Resonant optical pumping in dense atomic media is discussed, where the
absorption length is less than the smallest characteristic dimension of the
sample. It is shown that reabsorption and multiple scattering of spontaneous
photons (radiation trapping) can substantially slow down the rate of optical
pumping. A very slow relaxation out of the target state of the pump process is
then sufficient to make optical pumping impossible. As model systems an
inhomogeneously and a radiatively broadened 3-level system resonantly driven
with a strong broad-band pump field are considered.Comment: 6 pages, 5 figure

Fleischhauer, Michael

English

arXiv

Resonant optical pumping in dense atomic media is discussed,
where the absorption length is less than the smallest characteristic dimension of the sample.
It is shown that reabsorption and multiple scattering of spontaneous photons (radiation trapping) 
can substantially slow down the rate of optical pumping. 
A very slow relaxation out of the target
state of the pump process is then sufficient to make optical pumping 
impossible. As model systems an inhomogeneously and a radiatively
broadened 3-level system resonantly driven with a strong broad-band pump field
are considered

M. Fleischhauer

EDP Sciences OAI-PMH repository (1.2.0)

Optical pumping in dense atomic media: Limitations due 
to reabsorption of spontaneously emitted photons

Abstract: Resonant optical pumping in dense atomic media is discussed, where the absorption length is less than the smallest characteristic dimension of the sample. It is shown that reabsorption and multiple scattering of spontaneous photons (radiation trapping) can substantially slow down the rate of optical pumping. A very slow relaxation out of the target state of the pump process is then sufficient to make optical pumping impossible. As model systems an inhomogeneously and a radiatively broadened 3-level system resonantly driven with a strong broad-band pump field are considered

