Limit on Lorentz and CPT violation of the bound Neutron Using a Free Precession 3He/129Xe co-magnetometer
C. Gemmel, W. Heil, S. Karpuk, K. Lenz, Yu. Sobolev, K. Tullney, M. Burghoff, W. Kilian, S. Knappe-Grüneberg, W. Müller, A. Schnabel, F. Seifert, L. Trahms, U. Schmidt
aa r X i v : . [ g r- q c ] J a n Limit on Lorentz and CPT Violation of the Bound Neutron Using a Free Precession He/
Xe Co-magnetometer
C. Gemmel, W. Heil, ∗ S. Karpuk, K. Lenz, Yu. Sobolev, K. Tullney, M. Burghoff, W. Kilian, S. Knappe-Gr¨uneberg, W. M¨uller, A. Schnabel, F. Seifert, L. Trahms, and U. Schmidt Institut f¨ur Physik, Johannes Gutenberg-Universit¨at, 55099 Mainz, Germany Physikalisch-Technische-Bundesanstalt (PTB) Berlin, 10587 Berlin, Germany Physikalisches Institut, Universit¨at Heidelberg, 69120 Heidelberg, Germany (Dated: May 29, 2018)We report on the search for Lorentz-violating sidereal variations of the frequency difference of co-located spin species while the Earth and hence the laboratory reference frame rotates with respectto a relic background field. The comagnetometer used is based on the detection of freely precessingnuclear spins from polarized He and
Xe gas samples using SQUIDs as low-noise magnetic fluxdetectors. As result we can determine the limit for the equatorial component of the backgroundfield interacting with the spin of the bound neutron to be ˜ b n ⊥ < . · − GeV (95% C.L.).
PACS numbers: 06.30.Ft, 07.55.Ge, 11.30.Cp, 11.30.Er, 04.80.Cc, 32.30.Dx, 82.56.Na
A great number of laboratory experiments have beendesigned to detect diminutive violations of Lorentz in-variance. Among others, the Hughes-Drever-like experi-ments [1, 2] have been performed to search for anomalousspin coupling to an anisotropy in space using electronand nuclear spins with steadily increasing sensitivity [3–14]. Lorentz-violating theories should generally predictthe existence of privileged reference systems. In contrastwith the situation at the end of the 19th century, we havea rather unique choice nowadays for such a ”preferred in-ertial frame,” i.e., the frame where the Cosmic MicrowaveBackground (CMB) looks isotropic. Trying to measurean anomaleous coupling of spins to a relic backgroundfield which permeates the Universe and points in a pre-ferred direction in spacetime as a sort of new aether windis a modern analogue of the original Michelson-Morleyexperiment.The theoretical framework presented by Kosteleck´yand colleagues parametrizes the general treatment of
CPT - and Lorentz violating effects in a standard modelextension (SME) [15]. The SME was conceived to fa-cilitate experimental investigations of Lorentz and
CPT symmetry, given the theoretical motivation for violationof these symmetries. Although Lorentz-breaking inter-actions are motivated by models such as string theory[16, 17], loop quantum gravity [18–21], etc. (i.e., fun-damental theories combining the standard model withgravity), the low-energy effective action appearing in theSME is independent of the underlying theory. Each termin the effective theory involves the expectation of a tensorfield in the underlying theory. These terms are small dueto Planck-scale suppression and, in principle, are mea-surable in experiments. Predictions for parameters in theSME for a loop quantum gravity system with a preferredframe were discussed, e.g., in Ref. [22].The SME contains a number of possible terms that ∗ Corresponding author: [email protected] couple to the spins of standard model particles like theelectron, proton, and nucleon (mostly the bound neu-tron) [23]. These terms have set the most stringent lim-its on
