Angular momentum alignment-to-orientation conversion in the ground state of Rb atoms at room temperature
AAngular momentum alignment-to-orientation conversion in the ground state of Rbatoms at room temperature
A. Mozers, ∗ L. Busaite, D. Osite, and M. Auzinsh
Laser Centre, University of Latvia, Rainis Boulevard 19, LV-1586 Riga, Latvia (Dated: July 2, 2020)We investigated experimentally and theoretically angular momentum alignment-to-orientationconversion created by the joint interaction of laser radiation and an external magnetic field withatomic rubidium at room temperature. In particular we were interested in alignment-to-orientationconversion in atomic ground state. Experimentally the laser frequency was fixed to the hyperfinetransitions of D line of rubidium. We used a theoretical model for signal simulations that takes intoaccount all neighboring hyperfine levels, the mixing of magnetic sublevels in an external magneticfield, the coherence properties of the exciting laser radiation, and the Doppler effect. The experi-ments were carried out by exciting the atoms with linearly polarized laser radiation. Two oppositelycircularly polarized laser induced fluorescence (LIF) components were detected and afterwards theirdifference was taken. The combined LIF signals originating from the hyperfine magnetic subleveltransitions of Rb and Rb rubidium isotopes were included. The alignment-to-orientation con-version can be undoubtedly identified in the difference signals for various laser frequencies as wellas change in signal shapes can be observed when the laser power density is increased. We studiedthe formation and the underlying physical processes of the observed signal of the LIF componentsand their difference by performing the analysis of the influence of incoherent and coherent effects.We performed simulations of theoretical signals that showed the influence of ground-state coherenteffects on the LIF difference signal.
I. INTRODUCTION
Laser radiation by its very nature has speciallyanisotropic electric field distribution. First, the laserbeam defines a direction in space to which the vectorof the electric field of the light is always perpendicular.So it means that light electric field is always in a planeperpendicular to the beam direction. In addition, laserradiation very often is polarized. For example, if it ispolarized linearly, then there exists a direction perpen-dicular to the beam direction which defines direction oflaser radiation polarization.When such a radiation is absorbed by an ensemble ofatoms, it creates the spatial anisotropy of angular mo-mentum distribution in atoms. This angular momentumspatial distribution anisotropy has the same spatial sym-metry as the electric field vector of the exciting light.When the laser beam is linearly polarized, it createsalignment of the angular momentum of atoms in the ex-cited state. If the absorption is nonlinear, alignment iscreated in the ground state of the atoms as well. Angu-lar momentum alignment can be imagined as a double-headed arrow. If the angular momentum of the atoms isaligned along the quantization axis it is called the longitu-dinal alignment. In the case of longitudinal alignment thepopulations of magnetic sublevels with quantum number+ m F and − m F are equal, but the population is differ-ent for different | m F | states. But if the angular momen-tum is aligned perpendicularly to the quantization axis(it is called transverse alignment), it means that thereis a coherence created between magnetic sublevels with ∗ [email protected] quantum numbers that differ by ∆ m F = ±
2, for detailssee [1, 2].In a similar way we can introduce longitudinal andtransverse orientation of angular momentum. Usuallyorientation can be created by a circularly polarized laserexcitation. In the case of orientation of the angularmomentum, the spatial distribution can be representedsymbolically by a single-headed arrow, and in the caseof longitudinal orientation the magnetic sublevels withquantum numbers + m F and − m F in general have differ-ent populations. However, the case of transverse orien-tation corresponds to coherence between magnetic sub-levels with values that differ by ∆ m F = ± a r X i v : . [ phy s i c s . a t o m - ph ] J u l A magnetic field alone cannot create atomic angularmomentum orientation from an initially aligned ensem-ble. because it is an axial field that has even parity or itis symmetric under reflection in the plane perpendicularto the field direction.However, if the strength of magnetic field interactionwith an atom is comparable with hyperfine interaction inthe atom the joint action of both interactions can causealignment to orientation conversion [17].At the intermediate magnetic field strength the hy-perfine interaction can cause a nonlinear dependence ofthe energies of the magnetic sublevels on the magnitudeof the magnetic field—the nonlinear Zeeman or hyper-fine Paschen–Back effect (see Fig. 1 and Fig. 2). If, inaddition, we have such a linearly polarized excitation ra-diation that it simultaneously contains linear ( π ) andcircular ( σ ± ) polarization components in the referenceframe defined by the magnetic field direction (see Fig. 3),then ∆ m F = 1 coherences can be created, which leads tothe breaking of the angular momentum spatial distribu-tion symmetry [17] and creates the angular momentumorientation in a direction transverse to the magnetic fielddirection.AOC in an external magnetic field was first studiedtheoretically for cadmium [18] and sodium [19], and ob-served experimentally in cadmium [20] and in the D line of rubidium atoms [21]. Also the conversion in theopposite sense—conversion of an oriented state into analigned—is possible [22].It can be concluded that in general the