Nonlinear magneto-optical rotation in optically thick media

Nonlinear magneto-optical rotation in optically thick mediaS. M. Rochester 1 1, 2, ∗and D. Budker1 Department of Physics, University of California at Berkeley, Berkeley, California 94720-73002 Nuclear Science Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720(Dated: February 28, 2002)Nonlinear magneto-optical rotation is a sensitive technique for measuring magnetic fields. Here,the shot-noise-limited magnetometric sensitivity is analyzed for the case of optically-thick media andhigh light power, which has been the subject of recent experimental and theoretical investigations.PACS numbers: 33.55.Ad,07.55.GearXiv:physics/0202071 v1 28 Feb 2002I. INTRODUCTIONResonant nonlinear magneto-optical rotation (NMOR)[1, 2] has been the subject of extensive theoretical andexperimental studies because it provides a very sensitiveway to measure magnetic fields (see, e.g., Ref. [3]). WhileNMOR experiments are usually carried out with vaporsamples of moderate optical thickness (no more than ∼ 2absorption lengths), the intense research of recent yearson electromagnetically-induced transparency (EIT; see,e.g., Ref. [4] for a review) has motivated investigation ofoptically thick media [5, 6].Recently, NMOR in the vicinity of the D1 and D2lines by an optically thick vapor of rubidium was studied[7, 8, 9, 10, 11]. The authors of Ref. [8] used a5 cm-long vapor cell containing 87 Rb and a laser beamtuned to the maximum of the NMOR spectrum near theD1 line (laser power was 2.5 mW, beam diameter ∼ 2mm). They measured maximum (with respect to laserfrequency and magnetic field) polarization rotation ϕ maxas a function of atomic density. It was found that ϕ maxincreases essentially linearly up to n ≈ 3.5 × 10 12 cm −3 .At higher densities, dϕ max /dn decreases and eventuallybecomes negative. The maximum observed rotation was≈ 10 rad with the applied magnetic field of ≈ 0.6 G.(This shows the effect of power broadening on the magneticfield dependence; at low light power, the maximumrotation would occur at a magnetic field about an order ofmagnitude smaller.) Rotation slope dϕ/dB| B=0 , an importantparameter for magnetometry, was also measuredas a function of atomic density in Ref. [8].The goal of the present contribution is to analyze thescaling of optimal magnetometric sensitivity with respectto optical thickness of the sample. We analyze NMORfor a simple system—an isolated J = 1 → J = 0 transition—forwhich analytical solutions for the density matrixare readily obtained. We assume that the transversethickness of the sample is small, so that the trapping ofspontaneously-emitted radiation can be neglected. (Thisassumption may not be justified in actual experimentswith optically thick media [9].) We also neglect mix-∗ Electronic address: budker@socrates.berkeley.eduing between different velocity groups due to velocitychangingcollisions. Such mixing is important in the presenceof a buffer gas, or for anti-relaxation-coated cells.A theoretical treatment of NMOR in dense media hasbeen carried out earlier in Refs. [7] and [12]. The presentanalysis partially overlaps with this earlier work and extendsit in several ways: we extend the treatment tothe case of large Doppler broadening and the intermediateregime where power and Doppler widths are comparable;provide a qualitative discussion of the contributionto optical rotation as a function of optical depth inthe medium; discuss the role of the coherence and thealignment-to-orientation conversion effects [13]; discussscaling of magnetometric sensitivity with optical thicknessof the medium; and finally, give a general argumenton scaling based on the observation that an optimizednonlinear optical rotation measurement is a way to carryout a spin measurement of an atomic ensemble with sensitivitygiven by fundamental quantum noise limits [3, 12].II.DESCRIPTION OF THE DENSITY MATRIXCALCULATIONThe calculation is based on a standard density matrixapproach. The time evolution of the atomic density matrixρ is given by the Liouville equation (see, e.g., Ref.