAL REVIEW LETTERS

week ending 11 JULY 2003

<0:1 Hz=h [5], which makes COREs particularly well

suited for high-precision measurements such as relativity

tests [6,7]. For the MM experiment, it allows us to rely

solely on Earth’s rotation. This avoids the systematic

effects associated with active rotation, which previous

experiments had to use to overcome the creep of room

temperature resonators made from glass ceramics, e.g.,

ULE (ultralow expansion), on the time scale of a day.

A suitable theoretical framework for analyzing tests of

Lorentz invariance is the general standard model exten-

sion (SME) [8,9]. A Lagrangian formulation of the stan-

dard model is extended by adding all possible observer

Lorentz scalars that can be formed from known particles

and Lorentz tensors. In the Maxwell sector, the

Lagrangian

is

L

ˆ

1 4

FF

‡

1 4

…kF † F F

[10].

The tensor …kF† (the greek indices run from 0 –3)

has 19 independent components; they vanish, if SR is

valid. Ten of its components, that can be arranged into

traceless symmetric 3  3 matrices …~e‡†AB and …~oÿ†AB,
describe polarization-dependent effects. They are restricted to <2  10ÿ32 by polarization measurements

on light from astronomical sources [9] and are assumed

to be zero in the following. The remaining nine compo-

nents describe boost invariance and isotropy of c and can

be arranged into traceless 3  3 matrices …~eÿ†AB (symmetric) and …~o‡†AB (antisymmetric) plus one additional parameter. They lead to a shift  of the resonance

frequency of a cavity [9] with a characteristic time sig-

nature and can thus be measured in cavity experiments.

For our experiment (Fig. 1), we use two L ˆ 3 cm long

COREs that feature linewidths of 100 and 50 kHz, re-

spectively (Finesses 105), located inside a liquid helium

(LHe) cryostat with a liquid nitrogen (LN2) shield.

Refills and the evaporation of coolants cause mechani-

cal deformations of the cryostat, which change the reso-

nator position. An automatic beam positioning system

actively compensates for these movements (Fig. 1). The

frequencies las of two diode-pumped Nd:YAG lasers at 1064 nm are stabilized (‘‘locked’’) to resonances of the

COREs using the Pound-Drever-Hall method. The

Nd:YAG laser crystal strain (generated by a piezo at-

tached to the crystal) and temperature were used for

tuning the frequency. The phase modulation at frequen-

cies fm around 500 kHz is also generated by crystal strain modulation using mechanical resonances of the piezo.

The light reflected from the COREs is detected inside

the cryostat (Fig. 1) on Epitaxx 2000 InGaAs detectors.

Down-conversion to dc at 3fm rather than 1fm reduces the influence of residual amplitude modulation and gives

a higher signal-to-noise (S=N) ratio for the same circu-

lating laser power inside the cavity. For the detector

signal, amplifiers consisting of eight paralleled BF1009

dual-gate MOSFETs provide low current noise in spite of

the high detector capacitance (500 pF). At 80 nW laser

power impinging onptheCOREs, we achieve an error S=N ratio of 1:5  104 =s. Between 80 –200 nW have been

used, minimizing the change of cav due to laser heating

020401-2

of the COREs (  10 Hz=W). The error signal is generated using minicircuits ZHL-32 high-level amplifiers and SAY-1 23 dB m double-balanced mixers operating at 0 dB m rf amplitude and thus well below saturation. This provides highly linear operation and proves very important for a low sensitivity to systematic disturbances. On a time scale of minutes, we reach a minimum relative frequency instability of the lasers locked to the COREs of 7  10ÿ16, referred to a single CORE (Fig. 2). Such a level is reached by the best ULE-cavity stabilized lasers [11] only if a large linear drift 2 Hz=s is subtracted.
A number of systematic disturbances (e.g., residual amplitude modulation, parasitic etalons, or mixer offset voltages) that cause lock-point shifts are compensated for using a new technique: By an additional phase modulation of the laser beams, a part of the laser power is shifted into sidebands, thus reducing the amplitude of the useful part of the error signal. If the parasitic effects do not have sufficient frequency selectivity to discriminate these sidebands against the carrier, modulating the amplitude of the sidebands makes the lock-point shifts time dependent, so they can be identified and compensated for [12]. This proved instrumental in achieving the reliable and repeatable laser system performance required for accumulating enough data.
Except for a 10 d break around January 1, 2002, the COREs were operated continuously at 4.2 K over more than one year. Usable data (discounting data sets shorter than 12 h and data taken during adjustment or LHe refills) starts June 19, 2001, and was acquired over 390 days until July 13, 2002. A total of 146 data sets of 12 h to 109 h in length, totaling 3461 h, are available (Figs. 2 and 3). Forty-nine almost equally distributed data sets are longer than 24 h. For extracting results, simultaneous leastsquares fits with a constant offset, a linear drift, and the amplitude of a sinusoidal signal at fixed frequency and phase as suggested by the test theory are performed. Before fitting, the data sets are divided into subsets of 12 h (or 24 h for the 24 h signals). This drops a fraction of the data, but makes the resulting sinewave amplitudes independent of offsets in the data. We obtain 199 fits of 12 h data subsets (Fig. 3).