action of exter-nal perturbations can break the symmetry of the angu-lar momenta distribution and allow the linearly polarizedexciting radiation to produce orientation, which is mani-fested by the presence of circularly polarized fluorescence.At the beginning, AOC caused by the joint action ofexternal magnetic field and internal hyperfine interactionwas studied in the rubidium atoms in their excited state.An excitation with weak laser radiation in the linear ab-sorption regime was used [17].The magnetic sublevels of the excited-state angularmomentum hyperfine levels in Rb atoms in an externalmagnetic field start to be affected by the nonlinear Zee-man effect already at moderate field strengths of severaltens of Gauss. It should be noted that at this magneticfield strength the ground-state Zeeman effect is still closeto linear.However, many practical and experimental applica-tions require excitation with higher laser power density,in which case the absorption becomes nonlinear.Detailed study of alignment-to-orientation conversionin an excited state of Rb atoms in the case of nonlinearabsorption recently was carried out in [23].Strongest alignment-to-orientation conversion happensat the magnetic field strength at which coherently ex-cited magnetic sublevel pairs undergo level crossing dueto nonlinear magnetic sublevel splitting [17, 23]. Suchlevel crossings do not happen in the ground state of al-kali atoms. For this reason usually it is not analyzed if the AOC can still happen in the ground state. At thesame time it is known, see for example [17], that althoughlevel crossing strongly enhances AOC signals, conversioncan also happen without level crossing. The only require-ment is that excitation conditions are such that betweencertain magnetic sublevels coherences can be created, forexample due to finite absorption line-width and width ofthe laser radiation spectral profile.The aim of this study is to examine AOC processesin detail and to deconvolute AOC processes caused bydifferent effects – processes in excited state, processes inthe ground state, changes in the absorption probabilitiescaused by the magnetic sublevel scanning in the externalmagnetic field and the effect that several isotopes of anatom can interact with the same laser radiation simulta-neously.Although AOC effects usually are small, especiallythose that are caused by the ground state of atoms itis important to have a clear understanding of them notonly due to theoretical interest, but also due to practi-cal and fundamental applications where they can play animportant role.Not to go into great detail, we will mention here justone, but fundamentally interesting, example.Experiments with atoms and molecules allow to con-duct a search of the permanent electric dipole moment(EDM) of the electron with high sensitivity [24].Various physical effect can contaminate EDM searchsignals and lead to systematic errors. One such sourcefor signal contamination is alignment-to-orientation con-version, see [25] and references cited therein. For thatreason ground-state level participation in AOC processesand measured signals are important to take into account.All signals obtained from experiments were analysedby a numerical theoretical model based on optical Blochequations (see section “Theoretical model”). We carriedout experiments where two oppositely circularly polar-ized LIF components and afterwards their difference wereobserved (section “Experiment”). Experimental resultsclearly show AOC happening for various laser frequenciesas well as change in signal shapes is observed when themeasurement of laser power dependence is performed. Adetailed explanation of the observed experimental signalsis provided in section “Results & Discussion”. II. THEORETICAL MODEL
Prior to the experimental measurement, we made anassessment which hyperfine transitions in Rb atoms arethe most suitable to detect the alignment-to-orientationconversion in the atomic ground state. To estimate theexpected signal strength and to analyze the experiemen-tal signals, we use the Liouville or optical Bloch equations(OBEs) for the density matrix ρ . The atomic density ma-trix will be written in a basis defined by the whole man-ifold of hyperfine levels in the ground and excited state: | ξ, F i , m F i (cid:105) , where F i refers to the quantum number of Energy / h (MHz) M a g n e t i c f i e l d ( G a u s s ) F e = E x c i t e d S t a t e m J = + F e = m J = - FIG. 1. Frequency shifts of the magnetic sublevels m F ofthe excited-state fine-structure level 5 P / as a function ofmagnetic field for Rb. Zero frequency shift corresponds tothe excited-state fine-structure level 5 P / . Energy / h (MHz) M a g n e t i c f i e l d ( G a u s s )
G r o u n d S t a t e F g = F g = m J = + m J = - FIG. 2. Frequency shifts of the magnetic sublevels m F ofthe ground-state fine-structure level 5 S / as a function ofmagnetic field for Rb. Zero frequency shift corresponds tothe ground-state fine-structure level 5 S / . the hyperfine angular momentum in the ground ( i = g )or the excited ( i = e ) state, m F i denotes the respectivemagnetic quantum number and ξ represent all the otherquantum numbers which are not essential for the currentstudy.The time evolution of the density matrix is describedby the optical Bloch equations [26] i (cid:126) ∂ρ∂t = (cid:104) ˆ H, ρ (cid:105) + i (cid:126) ˆ Rρ, (1)where ˆ H is the total Hamilton operator of the system andˆ R is the relaxation operator. The full Hamiltonian can bewritten as ˆ H = ˆ H + ˆ H B + ˆ V , where ˆ H is unperturbedsystem Hamiltonian, ˆ H B describes the interaction with the external magnetic field, and ˆ V = − ˆ d · E ( t ) is theoperator which describes atom – laser field interaction inthe electric dipole