CPT and Lorentz violations. To determine theleading-order effects of a Lorentz violating potential V ,it suffices to use a non-relativistic description for the par-ticles involved given by [23] V = − ˜ b wJ · σ wJ (with J = X, Y, Z ; w = e, p, n ) . (1)The most sensitive tests were performed using a He-
Xe Zeeman maser to place an upper limit on theneutron coupling to the anomalous field of ˜ b n ⊥ = r(cid:16) ˜ b nX (cid:17) + (cid:16) ˜ b nY (cid:17) < − GeV [9, 10] and, recently, byuse of a K- He co-magnetometer thereby improving theprevious limit by a factor of 30 [4]. An essential assump-tion in these so-called clock comparison experiments isthat the anomalous field ˜ b wJ does not couple to magneticmoments but directly to the sample spins σ wJ . This directcoupling allows comagnetometry that uses two differentspin species to distinguish between a normal magneticfield and an anomalous field coupling.The comagnetometer used for the presented measure-ments is based on the detection of freely spin precess-ing nuclear spins from polarized He and
Xe samplesgas with SQUIDs as low-noise magnetic flux detectors.Like in [9, 10], we search for sidereal variations of thefrequency of colocated spin species while the Earth andhence the laboratory reference frame rotates with respectto a relic background field. The observable to trace pos-sible tiny sidereal frequency modulations is the combina-tion of measured Larmor frequencies given by∆ ω = ω L,He − γ He γ Xe · ω L,Xe . (2)By that measure the Zeeman term is eliminated andthus any dependence on fluctuations and drifts of themagnetic field. For the He/
Xe gyromagnetic ratioswe took the literature values [24, 25] given by γ He /γ Xe =2 . He/
Xe comag-netometer has been shown recently [26]. Briefly, in ourmeasurements, we used a low- T c DC-SQUID magnetome-ter system inside the strongly magnetically shielded roomBMSR-2 at PTB [27]. A homogeneous guiding magneticfield B of about 400 nT was provided by one of the twosquare coil pairs which were arranged perpendicular toeach other in order to manipulate the sample spins, e.g., π /2 spin flip by nonadiabatic switching. The maximumfield gradients were about 33 pT/cm. The He/
Xe nu-clear spins were polarized outside the shielding by meansof optical pumping. Low-relaxation spherical glass ves-sels (R=3 cm) were filled with the polarized He/
Xegases and placed directly below the Dewar as close as pos-sible to SQUID sensors, which detect a sinusoidal changein magnetic flux due to the spin precession of the gasatoms in the glass cell. In order to obtain a high commonmode rejection ratio, gradiometric sensor arrangementsare commonly used. For our analysis it was sufficient touse a first-order gradiometer in order to suppress envi-ronmental disturbance fields.Nitrogen was added as buffer gas to suppress the vander Waals spin relaxation of
Xe [28]. In the regimeof motional narrowing, i.e., at gas pressures of ordermbar and at low magnetic fields [29, 30], transverse spin-relaxation times T ∗ of up to 60 h have been measuredfor He. The actual limitation in the T ∗ , Xe of xenon isgiven by the relatively short wall relaxation time of 8 h 16 h. Therefore, the total observation time T offree spin-precession of our He/ Xe comagnetometer isset by this characteristic time constant. According to theCramer-Rao Lower Bound (CRLB) [31], the accuracy bywhich the frequency of a damped sinusoidal signal canbe determined is given by σ f ≥ √ π ) · SN R · √ f BW · T / × q C ( T, T ∗ ) . (3) SN R denotes the signal-to-noise ratio, f BW the band-width, and C ( T, T ∗ ) describes the effect of exponentialdamping of the signal amplitude with T ∗ . For observa-tion times T ≤ T ∗ , C ( T, T ∗ ) is of order one. Deviationsfrom the CRLB power law, due to noise sources inher-ent