beat frequency [Hz]
root Allan variance

20 10
0 -10 -20
555.75 555.90 556.05 time since Jan 01, 2000 [days]

1E-14

1E-15

1E-16 10

100 1000

integration time τ [s]

FIG. 2 (color online). Left: Typical data set fitted with a 12 h sinewave amplitude, a linear drift, and a constant offset. Peaks occur every few hours due to automatic LN2 refills. Right: Root Allan variance calculated from this data (upper curve), and from a quiet part between two LN2 refills (118 min starting at 555.87 days; lower curve).
020401-2

VOLUME 91, NUMBER 2

PHYSICAL REVIEW LETTERS

week ending 11 JULY 2003

residuals [Hz] amplitude [Hz]

20

10

0

-10

-20

20 A B

C

D

E

number of samples
10 20 30

with a ÿ b ˆ 8:9  4:9 (in the sun-centered celestial

equatorial reference frame of [9]). Likewise,

0 0
~o‡ ˆ B@ ÿ14  14

14  14 0

ÿ1:2  2:6 1 0:1  2:7 CA  10ÿ11

1:2  2:6 ÿ0:1  2:7

0

10

0

600

700

800

900

time since Jan 01, 2000 [days]
p FIG. 3. Fit results (upper part; dots) and fit residuals  2 (lower part; crosses) for the amplitude of the cosine component of an assumed isotropy violation sigpnal at 2!. Strongly perturbed data sets are omitted when  2 > 20 Hz (25 subsets). In eight of these cases, a good fit was achieved after discarding the first 1–3 h (which likely contain residual perturbations from a preceding LHe refill); these were then reintroduced to the analysis. The histograms (bin size ˆ 1 Hz) on the right-hand side give the distribution of the fit results and residuals.

The individual fit results (Fig. 3) are combined for the

final result by coherent (vector) averaging. In the SME,
Pthe hypothetical Lorentz violation signal …x ÿ y†= ˆ iASi sin…!iT† ‡ ACi cos…!iT†, where  ˆ …x ‡ y†=2,
has Fourier components at six frequencies !i. As defined in [9], T ˆ 0 on March 20, 2001, 11:31 UT. The signal
components as given in Table I are derived [13] from

Eq. (39) in [9] (the cavity length L is not substantially

affected by the hypothetical Lorentz violation [14]).

Because our data extends over more than one year, in

the vector average we can resolve the six signal frequen-

cies [15] and extract elements of the traceless symmetric

matrix 0
~eÿ ˆ B@

a 1:7  2:6

1:7  2:6 ÿ6:3  12:4 1

b

3:6  9:0 CA

ÿ6:3  12:4 3:6  9:0 ÿ…a ‡ b†

 10ÿ15;

for the antisymmetric matrix. One sigma errors are

quoted. The limits on ~o‡ are weaker because its elements enter the experiment suppressed by   10ÿ4, Earth’s

orbital velocity. All elements of ~o‡ and all but one element of ~eÿ are obtained. Compared to [16], we improve the accuracy by about 2 orders of magnitude.

Moreover, while [16] give limits on linear combinations

of the elements of ~o‡, the present experiment allows individual determination, because of the > 1 yr span of

our data.

For comparison to previous work (e.g., [4,6,17,18]), we

also analyze our experiment within the Robertson-

Mansouri-Sexl test framework [19]. It assumes general-

ized Lorentz transformations that contain parameters

; , and . In a preferred frame  (usually the cosmic

microwave background), the speed of light c0 ˆ const. In a frame S moving with the velocity v~ with respect to ,

c=c0 ˆ 1 ÿ …A ‡ Bsin2†v2=c20, where  is the angle between the direction of c and v~. Both A ˆ  ÿ  ‡ 1

and

B

ˆ



ÿ



ÿ

1 2

vanish

in

SR.

