Cascade coherence transfer and magneto-optical resonances at 455 nm excitation of Cesium
Marcis Auzinsh, Ruvin Ferber, Florian Gahbauer, Andrey Jarmola, Linards Kalvans, Aigars Atvars
aa r X i v : . [ phy s i c s . a t o m - ph ] D ec Cascade coherence transfer and magneto-optical resonances at 455 nm excitation ofcesium
M. Auzinsh, ∗ R. Ferber, F. Gahbauer, A. Jarmola, and L. Kalvans
The University of Latvia, Laser Centre, Rainis Blvd., LV-1586 Riga, Latvia
A. Atvars
Institute of Physical Research and Biomechanics, Maskavas Str. 22-1, LV-4604 Rezekne, Latvia (Dated: October 29, 2018)We present an experimental and theoretical study of nonlinear magneto-optical resonances ob-served in the fluorescence to the ground state from the 7 P / state of cesium, which was populateddirectly by laser radiation at 455 nm, and from the 6 P / and 6 P / states, which were populatedvia cascade transitions that started from the 7 P / state and passed through various intermediatestates. The laser-induced fluorescence (LIF) was observed as the magnetic field was scanned throughzero. Signals were recorded for the two orthogonal, linearly polarized components of the LIF. Wecompared the measured signals with the results of calculations from a model that was based onthe optical Bloch equations and averaged over the Doppler profile. This model was adapted from amodel that had been developed for D and D excitation of alkali metal atoms. The calculationsagree quite well with the measurements, especially when taking into account the fact that someexperimental parameters were only estimated in the model. I. INTRODUCTION
The technique of populating atomic states via cas-cade transitions from higher-lying states has been usedfor many years to study atomic properties and quantumphenomena. The excited-state Hanle effect, or zero-fieldlevel crossing in the case of weak excitation, has beenused to measure lifetimes and hyperfine structure (hfs)parameters of atomic states. For example, Tsukakoshiand Shimoda [1] and Carrington [2] used discharge lampsto observe the cascade Hanle effect in order to study de-cay times of atomic levels in xenon and neon, respec-tively. The cascade Hanle effect has been used to studylifetimes of alkali metal S states [3] and, together withother cascade techniques, to measure lifetimes and hfsparameters of atomic states in alkali metal D states [4].In some cases, the motivation for populating states viacascade transitions was to populate states that were oth-erwise unreachable via direct excitation [5, 6]. Mean-while, the ground-state Hanle effect was first observedby Lehmann and Cohen-Tannoudji [7]. Schmieder [8] andlater Alzetta [9] observed dark resonances, where the flu-orescence is at a minimum at zero magnetic field, whenexciting D or D transitions in alkali metal atoms bymeans of discharge lamps. Similar resonances were ob-served by means of laser excitation by Ducloy et al. [10] influorescence signals and Gawlik et al. [11] in connectionwith the nonlinear Faraday effect. Much later, Dancheva et al. [12] observed bright resonances, which have a flu-orescence maximum at zero magnetic field, in the D and D transitions of rubidium atoms in a vapor cell.Recently, Gozzini and co-workers observed the narrowmagneto-optical resonances associated with the ground- ∗ [email protected] state Hanle effect in the fluorescence from states thatwere populated by cascade transitions from higher-lyingstates [13]. They excited the second resonance line ofpotassium with linearly and circularly polarized light andobserved nonlinear magneto-optical resonances in the un-polarized fluorescence from the 4 P / and 4 P / transi-tions, which had been populated from the 5 P / state viaspontaneous cascade transitions through various inter-mediate states. Measurements were obtained at varioustemperatures, but no theoretical description was given.In the present article, we describe an experimentalstudy of nonlinear magneto-optical resonances observedin the fluorescence to the ground-state via various de-excitation pathways from the 7 P / state (second res-onance line) of cesium together with theoretical calcu-lations to describe the observed signals. In addition,we monitor the transfer of coherence through these cas-cades by measuring the polarization degree of the fluo-rescence radiation, observed after excitation with linearlypolarized radiation, and compare these measurementswith theoretical calculations. Observations of nonlin-ear magneto-optical resonances in the fluorescence fromstates that are populated via cascades could be partic-ularly interesting for magnetometry, because the reso-nances are narrow and can be observed at a wavelengthfar removed from the wavelength of the exciting laserradiation, which is the main source of noise in such mea-surements. Therefore, it seemed important to be able tostudy a system both experimentally and theoretically.The basic theory of the fluorescence from a state pop-ulated from above via cascade transitions to a state otherthan the ground state was given by Gupta et al. [14] forlinear excitation. The theory was based on the opticalBloch equations for the density matrix. In 1978 Picqu´eused the optical Bloch equations to describe accuratelydark resonances that arose in nonlinear excitation of onehyperfine component of the D transition in a stronglyexcited beam of sodium atoms [15]. In recent years, suchmodels have achieved very good agreement for the D transitions of cesium [16] and rubidium [17] when aver-aging over the Doppler profile and taking into accountthe coherence properties of the laser radiation, as well asall adjacent hyperfine states and even the small effect ofthe mixing of magnetic sublevels in the magnetic field.In the context of the present study it was necessaryto adapt the theoretical model developed for magneto-optical resonances in the D lines of alkali metal atomsunder nonlinear excitation in Refs. [16, 17] to the cascadetransitions that result when the second resonance line ofalkali atoms was excited. We compared experimentallymeasured signals with the results of calculations of theintensity of direct flourescence from the 7 P / state ofcesium, as well as fluorescence from the 6 P / state ( D line) and the 6 P / state ( D line). The calculations werebased on an extension of a theoretical model developedfor the first resonance line of alkali metal atoms. How-ever, in the case of cascade transitions, the large numberof decay channels leads to a density matrix that is sub-stantially larger than in the case for the first resonanceline. We studied the unpolarized fluorescence intensityemitted along the direction of the scanned magnetic fieldas well as the polarized fluorescence intensity and thepolarization degree. Figure 1 shows the atomic statesinvolved in our experiment. The figure includes the ex-citing line, the cascade pathways, and the observed fluo-rescence lines. The theoretical model took into accountthe population and coherence transfer of all possible de-excitation paths. As a result, it was necessary to solvevery large systems of equations, which is computationallyintensive and, thus, time-consuming. Therefore, insteadof searching for the optimal parameters needed to de-scribe the experimental signals in detail, we aimed to re-produce and understand the experimental features usingestimated values for the model parameters. FIG. 1. (Color online) Level diagram. Excitation takes placeat 455 nm. Fluorescence is observed at 455 nm, 894 nm, and852 nm. The numbers in square brackets correspond to thelabeling scheme used in the equations of Sec. III B.