in the comagnetometer itself, did not show up in Al-lan standard deviation plots used to identify the power-law model for the phase noise spectrum of our runs with T ≈ 14 h, typically [26].The recorded signal is a superposition of the He and Xe precession signals at Larmor frequencies ω He = γ He · B ≈ π · . ω Xe = γ Xe · B ≈ π · . j = 1, . . . ,7) were divided into - - - H s L S Q U I D s i gna l H p T L FIG. 1. Typical subdata set of 3.2 s length showingthe recorded SQUID gradiometer signal of the precessing He/ Xe sample spins (sampling rate: r s =250 Hz). Theuncertainty at each data point is ± 34 fT ( k =1) and thereforeless than the symbol size. The signal amplitudes at the begin-ning of each run were typically S He ≈ 13 pT and S Xe ≈ sequential time intervals ( i ) of τ = 3.2 s ( i = 1, . . . , N j ).The number of obtained subdata sets laid between 13350 < N j < T j ofcoherent spin precessions in the range of 12 h < T j < 16 h.For each subdata set a χ minimization was performed,using the fit function A i ( t ) = A i He · sin (cid:0) ω i He t (cid:1) + B i He · cos (cid:0) ω i He t (cid:1) + A i Xe · sin (cid:0) ω i Xe t (cid:1) + B i Xe · cos (cid:0) ω i Xe t (cid:1) +( c i + c i lin · t ) (4)with a total of 8 fit parameters. Within the relativelyshort time intervals, the term ( c i + c i lin · t ) representsthe adequate parameterization of the SQUID gradiome-ter offset showing a small linear drift due to the elevated1/ f noise at low frequencies ( < χ min-imization. The sum of sine and cosine terms are chosento have only linear fitting parameters for the subdata setphases which are given by ϕ i = arctan ( B i /A i ) . (5)The normalized χ ( χ /d.o.f) of most subdata sets ( i ) isclose to 1 which is consistent with the assigned uncer-tainty to each data point of ± 34 fT ( k =1); see FIG. 1.The latter value is the typical noise signal N s derivedfrom the mean system noise ¯ ρ s ≈ / √ Hz in therecorded effective bandwidth of 100 Hz. Jumps in theSQUID signal in the order of 1 pT caused by externaldisturbances gave χ /d.o.f ≫ χ /d.o.f ≥ < ≤ ( t − t i − ,j ) ≤ +1.6 s (see FIG. 1), we finally obtain numbersfor the respective fit parameters ω i He , ω i Xe , ϕ i He , ϕ i Xe in-cluding error bars.In a further step, we can deduce values for the averagefrequency ¯ ω j = N j P N j i =1 ω i for each run. The accumu-lated phase (omitting the index j ) is then determined tobe Φ ( t = mτ ) = Φ ( t = ( m − τ ) + ¯ ω · τ + ϕ m − mod [Φ ( t = ( m − τ ) + ¯ ωτ ; 2 π ] (6)with m =1,...,N-1 and Φ( t = 0) = ϕ being the phaseoffset of the first time interval. Following Eq. (2) the ex-tracted phase difference ∆Φ (1) ( t ) = Φ (1)He ( t ) − ( γ He /γ Xe ) · Φ (1)Xe ( t ) is plotted for run 1. ∆Φ (1) ( t ) is expected tobe constant if there is no sidereal modulation of thespin-precession frequency and/or no other drifts andnoise sources. Nevertheless, in addition to an arbitraryphase offset an almost linear time dependence is seen inFIG. 2(a). The dominant contribution is caused fromthe Earth’s rotation, i.e., the rotation of the SQUID de-tector with respect to the precessing spins. For the loca-tion of the PTB Berlin, Germany ( θ =52.5164 ◦ north) andthe angle between north-south direction and the guidingmagnetic field ( ρ =28 ◦ ), the linear term in the weightedphase difference due to Earth’s rotation is given by [26]Φ Earth = − Ω SD · (1 − γ He /γ Xe ) · cos ρ · cos θ · t = 6 . × − rad/s · t . (7)Ω SD is the angular frequency of the sidereal day withΩ SD = 2 π/T SD = 2 π/ (23 h : 56 min : 4 . s ). Subtract-ing this term from ∆Φ (1) ( t ), we get the corrected phase∆Φ (1)corr ( t ) which is plotted in FIG. 2(a), too. Let us as-sume that there is no sidereal