ISSN 001-3640, JETP Letters, 010, Vol. 91, No. 1, pp. 65 69. Pleiades Publishing, Inc., 010. Original Russian Text D.V. Brazhnikov, A.V. Taichenachev, A.M. Tumaikin, V.I. Yudin, I.I. Ryabtsev, V.M. Entin, 010, published in Pis ma v Zhurnal Éksperimental noі i Teoreticheskoі Fiziki, 010, Vol. 91, No. 1, pp. 694 698. Effect of the Polarization of Counterpropagating Light Waves on Nonlinear Resonances of the Electromagnetically Induced Transparency and Absorption in the Hanle Configuration D. V. Brazhnikov a, b, A. V. Taichenachev a, b, A. M. Tumaikin a, V. I. Yudin a c, I. I. Ryabtsev b, d, and V. M. Entin d a Institute of Laser Physics, Siberian Branch, Russian Academy of Sciences, pr. Akademika Lavrent eva 13/3, Novosibirsk, 630090 Russia e-mail: llf@laser.nsc.ru b Novosibirsk State University, ul. Pirogova, Novosibirsk, 630090 Russia c Novosibirsk State Technical University, ul. Pirogova, Novosibirsk, 63009 Russia d Institute of Semiconductor Physics, Siberian Branch, Russian Academy of Sciences, pr. Akademika Lavrent eva 13, Novosibirsk, 630090 Russia e-mail: ventin@isp.nsc.ru Received April 1, 010 A polarization method has been proposed for transforming the nonlinear electromagnetically induced transparency resonance into the electromagnetically induced absorption resonance (and vice versa) in the Hanle configuration in the field of counterpropagating waves. Numerical calculations have been performed for the F g = F e = 1 and F g = F e = 3 transitions in the 87 Rb atom. The qualitative analysis of the effect has been performed for the three-level Λ scheme of the atomic energy levels. The main theoretical conclusions are in qualitative agreement with our experimental data for 87 Rb. The results can be applied in magnetometry and nonlinear optics. DOI: 10.1134/S00136401010039 1. Nonlinear interference effects based on atomic coherent states are currently under active investigation. The nonlinear electromagnetically induced transparency (EIT) [1] and electromagnetically induced absorption (EIA) [] resonances are clear examples of such effects. The first effect is based on the coherent population trapping, when there is a specific ( dark ) state such that the atoms collected in it do not absorb the energy from the external field [3]. The EIA resonance with sign opposite to the EIT resonance is associated with the spontaneous transfer of coherence from the excited state to the ground state [4]. The width of the EIT and EIA resonances can be much smaller than the natural width and can be hundreds and tens of hertz. For this reason, we use the common term ultranarrow resonances for such resonances. EIT and EIA are applied in nonlinear optics [5], optical communications [6], metrology [7], etc. EIA and EIT resonances are often observed in a two-frequency field formed by copropagating laser beams. In this case, a contrast dip (EIT) or peak (EIA) appears in the absorption of the medium as a function of the wave frequency difference. However, the experimental observation of these resonances is simpler in the Hanle configuration, when resonance is formed in the absorption signal of a single traveling light wave as a function of the longitudinal magnetic field applied to a cell with a gas [8] and is associated with crossing of magnetic sublevels in the ground state (level-crossing resonance). In addition to the analysis of the parameters of ultranarrow resonances (amplitude, width, and shift), the problem of the mutual transformation of EIT resonances to EIA resonances and vice versa is important. It is known that the sign (type) of the resonance can depend on various factors: the configuration of the atomic transition levels [, 9], collisions of atoms in the gas [10] or with the walls of the cell [11] and the corresponding depolarization of the excited state, and external static [1] or microwave fields [13]. Zhukov et al. [14] recently investigated the contrast and sign of EIA resonance in the absorption of a probe wave in the Hanle configuration in dependence on the intensity of the counterpropagating wave exciting an adjacent atomic transition (V-scheme excitation). The effect of the polarization of light waves (the ellipticity parameter and the orientation of the ellipses) on the sign of the ultranarrow resonance has been poorly studied. At the same time, the strong effect of the elliptic polarization of a light wave on the shape and amplitude of the EIA resonance in the Hanle configuration was demonstrated in [1, 15]. For 65