II. EXPERIMENTAL DESCRIPTION
In the experiment, a Toptica TA-SHG110 laser at 455nm was used to excite cesium atoms in a vapor cell. Thecell was home-made and kept at room temperature atthe center of a three-axis Helmholtz coil system. Twosets of coils were used to cancel the ambient magneticfield, while the third set was used to scan the magneticfield B from -7 Gauss to +7 Gauss by means of a KepcoBOP-50-8M bipolar power supply. The laser was usuallytuned to the frequency for which the fluorescence at zeromagnetic field was at a maximum for a given transition,except in the case of certain studies where it was delib-erately detuned from this frequency by a known amount.The laser frequency was monitored using a High-FinesseWS7 wavemeter to ensure that the frequency did notdrift significantly during an experiment. During a givenmeasurement, the laser frequency did not drift by morethan 10 MHz.The geometry of the polarization vector of the excitinglaser radiation, the magnetic field, and the direction offluorescence observation are given in Fig. 2. The fluo-rescence light was focused with a lens onto a polarizingbeam splitter, which directed two orthogonally polarizedcomponents of the fluorescence radiation to two separatephotodiodes (Thorlabs FDS-100). In front of the polariz-ing beam-splitter, interference filters were used to selectfluorescence at 455 nm, 852 nm, or 894 nm. Two differ-ent polarizing beam splitters were used, depending on thewavelength of the fluorescence radiation being observed:one was used for observations at 852 and 894 nm, whileanother was used for observations at 455 nm. The pho-todiode signals were amplified and recorded separatelyon an Agilent DSO5014A oscilloscope. To balance theamplifiers of the two photodiodes, the laser beam polar-ization was turned in such a way that the polarizationvector of the laser radiation was parallel to the magneticfield. The difference signal ( I x − I y ) in this case shouldbe zero when the amplifications of the photodiodes areproperly balanced. Differences in sensitivity to unpolar-ized light and electronic offsets present in the absence ofany light were also checked and taken into account.The cross-section of the laser beam determines thetransit relaxation rate, and it was 3.2 mm . The beamcross-section was determined by considering the area ofthe beam where the power density was within 50% of themaximum power density. The beam profile, which wasapproximately circular, was characterized by means ofa Thorlabs BP104-VIS beam profiler. Different powerswere selected by means of neutral density filters. Unlessotherwise specified, the results presented in this articlewere obtained with a laser beam whose cross-sectionalarea was 3.2 mm and whose total laser power was 40mW. For some experiments, diminished laser powers wereobtained using neutral density filters: 10 mW, 2.5 mW,and 0.625 mW. The signal background was determinedby tuning the laser away from the resonance. No addi-tional background from scattered laser-induced fluores-cence (LIF) was taken into account in the analysis. FIG. 2. Experimental geometry. The relative orientation ofthe laser beam ( exc ), laser light polarization ( E exc ), magneticfield ( B ), and observation direction ( obs ) are shown. I x and I y are the linearly polarized components of the LIF intensity. III. THEORETICAL MODELA. Outline of the model
In order to build a model of the nonlinear Hanle ef-fect in alkali atoms confined to a cell, we used the den-sity matrix of an atomic ensemble. The diagonal ele-ments of the density matrix ρ ii of an atomic ensembledescribe the population of a certain atomic level i , andthe non-diagonal elements ρ ij describe coherences cre-ated between the levels i and j . In our particular casethe levels in question are magnetic sublevels of a certainhfs level. If atoms are excited from the ground state hfslevel g to the excited state hfs level e , then the densitymatrix consists of elements ρ g i g j and ρ e i e j , called Zee-man coherences, as well as ”cross-elements” ρ g i e j , calledoptical coherences.The time evolution of the density matrices is describedby optical Bloch equations (OBEs), which can be writtenas [18, 19]: i ~ ∂ρ∂t = h b H, ρ i + i ~ b Rρ, (1)where the operator b R represents the relaxation matrix.If an atom interacts