Quantum limit of optical magnetometry in the presence of ac-Stark shifts M. Fleischhauer1, A.B. Matsko2, and M. O. Scully2,3 1Sektion Physik, Universit¨at Mu¨nchen, D-80333 Mu¨nchen, Germany 2Department of Physics, Texas A&M University, College Station, Texas 77843-4242, 3Max-Plank-Institut fu¨r Quantenoptik, Garching, D-85748, Germany, (February 1, 2008) factor”ofthesystem,isdeterminedbyaneffectivewidth Weanalyzesystematic(classical) andfundamental(quan- of the magnetic resonance. The ultimate goal of magne- tum) limitations of the sensitivity of optical magnetometers tometer design is to minimize the noise level and the resulting from ac-Stark shifts. We show that in contrast to effective width at the same time. absorption-based techniques, the signal reduction associated 0 The width of magnetic resonances in optical magne- 0 withclassicalbroadeningcanbecompensatedinmagnetome- tometers is subject to two types of broadening: reso- 0 ters based on phase measurements using electromagnetically nant power-broadening due to the coupling of the opti- 2 induced transparency (EIT). However due to ac-Stark asso- cal fields to the probe-transition and a broadening due ciated quantum noise the signal-to-noise ratio of EIT-based n to ac-Stark shifts resulting from non-resonant couplings magnetometersattainsamaximumvalueatacertainlaserin- a to other transitions. As shown in [4] and [5] power- J tensity. This value is independent on the quantum statistics ofthelight anddefinesastandard quantumlimit ofsensitiv- broadeninglimitsthesimultaneousminimizationofnoise 0 −1 2 ity. We demonstrate that an EIT-based optical magnetome- and dS/dω in absorption based magnetometers. In ter in Faraday configuration is the best candidate to achieve suchdevicesincreasingtheprobelaserpowerreducesthe (cid:0) (cid:1) 1 the highest sensitivity of magnetic field detection and give a shot-noise but does reduce the signal at the same time. v detailed analysis of such a device. Asaconsequencethesensitivitysaturatesataratherlow 2 power level. On the other hand, as shown in [4] and [5], 7 this effect can be compensated in a magnetometer that 0 detects phase shifts of the probe electromagnetic wave 1 0 propagating in an optically thick atomic medium under 0 conditions of electromagnetically induced transparency I. INTRODUCTION 0 (EIT) [7]. Theoretically a complete elimination is possi- / ble in a 3-level Λ-type system. h The detection of magnetic fields by optical means is a p In any real atomic system, however, there are non- well developed technique with applications ranging from - resonant couplings to additional levels which lead to ac- t geology and medicine [1] to fundamental tests of viola- n Stark shifts and an additional broadening of the mag- tions of parity and time-reversal symmetry [2]. a neticresonanceproportionaltothelaserintensity. Inthe u In spite of their great variety, optical magnetometers present paper we analyze the influence of ac-Stark shifts q can be divided in two basic classes. In the first class and show that they (i) can diminish the magnetometer : light absorption at a magnetic resonance is used to de- v signaland(ii) leadtoadditionalnoisecontributions. We tectZeemanlevelshifts,whilethesecondclassmakesuse i show that in absorption based devices ac-Stark broaden- X of the associated changes of the index of refraction. So ingleads toa furtherreductionofthe signal. Incontrast r called optical pumping magnetometer (OPM) [1] as well a itonlygivesrisetoabiasphaseshiftinanphase-sensitive as dark-state magnetometers based on absorption mea- EIT magnetometer. This bias phase shift can be cali- surements [3] belong to the first class. The recently de- bratedbut is stilla majorsourceof systematic errors. It veloped magnetometers based on phase-coherent atomic canbeeliminated,ifanEITmagnetometerwithFaraday media[4,5]andthemean-fieldlasermagnetometerofref. configuration is considered. [6] belong to the second class. However, in both, absorptive and dispersive type de- If systematic measurement errors can be avoided, vices, ac-Stark shifts give also rise to fundamental noise which in practice can be a challenging task, the smallest contributions which increase with the laser power more detectable Zeeman shift (in units of frequency) is deter- rapidly than shot noise. Hence the magnetometer sen- minedbytheratioofthe noiselevelofthe signalS toits sitivity decreases above a certain power level. The rate of change with respect to frequency maximum value of sensitivity constitutes the standard S quantum limit. For an EIT magnetometer based on noise ∆ω = . (1) min phase-shift measurements this limit is determined by dS/dω the dispersion-absorption ratio of the medium and the (cid:12) (cid:12) intensity-phasenoisecouplingduetotheself-phasemod- A fundamental lower limit(cid:12)of S (cid:12) results from photon (cid:12) nois(cid:12)e ulation associated with ac-Stark shifts. countingerrorsduetoshot-noiseoftheprobeelectromag- Wealsodiscussthepossibilityoffurtherincreasingthe −1 netic wave. dS/dω , which characterizes a “quality (cid:0) (cid:1) 1 sensitivitybymeansofnon-classicallightfieldsandshow α is some numerical pre-factor of order unity that de- that the maximum sensitivity is essentially independent pends on the specific model [5,9]. This broadening effect of the light