Huss et al. Vol. 23, No. 9 / September 2006 / J. Opt. Soc. Am. B 1729 Polarization-dependent sensitivity of level-crossing, coherent-population-trapping resonances to stray magnetic fields Arno Huss, Roland Lammegger, and Laurentius Windholz Institute of Experimental Physics, Technical University of Graz, Petersgasse 16, Graz 8010, Austria Emilia Alipieva, Sanka Gateva, Lubomir Petrov, Elena Taskova, and Georgy Todorov Institute of Electronics, Bulgarian Academy of Sciences, Tzarigradsko Chaussee 72, Sofia 1784, Bulgaria Received January 6, 2006; revised May 8, 2006; accepted May 8, 2006; posted May 18, 2006 (Doc. ID 67052) Coherent-population-trapping resonances within the degenerate two-level system of the F = 2 → F⬘ = 1 transition of the 87Rb D1 line were investigated in an uncoated Rb vapor cell by means of level-crossing-type experiments. Tuning over the two-photon resonance is achieved sweeping a magnetic field around zero value. The influence of transverse magnetic fields on the amplitude and the width of the resonances, recorded in fluorescence and absorption, were investigated in the cases of excitation with linear, circular, and elliptical laser light polarization. A theoretical analysis was performed for the case of linearly polarized excitation, the results of which are in good agreement with the experiment. © 2006 Optical Society of America OCIS codes: 020.1670, 020.7490, 030.1670. 1. INTRODUCTION Coherent-population-trapping (CPT) resonances arise in a three-level atomic system (⌳ system) as a result of destructive quantum interference of two resonantly excited transitions.1 Owing to very interesting opportunities for application, the CPT resonance has recently been widely investigated. Usually, two ground levels are coupled to a common excited state by means of two coherent laser fields. When the frequency difference of the two coupling laser fields equals the frequency separation between the two ground levels (two-photon resonance), the atoms are prepared in the nonabsorbing dark state. Detecting the fluorescence light emitted by the atoms, CPT is observed experimentally as a narrow-width dip (possibly orders of magnitude narrower than the natural line width of the transition) when the frequency of one of the coherent fields is scanned. A CPT resonance can also be observed in single-frequency excitation in the so-called Hanle configuration.1,2 Here, the Zeeman sublevels of a single hyperfine level are excited by the ␴+ and ␴− components of linearly polarized laser light, with the polarization direction orthogonal to the quantization axis given by the scanned magnetic field (MF). In this case, instead of tuning the laser frequency, the CPT resonance is detected via scanning the MF and changing the energy of the Zeeman sublevels. At zero MF (level crossing), the two-photon resonance is satisfied and the resonance is observed. Magneto-optical effects are playing a major role in the field of magnetometry. Very high resolution (1 fT Hz−1/2) has been reached in nonlinear magneto-optical rotation (NMOR) experiments3 or with optical pumping magnetometer (OPM) devices.4–6 Recently, an optically pumped magnetometer was applied successfully to measuring the weak cardiomagnetic field.7 However, the interest in new 0740-3224/06/091729-8/$15.00 MF detection systems based on magneto-optical effects continues to be strong, the CPT-based magnetometer being a promising candidate. In a MF, the CPT resonance, obtained by two laser fields on two different hyperfine transitions within the hyperfine structure of alkali metal atoms, splits into several components owing to the Zeeman effect. The relative position of these components are used for MF measurements.8 Another method used for MF measurements is based on excitation of a single hyperfine transition with laser light, which is additionally modulated in the kilohertz band.9 In this