We show that the effective decay rate of Zeeman coherence, generated in a
Rb87 vapor by linearly polarized laser light, increases significantly with the
atomic density. We explain this phenomenon as the result of radiation trapping.
Our study shows that radiation trapping must be taken into account to fully
understand many electromagnetically induced transparency experiments with
optically thick media

Matsko, A. B.

Novikova, I.

Scully, M. O.

Welch, G. R.

English

arXiv

Texas A&M University

Physical Review Letters

Radiation Trapping in Coherent Media

OAKTrust Digital Repository (Texas A&M Univ)

arXiv:quant-ph/0101147v1  31 Jan 2001Radiation TrappingRadiation trapping in coherent mediaA. B. Matsko, I. Novikova, M. O. Scully, and G. R. WelchDepartment of Physics, Texas A&M University, College Station, Texas 77843-4242,(May 31, 2018)AbstractWe show that the effective decay rate of Zeeman coherence, generated ina 87Rb vapor by linearly polarized laser light, increases significantly with theatomic density. We explain this phenomenon as the result of radiation trap-ping. Our study shows that radiation trapping must be taken into account tofully understand many electromagnetically induced transparency experimentswith optically thick media.Typeset using REVTEX1Coherent population trapping (CPT) was first observed in experiments establishing Zee-man coherence in sodium atoms [1]. In these experiments, explained in terms of a three-levelΛ-type level scheme, a laser field was used to create superpositions of the ground state sub-levels. One of these superpositions, referred to as the “bright” state, can interact with thelaser field while the other superposition does not and is referred to as the “dark” state [2].All the population in the system is eventually optically pumped into the dark state, andresonant absorption of the electromagnetic field almost disappears. This phenomenon is onemanifestation of electromagnetically induced transparency (EIT) [2,3].EIT is particularly interesting because it offers a wide variety of applications rangingfrom lasers without population inversion [4,5] to new trends in nonlinear optics [6]. Oneof the most striking phenomena connected with EIT is that atoms prepared in a coherentsuperposition of states can produce a steep dispersion and a large resonant index of refractionwith vanishing absorption [7]. Preparation of matter in such a state (which has been dubbed“phaseonium” [5]) provides us with a new type of optical material of interest both in its ownright, and in many applications to fundamental and applied physics.A common condition for applications of EIT is a high optical density of the resonantmedium. For example, in experiments demonstrating enhancement of index of refraction(χ′ ∼ 10−4) the density of the particles was N ∼ 1012 cm−3 [7]. Also, for highly sensitivemagnetometry based on atomic phase coherence the density of atoms is estimated to beN ∼ 5× 1012 cm−3 [8].For optically thick media, reabsorption of spontaneously emitted photons can becomeimportant. This process, called radiation trapping, has been studied extensively in astro-physics, plasma physics, and atomic spectroscopy [9]. Radiation trapping has been predictedand demonstrated to have a destructive effect on the orientation produced by optical pump-ing [10–12]. Because the spontaneously emitted photons are dephased and depolarized withrespect to the coherent fields creating the atomic polarization, the effect of radiation trappingcan be described as an external incoherent pumping of the atomic transitions [12]. Underthe conditions of EIT, there are not many atoms undergoing spontaneous emission. How-2ever, these spontaneous photons destroy the atomic coherence in the same way as incoherentpumping. This effect can change the results of CPT and EIT experiments significantly.In this Letter we report the observation and analysis of the increase of the effective decayrate of Zeeman coherence due to radiation trapping. We create the coherence between groundstate sublevels of the D1 