[14]):dρdt = 1i¯h [H, ρ] − 1 {Γ, ρ} + Λ, (1)2where the square brackets denote the commutator andthe curly brackets the anti-commutator, and the totalHamiltonian H is the sum of light-atom interactionHamiltonian H L = −d ⃗ · ⃗E (where E ⃗ is the electric fieldvector, and d ⃗ is the electric dipole operator), the magneticfield-atom interaction Hamiltonian H B = −⃗µ · ⃗B(where B ⃗ is the magnetic field and ⃗µ is the magnetic moment),and the unperturbed Hamiltonian H 0 . Γ is therelaxation matrix (diagonal in the collision-free approximation)〈ξJm|Γ|ξJm〉 = γ + γ 0 δ(ξ, ξ e ), (2)where γ is the ground state depolarization rate (e.g., dueto transit of atoms through the laser beam), γ 0 is the

spontaneous decay rate from the upper state, and ξ representsthe quantum number distinguishing the groundstate (ξ g ) from the excited state (ξ e ). Λ = Λ 0 + Λ repop isthe pumping term, where the diagonal matrixdescribes incoherent ground state pumping (ρ 0atomic density), and2is the〈ξ g J g m|Λ 0 |ξ g J g m〉 =γρ 0(2J g + 1)(3)〈ξ g J g m|Λ repop |ξ g J g m ′ 〉 = γ 0∑m e,m ′ e ,q 〈J g , m, 1, q|J e , m e 〉〈J g , m ′ , 1, q|J e , m ′ e 〉ρ ξ eJ em eξ eJ em ′ e , (4)describes repopulation due to spontaneous relaxationfrom the upper level (see, e.g., Ref. [15]). Here 〈. . .| . . .〉are the Clebsch-Gordan coefficients.The electric field vector is written (see, e.g., Ref. [16])⃗E = 1 []E 0 e iφ (cos ϕ cos ɛ − i sin ϕ sin ɛ)e i(ωt−kz) + c.c. ˆx2+ 1 []E 0 e iφ (sin ϕ cos ɛ + i cos ϕ sin ɛ)e i(ωt−kz) + c.c. ŷ,2(5)where ω is the light frequency, k = ω/c is the vacuumwave number, E 0 is the electric field amplitude, ϕ is thepolarization angle, ɛ is the ellipticity (arctangent of theratio of the major and minor axes of the polarizationellipse), and φ is the overall phase. By substituting (5)into the wave equation( ) ω2c 2 + d2 ⃗Edz 2 = − 4π d 2c 2 dt ⃗ 2 P, (6)where ⃗ P = T r(ρ ⃗ d) is the polarization of the medium, theabsorption, rotation, phase shift, and change of ellipticityper unit distance for an optically thin medium canbe found in terms of the density matrix elements (theseexpressions are given in Ref. [17]). Once the solutionsfor the density matrix are obtained, we will perform anintegration to generalize the result to media of arbitrarythickness.III.THE DOPPLER-FREE CASEWe consider the case of a J = 1 → J = 0 transition,and linearly-polarized incident light, with a magneticfield directed along the light propagation direction(Faraday geometry). Using the rotating wave approximation,the solution of Eq. (1) is obtained, and from this,analytic expressions for thin medium absorption and rotationare found. These expressions can be simplified byassuming that γ ≪ γ 0 . We first consider the case wherethe power-broadened line width is much greater than theDoppler width. In this case, the absorption coefficientper unit length α is found to beα ≈α 0(2∆/γ 0 ) 2 + 2κ/3 + 1 , (7)where ∆ is the light-frequency detuning from resonance,κ = d2 E 2 0¯h 2 γγ 0(8)is the optical pumping saturation parameter, andα 0 ≈ 16π λ2 n (9)is the unsaturated absorption coefficient on resonance,where λ is the transition wavelength and n is the atomicdensity. For Ω ≪ γ, where Ω = gµB is the Larmorfrequency (g is the Landé factor, and µ is the Bohrmagneton), the slope of optical rotation per unit length,dϕ/(dΩdx), (proportional to rotation for small magneticfields) is found to bedϕdΩdx ≈ 1 α 0γ (2∆/γ 0 ) 2 + 2κ/3 , (10)where we have neglected linear optical rotation. Ingeneral, optical rotation can be induced by either lineardichroism or circular birefringence. Analysis of thesteady-state polarization of the ground state shows thatboth processes contribute here: the contribution due tolinear dichroism induced in the medium is given bydϕdΩdx∣ ≈ 1 1α 0dichr.