Magnetic field--induced modification of selection rules for Rb D_2 line monitored by selective reflection from a vapor nanocell
Emmanuel Klinger, Armen Sargsyan, Ara Tonoyan, Grant Hakhumyan, Aram Papoyan, Claude Leroy, D Sarkisyan
aa r X i v : . [ phy s i c s . a t o m - ph ] F e b Magnetic field–induced modification of selection rules for Rb D line monitored by selective reflection from a vapor nanocell E. Klinger , , A. Sargsyan , A. Tonoyan , G. Hakhumyan , A. Papoyan , C. Leroy , & D.Sarkisyan Institute for Physical Research, NAS of Armenia, Ashtarak-2, 0203, Armenia Laboratoire Interdisciplinaire Carnot de Bourgogne, UMR CNRS 6303, Universit´e Bourgogne - Franche-Comt´e, BP47870, 21078 Dijon Cedex, France Institute of Physics and Technology, National Research Tomsk Polytechnic University, Tomsk 634050, Russia
Key words: Atomic physics, Intensities and shapes of atomic spectral lines, Zeeman and Starkeffects.
Abstract – Magnetic field-induced giant modification of the probabilities of five transitions of5 S / , F g = 2 → P / , F e = 4 of Rb and three transitions of 5 S / , F g = 1 → P / , F e = 3of Rb forbidden by selection rules for zero magnetic field has been observed experimentally anddescribed theoretically for the first time. For the case of excitation with circularly-polarized ( σ + )laser radiation, the probability of F g = 2 , m F = − → F e = 4 , m F = − Rb F g = 2 → F e = 1 , , , F g = 1 , m F = − → F e = 3 , m F = 0 transition becomes the largest amongthe nine transitions of Rb F g = 1 → F e = 0 , , , Introduction. –
Alkali atoms (Rb, Cs, K, Na) arewidely used in atomic spectroscopy due to strong atomictransitions from the ground state with wavelengths in vis-ible and near-infrared regions (600 – 900 nm), where cwnarrow-band smoothly tunable diode lasers are available.Application areas of rubidium atoms include laser cool-ing experiments, information storage, spectroscopy, mag-netometry, laser frequency stabilization, etc. [1, 2]. Thatis why any new knowledge of the behavior of Rb atomictransitions, particularly, exposed to an external magneticfield, is of high importance.It is well known that in quite moderate magnetic field B the splitting of atomic energy levels to Zeeman sublevelsdeviates from the linear behavior, and the atomic transi-tion probabilities undergo significant changes [2–8]. Themost simple technique to monitor and study such mod-ification is laser spectroscopy of atoms contained in anatomic vapor cell. For B up to ∼ = 1000 G, the Zeemansplit hyperfine transitions remain overlapped because ofDoppler broadening, and sub-Doppler techniques are re-quired to spectrally resolve transition probabilities of in-dividual components [3]. Coherent Doppler narrowing ina thin vapor cell has been attained in [9]. As it was demonstrated recently [10, 11], strong linenarrowing can be achieved in Derivative Selective Reflec-tion (DSR) spectra using an atomic vapor cell of nearlyhalf-wavelength thickness ( L ≈ λ/
2, where λ is the res-onant wavelength of laser radiation). These studies havebeen done for the case of D lines of Rb and Cs. Amongthe advantages of DSR technique is proportionality of therecorded signal to atomic transition probability. In addi-tion, the DSR resonance linewidth is practically immuneagainst 10% deviation of the cell thickness. These benefitsmake it convenient to use the DSR-method for studies ofclosely-spaced individual atomic transition components ina magnetic field.In the present paper we demonstrate that the DSRlinewidth for Rb D line is ∼
50 MHz FWHM (opposedto ∼
500 MHz Doppler absorption linewidth in ordinarycells), which allows frequency separation of individualZeeman components of hyperfine transitions and study-ing their transition probabilities in an external magneticfield. The DSR technique was employed to study for thefirst time, both experimentally and theoretically, the dy-namics of giant modification of transition probabilities of5 S / , F g = 2 → P / , F e = 4 and 5 S / , F g = 1 →