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arXiv:quant-ph/9811017   6 Nov 1998Optical pumping in dense atomic media: Limitations due to reabsorption ofspontaneously emitted photonsMichael FleischhauerSektion Physik, Ludwig-Maximilians Universitat Munchen, D-80333 Munchen, GermanyResonant optical pumping in dense atomic media is discussed, where the absorption length isless than the smallest characteristic dimension of the sample. It is shown that reabsorption andmultiple scattering of spontaneous photons (radiation trapping) can substantially slow down therate of optical pumping. A very slow relaxation out of the target state of the pump process isthen sucient to make optical pumping impossible. As model systems an inhomogeneously anda radiatively broadened 3-level system resonantly driven with a strong broad-band pump eld areconsidered.I. INTRODUCTIONOptical pumping is an established technique in atomic and molecular physics to selectively populate or depopulatespecic states or superpositions [1,2]. It is based on the absorption of photons of a specic mode and subsequentspontaneous emission into many modes. The dissipative nature of the latter part makes it possible to transform mixedinto pure atomic states. From this results the importance of optical pumping for state preparation in systems with athermal distribution of population and for laser cooling [3].The maximum achievable rate of pumping is determined by the escape time of the emitted photons, which inoptically thin media is given by the free-space radiative lifetime. When the medium becomes optically thick, however,i.e. when the absorption length becomes smaller than the smallest sample dimension, the escape time of photons canbe substantially reduced. This phenomenon, known as radiation trapping [4], is due to reabsorption and multiplescattering of spontaneously emitted photons and can drastically reduce the rate of optical pumping in dense media.These limitations could be of major importance in many dierent elds as for instance near-resonance linear andnonlinear optics in dense media [5,6] or the realisation of Bose condensation by velocity selective coherent populationtrapping (VSCPT) [7].To describe the reabsorption and multiple scattering of photons we here utilize a recently developed approach toradiative interactions in dense atomic media [8]. In this approach a nonlinear and nonlocal single-atom density matrixequation is derived which generalizes the linear theory of radiation trapping [4] to the nonperturbative regime. As amodel system a 3-level  conguration driven by a strong broad-band eld is considered and the limits of (i) largeinhomogeneous and (ii) purely radiative broadening are studied.Let us consider the -type system shown in Fig. 1. A strong driving eld with (complex) Rabi-frequency (t)couples the lower state jci to the excited state jai, which spontaneously decays into jci and jbi. Since jbi is notcoupled by the driving eld, this results in optical pumping from jci to jbi. We also take into account a possible nitelifetime of the target state described by a population exchange between the lower states at rate 0.acΩ γ γ Γγ 0 bFIG. 1. Optical pumping in a  system.It was shown in [8] that the eect of the incoherent background radiation can be described by additional (nonlinearand nonlocal) pump and relaxation rates and level shifts in the single-atom density matrix equation. If we assumeorthogonal dipole moments or suciently dierent frequencies of the two optical transitions, the level shifts arenegligible. Also if the driving eld is strong, the incoherent photons do not aect the pump transition a $ c. Thus1we are left with a pump and decay rate  (t) on the a$ b transition and the eective single-atom equations of motionread in a rotating frame:_aa =  ( + 0 +  )aa +  bb + i(ac   c:c); (1)_cc = 0aa + 0bb   0cc   i(ac   c:c); (2)_ac =  (iac +  ac)ac + i(aa   cc): (3)ac is the detuning of the drive eld from resonance and  ac is the respective coherence decay rate. It should benoted, that   is a function of the density matrix elements of all other atoms, and hence the Eqs. (1-3) are nonlinearand nonlocal.We are here interested in genuine optical pumping and therefore consider a broad-band pump [9], i.e. (t) is assumedto have a vanishing mean value and Gaussian -like correlations(t)(t0)= R(t t0): Formally intergating Eq.