line of87Rb vapor, and study the effect of CPT as a function of thevapor density using polarization spectroscopy based on nonlinear magneto-optic rotation(NMOR) [13–15]. We see that the the relaxation time of the coherent state is determinednot only by the time-of-flight of the atom through the laser beam, but also by the densityof the atomic vapor. For atomic densities N ≈ 5 × 1012 cm−3 the effective coherence decayrate increases by several times compared with the decay rate for N ≈ 5 × 1011 cm−3. Thisdemonstrates the importance of radiation trapping for experiments with optically thickcoherent media, and the need to account for it to understand the experimental results.Our experimental setup is shown schematically in Fig. 1. An external cavity diode laseris tuned in the vicinity of the F = 2 → F ′ = 1 transition of the 87Rb D1 line. The laserbeam passes through a high-quality polarizer P1, and then through a cylindrical glass cellcontaining isotopically enhanced 87Rb. The length of the cell is L = 5.0 cm and its anddiameter is D = 2.5 cm. The laser power after the polarizer is P = 2.5 mW, and the beamdiameter is d = 2 mm. The glass cell is placed inside a two-layer magnetic shield to suppressthe laboratory magnetic field. A homogeneous longitudinal magnetic field is created by asolenoid mounted inside the inner magnetic shield. The density of the Rb vapor is controlledwith the temperature of the cell. A second polarizer P2 (a polarizing beam splitter) is placedafter the cell and is tilted at 45◦ with respect to the first polarizer. The two beams emergingfrom the polarizing beam splitter are detected with detectors D1 and D2. A simple analysisof the signals from the two channels gives the angle of rotation of the polarization φ and thetransmitted intensity Iout.We consider the linearly polarized light as two circular components E+ and E− whichgenerate a coherent superposition of the Zeeman sublevels (a dark state). To study thisdark state we apply a longitudinal magnetic field in the direction of light propagation which3leads to a splitting of the |b±〉 states of magnitude h¯δ0/2 = −µBB where B is the mag-netic field, µB is the Bohr magneton. Because the two circular components interact withground-state sublevels of oppositely signed magnetic quantum number, the changes in theindex of refraction for the two components have opposite sign. As a result of the indexchange, the components acquire a relative phase shift φ which leads to a rotation of thepolarization direction which is proportional to the magnetic field for small fields. Thus, bymeasuring the rotation as a function of applied magnetic field, we obtain information aboutthe dispersive properties of the medium. As the magnetic field increases, the absorption alsoincreases because the splitting between the sublevels destroys the two-photon resonance, sothe detection of the transmitted electromagnetic field intensity as a function of the magneticfield allows us to study the absorptive properties of the medium. Hence, using the nonlinearFaraday technique we can easily study both dispersive and absorptive properties of the EITresonance simultaneously.We have measured the polarization rotation slope dφ/dB and the transmission Iout/Iinof the cell. The detailed shape depends on the particular cell and laser beam size, but atypical result is shown in Fig. 2. The individual points on this plot correspond to differentatomic densities. These data cannot be fit by existing theoretical considerations for thissystem assuming a constant decay rate γ0 [15]. This fact is demonstrated by extrapolatingthe low density end of the curve (where Iout/Iin is nearly 1) with a constant decay rate, asshown in the dashed curve and inset. However, taking into account the effect of radiationtrapping we can understand this data quite well.We first analyze these data in terms of a simple theoretical model and then by a detailednumerical simulation. To simplify the analysis we neglect the process of optical pumpingand assume a closed system. We include a