Atomic transitions of Rb, D 2 line in strong magnetic fields: hyperfine Paschen-Back regime
A. Sargsyan, A. Tonoyan, G. Hakhumyan, C. Leroy, Y. Pashayan-Leroy, D. Sarkisyan
aa r X i v : . [ phy s i c s . a t o m - ph ] M a y Atomic transitions of Rb, D line in strong magnetic fields: hyperfinePaschen-Back regime A. Sargsyan , A. Tonoyan , , G. Hakhumyan , C. Leroy , Y. Pashayan-Leroy , D. Sarkisyan Institute for Physical Research, 0203, Ashtarak-2, Armenia Laboratoire Interdisciplinaire Carnot de Bourgogne, UMR CNRS 6303, Universit´e de Bourgogne, 21078 Dijon Cedex,France
Abstract
An efficient λ/ λ is the resonant wavelength of laser radiation) based on nanometric-thicknesscell filled with rubidium is implemented to study the splitting of hyperfine transitions of Rb and Rb D lines in an external magnetic field in the range of B = 3 kG – 7 kG. It is experimentally demonstratedthat at B > B -field in Rb ( Rb) spectra in thecase of σ + polarized laser radiation there remain only 12 (8) which is caused by decoupling of the totalelectronic momentum J and the nuclear spin momentum I (hyperfine Paschen-Back regime). Note thatat B > . Rb and Rb labeled 19 and 20 (for low magnetic field they could be presented as transitions F g = 3 , m F = +3 → F e = 4 , m F = +4 and F g = 2 , m F = +2 → F e = 3 , m F = +3, correspondingly) isstressed. The experiment agrees well with the theory. Comparison of the behavior of atomic transitions for D line compared with that of D line is presented. Possible applications are described.
1. Introduction
It is well known that in an external magnetic field B the energy levels of atoms undergo splitting intoa large number of Zeeman sublevels which are strongly frequency shifted, and simultaneously, there arechanges in the atomic transition probabilities [1, 2]. Since ceasium and rubidium are widely used for in-vestigation of optical and magneto-optical processes in atomic vapors as well as for cooling of atoms, forBose-Einstein condensation, and in a number of other problems [3, 4], therefore, a detailed knowledge of thebehavior of atomic levels in external magnetic fields is of a high interest. The implementation of recentlydeveloped technique based on narrowband laser diodes, strong permanent magnets and nanometric-thicknesscell (NTC) makes the study of the behavior of atomic transitions in an external strong magnetic field simpleand robust, and allows one to study the behavior of any individual atomic transition of Rb and Rb Preprint submitted to Optics Communications June 26, 2018 toms for D line [5, 6].Recently, a number of new applications based on thin atomic vapor cells placed in a strong magnetic fieldhave been demonstrated: i) development of a frequency reference based on permanent magnets and micro-and nano-cells widely tunable over the range of several gigahertz by simple displacement of the magnet;ii) optical magnetometers with micro- and/or nano-metric spatial resolution [5, 6]; iii) a light, compactoptical isolator using an atomic Rb vapor in the hyperfine Paschen-Back regime is presented in [7, 8]; iv)it is demonstrated that the use of Faraday rotation signal provides a simple way to measure the atomic re-fractive index [9]; v) widely tunable narrow optical resonances which are convenient for a frequency lockingof diode-laser radiation [10].Strong permanent magnets produce non-homogeneous magnetic fields. In spite of the strong inhomo-geneity of the B -field (in our case it can reach 15 mT/mm), the variation of B inside the atomic vaporcolumn is by several orders less than the applied B value because of a small thickness of the cells. In caseof micrometer thin cells with the thickness L in the range of 10 − µ m the spectral resolution is limitedby the absorption Doppler line-width of an individual atomic transition (hundreds of megahertz). If thefrequency distances between Zeeman sublevels are small a big number of atomic transitions are stronglyoverlapped and it makes absorption spectra very complicated. Fortunately, as demonstrated for Cs D line, at strong ( B >
B > Rb and Rb D linesare strongly overlapped, so pure isotope Rb and 1 mm-atomic vapor cell have been used to separate eightZeeman transitions [7, 8, 9]. However with this technique even in the case of using pure isotope Rb, theatomic lines will be strongly overlapped.Although, the HPB regime was discovered many decades earlier (see Refs. in [2, 12, 13]), however theimplementation of recently developed setup based on narrowband laser diodes, strong permanent magnetsand NTC makes these studies simple and robust, and allows one to study the behavior of any individualatomic transition of Rb and Rb atoms; the simplicity of the system also makes it possible to use it fora number of applications.In this paper we present (for the first time to our best knowledge), the results of experimental and theo-retical studies of the Rb D line transitions (both Rb and Rb are presented) in a wide range of magneticfields, namely for 3 kG < B <
B > . σ + polarized laser radiation, there remain only 20 Zeeman transitions. In the absorption spectrum these 20transitions are regrouped in two separate groups each of 10 atomic transitions (HPB regime), while there2re 60 allowed Zeeman transitions at low B -field.