1. Klinger et al. R R R (SR) laser L QP 3FI SR = 780 nm Ref.
Fig. 1: Layout of the experimental setup: (ECDL) extendedcavity diode laser; (FI) Faraday isolator; (1) nanocell filledwith Rb inside the oven; (2) strong permanent magnet on atranslation stage to produce
B >
200 G field (for B = 1 – 200 G,the magnet was replaced by Helmholtz coils - not shown in thefigure); (3) photodetector; (Ref.) Auxiliary unit for formationof frequency reference spectrum; (4) oscilloscope; (SR) selectivereflection beam; (QP) quarter-wave plate. Upper left panel:photograph of the NC; interference fringes formed in the lightreflected from the inner surfaces of the windows are seen; theoval marks the thickness region 300 nm ≤ L ∼ λ/ ≤
400 nm.Upper right panel: the geometry of three beams reflected fromthe NC; the (SR) beam propagates along R . P / , F e = 3 transitions of Rb D line induced by amagnetic field varied in a wide range (up to 1000 G).These transitions are forbidden at B = 0 according to∆ F = 0 , ± Experimental. –
A nanometric-thin cell (NC) filledwith Rb has been used in our experiment, allowing toobtain sub-Doppler spectra, and thus resolving a largenumber of hyperfine and transitions and their Zeemancomponents. The general design of NC is similar to thatdescribed in [8, 12]. Compact oven was used to set theneeded temperature regime of 110 ◦ C, which correspondsto the number density of Rb atoms N ∼ = 10 cm − .Adjustment of needed vapor column thickness withoutvariation of thermal conditions was attained by smoothvertical translation of the cell-and-oven assembly.Schematic diagram of the optical part of experimentalsetup is shown in Fig. 1. A circularly polarized laser ra-diation beam ( λ = 780 nm, P L = 2 mW, ∆ ν L = 1 MHz)resonant with Rb D line was focused ( ∅ = 0 . L < λ/ ≈
300 nm. This optimumthickness was chosen to combine high spatial resolution, which is very important when using high-gradient fieldfrom permanent magnet, with acceptable broadening ofthe SR spectral linewidth (the latter increases with reduc-tion of L ). A calibrated strong permanent neodymiummagnet placed near the rear window of the NC was usedto produce strong longitudinal magnetic field controllableby changing the distance to the window. Noteworthy, ex-tremely small thickness of the NC is advantageous for ap-plication of strong magnetic field produced by a perma-nent magnet: in spite of high field gradient, the variationof B -field inside the cell is several orders less than theapplied value. To record DSR spectra, the laser radia-tion frequency was linearly scanned within up to 7 GHzspectral region covering the studied group of transitions.The nonlinearity of the scanned frequency ( ∼
1% through-out the spectral range) was monitored by simultaneouslyrecorded transmission spectra of a Fabry-P´erot etalon (notshown). About 30% of the pump power was branched to 3cm-long Rb cell providing reference saturation absorptionspectrum for B = 0. All the spectra were detected by pho-todiodes with amplifiers, and recorded by a four-channeldigital storage oscilloscope Tektronix TDS 2014B.The diagram of relevant σ + components (∆ m F = +1)of Rb D line transitions F g = 2 → F e = 3 labeled 4– 8, and F g = 2 → F e = 4 labeled 4 − − F = 0 , ±
1. Thediagram of relevant σ + components (∆ m F = +1) of Rb D line transitions F g = 1 → F e = 2 labeled 1 – 3, and F g = 1 → F e = 3 labeled 1 (cid:13) − (cid:13) are shown in Fig. 2(b).The transitions 1 (cid:13) − (cid:13) are forbidden for zero magneticfield because of selection rule ∆ F = 0 , ±
1. The transi-tions F g = 2 → F e = 1 , Rb) and F g = 1 → F e = 0 , Rb) are not shown in Fig. 2, since for
B >
