Nonlinear spectroscopy of high-spin fluctuations
A. A. Fomin, M. Yu. Petrov, I. I. Ryzhov, G. G. Kozlov, V. S. Zapasskii, M. M. Glazov
NNonlinear spectroscopy of high-spin fluctuations
A. A. Fomin, M. Yu. Petrov, I. I. Ryzhov,
2, 1
G. G. Kozlov, V. S. Zapasskii, and M. M. Glazov
3, 1 Spin Optics Laboratory, St. Petersburg State University, 198504 St. Petersburg, Russia Photonics Department, St. Petersburg State University, Peterhof, 198504 St. Petersburg, Russia Ioffe Institute, 194021 St. Petersburg, Russia
We investigate theoretically and experimentally fluctuations of high spin ( S > / ) beyond thelinear response regime and demonstrate dramatic modifications of the spin noise spectra in the highpower density probe field. Several effects related to an interplay of high spin and perturbation arepredicted theoretically and revealed experimentally, including strong sensitivity of the spin noisespectra to the mutual orientation of the probe polarization plane and magnetic field direction,appearance of high harmonics of the Larmor frequency in the spin noise and the fine structure ofthe Larmor peaks. We demonstrate the ability of the spin-noise spectroscopy to access the nonlineareffects related to the renormalization of the spin states by strong electromagnetic fields. Introduction . Spin manipulation by optical means in-cluding optical orientation effect, spin readout by Kerrand Faraday effects, inverse Faraday effect, coherent con-trol, etc. are of high importance for the growing field ofspintronics [1–3]. The spin- / state serves as a proto-type qubit two-level system for various applications, alsobeing the simplest and most studied model for coherentcontrol of quantum states [4, 5]. Regarding the quan-tum processing of information the use of high spins, i.e., S > / , offers serious advantages due to a higher numberof controlled degrees of freedom [6–10]. While the finestructure of the spin states can be tuned by a magneticfield, the non-magnetic spin control is one of the challeng-ing tasks offering significant advantages. One example ofsuch kind phenomena is the spin-orbit interaction whichcouples spin and orbital degrees of freedom and, con-sequently, allows one to control the electron spins elec-trically [11, 12]. Another is an optical quantum statecontrol, which was demonstrated to be possible not onlyfor trapped cold atoms, but even for strongly interactingsystems [13].Another possibility is to use the nonlinear interactionsof spin systems with electromagnetic fields. Tailored pe-riodic driving of quantum systems, also known as Floquetengineering, provides vast playground for developing thesystems with novel properties and applications in vari-ous areas of physics including ultrafast spectroscopy andhigh-speed spintronics. In particular, due to the inverseFaraday effect [15, 16], the circularly polarized light fieldproduces effective, optically induced, magnetic field act-ing on the spins [17]. For a two-level spin- / systemany perturbation can be reduced to an effective magneticfield [18].The situation becomes very rich for the S > / . Firstly,there are higher-order spin arrangements which are notreduced to the spin polarization, e.g., the spin alignmentwhere the average spin orientation is absent, but the spinsare predominantly aligned parallel or antiparallel to acertain axis (for more details on this effect, see the prece-dent work of our group [32]). Accordingly, one may seekthe ways to control high spins not only by the circularly polarized, but also by a linearly polarized light.In this paper we investigate, both theoretically andexperimentally, the spin fluctuations in the atomic gasin the presence of strong linearly polarized probe beam.The spin noise spectroscopy has initially emerged as vir-tually perturbation-free technique to study the spin dy-namics and fine structure of the spin states [19–25]. In-creasing the probe beam intensity makes it possible to ad-dress a plethora of non-equilibrium phenomena includingincoherent effects related with relaxation processes andalso the coherent phenomena [26–30]. Here we demon-strate that the renormalization of the spin S > / statesby the electromagnetic field causes a number of non-linear effects, which include: the light-induced fine split-tings of the spin states, induced anisotropy of the spinsystem, i. e., the dependence of the spin noise spectra onthe orientation of the probe beam with respect to themagnetic field, and appearance of high harmonics of theLarmor precession frequency in the spin noise. We il-lustrate experimentally theoretical predictions taking Csatomic vapour as testbed, because in this case the mul-tiplet corresponding to the total spin S = 3 , Theory.