statistics. leadstoasubstantiallimitationofthesignalinanoptical The paper is organized as follows: In Sec. II we dis- pumping magnetometer, as shown in [5] and [9]. cuss the fundamental broadening mechanisms of mag- netic resonances, power-broadening and ac-Stark associ- atedbroadenings. ItisshowninSec.IIIthattheclassical B. Broadening due to ac-Stark shifts signalreductionduetothesebroadeningscanbecompen- sated in phase-sensitive EIT magnetometers in contrast The second broadening mechanism is due to non- to absorption-based techniques. In Sec. IV fundamen- resonant couplings of the probe electromagnetic wave tal quantum noise sources are discussed and the stan- with other than the probe transition and the associated dardquantumlimitofmagnetometersensitivityderived. ac-Starkshifts. Theac-Starkeffectleadstoashiftofthe A detailed analysis of an EIT-Faraday magnetometer is magnetic resonance of given in Sec. V and the prospects of using non-classical input states are discussed. Ω2 ∆ω = | | (4) ac−Stark ∆ 0 II. BROADENING OF MAGNETIC where∆0issomeeffectivedetuningofnon-resonanttran- RESONANCES sitions from the frequency of the probe field weighted with relative oscillator strengths. Ω is again the Rabi- Opticalmagnetometersmeasureinessencetheposition frequencyoftheprobefieldcorrespondingtotheresonant ofcertainresonanceswhicharesensitivetomagneticlevel probetransition. (∆0isofcoursejustamodel-dependent shifts. Animportantquantitythatdeterminesthesignal coupling parameter. We haveusedthis notationhere for strengthofsucha measurementis the width of the mag- simplicity of the discussions.) netic resonance. As a rule the narrower the resonance, In the classical limit and for a homogeneous laser in- the easier it is to detect level shifts. tensitythroughouttheatomicvapor,thereisonlyacon- Magnetic resonances with small natural width can be stantfrequencyshiftduetotheac-Starkeffect. Thisshift obtained e.g. by coupling Zeeman or hyperfine compo- can be calibrated. However, maximum signal is usually nents of ground states in atoms either with an RF field achieved when the atomic density is chosen such that or via an optical Raman transition. In an optical mag- thereisasubstantialabsorptionoftheprobefield. Hence netometertheseground-statesub-levelsarethencoupled when the probe Rabi-frequency exceeds the value by laser fields to excited atomic states. The optical cou- pling is also used to detect energy shifts of the ground- Ω(c2ri)t ∼ ∆0γ0 (5) state sub-levels induced by a magnetic field. However, p the resonance frequency changes as a function of prop- at the same time this coupling leads to a broadening of agation through the medium. This leads to an effective the magnetic resonances via two mechanisms: (i) power- inhomogeneous broadening of the magnetic resonance. broadening and (ii) broadening due to ac-Stark shifts. For example, the transmissionof a cell containing atoms withaLorentzianmagneticresonancesubjecttoac-Stark shifts is determined by the integrated imaginary part of A. Power-broadening the susceptibility (χ′′ =Im[χ]) The first mechanism is power-broadening due to the L L γ resonant interaction with the probe transition. When dzχ′′(z) dz 0 . (6) ∼ γ2+(∆+ Ω(z)2/∆ )2 the Rabi-frequency Ω of the optical probe field exceeds Z0 Z0 0 | | 0 the value Ω(z)2 characterizesthez-dependentpoweroftheprobe | | Ω(1) √γγ , (2) field and ∆ the detuning from the un-shifted transition crit ∼ 0 frequency. It is easy to see, that there is a broadeningof where γ is the unbroadened width of the magnetic res- the magnetic resonance depending on the magnitude of 0 onance and γ the homogeneous linewidth of the opti- the ac-Starkshifts and the details ofthe absorptionpro- cal transition, the magnetic resonance becomes power- cess. Animportantfeatureisthatthisbroadeningispro- broadened. (Here and below we assume that γ portionaltothesquareoftheRabi-frequencyorthelaser γ .) The effective width scales linearly with the Rab≫i- power. Thus above a certain power level, determined by 0 frequency Ω of the optical field or the square root of the Eq.(5) ac-Stark associated broadening can exceed power corresponding power broadening, which leads e.g. to further reduction of the signal in an optical pumping magnetometer. γ 0 Γ =γ +α Ω + . (3) eff 0 γ | | ··· r 2 III. COMPENSATION OF BROADENING Weconcludethissectionbyemphasizingthatinphase- EFFECTS IN EIT MAGNETOMETER detection schemes based on EIT the detrimental (classi- cal) effects of power-broadening and ac-Stark associated We here demonstrate that the classical broadening broadening are eliminated. In the following section we mechanismsdiscussedintheprevioussectiondonotnec- will discuss the fundamental quantum noise sources of essarily lead to a reduction of the magnetometer signal such magnetometer schemes. if phase measurement techniques are used. It has been shownindetailin[5]and[8],thatpower-broadeningcan be completely compensated in a phase measurement by IV. QUANTUM-NOISE LIMIT OF MAGNETIC making use of EIT in optically dense Λ-type systems. FIELD DETECTION VIA OPTICAL PHASE SHIFTS IN THE PRESENCE OF AC-STARK The3-levelΛconfigurationofanEITmagnetometeras EFFECTS wellastheassociatedlinearsusceptibilityspectrumofthe probe fieldare shownin Fig.1. Here andin the following we consider closed systems i.e. we assume that there The problem of