66 BRAZHNIKOV et al. F e F g mg = 0 Fig. 1. (a) The scheme of the energy levels of the atom in the longitudinal magnetic field. The electromagnetically induced transitions for the wave traveling along the z axis and the counterpropagating wave are indicated by the solid and dashed lines, respectively. (b) The electromagnetic field configuration. this reason, it can be expected that the polarization parameters can be tools for controlling the sign of the ultranarrow resonance. In particular, the transformation of the EIT resonance to the EIA resonance with a continuous change in the polarization from linear to circular was observed in [16], where the F g = F e = 1 transition (F g and F e are the total angular momenta of the atom in the ground and excited states, respectively) on the D 1 line in the 87 Rb atom excited by a traveling light wave in the Hanle configuration was investigated. The change in the sign of the ultranarrow resonance was attributed to the residual transverse magnetic field. The width and sign of the EIT resonances in the Hanle configuration in the spatially separated copropagating linearly polarized laser beams, which excite the same transition, were studied in [17]. In particular, it was shown that the transparency resonance in the absorption of the probe beam is transformed into the absorption resonance as the angle between the linear polarizations of the waves increases. In this work, we propose a simpler configuration that allows for the observation of ultranarrow resonances, as well as a change in their sign on both of the darktype transitions (e.g., the F g = F e = 1 transition), where coherent population trapping is possible, and on bright type transitions where coherent population trapping is absent. To this end, spatially matched counterpropagating laser beams of the same frequency, which excite a single optical transition in the atom, are used. The change of the sign of ultranarrow resonance is attributed exclusively to the polarization parameters of the counterpropagating waves and can occur due to the variation of either the angle between the linear polarizations or the ellipticity degree (at the constant angle between the major axes of the ellipses). It is also worth noting that the proposed configuration makes it possible to qualitatively interpret the effect of the transformation of ultranarrow resonances and does not require the consideration of the dynamics of the atomic system in contrast to the configuration proposed in [17].. We consider the interaction of the F g F e transition with the light field of elliptically polarized counterpropagating plane waves (see Fig. 1a). The wave vectors and the external static magnetic field are directed along the z axis (see Fig. 1b). According to Fig. 1a, the excitation scheme forms a closed contour. In this case, as was shown in [18], it can be expected that nonlinear effects are sensitive to the phases and intensities of the waves (in our case, to the angle φ and ellipticity parameters ε 1 and ε ). We perform a theoretical analysis using the standard semiclassical approach, where the field is considered as classical and the medium is described quantum-mechanically using the equation for the atomic density matrix: --- + v i ρˆ = - [( Ĥ t 0 + Ĥ E + Ĥ B ), ρˆ ] + ˆ { ρˆ }. (1) Here, v is the velocity of the atom, Ĥ 0 is the Hamiltonian of the unperturbed atom, and the operators Ĥ E and Ĥ B describe the interactions of the atom with the light waves and static magnetic field, respectively. The operator ˆ describes various processes of the relaxation of the atomic system: the spontaneous relaxation rate is γ and the passage of the atom through the cross section of the light beams is described by the relaxation constant Γ; therefore, the operator in Eq. (1) has only the z components. Note that such an approach is often used, because it strongly simplifies the theoretical analysis, retaining in most cases the qualitative features of the spectroscopic signal. We assume that the time of the interaction of the gas with the field is much longer than the characteristic relaxation times over the internal degrees of freedom in the atom. For this reason, we seek the stationary solution, omitting the time differentiation operator / t in Eq. (1) in the rotating basis. Moreover, the light fields are assumed to be weak or moderate so that R 1, Δ D, where R 1, = de 1, / are the Rabi frequen-