Let us consider a spin S > / system, for ex-ample, an atom in the presence of the intense monochro-matic laser beam (probe). We assume that the frequencyof the laser ω exceeds by far all characteristic frequenciesof the spin system, particularly, the Larmor frequency,but can be close to one of the optical resonances. Weassume, however, that the probe beam is not perfectlyresonant with any of transitions and neglect in in thiswork possible absorption of the light related to real tran-sitions between atomic states. Such situation is typicalin the spin noise spectroscopy [19, 32].We are interested in the effects of the intense probebeam on the spin noise spectrum. Since the beam isoff-resonant and the absorption is disregarded, only vir- a r X i v : . [ phy s i c s . a t o m - ph ] F e b tual transitions are possible and the description of theprobe beam effects can be reduced to an effective Flo-quet Hamiltonian which describes low-energy dynamicsof the spin system, while the energy scales of (cid:126) ω andhigher are integrated out. The general form of the effec-tive Hamiltonian can be determined from the symmetryarguments. The effective (dressed) Hamiltonian for thespin F in the presence of the external magnetic field B and the probe beam with the electric field E in the form E = E e − i ωt + c . c ., (1)with E being the complex amplitude, in quadratic inthe | E | approximation reads H = γ ( B · F ) + i B ([ E × E ∗ ] · F ) + C{ ( E · F )( E ∗ · F ) } s . (2)Here B and C are the parameters, γ is the gyromagneticratio, the non-linear in B contributions to the Hamilto-nian (2) are disregarded. The first term here is the Zee-man effect caused by the external field. The derivationof the remaining terms is based on the method of invari-ants in the group theory and relies on the fact that theHamiltonian is invariant under the point-group transfor-mations of the system [14]. In our case, this is the groupof all transformations of the three-dimensional space, i.e., SO (3) × I : the transformations include all the rotations,inversion and their compositions. Thus, to derive Eq. (2)we need to build the invariant combinations of the spincomponents and probe field. We restrict the considera-tion by bi-linear in E , E ∗ effects, i.e., the contributionsto the Hamiltonian linear in the light intensity. Corre-sponding field combinations transform either as a pseu-dovector, [ E × E ∗ ] (i.e., asymmetric second-rank tensorwhich is dual to a pseudovector), and as the symmetricsecond-rank tensor, { E ,α E ∗ ,β } s . The former ones cou-ple with the pseudovector F and provide an invariant as[ E × E ∗ ] · F [second term in Eq. (2)], while the latter onescouple with symmetrized combinations { F α F ∗ β } s [thirdterm in the Hamiltonian (2)]. The term i B ([ E × E ∗ ] · F )describes the inverse Faraday effect [15–17]: The circu-larly polarized probe induces the effective Zeeman split-ting of the atomic levels. Indeed, the product i[ E × E ∗ ]transforms as a pseudovector and describes the helicityof the photon, thus it couples with spin just like an exter-nal magnetic field. The term C{ ( E · F )( E ∗ · F ) } s withcurly brackets standing for symmetrization of the opera-tors describes the renormalizations of the energies due tothe linearly polarized light. Note that while individualfactors ( E · F ) and ( E ∗ · F ) are pseudoscalars, their prod-uct is scalar. For spin S = / such symmetrized prod-uct is unimportant, since the products of spin- / com-ponents are reduced to the first powers of spins. Thus,the term ∝ C appears for S > / only, it is somewhatsimilar to the quadrupole splittings of the nuclear statesby the strain or electric field gradients [5].The parameters B and C can be calculated in the Figure 1. (a) Energy spectrum of the dressed Hamiltonian (2)for S = 3 at the fixed magnetic field for various probe beamintensities and θ = 0. Arrows indicate transition frequencies.(b) Transition frequencies Ω M,M +1 calculated after Eq. (11)(black dashed lines) and by numerical diagonalization of theHamiltonian (2) (red solid lines) at C| E | /ω L = 1 /
20. Arrowindicates the magic angle θ m . framework of the time-dependent perturbation the-ory [35] and can be estimated as B , C ∼ | d | / ∆ = (cid:36) R / ( | E | ∆), where d is the dipole matrix element ofthe optical transition, (cid:36) R is the Rabi frequency, and∆ = ω − ω is the probe detuning from the optical reso-nance ω [36].Equation (2) demonstrates the key effects of the in-tense probe beam. To begin with, let us briefly considercircularly polarized radiation where the symmetric com-binations E ,α E ∗ ,β + E ∗ ,α E ,β vanish andi[ E × E ∗ ] = P c n | E | , (3)where n is the unit vector along the light propagationdirection, P c is the degree of circular polarization. Thus,the second term in Eq. (2) can be presented in the Zee-man form