variation of the He/ Xefrequencies induced by Lorentz-violating couplings, andthen ∆Φ ( j )corr ( t ) can be described best byΦ ( j )fit ( t ) = Φ ( j )0 + ∆ ω ( j )lin ( t − t ,j )+ E ( j )He · exp − ( t − t ,j ) T ∗ ( j )2 , He ! + E ( j )Xe · exp − ( t − t ,j ) T ∗ ( j )2 , Xe ! (8)withΦ ( j )fit ( t ) = (cid:26) Φ ( j )fit ( t ) for t ,j ≤ t ≤ ( t ,j + N j · τ )0 elsewhere. t ,j is the absolute starting time of each run. Our inter-pretations of the terms are as follows: Φ ( j )0 is a generalphase offset and ∆ ω ( j )lin ( t − t ,j ) is an additional linearphase shift mainly arising from deviations of the gyro-magnetic ratios of He and Xe from their literaturevalues due to chemical shifts and uncertainties in the sub-traction of Φ Earth [26]. The two exponential terms withamplitudes E ( j )He and E ( j )Xe reflect the respective phase As the maximal frequency deviation ∆ ω from the mean ¯ ω j wassmaller than 5 · − rad/s in the course of one run [26], we hadat all times ∆ ω · τ ≪ π . DF H L H t L = F He H L H t L - Γ He Γ Xe F Xe H L H t L DF corr H L H t L = DF H L H t L - F Earth H t L a. L t - t H h L pha s e H r ad L t - t H h L b. L - - - pha s e H r ad L FIG. 2. (a) Measured phase differences ∆Φ (1) ( t ) for run 1 andthe corresponding corrected phases ∆Φ (1)corr ( t ) after subtractionof the effect of the Earth’s rotation. (b) Phase residuals aftersubtraction of phase drifts given by the fit model of Eq. (8)(one data point comprises 20 subdata sets, i.e., ∆ t = 64 s). shift due to demagnetization fields in a nonideal sphericalcell seen by the spin ensembles (self-shift). These phaseshifts are directly correlated to the decay times T ( j )2 , He and T ( j )2 , Xe of the respective signal amplitude of the precessinghelium and xenon spins [26]. As the T ∗ ( j )2 times canbe determinded independently for both spin species fromthe experiment, four fit parameters are left for each run,such that the fit model is basically a linear function inparameters.Fitting the corrected phase difference ∆Φ ( j )corr ( t ) toEq. (8) and subtracting the fit function from ∆Φ ( j )corr ( t ) re-sults in the phase residual as shown for run 1 in FIG. 2(b).Because of the exponential decay of the signal ampli-tudes, mainly that of xenon with the shorter T ∗ , Xe ofonly 4-5 h, the SN R decreases resulting in an increase ofthe residual phase noise, i.e., σ Φ , res ∝ exp (cid:16) t/T ∗ ( j )2 , Xe (cid:17) [26].In the last step, a piecewise fit function was defined,which is a combined fit to all seven runs, now includingthe parameterization of the sidereal phase modulation DF corr H L H t L DF corr H L H t L DF corr H L H t L DF corr H L H t L DF corr H L H t L DF corr H L H t L DF corr H L H t L t - t H h L pha s e H r ad L FIG. 3. Corrected phase differences ∆Φ ( j )corr ( t ) with combinedfit function Φ SDfit ( t ) (white solid line) for the seven runs (onedata point comprises 20 sub-data sets, i.e., ∆ t = 64 s). Inorder to present these results in a common plot, the generalphase offset Φ ( j )0 was subtracted from ∆Φ ( j )corr ( t ) for each run. Φ SDfit ( t ) = X j =1 Φ ( j )fit ( t )+ { a s · sin (Ω SD ( t − t , ) + ϕ SD ) − a c · cos (Ω SD ( t − t , ) + ϕ SD ) } (9) ϕ SD = 2 π · t SD represents the phase offset of sidereal mod-ulation at the local sidereal time (vernal equinox J2000.0)on 2009 March 21 at 20:52 UT (universal time) which isthe starting time t , of the first run. Neglecting multi-ples of 24 h the local sidereal time is 9.7 h, which in unitsof sidereal day gives t SD = 0 . ( j )corr ( t )together with the fit function Φ SDfit ( t ) (white solid line) forall seven runs. The χ /d.o.f of the fit gave 1.868, whichshows that the phase model of