γ 1 + (2∆/γ 0 ) 2 (2∆/γ 0 ) 2 + 2κ/3 , (11)and the contribution due to circular birefringence (arisingdue to alignment-to-orientation conversion in the presenceof both the magnetic field and the strong electricfield of the light [13]) is given bydϕdΩdx∣ ≈ 1 (2∆/γ 0 ) 2 α 0biref.γ 1 + (2∆/γ 0 ) 2 (2∆/γ 0 ) 2 + 2κ/3 . (12)

3The sum of the two contributions produces theLorentzian line shape of Eq. (10).We now generalize the formulas for absorption androtation to the case of thick media, and find the magnetometricsensitivity. In the Doppler-free case we arepresently considering, we can further simplify the expressionsby assuming ∆ = 0 (we will have to includenon-zero detunings in the discussion of the Dopplerbroadenedcase below), and κ ≫ 1. (We will see fromthe final result that this holds everywhere in the mediumwhen the input light power is optimal.) Generally, themedium in the presence of a magnetic field produces lightellipticity as well as rotation; however, the ellipticity isan odd function of detuning and is zero on resonance.The rate of change of the saturation parameter as lighttravels through the medium is given bydκ(x)dx= −ακ(x)≈ − 3 2 α 0, (13)so solving for the saturation parameter as a function ofposition,κ(x) ≈ κ 0 − 3 2 α 0x, (14)where κ 0 is the saturation parameter at x = 0.The contribution to the small-field optical rotation ofthe “slice” of the medium at position x is found by substitutingEq. (14) into Eq. (10):dϕdΩdx (x) ≈ 1 α 0γ 2κ 0 /3 − α 0 x . (15)This contribution is plotted as a function of position inFig. 1. The plot illustrates that the part of the mediumnear its end contributes significantly more to the overallrotation than the part near its beginning. This is becauselight power, and correspondingly, the light broadening ofthe resonance, is lower at the end of the medium [theratio of the rotation per unit length is approximatelyκ(l)/κ 0 ≈ 1 − (3/2)α 0 l/κ 0 ]. Integrating over the lengthl of the medium gives the total slope:∫dϕ ldΩ =0∫ ldϕdΩdx (x)dxα 0≈ 1 γ 0 2κ 0 /3 − α 0 x dx= 1 (γ ln κ 0κ 0 − 3α 0 l/2). (16)The slope (16) is plotted as a function of α 0 l in Fig. 2.The photon shot-noise-limited magnetometric sensitivityδB [22] is given in terms of the number N γ of transmittedphotons per unit time, the slope of rotation withdφddxΓ{5432100 0.2 0.4 0.6 0.8 1x{FIG. 1: Normalized contribution to small-field slope (in radians)of a slice of the medium of width dx at position x, givenby Eq. 15. We have set κ 0 = 1.8α 0l, the value which is laterseen to produce the greatest magnetometric sensitivity.dφdΓ5432100 0.2 0.4 0.6 0.8 1q 1FIG. 2: Normalized slope (in radians) of NMOR as a functionof optical thickness of the medium [q −1 = (3/2)α 0l/κ 0].The value of q −1 that produces the greatest magnetometricsensitivity is shown with a dashed line.respect to B, and the measurement time t:(δB) −1 = 2 dϕ √Nγ tdBdϕ= gµω¯hc≈ gµω¯hc= gµω¯hcdΩ√√3π√2Atπ γκ(l)2At κ 0 − 3 2 α 0llnπ γ((Atα 0 l√ q − 1 lnγκ 0κ 0 − 3α 0 l/2qq − 1)), (17)where A is the cross-sectional area of the light beam andwe have made the change of variables κ 0 = 3qα 0 l/2. Thefactor√q − 1 ln(qq − 1reaches a maximum of ∼ 0.8 at q ≈ 1.2 so we see that formedia of sufficient thickness, i.e. where α 0 l ≫ 3, and for)