approximation. The operator includeselectric field of excitation light E ( t ) and an electric dipoleoperator ˆ d .The general OBEs (1) can be transformed into ex-plicit rate equations for the Zeeman coherences withinthe ground ( ρ g i g j ) and excited ( ρ e i e j ) states by apply-ing the rotating-wave approximation, averaging over anddecorrelating from the stochastic phases of laser radia-tion, and adiabatically eliminating the optical coherences[27–30]: ∂ρ g i g j ∂t = (cid:16) Ξ g i e m + Ξ ∗ g j e k (cid:17) (cid:88) e k ,e m d ∗ g i e k d e m g j ρ e k e m −− (cid:88) e k ,g m (cid:16) Ξ ∗ g j e k d ∗ g i e k d e k g m ρ g m g j ++Ξ g i e k d ∗ g m e k d e k g j ρ g i g m (cid:17) − iω g i g j ρ g i g j − + (cid:88) e k e l Γ e k e l g i g j ρ e k e l − γρ g i g j + λδ ( g i , g j ) (2a) ∂ρ e i e j ∂t = (cid:0) Ξ ∗ g m e i + Ξ g k e j (cid:1) (cid:88) g k ,g m d e i g k d ∗ g m e j ρ g k g m −− (cid:88) g k ,e m (cid:16) Ξ g k e j d e i g k d ∗ g k e m ρ e m e j ++Ξ ∗ g k e i d e m g k d ∗ g k e j ρ e i e m (cid:17) − iω e i e j ρ e i e j −− (Γ + γ ) ρ e i e j . (2b)The equation Eq. (2a) describes time evolution of thepopulations and Zeeman coherences in the manifold ofmagnetic sublevels of all the hyperfine levels in the atomicground state. The first term describes the repopulation ofthe ground state and the creation of Zeeman coherencesdue to induced transitions, Ξ g i e m and Ξ ∗ g j e k representsthe atom-laser field interaction strength. The matrix el-ement d g i e j = (cid:104) g i | ˆd · e | e j (cid:105) is the dipole transition matrixelement for transition between hyperfine level magneticsublevels g i and e j when excited by laser radiation withpolarization e . The second term denotes the changes inground-state Zeeman sublevel population and the cre-ation of ground-state Zeeman coherences due to light ab-sorption. The third term describes the destruction of theground-state Zeeman coherences by the external mag-netic field and hyperfine splitting, where ω g i g j is the en-ergy difference between magnetic sublevels | i (cid:105) and | j (cid:105) .The fourth term describes the repopulation and trans-fer of excited-state coherences to the ground state due tospontaneous transitions. The transtions are closed withinhyperfine structure, so that (cid:80) e i e j Γ e i e j g i g j = Γ. The fifthand sixth terms in (2a) describe the transit relaxationin the ground state. The fifth term accounts for the pro-cess of thermal motion in which atoms leave the region ofatom interaction with a spatially restricted laser beam.The rate of this process is characterized by the constant γ . It is assumed that the atomic equilibrium density ma-trix outside the interaction region is the unit matrix ˆ1divided by the total number of the magnetic sublevels n g in the ground-state hyperfine manifold. Therefore repop-ulation rate is connected to transit relaxation rate γ as λ = γ/n g , where n g is the total number of the magneticsublevels in the ground state. The transit relaxation rate γ can be roughly estimated as the inverse to the averagetime that atoms spend in the laser beam when they aretraversing it due to thermal motion.Similarly to the Eq. (2a), in Eq. (2b) the first termdenotes the changes in the excited state density matrixcaused by the light absorption, the second term describesinduced transitions to the ground state, the third standsfor the destruction of the excited-state Zeeman coher-ences in the external magnetic field and due to hyperfinesplitting, where ω e i e j is the splitting of the excited-stateZeeman sublevels. Finally, the fourth term describes thecombined rate of spontaneous decay and transit relax-ation (atoms are leaving the region where they interactwith laser radiation due to thermal motion) of the excitedstate.The atom-laser radiation interaction strength Ξ g i e j ,used in (2), is expressed as:Ξ g i e j = Ω R Γ+ γ +∆ ω + ˙ ı (cid:0) ¯ ω − k ¯ ω · v + ω g i e j (cid:1) , (3)where Ω R is the reduced Rabi frequency, Ω R being pro-portional to the laser power density. ∆ ω is the finitespectral width of the exciting radiation, ¯ ω is the centralfrequency of the exciting radiation, k ¯ ω the wave vectorof exciting radiation, and k ¯ ω v is the Doppler shift expe-rienced by an atom moving with velocity v .As far as the experiments were conducted at continu-ous wave excitation conditions, we are interested in thestationary solution of the equations (2) ∂ρ g i g j ∂t = ∂ρ e i e j ∂t = 0 , (4)reducing the differential equations (2) to the system oflinear equations. The solution of the system yields den-sity matrices for the ground and excited states.The observed fluorescence intensity of polarization e fl can be then calculated from excited state density matrixelements as I fl ( e fl ) = ˜ I (cid:88) g i ,e j ,e k d ∗ ( ob ) g i e j d ( ob ) e k g i ρ e j e k , (5)where d ( ob ) e i g j are the dipole transition matrix elements forthe radiation with specific polarization observed in a cho-sen direction. ˜ I is the constant of proportionality.The thermal motion of the atoms was accounted forby signal averaging over the thermal velocity distributionof atoms. This averaging was performed by solving theEqs. (2) for each velocity group, accounting for relativeprobability for atoms to have this velocity and averagingthe fluorescence intensity (5) over this distribution. To simulate expected signals and to fit experimentalresults, as the first approximation we estimated the val-ues of several parameters.The transit relaxation rate can be estimated from themean thermal velocity v th of the atoms projected ontothe plane perpendicular to the laser beam and from thelaser-beam diameter d : γ = v th d , (6)For d = 1400 µ m and T = 293 K we estimate γ =2 π · (0 .