with laser light and an external dc magnetic field, the Hamiltonian can be expressed as b H = b H + b H B + b V . b H is the unperturbed atomicHamiltonian, which depends on the internal atomic coor-dinates, b H B is the Hamiltonian of the atomic interactionwith the magnetic field, and b V = − b d · E ( t ) is the interac-tion operator with the oscillating electric field in dipoleapproximation, where b d is the electric dipole operatorand E ( t ), the electric field of the excitation light.When using the OBEs to describe the interaction ofalkali atoms with laser radiation in the presence of a dc magnetic field, we describe the light classically as a timedependent electric field of a definite polarization e : E ( t ) = ε ( t ) e + ε ∗ ( t ) e ∗ (2) ε ( t ) = | ε ω | e − i Φ( t ) − i ( ω − k ω v ) t , (3)where ω is the center frequency of the spectrum and Φ ( t )is the fluctuating phase, which gives the spectrum a finitebandwidth. In this model the line shape of the excitinglight is assumed to be Lorentzian with line-width ∆ ω .As each atom moves with a particular velocity v , it ex-periences a shift ω − k ω v in the laser frequency due tothe Doppler effect, where k ω is the wave vector of theexcitation light. The treatment of the Doppler effect isdescribed in Sec. III B.The matrix elements of the dipole operator b d that cou-ple the i sublevel with the j sublevel can be written as: d ij = h i | b d · e | j i . In the external magnetic field, sublevelsare mixed so that each sublevel i with magnetic quantumnumber M labeled as ξ is a mixture of different hyperfinelevels | F M i with mixing coefficients C i,F,M : | i i = | ξM i = X F C i,F,M | F M i . (4)The mixing coefficients C i,F,M are obtained as the eigen-vectors of the Hamiltonian matrix of a fine structure statein the external magnetic field. Since some of the upperstates in the cascade system have rather small hyperfinesplittings, the mixing of magnetic sublevels can be rathersignificant. For example, at a magnetic field of 7 Gauss,the mixing coefficients for the 7 P / state are on the orderof 10%, while in the 5 D / state the magnetic sublevelsare fully mixed.The dipole transition matrix elements h F k M k | d · e | F l M l i should be expanded further using angular mo-mentum algebra, including the Wigner – Eckart theoremand the fact that the dipole operator acts only on theelectronic part of the hyperfine state, which consists ofelectronic and nuclear angular momentum (see, for ex-ample, Refs. [19, 20]). B. Rate equations
The rate equations for Zeeman coherences are devel-oped by applying the rotating wave approximation tothe optical Bloch equations with an adiabatic elimina-tion procedure for the optical coherences [18] and thenaccounting realistically for the fluctuating laser radiationby taking statistical averages over the fluctuating lightfield phase ( the decorrelation approximation ) and assum-ing a specific phase fluctuation model: random phasejumps or continuous random phase diffusion. As a resultwe arrive at the rate equations for Zeeman coherences forthe ground and excited state sublevels of atoms [21]. Inapplying this approach to a case in which atoms are ex-cited only in the finite region corresponding to the laserbeam diameter, we have to take into account transit re-laxation.In Ref. [21], only resonant excitation at the D lines wasconsidered with one ground state and one excited state.In the case of the cascade transitions considered here wehave more than two states, and so they are denoted asfollows (see Fig. 1): the 6S / state is denoted by ’[0]’ and the part of the density matrix related to this level isrepresented by ρ [0] . The 7P / state is denoted by ’[1]’and the part of the density matrix that corresponds toit as ρ [1] . Similarly, the 6P / state is indicated by ’[2]’,the 6P / state by ’[3]’, the 7S / state by ’[4]’, the 5D / state by ’[5]’ and the 5D / state by ’[6]’. If the abovetreatment of the OBEs is applied to the level scheme indiscussion, we obtain the following rate equations: ∂ρ [0] g i g j ∂t = − iω g i g j ρ [0] g i g j − γρ [0] g i g j + X e k e m Γ [1] e k e m [0] g i g j ρ [1] e k e m + X e k e m Γ [2] e k e m [0] g i g j ρ [2] e k e m + X e k e m Γ [3] e k e m [0] g i g j ρ [3] e k e m ! + | ε ω | ~ X e k ,e m (cid:18) R + i ∆ e m g i + 1Γ R − i ∆ e k g