dephasing of the ground-state coherence with rateγ0 and neglect population exchange between the ground states. As in previous treatments,we model the effect of radiation trapping by introducing an incoherent pumping rate R fromground states |b±〉 to excited state |a〉. R is not a constant, but a function of all parametersof the system [12]. In particular, R depends on the excited-state population, because as4more population is transferred to the excited state, more radiation will be produced thatcan eventually cause this incoherent excitation. We assume that R may be of the same orderof magnitude as γ0 but that it is much less then the radiative decay rate γr of transitions|a〉 → |b±〉.We can understand the origin of this incoherent pumping by considering a two-levelsystem coupled to a radiation reservoir. The reduced density matrix operator derived inWeisskopf-Wigner approximation [16] has the formρ˙(t) = −n¯thγr [σˆ−σˆ+ρ(t)− σˆ+ρ(t)σˆ−]−(n¯th + 1)γr [σˆ+σˆ−ρ(t)− σˆ−ρ(t)σˆ+] + H.c. , (1)where n¯th is thermal average photon number in the reservoir, γr is the atomic decay rate ofthe upper level, σˆ− = |b〉〈a| and σˆ+ = |a〉〈b|. To model atomic excitation by the incoherentradiation in the reservoir, the incoherent pumping rate can be written as R = 2γrn¯th . FromEq. (1) we find the equations of motion for the excited state population:ρ˙aa = −2γr(n¯th + 1)ρaa + 2γrn¯thρbb . (2)In an optically thin atomic medium the probability of photon reabsorption is small andn¯th = 0. However, in optically thick media photons diffuse slowly and n¯th 6= 0. The value ofn¯th can be estimated from a rate equation˙¯nth = −ren¯th + raρaa , (3)where re is the photon escape rate and ra is the pumping rate due to the atomic decay. Bothre and ra depend on the geometry of the system and on the atomic density. In steady state,n¯th = raρaa/re , and it follows from Eq. (2) and ρaa + ρbb = 1 that re > ra.It is convenient to formally introduce a function f(N) defined by ra/re = f/(1 + f)which characterizes the radiation trapping, such that f(N) ≥ 0 and f(N = 0) = 0. Inthe case when most population is in the ground state, ρaa ≪ 1, we see from Eq. (2) thatρ˙aa ≈ −2γrρaa/(1 + f). In the limit of low light intensity |Ω| ≪ γr and weak radiation5trapping n¯th ≪ 1 the cw light propagation obeys d|Ω|2/dz ≃ −2κγrρaa, where Ω is the Rabifrequency of the transition, κ = (3/8pi)Nλ2γr , λ is the wavelength, and z is the distanceof propagation through the medium. We thus have a simple equation for the incoherentpumping rate due to radiation trapping:R = −1κf(N)1 + f(N)ddz|Ω|2 . (4)This is a very intuitively appealing model: radiation trapping can exist only if the coherentradiation is absorbed by the system and is scattered due to spontaneous emission.We now return to the problem of radiation trapping in a three-level system with EIT.Because the Doppler distribution depends on atomic density (temperature), it is simplest tostudy radiation trapping in the Doppler-free limit of EIT, i.e., when the absorption and thedispersion do not depend on the width of the Doppler distribution Wd . Doppler averagingshows that this condition is fulfilled for relatively large light intensities |Ω(z)| ≫Wd√γ0/γrfor any z.The stationary propagation of the right and left circular polarized electric field com-ponents through the atomic vapor is described by Maxwell-Bloch equations in the slowly-varying amplitude and phase approximation [16]. We solve the equations by considering onlythe lowest order in γ0, R and δ0, assuming |Ω−(z)|2 ≈ |Ω+(z)|2, where Ω± are the complexRabi-frequencies of the two optical fields. We separately consider the spatial evolution of theamplitudes and phases of these complex Rabi-frequencies by writing Ω±(z) = |Ω±(z)|eiφ±(z),and derive equations for the total intensity |Ω|2 and the relative phase φ = φ− − φ+ddz|Ω|2 = −κ(γ0 +R) , (5)ddzφ = δ0κ|Ω|2. (6)To solve these equations we must specify the functional form of the incoherent pumping rateR in the three level configuration. From the general properties of radiation trapping [9] andfrom the results from radiation trapping in a two-level system, we assume that in the caseof Doppler-free EIT the incoherent pumping can be modeled by Eq. (4) as in the case of a6two-level system. As we shall see, this model works very well.With this form for R, Eq. (5) can be easily solved and we arrive at∣∣∣∣∣Ω(z)Ω(0)∣∣∣∣∣2= 1−γ0κz|Ω(0)|2(1 + f(N)) (7)so from Eq. (4) we have R = f(N)γ0. Integration of Eq. (6) for the phase yieldsdφ(z)dB∣∣∣∣∣B→0=2µBh¯ (γ0 +R)ln∣∣∣∣∣Ω(0)Ω(z)∣∣∣∣∣2. (8)Detection of |Ω(L)/Ω(0)|2 and dφ(L)/dB, allows us to infer the value of the coherence decayrate as a function of the atomic density and estimate the radiation trapping effect. Thus wesee that for optically thick media the coherence decay rate increases with the density.For smaller intensities |Ω(z)| ≪ Wd√γ0/γr , Doppler-free EIT is not established, so theapproximation Eq. (4) is not valid and we do not discuss this regime here.Based on the low-density data in Fig. 2 (for which radiation trapping is negligible)Eqs.(7) and (8) allow us to determine the coherence decay rate to be γ0 ≈ 0.004γr . Giventhis value, we can then use the high density set of these data and Eq. (8) to obtain theincoherent pumping rate R due to radiation trapping. The dependence is shown by the dotsin Fig. 3. In general, the functional form of R is not an “absolute” and it changes if thecell geometry or laser beam size changes, which is a key signature of the effect of radiationtrapping.The probability of photon reabsorption becomes significant when the medium becomesoptically thick on the length scale of the atomic cell size [9] (under the Doppler-free EITcondition almost all atomic population is in the ground state), or38piNλ2dγrWd> 1 . (9)For our experiment γr/∆D ≈ 0.01, so Eq. (9) is fulfilled for N > 5 × 1010 cm−3. Fordensities less than this, radiation trapping is negligible and we have R ≈ 0. Above thisvalue there are two distinct regimes of behavior, with both seen in Fig. 3. At low density wehave that R increases linearly with density due to photon absorption and emission within7the cell. If the atomic beam is narrower than the radius of the atomic cell, as we have inour experiment (d ∼ 0.1D), the next regime occurs for densities when photon reabsorptionbecomes significant inside the laser beam. In our case this is N > 5× 1011 cm−3.To confirm our simple analytical calculations we have also made detailed numerical simu-lations of the experiment. We have considered light propagation in a thirteen-level Doppler-broadened system corresponding to the F = 2 → F ′ = 1, 2 transition in 87Rb. The decayof the atomic coherence was modeled by finite time of the flight through the laser beam(an open system). We solved the density matrix equations in steady state using the coher-ence decay rate as a fit parameter. In other words, we choose the effective coherence decayrate γ0 +R in such a way that our numerical points for the dispersion dφ/dB and intensityIout/Iin corresponds to the experimental results. This is shown in the solid line in Fig. 2. Thedependence for R/γ0 obtained this way is shown in the solid line in Fig. 3. We see that thesimple analytical analysis of the data coincides with the simulations for low atomic densitiesand diverges slightly for high densities. We explain this difference by inadequate intensityof the laser light. The maximum intensity of our laser (∼ 100 mW/cm2) corresponds to aRabi frequency |Ω0| ∼ 3.6γr , which lies on the edge of Doppler-free region determined by|Ω0| ≥ Wd√γ0/γr ≈ 6γr . The absorption further decreases the intensity resulting in theDoppler broadening becoming important, unlike in our simplified calculations.Finally, we note that the observations reported here cannot be explained by spin exchangecollisions between the atoms. The collisional cross section for Rb atoms is approximately2× 10−14 cm2 [10] which results in a coherence decay rate γ0 ≈ 2× 10−5γr for the densitiesreported here. This is approximately two orders of magnitude less than the time-of-flightlimited coherence decay rate