Nonlinear magneto-optical rotation with frequency-modulated light D. Budkera,b, D. F. Kimballa, V. V. Yashchuka, and M. Zolotorevc a Department of Physics, University of California, Berkeley, CA 94720-7300 e-mail: [email protected] b Nuclear Science Division, Lawrence Berkeley National Laboratory, Berkeley CA 94720 2 c Center for Beam Physics, 0 Lawrence Berkeley National Laboratory, Berkeley CA 94720 0 (Dated: February 2, 2008) 2 A magnetometric technique is demonstrated that may be suitable for precision measurements n of fields ranging from the sub-microgauss level to above the Earth field. It is based on resonant a J nonlinear magneto-optical rotation caused by atoms contained in a vapor cell with anti-relaxation wall coating. Linearly polarized, frequency-modulated laser light is used for optical pumping and 1 probing. Ifthetime-dependentopticalrotationismeasuredatthefirstharmonicofthemodulation 1 frequency, ultra-narrow ( a few Hz) resonances are observed at near-zero magnetic fields, and ] at fields where the Larmo∼r frequency coincides with half the light modulation frequency. Upon h optimization, thesensitivity of the techniqueis expected toexceed 10−11G/√Hz. p - m Whenlightnear-resonanttoanatomictransitionprop- 1kHz,andtherotationangleisdeterminedbymeasur- o agates through an atomic medium immersed in a mag- ∼ing the firstharmonic ofthe signalfroma photodetector t a netic field, the light polarization can be affected. For in the dark output of the analyzer. Such a method al- . example,whenamagneticfieldisappliedalongthedirec- lows detection of signals at high frequencies, eliminating s c tion of light propagation,there is light-power-dependent sensitivity to excess low-frequency noise. However, the i s rotation of the polarization plane known as nonlinear polarizationmodulationtechniquehasanumberofshort- y magneto-optical (Faraday) rotation (NMOR). Recently comings. Importantly, it is not immune to the problem h [1], we observed ultra-narrow ( 1 Hz) zero-field res- of “drifting zero” – any relative rotation of the polarizer p ∼ onances in NMOR with rubidium atoms contained in and analyzer, or a change in the birefringence of optical [ vapor cells with high-quality anti-relaxation coating [2]. elements, etc. is detected as a rotation signal indistin- 1 These resonances arise due to preservation of atomic guishable from rotation produced by the atoms. In or- v polarization over thousands of collisions with the walls dertoavoidsuchproblems,alightfrequencymodulation 5 2 of the cell. The sensitivity of an NMOR-based mag- technique was introduced in Refs. [4, 5] and was applied 0 netometer for sub-microgauss fields could, in principle, to measurements of linear magneto-optical rotation and 1 exceed 10−11 G/√Hz [3], approaching the fundamen- parity-violating optical rotation. This technique allowed 0 tal shot-noise limit (given the number of atoms in the detection of optical rotation at the light modulation fre- 2 cell and the polarization relaxation rate). In the present quency without introducing additional optical elements 0 / work,wedescribe anew techniquethatinvolvesbothin- (suchasaFaradaymodulator)betweenthepolarizerand s ducing and measuring NMOR with a single frequency- analyzer, considerably reducing spurious rotations. Sen- c i modulated light beam. Additional ultra-narrow reso- sitivitytomostspuriousrotations(includingthe“driftof s y nancesareobservedwhenthelightmodulationfrequency zero”)isfurtherreducedbecausesucheffectsgenerallydo h coincideswithtwicetheLarmorfrequency,2ΩL(andalso not lead to spectral features as sharp as an atomic reso- p with Ω , etc.), the use of which can extend the dynamic nance. Opticalpumping with frequency-modulatedlight L v: range of an NMOR-based magnetometer to above the was studied and applied to 4He resonance magnetome- i Earth field range ( 1 G) of interest in many applica- ters [6, 7, 8], but in that work transmission was moni- X ∼ tions. tored, in contrast to the measurements of optical rota- r a The high sensitivity ofNMOR magnetometryis based tion performed in the present work. The performance of inpartontheabilityofthepolarimetertomeasuresmall the 4He magnetometer is limited by laser intensity noise optical rotation angles. The sensitivity of polarimeters [8], which is not expected to limit the performance of without modulation is limited by low-frequency noise. NMOR-based magnetometers. Therefore, the most successful polarimetric