(3),substituting the result back into Eqs.(1) and (2), and averaging over the Gaussian distribution of the pump eld leadsto the rate equations_aa =  ( + 0 +  )aa +  bb  R(aa   cc); (4)_cc = 0aa + 0bb   0cc + R(aa   cc): (5)II. COLLECTIVE DECAY RATEWe now have to determine the collective rate  .   is proportional to the spectrum of the incoherent eld at theposition ~r0 and the resonance frequency ! of the atom under consideration [8] (!; t) =}2h2eD(~r0; !; t) = }2h2Z 1 1d hhÊ (~r0; t)Ê+(~r0; t+  )ii ei! : (6)Here Ê are the positive and negative frequency parts of the eld operators, } is the dipole matrix element of theatomic transition, and hhABii  hABi   hAihBi. eD(!) can be obtained by summing the spontaneous emissioncontributions of all atoms propagated through the medium [8]D(1; 1) =ZZd3 d4Dret(1; 3)Dret (1; 4) s(3; 4): (7)Here Dret(1; 2) is the retarded propagator of the electric eld inside the medium, which obeys a Dyson-equation inself-consistent Hartree approximation:Dret(1; 2) = Dret0 (1; 2) ZZd3 d4Dret0 (1; 3)ret(3; 4)Dret(4; 2): (8)In Eqs.(7) and (8) the numbers 1; 2 : : : stand for f~r1; t1g; f~r2; t2g : : :, and the intergrations extend over time from 1 to +1 and over the whole sample volume. Dret0 is the free-space retarded propagator of the electric eld. Forsimplicity we here have disregarded polarisation. We also have introduced the atomic source correlation s(~r1; t1;~r2; t2) =}2h2Xjyj (t1)j(t2)(~r1   ~rj) (~r2   ~rj) (9)and the atomic response functionret(~r1; t1;~r2; t2) =}2h2(t1   t2)Xjyj(t1); j(t2)(~r1   ~rj) (~r2   ~rj); (10)where j = jbijjhaj is the spin-ip operator of the jth atom and  is the Heaviside step function. In terms ofthe 's the dipole operator of the jth atom reads dj = }(j + yj). The names reect the physical meaning of thequantities (9, 10). The Fourier-transform of  s is proportional to the spontaneous emission spectrum of the atomsand that of ret gives the susceptibility of the medium. Eqs.(7) and (8) represent a nonperturbative summation of the2spontaneous radiation contributions of all atoms propagated through the medium. It assumes a Gaussian statistics,which is however a good approximation for the background radiation.The Dyson-equation (8) was solved in [8] with some approximations in a macroscopic (continuum) limit where(~r1; t1;~r2; t2) =Rd3~r P (~r; t1; t2) (~r1   ~r) (~r2   ~r). This yielded for the collective decay rate (!; t) =}2!4(6)220c4ZVd3~re2q000(~r;!;t)rr2eP s(~r; !; t); (11)where r = j~r0   ~rj is the distance bewteeen the source and the probe atom. The probability that a photon reachesthe probe atom is determined by the absorption coecientq000 (~r; !; t) =h!30cReh eP ret(~r; !; t)i : (12)One can easily calculate the atomic source and response functions for the -system of Fig. 1.eP ret(~rj ; !; t) = }2h2Njaa(t)   jbb(t) ab + i(!   !jab); (13)eP s(~rj ; !; t) = 2}2h2Njaa(t) ab( ab)2 + (!   !jab)2; (14)where N is the density of atoms, !jab is the resonance frequency of the jth atom,  ab the coherence decay rate of thecorresponding transition and the overbar denotes averaging over a possible inhomogeneous distribution of frequencies.At this points we shall distinguish two limiting cases. We rst consider the limit of large Doppler-broadening andsecondly the case of purely radiative broadening.III. INHOMOGENEOUSLY BROADENED SYSTEMThe approach of [8] is based on the Markov approximation of a spectrally broad incoherent radiation. This ap-proximation is justied for example in an inhomogeneously broadened system. We therefore discuss rst the case oflarge Doppler-broadening. If we are interested in the population dynamics on a time scale slow compared to velocitychanging collissions, we may set j(t) = j(t)  (~rj; t) and thus have the same population dynamics in allvelocity classes. Since   depends on the populations of all atoms, Eqs.(4) and (5) are nonlocal. In the case of aconstant density of atoms and a homogeneous pump eld,   and hence all density matrix elements will be approx-imately homogeneous. We therefore make a simplifying approximation and disregard the space dependence. Thevolume integral is then carried out by placing the probe atom in the center of the sample. This yields for a GaussianDoppler-distribution of width D   (!; t)=aa(t)bb(t)  aa(t)h1  exp H(t)e 2=22Di; (15)where  = !   !0ab is the detuning from the atomic resonance at rest, and H(t) = K [bb(t)  aa(t)]: K = g N2dewith g = =p2D characterizes the number of atoms within one relevant velocity class in a volume given by thewavelength squared and the eective escape distance de . In deriving (15) we have used the relation between thefree-space radiative decay rate  and the dipole moment }: }2 = 3h0c3=!3 [10]. de corresponds for a longcylindrical slab to the cylinder radius; for a thin disk to its thickness and for a sphere to its radius.Averaging over the inhomogeneous velocity distribution of the atoms eventually yields (t) =  (!; t) =Z 1 1d!1p2De 2=22D  (!; t)= aa(t)bb(t)  aa(t)1pZ 1 1dy e y2h1  exp H(t)e y2i: (16)In Fig. 2a we have shown the population in the target state jbi as function of time starting from equal populationsof levels jci and jbi at t = 0. We here have assumed that the target state is stable, i.e. 0 = 0. One recognizes that3optical pumping is considerably slowed down already for values of K on the order of 10, which usually correspondsto much less than one atom per 3. The slow-down of pumping is further illustrated in Fig. 2b, where the eectivepump rate dened as p    ddtln[aa + cc] (17)is plotted normalized to the value in an optically thin medium ( 0p = =2). One can see that the optical pump rateapproaches a constant asymptotic value, which for K  1 and large pump rates R is given by asp =2K  lnK1=2  2 : (18)tγ tγ0 100 200 300 400 5000.50.60.70.80.910 100 200 300 400 5002.5-21.5-10.5K= 0     50       5     10       1K= 1       5     10     50(t) pΓpΓ 010logbbρFIG. 2. (a) Time evolution of population of level jbi for 0 = 0, R= = 10 and 0= = 1 for dierent density parametersK = g N2de , g = =p2D. (b) Corresponding eective rate of optical pumpingSince we have assumed in the plots of Fig. 2 an innitely long-lived target state (0 = 0), all populations eventuallyends up in jbi. However if 0 is nonzero and in particular if it becomes comparable to the asymptotic rate  asp , thesteady-state populations of all states equalize. In this case optical pumping is less and less ecient and becomeseventually impossible. This is illustrated in Fig. 3, where the stationary population in state jbi is shown as a functionof the density parameter K for dierent values of 0.0 1 2 3 4 5.4.5.6.7.8.91= 10 10 10 10-2 -3 -4 -5γ0γbbρssK10logFIG. 3. Stationary population in level jbi for R= = 10, 0= = 1 and dierent values of 0 as function of density parameterK.IV. RADIATIVELY BROADENED SYSTEMWe now discuss the case of a radiatively broadened system. In analogy to the case of inhomogeneous broadening,we nd for the spectral distribution (!; t)=aa(t)bb(t)   aa(t)1  exp H(t) ab ab 2ab +2; (19)4where  = ! !ab,  ab = ab+ , and ab = (+ 0 +R+ 0)=2, and H(t) = eK [bb(t)  aa(t)]: Here eK = eg N2dewith eg = =2ab. As opposed to the corresponding relation in the inhomogeneous case, Eq.(19) determines thecollective decay rate only implicitly, and   needs to be calculated self-consistently. For small atomic densities oraa  bb the exponential function in Eq.(19) can be expanded into a power series. The rst nonvanishing termfound from this has the same spectral shape than the single-atom response function. In such a case the Markovapproximation used in [8] is no longer valid and the approach is quantitatively incorrect. We shall nevertheless use itand discuss the range of validity afterwards.We nd that in the case of radiative broadening the rate of optical pumping decreases exponentially with the densityparameter as opposed to [N2de] 1 in the inhomogeneous case. For suciently large pump rates R and stable targetstate (0 = 0) the asymptotic rate of optical pumping is here asp =2exp  eK	: (20)Physically this is due to the fact that here the incoherent photons are in resonance with all atoms, which drasticallyincreases the scattering probability. As a consequence much smaller decay rates 0 out of the target state are sucientto make optical pumping impossible. This is illustrated in Fig. 4, where we have plotted the stationary population instate jbi as function of the density parameter K0 = N2de=2 for dierent values of 0.-1 0 1 2 3 40.40.50.60.70.80.91log K10-210-3-410-510-610γ0γ = 10ρbbss0FIG. 4. Same as Fig.3 for radiatively broadened system; K0 = N2de=2, R= = 10, 0= = 1In order to check the validity of the Markov approximation, we have shown in Fig. 6 the stationary normalizedspectral distribution  (!)