Eq. (8) may be somewhatincomplete or, what is more likely, the phase errors areunderestimated in the analysis of the subdata sets. Inorder to take an (unknown) uncertainty into account, theerrors on the phases were scaled to obtain a χ /d.o.f ofone, as recommended, e.g., by Refs. [33, 34]. In Table(2nd row) the fit results for the amplitudes a c and a s ofthe sidereal phase modulation are shown together withtheir correlated and uncorrelated 1 σ errors.It is noticeable that the uncorrelated error which rep-resents the integrated measurement sensitivity of our He/ Xe comagnetometer is about a factor of 50 lessthan the correlated one. The big correlated error on a s and a c is caused by a piecewise similar time structure ofΦ ( j )fit ( t ) and the sidereal phase modulation in the fit func-tion of Eq. (9). On a closer look, this can be traced backto the relatively short T ∗ ( j )2 , Xe times (compared to T SD ) thatenter in the argument of the exponential terms of Eq. (8).Therefore, the present sensitivity limit of our He/ Xecomagnetometer is set by the correlated error. In order tosubstantiate that more clearly, we changed the fit modelof Eq. (9) by taking multiples of Ω SD (Ω ′ SD = g · Ω SD ), i.e.,replacing T SD by T ′ SD = T SD /g . The results show that thecorrelated error approaches the uncorrelated one already for g ≥ SD = p a + a , yielding(2.25 ± CPT -violating effects,giving reasonable assumptions about the probability dis-tribution for Φ SD [35].In terms of the SME [23] we can express the siderealphase amplitudes according to a s = 2 π Ω SD · δν X and a c = 2 π Ω SD · δν Y (10)with2 π | δν X,Y | · ~ = (cid:12)(cid:12)(cid:12) · (1 − γ He /γ Xe ) · sin χ · ˜ b nX,Y (cid:12)(cid:12)(cid:12) . (11) χ is the angle between the Earth’s rotation axis and thequantization axis of the spins with χ = arccos (cos θ · cos ρ ) = 57 ◦ . Within the Schmidt model [36], the valenceneutron of He and Xe determines the spin and themagnetic moment of the nucleus. Thus, our He/ Xecomagnetometer is sensitive to the bound neutron pa-rameters ˜ b nX,Y . From that, we can deduce numbers for˜ b nX,Y : ˜ b nX = (3 . ± . · − GeV (1 σ ) , (12)˜ b nY = (1 . ± . · − GeV (1 σ ) , (13)which can be interpreted as (cid:12)(cid:12)(cid:12) ˜ b n ⊥ (cid:12)(cid:12)(cid:12) < . · − GeV at95% confidence level for the upper limit of the equatorialcomponent of the background tensor field interactingwith the spin of the bound neutron. For the calculationof the upper limit on ˜ b n ⊥ Eq. (10) and Eq. (11) were usedputting in the 95% C.L. for the rms value of the siderealphase amplitude Φ SD .Further improvements for Lorentz and CPT testsusing the free spin-precession He/ Xe comagnetome-ter can be achieved via two mayor steps: First, therelatively short wall relaxation time of Xe limitingthe total observation time T of free spin precessionhas to be increased considerably (T , wall ≈ T SD ) suchthat we approach the measurement sensitivity given bythe uncorrelated error. Since the latter one follows the ∝ T − / power law according to CRLB of Eq. (3), thelonger observation time T will lead to an additionalincrease in sensitivity. Second, the number of measure-ment runs has to be increased to a period of 100 days.Besides gain in statistics, the long time span providesan important separation between sidereal and possiblediurnal variations.This work was supported by the Deutsche Forschungs-gemeinschaft (DFG) under Contract No. BA 3605/1-1. TABLE I. Results for the sidereal phase amplitudes a c and a s together with theircorrelated and uncorrelated 1 σ errors (2nd row) determined by a χ minimizationusing the fit model of Eq. (9). 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