techniques A simplified diagram of the experimental apparatus is alwaysinvolvesome type of fastmodulation. In ourpre- shown in Fig. 1. A paraffin-coated vapor cell (diameter vious work [1, 3], we used polarization-modulation po- 10 cm) containing an isotopically enriched sample of Rb larimetry, where in addition to the vapor cell under in- atoms( 94%of87Rb;atomicdensity 7 109cm−3 at ◦ ≈ ≈ × vestigation, a Faraday rotator is inserted between the 20 C)isplacedinamulti-layermagneticshieldequipped crossed polarizer and analyzer. The Faraday rotator with a system of magnetic coils within the inner shield modulates the polarization direction at a frequency of that are used to compensate residual magnetic fields 2 (to a level of 0.1 µG), and to apply well-controlled, Additional resonances are seen at magnetic field mag- ∼ arbitrarily-directed fields to the cell [1, 3]. The cell is nitudes of 714.4 µG (Ω =2Ω ) and 1428.9 µG (Ω = m L m placedbetweenapolarizerandananalyzerorientedat Ω )forthefirstandsecondharmonics,respectively(Fig. L ◦ ≈ 45 with respect to each other (a balanced polarimeter). 2). For these resonances, there are both dispersively- The frequency of the laser (tuned near the D1-line) is shaped in-phase signals (Fig. 2(a,c)) and π/2 out of modulated with anamplitude typicallyof a few hundred phase (quadrature)components peaked at the centers of MHz. For the measurements presented here, we have these resonances (Fig. 2(b,d)). The origin of the reso- chosen modulation frequencies up to Ω = 2π 3 kHz nances can be understood as follows (Fig. 3). As the m × achievedby modulating a piezo actuator [9] in the diode laser frequency is modulated, the optical pumping rate laser (New Focus, Vortex 6000). The amplitudes at the changes depending on the instantaneous detuning of the first and second harmonics of Ω of the difference sig- laser from the atomic transition. This occurs with peri- m nal between the two photodiodes at the outputs of the odicitydictatedbythemodulationfrequencyΩ . Onthe m analyzer are detected with a lock-in amplifier (Stanford otherhand,the stateofatomicpolarizationalsochanges Research Systems, SR810DSP). Upon normalization by periodically due to the presence of the magnetic field. the time-averaged sum of the photodiode outputs, this Inthe caseofalignmenttransversetothe magneticfield, represents the signal measured in this experiment (i.e. thischangeoccursatfrequency2Ω (alignmentprecesses L the effective amplitude ofthe firstorsecondharmonicof at Ω , but it returns to the same state after a half of L the time-dependent polarization rotation angle). the corresponding period). If the optical pumping rate The signals at the first and second harmonics of Ω is synchronized with Larmor precession, a resonance oc- m asafunctionoflongitudinal(alongthelightpropagation curs, and the atomic medium is optically pumped into direction) magnetic field are shown in Fig. 2. The laser an aligned state whose axis rotates at ΩL. The optical frequencyismodulatedatΩ =2π 1kHzwithmodula- properties of the medium are thus being modulated at m × tionamplitude∆ω =2π 220MHz. Forthefirstandthe 2ΩL,causingrotationofthelightpolarizationatthisfre- × second harmonic signals, the laser is tuned to where the quency. This leads to dispersively-shaped resonances in respective signalsareofmaximalsize: the low-frequency opticalrotationthataredetectedasthesignalattheap- slope of the F = 2 F′ = 1 resonance (F,F′ are the propriate harmonic of Ωm. The quadrature components → total angular momenta of the lower and upper state) for (Fig. 2(b,d)) arise because exactly on resonance, the the first harmonic (Fig. 4(a)), and to the center of the aligned atoms produce maximum optical rotation when F =2 F′ =1 resonance for the second harmonic. the alignment axis is at an angle of π/4 to the direction → Forbothharmonics,anarrow,dispersively-shapedres- of the light polarization. (There is no quadrature com- onance is seen in Fig. 2 near B =0. Its origin is similar ponent at zero magnetic field because atomic alignment to that of the ultra-narrow resonances observed in the doesnotrotateatallinthiscase.) Sincethewidthsofthe previous work [1]: the atoms are pumped by linearly- additional resonances at Ωm = 2ΩL and Ωm = ΩL are polarized light into a state aligned [10] along the light determined by γrel as for the zero-field resonance, they polarization;optically-pumpedatomicalignmentevolves are similarly narrow. In addition to these resonances, due to the applied magnetic field; the resulting atomic therearealsosmallerresonancesinthefirst-andsecond- polarization [11] is “probed” by the light, leading to a harmonic signals, generally occurring for commensurate modified light polarization at the output of the cell. For Ωm and ΩL. There are several mechanisms that lead to the zero-field resonances in Fig. 2, the modulation fre- these resonances, which will be described in detail else- quencyismuchfasterthantheLarmorfrequencyandthe where. opticalpumping rateforthe cell,sothatfrequencymod- TheoverallslopeofthecurvesinFig. 