2. EXPERIMENTAL DETAILS
The design of a nanometric-thin cell NTC is similar to that of extremely thin cell described earlier [14].The modification implemented in the present work is as follows. The rectangular 20 mm ×
30 mm, 2 . < O ),which is chemically resistant to hot vapors (up to 1000 ◦ C) of alkali metals. The wafers are cut across the c -axis to minimize the birefringence. In order to exploit variable vapor column thickness, the cell is verticallywedged by placing a 1 . µ m-thick platinum spacer strip between the windows at the bottom side prior to glu-ing. The NTC is filled with a natural rubidium (72 . Rb and 27 . Rb). A thermocouple is attachedto the sapphire side arm at the boundary of metallic Rb to measure the temperature, which determines thevapor pressure. The side arm temperature in present experiment was 120 ◦ C, while the windows temperaturewas kept some 20 ◦ C higher to prevent condensation. This temperature regime corresponds to the Rb atomicnumber density N = 2 · cm − . The NTC operated with a special oven with two optical outlets. Theoven (with the NTC fixed inside) was rigidly attached to a translation stage for smooth vertical movement toadjust the needed vapor column thickness without variation of thermal conditions. Note, that all experimen-tal results have been obtained with Rb vapor column thickness L = λ/ Figure 1 presents the experimental scheme for the detection of the absorption spectrum of the nano-cell filled with Rb. It is important to note that the implemented λ/ L = λ/ D line ( L = λ/ ≈ ∅
50 mm permanent magnets (PM) with 3 mm holes (toallow the radiation to pass) placed on the opposite sides of the NTC oven and separated by a distance that3as varied between 40 and 25 mm (see the upper inset in Fig. 1). The magnetic field was measured by acalibrated Hall gauge. To control the magnetic field value, one of the magnets was mounted on a micrometrictranslation stage for longitudinal displacement. In the case where the minimum separation distance is of25 mm, the magnetic field B produced inside the NTC reaches 3600 G. To enhance the magnetic field up to6 kG, the two PM were fixed to a metallic magnetic core with a cross section of 40 mm x 50 mm. Additionalform-wounded Cu coils allow for the application of extra B -fields (up to ± Figure 1: Sketch of the experimental setup. DL - tunable diode laser, FI - Faraday isolator, 1 - λ/ B = 0 reference spectrum, PM - permanentmagnets, 3 - photo-detectors, 4 - metallic magnetic core. The beam with σ + circular polarization was formed by a λ/ F = 20 cm) on the NTC to create a spot size (1 /e diameter, i.e. the distance where the power drops to13 . d = 0 . L = λ/
2. The absorption spectrumof the latter at the atomic transition F g = 1 → F e = 1 , F g = 1 → F e = 0 is not well seen) [10].