EFFECT OF THE POLARIZATION OF COUNTERPROPAGATING LIGHT WAVES 67 Fig. 3. Λ scheme of the energy levels of the atom in the (a) basis { 1, 1, 1 } and (b) new basis given by Eqs. () and (3). The electromagnetically induced transitions for the wave traveling along the z axis and the counterpropagating wave are indicated by the solid and dashed lines, respectively. Fig.. Magnetic field dependence of the absorption of the probe wave E calculated for the (a d) F g = F e = 1 and (e h) F g = F e = 3 transitions for the (a c) and (e g) linearly polarized waves and (d) and (h) elliptically polarized waves with ε 1 = ε = 5, the Rabi frequencies R 1 = (a g) 0.5γ and (h) γ and R = 0.1γ, and the time-offlight relaxation rate Γ = 0.0γ under the exact-resonance condition δ = 0. cies (d is the electric dipole moment) and Δ D is the Doppler line width. This condition means that the nonlinear interference effects between the counterpropagating waves can be neglected [19]. Figure shows the magnetic field dependence of the absorption of the probe wave E obtained by numerically solving Eq. (1) with the averaging over the Maxwell velocity distribution. The outer contour of the resonance curve undergoes Doppler broadening (its FWHM under the conditions of Fig. a is 450 G). The central (narrowest) structure is associated with the crossing of magnetic sublevels in the ground state (in Fig. a, this resonance visually merges with the natural-width resonance). The remaining intra-doppler structures of the signals are attributed to the saturation, optical pumping, and crossing of magnetic sublevels in the excited state. It is seen in the figures that the change of the angle between the linear polarizations from 0 to π/ on the dark F g = F e = 1 transition is accompanied by the change of the sign of the ultranarrow resonance (the EIT resonance is transformed into the EIA resonance, see Figs. a, b). On the contrary, on the bright F g = F e = 3 transition, the EIA resonance is transformed into the EIT resonance (see Figs. e, f). An asymmetric shape of the ultranarrow resonance is observed at the angles 0 < ϕ < π/ (see Figs. c, g). At first glance, this fact is not obvious for the case of linear polarizations, e.g., because it does not directly follow from the analysis of Fig. 1a for the opposite directions of the magnetic field. However, the general symmetry analysis of the geometry of the problem similar to that performed in [0] does not exclude the observation of such an asymmetry. The further calculations show that the sign of the nonlinear resonance can be controlled by the variation of the ellipticity parameter of the fields at a fixed angle ϕ. For example, the observation of the EIT resonance on the F g = F e = 1 transition is expected at the parallel major axes of the polarization ellipses (ϕ = 0). As the difference ε 1 ε increases, the EIT resonance can be transformed to the EIA resonance (see Fig. d). As is seen in Fig. h, the situation on the bright transition is opposite. 3. To qualitatively explain the effect, we consider the transformation of the EIT resonance to the EIA resonance using the three-level Λ scheme of the atom (see Fig. 3a). Under stationary conditions and in the absence of time-of-flight effects, such a scheme corresponds to the closed F g = 1 F e = 1 transition. To take into account polarizations of the light waves, we assume that the cyclic components of the elliptic polarizations are responsible for the excitation of the corresponding arms of the Λ scheme. The splitting of the magnetic sublevels (denoted as 1 and ) of the ground state is proportional to the magnetic field. For