j (cid:19) d ∗ g i e k d e m g j ρ [1] e k e m − | ε ω | ~ X e k ,g m (cid:18) R − i ∆ e k g j d ∗ g i e k d e k g m ρ [0] g m g j + 1Γ R + i ∆ e k g i d ∗ g m e k d e k g j ρ [0] g i g m (cid:19) + λδ ( g i , g j ) (5) ∂ρ [1] e i e j ∂t = − iω e i e j ρ [1] e i e j − (cid:16) γ + Γ [1] (cid:17) ρ [1] e i e j + (0)+ | ε ω | ~ X g k ,g m (cid:18) R − i ∆ e i g m + 1Γ R + i ∆ e j g k (cid:19) d e i g k d ∗ g m e j ρ [0] g k g m − | ε ω | ~ X g k ,e m (cid:18) R + i ∆ e j g k d e i g k d ∗ g k e m ρ [1] e m e j + 1Γ R − i ∆ e i g k d e m g k d ∗ g k e j ρ [1] e i e m (cid:19) , (6) ∂ρ [2] f i f j ∂t = − iω f i f j ρ [2] f i f j − (cid:16) γ + Γ [2] (cid:17) ρ [2] f i f j + X e k e m Γ [4] e k e m [2] f i f j ρ [4] e k e m + X e k e m Γ [6] e k e m [2] f i f j ρ [6] e k e m ! , (7) ∂ρ [3] f i f j ∂t = − iω f i f j ρ [3] f i f j − (cid:16) γ + Γ [3] (cid:17) ρ [3] f i f j + X e k e m Γ [4] e k e m [3] f i f j ρ [4] e k e m + X e k e m Γ [5] e k e m [3] f i f j ρ [5] e k e m + X e k e m Γ [6] e k e m f i f j ρ [6] e k e m ! , (8) ∂ρ [4] f i f j ∂t = − iω f i f j ρ [4] f i f j − (cid:16) γ + Γ [4] (cid:17) ρ [4] f i f j + X e k e m Γ [1] e k e m [4] f i f j ρ [1] e k e m , (9) ∂ρ [5] f i f j ∂t = − iω f i f j ρ [5] f i f j − (cid:16) γ + Γ [5] (cid:17) ρ [5] f i f j + X e k e m Γ [1] e k e m [5] f i f j ρ [1] e k e m , (10) ∂ρ [6] f i f j ∂t = − iω f i f j ρ [6] f i f j − (cid:16) γ + Γ [6] (cid:17) ρ [6] f i f j + X e k e m Γ [1] e k e m [6] f i f j ρ [1] e k e m . (11)Here g i denotes the ground state ’0’ magnetic sublevel, while e i and f i denote magnetic sublevels of states ’1’, ’2’,’3’, ’4’, ’5’, or ’6’ according to the associated index, with e i always referring to the level with higher energy. For ex-ample, f i in the expresion ρ [5] f i f j belongs to level ’5’. Theterm, ∆ ij = ¯ ω − k ¯ ω v − ω ij expresses the actual laser shiftaway from the resonance for transitions between levels | i i and | j i for atoms moving with velocity v . The total re-laxation rate Γ R is given by Γ R = Γ [1] + ∆ ω + γ , whereΓ [ k ] is the relaxation rate of the level ’k’, γ is the transitrelaxation rate, and λ is the rate at which ”fresh” atoms move into the interaction region. The rate γ can be esti-mated as 1 / (2 πτ ), where τ is time it takes for an atom tocross the laser beam at the mean thermal velocity v th . Itis assumed that the atomic equilibrium density outsidethe interaction region is normalized to 1, which leads to λ numerically equal to γ , since λ = γn , where n is thedensity of atoms. The term Γ e i e j f i f j is the rate at whichexcited state population and coherences are transferredto the lower state as a result of spontaneous transitionsand it is obtained as follows [19]:Γ e i e j f i f j = Γ s ( − F e − M ei − M e (2 F f + 1) X q (cid:18) F e F f − M e i q M f j (cid:19) (cid:18) F e F f − M e j q M f i (cid:19) (12)If the system is closed, all excited state atoms re-turn to the initial state through spontaneous transitions, P e i f j Γ [ s ] e i e i [ r ] f j f j = α ( s, r )Γ [ s ] . where α ( s, r ) is the branchingratio of spontaneous emission from level ’s’ to level ’r’.Furthermore, P r α ( s, r ) = 1.Equations (5)—(11) describe the time evolution of theparts of the density matrix for states [ i ] = [0]—[6], re-spectively. The first term on the right-hand side of eachequation describes the destruction of the Zeeman coher-ences due to magnetic sublevel splitting in an externalmagnetic field ω ij = ( E i − E j ) / ~ . The second term char-acterizes the effects of the transit relaxation rate ( γ [ i ] )and the spontaneous relaxation rate (Γ [ i ] ), with the lat-ter being absent for the ground state ’[0]’. The next termshows the transfer of population and coherences from theupper state [ j ] to the state [ i ] described by a particu-lar equation due to spontaneous transitions; this term isequal to zero in equation (6), which describes the ’[1]’level, as no levels above this one are excited. For equa-tions (5) and (6) the fourth term describes the popula-tion increase in the level due to laser-induced transitions,while the fifth term stands for the population driven awayfrom the state via laser-induced transitions. Finally, thesixth term in equation (5) describes how the populationof ”fresh atoms” is supplied to the initial state from thevolume