2isduetotheso- ulationisequivalenttospectralbroadeningofthepump- called transit effect (see e.g. [1] and references therein), ing light, which does not significantly affect the pump- for which the peaks occur at B = γtr/2gFµ 50 mG, ± ∼ ing process. On the other hand, modulation is essential where γtr is the rate of atoms’ transit through the laser at the probing stage. As the laser frequency is scanned beam. throughthe atomic resonance,there is a time-dependent The spectraldependences of various signals are shown opticalrotation,sothesignalcontainsvariousharmonics in Fig. 4. Fig. 4(a) depicts the spectrum of the first of Ω . The peaks in the magnetic field dependence of harmonic signal obtained when the magnetic field was m the signal occur at ΩL gFµB = γrel/2, where gF is fixedat 2.2µG (correspondingto maximumsignal),and the Land´e factor(g =≡1/2forthe F±=2 state of87Rb), the centralfrequency ofthe laser wasscannedacrossthe F µ 2π 1.40 MHz/G is the Bohr magneton, and γrel hyperfine structure of the Rb D1-line (the transmission ≈ · ( 2π 1 Hz for the data in Fig. 2)is the relaxationrate spectrum is shown for comparison in Fig. 4(d)). Com- ∼ · of atomic polarization. The zero-field resonances can be paringthisspectrumwiththeNMORspectrumobtained appliedtolow-fieldNMOR-magnetometry[3],withallof without frequency modulation (as in earlier work [1, 3]) theadvantagesofthefrequency-modulationtechniqueas shown in Fig. 4(c), it is seen that the spectral pro- in the case of linear optical rotation [4, 5]. file of the signal with frequency modulation is similar 3 to dϕ(ν)/dν, where ϕ(ν) is the NMOR angle and ν is difference between the Land´e factors for Rb and Cs, this the laser frequency. Fig. 4(b) shows the spectrum ob- avoids “locking” of their polarizations to each other. In tained with the magnetic field of B = (714.4+2.2) µG, addition, presence of a bias field eliminates the effects of corresponding to an Ω =2Ω resonance (Fig. 2). The stray transverse fields. m L spectraldependence of the quadraturecomponentof the TheauthorsaregratefultoE.B.Alexandrovforstim- first harmonic for B = 714.4 µG (not shown) is similar ulating discussions. This work is supported by the Of- to that of the in-phase component. The signals acquired fice of Naval Research (grant N00014-97-1-0214). D. B. at the second harmonic of Ωm (not shown) have spectra alsowishestoacknowledgesupportbyanNSFCAREER resembling d2ϕ(ν)/dν2. grant. We haveinvestigatedthe dependence ofthe signals on the modulation amplitude ∆ω and on the light power, and found similar behavior for the zero-field and the Ω =2Ω resonances. Themaximumfirstharmonicam- m L plitudeisachievedformodulationamplitude∆ω compa- [1] D. Budker, V. Yashchuk, and M. Zolotorev, Phys. Rev. rable to the Doppler width of the transition 300MHz. Lett. 81(26), 5788 (1998). ∼ [2] E. B. Alexandrov, M. V. Balabas, A. S. Pasgalev, A. K. The signals are proportional to light power at low pow- Vershovskii, and N. N. Yakobson, Laser Phys. (Russia) ers, peak at around 30 µW, and decrease with further p. 244 (1996). powerincreaseduetolightbroadening. Subsequentwork [3] D. Budker, D. F. Kimball, S. M. Rochester, V. V. willincludesystematicoptimizationofallparametersfor Yashchuk,andM.Zolotorev,Phys.Rev.A62(4),043403 highest magnetometric sensitivity, as it was done earlier (2000). [3] for low-field NMOR with conventional polarization- [4] L. M. Barkov and M. S. Zolotorev, Pis’ma v ZhETF modulation polarimetry. Additional improvement may 28(8), 544 (1978), (JETP Lett. 28(8), 503 (1978).). [5] L. M. Barkov, M. Zolotorev, and D. Melik-Pashaev, be expected if the pump and probe laser beams are sep- Pis’mavZhETF,48(3),134(1988), (JETPLett.48(3), arated, and the modulation method is optimized (e.g., 144 (1988).). a combination of frequency and amplitude modulation). [6] B.Cheron,H.Gilles, J.Hamel,O.Moreau,andE.Noel, Withthepresentunoptimizedparametersandset-up,we Opt. Commun. 