sensitive detection of phase shifts is are no effective decay mechanism due to time-of-flight commonin optics. Onthe quantum level,the sensitivity limitations. The upper level of the probe-field transition ofsuchkindofmeasurementsisrestrictedby(i) vacuum a b is coupled to a meta-stable lower level c by a fluctuations in the system and (ii) self-action noise due | i↔| i | i coherent and strong driving field of Rabi-frequency Ω . to nonlinearitiesin the system,as for example causedby d The probe field Rabi-frequency is denoted as Ω (Ω ac-Starkshifts. Thesimultaneouspresenceofbothnoises p p ≪ Ω )andthe coherencedecay rateofthe probe transition usually leads to an absolute limit of the sensitivity. d as γ. ∆ is the one-photon detuning of the drive field Let us discuss this problem for the particular case and δ the two-photon detuning. The transverse decay of optical magnetometry based on phase-shift measure- rate of the two-photon resonance (magnetic resonance) ments in an atomic medium. The ultimate limit for the is denoted as γ . It is assumed that the corresponding smallest detectable phase shift is set by the generalized 0 populationexchangebetweentheground-statesub-levels uncertainty relation [10] between phase- ∆φ φ φ ≡ −h i is small and will be neglected [11]andphoton-numberfluctuations∆n n n ofthe ≡ −h i As in the case of an OPM there is power-broadening output field. uinniaqnueEpITropmeartgyneotfomaneteErITasrseosoonnaansce|Ωisd|h>ow√evγeγr0.thaAt ∆φ2 ∆n2 1+ 1 ∆φ,∆n 2, (10) the dispersion-absorption ratio of the optical transition h ih i≥ 4 { } is given by the inverse of the width of the ground-state (cid:10) (cid:11) where , denotes the anti-commutator. If phase- and transition γ and is independent on the drive power if { } 0 photon-numberfluctuationsareuncorrelated,thesecond |Ωd|>√γγ0. Under conditions of one-photon resonance term on the r.h.s. vanishes and one recoversthe familiar (∆=0) one finds for small two-photon detuning Heisenberg relation. In any real magnetometer schemes δ phase and intensity fluctuations are however correlated χ′ Re[χ] , (7) due to e.g. ac-Stark shifts (self phase modulation), and ≡ ∼− Ω 2+γγ | d| 0 thus the second term in Eq.(10) is in general nonzero. γ χ′′ Im[χ] 0 . (8) When the intensity-phase coupling is small, it can be ≡ ∼ Ω 2+γγ | d| 0 characterized by a linear coupling coefficient β in the The residual absorption at the EIT resonance decreases form∆φ=∆φ0+β∆n,where∆φ0 denotesphasefluctu- with increasing laser power in the same way as the dis- ations not correlated to intensity fluctuations. Thus we persion. Thusinaphaseshiftmeasurementpowerbroad- find ening can be compensated by increasing the density and 1 keeping a constant optical depth of the medium. ∆φ2 +β2 ∆n2 . (11) h i≥ ∆n2 h i Similarlyonefindsthataslongasthedrive-fieldRabi- h i frequency is large compared to probe-induced ac-Stark The signal phase accumulated during the propagation shifts, which is very well satisfied, ac-Stark shifts of the through an atomic vapor cell is proportional to the Zee- magnetic resonance (eq.(4)) lead only to a bias phase man splitting ∆ω , the length of the cell L, and the B shift. dispersion of the real part of the susceptibility at the L Ω(z)2 laser frequency dχ′/dω. The cell length is restricted by ∆φac−Stark dz | | , (9) the absorption at the laser frequency, and a reasonable ∼ ∆ Z0 0 upper limit for L is the (amplitude) absorption length where L is the length of the atomic vapor cell. This L =(πχ′′/λ)−1. abs phase shift can in principle be calibrated but gives rise Thus the maximum phase shift is to systematic errors. As will be discussed in detail later on,thereisnosuchbiasphaseshiftinaresonantFaraday 1 dχ′ ∆φ = ∆ω . (12) configuration. |max χ′′ dω B 3 One recognizes, that the sensitivity of phase measure- presence of the intensity-dependent dark resonance gen- ments to Zeeman shifts is determined by the dispersion- eratedbytheactionofthestronglaserfieldasopposedto absorption ratio 1/χ′′ dχ′/dω. ausualabsorptionresonanceintheweak-fieldlimit. The The limit for the smallest detectable Zeeman shift is rotationoftheplaneofpolarizationattheoutputcanbe (cid:0) (cid:1) therefore given by measuredbydetectingthe intensitydifferenceoftwolin- ear polarized components 45o rotated with respect to 1 dχ′ −1 1/2 the input polarization. ± ∆ω = ∆n2 −1+β2 ∆n2 . (13) B min χ′′ dω h i h i An aspect of the system, which becomes particularly (cid:12) (cid:20) (cid:21) h i important when strong fields are considered, are non- Under(cid:12)the condition, that the dispersion-absorption ra- resonant couplings of the two circular components to tio is independent on the laser power, the r.h.s. of this other levels, which to lowest order give rise to ac-Stark expression is minimized when h∆n2i|opt = β−1. There- shifts of the states |b±i. In a Faraday configuration the fore there is an absolute lower limit or “quantum limit” ac-Stark shifts of b and b are exactly equal and + − | i | i of magnetic fielddetection via phase-shift measurements opposite in sign due to symmetry and thus there is no independent on the photon-number fluctuations average effect on the signal and no bias phase shift or rotation. Thus the Faraday magnetometer is not sub- 1 dχ′ −1 ject to systematic errors associated with ac-Stark shifts. ∆ω = 2β. (14) B min χ′′ dω However, as