68 BRAZHNIKOV et al. Here, A 1 = cos( ϕ) cos( ε 1 ε ) i sin( ϕ) sin( ε 1 + ε ), (8) A = cos( ϕ)sin( ε 1 ε ) + i sin( ϕ) cos( ε 1 + ε ). (9) According to Eqs. (4) and (6), the coherent state NC in zero magnetic field does not evolve and, moreover, is a dark state for the wave E 1 at any ellipticity parameter ε 1 (see Fig. 3b); i.e., Vˆ 3 1 NC = 0. (10) In the stationary regime, atoms are collected in this state and the medium becomes transparent for the field E 1. The interaction of the probe wave E with the atoms collected in the NC state is described by the expression Fig. 4. Measured absorption of the probe light wave versus the longitudinal magnetic field for the (a) parallel, E pump E probe, and (b) perpendicular, E pump E probe, mutual orientations of the counterpropagating waves. a qualitative analysis, it is convenient to pass from the basis of the eigenstates of the Hamiltonian of the unperturbed atom { 1,, 3 } to the new orthonormalized basis { C, NC, 3 }, where C = cos( ε 1 π/4) 1 sin( ε 1 π/4), () NC = sin( ε 1 π/4) 1 + cos( ε 1 π/4) (3) (see also Fig. 3b). The Hamiltonian of the atom in the magnetic field in the rotating wave basis has the form Ĥ = Ω sin( ε 1 )( NC textnc C C ) (4) Ω cos( ε 1 )( NC C + C NC ) δ 3. 3 Here, Ω is the Larmor frequency and δ = ω ω 0 is the detuning of the frequency of the light field from the optical transition frequency. In the dipole approximation and in the rotating wave basis, the operator of the interaction of the atom with the field of counterpropagating waves is represented in the form Ĥ E = Vˆ 1e ikz + Vˆ e ikz + h.c., (5) where Vˆ 1 = ------- 1 3 C, (6) Vˆ = ------- 3 ( A 1 C A NC ). (7) Vˆ 3 NC = ------- A. (11) It follows from Eqs. (9) and (11) that the NC state for the same elliptic polarizations and ϕ = 0 is also dark for the probe wave. In this case, the absorption of the probe wave E near zero magnetic field exhibits the EIT resonance as is seen in Fig. a. The probe field for ϕ = π/ interacts with both NC and C states and undergoes absorption. Moreover, the absorption is particularly strong when ε 1 = ε, because the probe field E interacts exclusively with the atoms accumulated in the NC state owing to the E 1 field. The absorption of the probe wave in a nonzero magnetic field, when coherence between NC and C appears owing to the off-diagonal elements of Hamiltonian (4), decreases; i.e., the EIA resonance is formed when scanning the magnetic field near zero (see Fig. b). 4. We carried out the experiment under the following conditions. The radiation from a semiconductor laser (λ = 795 nm) with an external cavity was guided through the vacuum cell with the natural mixture of Rb isotopes at room temperature from two sides; the probe and pump laser beams were carefully matched. The linear polarization of the radiation was formed by means of two Glan prisms. The laboratory magnetic field in the cell was compensated using three pairs of the Helmholtz coils to a value of about 50 mg. The relative change in the absorption of the probe laser beam was detected by means of a photodetector. The configuration of the optical scheme made it possible to vary the angle between the polarizations of the probe (E probe = E ) and pump (E pump = E 1 ) waves. The longitudinal magnetic field in the cell was created by a solenoid and was swept by a sweep generator. The transmission signal was detected using a digital oscilloscope in the data storage regime synchronously with the variation of the solenoid current. Figure 4 shows the experimental records of the absorption signal of the probe radiation tuned to the