particular case, the condition for CPT is satisfied if the modulation frequency matches the frequency difference between two of the split hyperfine sublevels. A change in the registered fluorescence signal is therefore observable. The single-frequency level crossing CPT resonances do not split in an external MF, because for each couple of Zeeman sublevels the resonance condition is fulfilled only at a zero MF level. This resonance phenomenon is well known also as the groundstate Hanle effect.1 As this CPT, resonances are created by the coupling of Zeeman sublevel pairs; any external MF influences the amplitude, width, and shape of the observed signal. Although this effect is well known, it has not been studied systematically for this case. Generally, the CPT resonances can be detected either as a decrease of the fluorescence intensity, or as an enhancement of the transmitted intensity. In the latter case, the transmission properties of the coherently excited medium are mainly determined by the electromagnetic field parameters; thus, the effect is called electromagnetically induced transparency (EIT).10 In this paper, we report studies on both the fluorescence (CPT) and the transmitted (EIT) intensity signals, in or© 2006 Optical Society of America 1730 J. Opt. Soc. Am. B / Vol. 23, No. 9 / September 2006 Huss et al. der to explore the possibilities for applying Hanle-type CPT resonances to MF measurements. In this work, we investigate the width and contrast of single-frequency level-crossing CPT resonances in the presence of an additional probe MF orthogonal to the laser propagation direction, applying linearly, circularly, or elliptically polarized light. All measurements were performed on the 5 2S1/2 → 5 2P1/2 transition F = 2 → F⬘ = 1 of the 87Rb D1 line. The theoretical description of the transverse MF on the CPT resonances in the case of linearly polarized excitation is in good qualitative agreement with the corresponding experimental results. 2. EXPERIMENTAL SETUP The experimental setup is shown in Fig. 1. An extendedcavity diode laser is frequency stabilized to the F = 2 → F⬘ = 1 87Rb D1 transition at a 795 nm wavelength by a dichroic atomic vapor laser lock (DAVLL) setup.11 A part of the laser light is split off and sent through a 87Rb vapor cell. The cell is mounted inside a solenoid that creates a MF B0 up to 170 G oriented parallel to the laser beam propagation direction. The current driving the solenoid additionally enables us to heat up the cell to 40°C, which increases the vapor pressure and therefore the absorption signal. The linearly polarized beam excites. ␴+ and ␴− transitions, which show their maximum absorption at different laser frequencies owing to the energy shift of the Zeeman sublevels in the MF (magnetically induced dichroism). A quarter-wave plate after the absorption cell transforms the two circularly polarized components of the laser beam into two linearly polarized beams (orthogonal to each other) that are separated by a polarizing beam splitter and detected independently by two fast photodiodes. The difference between these photodiode signals is a dispersive shaped signal, suitable for laser stabilization purposes via a fast proportional integral differential (PID) servo unit (see Fig. 1). The laser mode spectrum is additionally observed and controlled by an optical spectrum analyzer. A part of the laser light (about 100 ␮W; a laser beam expanded to a diameter of about 1.5 cm) passes through an uncoated 87Rb vacuum cell kept at room temperature. The cell is placed inside a solenoid, through Fig. 1. Experimental setup. OSC, oscilloscope; PD, photodiode; FPI, Fabry–Perot etalon; ␭ m, lambda meter; BS, beam splitter; FG, triangle wave-frequency generator, ␭ / 4 quarter-wave plate; PBS, polarizing beam splitter; B0, static MF. Fig. 2. CPT (dots) and EIT (curve) resonances obtained on 5 2S1/2-5 2P1/2 