with the effective – optical – magnetic field B eff = γ − P c B n . (4)This field combines with the external field resulting in ashift of the Larmor frequency and corresponding modi-fications of the spin noise spectra. The effect of “opti-cal” magnetic field created by elliptically polarized lightwas studied in detail theoretically and demonstrated ex-perimentally on a semiconductor system in a precedingwork [17] of our group.In what follows we focus on the effects of linearly po-larized probe where the circular polarization vanishes, P c = 0, and the Hamiltonian along with the Zeeman con-tribution of the external field contains also the quadraticin spin F terms. Let θ = ∠ B , E be the angle be-tween the magnetic field and the polarization vector ofthe probe. To illustrate the effect, we assume that C| E | (cid:28) | γ B | . (5)Thus, the third term in the Hamiltonian (2) can beconsidered as a small perturbation. Accordingly, theeigenstates are characterized by a projection M = (a) (b) (c)ellipticityFaraday rotation Figure 2. Noise spectra of Faraday rotation N r ( ν ) and ellipticity N e ( ν ) calculated using Eqs. (9) and (10). Panels (a), (b),and (c) correspond to θ = 0, π/
4, and π/
2, respectively. The parameters of calculation are: ω L = 20 MHz, C| E | /ω L = 1 / . − S, − S + 1 , . . . , S − , S of F onto the B -axis. Mak-ing use of the first-order perturbation theory, we evalu-ate the matrix elements of the last term in Eq. (2) onthe basic functions with the definite spin projection M on the magnetic field (eigenfunctions of the first termin the Hamiltonian) we evaluate E M . As a result, thedifferences between the eigenenergies E M or transitionfrequencies acquire the formΩ MM (cid:48) ≡ E M (cid:48) − E M (cid:126) = ω L ( M (cid:48) − M )++ 12 C| E | (cid:0) θ − sin θ (cid:1) ( M (cid:48) − M ) . (6)Here ω L = γB, (7)is the Larmor spin precession frequency in the magneticfield B . Thus, due to the probe-induced effect, the spec-trum of the spin in the magnetic field is no longer equidis-tant and the transition frequencies are no longer multi-ples of the ω L , but demonstrate a fine structure deter-mined by the second term in Eq. (6), as schematically il-lustrated in Fig. 1(a). The fine structure depends on theorientation of the probe beam polarization plane with re-spect to the magnetic field and described, in accordancewith the symmetry requirements, by the second angularharmonics of the angle θ . Interestingly, the fine structurein the quadratic in the field approximation vanishes at a“magic” angle θ m = arctan √ . (8)The numerical diagonalization of the Hamiltonian (2)demonstrates that the suppression of splittings takesplace only in the | E | order, compare solid and dashedcurves in Fig. 1(b).These features in the energy spectrum are clearly re-vealed in the spectrum of spin fluctuations. Using theeffective Hamiltonian we can formally define the fluctua-tion spectrum of any observable O as( O ) ν = 2Γ (cid:126) S + 1 (cid:88) ij | O ij | ( (cid:126) ν − E j + E i ) + (cid:126) Γ , (9) where i and j enumerate the eigenstates of the system, Γis the phenomenological broadening. For example, in thegeometry where the magnetic field B (cid:107) x , for the spin- z component fluctuations we have( F z ) ν = Γ2(2 S + 1) S − (cid:88) M = − S ( S + M + 1)( S − M )( ν − Ω M,M +1 ) + Γ + { ν → − ν } , (10)Therefore, the intense probe beam results in the finestructure of the spin fluctuation spectrum: Instead ofa single peak at the Larmor frequency the system wouldbe characterized with a series of 2 F peaks with the fre-quencies [cf. Eq. (6)]:Ω M ≡ Ω M,M +1 = ω L + 12 C| E | (cid:0) θ − sin θ (cid:1) (2 M +1) . (11)The fluctuations of the spin z -component are revealedby the fluctuations of the Faraday rotation angle of theprobe beam passing through the atomic vapor in thetransparency region [5, 19, 32] N r ( ν ) ∝ ( F z ) ν . (12a)If, instead of the Faraday rotation, the fluctuations ofellipticity are detected, then, for S > / the noise of thespin alignment N e ( ν ) ∝ sin θ (cid:2) ( F x − F y ) (cid:3) ν +2 cos θ (cid:0) { F x F y } s (cid:1) ν (12b)is detected, see Refs. [32, 37] for details. The fluctuationspectra of the Faraday rotation and ellipticity calculatedafter general Eqs. (9), (10), and (12) are demonstratedin Fig. 2. Figure 3 shows the same spectra as functionsin the form of density plots. The positions of peaks arewell reproduced by analytical formula (11), as indicatedby the dashed lines in Fig. 3.It follows from Eq. (2) that, due to the term C{ ( E · F )( E ∗ · F ) } s , the states with different components M are mixed at θ (cid:54) = 0 , π . It gives rise to the higher harmon-ics of the Larmor frequency in the spin noise spectra, Figure 3. Calculated noise spectra of Faraday rotation N r ( ν )(a) and ellipticity N e ( ν ) (b) as a function of probe azimuth θ and frequency ν . The transition frequencies calculated afterEq. (11) are shown in white dotted lines, which are madetransparent in the vicinity of magic angle not to hinder thespectral features). The parameters of calculation are: ω L =20 MHz, C| E | /ω L = 1 /