outside the laser beam in a process of transit re-laxation.For a multilevel system that interacts with laser radi-ation, we can define the effective Rabi frequency in theform Ω = | ε ω | ~ h J e k d k J g i , where J e is the angular mo-mentum of the excited state ’1’ fine structure level, and J g is the angular momentum of the ground state ’0’ finestructure level. The influence of the magnetic field ap-pears directly in the magnetic sublevel splitting ω ij andindirectly in the mixing coefficients C i,F k ,M i and C j,F l ,M j of the dipole matrix elements d ij .We look at quasi-stationary excitation conditions so that ∂ρ [0] g i g j /∂t = ∂ρ [1] e i e j /∂t = ∂ρ [2] f i f j /∂t = ∂ρ [3] f i f j /∂t = ∂ρ [4] f i f j /∂t = ∂ρ [5] f i f j /∂t = ∂ρ [6] f i f j /∂t = 0.By solving the rate equations as an algebraicsystem of linear equations for ρ [0] g i g j and ρ [1] e i e j , ρ [2] e i e j , ρ [3] e i e j , ρ [4] e i e j , ρ [5] e i e j , ρ [6] e i e j we obtain the matrix of pop-ulations and Zeeman coherences for all levels involved(’0’—’6’). This matrix allows us to obtain immediatelythe intensity of the observable fluorescence characterizedby the polarization vector ˜e [19, 20]. Fluorescence that istransmitted from the excited-state level ’ i ’ to the ground-state level ’ j ’ is obtained as: I [ i ] ( ˜e ) = ˜ I i ] X g i ,e i ,e j d ( ob ) ∗ g i e j d ( ob ) e i g i ρ [ i ] e i e j , (13)where ˜ I i ] is a proportionality coefficient. The dipoletransition matrix elements d ( ob ) e i g j characterize the dipoletransition from the excited state e i to some ground state g j for the transition on which the fluorescence is observed.To calculate the fluorescence produced by an ensembleof atoms, we have to treat the previously written expres-sion for the fluorescence as a function of both the polar-ization vector of the fluorescence and the atomic velocity, I [ i ] ( ˜e ) = I [ i ] ( ˜e , k ω v ) and average it over the Doppler pro-file while taking into account the different velocity groups k ω v with their respective statistical weights. If the un-polarized fluorescence without discrimination of the po-larization or frequency is recorded, one needs to sum thefluorescence over the two orthogonal polarization compo-nents and all possible final state hfs levels. C. Model parameters
In order to perform theoretical simulations with themethods described in the previous section, a numberof theoretical parameters and atomic constants hadto be used. Some parameters are known rather pre-cisely. Thus, the hyperfine splitting constants for atomiclevels involved in the cesium D lines were obtainedfrom Ref. [22], while the magnetic dipole and electricquadrupole constants for the remaining energy levelswere taken from Ref. [23]. The natural linewidths forlevels not involved in the D lines were obtained fromRef. [24]. The branching ratios for the cascade transi-tions are available from the NIST database [25, 26] andin Ref. [27].For the parameters that were related to the experi-mental conditions, we used reasonable estimates basedon measurements of the laser beam parameters and ourprevious experience. The transit relaxation rate is theinverse of the mean time that an atom spends in thelaser beam as it moves chaotically in the vapor cell witha thermal velocity. For a laser beam diameter of 2 mm(full width at half maximum of the intensity profile) androom temperature (293K), we used a value of 0.02 MHz.To estimate the Rabi frequency to be used in the simu-lations, we calculated the saturating laser power densityfor the excitation transition using its natural linewidthand then related this value to the power densities usedin the experiments. The saturating laser power densityis the laser power density at which the stimulated emis-sion equals the spontaneous decay rate, and it can beobtained from the formula [28]: I sat = 4 hcλ eg Ω sat Γ [1] In such a way the saturating Rabi frequency Ω sat in ourexperiment was estimated to be about 5 MHz. In the cal-culations, the results were averaged over the Doppler pro-file with the appropriate weighting factor and a step-sizeof 2.5 MHz. Another parameter that had to be estimatedwas the laser frequency. For the experiment, the refer-ence frequency for a given transition was the frequencyat which maximum fluorescence was observed.