mentioned before, ac-Stark shifts cause a (cid:20) (cid:21) (cid:12) p coupling between intensity and phase fluctuations which The absorption-(cid:12)dispersionratio of a magnetic resonance need to be taken into account. isusuallygivenbyitsnaturalwidth,whichcanberather small if a two-photon Raman process between Zeeman- or hyperfine components is used as in an EIT magne- A. Detection scheme tometer. We will show later on that different measurement WehereconsiderthedetectionschemeshowninFig.3. strategies as well as the use of non-classical light fields A strong linear polarized field initially polarized in x di- do in general not improve this result. rection propagates through a cell of length L with the magneto-optic medium. Due to the nonlinear Faraday effect the plane of polarization is rotated by an angle V. EIT-BASED FARADAY MAGNETOMETER φ/2. In order to detect this angle the intensity difference of Let us now discuss in detail an EIT magnetometer in thetwoorthogonaloutputdirections1and2ismeasured. resonant nonlinear Faraday configuration. For this we The operator for the number of counts is given by consider the propagation of a strong, linear polarized light field through an optically dense medium, consist- nˆ =C dt Eˆ−(t)Eˆ+(t) Eˆ−(t)Eˆ+(t) . (15) ing of resonant Λ-type systems (atoms, quantum wells 2 2 − 1 1 etc.) as shown in Fig. 2. For simplicity we ignore opti- Ztm (cid:16) (cid:17) cal pumping into lower states other than those shown in where Eˆ± denote the positive and negative frequency the figure and assume a closed system. For a resonant part of the output electric field operators, t is m J = 1 J = 0 transition (say), optical pumping into the measurement time, and C = 2ǫ cA/¯hν , A be- → 0 0 the lower mJ = 0 state depletes both states mJ = ±1 ing the beam cross-section and ν0 the resonance fre- in the same way and thus effectively diminishes the op- quency. Making use of the field commutation relations tical density but does not affect the signal. Symmetric [Eˆ+ (L,t),Eˆ− (L,t′)] =C−1δ(t t′) and [Eˆ±,Eˆ±]= 0, re-pumping can be used to maintain the population in we1,c2anexpre1s,s2the meannumber−ofcountsas1wel2lasthe the relevant sub-system without affecting the detection fluctuations in terms of normal-orderedcorrelationfunc- scheme. We include a dephasing of the ground-state co- tions. The latter allows to apply a c-number approach herence with rate γ0 and a population exchange rate be- where the operators Eˆ are approximated by stochastic tween the ground states γ0r. complex functions E. The two circularcomponents E andE of the linear − + polarized light generate a coherent superposition (dark nˆ = n n , (16) 2 1 h i h i−h i state) of the states b± J = 1,mJ = 1 . A mag- ∆nˆ2 = ∆n2 + n + n . (17) | i ≡ | ± i 1 2 netic field parallel to the propagation axis leads to an h i h i h i h i anti-symmetric level shift of |b±i and thus by virtue of where n1,2 follows form Eq.(15) by replacing the field the largelineardispersionatanEIT-resonanceto anop- operators by c-numbers posite change in the index of refraction for both compo- nents. As a result the polarization direction is rotated, n =C dtE− (L,t)E+ (L,t). (18) which is the so-called resonant nonlinear Faraday effect 1,2 1,2 1,2 Ztm [12]. The difference to the linear Faraday effect is the 4 In the usual configuration only the x-polarized com- obtained from the stationary solution of the c-number ponent of the input field is excited and we will restrict Bloch equations for the atomic populations the discussion to a vacuum input of the y-polarized component. The propagation of the field through the σ˙b−b− = γ0r(σb−b− σb+b+)+γrσaa − − magneto-optical medium is most conveniently described i(Ω σ c.c.), − ab− − − in terms of right and left circular components E = ± σ˙ =γ (σ σ )+γ σ (1/√2) E iE , and we therefore have b+b+ 0r b−b−− b+b+ r aa x y ± i(Ω σ c.c.), + ab+ − − (cid:0) (cid:1) n= iC dt E−E+ E−E+ . (19) and polarizations − − + − + − Ztm (cid:16) (cid:17) σ˙ = Γ σ iΩ∗(σ σ ) Thepropagationofthecircularcomponentscanbechar- ab± − ab± ab±− ± b±b±− aa iΩ∗σ , (25) acterized by two parameters, the intensity transmission − ∓ b∓b± coefficientηandthephaseshiftφ±(L,t)oftherespective σ˙b−b+ =−Γb−b+σb−b+−iΩ−σab+ +iΩ∗+σb−a, (26) component at the output. where E+(L,t)=E+(0,t)√ηeiφ±(L,t). (20) ± ± γ δ 0r 0 Γ γ+ +i ∆+δ , (27) ab± ± In the limit of small magnetic fields the absorption of ≡ 2 ± 2 (cid:18) (cid:19) both circular components is identical for symmetry rea- Γ γ +γ +i(δ +δ δ ). (28) b−b+ 0 0r 0 + − ≡ − sons i.e. there is no dichroism. With this we obtain for cw-input fields γr istheradiativelinewidthofthetransitions a b± , | i→| i andγ isthehomogeneoustransverselinewidthoftheop- nˆ =η n sinφ η n φ , (21) tical transitions a b . δ is the Zeeman splitting h i h xiin sig ≈ h xiin sig | i → | ±i 0 and δ are the ac-Stark shifts of levels b . Ω are ± ± ± where φsig = φ+(L) φ−(L) is the (stationary) signal the complex Rabi-frequencies of the two|optiical fields, − phaseshift. Similarlywecanestimatethefluctuationsin Ω = ℘ E−/¯h. We have disregarded Langevin noise ± ± ± lowest order of the small rotation angle φ in the case of forces in Eqs.