20, and Γ = 0 . particularly, it gives rise to the 2 ω L and 3 ω L peaks, seeFig. 2(b,c) and Fig. 3. The higher-order in E terms(omitted in the Hamiltonian (2)) give rise to the higherharmonics.To conclude, let us summarize the main effects of theintense probe beam. • Circularly polarized beam gives rise to an effectiveoptical magnetic field, Eq. (4) (see [17] for details). • Linearly polarized probe beam results in the finestructure of the transition energies grouped in thevicinity of the Larmor frequency, Eq. (6). • Accordingly, it results in the fine structure of theLarmor peak in the spin noise spectrum. • The effective energy spectrum and, correspond-ingly, the spin noise spectrum depends on the mu-tual orientation of the probe beam polarization andthe magnetic field. • The higher Larmor harmonics in the spin noisespectra appear as a result of the symmetry break-ing by the intense probe beam.It should be noted that in high-precision applications (forexample, for accurate measurements of the Earth mag-netic field) one should take into account not only thedescribed above features, but also other effects affectingthe shape of the EPR spectral lines, in particular, thenonlinearity of the Zeeman effect and the accompanyingheading-error effect [38]. In the framework of this work,we omit these fine effects due to the fact that under ourconditions the width of the spin noise line significantlyexceeded the value of the splittings caused by the nonlin-ear Zeeman effect. The linewidth in our case was deter-mined not only by the field inhomogeneity, but also by
Figure 4. Energy-level diagram of Cesium atom in the rangeof D2 line. The spectral position of the probe light, whichwas slightly detuned from the center of D2 optical transition,is shown schematically. the optically induced line broadening which origin willbe discussed elsewhere.
Experimental illustrations.
As a high-spin system weused cesium atom with its ground state comprised of twohyperfine (hf) components with total angular momenta S = 3 and S = 4. Energy level diagram of Cs atom isshown schematically in Fig. 4. The measurements wereperformed at long-wavelength component of the D2 linecorresponding to the transition S / ( S = 4) → P / ( S = 3 , , ×
20 mm in size with cesium vapor at a tem-perature of 60. . . 70 ◦ C, was mounted inside a pair of theHelmholtz-type coils, creating magnetic field of around1.6 mT aligned across the laser light propagation.Design of the experimental setup was common for spinnoise spectra measurement. Briefly, the linearly polar-ized collimated laser beam, 4 mm in diameter, after pass-ing through the cell, was detected by a standard dif-ferential polarimetric photoreceiver, with its output sig-nal processed in real time by a broadband fast Fouriertransform radio-frequency spectrum analyzer TektronixRSA5103A. The light power density on the sample was0.08 W/cm for the total laser beam intensity of 10 mW.Placing a quarter-wave plate after the cell with one of itsprincipal axes aligned along the light beam polarizationallowed us to detect fluctuations of the light ellipticity,rather than the noise of the Faraday rotation. The po-larization plane azimuth of the incident beam was con-trolled by rotation of a half-wave plate installed before Figure 5. Azimuthal dependence of the ellipticity-noise spectra of Cs atoms, under conditions of resonant probing: experimentaldata (a) and model (b). The experimental conditions and the parameters of calculation are shown at the top of panels (a) and(b), respectively. the sample. All the measurements were performed withthe polarimetric photoreceiver set to the balance.Figure 5(a) shows the measured ellipticity noise spec-trum vs the angle between the light polarization planeand magnetic field direction. To compare these resultswith predictions of our theoretical treatment, we have totake into account that conditions of our measurementswere far from the idealized assumptions of the theory.Specifically, our experiments did not meet the strict re-quirements of sufficiently large detuning accepted in thetheory: In the experiment a combination of N e ( ν ) and N r ( ν ), as well as real processes of the probe beam absorp-tion may affect the results of the measurements. Whatis, perhaps, more important is that, because of inhomo-geneity of the light power density over the probe beamcross-section, the fine structure of the spectra could notbe as pronounced as in theory. So, to make our calcula-tions more realistic, we enhanced the relative broadeningof the spin-noise spectrum, Γ /ω L ≈ .