019 MHz).The reduced Rabi frequency is estimated asΩ R = k R || d || · | ε | (cid:126) = k R || d || (cid:126) (cid:114) I(cid:15) nc , (7)where k R is a dimensionless fitting parameter, || d || = 4 . ea [1] is the reduced dipole matrixelement for D transition, where e is the electron chargeand a is the Bohr radius [31], I is the power density(directly related to the amplitude of the electric field | ε | ), (cid:15) is the electric constant, n is the refractive index,and c is the speed of light.In practise, the power density I is not constant acrossthe laser beam, so that the estimation of the parameter k R is not straightforward. The theoretical model uses aconstant value for power density instead of actual powerdistribution. As the power density is increased, Ω R can-not be related to the square root of the power density I by the same constant k R as for the lower power densi-ties [30, 32], if one merely assumes that the laser powerdensity distribution within the beam is Gaussian.This leads to the more complex relationship between I and Ω R which has a simple explanation. Our ex-periment was performed in the regime of nonlinear ab-sorption, which leads to strong depletion of ground-state population for large laser power densities. For lowlaser power density, the ground-state population is onlyslightly changed even at the center of the beam, wherethe light is most intense. However, when the laser poweris increased, the atoms in the center of the beam are moreactively excited, leaving a low ground-state population inthe center of the beam. When the laser power density isincreased even more, the region of population depletionexpands to the “wings” of the Gaussian power densitydistribution.Due to this spatially dependent population depletion,the main contribution to the signal for weaker laser ra-diation comes from the central parts of the laser beamwhere the power density is the highest, and the theoret-ical proportionality of Ω R to the square root of powerdensity continues to hold. However, for stronger laserradiation power density, the peripheral parts of the laserbeam, where power density is lower, start to play a largerrole in the absorption process, because ground-state pop-ulation there is more significant than at the center of thebeam. Therefore, when increasing the laser power den-sity, the different parts of the laser beam play a dominantpart in the absorption process, and it should be relatedto the Rabi frequency Ω R in the theoretical model. Weaccount for this effect by adjusting the value of the co-efficient k R in the theoretical model to achieve bettercorrespondence between the experimental measurementsand theoretical calculations.An appropriate estimate of the spectral width used inthe theoretical model was found to be ∆ ω = 2 π · (2 MHz),which is close to the value given by the manufacturer ofthe laser. III. EXPERIMENT
The experiments were performed on atomic rubidiumvapor at room temperature. The cylindrical (diameter25mm, length 25mm) Pyrex atomic vapor cell with opti-cal quality windows from Toptica AG contained the nat-ural abundance of rubidium isotopes. The atoms wereexcited with linearly polarized light with its polarizationvector E in the y-z plane and in a π/ z defined by the externalmagnetic field B as shown in Figure 3. The two cir-cularly polarized laser-induced fluorescence (LIF) com-ponents I L and I R were observed along the x -direction.The LIF passed through a pair of convex lenses whilethe discrimination between I L and I R was achieved bychanging the relative angle between the fast axis of azero-order quarter-wave plate (Thorlabs WPQ10M-780)and the polarization axis of a linear polarizer (LPVIS050-MP). These optical elements were aligned in a lens tubewhile the rotation of the linear polarizer was achieved bya rotation mount (CLR1/M). 𝐵𝐸 𝜋
𝐸𝑥𝑐.0 𝜎 𝐸𝑥𝑐. − 𝜎 𝐸𝑥𝑐. + 𝐼 𝐿 𝐼 𝑅 𝑧 𝑦 𝑥 FIG. 3. Excitation and observation geometry. The linearlypolarized excitation laser light (cid:126)E can be split into two cir-cularly polarized excitation light components σ ± Exc. and onelinearly polarized excitation light component π Exc. . An external cavity, grating-stabilized, tuneable, single-mode diode laser DL 100, produced by Toptica AG, witha wavelength of 794.98 nm (D line of Rb) and a typi-cal linewidth of a few MHz, was used in all the experi-ments. The DTC110 and DCC110 modules from TopticaAG were used for temperature and current control of the Diode Laser LinearpolarizerDiaphragmSAS WS/7 Rb LensesLinear polarizer λ/4 plate
Photodiode 𝑰 𝑳 or 𝑰 𝑹
795 nm X Z Y Electromagnet