(25-26) associated with spontaneous emis- an initially coherent field sion and collisional decay processes, since it was shown in [5] that atomic noises have a negligible effect on the ∆nˆ2 =η n +η2 n 2 δφ2 . (22) h i h xiin h xiinh i magnetometer sensitivity. We calculate the stationary solutions of the Bloch- Thefirsttermcorrespondstothevacuumnoiseleveland equationsbyconsideringonlythelowestorderinγ ,γ , the second term proportionalto 0 0r δ and δ . In this limit we find 0 ± 1 δφ2 = dtdt′ δφ(t),δφ(t′) (23) iΩ (γ Ω 2+γ Ω 2) h i t2 h i σ = ± 0| ∓| 0r| ±| m ZZ ab± Ω2(2γ(2γ +γ )+ Ω2) 0r 0 | | | | describes fluctuations due to an intensity-phase noise δ 2Ω Ω 2 0 ± ∓ couInpltinhge fionlltohweinmgedwieumca.lc(uhlaa,tbeit≡heh[laos−s fhaacit][obr−η,hbtih]ei)sig- −(cid:16)δ±± 2 (cid:17)|Ω|2(2γ(2γ0r|+γ|0)+|Ω|2) (29) nalphase shiftφsig andthe fluctuations hδφ2i due to the +∆Ω± γ0r(|Ω+|4+|Ω−|4)+2γ0|Ω∓|2|Ω|2 intensity-phase noise coupling for the EIT-Faradaymag- γ Ω4(2γ(2γ +γ )+ Ω2) (cid:2) | | 0r 0 | | (cid:3) netometer. where Ω(z)2 = Ω (z)2 + Ω (z)2. Usually the co- − + | | | | | | herence decay between the ground levels dominates the B. Medium susceptibility and field propagation population exchange and thus γ0 γ0r. ≫ It is convenient to separately consider the spatial evo- lution of amplitudes and phases of the complex Rabi- The stationary propagation of the right and left cir- frequencies Ω (z) = Ω (z)eiφ±(z). The intensities of cular polarized electric field components through the ± | ± | the two fields are attenuated in the same way atomic vapor is described by Maxwell equations in slowly-varying amplitude and phase approximation d γ γ Ω 2 Ω 2 Ω 2 = κ 0 r | +| | −| , (30) d E+(z)= iν0 ℘ Nσ (z). (24) dz| ±| − |Ω|2 (2γ0γ+|Ω|2) dz ± 2cǫ ± b±a 0 where κ=(3/4π)Nλ2. N is the atomic number density, ℘ are the dipole mo- Eq. (30)canbe easilysolvedwhenthe lengthL ofthe ± ments of the respective transitions, and σb±a are the cell is small enough, such that |Ω(L)|2 ≫ 2γγ0. In the c-number analogues of the atomic lowering operators Faraday set-up discussed here Ω (0) = Ω(0)/√2, and ± σˆ = b a. Analytic expressions for σ can be therefore Ω (z)2 = Ω(z)2/2. We thus arrive at b±a ± b±a ± | ih | | | | | 5 γ γ κz |Ω(z)|2 =|Ω(0)|2 1− 2Ω0(r0)2 ∆ofatbh+e=aωab+b−ν0ananddc∆j =bωcjb+tr−anνs0itaiornest.he detunings (cid:16) | | (cid:17) | i−| +i | ji−| +i = Ω(0)2 1 α z . (31) | | − 0 H = ℘ a b Eˆ+ ℘ a b Eˆ+ (cid:16) (cid:17) S − −| ih −| − − +| ih +| + It is interesting to note that under conditions ofEIT the ℘ c b Eˆ++℘ c b Eˆ+ +adj. (39) − j+| jih +| + j−| jih −| − residual absorption is not exponential but linear. The Xj (cid:16) (cid:17) intensity transmission coefficient is then given by describestheresonantandnon-resonantcouplingsofthe quantizedfieldstotheatom. Thenon-resonantcouplings η = 1 α L . (32) 0 − to the excited states c cause ac-Stark shifts. We here j (cid:16) (cid:17) haveassumedthatbo|thifields arenearlymonochromatic The approximation Ω(L)2 2γγ sets an upper limit for the losses, such t|hat 1| ≫η 2γ0γ /Ω(0)2. and have set the energy of level |b+i equal to zero. ℘j± ≥ ≫ 0 | | are the dipole moments of the transitions cj b± . Similarly we find the phase equations | i→| i We proceed by formally eliminating the excited states c by means of a canonical transformation in second d κγr∆γ0+γ(δ0/2 δ−) | ji dzφ− = 2γ 2γ γ+ Ω−2 , (33) order perturbation 0 | | d κγ ∆γ γ(δ /2+δ ) H˜ =exp(S)H exp( S) H +[S,H]+[S,[S,H]], (40) φ = r 0− 0 + . (34) − ≃ dz + 2γ 2γ γ+ Ω2 0 | | where S obeys the equation The contributions from the one-photon detuning ∆ can- [S,H ]= ℘ c b Eˆ++℘ c b Eˆ++adj. cel when the relative phase φ=φ+ φ− is considered 0 j+| jih +| + j−| jih −| − − Xj (cid:16) (cid:17) d κγ δ δ δ φ= r 0 + +− − . (35) (41) dz − 2 Ω2 Ω2 (cid:16)| | | | (cid:17) Under conditions of exact two-photon resonance for the The first term describes the signal-phase shift due to a fields we obtain the transformation operator magnetic field and the second term the ac-Stark contri- bution. Integration of Eq.(35) yields for the signal S = ℘j+ c b Eˆ++ ℘j− c b Eˆ+ adj. . ∆ | jih +| + ∆ | jih −| − − δ Ω(0) 2 Xj (cid:18) j j (cid:19) 0 φ = ln (36) sig −γ Ω(L) (42) 0 (cid:12) (cid:12) (cid:12) (cid:12) and the ac-Stark contribution(cid:12) (cid:12) Assuming that the population of all excited levels is (cid:12) (cid:12) small, we eventually find for the transformed Hamilto- κγ L δ (z,t) δ (z,t) nian r + − δφ(t)= dz − . (37) − 2 Ω(z)2 Z0 | | H˜ ≃H0−℘+|aihb+|Eˆ++−℘−|aihb−|Eˆ−+ ℘2 ℘2 j+ b b Eˆ−Eˆ++ j− b b Eˆ−Eˆ+ C. Ac-Stark shifts and associated noise − j ¯h∆j| +ih +| + + ¯h∆j| −ih −| − −! X ℘ ℘ j+ j− b b Eˆ−Eˆ++ b b Eˆ−Eˆ+ . (43) Let us now discuss the average ac-Stark shift and the − ¯h∆j | +ih −| + − | −ih +| − + corresponding noise contributions. For this we first con- Xj (cid:16) (cid:17) sider the effect of an off-resonant quantized field on the Let us assume now, that ∆ is much larger than the j energy of a single atom in lowest non vanishing order of natural width of the excited states, and therefore the perturbation. Wethengeneralizetheresultsfortheaver- population transfer due to the non-resonant coupling is age ac-Stark shift and its fluctuations to an ensemble of negligible. We identify ℘2 /∆ ℘2/∆ , where ∆ j j± j → 0 0 atoms by making the physically reasonable assumption is someeffective detuning. The dipole moments℘ and j+ P that ac-Stark shifts of different atoms are uncorrelated. ℘j− have usually alternating signs for different excited We decompose