07. Under theseconditions, the Γ exceeded the scale of the light-inducedsplittings, Γ / ( C| E | ) ≈ . θ = 0 ◦ and 90 ◦ , the splitting is the greatest while at someintermediate angle this splitting tends to vanish. Withinthe accuracy of our measurements, position of the lat-ter point well correlates with predictions of the theory.At the double Larmor frequency, we can see the peak ofthe spin-alignment noise with its azimuthal dependencebeing in agreement with the model.To conclude, we have demonstrated both theoreticallyand experimentally that for S > / the interaction of spinwith electromagnetic field results in renormalization of its energy spectrum. Particularly, the linearly polarizedintense probe beam results in the appearance of the finestructure of the spin levels which strongly depends onthe mutual orientation of the light polarization plane andthe magnetic field. The effect can be straightforwardlydetected in the spin noise spectroscopy where the finestructure of Larmor precession peak in the transversemagnetic field is revealed along with higher harmonicsof the Larmor precession frequency. The developed an-alytical theory is illustrated by experimental data on Csvapors. The uncovered phenomena open up the possibil-ities to control the spin states non-magnetically via theFloquet engineering of the Hamiltonian. One should Acknowledgements.
Theoretical work of M. M. G. waspartially supported by Saint-Petersburg State Univer-sity, research Grant No. 51125686. The work was ful-filled using the equipment of the SPbU resource cen-ter ‘Nanophotonics’. Experimental investigations werefunded by RFBR grant No. 19-52-12054 which is highlyappreciated. I. I. R. acknowledges the support of experi-mental work by Presidential Grant No. MK-2070.2018.2. [1] A. Eckardt, Rev. Mod. Phys. , 011004 (2017).[2] N. Goldman and J. Dalibard, Phys. Rev. X , 031027(2014).[3] T. Oka and S. Kitamura, Annual Review of CondensedMatter Physics , 387 (2019).[4] S. A. Wolf, D. D. Awschalom, R. A. Buhrman, J. M.Daughton, S. von Molnar, M. L. Roukes, A. Y.Chtchelkanova, and D. M. Treger, Science , 1488(2001).[5] M. Glazov, Electron & Nuclear Spin Dynamics in Semi-conductor Nanostructures , Series on Semiconductor Sci- ence and Technology (OUP Oxford, 2018).[6] D. Kaszlikowski, D. K. L. Oi, M. Christandl, K. Chang,A. Ekert, L. C. Kwek, and C. H. Oh, Phys. Rev. A ,012310 (2003).[7] B. P. Lanyon, M. Barbieri, M. P. Almeida, T. Jennewein,T. C. Ralph, K. J. Resch, G. J. Pryde, J. L. O’Brien,A. Gilchrist, and A. G. White, Nature Physics , 134(2008).[8] P. B. R. Nisbet-Jones, J. Dilley, A. Holleczek, O. Barter,and A. Kuhn, New Journal of Physics , 053007 (2013).[9] A. R. Shlyakhov, V. V. Zemlyanov, M. V. Suslov, A. V.Lebedev, G. S. Paraoanu, G. B. Lesovik, and G. Blatter,Phys. Rev. A , 022115 (2018).[10] V. A. Soltamov, C. Kasper, A. V. Poshakinskiy, A. N.Anisimov, E. N. Mokhov, A. Sperlich, S. A. Tarasenko,P. G. Baranov, G. V. Astakhov, and V. Dyakonov, Na-ture Communications , 1678 (2019).