FIG. 4. Side view of the experimental setup. The beamenters the coils at an angle of 45 ◦ with respect to the y axisin yz plane (axes shown in inset). laser. During the experiments the laser frequency wasfixed to a saturation absorption spectrum (SAS) signalcoming from another atomic rubidium vapor cell, whichwas placed in a three-layer µ -metal shield. The lockingof the laser frequency was established using the SC110and DigiLock modules and software by Toptica AG. Byusing this feedback controlled loop it was possible to lockthe laser frequency to a particular peak of the SAS, i.e.to a particular hyperfine transition.Figure 4 shows a schematic of the experimental setup.The magnetic field was applied using an electromagnetwith an iron core (diameter 10.0 cm), the separation be-tween the surfaces of the poles was 4.3 cm. The inho-mogeneity of the field in the center of the poles was esti-mated to be not more that 0.027%. The current for theelectromagnet was supplied by a KEPCO BOP20-10MLbipolar power supply and the symmetrical triangular cur-rent wave scan was generated by a function generatorfrom TTi (TG 5011). The frequency of the magneticfield scan was 2.00 mHz with a maximum scan ampli-tude resulting in a magnetic field range from − µ m as full width at half maximum (FWHM) of the Gaus-sian fit by a beam profiler (Cohorent Inc. LASERCAMHR). The laser power was adjusted by a half-wave Fres-nel rhomb retarder (FR600HM), followed by a linear po-larizer (GTH10M). This enabled us to vary laser powervalues from 10 µ W to 600 µ W tranlating into laser powerdensity from 0.36 mW/cm to 28.6 mW/cm . The LIFwas detected with a photodiode (Thorlabs SM1PD1A)which was placed at the end of and fixed into the obser-vation lens tube. The LIF from each circularly polarizedcomponent was detected independently i.e. one at a time.The signal from the photodiode was amplified by a tran-simpedance amplifier based on a TL072 op-amp (Roith-ner multiboard) with a gain of 10 , followed by a voltageamplifier with a gain of 10 . Every scan was acquiredwith the use of a digital oscilloscope Agilent DSO5014Aand transferred to a PC with a minimum of 16 scans intotal for each component.Then the experimental signals of each LIF circularlycomponent were averaged over multiple scans. To elim-inate any residual asymmetry in the signal, an averag-ing over the negative and positive values of the magneticfield was performed. When comparing the experimen-tal signals to theory, the constant background was sub-tracted, before the signals were normalized to the maxi-mum of each component. The background was measuredby blocking the laser beam and recording the signal fromthe photodiode. In data processing we allowed the back-ground value to vary for different laser power densities inorder to achieve a better agreement between the experi-ment and the theory, but the variation of the backgroundvalue never exceeded 3%, which is within the measure-ment error of the measured background value. As the LIFcomponent signals were relatively large in comparison totheir difference and circularity signals, a SavitzkyGolaysmoothing filter [33] was applied to the LIF differencesignals. IV. RESULTS & DISCUSSION I L ( I e x c = 0 . 3 6 m W / c m ) W R = 1 . 0 0 M H z D D = 0 M H z ( R b ) D D = 0 M H z ( R b ) D D = - D D = 5 0 M H z D D = - D D = 1 0 0 M H z D D = - D D = 1 5 0 M H z D D = - D D = 2 0 0 M H z D D = - D D = 2 5 0 M H z M a g n e t i c F i e l d ( G )
LIF of IL (arb.u.) - 6 - 6 - 6 - 6 - 6 - 6 - 6 R b F g = 2 fi F e = 2 LIF of IL (w/o Doppler) (arb.u.) FIG. 5. LIF of a single circularly polarized light component( I L ): black dots – experimental data; red line – theoreticaldata (left axis); colored lines – theoretical data without aver-aging over the Doppler profile (right axis). Magenta: centralvelocity group of Rb; orange: central velocity group of Rb;blue curves: negative velocity shift; green curves: positive ve-locity shift.
Let us start with the analysis of the general structureof the observed signals. In this paper as an example wewill show only one of the components ( I L ), as the dif-ferences in the two oppositely circularly polarized LIFcomponents are small and can be barely seen when thetwo observed circularly polarized LIF components aredepicted side by side. Figure 5 shows a typical resultfor the measurement of a single circularly polarized LIFcomponent, when the laser frequency was locked to the F g = 2 → F e = 2 transition of the Rb. At zero mag-netic field an initial relative minimum of the LIF signalcan be observed. The increase of the magnetic field liftsthe degeneracy and the LIF signal rises because of other
R b F g = 2 fi F e = 2 Dn (MHz) M a g n e t i c f i e l d ( G ) m F g = 2 = 0 fi m F e = 2 = 1 m F g = 2 = - fi m F e = 2 = - m F g = 2 = - fi m F e = 2 = - m F g = 2 = - fi m F e = 2 = 0 m F g = 3 = - fi m F e = 3 = - m F g = 2 = - fi m F e = 3 = - m F g = 2 = - fi m F e = 2 = - m F g = 2 = - fi m F e = 2 = - m F g = 2 = - fi m F e = 2 = -
1 (
R b ) m F g = 3 = - fi m F e = 2 = - m F g = 3 = - fi m F e = 3 = - FIG. 6. The colored lines represent the dependence of energydifference between various pairs of magnetic sublevels on mag-netic field. ∆ ν = 0 corresponds to laser frequency equal tothe F g = 2 → F e = 2 hyperfine transition of the Rb. (Onlypairs crossing ∆ ν = 0 are shown) atoms with different velocity coming into resonance. TheLIF signal starts to fall after approximately 250 G. Thediminishing of the signal is caused by the same fact as theincrease in signal, but now the contrary happens – thenonlinear Zeeman effect both for the ground and excitedstate lead to a decrease in the number of atoms that in-teract with the laser light. A pronounced feature can beseen at approximately 1500 G (and another at 2800 G) –an increase in the LIF signal. This is caused by magneticsublevels coming back into resonance with the excitationlaser radiation. In particular, one can deduce exactlywhich magnetic sublevels are the ones that are interact-ing with the laser field from