the Hamiltonian of the single atom in- states c . We therefore set ℘ ℘ /∆ = 0. Then | i j j+ j− j teracting with the quantized field in a rotating frame in the ac-Stark shift of the single atom can be represented P the form H = H +H , where H is the unperturbed by the operator expression 0 S 0 part ℘2 δˆl (t)= Eˆ−(z ,t)Eˆ+(z ,t), (44) H0 =H0field+h¯ωb−b+|b−ihb−| ± ¯h2∆0 ± l ± l +h¯∆ab+|aiha|+h¯ ∆j|cjihcj|. (38) where l specifies the atom and zl its location. Thus we j find for the average ac-Stark shift X 6 ℘2 Ω(z ,t)2 L℘4 δˆl (t) = Eˆ−(z ,t)Eˆ+(z ,t) = | l | , (45) δˆ (z,t),δˆ (z′,t′) = δ(z z′) h ± i ¯h2∆0h ± l ± l i 2∆0 h ± ± i ¯h4∆20 − where ℘2 Eˆ±(zl,t) 2/¯h2 = ℘2 E(zl,t) 2/2h¯2 = × hE±−(z,t)E±+(z,t),E±−(z,t′)E±+(z,t′)i Ω(zl,t)2/|2h. Simiil|arly we obtai|nh for tih|e second- h δ(t t′) o|rder m| oments of the ac-Stark shifts hxˆ,yˆi ≡ + C− hE±−(z,t)E±+(z,t)i , (53) (xˆ xˆ )(yˆ yˆ ) i h −h i −h i i and δˆl (t)δˆl (t′) h ± ± i L℘4 = ℘4 Eˆ−(z ,t)Eˆ+(z ,t)Eˆ−(z ,t′)Eˆ+(z ,t′) , (46) hδˆ+(z,t),δˆ−(z′,t′)i= ¯h4∆2δ(z−z′) ¯h4∆2h ± l ± l ± l ± l i 0 0 E−(z,t)E+(z,t),E−(z,t′)E+(z,t′) . (54) δˆl (t)δˆl (t′) × h ± ± ± ± i h + − i h i = ℘4 Eˆ−(z ,t)Eˆ+(z ,t)Eˆ−(z ,t′)Eˆ+(z ,t′) , (47) We here have used that in continuum approximation for ¯h4∆2h + l + l − l − l i any smooth function f(z) holds 0 or after normal ordering L δ(z z )δ(z′ z )f(z )=δ(z z′)f(z). (55) l l l − − − δˆl (t)δˆl (t′) = Xl h ± ± i ℘4 It is now straight forward to evaluate the quadratic E−(z ,t)E+(z ,t)E−(z ,t′)E+(z ,t′) deviation of the relative ac-Stark shift ¯h4∆2 h ± l ± l ± l ± l i 0 h δ(t t′) δˆ (z,t) δˆ (z,t) δˆ (z′,t′) δˆ (z′,t′) + − E−(z ,t)E+(z ,t) , (48) + − − , + − − = C h ± l ± l i 2Ω(z)2 2Ω(z′)2 * ! !+ hδˆ+l (t)δˆ−l (t′)i= i δ(z |z′)δ(|t t′) ℘2L | . | (56) ¯h4℘∆42 hE+−(zl,t)E++(zl,t)E−−(zl,t′)E−+(zl,t′)i . (49) − − 2h¯2C∆20|Ω(z)|2 0 h i We note that the classical excess noise contributions ex- ThefirsttermsinEqs.(48)and(49)correspondtoclassi- actlycancelandonlythevacuumcontributionisleftover. calfluctuations, while the secondtermin(48)is vacuum Due to the intrinsic balancing in the EIT-Faraday mag- orshotnoise. If the appliedfields areina coherentstate netometer excess noise contributions are automatically only the shot noise term survives. In any practical re- canceled. This is an important advantage of the Fara- alizations there are however large excess noise contribu- day configuration as compared to the asymmetric EIT- tions and the first terms are usually the dominant ones. magnetometer discussed in [4] and [5]. We will show that all excess noise contributions are can- Using Eqs.(23), (37) and (56) we eventually find for celed in a Faraday magnetometer and only the vacuum the phase fluctuations due to ac-Stark shifts contribution survives. We generalize the above single-atom results to an en- 1 κ2γ2℘2L L 1 δφ2 = r dz (57) sembleofatomsassumingindependentfluctuationsofthe h i tm 4∆20 ¯h2C Z0 |Ω(z)|2 ac-Stark shifts of different atoms, i.e. δˆjδˆk δ , (50) h µ νi∼ jk D. Signal-to-noise ratio and minimum detectable where µ,ν +, . We introduce the continuous Zeeman shift { } ∈ { −} variable The minimum detectable Zeeman shift is obtained by δˆ (z,t)=L δ(z z )δˆj (t). (51) ± − j ± setting the mean number of counts j X δ nˆ =η n φ = η n 0ln η−1 (58) In a continuum approximation, (1/L) dz, and x in sig x in j → L h i h i − h i γ0 we have (cid:2) (cid:3) P R equal to the quantum mechanical uncertainty ℘2 Ω(z,t)2 δˆ (z,t) = Eˆ−(z,t)Eˆ+(z,t) = | | . (52) h ± i ¯h2∆0h ± ± i 2∆0 ∆nˆ2 1/2 = η n +η2 n 2 δφ2 1/2 h i h xiin h xiinh i Similarly h Ω(0)4 i 1/2 = η n 1+ | | η (1 η) ln(η−1) (59) h xiin ∆2γ2 − (cid:20) 0 0 (cid:21) p 7 This yields the signal-to-noise ratio Inthecaseofanopticalmagnetometer,ac-Starkshifts appear due to non-resonant nonlinearities and it would δ2 1/2 seem that these shifts can in principle be compensated 0 n η ln2(η−1) γ2h xiin byanadaptedmeasurementstrategyandtheuseofnon- SNR= 0  , (60) classicallight. Anessentialconditionforsuchmethodsis Ω(0)4 1+ | | η (1 η) ln(η−1) howeverthatthesystemisnearlylosslessinordertopre-  ∆2γ2 −   0 0  serve the non-classicalstate of light. On the other hand,   as discussed above, the maximum signal in an optical whichismaximizedforanoptimalpowerofthefieldcor- magnetometer is achieved under conditions of substan- responding to tial absorption. (We note that the SNR is proportional to ln(η)2.) We will show in the following with simple es- ∆2γ2 Ω(0)2 = 0 0 ∆ γ . (61) timates that this feature makes it impossible to increase | |opt sη (1 η) ln(η−1) ∼ 0 0 the sensitivity by using non-classicallight. − Let us consider the simplest example of compensation Substituting the optimum Rabi-frequency (61) into (60) of ac-Stark associated noise by non-classical light. We yields a maximum SNR for η 0.06. Thus we find the assume, that the slowly varying field operators in the ≈ quantum limit for the detection of Zeeman level shifts Heisenberg picture are represented in the form Eˆ = ± Eˆ +eˆ , where eˆ is the fluctuation part. To discuss δSQL =γ f γr 3 λ2 1 1/2, (62) thheicom±pensation o±f ac-Stark effects let us disregardthe 0 0 (cid:18)∆08π A γ0tm(cid:19) resonant coupling with the medium and the associated absorption. Then we find that the field fluctuations at where the end of the vapor cell can be written as 1/4 1 η κγ L f − (63) eˆ (t,L)=eˆ (t,0)+i r eˆ (t,0)+eˆ+(t,0) , (64) ≡ η ln3(η−1) − − 2∆ − − (cid:18) (cid:19) 0 h i κγ L is a numerical factor which varies between 1 and 2 for eˆ+(t,L)=eˆ+(t,0)−i2∆r eˆ+(t,0)+eˆ++(t,0) . (65) η = 0.01 0.8. (Note that η is the transmission coeffi- 0 h i ··· cientunderconditionsofEIT.WithoutEITthe medium The second terms in these equations are due to ac-Stark would be totally opaque.) In Fig. 4 we have shown the shifts. One can see that the uncertainty of the phase minimum detectable Zeeman splitting (proportional to difference increases as a result of ac-Stark shifts, which the magnetic field) as function of the laser input power leads to the sensitivity restriction, discussed above. for different transmission coefficients. Let us assume now, that the incident field is squeezed One clearly sees that for small laser powers shot-noise insuchaway,thatthe operatorsofthe fieldfluctuations isdominant,whileforlargerlaserpowersac-Starkassoci- at the input obey the relations atedfluctuations takeover. Alsoshownisthe saturation behavior of an OPM [5]. Due to power broadening the eˆ (t,0)=e˜ iκγrL(e˜ +e˜+), (66) sensitivity of an OPM saturates as soon as the Rabi- − −− 2∆ − − 0 mfreaqguneentocymeretearc,heosntthheevoatluheer√hγaγn0d., ItnhethoeptEimITu-mFarRaadbaiy- eˆ+(t,0)=e˜++iκ2γ∆rL(e˜++e˜++). (67) 0 frequency corresponding to the quantum limit is of the order of √∆0γ0. Since ∆0 ≫γ much higher sensitivities Here e˜± are free-field operators(the correspondingstate can be achieved here. is the field vacuum), which obey the commutation rela- tions[e˜+(t),e˜−(t′)]=C−1δ(t t′)and[e˜±(t),e˜±(t′)]=0. ± ± − ± ± Then, in the absence of losses, the effects of ac-Stark E. Compensation of ac-Stark associated noise by use shifts are completely compensatedinthe output andthe of non-classical input fields output fields are coherent. It is well known, that the effect of self-phase modula- eˆ−(t,L)=e˜−, (68) tiondue torefractivenonlinearitiescanbe compensated, eˆ (t,L)=e˜ . (69) + + at least in part, by means of an optimum detection pro- cedure(for example,by measuringnotthe phase,but an The sensitivity of the phase measurement would thus be appropriately chosen quadrature amplitude of the probe determined by shot-noise only, ∆φ=1/ n . h i electromagneticwave)and/orbyusingnon-classicallight In the absence of losses, the sensitivity of the detec- p [13,14]. Thepropertiesoftheinputquantumstateinthe tion can even be better than the shot-noise limit, if the methods utilizing non-classical light are thereby chosen initialstateofthefieldisappropriatelychosen[14]. Mak- such that after the interaction the probe wave is in the ing use of a SU(2) Lie-group description, Yurke showed coherent or phase-squeezed state. that the sensitivity of a phase shift measurement in a 8 Mach-Zehnderinterferometer canapproachthe so-called certain optimum intensity the fundamental signal-to- Heisenberglimit∆φ 1/ n ,where n isthetotalnum- noise ratio attains a maximum value which represents ≃ h i h i ber of registered quanta [15,16]. the standard quantum limit of optical magnetometer However, in the presence of losses resulting from the based on phase-shift measurements. This quantum limit resonant coupling the noise compensation by means of is determined by the dispersion-absorption ratio of the non-classicallight is only partial due to unwantednoises atomic medium and the strength of the intensity-phase added by the medium. Taking into account linear losses noise coupling. The unique property of EIT is to pro- and assuming, that the entrance field is squeezed in the vide a dispersion-absorption ratio which is independent waydiscussedabove,we canrewritethe equationforthe of power-broadening and is given by the lifetime of a residual noises in the phase as follows: ground-state coherence. The minimum magnetic level shift corresponding to the quantum limit of EIT magne- κγr L δ+(z,t) δ−(z,t) tometers can thus be orders of magnitude smaller than δφ(t)= dz − 1 η(z). (70) − 2 Ω(z) 2 − that of optical pumping devices. Z0 |h i| p We have shown that the best candidate to reach the η(z) = 1 α0z is the z-dependent transmission coeffi- standard quantum limit is a magnetometer in Faraday − cient. The expression indicates, that for small losses in configuration, which has been analyzed in detail. In an the medium, the noise can be almost completely sup- EIT-Faraday magnetometer the signal reduction due to pressed. A maximum signal is achieved however when power-andac-Starkbroadeningsiscompensatedbylarge η 1 and thus the use of non-classical light only leads densities of the atomic vapor. The influence of classical ≪ to a marginalreduction of the ac-Stark associatednoise. excess noise is completely eliminated due to symmetry Thisisincontrasttothemeasurementschemesdiscussed andtherearemuchlesssourcesforsystematicerrors. We in [13,14] which utilize squeezing to improve sensitivity. havealsoshownthattheuseofnon-classicallightanddif- The change of the expressionfor the ac-Stark associated ferentdetectiontechniquesonlymarginallyimprovesthe noiseleadsto achangeofthe sensitivityfactorf accord- attainable sensitivity since a maximum signal is associ- ing to ated with substantial losses in the atomic medium. (1 η)(ln(η−1)+η 1) 1/4 f f˜= − − . (71) −→ η ln4(η−1) ACKNOWLEDGEMENTS (cid:18) (cid:19) It is easy to see, that f˜ f for all relevant values of η, The authors would like to thank M. Lukin for stim- ≃ which means that squeezing does not improve the sensi- ulating discussions on the role of ac-Stark shifts. A.M. tivity of the detection. andM.O.S.gratefullyacknowledgefurther usefuldiscus- The same conclusioncanbe drawnfor any kind of op- sions with Y. Rostovtsev and the support from the Of- timal strategy of measurement to compensate ac Stark fice of NavalResearch,the NationalScience Foundation, shifts. The main reason for this is that both, the mag- theWelchFoundation,theTexasAdvancedResearchand nitude of the signal and absorption losses increase with Technology Program and the Air Force Research Labo- the density-length product of the atomic vapor cell. ratories. M.F. gratefully acknowledgesthe financialsup- portoftheAlexander-von-Humboldtfoundationthrough the Feodor-Lynen Program. VI. SUMMARY We have discussed the influence of ac-Stark shifts on thesensitivityofopticalmagnetometers. Wehaveshown that these shifts cause a broadening of the relevantreso- nancesandgiverisetoadditionalnoisecontributions. In [1] for a review on optical pumping magnetometers see: E. absorption-type magnetometers, such as OPMs, the ac- B.AlexandrovandV.A.Bonch-Bruevich,Opt.Eng.31, Starkassociatedbroadeningaswellaspower-broadening 711(1992);E.B.Alexandrovetal.,LaserPhysics6,244 leadtoareductionofthesignal. Wehaveshownthatthe (1996). classical part of these effects can be completely compen- [2] E.A.Hinds,AtomicPhysicsVol.11(1988),S.Haroche,J. sated in an EIT magnetometer in Faraday configuration C. Gray and G. Gryndberg, Eds.; L. R. Hunter, Science where polarization rotation or, equivalently, the relative 252, 73 (1991); D. Budker, V. Yashchuk, and M. Zola- phase shift of two circular components is measured. torev,Phys.Rev.Lett.81,5788(1998);V.Yashuketal. In a magnetometer based on phase measurements ac- preprint LBNL-42228 (1998). Stark shifts lead also to a coupling between intensity [3] A. Nagel et al., Europhys.Lett. 44, 31 (1998). and phase fluctuations. As a result there are addi- [4] M. O. Scully and M. Fleischhauer, Phys. Rev. Lett. 69, tional, ac-Stark associated fluctuations which dominate 1360 (1992). over shot noise beyond a critical laser power. For a 9 [5] M.FleischhauerandM.O.Scully,Phys.Rev.A49,1973 a ∆ (1994). [6] F. Bretenaker, B. L´epine, J. C. Cotteverte, and A. Le drive probe Floch, Phys. Rev.Lett. 69, 909 (1992). Ω Ω d p [7] fora reviewon EITsee: S.E. Harris, PhysicsToday 50, δ c 7, 36 (1997). b [8] M.D.Lukin,M.Fleischhauer,A.S.Zibrov,H.G.Robin- son, V. L. Velichansky, L. Hollberg, and M. O. Scully, probe-field susceptibility Phys.Rev.Lett. 79, 2959 (1997). 1 [9] M. Fleischhauer and M. O. Scully, Quantum Semiclass. 0.8 Opt.7, 297 (1995). 0.6 χ [10] H. P. Robertson, Phys. Rev. A 35, 667 (1930); E. 0.4 Schr¨odinger, “Zum Heisenbergschen Unsch¨arfeprinzip”, 0.2 0 Ber. Kgl. Akad.Wiss., Berlin, p.296, 1930; [11] Although there is no rigorous definition of a hermitian -0.2 χ -0.4 phase operator, we nevertheless use it here noting that -3 -2 -1 0 1 2 3 werestrictourselvestostatesoftheradiationfieldwhich δ / γ havenosignificantoverlapwiththevacuum.Inthiscase phaseandphotonnumberoperatorsnearlycoincidewith FIG. 1. Principle of a drive-probe EIT magnetome- thephase and amplitude quadratureoperators. ter. Strong drive field in 3-level Λ system (top) leads to [12] W. Gawlik, J. Kowalski, R.Neumann, and F. Tr¨ager, transparency of probe field and linear dispersion around Phys. Lett. A48, 283 (1974); W. Gawlik, in Modern two-photon resonance δ = 0 (bottom). Lower plot shows χ′ Nonlinear Optics, part 3, M. Evans and S. Kielich, eds., andχ′′ (realandimaginarypartofprobe-fieldsusceptibility) Wiley (1994); K. H. Drake, W. Lange, and J. Mlynek, in arbitrary unitscharacterizing refractive indexand absorp- Opt. Comm. 66, 315 (1988); S. Giraud-Cotton et al. tion.Drive-fieldRabi-frequencyequalsnaturalwidthofprobe Phys.Rev.A 32, 2211 (1985), ibid 2223 (1985); L. M. transition. Barkov et al. Opt.Comm. 70, 467 (1989); F. Schuller et al. ibid71, 61 (1989). [13] C. M. Caves, Phys.Rev. D 23, 1693 (1981). c [14] W. G. Unruh, in Quantum Optics, Experimental Grav- i itation, and Measurement Theory, eds. P. Meystre and ∆ i M. O. Scully, (Plenum, 1982), p. 647; M. T. Jaekel and S. Reynaud, Europhys. Lett. 13, 301 (1990); D. a V. Kupriyanov and I. M. Sokolov, Quantum Opt. 4, 55 ∆ (1992); A.F. Pace, M. J. Collett and D.F. Walls, Phys. Rev. A 47, 3173 (1993); S. P. Vyatchanin and A. B. γ Matsko,JETP77,218(1993);G. J.Milburn,K. Jacobs, r E γ - E r and D. F. Walls, Phys.Rev.A 50, 5256 (1994). + [15] B. Yurke,S. L. McCall, J. R. Clauder, Phys.Rev.A 33, 4033 (1986). b- δ γ 0 [16] C. Brif and A. Mann, Phys. Rev. A 54, 4505 (1996); 0 b + T. Kim, O. Phister, M. J. Holland, J. Noh, and J. L. Hall, Phys. Rev. A 57, 4004 (1998) and references FIG.2. Λ-systeminFaradayconfiguration,γrareradiative therein. (longitudinal) decay rates, γ0 therate of ground-state coher- encedecay(transversaldecay);∆denotesone-photon-andδ0 magnetic-fieldinducedtwo-photondetuning;E±describeleft- and right-circularpolarized fieldcomponents. Population ex- change (longitudinal decay) between ground-state sub-levels is disregarded. Also shown are non-resonant couplings to ex- cited states |cii causing ac-Stark shifts. φ/2 x B 1 2 45ο y L 10