[11] M. I. Dyakonov, ed., Spin physics in semiconductors , 2nded., Springer Series in Solid-State Sciences 157 (SpringerInternational Publishing, 2017).[12] E. I. Rashba and A. L. Efros, Phys. Rev. Lett. , 126405(2003).[13] J. Kong, R. Jim´enez-Mart´ınez, C. Troullinou, V. G. Lu-civero, G. T´oth, and M. W. Mitchell, Nat. Commun. ,2415 (2020).[14] G. L. Bir and G. E. Pikus. Symmetry and Strain-inducedEffects in Semiconductors . (Wiley/Halsted Press, 1974).[15] P. S. Pershan, J. P. van der Ziel, and L. D. Malmstrom,Phys. Rev. , 574 (1966).[16] L. P. Pitaevskii, JETP , 1008 (1961).[17] I. I. Ryzhov, G. G. Kozlov, D. S. Smirnov, M. M. Glazov,Y. P. Efimov, S. A. Eliseev, V. A. Lovtcius, V. V. Petrov,K. V. Kavokin, A. V. Kavokin, and V. S. Zapasskii, Sci.Rep. , 21062 (2016).[18] L. Allen, A. Allen, and J. Eberly, Optical Resonanceand Two-level Atoms , Ballard CREOL collection (Wiley,1975).[19] E. Aleksandrov and V. Zapasskii, JETP , 64 (1981).[20] J. L. Sørensen, J. Hald, and E. S. Polzik, Phys. Rev.Lett. , 3487 (1998).[21] T. Mitsui, Phys. Rev. Lett. , 5292 (2000). [22] S. A. Crooker, D. G. Rickel, A. V. Balatsky, and D. L.Smith, Nature , 49 (2004).[23] V. S. Zapasskii, Adv. Opt. Photon. , 131 (2013).[24] J. H¨ubner, F. Berski, R. Dahbashi, and M. Oestreich,physica status solidi (b) , 1824 (2014).[25] M. M. Glazov and V. S. Zapasskii, Opt. Express ,11713 (2015).[26] H. Horn, G. M. M¨uller, E. M. Rasel, L. Santos, J. H¨ubner,and M. Oestreich, Phys. Rev. A , 043851 (2011).[27] P. Glasenapp, N. A. Sinitsyn, L. Yang, D. G. Rickel,D. Roy, A. Greilich, M. Bayer, and S. A. Crooker, Phys.Rev. Lett. , 156601 (2014).[28] M. M. Glazov, JETP , 472 (2016).[29] D. S. Smirnov, P. Glasenapp, M. Bergen, M. M. Glazov,D. Reuter, A. D. Wieck, M. Bayer, and A. Greilich,Phys. Rev. B , 241408(R) (2017).[30] J. Wiegand, D. S. Smirnov, J. H¨ubner, M. M. Glazov,and M. Oestreich, Phys. Rev. B , 081403(R) (2018).[31] M. Y. Petrov, I. I. Ryzhov, D. S. Smirnov, L. Y. Belyaev,R. A. Potekhin, M. M. Glazov, V. N. Kulyasov, G. G.Kozlov, E. B. Aleksandrov, and V. S. Zapasskii, Phys.Rev. A , 032502 (2018).[32] A. A. Fomin, M. Y. Petrov, G. G. Kozlov, M. M. Glazov,I. I. Ryzhov, M. V. Balabas, and V. S. Zapasskii, Phys.Rev. Research , 012008 (2020).[33] A. V. Poshakinskiy and S. A. Tarasenko. Phys. Rev. B, , 075403 (2020).[34] M. M. Sharipova, A. N. Kamenskii, I. I. Ryzhov, M. Yu.Petrov, G. G. Kozlov, A. Greilich, M. Bayer, and V. S.Zapasskii. J. Appl. Phys. 126, 143901 (2019).[35] C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Atom-photon interactions. Basic processes and applica-tions (Wiley, 2004).[36] The detailed microscopic theory will be presented else-where.[37] G. G. Kozlov, A. A. Fomin, M. Y. Petrov, and V. S.Zapasskii, arXiv:2002.06035.[38] G. Bao, A. Wickenbrock, S. Rochester, W. Zhang, andD. Budker. Phys. Rev. Lett.120