87Rb 共Fg = 2 → Fe = 1兲 transition. Ilas = 57 ␮W / cm2 (laser beam diameter 1.5 cm); compensated laboratory MF. which a current is driven by a triangular wave frequency generator to produce a MF scanned around zero value. The transmitted beam is detected by a photodiode, and the CPT signal is registered by a digital data storage oscilloscope, which is read out by a personal computer. The fluorescence light is detected by a second photodiode, directly mounted on the sidewall of the vapor cell. This signal is observed on the same oscilloscope. Thus the CPT resonance signals recorded in fluorescence and transmission are directly comparable. To investigate the influence of probe MFs on the amplitude and width of the CPT resonance signals, the 87Rb vapor cell is placed inside three pairs of mutually orthogonal Helmholtz coils, each of them wired twofold. The Earth’s MF is thus compensated for, and additional probe MFs are applied. 3. EXPERIMENTAL RESULTS A. Excitation with Linearly Polarized Light The dependence of the fluorescence and the transmitted intensity on the scanned magnetic field Bscan is shown to be Lorentzian for an expanded laser beam and low laser power density—up to a few mW/ cm2 (Refs. 12 and 13). The experimental signal for CPT and EIT resonances obtained with a laser power density of 57 ␮W / cm2 and a laser beam diameter of 1.5 cm is shown in Fig. 2. The relaxation processes in the Rb cell mainly determine the CPT resonance width. At room temperature, the Rb vapor pressure is about 3 ⫻ 10−5 Pa, and collisions do not play a significant role. The decay rate of the grounds-state coherence is very low, so that its effective lifetime is determined by the laser power broadening and the transit time of the atoms crossing the laser beam diameter. When scanning the field Bscan without any additional field Badd applied perpendicular to Bscan, a FWHM of the CPT resonance of ⌫0 = 40 mG was measured, which for the D1 line of 87Rb corresponds to 0.7⫻ 40= 28 kHz. For the experimental conditions T = 300 K, the laser beam diameter was 1.5 cm, the estimated time-of-flight broadening was ⬍10 kHz, and the power broadening was approximately 30 kHz. The power broadening was calculated using the formula obtained by Javan et al.14 for low laser intensity. Huss et al. The presence of a stray MF changes the Zeeman splitting and disturbs the coherence created after laser–atom interaction. To investigate this effect on the shape and width of the CPT resonance, we applied additional MFs Badd in two orthogonal directions (parallel and perpendicular to the vector of the laser polarization Elas). These spatial directions were chosen in our investigations because, for these directions, the shape of the resonance signal is preserved. 1. Magnetic Field Perpendicular to the Laser Polarization The experimental geometry and the level scheme used for the level-crossing CPT resonance investigations are presented in Fig. 3. The linearly polarized laser light connects all Zeeman sublevels of the F = 2 → F⬘ = 1 transition to ⌳ schemes via simultaneous ␴+ and ␴− excitation (quantization axis parallel to Bscan). At zero MF, the atoms are prepared in a nonabsorbing state; as a result the fluorescence intensity is reduced. The MF Bscan scanned along the laser beam propagation axis splits the Zeeman sublevels and destroys the coherence. The atoms can interact with the resonant light again, so the fluorescence intensity increases. The same results are obtained by choosing the quantization axes parallel to Elas (see Fig. 3), because this kind of excitation creates a longitudinal alignment only. Bscan as well as Badd are both destroying this alignment. The result is again an increased fluorescence intensity. The dependencies of the CPT signal am- Vol. 23, No. 9 / September 2006 / J. Opt. Soc. Am. B 1731 Fig. 5. (Color online) Excitation scheme for level-crossing CPT