EFFECT OF THE POLARIZATION OF COUNTERPROPAGATING LIGHT WAVES 69 5 S 1/, F = 5 P 1/, F = 1 transition corresponding to the D 1 line of 87 Rb (γ = π 5.57 MHz). The powers of the pump and probe beams mm in diameter were 10 and 5 μw. As is seen in the figure, the EIT resonance and EIA resonance are observed for the parallel and orthogonal linear polarizations, respectively (see Figs. 4a, 4b, respectively). The asymmetry of the shape of the resonance is due to the presence of the residual inhomogeneous magnetic fields through the length of the cell. The presence of side structures near the dip of the EIT in Fig. 4a is associated with the residual detuning of the laser frequency from the resonance frequency. 5. To conclude, we formulate the main result of the work. A purely polarization method has been proposed for transforming the EIT resonances to the EIA resonances and vice versa in the Hanle configuration. In this method, counterpropagating light waves with the same frequency, which excite a common optical transition in the atom, are used. It has been shown that the sign of the resonance depends both on the angle between the polarizations of the waves and on their ellipticity parameters. The calculations have demonstrated the change of the sign for both dark (namely, F g = F e = 1) and bright (F g = F e = 3) transitions. To confirm some of the main theoretical conclusions, we have performed the experiments on the D 1 line of the 87 Rb atom. Note that the proposed method makes it also possible to observe the EIA resonance for the case where the cell contains a buffer gas destroying the anisotropy in the excited state. The results obtained in this work can be used in the fields of laser physics, metrology, and nonlinear optics, where the EIT and EIA ultranarrow resonances are used (magnetometry, propagation of light pulses in resonant media, etc.). This work was supported by the Ministry of Education and Science of the Russian Federation (programs Scientific and Pedagogical Staff of Innovative Russia for 009 013 and Development of the Scientific Potential of Higher Education for 009 010); by the Russian Foundation for Basic Research (project nos. 10-0-90717, 10-0-00987, 10-0 00406, 09-0-9047, 09-0-948, 08-0-01108, 08-07-0017, and 08-0-00730); and by the Presidium of the Siberian Branch, Russian Academy of Sciences. REFERENCES 1. G. Alzetta, A. Gozzini, L. Moi, and G. Orriols, Nuovo Cim. B 36, 5 (1976).. A. M. Akulshin, S. Barreiro, and A. Lezama, Phys. Rev. A 57, 996 (1998). 3. E. Arimondo and G. Orriols, Lett. Nuovo Cim. 17, 333 (1976). 4. A. V. Taichenachev, A. M. Tumaikin, and V. I. Yudin, Pis ma Zh. Eksp. Teor. Fiz. 69, 776 (1999) [JETP Lett. 69, 819 (1999)]. 5. D. Budker, W. Gawlik, D. F. Kimball, et al., Rev. Mod. Phys. 74, 1153 (00); V. M. Entin, I. I. Ryabtsev, A. E. Boguslavsky, and Yu. V. Brzhazovsky, Opt. Commun. 07, 01 (00). 6. A. M. Akul shin, A. Cimmino, and G. I. Opat, Kvant. Elektron. 3, 567 (00) [Quantum Electron. 3, 567 (00)]; L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, Nature 397, 594 (1999). 7. J. Vanier, Appl. Phys. B 81, 41 (005); M. Stähler, S. Knappe, C. Affolderbach, et al., Europhys. Lett. 54, 33 (001). 8. Y. Dancheva, G. Alzetta, S. Cartaleva, et al., Opt. Commun. 178, 103 (000). 9. V. I. Yudin, Doctoral Dissertation in Mathematical Physics (Novosib. Gos. Univ., Novosibirsk, 000); A. S. Zibrov and A. B. Matsko, Pis ma Zh. Eksp. Teor. Fiz. 8, 59 (005) [JETP Lett. 8, 47 (005)]. 10. H. Failache, P. Valente, G. Ban, et al., Phys. Rev. A 67, 043810 (003); D. V. Brazhnikov, A. V. Taichenachev, A. M. Tumaikin, and V. I. Yudin, J. Opt. Soc. Am. B, 57 (005). 11. C. Andreeva, A. Atvars, M. Auzinsh, et al., Phys. Rev. A 76, 063804 (007). 1. K. Nasyrov, S. Cartaleva, N. Petrov, et al., Phys. Rev. A 74, 013811 (006). 13. M. D. Lukin, S. F. Yelin, M. Fleischhauer, and M. O. Scully, Phys. Rev. A 60, 35 (1999); G. S. Agarwal, T. N. Dey, and S. Menon, Phys. Rev. A 64, 053809 (001). 14. A. A. Zhukov, S. A. Zibrov, G. V. Romanov, et al., Phys. Rev. A 80, 033830 (009). 15. D. V. Brazhnikov, A. V. Taichenachev, A. M. Tumaikin, et al., Pis ma Zh. Eksp. Teor. Fiz. 83, 71 (006) [JETP Lett. 83, 64 (006)]. 16. Y. J. Yu, H. J. Lee, I.-H. Bae, et al., Phys. Rev. A 81, 03416 (010). 17. Z. D. Gruji ć, M. Mijailovi ć, D. Arsenovi ć, et al., Phys. Rev. A 78, 063816 (008). 18. E. A. Korsunsky and D. V. Kosachiov, Phys. Rev. A 60, 4996 (1999). 19. S. G. Rautian, G. I. Smirnov, and A. M. Shalagin, Nonlinear Resonanses in Spectra of Atoms and Molecules (Novosibirsk, Nauka, 1979) [in Russian]. 0. D. V. Brazhnikov, A. V. Taichenachev, A. M. Tumaikin, et al., Zh. Eksp. Teor. Fiz. 136, 18 30 (009) [JETP 109, 11 (009)]. Translated by R. Tyapaev