115(1-2), 71 (1995). estimatethattheshot-noise-limitedsensitivityofthe de- [7] B.Cheron,H.Gilles, J.Hamel,O.Moreau,andE.Noel, viceis 10−11 G/√Hz (comparableorsuperior,e.g.,to J. Phys.II (France) 6(2), 175 (1996). themo∼stsensitivesuperconductingquantuminterference [8] H. Gilles, J. Hamel, and B. Cheron, Rev. Sci. Instrum. 72(5), 2253 (2001). (SQUID) sensors). Also note that shot-noise-limited op- [9] Wehavealso carried out measurementswhere laser fre- erationin the Earth-field rangewill be facilitated by the quency was modulated by changing the diode laser cur- reduced laser noise at high frequencies, and by the fact rent at frequencies up to 100 kHz. (Modulation at thatthismethodusesresonantopticalrotationwhichal- frequencies up to 100 MHz∼is possible with this laser.) lows suppression of the common mode noise of the two At low modulation frequencies, results for both type of photodetectors and spurious rotations. Due to the ab- laser modulation are similar, although for a quantita- sence of inertia in magnetic spin-precession, a magne- tive analysis, it is necessary to take into account non- negligiblelightamplitudemodulationwhichaccompanies tometer based on this technique (which employs trans- frequency modulation when current modulation is used. verse polarization) could, in principle, also have a high [10] Alignment is a particular case of atomic polarization band width ultimately limited by electronics. corresponding to an anisotropy characterized by a pre- In conclusion, we have demonstrated a novel magne- ferred axis, rather than by a preferred direction as in tometric technique based on nonlinear optical rotation the case of orientation. Alignment is represented by a withfrequency-modulatedlight. Thistechnique isuseful rank-two (quadrupole) irreducible tensor component of fornearzero-fieldmagnetometryanditalsoopensapos- the density matrix, while orientation is represented by a rank-onecomponent. sibility for ultra-sensitivemagnetometerswith broaddy- [11] The resulting polarization is alignment rotated due to namicrangethatincludestheEarthfieldrange,ofimpor- Larmor precession at low light powers, and a more com- tancetomanyapplications,suchasgeophysics,magnetic plicated atomic polarization state when optical pump- prospecting and navigation. Possible new applications ing is saturated (see D. Budker, D. F. Kimball, S. M. alsoincludenuclearmagneticresonancespectroscopyand Rochester,andV.V.Yashchuk,Phys.Rev.Lett.85(10), imaging using weakly-polarized samples that would not 2088 (2000).). require any additional magnet, except the Earth itself [12] A.H.Trabesinger,R.McDermott,M.Muck,E.L.Hahn, J. Clarke, and A. Pines, in 42-nd Experimental NMR [12]. Narrow magneto-optical resonances may also find Conference, Orlando, FL, USA; March Meeting of the applications in atomic tests of fundamental symmetries APS, Seattle, WA, USA (2001), in Books of Abstracts. [13]. For example, it may be advantageous to carry out [13] V.Yashchuk,D.Budker,andM.Zolotorev(1999),p.177, a search for a parity- and time-reversal invariance vio- Trapped Charged Particles and Fundamental Physics lating electric dipole moment of Cs in the presence of a Asilomar, CA, USA 31 Aug.-4 Sept. 1998 AIP Conf. Rbco-magnetometerinabiasmagneticfield. Duetothe Proc. #457. 4 FIG. 1: Simplified schematic of the apparatus. FIG. 2: Signals detected at the first harmonic (a,b) and second harmonic (c,d) of Ωm as a function of longitudinal magnetic field. The laser power was 15 µW, beam diameter 2 mm, Ωm = 2π 1 kHz, ∆ω = 2π 220 MHz. Traces ∼ × × (a,c)and(b,d)correspondtothein-phaseandthequadrature outputsof the signals from thelock-in detector, respectively. 5 FIG. 3: Origin of the NMOR resonances at Ωm = 2ΩL,ΩL. (a)AtomicalignmentprecesseswithLarmorfrequency(line). Linearly-polarizedlightperiodicallyinteractswiththeatoms. (b) Optical rotation of an unmodulated probe beam occurs at frequency 2ΩL. FIG. 4: Spectral dependences of various signals. (a) First harmonic amplitude measured at B = 2.2 µG (the narrow zero-field resonance), light power 15 µW; (b) same at B = ∼ 714.4+2.2µG(theΩm =2ΩL resonance),samelightpower, Ωm =2π 1kHz,∆ω=2π 220MHz;(c)nonlinearFaraday × × rotationrecordedwithoutfrequencymodulation(asin[1,3]), B =2.2 µG, same light power; (d) transmission through the vapor cell for low light power (0.4 µW). For all traces, light beam diameter 2 mm. ∼