resonances—polarization parallel to the MF Badd. The relative transition probabilities are quoted in the insets. Fig. 6. Amplitude and width of the CPT resonances in dependence on Badd (parallel to the laser polarization). Fig. 3. (Color online) Excitation scheme for level-crossing CPT resonances. Polarization perpendicular to the MF Badd. Fig. 4. Amplitude and width of the CPT resonances in dependence on Badd (perpendicular to the laser polarization Elas); squares, fluorescence signal; circles, transmission signal; lineLorentzian fit. plitude and width on the value of the additional MF Badd are presented in Fig. 4. The resonances measured in fluorescence and transmission show the same shape and width. The signal amplitude is taken at Bscan = 0 mG. The CPT signal amplitude in dependence on the MF Badd behaves like a Lorentzian function. MFs of the order of 2 – 3 ⌫0 共80– 20 mG兲 destroy the ground-state coherence, and a CPT resonance is no longer observed. The MF Badd increases the width of the resonance in a nonlinear manner. 2. Magnetic Field Parallel to the Laser Polarization The experimental geometry and the corresponding level scheme valid for this case are presented in Fig. 5. The dependencies of the CPT resonances’ amplitude and width on the value of the additional MF parallel to the laser polarization Elas are shown in Fig. 6. As in the previous case, the CPT amplitude dependence on Badd has a Lorentzian shape (but with opposite sign). The additional MF in this direction increases the resonance amplitude. The resonance width dependence on Badd can be approximated as being linear at a rate [0.98(2) G/G] for fields stronger than 1–2 ⌫0, i.e., 40– 80 mG. In this configuration, the increasing additional MF destroys the coherence more slowly, and a resonance can be recorded in a wide region, until MFs corresponding to 25–30 ⌫0 (i.e., 0.9– 1.0 G) are applied. A qualitative explanation of the dependence observed can be given considering the atomic 1732 J. Opt. Soc. Am. B / Vol. 23, No. 9 / September 2006 Fig. 7. (Color online) Excitation scheme for level-crossing MF resonances in case of circular polarization. system in the vicinity of the zero value of the scanned MF (Fig. 5). If we choose a quantization axis directed along the laser field polarization Elas, the laser-atom interaction creates ␲ transitions 共⌬mF = 0兲. Through the spontaneous emission, atoms are pumped into the states mF = + 2 and mF = −2. This process is very effective, because the relative transition probabilities are the largest for these transitions. The MF Badd together with the scanned MF Bscan cause a redistribution of these populations and, moreover, they influence the optical coherence. As a result, the fluorescence and transmission signals are enhanced (Fig. 6). A more detailed consideration for the case of excitation with linearly polarized laser light, based on numerical solution of the system of density matrix equations, is presented in Section 4. B. Excitation with Circularly Polarized Light In the case of Badd = 0, no ⌳ systems are formed in the excitation scheme of Fig. 7, and thus the conditions for CPT are not fulfilled. Thus in case of exact circular polarization, there is no quantum interference effect at all and, intuitively, the intensity of the fluorescence signal must not show any sharp dependence on the MF Bscan. However, around the zero MF a distinct bright level-crossing MF resonance due to optical pumping is observed. This resonance shows a strong dependence on the additional probe MF Badd. The circularly polarized laser beam excites, e.g., ␴+ transitions and pumps the atoms into two of the ground-state hyperfine sublevels (Fig. 7). The atomic magnetic momentum created due to the atom-laser interaction is directed collinearly to the MF Bscan and does not destroy the orientation of the atoms. The signal obtained is due entirely to an