Figure 6. Figure 6 showsthe dependence of energy difference ∆ ν between pairs ofmagnetic sublevels on the external magnetic field. When∆ ν is equal to 0, the energy difference between pairsof magnetic sublevels coincides with the laser frequency.Thus it can be easily deduced that the pair of magneticsublevels corresponding to the increase in LIF at 1500 Gare m F g =2 = − → m F e =3 = − Rb as well. Inorder to combine the LIF signals from both isotopes, thesignals were weighed according to the difference in iso-tope abundance and line strength [1].The width of these non-zero field structures in the ob-served signal can be attributed to the fact that the in-teracting atoms are in thermal motion, thus the Dopplereffect plays a large role in the formation of these line-shapes. The curves below the experimental and isotopi-cally combined LIF signal are the data from LIF signalsimulations, where the averaging over the Doppler pro-file was omitted and a single velocity group was selectedfrom the Doppler profile. Now the width of the shapesin the experimental data as well as in the simulated redcurve in Figure 5 can be interpreted as LIF coming fromdifferent velocity groups. The width of the narrow peaksappearing in the simulated LIF curves for single velocitygroups is related to the combined width coming from thenatural line-width and excitation laser line-width. Thedifferent relative amplitudes of these peaks in LIF sig-nals from different velocity groups are related to transi-tion probabilities between magnetic sublevels, i.e. whenan external magnetic field is applied the wave functionsof magnetic sublevels mix and their transition proba-bilities change [34]. The summation over all of theseLIF curves from different velocity groups (Doppler com-ponents) would yield the complete LIF simulated curve(Fig. 5 red).A rather counter-intuitive feature can be noticed atapproximately 1250 G. The zero velocity group LIF curve(Fig. 5 ∆ D = 0 MHz magenta) shows an increase inthe signal and in Figure 6 magnetic sublevels m F g =2 = − → m F e =2 = − ν between pairs ofmagnetic sublevels in Figure 6. For a peak to appearin the observed signal the shift in transition frequencybetween two magnetic sublevels (change in ∆ ν ) shouldbe larger than the change in the absolute value of theapplied magnetic field (change in B ). When a rather largechange in B is necessary to achieve the same change in∆ ν , the Doppler components get spread out more andthis flattens the overall signal, thus leading to a relativeminimum in the observed signal.We performed an analogous examination of all mea-sured circularly polarized LIF signals for different ex-citing laser frequencies and power densities. All experi-mentally obtained signals were fitted to simulated curves.Figure 7 shows all of the Rabi frequency values (squared)obtained from the data fitting procedure vs laser powerdensity. Different colors in Figure 7 correspond to dif-ferent laser frequencies. The data points are in goodagreement with a linear fit described by Eq. (7). We al-lowed the fitting parameter k R to vary in order to achievebetter agreement between the experiment and the theoryleading to a symmetric distribution of data points in Fig-ure 7. It can be seen that the data points are in goodagreement and within the margins of error. Neverthe-less, in closer examination the data points show a ten-dency to fall below the linear fit at laser power densitiesabove 20 mW/cm . This happens because the theoreticalmodel does not take into account the spatial distributionof the exciting optical field. The influence of differentlaser power density in different spatial positions on thefluorescence signal that causes the atoms to interact dif-ferently with the laser beam has been studied in [35].The LIF signal dependence on laser detuning was anal-ysed in terms of the difference between the two observedLIF components defined as I L − I R . We show the differ-ence signals as it depends only on the angular momentumtransverse orientation besides the other measure of ori- L i n e a r F i t F g = 3 fi F e = 2 F g = 3 fi F e = 3 F g = 2 fi F e = 3 F g = 2 fi F e = 2 W P o w e r d e n s i t y ( m W / c m ) W P o w e r d e n s i t y ( m W / c m ) FIG. 7. Dependence of the Rabi frequency squared Ω R onthe laser power density I together with a linear fit for all thefixed laser frequencies used in the experiment (colored datapoints). entation – circularity, which had much the same shapebut which was slightly influenced by the dependence onthe angular momentum alignment as well [2].Figure 8 shows the dependence of I L − I R on the ex-ternal magnetic field for different laser frequencies. Asthe laser frequency is increased (from a to d in Fig. 8),a change in lineshapes can be observed. The circu-larity signal when the laser frequency was fixed to the F g = 2 → F e = 3 transition of the Rb changes fromslightly positive ( ≈ +0 . ≈ − . ≈