4the optimum initial saturation parameter κ 0 ≈ 1.8α 0 l,we have from Eq. (14) that κ(x) ≫ 1 for all x. For theoptimum light intensity, then,√(δB) −1 ≈ 0.8 gµω Atα 0 l¯hc≈ 1.1 gµ¯hγ√Alntγ . (18)This result is consistent with the general observation [3]that optimized NMOR provides a method for measuringa spin-system at the standard quantum limit (SQL) givenby spin-projection noise [23]. The sensitivity is then expectedto scale as the square root of the product of thenumber of available atoms and the spin-relaxation time(see, e.g., Ref. [18], Sec. 3.1.1), which is indeed the result(18).IV.THE DOPPLER-BROADENED CASENow we consider the case where the Doppler width Γis much greater than the power-broadened line width. Inthis case, we need to average over the over the atomic velocitydistribution, which is equivalent to averaging overthe Doppler-free spectral profiles. On resonance with theDoppler broadened transition, the absorption coefficientis given byα DB ≈ 1 √ πΓ∫ ∞−∞≈ α ∫ ∞0√ πΓ=√3π8γ 0Γ−∞α(∆)d∆d∆(2∆/γ 0 ) 2 + 2κ/3α 0√ κ. (19)In this case the Doppler-broadened unsaturated absorbtioncoefficient is given in terms of the Doppler-free unsaturatedabsorbtion coefficient by√ πα 0 | DB=2γ 0Γ α 0. (20)Comparing Eqs. (19) and (7), we see that we have reproduceda well-known result (see, e.g., Ref. [19], Sec.7.2.1) that resonant absorption falls as 1/κ for Dopplerfreemedia and as 1/ √ κ for Doppler-broadened mediawhen κ ≫ 1. The change in κ per unit length isdκ(x)dx= −α DBκ(x)√6π γ 0≈ −4 Γ α √0 κ(x), (21)and solving for κ as a function of position,( √ ) 2√ 6π γ 0κ(x) ≈ κ0 −8 Γ α 0x . (22)Note that the behavior of the saturation parameter as afunction of distance is different in the Doppler-broadenedcase compared to the Doppler-free case [cf. Eq. (14)]where the saturation parameter falls approximately linearlywith distance.Taking the average of small-field rotation over theDoppler distribution givesdϕdΩdx∣ ≈ 1 ∫ ∞√DBπΓ −∞≈ α ∫ ∞0√ πγΓ≈√6π4γ 0Γ−∞dϕdΩdx (∆)d∆1(2∆/γ 0 ) 2 + 2κ/3 d∆α 0γ √ κ . (23)We see that the rotation per unit length scales as 1/ √ κ,in contrast to the 1/κ scaling for the Doppler-free case[similar to the situation with absorption, Eqs. (7,19)].This is because the number of atoms producing the effectis, in a sense, not fixed; with increasing light power, alarger fraction of the Doppler distribution is involved.Substituting (22) into Eq. (23) to find the contributionto the slope as a function of position givesdϕdΩdx (x) ∣ ∣∣∣DB≈ 2 γα 08 √6πΓγ 0√κ0 − α 0 x . (24)It is interesting to note that while the light power andthe rotation slope per unit length behave differently inthe Doppler-broadened case compared to the Dopplerfreecase, Eq. (24) has the same functional form as forthe Doppler-free case [Eq. (15)].Integrating over the length of the medium, we obtain∫dϕl∣ dϕ ∣∣∣DBdΩ∣ ≈DBdΩdx (x) dx≈ 2 γ0∫ l0√8Γ6π γα 0√κ0 − α 0 x dx(= 2 √γ ln κ0√κ0 − √ 6π γ 0Γ8α 0 l). (25)The behavior of (25) as a function of α 0 l is qualitativelysimilar to that of Eq. (16) shown in Fig. 2. However, thedependence on κ 0 is different.The magnetometric sensitivity is given by(δB) −1DB= gµω¯hc≈ gµω¯hc= gµω¯hc√dϕ2AtdΩ∣ DBπ γκ(l)√8Atπγ√ √32( √ ) ( √ )√ 6π γ 0κ0 −8 Γ α κ00l ln √κ0 − √ 6π γ 0Γ8α 0 l( )At γ 0pγ Γ α 0l(p − 1) ln , (26)p − 1

5with the change of variablesThe factorκ 0 = 3π (p γ )0 2.64 Γ α 0l(27)( ) p(p − 1) lnp − 1goes to unity as p goes to infinity; it is approximately 0.9at p = 5. (Further gain from increased power is minimal,and if the power becomes too high, the approximationof the Doppler width being much larger than the powerbroadenedwidth breaks down.) Thus, in this case,√(δB) −1DB ≈ 0.8gµω ¯hc≈ 0.3 gµ¯h λlnγ 0ΓAt γ 0γ Γ α 0l√Atγ , (28)for sufficiently high κ 0 .This result, where sensitivity increases linearly withoptical thickness, holds for the case where the powerbroadenedwidth ∼ γ 0√κ(x) is smaller than the Dopplerwidth for all x within the sample, i.e.1 ≪ κ 0 ≪ (Γ/γ 0 ) 2 andΓ/γ 0 ≪ α 0 l ≪ (Γ/γ 0 ) 2 , (29)since α 0 l is related to κ 0 by Eq. (27). As power andoptical thickness are increased beyond this range, i.e.κ 0 ≫ (Γ/γ 0 ) 2 andα 0 l ≫ (Γ/γ 0 ) 2 , (30)we obtain the Doppler-free case, where sensitivity increasesas the square root of the thickness [Eq. (18)].V. THE GENERAL CASEA numerical result can be obtained for the general casewhere the restrictions (29,30) on κ and α 0 l are removed.In a typical experiment, light power is ∼ 1 mW, the laserbeam diameter is ∼ 0.1 cm, λ ≈ 800 nm, and Γ/γ 0 ≈ 60.Thus the effective ground state relaxation rate due to thetransit of atoms through the laser beam is γ ≈ 2π·50 kHzand the initial saturation parameter is κ 0 ≈ 4 × 10 3 .Here, as in a typical experimental procedure, the opticaldepth is varied (by changing atomic density) whilethe laser power is kept constant. Normalized transmission,differential small-field rotation, total small-field rotation,and magnetometric sensitivity are plotted in Fig.3 as a function of optical depth. For small optical depth,κ > (Γ/γ 0 ) 2 and the medium is effectively Doppler-free.Transmission (∝ κ) falls linearly until the transition tothe Doppler-broadened case is made (dashed line). ThendφddΑ0{Γ Κ0310 3a210 3110 3310 30b210 3110 306c42040d2000 210 3 410 3 Α610 3{810 3 1010 3410 3dφdΓ Κ dφdΓFIG. 3: Normalized (a) saturation parameter, (b) differentialsmall-field rotation, (c) total small-field rotation, and (d) inversemagnetometric sensitivity as a function of optical depthwith initial saturation parameter κ 0 = 4×10 3 and Γ/γ 0 = 60.Plots (b), (c), and (d) are in units of radians. The dashed lineindicates the transition from the Doppler-free to the Dopplerbroadenedregime [κ = (Γ/γ 0) 2 ], and the dotted line indicatesthe point at which non-linear effects begin to turn off (κ = 1).Linear optical rotation is neglected in this plot. The solid lineindicates the optical depth at which maximum sensitivity isachieved.transmission falls quadratically until the linear regime isreached (dotted line), after which it falls exponentially.Differential small-field rotation (∝ dϕ/[dΩd(α 0 l)]γ) initiallyrises, as κ falls and power broadening is reduced,until non-linear effects begin to turn off. (Linear opticalrotation is neglected in this plot.) Since magnetometricsensitivity depends both on total optical rotation andtransmission, an intermediate value for the optical depthproduces the greatest sensitivity (solid line). Multiplyingthe normalized inverse sensitivity dϕ/dΩ · γ √ κ by2 √ 2π gµ¯h√A(λ √ γ ≈ 108 G/ √ ) −1Hz (31)

maxΓ φmax rad20151050302010ab00 210 3 410 3 610 3 810 3 1010 3Α 0 {The maximum optical rotation with respect to magneticfield can be determined with a numerical calculation.Maximum rotation is plotted as a function of opticaldepth in Fig. 4, for the same parameters as usedfor Fig. 3. The rotation initially rises linearly, the samebehavior seen in the experiment [8] mentioned in the introduction.For large optical depth, however, rotation inFig. 4 then begins to rise more quickly, and finally saturates,whereas experimentally a slower increase and thena decrease in rotation is seen. This is evidence for an additionalrelaxation mechanism not accounted for in thepresent theory; in Ref. [8] it is attributed to the effect ofradiation trapping.6FIG. 4: (a) Maximum optical rotation and (b) normalizedmagnetic field at which rotation is maximum as a function ofoptical depth. Parameters are the same as those used for Fig.3.VII.CONCLUSIONgives the absolute magnitude of sensitivity, ∼ 2 ×10 −10 G/ √ Hz (we assume g = 1). Although this sensitivityis not as high as could be achieved with lowatomic density paraffin-coated cells (∼ 3×10 −12 G/ √ Hz[3]), it is, nevertheless, sufficiently high to be of interestin practical applications [10]. In particular, the powerbroadeningof the magnetic field dependence of opticalrotation at high light power provides