uncompensated MF in transverse direction, which mixes the ground-state sublevels and causes redistribution of the ground-states population,15,16 providing evidence of the magnetic momentum created by the laser field. A precise compensation of the MF Badd cancels this signal. The dependence of the resonance amplitude and width on the additional MF Badd is depicted in Fig. 8. Again the resonance amplitude dependence on the transverse MF has a Lorentzian shape. As in Subsection 3.A.2, a redistribution of the ground-state population in mF = 1 , 2 (Fig. 7), caused by the transverse field, increases the amplitude of the resonance. The dependence of the resonance width on the additional MF Badd shows a linear dependence with a slope of 1.00共2兲 mG/ mG. Such linear behavior is easily applicable for magnetometry, as it allows measurement of the value of the transverse MF by determining the resonance width. Huss et al. C. Excitation With Elliptically Polarized Light In the realistic experimental setup, the laser light polarization is not perfectly linear or circular. Consequently, in this subsection, we investigate how a CPT resonance obtained by imperfectly polarized laser light behaves in the presence of a probe MF. Experimental results for CPT resonances are presented for two cases, namely, a small deviation from a purely linear or purely circular polarization. These measurements were performed in flurescence with 8 mW laser power and a laser beam diameter of 2 mm. In the presence of an additional MF Badd, a deviation of the laser light polarization from a purely linear or circular one by less than 10% causes a distortion of the Lorentzian resonance signal shape. The geometry of the experiment is the same as in Fig. 3. First we discuss a deviation from linear polarization. The ellipticity of the light is Ey / Ex = 8%. The shape and width of the resonances were investigated in the presence of probe MFs applied parallel and perpendicular to the axis Ex of the laser beam polarization. The additional MF perpendicular to Ex widens the contour, as it does in the case of purely linear polarization. The MF directed along Ex causes a split of the CPT resonance that is arranged symmetrically around the zero value of the scanned MF Bscan. An example of a CPT resonance, obtained with elliptical polarization Ey / Ex = 8% in the presence of Badd = 1.2 G, is shown in Fig. 9. The characteristic splitting shows a linear dependence on the amplitude of the additional MF (Fig. 10) with a slope of 0.46 mG/ mG. It could well be Fig. 8. Amplitude and width of the level-crossing MF resonances in dependence on Badd in case of circular laser polarization: squares, fluorescence; circles, transmission. Fig. 9. CPT resonance in case of elliptical polarization for Badd = 1 , 2 G; lower curve, polarization ellipse oriented vertically; upper curve, polarization ellipse tilted around the z axis by 15°. Huss et al. Vol. 23, No. 9 / September 2006 / J. Opt. Soc. Am. B 1733 Brazhnikov et al. conclude that the amplitude of electromagnetically induced absorption (EIA) increases by more than 1 order of magnitude in optimal elliptically polarized light in comparison with excitation with linear polarization. For EIT or dark resonances, the maximum amplitude is obtained for pure linear polarization, which was confirmed by our experiment. The conclusions for the symmetry of the signal are proved by our experiment— the line shape is symmetric in the case of exact resonance. Fig. 10. Splitting of the CPT resonance obtained with elliptical polarization in dependence on the additional MF Badd The arrow indicates the splitting of the CPT resonance shown in Fig. 9 (lower trace). Fig. 11. Shape of the level-crossing MF resonance obtained in circular polarization and the CPT resonances in case of elliptical polarization with 8% deviation from the circular one. used for evaluation of MFs in