3. EXPERIMENTAL RESULTS and DISCUSSIONS
B < kG In case of relatively low magnetic fields ( ∼ Rb, and 38 transitions belongingto Rb D line. These numerous atomic transitions are strongly overlapped and can be partially resolved4n case of using Rb or Rb isotope. When using natural Rb, the implementation of λ/ B ≥ D line are presented in [19]). Note, that for B ≫ B the number of allowed transitionscan be simply obtained from the diagrams shown in Fig. 4.In Fig. 2 the absorption spectrum of Rb NTC with L = λ/ B = 3550 G and σ + laser excitation is ’ Rb, m
J mJ Rb, m
J mJ Rb, m
J mJ
Laser frequency detuning (MHz) A b s o r p t i on ( a r b . un . )
1 2 ’ Rb, m
J mJFR
Figure 2: Absorption spectrum of Rb NTC with L = λ/ B = 3550 G and σ + laser excitation. The bottom curve (FR)is the absorption spectrum of the reference NTC showing the positions of Rb 1 → ′ , ′ transitions for B = 0 (the frequencyseparation is 157 MHz). In the upper corner the corresponding atomic transitions are indicated. The absolute value of thepeak absorption of the transition labeled as 1 is ∼ . shown. The laser power is 10 µ W. For Rb D line there are 20 atomic absorption resonances located at theatomic transitions. Among these transitions 12 belong to Rb, and 8 transitions belong to Rb.The atomic transition pairs labeled (19 ,
18) and (7 ,
5) are strongly overlapped (although in the caseof strongly expanded spectrum the peaks belonging to the corresponding transitions are well detected),while the other 16 transitions are overlapped partially, and the positions of the absorption peaks are wellseen. Thus, the fitting of the absorption spectrum with 20 atomic transitions is not a difficult problem.The vertical bars presented in Fig. 2 indicate the frequency positions and the magnitudes for individualtransitions between the Zeeman sublevels as given by numerical simulations using the model describedbelow. The corresponding atomic transitions presented by the vertical bars are indicated in the uppercorner of Fig. 2. 5 ’ A b s o r p t i on ( a r b . un . ) Laser frequency detuning (MHz)
Shift FR ’ (a) A b s o r p t i on ( a r b . un . ) Laser frequency detuning
10 9 8 7 6 5 4 3 2 1 (b) A b s o r p t on ( a r b . un . ) Laser frequency detuning (c)Figure 3: a) Absorption spectrum of Rb NTC with L = λ/ B = 6850 G and σ + laser excitation. For the transition labels,see Fig. 4. The left curve is the absorption spectrum of the reference NTC showing the positions of Rb 1 → ′ , 2 ′ transitionsfor B = 0 (the frequency separation is 157 MHz) and the frequency shift of the atomic transition labeled 20 with respectto 1 → ′ . b) The fragment of the absorption spectrum presented in Fig. 3(a). The group contains the atomic transitionslabeled 1 −
10 which are fitted with the pseudo-Voigt profiles, with the line-width (Full Width Half Maximum) of 250 MHz;the inset shows an expanded view of the part of the experimental results limited by the dashed rectangle. c) The fragmentof the absorption spectrum presented in Fig. 3(a) containing the atomic transitions labeled 11 −
20 which are fitted with thepseudo-Voigt profiles.
B > kG : hyperfine Paschen-Back (HPB) regime In case of strong (
B > λ/ . B > . L = λ/ B = 6850 G and σ + laser excitation6s shown. There are still 20 atomic absorption resonances of Rb D line located at the atomic transitions.Among these transitions, 12 belong to Rb, and 8 transitions belong to Rb. Atomic transition pairslabeled (19 , ,
14) and (3 ,
2) are overlapped (although in the case of strongly expanded spectrum thepeaks belonging to the corresponding transitions are well detected, see Fig. 3(c)) while the other 14 tran-sitions are overlapped partially, and the positions of the absorption peaks of the individual transitions arewell detected. The left curve presents the absorption spectrum of the reference NTC with L = λ/ Rb, F g = 1 → F e = 1 , B = 0 (the frequency shift of the transitionsis determined with respect to 1 → ′ transition).Figure 3(b) shows the fragment of the spectrum (presented in Fig. 3(a)) for the atomic transitions la-beled 1 →
10, where the transitions labeled 1 , , Rb, while the transitions labeled3 − ,
8, and 9 belong to Rb. The fitting (with the pseudo-Voigt profiles [11]) is justified through thefollowing advantageous property of the λ/ A of an individual transition component is proportional to σN L , where σ is the absorption cross-sectionproportional to d (with d being the matrix element of the dipole moment), N is the atomic density, and L is the thickness. Measuring the ratio of A i values for different individual transitions, it is straightforwardto estimate their relative probabilities (line intensities).The fragment of the spectrum (presented in Fig. 3(a)) is shown in Fig. 3(c) for the atomic transitionslabeled 11 −