300 G theirprobabilities strongly reduce, making them practically un-detectable in the DSR spectra. Note, that in the case of σ − excitation, the F g = 2 → F e = 4 transitions of Rband F g = 1 → F e = 3 transitions of Rb have smallerDSR amplitudes than for the case of σ + for the same B values (the reason of this distinction is out of scope of thispaper, and will be presented elsewhere).The recorded DSR (real time derivative selective reflec-tion) spectra of the Rb NC with thickness L ≈
300 nm for σ + laser excitation ( P L = 10 µ W) and B ≈
335 G, 450G, 580 G and 855 G are shown in Fig. 3 (red solid lines).The dashed blue lines represent the results of theoreticalmodeling (see below). The sixteen transition components(for labeling, see Fig. 2) appear with ≈
60 MHz linewidth,being very well resolved. As it is seen, up to B ≈
600 Gthe amplitudes of transitions 7 and 8 are the strongestamong the transitions of Rb, F g = 2 → F e = 1 , , , B ≈
950 G) and otherwise invariableexperimental conditions is presented by a red solid line inthe lower panel of Fig. 4, together with the modeled DSR2agnetic field induced modification F e = 2F e = 1F e = 3m F g m F e Rb +2+10 +4+3+2+10 F e = 4F g = 2 (a) F e = 1 F e = 0 F e = 2 m F g m F e Rb +10 +3+2+10 F e = 3 F g = 1 (b) Fig. 2: Diagram of the relevant transitions between the Zeeman sublevels of Rb D line with σ + (left-circular) laser excitationfor the case of (a) Rb (nuclear spin I = 5 / Rb (nuclear spin I = 3 / Laser frequency detuning (MHz)3000 4000 5000 6000 7000 8000 900000.010.020.03
B = 855 G
TheoryExperiment
B = 580 G
TheoryExperiment D S R s i gn a l ( a r b . un it s ) B = 450 G
TheoryExperiment
Laser frequency detuning (MHz)3000 4000 5000 6000 7000 8000 900000.020.040.060.08
B = 335 G
TheoryExperiment Fig. 3: Red solid lines: the DSR spectra for the Rb NC with the thickness L ≈
300 nm for σ + laser excitation ( P L = 10 µ W)experimentally recorded at 4 values of the applied B -field: 335 G, 450 G, 580 G and 855 G. For labeling, see Fig. 2. Dashedblue lines: calculated spectra. Fragmentary presentation of experimental spectra is caused by mode hop–restricted continuityof the laser frequency scanning, verified by the Fabry-P´erot cavity signal.
3. Klinger et al. spectrum (dashed blue line). All the sixteen labeled tran-sition components are completely frequency resolved (ex-cept transitions 2 and 4 , which are overlapped). The up-per panel shows the theoretical DSR spectrum calculatedfor effective broadening of Γ eff = 6 MHz correspondingto the natural transition linewidth, which is the minimumattainable width for the single laser beam experimentalconfiguration.It is well known that the effect of applied field B onthe hyperfine structure is characterized by the parameter B = A hfs /µ B , where A hfs is the hyperfine coupling con-stant for 5 S / and µ B is the Bohr magneton [14]. For Rb, B ≈
700 G, and for Rb, B ≈ B ≪ B (such fields are treated as moderate), the splittingof energy levels is described by the total momentum of theatom F = J + I and its projection m F , where J = L + S is the total momentum of the electron, and I is the spinmomentum of the nucleus. When B ≫ B , the couplingof J and I ceases, and the splitting is described by theprojections m J and m I . The latter manifests the onset ofhyperfine Paschen-Back (HPB) regime characterized by anumber of peculiarities [14–16]. Particularly, probabilitiesof the eight above mentioned transitions forbidden at B =0 reduce back to zero when B ≫ B , so they are absent inthe m J and m I basis. Note that the probabilities of Rbtransitions labeled 4 − B than thetransitions 1 (cid:13) − (cid:13) of Rb (see Fig. 5). This is caused bythe fact that B value for Rb is larger than the one for Rb, consequently HPB regime for Rb occurs at largermagnetic fields. In total 20 atomic transitions remain inthe complete HPB regime, 12 belonging to Rb, and 8belonging to Rb [16].