IV. RESULTS AND DISCUSSION
Both orthogonal, linearly polarized polarization com-ponents I x,y were recorded in all experiments. To visual-ize the data, three quantities were considered: the unpo-larized fluorescence ( I x + I y ), the polarized fluorescence( I x and I y ), and the polarization degree [( I x − I y ) / ( I x + I y )]. A. Unpolarized Fluorescence
Figure 3 displays results obtained by exciting the 7 P / state from the F g = 3 ground-state level and observ-ing direct, unpolarized fluorescence to the ground statefrom the 7 P / state as well as fluorescence to the groundstate from the 6 P / ( D line) and 6 P / ( D line) states,which had been populated by cascades transitions. In all cases, a narrow, dark, Hanle-type resonance was ob-served. The experiment showed, and theoretical calcula-tions confirmed, that the shape of the resonance does notdepend on which fluorescence line to the ground state isobserved [see Fig. 3(a)] and Fig. 3(b)]. In fact, the threeexperimental curves in Fig. 3(a) are practically indistin-guishable, and the same is true for the three theoreticalcurves in Fig. 3(b). Fig. 3(c) shows the unpolarized flu-orescence intensity observed from the 6 P / state popu-lated via cascades versus magnetic field. Experimentalmeasurements and results from calculations are shownon the same plot. Because the calculations are extremelytime-consuming, it was not possible to vary the modelparameters in order to find those experimental parame-ters that could not be measured directly. Nevertheless,by making reasonable estimates of the parameter valuesbased on the experience gained in Refs. [16, 17], it waspossible to obtain almost perfect agreement between ex-periment and theory (see Sec. III C). The theoreticalcalculation assumed that the laser frequency was tunedto the F g = 3 → F e = 3 transition.Figure 3(d) shows the measured resonance contrastsfor each observed fluorescence line as a function of laserpower density. Similar to the case of nonlinear magneto-optical resonances in D line excitation, the contrast in-creased with increasing laser power density up to somemaximum and then decreased (see, for example, Fig. 8in Ref. [17]).Figure 4 shows similar results as Fig. 3, except that inFig. 4, the atoms were excited from the F g = 4 ground-state level. One notable difference with the case of exci-tation from the F g = 3 level is that the resonance shapesdid depend on which fluorescence line was observed (seeFig. 4(a). The theoretical calculations in Fig. 4(b) con-firm that the resonance shape, in particular the contrast,depends on the fluorescence line that is observed. Thetheoretical calculations do not reproduce the experimen-tal signals extremely well at fields larger than severalGauss. However, the theoretical curve in Fig. 4(c) de-scribes quite well the narrow portion of the resonance upto a magnetic field of up to about ± F g = 4 → F e = 4 transition. B. Polarized Fluorescence
Figure 5 depicts as a function of magnetic field the in-tensities of two orthogonally polarized fluorescence com-ponents from the de-excitation of the 7 P / state directlyto the ground state for various values of the laser detun-ing. The value of zero detuning in Fig. 5(a) was cho-sen to correspond to the frequency at which the value ofthe observed fluorescence intensity was at a maximum.As can be seen in Fig. 5(b), the shape of the measuredresonances at magnetic field values up to several Gausswas not very sensitive to detuning. However, the con-trast of the narrow nonlinear magneto-optical resonance -4 -3 -2 -1 0 1 2 3 40,800,850,900,951,00 Observation: 455nm Observation: 852nm ( D line ) F l uo r e sc en c e I n t en s i t y ( a r b . un i t s ) Magnetic field (G)
Observation: 894nm ( D line ) (a) Experiment -4 -3 -2 -1 0 1 2 3 40,850,900,951,00 F l uo r e sc en c e I n t en s i t y ( a r b . un i t s ) Magnetic field (G)
Observation: 852nmObservation: 894nm Observation: 455nm (b)
Theory -4 -3 -2 -1 0 1 2 3 4
Theory F l uo r e sc en c e I n t en s i t y ( a r b . un i t s ) Magnetic field (G)
Experiment (c)
Observation: 894 nm( D line)
10 100 10000,060,080,100,120,140,160,180,20
Observation: 455nmObservation: 894nm ( D line) Observation: 852nm ( D line) C on t r a s t Laser power density (mW/cm ) (d) FIG. 3. (Color online) Intensity of the non-polarized fluorescence to the ground state versus magnetic field for excitationof the 6 S / ( F g = 3) → P / transition at 455 nm. (a) Observed fluorescence from the 7 P / (direct), 6 P / (cascade),and 6 P / (cascade) states. The three curves are practically indistinguishable. (b) Theoretical calculations corresponding tothe observations in (a). The three curves almost coincide. (c) Experimental observation and theoretical calculation of thefluorescence from the 6 P / state ( D transition). (d) Observed contrast as a function of the laser power density for thefluorescence from the three levels mentioned in (a). in I x near zero magnetic field decreased noticeably as thelaser detuning was scanned from -300 to +300 MHz. Thereason is that at larger detuning the radiation tends toexcite more strongly those transitions in which the totalground-state angular momentum F g is less than the to-tal excited state angular momentum F e . Resonances attransitions with F g < F e should be bright rather thandark [29, 30]. Fig. 5(c) shows theoretical calculations forthe fluorescence observed directly from the 7 P / state,with the assumption that the laser is tuned exactly to thefrequency of the F g = 4 → F e = 4 transition. The darkresonance that was observable in I x in Fig. 5(b) is not ap-parent in the theoretical calculations. One can supposethat the exact frequency of the F g = 4 → F e = 4 tran-sition lies closer to the laser frequency that was detunedby +300 MHz from the