γ0 ≈ 4× 10−3γr that we measured.In conclusion, we have shown both experimentally and theoretically that the effect ofradiation trapping enhances the decay rate of the atomic coherence established by linearlypolarized laser radiation between Zeeman sublevels. This effect leads to significant increaseof the residual absorption in EIT experiments with optically thick atomic vapors.The authors gratefully acknowledge useful discussions with D. Budker, M. Fleischhauer,8L. Holberg, E. E. Mikhailov, Y. V. Rostovtsev, V. A. Sautenkov, V. L. Velichansky, R.Wynands, and V. Yashchuk, and the support from the Office of Naval Research, the NationalScience Foundation, and the Welch Foundation.9REFERENCES[1] E. Arimondo, G. Orriols, Nuovo Cimento Lett. 17, 333 (1976); H. R. Gray, R. M.Whitley, C. R. Stroud, Opt. Lett. 3, 218 (1978); H. I. Yoo, J. H. Eberly, Phys. Reports118, 239 (1985).[2] E. Arimondo, in Progress in Optics edited by E. Wolf, Vol. XXXV, p.257 (ElsevierScience, Amsterdam, 1996).[3] For reviews on EIT see S. E. Harris, Phys. Today p. 36, June (1997); J. P. Marangos,J. Mod. Opt. 45, 471 (1998).[4] A. S. Zibrov et al., Phys. Rev. Lett. 75, 1499 (1995); G. G. Padmabandu et al., ibid.,76, 2053 (1996). J. Mompart and R. Corbalan, Quantum Semiclass. Opt. 2, R7 (2000).[5] M. O. Scully, Phys. Rep. 219, 191 (1992).[6] S. E. Harris, J. E. Field, and A. Imamoglu, Phys. Rev. Lett. 64, 1107 (1990); K. Hakuta,L. Marmet, and B. P. Stoicheff, ibid., 66, 596 (1991); P. Hemmer et al., Opt. Lett. 20,982 (1995); M. Jain et al., Phys. Rev. Lett. 77, 4326 (1996); A. S. Zibrov, M. D. Lukin,and M. O. Scully, ibid., 83, 4049 (1999); A. V. Sokolov et al., ibid., 85, 562 (2000).[7] M. O. Scully, Phys. Rev. Lett. 67, 1855 (1991); A. S. Zibrov et al., ibid., 76, 3935 (1996).[8] M. Fleischhauer, M. O. Scully, Phys. Rev. A 49, 1973 (1994); H. Lee, M. Fleischhauer,M. O. Scully, ibid., 58, 2587 (1998).[9] A. F. Molisch, B. P. Oehry, Radiation Trapping in Atomic Vapours (Clarendon Press,Oxford), 1998.[10] W. Happer, Rev. Mod. Phys. 44, 169 (1972).[11] D. Peterson, L. W. Anderson, Phys. Rev. A 43, 4883 (1991); T. Scholz et al., ibid., 53,2169 (1996).10[12] G. Ankerhold et al., Phys. Rev. A 48, R4031 (1993); M. Fleischhauer Europhys. Lett.45, 659 (1999).[13] W. Gawlik, inModern Nonlinear Optics edited by M. Evans and S. Kielich, Vol. LXXXVof Advances in Chemical Physics Series (Wiley, New York, 1994), part 3, p. 733; D.Budker, D. J. Orlando, V. Yashchuk, Am. J. Phys. 67, 589 (1999); V. A. Sautenkov etal., Phys. Rev. A 62, 023810 (2000); F. Schuller, D. N. Stacey, ibid., 60, 973 (2000).[14] D. Budker et al., Phys. Rev. A 62, 043403 (2000).[15] M. Fleischhauer, A. B. Matsko and M. O. Scully, Phys. Rev. A 62, 013808 (2000).[16] M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge University Press,Cambridge) 1996, p. 252.11FIGURESFIG. 1. Diagram showing the experimental setup; inset: Idealized three-level Λ-scheme con-sidered in the theoretical calculations.FIG. 2. The dependence of rotation rate dφ/dB on transmission through the system Iout/Iin :experimental (dots), previous theory with γ0 = 0.004γr (dashed line), and obtained by numericalsimulation including radiation trapping. (solid line).FIG. 3. The incoherent pumping rate R/γ0 due to radiation trapping as a function of atomicdensity N : calculated by applying Eq. (8) to the data (dots) and obtained by numerical simulation(solid line).12Laser beamPolarizer P1Polarizer P2(polarizingbeam-splitter)SolenoidGlass cell withRb vaporPhoto-detector D1Photo-detector D2a+b-bE+E-γ0Radiation trapping in oherent mediaMatsko et al. | Fig. 1130.0 0.2 0.4 0.6 0.8 1.0051015202530Rotation rate dφ  /dB, rad/GTransmission Iout/Iin0.96 0.98 1.000.00.51.01.5Radiation trapping in oherent mediaMatsko et al. | Fig. 2140 1 2 3 4 5 6012345Incoherent pump rate R/γ 0Atomic density, 1012 cm-3Radiation trapping in oherent mediaMatsko et al. | Fig. 315

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Radiation trapping in coherent media

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