In

this

formalism,

SR follows from (i) Kennedy-Thorndike (KT), (ii) MM,

and (iii) Doppler shift experiments. The latter determine

the time

1 2

j

<

8



dilation coefficient 10ÿ7 [18]. KT exper

 (ˆ iments

ÿ

1 2

test

in SR) to j ‡ velocity invari-

ance of c, using the periodic modulation of v provided

by Earth’s orbit [6] or rotation [17]. Measuring the

frequency of a cryogenic microwave cavity (that is

proportional to c) against an H maser, [17] obtained A ˆ …ÿ3:1  6:9†  10ÿ7.

While all three experiments are required for a com-

plete verification of SR within this framework, the MM

experiment currently offers the highest resolution. In case

of a violation of isotropy, B Þ 0, for our experiment we

obtain a periodic change of the beat frequency x ÿ y at

TABLE I. Signal amplitudes ASi ; ACi from the SME, based on [9], and fit results. !  2=…23 h 56 min† and 
 ˆ 2=1 yr denote the angular frequencies of Earth’s sidereal rotation and orbit;   42:3 is the colatitude of Konstanz,   23:4 the angle
between the ecliptic and Earth’s equatorial plane. The signal includes contributions of order 1, or suppressed by either Earth’s orbital velocity   10ÿ4 or by the velocity of the laboratory due to Earth’s rotation L  10ÿ6. For each !i, we include only the largest term, in effect dropping all terms proportional to L. A fit result that has been used to extract a parameter of the SME is set in bold face. The unused fit results lead to additional (but weaker) limits on the elements of ~o‡.

!i
! ÿ 
 !
! ‡ 
 2! ÿ 

2! 2! ‡ 


ASi

 2

sin

cos‰…cos

‡

1†…~o‡†XY

ÿ

cos…~o‡†XZŠ

ÿ sin cos…~eÿ†YZ

 2

sin cos‰…cos ÿ 1†…~o‡†XY

ÿ

 4

…1

‡

cos2

†…cos

‡

ÿ sin…~o‡ 1†…~o‡†YZ

†XZ

Š

 4

…1

1 2
‡

…1 ‡ cos2 cos2†…1

†…~eÿ†XY ÿ cos†…~o‡

†YZ

Fit (Hz)
1:82  1:91 ÿ0:51  1:26 ÿ0:43  1:68 ÿ0:01  0:57
0:37  0:56 0:50  0:55

ACi

 2

sin

cos

sin…~o‡

†YZ

ÿ sin cos…~eÿ†XZ

ÿ

 4

 2
…1

sin cos sin…~o‡†YZ ‡ cos2†…1 ‡ cos†…~o‡

†XZ

144…1…1‡‡cocso2s2†‰†……~1eÿÿ†XcXosÿ†……~~eoÿ‡††YXYZŠ

Fit (Hz)
0:67  1:58 0:89  1:74 1:83  1:83 0:25  0:55 0:97  0:53 ÿ0:75  0:58

020401-3

020401-3

VOLUME 91, NUMBER 2

PHYSICAL REVIEW LETTERS

week ending 11 JULY 2003

2! [20]. For such a signal, the experiment yields an amplitude 1:03  0:53 Hz, or B ˆ …ÿ3:1  1:6†  10ÿ9. As the quality of the data is not uniform (Fig. 3), taking a weighted vector average is more appropriate here [15]. We divide the data into the intervals A–E (Fig. 3) with approximately uniform data quality within each. The averages over the intervals are then combined to the final result, weighted according to their standard error. This leads to a signal of 0:73  0:48 Hz, or

B

ˆ



ÿ



ÿ

1 2

ˆ

…ÿ2:2



1:5†



10ÿ9;

which we regard as the final result within the RMS framework. It has an inaccuracy about 3 times lower than the best previous result B ˆ …3:0  4:9†  10ÿ9 [4,22].
In summary, we performed a modern MichelsonMorley experiment by comparing the frequencies of two crossed cryogenic optical resonators subject to Earth’s rotation over a period spanning more than one year. Within the Robertson-Mansouri-Sexl framework, our limit on the isotropy violation parameter is about 3 times lower than that from the classic experiment of Brillet and Hall [4]. Moreover, we obtain limits on seven parameters from the comprehensive extension of the standard model [8,9], down to 10ÿ15, about 2 orders of magnitude lower than the only previous result [16].
While the high long-term stability of COREs allows one to use solely Earth’s rotation, active rotation could still improve the accuracy significantly. Because of the low drift of COREs compared to ULE cavities, the optimum rotation rate would be relatively slow, which is desirable for minimizing the systematics. At a rate of 0:2=min (  5= min was used in [4]), one could utilize the optimum 7 
 10ÿ16 frequency stability (Fig. 2) of the COREs, more than 10 times better than on the 12 h time scale used thus far. Since 500 measurements (2 per turn) could be accumulated per day, one should thus be able to reach the 10ÿ17 level of accuracy. Further improvements include fiber coupling and COREs of higher finesse. Finally, space based missions with resonators are currently studied (OPTIS [23] and SUMO [24]).
We thank Claus La¨ mmerzahl for many valuable discussions and Ju¨ rgen Mlynek for making this project possible. This work has been supported by the Deutsche Forschungsgemeinschaft and the OptikZentrum Konstanz.