the region of 0 – 1 G. The splitting of the resonance can be compensated by tilting the principal axis of the polarization ellipse around the z direction (Fig. 9). The change of a purely circular polarized laser light to a slightly elliptical one also has influences on the shape of the resonance—an optical pumping signal with opposite sign is superimposed to the resonance (Fig. 11). Even though the optical pumping signal is not symmetric, the distinct MF resonance peak is always centered at the point of zero-scanning MF. Our investigations demonstrated that to apply level-crossing MF resonances (like CPT) to magnetometry one needs to strictly maintain the laser beam polarization quality and direction. Any deviation from perfect polarization causes signal distortion that would immediately result in distinct systematic measurement errors. Matsko et al.17 showed that in NMOR signals obtained with elliptically polarized light, similar peculiarities are due to the creation of high-order ground-state coherence. In our experiment, an M-scheme excitation connects the m = + 2 with the m = −2 Zeeman sublevel via a multiphoton process [hexadecapole moment 共⌬m = 4兲]. The created coherency is observed in transmission. The CPT signal will not be sensitive to the ellipticity of the exciting light if it is formed on the levels connected in a pure ⌳ scheme (for instance Fg = 1 − Fe = 0). New results for magnetooptical resonances in a elliptically polarized field for a closed F → F + 1 transition were published recently.18,19 4. THEORETICAL ANALYSIS To analyze in detail the experimental results described in Subsection 3.A we carried out numerical calculations concerning the influence of the additional MF on the CPT resonances. The theoretical description in the present work proceeds from the equations in Refs. 20 and 21, which describe the interaction of an atomic system with a constant MF H0 and a resonant laser radiation. A concrete model was developed for the case of orthogonal MFs, continuous (scanned) H0, an additional arbitrarily oriented probe field H⬘, and a linearly polarized laser field E共␻0 , t兲. The irreducible tensor operator formalism was used. The advantages of this representation, together with a clear physical meaning of the tensor component, are related to the diagonalization of the relaxation matrix ⌫. A necessary condition is that binary collisions and radiation trapping are assumed to be the main relaxation processes in the atomic system. Then the relaxation constants ⌫␳共k兲 will depend only on the rank k of the statistic operator ␳qk 共␳ = f , ␸ , ␰兲.21,22 The set of equations describing the ground-state ␸ with quantum number F␸, the excited statef with quantum number Ff, and the optical coherency ␰, can be quoted for arbitrary angular momentum: 再 ḟqk + ⌫f共k兲fqk = i␮Bgfh−1 qH0fqk 冋 冋 1 + 2 − 册 册 1/2 共k + q兲共k − q + 1兲 1 2 k H1fq−1 1/2 共k − q兲共k + q + 1兲 k H−1fq+1 冎 + Lqk + 共2Ff + 1兲NfW共v兲␦k0␦q0 , 再 共1a兲 ␸˙ qk + ⌫␸共k兲␸qk = i␮Bg␸h−1 qH0␸qk + − 冋 冋 1 2 册 册 1/2 共k + q兲共k − q + 1兲 1 2 共k − q兲共k + q + 1兲 k H1␸q−1 1/2 k H−1␸q+1 冎 + Mqk + 共2F␸ + 1兲N␸W共v兲␦k0␦q0 + ⌫f␸共k兲fqk , 共1b兲 1734 J. Opt. Soc. Am. B / Vol. 23, No. 9 / September 2006 Huss et al. ⌫f␸共␬兲 = 共− 1兲Ff+F␸+␬+1⌫f␸共0兲共2Ff + 1兲共2F␸ + 1兲共2Jf + 1兲 ␰˙ qk + 关⌫␰共k兲 + i␻0兴␨qk = ih−1 ⫻ 冉 兺 共− 1兲 Q H−Q共− 1兲Jf+J␸+q共2k⬘ + 1兲 k⬘kQ k⬘ 1 k − q⬘ Q q 冊冉 ␮Bg␸共␸储j␸储␸兲 冉 + 共− 1兲k+k⬘ ␮Bgf共f储jf储f兲 再 再 ⫻ Ff F␸ k⬘ 1 k F␸ F␸ Ff k⬘ 1 k Ff 冎冊 冎 k ␰q⬘ + Gqk . 