an increased dynamicrange for magnetometry over the low-power case.There are, however, techniques for shifting the narrowresonance obtained with a paraffin-coated cell to highermagnetic fields, e.g., frequency modulation of the laserlight [20].VI.LARGE-FIELD OPTICAL ROTATIONIn conclusion, we have analyzed magnetometric sensitivityof NMOR measurements optimized with respectto light intensity in the case of negligible Doppler broadening,and in the case of large Doppler broadening. Inthe former case, we find that the sensitivity improves asthe square root of optical density, while in the latter, itimproves linearly. In the present discussion, we have neglectedthe effect of velocity-changing collisions, whichmakes this analysis not directly applicable to buffer-gasand anti-relaxation-coated cells. However, since there isfull mixing between velocity components in these cells,one can expect that the sensitivity should scale as squareroot of optical density (if this quantity can be varied independentlyof the ground state relaxation rate).The authors are grateful to D. F. Kimball, I. Novikova,V. V. Yashchuk, and M. Zolotorev for helpful discussions.This work has been supported by the Office of NavalResearch (grant N00014-97-1-0214).[1] W. Gawlik, in Modern Nonlinear Optics, edited byM. Evans and S. Kielich (Wiley, New York, 1994), vol.LXXXV of Advances in Chemical Physics, p. 733.[2] D. Budker, D. J. Orlando, and V. Yashchuk, Am. J. Phys.67(7), 584 (1999).[3] D. Budker, D. F. Kimball, S. M. Rochester, V. V.Yashchuk, and M. Zolotorev, Phys. Rev. A 62(4),043403/1 (2000).[4] S. E. Harris, Phys. Today 50(7), 36 (1997).[5] M. O. Scully and M. Fleischhauer, Phys. Rev. Lett.69(9), 1360 (1992).[6] M. Fleischhauer and M. O. Scully, Phys. Rev. A 49(3),1973 (1994).[7] V. A. Sautenkov, M. D. Lukin, C. J. Bednar, I. Novikova,E. Mikhailov, M. Fleischhauer, V. L. Velichansky, G. R.Welch, and M. O. Scully, Phys. Rev. A 62(2), 023810/1(2000).[8] I. Novikova, A. B. Matsko, and G. R. Welch, Opt. Lett.26(13), 1016 (2001).[9] A. B. Matsko, I. Novikova, M. O. Scully, and G. R. Welch,Phys. Rev. Lett. 87(13), 133601/1 (2001).[10] I. Novikova and G. R. Welch, J. Mod. Opt. 49, 349(2002).[11] A. B. Matsko, I. Novikova, and G. R. Welch, J. Mod.Opt. 49, 367 (2002).[12] M. Fleischhauer, A. B. Matsko, and M. O. Scully, Phys.Rev. A 62(1), 013808/1 (2000).[13] D. Budker, D. F. Kimball, S. M. Rochester, and V. V.Yashchuk, Phys. Rev. Lett. 85(10), 2088 (2000).[14] S. Stenholm, Foundations of laser spectroscopy, Wiley

7series in pure and applied optics (Wiley, New York, 1984).[15] S. M. Rautian and A. M. Shalagin, Kinetic problemsof non-linear spectroscopy (North-Holland, Amsterdam,1991).[16] S. Huard, Polarization of light (Wiley, New York, 1997).[17] S. M. Rochester, D. S. Hsiung, D. Budker, R. Y. Chiao,D. F. Kimball, and V. V. Yashchuk, Phys. Rev. A 63(4),043814/1 (2001).[18] I. B. Khriplovich and S. K. Lamoreaux, CP violationwithout strangeness : electric dipole moments of particles,atoms, and molecules, Texts and monographs inphysics (Springer-Verlag, Berlin, 1997).[19] W. Demtröder, Laser spectroscopy : basic concepts andinstrumentation (Springer, Berlin, 1996), 2nd ed.[20] D. Budker, D. F. Kimball, V. V. Yashchuk, and M. Zolotorev,Submitted (2002).[21] D. Ulam-Orgikh and M. Kitagawa, Phys. Rev. A 64(5),052106/1 (2001).[22] In the general case, there may exist an additional sourceof noise due to AC-Stark shift associated with offresonantlevels [12]; however, this source of noise is absentfor an isolated transition such as the one considered here.[23] The use of so-called spin-squeezed quantum states of light(see, e.g., Ref. [21] and references therein) can, in principle,allow one to overcome the SQL. We consider onlynon-squeezed states of atoms and light here.