20, where the transitions labeled 11 , ,
18 and 20 belong to Rb, while the transitions labeled12 , , , ,
17, and 19 belong to Rb.It is important to note that, as seen from Fig. 3, the absorption peak numbered 1 is the most convenientfor magnetic field measurements, since it is not overlapped with any other transition in the range of 1 −
10 kG(see also Fig. 4), while having a strong detuning value in the range of 2 − . The magnetic field required to decouple the electronic total angular momentum J and the nuclear mag-netic momentum I is given by B ≫ B = A hfs /µ B . For Rb and Rb it is estimated to be approximatelyequal to B ( Rb) ≈ B ( Rb) ≈ . A hfs is the ground-state hyperfine couplingcoefficient for Rb and Rb and µ B is the Bohr magneton [2, 16].For such strong magnetic fields when I and J are decoupled (HPB regime), the eigenstates of the Hamil-tonian are described in the uncoupled basis of J and I projections ( m J ; m I ). Fig. 4(a) shows 12 atomictransitions of Rb labeled 3 − , , , , , , ,
17 and 19 for the case of σ + polarized laser excitation inthe HPB regime and 8 transitions of Rb labeled 1 , , , , , ,
18 and 20.Simulations of magnetic sublevel energy and relative transition probabilities for Rb D line are wellknown, and are based on the calculation of dependence of the eigenvalues and eigenvectors of the Hamilto-7 mJ = -3/2mJ = +1/2mJ = -1/2mJ = +3/2mJ = +1/2mJ = -1/2 m I -5/2 -3/2 -1/2 1/2 3/2 5/2 Rb, D ( I = 5/2) (a) mJ = -3/2mJ = -1/2 m I -3/2 -1/2 1/2 3/2mJ = +1/2mJ = +3/2mJ = -1/2
10 25P3/2 mJ = +1/2 Rb, D ( I = 3/2) + (b)Figure 4: a) Diagram of Rb, D line ( I = 5 /
2) transitions for σ + laser excitation in HPB regime. The selection rules:∆ m J = +1; ∆ m I = 0. Therefore, there are 12 atomic transitions marked by the respective numbers 3 − , , , , , , , Rb, D line ( I = 3 /
2) transitions for σ + laser excitation in HPB regime. Due to the selection rulesthere are 8 atomic transitions marked by the respective numbers 1 , , , , , ,
18, and 20. nian matrix on magnetic field for the full hyperfine structure manifold [2, 4, 5, 17, 18, 19]. The calculateddependence of atomic transition probabilities (shown in Fig. 6) and frequency shifts vs B (shown in Fig. 5)are obtained using formulas (1)-(7) from work [11] and are omitted in the paper due to their bulkiness.Particularly, recently it has been demonstrated that the calculations using the above mentioned formulasperfectly well describe the experimental observation of a giant modification of the atomic probabilities ofthe Cs D line F g = 3 → F e = 5 transitions (which are forbidden for B = 0) in strong magnetic fields [20].It is important to note that for B ≫ B the energy of the ground 5 S / and upper 5 P / levels for Rb D line is given by the following equation [16]: E | Jm J Im I i = A hfs m J m I + B hfs m J m I ) + m J m I − I ( I +1) J ( J +1)2 J (2 J − I − + µ B ( g J m J + g I m I ) B z . (1)The values for nuclear ( g I ) and fine structure ( g J ) Land´e factors and hyperfine constants A hfs and B hfs are given in [16]. Note, that Eq. (1) gives correct frequency positions of the components 1 −
20 with aninaccuracy of 2% practically when B ≥ B , i.e. B ≥ − Rb and B ≥
20 kG for Rb [19].Fig. 5 illustrates the frequency positions (i.e. frequency shifts) of the components 1 −
20 as functions ofthe magnetic field B . The theoretical curves are shown by solid lines. The black squares are the experimentalresults which are in a good agreement with the theoretical curves (with an error of 3%). As seen, the atomictransitions are regrouped at B > B . The dashed line denotes the frequencyposition of the Rb, F g = 1 → F e = 2 transition for B = 0.The experimental values of the slopes (the frequency shift with respect to the magnetic field) at8
000 4000 5000 6000 7000 -200002000400060008000100001200014000
Magnetic field (G) T r an s i t i on s h i ft ( G H z ) Rb, F g = 1 F e = 2
21 171618 1915141312986 54 3711201021
Figure 5: Frequency positions of the Rb, D line atomic transitions 1 −
20 versus the magnetic field. Solid lines are thecalculated curves and black squares are the experimental results (with an error of 3%). At
B > . B ≫ B the frequency slope of the 1-st group (1 −
10 transitions)is s ≈ .