*Electronic address: holger.mueller@physik.hu-berlin.de †Electronic address: achim.peters@physik.hu-berlin.de ‡URL: http://qom.physik.hu-berlin.de/
[1] V. A. Kostelecky´ and S. Samuel, Phys. Rev. D 39, 683
(1989).

[2] R. Gambini and J. Pullin, Phys. Rev. D 59, 124021

(1999).

[3] A. A. Michelson, Am. J. Sci. 22, 120 (1881); A. A.

Michelson and E.W. Morley, ibid. 34, 333 (1887);

Philos. Mag. 24, 449 (1897).

[4] A. Brillet and J. L. Hall, Phys. Rev. Lett. 42, 549

(1979).

[5] S. Seel et al., Phys. Rev. Lett. 78, 4741 (1997); R. Storz

et al., Opt. Lett. 23, 1031 (1998).

[6] C. Braxmaier et al., Phys. Rev. Lett. 88, 010401 (2001).

[7] H. Mu¨ller et al., Int. J. Mod. Phys. D 11, 1101 (2002).

[8] D. Colladay and V. A. Kostelecky´, Phys. Rev. D 55,

6760 (1997); 58, 116002 (1998); R. Bluhm et al., Phys.

Rev. Lett. 88, 090801 (2002), and references therein.

[9] V. A. Kostelecky´ and M. Mewes, Phys. Rev. D 66, 056005

(2002).

[10] Additional parameters …kAF† are strongly bounded by other experiments [9] and assumed to be zero here.

[11] B. C. Young et al., Phys. Rev. Lett. 82, 3799 (1999).

[12] H. Mu¨ller et al. (to be published).

[13] For our cavity orientation (one cavity pointing north, the

other west), the terms proportional to A (that in part

depend on the transverse relative permittivity  inside

the resonator) and B, as defined in [9], drop out. We

substitute C0 . . . C4, as given in appendix E of [9] into Eq. (39) of that paper. The difference T between the two

time scales T and T used in [9], 2 h for our experi-
ment, gives rise to additional terms proportional to sin
T  10ÿ3 that can be neglected.
[14] H. Mu¨ller et al., Phys. Rev. D 67, 056006 (2003).

[15] In the SME analysis, unweighted averages are taken,

since weighted averages here would increase the

interdependence of the fit results, typically 0:15 and

in all cases  0:36. We can neglect it for the present

purpose.

[16] J. A. Lipa et al., Phys. Rev. Lett. 90, 060403 (2003).

[17] P. Wolf et al., Phys. Rev. Lett. 90, 060402 (2003).

[18] R. Grieser et al., Appl. Phys. B 59, 127 (1994).

[19] H. P. Robertson, Rev. Mod. Phys. 21, 378 (1949); R. M.

Mansouri and R. U. Sexl, Gen. Relativ. Gravit. 8, 497

(1977); 8, 515 (1977); 8, 809 (1977); see also

C. La¨ mmerzahl et al., Int. J. Mod. Phys. D 11, 1109

(2002).

[20]

…x

ÿ

y†=

ˆ

ÿB

1 2

…v=c0†2…1

‡

cos2

†cos2



cos2!TMS

(plus negligible Fourier components), obtained by ex-

pressing the angles x and y between the cavity axes and v~ in terms of , the declination   ÿ7 of v~, and jv~j ˆ

369 km=s [21]. TMS ˆ 0 on March 8, 2001, 11:34 UT, when the west-pointing resonator is orthogonal to the

projection of v~ onto the equatorial plane.

[21] C. H. Lineweaver et al., Astrophys. J. 470, 38 (1996).

[22] The result is quoted here assuming v ˆ 369 km=s [21]

and taking into for the B ˆ 50

account a colatitude

1 4

‰1

of t

‡ he

cos…B†Š2cos2 factor laboratory in Boulder.

[23] C. La¨ mmerzahl et al., Classical Quantum Gravity 18,

2499 (2001).

[24] S. Buchman et al., Adv. Space Res. 25, 1251 (2000).

020401-4

020401-4