共1c兲 In Eqs. (1), the first term on each right-hand side describes the magnetodipole interaction of the atomic system. Here Hq 共q = 0 , ± 1兲 are the circular components of the MF H. The interaction with the laser field is described by excitation tensors Lqk, Mqk, and Gqk, which are defined as Lqk = ih−1共2F␸ + 1兲−1/2 兺E kk⬘ −QCqq⬘Q ⫻关d␰q⬘⬘ + d共␰−q⬘ ⬘兲共− 1兲k+k⬘+q⬘兴, k k 共2a兲 with a geometrical coefficient Cqq⬘⬘Q = 共− 1兲2J␸+q⬘共2Ff + 1兲1/2共2k⬘ + 1兲 kk ⫻ 冉 1 k⬘ − q⬘ k Q q Mqk = ih−1共2F␸ + 1兲−1/2 冊 再 k⬘ 1 k Ff Ff F␸ 共3a兲 , 兺E kk⬘ −QBqq⬘Q共− 1兲␬+k⬘ ⫻关d␰q⬘⬘ + d共␰−q⬘ ⬘兲共− 1兲k+k⬘+q⬘兴, k 冎 k 共2b兲 with Bqq⬘⬘Q = 共− 1兲2Jf+q⬘共2F␸ + 1兲1/2共2k⬘ + 1兲 kk ⫻ 冉 k⬘ 1 k − q⬘ Q q 冊 k⬘ 1 k F␸ F␸ Ff 冎 共3b兲 , Gqk = ih−1共2Ff + 1兲−1/2d kk k ⫻关Sqq⬘⬘Qfq⬘⬘ 再 兺E −Q + 共− 1兲k+k⬘Rqq⬘Q␸q⬘⬘兴, k 共2c兲 with Rqq⬘⬘Q = 共− 1兲2J␸+1共2Ff + 1兲1/2共2k⬘ + 1兲 kk ⫻ 冉 k⬘ 1 k −q Q q 冊 , 再 k⬘ 1 k Ff F␸ F␸ 冎 共3c兲 Sqq⬘Q is obtained from Rqq⬘Q with substitution of f ↔ ␸, ⬘ ⬘ and 共␳ 储 j 储 ␳兲 = 关共2F␳ + 1兲共F␳ + 1兲F␳兴1/2. The last term in Eq. (1b) describes the transfer of population 共k = 0兲 and coherence 共k = 2兲 from the excited state 共Ff兲 to the ground state 共F␸兲. The relaxation constant ⌫f␸共k兲 representing this transfer is given by23 kk kk 再 Ff Ff ␬ F␸ F␸ 1 冎再 Ff F␸ 1 J␸ Jf I 冎 2 , 共4兲 We should note that the relaxation constant ⌫f␸共k兲 characterizes the losses in the channel Ff → F␸. In the case of a branching-ratio ⌫f␸共0兲 / ⌫f共0兲 close to 1, the atomic system is closed, and thus the losses are minimal. On the other hand, if this ratio is close to 0, the atomic system becomes an open one, and therefore the losses reach a maximum. In the particular case of the 87Rb D1-line Ff = 1 → F␸ = 2 and Ff = 1 → F␸ = 1 transitions, the branching ratio can be derived from Eq. (4): 关⌫1−2共0兲 / ⌫1−1共0兲 = 5兴. The notations in Eqs. (1) are (as commonly used): ␮B, Bohr’s magneton; ប, Planck constant; gf and g␸, Landé factors for the corresponding levels; ␻0, resonance frequency for the given transition; W共v兲, velocity distribution function; Nf and N␸, population of the excited and ground states, and d, reduced matrix element of the dipole transition. Parentheses denote 3j-Wigner symbols, and brackets denote 6j-Wigner symbols. Using the basic equations, Eqs. (1a)–(1c), one can write a concrete system of equations for the chosen atomic transition scheme in a certain geometry and experimental condition. In these calculations, the quantization axis was chosen to be parallel to the electric vector Ez of the laser field, and the scanned MF Hx was assumed to be perpendicular to this axis. Only the additional MF vectors’ component perpendicular to the scanned MF is of importance, and it can be resolved to Hz (parallel to Ez) and Hy (perpendicular to Ez). Using the rotating-wave approximation and assuming a monochromatic laser field propagating in the x direction 共␻L , kx兲, i.e., EQ共␻L , t兲 = eQE exp兵−i共␻Lt − kxx兲其 + c . c., where the circular components are designated with eQ, the system of equations can be reduced to an algebraic one. The additional but not compulsory condition ␻L − ␻0 ≪ ⌫␰共k兲 was assumed to simplify the calculations. The equations written under these conditions for the transition J␸ = 2 → Jf = 1 were solved numerically with the aim of examining the solution behavior as a function of parameters: the Rabi frequency df␸E / h, the spontaneous emission transfer rate ⌫f␸共k兲, and the stray field components Biadd. All these parameters and the MFs were expressed in units of ⌫␰. We present here the results concerning the stray MF influence on the resonance amplitude and width only. The values of the parameters were chosen in accordance with our experimental conditions. The radiative width of the upper level is known to be 6 MHz, while the evaluated time-of-flight width of the lower level is ⬍10 kHz, so the ratio of the upper to lower level decay rate constants is 500. All experimental investigations were performed with laser power density of 57 ␮W / cm2, providing for the parameter24 共df␸E兲2 / h2⌫f⌫␸ approximately 5. By virtue of this estimation, the Rabi frequency was chosen to be 100 ⌫ ␸. The model calculations for the geometry chosen in the experiment showed that the longitudinal alignment of the ground state plays a major role. An additional MF paral- Huss et al. Vol. 23, No. 9 / September 2006 / J. Opt. Soc. Am. B 1735 lel to E changes the optical