33 MHz/G, and for the second group (11 −
20 transitions) s ≈ .
39 MHz/G. Two upper curves show 1 ′ and 2 ′ belonging to the Rb, D , F g = 1 → F e = 3 transitions, with the probability reducing to zero for B > B = 0) has been demonstrated in [18, 20]. IVIIIII ( Rb, m J = -1/2 m J = +1/2) ( Rb, m J = 1/2 m J = 3/2) ( Rb, m J = -1/2 m J = +1/2) T r an s i t i on i n t en s i t y ( a r b . un ) Magnetic field (G)
20 181411 ( Rb, m J = 1/2 m J = 3/2) I Figure 6: Intensity (probability) of the atomic transitions: ( I -st group) Rb, D line transitions labeled 11 , ,
18, and 20;( II -nd group) Rb, D line transitions labeled 12 , , −
17, and 19; (
III -rd group) Rb, D line transitions labeled 1 , , IV -th group) Rb, D line transitions labeled 3 , , , , B -fields but tend to the same value within the group at B ≫ B . Two lower curves show 1 ′ and 2 ′ belonging to the Rb, D , F g = 1 − F e = 3 transitions, with the probability reducing to zero for B > = 7 kG are s ≈ .
29 MHz/G (for Rb transitions this value is slightly larger, while for Rb this valueis slightly smaller) and s ≈ .
42 MHz/G (for Rb transitions this value is slightly smaller, while for Rbthis value is slightly larger) for the groups 1-10 and 11-20, respectively. It is noteworthy that the slope valuefor Rb and Rb (inside the same group, see Fig. 4) are nearly equal to each other when the initial andfinal energy levels are the same (see below).The slopes of the transitions for the fields B ≫ B can be easily found from Eq. (1) as s = [ g J ( P / ) m J − g J ( S / ) m J ] µ B /B ≈ .
33 MHz/G and s = (cid:2) g J ( P / ) m J − g J ( S / ) m J (cid:3) µ B /B ≈ .
39 MHz/G for thegroups 1 −
10 and 11 −
20, respectively (as g I ≪ g J , we ignore g I m I contribution). Consequently, at B ≥
20 kG (when the condition of HPB is fully satisfied for Rb atoms, too), the slope for the group 1 − s ), while the slope for the group 11 −
20 decreases slightly to s . In addition, one caneasily find from Eq. (1) the frequency intervals between the components within each group.Figure 6 presents the theoretical values of “1 −
20” atomic transitions probabilities (intensities) in thefields of 5-7 kG. Let us compare the experimental results presented in Fig. 3 obtained for B = 6850 G withthe theoretical calculations of the atomic transitions probabilities shown in Fig. 6. The atomic transitionslabeled 3, 4, 5, 6, 8 and 9 of Rb shown in Fig. 3(b) have the same amplitudes (probabilities) withinaccuracy less than 5% and this is in a good agreement with the theoretical curves shown in the IV-thgroup in Fig. 6. The atomic transitions labeled 12, 13, 15 – 17, and 19 shown in Fig. 3(c) have the sameamplitudes (probabilities) with inaccuracy less than 5% and this is in a good agreement with the theoreticalcurves shown in the II-nd group in Fig. 6. It is easy to see that there is a similar good agreement between theamplitudes (probabilities) for the atomic transitions of Rb shown in Fig. 3(b) and (c) with the theoreticalcurves shown in the I-st and III-rd groups presented in Fig. 6. Note, that the probabilities of the transitionsfor Rb inside the same group 1-10 or 11-20 are nearly two times larger than the probabilities for Rb insidethe same group. However, since for natural rubidium the atomic density ratio N ( Rb) /N ( Rb) ≈ .