4% was observed when the laser frequency wasset to the F g = 2 → F e = 2 transition of the Rb.For the purpose of this study it is important to exam-ine how the difference signal for the two circularly po-larized fluorescence components (which in chosen geome-try of excitation – observation is directly proportional tothe angular momentum transverse orientation) dependson the power density of the excitation radiation. Thisdependence serves as one of the indicators that helpsto separate the effects of the atomic excited state thatare present even at linear absorption region, from theground-state effects that are intrinsically nonlinear withrespect to the light intensity and do not manifest them-selves at weak excitation laser power density.Figure 9 shows the signal dependence on laser powerdensity for the case when the laser frequency was fixed tothe F g = 2 → F e = 3 transition of the Rb. As the laserpower is being increased, the aforementioned change ofthe sign of circularity disappears for laser power densitiesgreater than 1.78 mW/cm i.e. for all magnetic field val-ues the circularity stays negative. In order to understand I L (cid:1) I R ( I e x c = 3 . 5 7 m W / c m ) W R = 2 . 0 0 M H z R b F e = 3 fi F g = 3 M a g n e t i c F i e l d ( G )
Dfference I L (cid:1) I R ( I e x c = 3 . 5 7 m W / c m ) W R = 1 . 8 0 M H z R b F e = 3 fi F g = 2 M a g n e t i c F i e l d ( G )
Dfference I L (cid:1) I R ( I e x c = 3 . 5 7 m W / c m ) W R = 2 . 3 0 M H z R b F e = 2 fi F g = 2 Difference
M a g n e t i c F i e l d ( G ) d )c )b )
R b F g = 2 fi F e = 3I L (cid:1) I R ( I e x c = 3 . 5 7 m W / c m ) W R = 1 . 2 0 M H z Difference
M a g n e t i c f i e l d ( G ) a )
FIG. 8. The dependence of the difference between the two LIF components on the hyperfine transitions that the laser frequencywas fixed to. Laser frequency was fixed to: a) F g = 2 → F e = 3, b) F g = 2 → F e = 2, c) F g = 3 → F e = 3, d) F g = 3 → F e = 2hyperfine transition of Rb. a)-d) laser frequencies are in descending order. the root cause of the circularity lineshapes, theoreticalsimulations omitting the averaging over the Doppler pro-file were carried out for various velocity groups of theDoppler profile (Fig. 10). The pronounced peak (struc-ture) seen in Figure 9d at approximately 1500 G wouldappear to be connected with magnetic sublvels cominginto resonance with the laser light, but the LIF compo-nent signals in Figure 9a-b-c clearly show a minimum at1500 G. The origin of this non-zero circularity can beunderstood by looking at Figure 10: as different velocitygroups come into resonance some shift the transverse ori-entation of the angular momentum in the positive direc-tion and some in the negative. When the summation overall of these contributing velocity groups are combined,only the ones that were not compensated by other veloc-ity groups contribute to the observed circularity signal.The arbitrary units in both vertical axis in Figure 10 aredirectly comparable as the ones on the left correspond tothe red line, which is the LIF signal difference obtainedby adding the difference signals from separate velocitygroups multiplied by the corresponding factor from theDoppler profile.As the transverse angular momentum AOC is a coher-ent effect, we wanted to distinguish between the ground-state coherent effects and the excited-state coherent ef-fects contributing to the signal. At low Rabi frequenciesthe effect of ground-state coherence transfer was minute,and because the underlying causes for the signal shapescan be better understood by analysing the LIF signalscoming from separate velocity groups (omitting the av-eraging over the Doppler profile), we show the simulatedcurves in Figure 11 for the central velocity group (withrespect to the exciting laser frequency) with large Rabifrequency.With the aim to distinguish which features in the sig-nal are caused by ground-state coherent effects, we setthe non-diagonal density matrix elements (in Eq. (2a))to zero. We did this by increasing the relaxation rate γ non - diagonal of only these elements with the ratio of γ non - diagonal /γ diagonal = 10 with respect to the γ diagonal which is the normal transit relaxation rate experiencedby diagonal elements. This allowed us to observe the in-fluence of transfer of coherences from the ground state to the excited state. Figure 11 shows the comparison ofthe two cases of simulated LIF signals from the centralvelocity group. The red curve (in Fig. 11) corresponds tothe case when the ground-state coherent effects were setto zero whereas the black curve – when all the elementsin the density matrix experience normal relaxation.As can be seen from the differences in the two curves(Fig. 11), some features, e.g. at approximately 1300 G,persist in both curves virtually unchanged, but somefeatures experience a dramatic change e.g. features at1000 G and 1750 G indicating that these features aredirectly connected to the ground-state coherent effects.Both features show a change in the direction of angularmomentum orientation – when the ground-state coher-ences were set to zero I L < I R , while the black curveshows the signal to be I L > I R when the parameters forall effects were set to normal values. When the averag-ing over the Doppler profile is included, these featuresbecome less pronounced as the signals from different ve-locity groups compensate each other, causing the over-all signal to approach zero (much like in the analysis ofFigure 10). This is partially verified by experimentallyobserved signals – the feature at 1000 G and 1750 G (seeFig. 9f) also exhibits a tendency of increase of I L − I R signal. V. CONCLUSION