frequency of maximum fluores-cence, than to the laser frequency detuned by -300 MHz,but the calculations are too time-consuming to verify thissupposition.Figure 6 shows the intensity versus magnetic field for two orthogonally polarized fluorescence components mea-sured from the 6 P / level, which was populated from the7 P / level via cascades, for various values of the laserdetuning. As in the previous figure, the value of zero de-tuning [Fig. 6(a)] was chosen for the laser frequency thatcorresponded to the maximum observed fluorescence in-tensity. Figure 6(b) shows the results for detunings of-375 and +375 MHz. It is interesting to note that ata detuning of -375 MHz a narrow dark resonance is ob-served in I x , whereas at a detuning of +375 MHz theresonance is bright. As the detuning is increased, thetransitions that are excited tend more towards transi-tions with F e > F g , which is the criterion for a brightresonance. Figure 6(c) shows the results of theoreticalcalculations made with the assumption that the laser istuned exactly to the F g = 4 → F e = 4 transition. Theagreement between the curves in Fig. 6(c) and the ex-periment is not too good, but it is qualitatively correct.In particular, both the theoretical curve for zero detun-ing and the experimental curves for detuning of +375 -4 -3 -2 -1 0 1 2 3 40,920,940,960,981,001,021,04 Observation: 455nm
Observation: 852nm ( D line ) F l uo r e sc en c e I n t en s i t y ( a r b . un i t s ) Magnetic field (G)
Observation: 894nm ( D line ) (a) Experiment -4 -3 -2 -1 0 1 2 3 40,940,960,981,001,02 F l uo r e sc en c e I n t en s i t y ( a r b . un i t s ) Magnetic field (G)
Obs.: 894nmObs.: 852nm Obs.: 455nm (b)
Theory -4 -3 -2 -1 0 1 2 3 40,940,960,981,001,021,04
Theory F l uo r e sc en c e I n t en s i t y ( a r b . un i t s ) Magnetic field (G)
Experiment (c)
Observation: 894 nm ( D line ) FIG. 4. (Color online) Intensity of the non-polarized fluorescence to the ground state versus the magnetic field for excitationof the 6 S / ( F g = 4) → P / transition at 455 nm. (a) Observed fluorescence from the 7 P / (direct), 6 P / (cascade), and6 P / (cascade) states. (b) Theoretical calculations corresponding to the observations in (a). (c) Experimental observation andtheoretical calculation of the fluorescence from the 6 P / state ( D transition). MHz from the frequency of maximum fluorescence showa bright resonance in I x . C. Degree of Polarization of the Fluorescence
The degree of polarization of the fluorescence ( I x − I y ) / ( I x + I y ) is plotted as a function of the magneticfield when the 7 P / state was excited from the ground-state level with total angular momentum F g = 3 and flu-orescence was observed back to the ground state directlyfrom the 7 P / state [Fig. 7(a)] or from the 6 P / state[Fig. 7(b)]. The agreement between theory and experi-ment in this case is quite good over a range of magneticfield values from − P / state suggeststhat at zero magnetic field there should have been a smallfeature with negative second derivative, which was notobserved in the experiment.Figure 8 shows the polarization degree of the fluores-cence as a function of magnetic field when the 7 P / statewas excited from the F g = 4 ground-state level. The re- sults for fluorescence observed from the 7 P / state to theground state are shown in Fig. 8(a), while Fig. 8(b) de-picts the results for fluorescence observed from the 6 P / state populated from above via cascade transitions. Re-sults from experimental observations are compared withthe results of theoretical calculations. In general, thetheoretical calculations qualitatively describe the exper-imentally measured curves, although the agreement inthe contrast is not very precise. The inevitable depolar-ization of the experimentally measured signals was nottaken into account in the theoretical calculations.The polarization degree ( I x − I y ) / ( I x + I y ) in Fig. 7and Fig. 8 is related to the polarization moments, whichappear in the multipole expansion of the density matrix(see, for example, Ref. [19]). By comparing the verticalscales of Fig. 7(a) and Fig. 7(b), one finds that the po-larization degree observed from the 7 P / level was anorder of magnitude higher than the that observed fromthe 6 P / level when the 7 P / level was excited from theground-state sublevel with F g = 3 and the 6 P / statewas populated by spontaneous cascade transitions fromthe 7 P / state through various intermediate states. It -6 -4 -2 0 2 4 60,750,800,850,900,951,001,05 I y I x F l uo r e sc en c e I n t en s i t y ( a r b . un i t s ) Magnetic field (G)
Experiment (a) fluorescence at 455 nm -6 -4 -2 0 2 4 60,750,800,850,900,951,001,05 fluorescence at 455 nm + 300 MHz- 300 MHz I y I x F l uo r e sc en c e I n t en s i t y ( a r b . un i t s ) Magnetic field (G) + 300 MHz (b)
Experiment -6 -4 -2 0 2 4 60,750,800,850,900,951,001,051,10 fluorescence at 455 nm I y I x F l uo r e sc en c e I n t en s i t y ( a r b . un i t s ) Magnetic field (G)
Theory (c)
FIG. 5. (Color online) Intensities of the orthogonally polarized components I x and I y of the fluorescence from the 7 P / stateto the ground state versus the magnetic field for excitation from the F g = 4 ground-state level for various laser detunings fromthe frequency of maximum observed fluorescence intensity. The intensities of the fluorescence with polarization vector parallelor perpendicular to the polarization vector of the exciting laser radiation are denoted as I x and I y , respectively (see Fig. 2).(a) Observations with the exciting laser (455 nm) tuned to the frequency that gave maximum fluorescence (0 detuning). (b)Observations with the exciting laser detuned by ±