Theoretical model and discussion. –
Below wesummarize the main relations used to calculate the the-oretical DSR spectra. The reader is invited to read thearticles [11, 17] for more detailed information. Startingfrom the propagation equations of electric field inside thecell, one deduces the reflected signal given by S r ≈ t E in ℜ n r (cid:2) − exp( − ikL ) (cid:3) I SR o / | Q | , (1)where t and r are transmission and reflection coefficientsrespectively, Q = 1 − r exp(2 ikL ) is the quality factorassociated to the nanocell, and I SR is given in the linearregime of interaction by I SR = (cid:2) r exp(2 ikL ) (cid:3) I linSR − r exp(2 ikL ) I linT , (2)which depends on the induced polarization in the vaporthrough the integrals I linSR and I linT , respectively expressedas I linSR = C Z + ∞−∞ W ( v ) h ( ω − ω eg , Γ , L, v ) dv (3a) I linT = C Z + ∞−∞ W ( v ) g ( ω − ω eg , Γ , L, v ) dv (3b) where v is the speed of atoms in the cell, and W ( v ) is thevelocity distribution assumed to be Maxwellian, defined as W ( v ) = (1 /u √ π ) exp( − v /u ) with u the thermal velocitygiven by u ( T ) = p k B T /m ( T is the vapor temperature, k B the Boltzmann’s constant, and m the atomic mass).The expressions for h and g are given in [17], being thefunction of transition frequency ω eg , the cell thickness L ,and transition width Γ, which includes the natural Γ nat ,collisional Γ col , and inhomogenous Γ inhom broadening co-efficients. In Eqs. (3a) and (3b), C is a function of theatomic density N and the dipole moment of transitionfrom state | g i to state | e i , and reads C = N t E in ~ ǫ Q |h e | D q | g i| , (4)where q denotes the standard component of the dipoleassociated to the scanning field polarization. Simulationsof the magnetic sublevels energy and relative transitionprobabilities for Rb F g = 2 → F e = 3 , Rb F g =1 → F e = 2 , D line are known, and arebased on calculation of the dependence of eigenvalues andeigenvectors of the Hamiltonian matrix on magnetic fieldfor the whole hyperfine structure manifold [2, 3, 7].Magnetic field dependences of the frequency shifts andthe relative probabilities of 4 − Rb, as well as of 1 (cid:13) − (cid:13) and 1 – 3 transitions of Rbfor the case of σ + excitation are shown in Fig. 5. The | g i → | e i transition dipole moment for an atom interactingwith longitudinal magnetic field is proportional [3] to |h e | D q | g i| ∝ X F e ,F g c F ′ e F e a ( F e , m F e ; F g , m F g ; q ) c F ′ g F g (5)with a ( F e , m F e ; F g , m F g ; q ) = ( − I + J e + F e + F g − m Fe × p J e + 1 p F e + 1 p F g + 1 × (cid:18) F e F g − m F e q m F g (cid:19) (cid:26) F e F g J g I J e (cid:27) , (6)where the parentheses and the curly brackets denote the3- j and 6- j coefficients, respectively.Good agreement between the theory and experimentis observed throughout the whole explored range of B -field (up to 1000 G), which proves complete con-sistency of the theoretical model. It should be notedthat the calculations performed for other forbiddentransitions of Rb D line ( F g = 3 → F e = 1 of Rb and F g = 2 → F e = 0 of Rb) show just a weak modificationof the transition probabilities (several orders less thanthat for F g = 2 → F e = 4 of Rb and F g = 1 → F e = 3of Rb), so that their experimental study is restricted byextremely small signal-to-noise ratio.The results of this study can be used for developinghardware and software solutions for wide-range opticalmagnetometers with nanometric (300 nm) local spatialresolution, as well as widely tunable frequency reference4agnetic field induced modification