coherence 共␰0k ; k = 1 , . . . , 3兲, thus enhancing the alignment. The calculated dependencies of the excited-level population f00 on stray MFs are proportional to the unpolarized fluorescence intensities observed. To compare the experimental data (see Figs. 4 and 6) with the model calculations a recalibration of ⌫␸ into units of a magnetic flux density (cgs system of units) was done. The results are presented in Figs. 12 and 13. A good qualitative agreement is seen, especially for the amplitude dependence on the MF Badd. A qualitative agreement between the CPT resonance measured width and the width calculated in our model can also be claimed (Figs. 12 and 13). The nonlinear behavior of the CPT resonance width at small field strengths is therefore well represented by the model calculations. It is worth mentioning that both effects (broadening of the CPT resonance and change of the CPT amplitude) in the presence of additional MFs parallel to the laser polarization direction are nonlinear effects. It should be also emphasized that the predictions of performed calculations have only qualitative character because, e.g., the real velocity distribution of atoms always present in thermal vapor cells was not taken into account. Fig. 12. Comparison of the calculated (curve) and experimentally obtained (squares) normalized amplitude and width of the fluorescence signal in dependence on the Badd applied perpendicular to the laser beam polarization Elas. 5. CONCLUSION The investigation of the influence of transverse MFs on CPT resonances in the cases of linear and elliptical polarization (and on the level-crossing MF resonances in the case of circular polarization) of the exciting laser light showed that the resonances obtained in such a singlefrequency level-crossing configuration are sensitive to probe MFs in the order of their width. The theoretical analysis of the influence of transverse MFs on the CPT resonances in the case of linearly polarized excitation is in good agreement with the experimental results. The results obtained can be used in MF measurements. The resonance obtained with circular polarized laser excitation shows a quite-good linear dependence of the signal width on a transverse probe magnetic field. This signal width (in milligrams) equals the value of the transverse field with good accuracy. A possible disadvantage of using the observed resonances for measuring MFs lies in the fact that such an experimental arrangement gives only the component in a plain perpendicular to the laser beam of the measured MF. The sensitivities of the resonances to MFs in the cases of excitation by linearly and circularly polarized laser light are comparable. The sensitivity can be increased by, e.g., using narrower structures in the CPT resonance25 and by improving the registration system’s noise performance. ACKNOWLEDGMENTS Fig. 13. Comparison of the calculated (curve) and experimentally obtained (squares) normalized amplitude and width of the fluorescence signal in dependence on the Badd applied parallel to the laser beam polarization Elas. This work was supported in part by FP5 (EU) project G6RD-CT-2001-00642 and the Bulgarian National Council for Scientific Research (grants F-1006/00 and F-1409/ 04). E. Alipieva’s e-mail address is alipieva@ie.bas.bg. 1736 J. Opt. Soc. Am. B / Vol. 23, No. 9 / September 2006 Huss et al. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. E. Arimondo, “Coherent population trapping in laser spectroscopy,” Prog. Opt. 35, 257–354 (1996). F. Renzoni, W. Maichen, L. Windholz, and E. Arimondo, “Coherent population trapping with losses observed on the Hanle effect of the D1 sodium line,” Phys. Rev. A 55, 3710–3718 (1997). D. Budker, W. Gawlik, D. Kimball, S. Rochester, V. Yashcuk, and A. Weis, “Resonant nonlinear magnetooptical effects in atoms,” Rev. Mod. 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