6, thepeak absorption of the atomic transitions for Rb is nearly 1 . Rb (Fig. 6).It is worth to note that the probability of the atomic transition for Rb labeled 20 (for low magneticfield it could be presented as transition F g = 2 , m F = +2 → F e = 3 , m F = +3) is the same in thewhole range of magnetic field from zero up to 10 kG. This is caused by the absence of Zeeman sublevelswith F g = 1 , m F = +2 and F e = 2 , m F = +3, since the perturbation induced by the magnetic fieldcouples only sublevels with ∆ m F = 0 which satisfies the selection rules ∆ L = 0, ∆ J = 0, ∆ F = 1,where L is the orbital angular momentum and F is the total atomic angular momentum (see formula (2)from [17]). Thus, probability modification is possible only for transitions between sublevels each of whichis coupled with another transition according to the selection rules presented above. Due to the similarreason the probability of the atomic transition for Rb labeled 19 (it could be presented as transition F g = 3 , m F = +3 → F e = 4 , m F = +4) is also the same in the range of magnetic fields from zero up to10 kG. Since for the transitions labeled 19 and 20 the absolute value of the probability could be calculated10rom [16], thus using the experimental results presented in Fig. 3 the absolute value of the probabilities forthe other atomic transitions (modified by magnetic field) can be calculated as well. Also, due to the abovementioned reason the frequency shifts of transitions labeled 19 and 20 as a function of magnetic field issimply linear with a fixed slope of s = 1 .
39 MHz/G. Note, that the reduction of the total number of Rb, D transitions to strictly 20 in strong magnetic fields (which are well described by the diagrams presentedin Fig. 4), as well as the behavior of the slopes s and s of Rb and Rb (which are close to the valuesobtained by Eq. 1) is the manifestation of the hyperfine Paschen-Back regime.Note, that there are three main distinctions in the behavior of atomic transitions for D line as comparedwith the behavior for D line [5, 6].i) At B > . D line there are only 10 atomic transitions forming onegroup.ii) There are two remarkable atomic transitions for D line: for Rb atom, the transition labeled 20 andfor Rb atom, the transition labeled 19. In a wide region of magnetic fields from zero up to 10 kG theprobabilities of these atomic transitions remain unchanged. Also, the frequency shifts of the transitionslabeled 19 and 20 are simply linear versus magnetic B -field. Such type of remarkable atomic transitions isabsent for D line in the case of circular polarized laser radiation.iii) In order to determine theoretically the frequency positions of atomic transitions in the case of D line( J = 1 /
2) the well-known Rabi-Breit formulas can be implemented [16], while they are not useful for D line ( J = 3 /
4. Conclusion
We present the results of experimental and theoretical studies of Rb and Rb D line transitions ina wide range of magnetic fields 3 kG < B < B > σ + polarized laser radiation, there remain only 20 Zeeman transitions while there are 60allowed Zeeman transitions at low B -field. In the case of B > . s and s correspondingly. The frequency separation between the two groups increaseswith the magnetic field. The above mentioned peculiarities are the manifestation of hyperfine Paschen-Backregime. The implemented theoretical model very well describes the experiment.Possible applications of the λ/ ±
15 GHz) shifted frequency with respect to the initial atomic levels of Rb with the use of a permanent magnetas well as frequency locking of diode-laser radiation to these resonance [10]; (ii) designing a magnetometer11or measuring strongly inhomogeneous magnetic fields with a high spatial resolution [5]; (iii) designing anoptical insulator based on Rb vapor with the use of the Faraday effect in strong magnetic fields ( ∼ B , we do not expect anylimitations for using the λ/ B >
10 kG.
5. Acknowledgement
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