300 MHz from the frequency at which the observed fluorescence intensitywas at a maximum. (c) Theoretical calculations for the laser tuned exactly to the F g = 4 → F e = 4 transition. should be noted that the reduction in polarization degreedepends on the external magnetic field and the shapeof the plot of polarization degree versus magnetic fieldmarkedly differs in Fig. 7(a) and Fig. 7(b). In the caseof excitation from the ground-state sublevel with F g = 4,the polarization degree observed from the 6 P / state isalso smaller than the polarization degree observed fromthe 7 P / state, but only by a factor of three. In this case,the shape of the plots in Fig. 8(a) and Fig. 8(b) do notdiffer as dramatically as in the case of excitation from theground-state sublevel with F g = 3, although the peak inFig. 8(b) is narrower than the peak in Fig. 8(a). The rea-sonable agreement between experimental measurementsand theoretical calculations in these figures suggests thatthe density matrix is well known in this system of manylevels connected by spontaneous cascade transitions.It should be noted that the transfer of polarizationfrom one level to another has been studied theoreticallyin Ref. [31]. There it was shown that the maximum po- larization moment rank κ of an excited state with unre-solved hyperfine structure that can be observed throughfluorescence is κ ≤ J e . In particular, this means thatthe polarization degree observed from the D line shouldbe zero, since a non-zero polarization degree in the fluo-rescence would imply a polarization rank κ = 2. In fact,we measured it to be zero at zero magnetic field and lessthan 0.7% over the range of magnetic field values from-7G to 7G. V. CONCLUSION
The ground-state, nonlinear magneto-optical reso-nances have been observed in the fluorescence from the7 P / state populated by linearly polarized 455 nm laserradiation and from the 6 P / and 6 P / states populatedvia cascade transitions from the 7 P / state. A theo-retical description of these effects has been furnished and0 -6 -4 -2 0 2 4 60,840,860,880,900,920,940,960,981,001,02 F l uo r e sc en c e I n t en s i t y ( a r b . un i t s ) Magnetic field (G)
Experiment I x I y (a) Observation at 852 nm( D ) -6 -4 -2 0 2 4 60,860,880,900,920,940,960,981,00 Observation at 852 nm( D ) + 375 MHz - 375 MHz F l uo r e sc en c e I n t en s i t y ( a r b . un i t s ) Magnetic field (G)
Experiment + 375 MHz I y I x (b) -6 -4 -2 0 2 4 60,840,860,880,900,920,940,960,981,001,02 Theory F l uo r e sc en c e I n t en s i t y ( a r b . un i t s ) Magnetic field (G) I x I y (c) FIG. 6. (Color online) Intensities of the orthogonally polarized components I x and I y of the fluorescence from the 6 P / stateto the ground state ( D ) versus the magnetic field for excitation from the F g = 4 ground-state level for various laser detuningsfrom the frequency of maximum observed fluorescence intensity. (a) Observations with the exciting laser (455 nm) tuned tofrequency that gave maximum fluorescence intensity. (b) Observations with the exciting laser detuned by ±
375 MHz fromthe frequency at which the observed fluorescence intensity was at a maximum. (c) Theoretical calculations for the laser tunedexactly to the F g = 4 → F e = 4 transition. -4 -3 -2 -1 0 1 2 3 40,000,050,100,150,20 ( I x - I y ) / ( I x + I y ) Observation: 455nm ( F g = 4) Magnetic field (G)
TheoryExperiment (a) -4 -3 -2 -1 0 1 2 3 40,000,020,040,060,080,10
Theory
Observation: D ( F g = 4 ) ( I x - I y ) / ( I x + I y ) Magnetic field (G)
Experiment (b)
FIG. 7. (Color online) Polarization degree of the fluorescence [( I x − I y ) / ( I x + I y )] for excitation of the 7 P / state from the F g = 3 ground-state level as a function of the magnetic field. (a) Experiment and theory for observation of the fluorescence tothe ground state from the 7 P / state. (b) Experiment and theory for observation of the fluorescence to the ground state fromthe 6 P / state ( D transition) populated from above via cascade transitions. -4 -3 -2 -1 0 1 2 3 40,000,020,040,060,080,100,120,14 Theory
Observation: 455nm ( F g = 3) ( I x - I y ) / ( I x + I y ) Magnetic field (G)
Experiment (a) -4 -3 -2 -1 0 1 2 3 40,0000,0050,0100,015
Theory
Observation: D ( F g = 3) ( I x - I y ) / ( I x + I y ) Magnetic field (G)
Experiment (b)
FIG. 8. (Color online) Polarization degree of the fluorescence [( I x − I y ) / ( I x + I y )] from excitation of the 7 P / state from the F g = 4 ground-state level as a function of the magnetic field. (a) Experiment and theory for observation of the fluorescence tothe ground state from the 7 P / state. (b) Experiment and theory for observation of the fluorescence to the ground state fromthe 6 P / state ( D transition) populated from above via cascade transitions. D -line excitation in alkali metal atoms andwas based on the optical Bloch equations with averag-ing over the Doppler profile. This model was modifiedto take into account the populations of all levels, includ-ing levels populated by cascade transitions. The modelalso accounted for the mixing of the magnetic sublevelsin an external magnetic field, which was significant inthe experiment for some of the higher states with smallhfs splittings. In general, the agreement between theobserved signals and the calculated curves was surpris-ingly good, especially taking into account that the exper-imental parameters were only estimated. In the future itwould be desirable to take advantage of improved algo-rithms and more powerful computers to be able to searchfor the values of the experimental parameters that couldnot be measured explicitly by varying the parameter val-ues in the model. Cascade techniques are interesting be- cause they provide a way to observe magneto-optical res-onances in fluorescence at a frequency far removed fromthe exciting laser radiation. ACKNOWLEDGMENTS
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