Laser frequency detuning (MHz)4000 5000 6000 7000 8000 9000 -0.00500.0050.010.0150.020.025 Laser frequency detuning (MHz)4000 5000 6000 7000 8000 9000 D S R s i gn a l ( a r b . un it s ) -0.01-0.00500.0050.010.015 TheoryExperiment D S R s i gn a l ( a r b . un it s ) Fig. 4: Lower panel: experimentally recorded (red solid line) and calculated (dashed blue line) DSR spectra for the Rb NCwith the thickness L ≈
300 nm for σ + laser excitation ( P L = 10 µ W) and B ≈
950 G. Upper panel: theoretical DSR spectrumcalculated with Γ eff = 6 MHz. For labeling, see Fig. 2. Magnetic field (G)0 200 400 600 800 1000 F r e qu e n c y s h i f t ( M H z ) T r a n s iti on i n t e n s iti e s ( a r b . un it s ) (a) (b) (c) Rb Rb
64 5 7 81 2 3 32 14 5 67 82 13 8 7 6 5 4 45678 123 Rb Rb Fig. 5: (a) Magnetic field dependence of frequency shifts for Zeeman components of F g = 2 → F e = 3 , Rb D line and F g = 1 → F e = 2 , Rb D line for the case of σ + excitation. (b) Evolution of probabilities for Zeemancomponents of F g = 2 → F e = 3 , Rb) versus B -field for σ + excitation. (c) Evolution of probabilities for Zeemancomponents of F g = 1 → F e = 2 , Rb) versus B -field for σ + excitation. For labeling, see Fig. 2.
5. Klinger et al. system, based on a NC and strong permanent magnets.Besides, formation of narrow (30 – 50 MHz) opticalresonances far away from the atomic transition (up to20 GHz for strong B -field) allows realization of widelytunable laser frequency lock [18, 19]. Owing to smalldivergence of selective reflection radiation beam, whichfollows the divergence of incident radiation, the DSRsignal can be easily detected at large distance ( ∼
10 m)from the nanocell, which can be used for the remoteoptical monitoring of magnetic field.
Conclusion. –
Summarizing, we have studied, bothexperimentally and theoretically, the dynamics of fre-quency shifts and modification of probabilities of individ-ual Zeeman components of Rb D line hyperfine transi-tions in an external magnetic field up to 1000 G for σ + laser excitation, employing derivative selective reflectiontechnique using L ∼
300 nm vapor nanocell. It wasshown, for the first time, that the probabilities of eighttransitions, which are forbidden for B = 0 (5 componentsfor Rb 5 S / , F g = 2 → P / , F e = 4 transitions and3 components for Rb 5 S / , F g = 1 → P / , F e = 3transitions), undergo significant increase when applyingmagnetic field. Moreover, it was revealed that for the caseof σ + excitation the intensity of F g = 2 , m F = − → F e =4 , m F = − Rb becomes the largest amongall the 25 Zeeman transitions of F g = 2 → F e = 1 , , , D lines of Cs, K and Na, as well as for applications inoptical magnetometry, metrology, and laser technologies. ∗ ∗ ∗ The authors are grateful to A. Sarkisyan for his valuablecontribution in fabrication of the NC. C.L. is grateful toProf. Oleg Ulenikov for valuable discussions. Researchconducted in the scope of the International AssociatedLaboratory IRMAS (CNRS-France & SCS-Armenia)
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