When the coherent effects in the manifold of atomicangular momentum magnetic sublevels, induced by in-teraction of atoms with laser radiation, are conceptuallydiscussed, very often the primer attention is paid to thecreation of coherent superposition of these sublevels dueto two factors. First, exctiaton light polarization compo-nents capable to excite coherently these sublevels are con-sidered and, second, transition probabilities determinedby the transition dipole moments between angular mo-mentum states are accounted for [1].In this paper we analyze in detail and show that ontop of these effects a very important role in this pro-cess is played by the magnetic scanning of the magneticsublevels in the external magnetic field. In the Paschen– f ) d ) c )b )a ) ( I L (cid:1) I R ) / ( I L + I R ) ( I e x c = 7 . 1 4 m W / c m ) W R = 2 . 3 0 M H z R b F g = 2 fi F e = 3 Circularity (%)
M a g n e t i c F i e l d ( G ) ( I L (cid:1) I R ) / ( I L + I R ) ( I e x c = 1 . 7 8 m W / c m ) W R = 1 . 0 0 M H z R b F g = 2 fi F e = 3 Circularity (%)
M a g n e t i c F i e l d ( G ) ( I L (cid:1) I R ) / ( I L + I R ) ( I e x c = 0 . 8 9 m W / c m ) W R = 0 . 7 0 M H z M a g n e t i c F i e l d ( G )
R b F g = 2 fi F e = 3 Circularity (%) I L ( I e x c = 7 . 1 4 m W / c m ) W R = 2 . 3 0 M H z LIF of IL (arb.u.) M a g n e t i c F i e l d ( G )
R b F g = 2 fi F e = 3 I L ( I e x c = 1 . 7 8 m W / c m ) W R = 1 . 0 0 M H z LIF of IL (arb.u.) M a g n e t i c F i e l d ( G )
R b F g = 2 fi F e = 3 I L ( I e x c = 0 . 8 9 m W / c m ) W R = 0 . 7 0 M H z M a g n e t i c F i e l d ( G )
R b F g = 2 fi F e = 3 LIF of IL (arb.u.) e ) FIG. 9. First row a)-c) shows the signal of a single circularly polarized LIF component I L dependence on laser power density.Second row d)-f) shows the corresponding circularity dependence on laser power density for the case when the laser frequencywas fixed to the F g = 2 → F e = 3 hyperfine transition of Rb. Black dots: experimental data; red curve: theoreticalcalculation. - 8 - 2 . 0 x 1 0 - 8 - 8 - 8 - 8 - 8
Difference (w/o Doppler) (- arb. u.) W R = 1 . 0 0 M H z D D = 0 M H z D D = - D D = + 5 0 M H z D D = - D D = + 1 0 0 M H z D D = - D D = + 1 5 0 M H z D D = - D D = + 2 0 0 M H z D D = - D D = + 2 5 0 M H z M a g n e t i c F i e l d ( G ) - 0 . 0 0 4- 0 . 0 0 3- 0 . 0 0 2- 0 . 0 0 10 . 0 0 0
R b F g = 2 fi F e = 3 ; I L (cid:1) I R Difference (with Doppler) (arb.u.)
FIG. 10. Red curve (left axis): theoretical data of the dif-ference between two circularly polarized light components( I L − I R ) with averaging over the Doppler profile. Coloredcurves (right axis): theoretical data of the difference betweentwo circularly polarized light components without averagingover the Doppler profile. Different colors represent various ve-locity groups from the Doppler profile. Magenta: central ve-locity group; blue curves: negative velocity shift; green curves:positive velocity shift. Back effect regime it leads to two effects: first, nonlin-ear magnetic sublevel splitting that can lead to angu-lar momentum spatial distribution symmetry breaking– the alignment-to-orientation conversion and, second, itcauses changes in the transition probabilities due to mag-netic sublevel mixing the magnetic field.And a second very important moment in the analysisof laser light–atom interaction is a necessity for clear sep- - 7 - 7 - 6
R b F g = 2 fi F e = 3 I L (cid:1) I R Difference (w/o Doppler) (arb.u.)
M a g n e t i c f i e l d ( G a u s s ) W R = W R = FIG. 11. I L − I R dependence on the magnetic field fromcentral velocity group – averaging over the Doppler profile isomitted. Red curve: theoretical data from Rb with Rabi fre-quency 100 MHz with large γ non − diagonal ; Black curve: the-oretical data from Rb with Rabi frequency 100 MHz withnormal γ non − diagonal . aration of incoherent (related to the populations distribu-tion of magnetic sublevels) and coherent (determined bya well defined phase relations of magnetic sublevel wavefunctions) contributions to the observed signals.In this paper we have shown that, for example, in Rbatoms used in this study, at a different magnetic fieldstrength not only different hyperfine transitions of a spe-cific isotope of an atom are coming into resonance with0laser radiation, but the same laser radiation at differentmagnetic field strength can excite hyperfine transitionsin different isotopes of rubidium atoms. This effect ap-pears due to magnetic sublevel scanning and primarily isincoherent effect, see Fig. 5 and the analysis of it.And, finally, based on the comparison of signals ob-tained in numerical model in which we are able to“switch-off” and “switch-on” different relaxation pro-cesses,we managed to get evidence that specific featuresin the observed signals are determined by the alignment-to-orientation conversion in the atomic ground state, seeFigs. 9 and 11 and analysis there. We believe thatalignment-to-orientation conversion in the ground stateof atoms has not been identified before. The clear un-derstanding of the presence of these effects is importantfor applications as well as for research in fundamental physics in the table top atomic physics experiments, forexample, in search of the permanent electric dipole mo-ment of an electron – EDM experiments. ACKNOWLEDGMENTS
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