=  for K0 = 1, 10 and 100 and 0= = 10 4. Also plotted is the atomic absorption spectrumfor K0 = 1 (solid line).-40 -20 0 20 4000.20.40.60.81∆ / γ1K  =100100FIG. 5. Spectral distribution of incoherent background radiation for R= = 10, 0 = , 0= = 10 4, and K0 = 1 (dotted),K0 = 10 (dashed) and K0 = 100 (dashed-dotted). Also shown is the normalized absorption spectrum for K0 = 1.One recognizes that spectrum of the background radiation has only a slightly larger width than the atomic responsefor K0 = 1. In this case the Markov approximation is not valid. The situation however improves when the density isincreased. Thus Fig. 4 has only qualitative character for lower densities.5V. SUMMARYWe have shown that resonant optical pumping in a dense atomic medium is substantially dierent from opticalpuming in dilute systems. When the absorption length of spontaneously emitted photons process becomes less thanthe minimum escape distance, these photons are trapped inside the medium and cause repumping of population. Thisleads to a considerable slow-down of the transfer rate and can make optical pumping impossible if the target stateof the pump process has a nite lifetime. The eect is much less pronounced in inhomogeneously broadened systemsdue to the reduction of the spectral density of background photons.These results may have some important consequences. It is practically impossile to use resonant optical pumpingin media with N3  1. This sets strong limits to the possibility to prepare pure states or coherent superpositions insystems with initial thermal occupation of states, such as Hyperne ground levels of alkali at room temperature. Eventhough the above analysis did not take into account quantum properties of the atoms and considers only resonantpumping, the results indicate, that it may be very dicult to achieve Bose Condensation via VSCPT in optical lattices[11]. Also the present results show that electromagnetically induced transparency (EIT) [12] in dense media cannot beunderstood as the result of optical puming into a dark state. Essential for EIT in dense media is an entirely coherentevolution [13] via stimulated adiabatic Raman passage [14]. Some of these aspects will be discussed in more detailelsewhere.ACKNOWLEDGEMENTThe author would like to thank C.M. Bowden and S.E. Harris for stimulating discussions.[1] A. Kastler, J. Phys. 11, 255 (1950).[2] C. Cohen-Tannoudji, \Optical pumping with lasers" in Atomic Physics IV p.589, ed. by G. zu Putlitz, E.W. Weber, A.Winnacker (Plenum, New York 1975).[3] see for example the special issuses: \ Laser Cooling and Trapping of Atoms", J. Opt. Soc. Am. B 6, 2020 (1989); \Opticsand Interferometry with Atoms", Appl. Phys. B 54, 319 (1992); \ Laser Cooling and Trapping", Laser Physics 4, 829(1994).[4] T. Holstein, Phys. Rev. 72, 1212 (1947); ibid. 83, 1159 (1951).[5] M. O. Scully, Phys. Rev. Lett.67, 1855 (1991); M. Fleischhauer et al. Phys. Rev. A .46, 1468 (1992).[6] M. Jain, H. Xia, G. Y. Yin, A. J. Merriam, and S. E. Harris Phys. Rev. Lett.77, 4326 (1996).[7] M. Olshanii, Y. Castin, and J. Dalibard in \Proceedings of the XII Conf. on Laser Spectroscopy" ed. by M, Inguscio,M. Allegrini, A. Sasso, (World Scientic, New York, 1995); Y. Castin, J. I. Cirac, and M. Lewenstein Phys. Rev. Lett. 80,5305 (1998).[8] M. Fleischhauer and S.F. Yelin, \Radiative atom-atom interactions in optically dense media: Quantum corrections to theLorentz-Lorenz formula" (unpublished, preprint quant-ph/9809087 29.Sept.1998).[9] A narrow-bandwidth coherent laser would cause Autler-Townes splitting and other interference eects in the atomicresponse and spontaneous emission spectrum. Furthermore in dense media the Lorentz-Lorenz corrections to the meanamplitude would have to be taken into account as well.[10] W. H. Lousiell, Quantum Statistical Properties of Radiation, (John Wiley & Sons, New York, 1973).[11] S.E. Hamann et al. Phys. Rev. Lett. 80, 4149 (1998); S. Friebel et al. Phys. Rev. A 57, R20 (1998).[12] For a recent review on EIT see: S.E. Harris, Physics Today 50, #7, 36 (1997).[13] S.E. Harris and Z.-F. Luo, Phys. Rev. A 52, R928 (1995).[14] For a review on STIRAP see: K. Bergmann, H.Theuer and B.W. Shore, Rev. Mod. Phys. 70, 1003 (1998).6

Optical pumping in dense atomic media: Limitations due to reabsorption of spontaneously emitted photons

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