AU0019012

Theoretical calculation of saturated absorption for multilevel

''*'<••, T.J. O'Kane, R.E. Scholten and P.M. Farrellf £ School of Physics, University of Melbourne, Parkville 3052, Australia ^ Ckptical Technology Research Laboratory, Victoria University, Footscray 8001, Australia x >" (November 16, 1998) We present the first theoretical saturated absorption spectra for general multi-level atoms, using a model based on extensions of the optical Bloch equations, and using Monte Carlo averaging of the absorption of individual atoms with random trajectories through a standing wave. We are for the first time able to accurately predict the merging of hyperfine and cross-over due to intensity dependent phenomena such as power broadening. Results for 20-level sodium and 24-level rubidium models are presented and compared to experiment, demonstrating excellent agreement.

32.70.-n, 32.80.Bx interactions with a standing wave field lie model for investigating optical pumping. However, for a at the heart of many problems ranging from the theory travelling wave the field-atom coupling is linear, thereby of the gas laser [1], to saturation spectroscopy [2] and removing the need for a true moving atom model. [3,4]. The calculation of the effect of these To accurately predict saturated absorption spectra, we interactions on the internal state and external motion of have developed a model for multi-level atoms, using op- the atoms, and on the laser field, has been approached tical pumping calculations for each individual atom as from several directions, from continued fractions [5], to it moves through a true standing wave. Our model cor- perturbation theory [6], rate equations [7] and quantum rectly predicts, for the first time, saturated absorption Monte Carlo methods [8]. We are particularly interested spectra for multi-level atoms, including the basic absorp- in saturated absorption spectra, as a problem which is tion peaks, power broadening, and merging of the hy- easily tested by experiment, yet incorporates all the fun- perfine and cross-over resonances. We present saturation damental aspects of the standing wave-atom interaction. spectra calculated using a Monte Carlo approach for the Earlier work on the calculation of such spectra has re- generation and averaging of atoms whose semiclassical quired approximations that ignore many of the impor- density matrix [17] has an explicit spatial and temporal tant physical processes. The continued-fraction method dependences. The individual nonlinear processes, statis- is limited to two-level atoms and weak field-atom cou- tically averaged in the calculations of saturated absorp- pling [5]. Perturbation theory [6,9] has also been used tion presented here, are discussed in detail elsewhere [18]. for two-level atoms, treating saturation effects as a first- We consider multi-level rubidium (Rb87) and sodium order correction to the travelling wave solution through (Na23) systems (Fig. 1). Such an ensemble is readily de- expansion of the matrix elements in a Fourier series with scribed by an N x N density matrix [17] whose diagonals slowly varying coefficients. This method is only appli- represent the populations of the respective energy lev- cable when one of the counter-propagating waves is well els; N is the number of states. The atom, with resonant below saturation intensity (i.e. strong pump, weak probe) transition frequency of wo, is subjected to two counter and in the low intensity regime where the adiabatic ap- propagating electromagnetic waves of the same angular proximation is valid [10]. frequency u>, propagating in opposite directions along the With two equal-intensity counter-propagating beams, z axis, with a+ polarization relative to a single direction. the atoms move through a spatially-varying field, and We extend the optical Bloch equations by applying the therefore experience a time-dependent Hamiltonian. The formalism of Allen and Eberly [19] to both 24-level rubid- strong nonlinearity of the atom-field coupling, even at ium and 20-level sodium systems, in the style of McClel- the low intensities typical of saturated absorption spec- land and Kelley [15].- We first define the matrix element troscopy, leads to sensitive dependence on the velocity, of the interaction Hamiltonian in the dipole approxima- laser intensity, and atom trajectory, illustrated by com- tion as plex limit cycle behavior, period doubling, and chaos [11,12]. Pegg and Schulz [13] introduced a technique to deal with this time dependence for two-level atoms, using f=±l,0 a series of transformations to rotating reference frames, but this approach was only valid for specific oscillation where the standing wave field is given by frequencies, and extension to multi-level atoms becomes unwieldy. More recently, Nakayama [7] developed a four E = Ecos(kz) (ev exp(-iwt) + e* exp(iwi)). (2) level rate equation model for the Na D2 line in saturation, Ee is the (real) electric field amplitude at polarization but this model neglects coherence terms. Balykin [14], u e , where v = ±1,0 for cr± circular, linear laser polariza- McClelland and Kelley [15], and later Farrell et al. [16], u tion respectively, z is the distance along the laser propa- developed and extended the multi-level travelling wave v gation axis, C ap is the Clebsch-Gordan coefficient for the a —>• (3 transition, and JX is the dipole transition matrix el- Excited state diagonals: ement. The quantity fj,E/h = T\/Io/2Isat for an atomic decay rate of F, a travelling wave of peak intensity /o, (7) and a saturation intensity of Isat. For a standing wave, -900 we replace h with /, the periodic spatially-dependent intensity given by: Off diagonals:

2-KZ ifiE(z) = 470 W2 (3) ~ P0(i) diara dt

where XiaSer is the wavelength and Idiam the beam diam- (8) eter. The time evolution of the density matrix elements is given by where p@a are the matrix elements in the rotating frame. These equations are solved via Runge-Kutta tech- (4) niques with the initial populations evenly distributed dt among the ground state diagonals and al] other elements of the density matrix set to zero. The time step through to which we add phenomenological decay terms in the the numerical routine is bounded by the radiative pro- semi-classical style [19]. F = 1/r is determined from the cesses, i.e. dt « F"1, Q,~l, A.~1 where Q = ^B/h is the lifetime of the transition, r = 16.237ns for sodium [20] Rabi frequency and A the total detuning (including any and 25.8 ns for rubidium [21]. Doppler component). The time evolution equations with decay terms pro- In a typical saturated absorption spectrum experi- duce 576 coupled differential equations, with coefficients ment, atom-atom interactions can be neglected because that have a cosinusoidal dependence on time due to the of the low temperature and pressure in the vapor cell [22]. standing wave field [3]. We make several approxima- The observed structure is therefore a statistical average of tions to reduce the numerical task, firstly invoking the the spectra of many individual, mutually noninteracting rotating wave approximation [19] thereby eliminating all atoms. To calculate this structure we first generate a pair terms that oscillate at twice the optical frequency. We of random deviates with zero mean and unit variance, exclude all off-diagonal terms that do not couple an ex- scaled to produce a Maxwellian distribution of speeds. cited state to a ground state (Fig. 1). Farrell et al. [16] A random angle is then generated to define an atom's have developed a full Q.E.D. model, in which all relevant trajectory through the 2-dimensional (cylindrically sym- off-diagonal matrix elements within degenerate manifolds metric) Gaussian standing wave field. were identified, but these extra terms were not significant Each atom moves through the field with a constant at the low intensities typical of saturated absorption ex- velocity thereby experiencing a changing intensity de- periments. pendent on each new position in the field. A 4fh-order We also ignore coupling between magnetic sublevels of Runge-Kutta method is employed to track the position the same F level as well as off-diagonal pumping terms and the changing field intensity thus allowing the atom- that couple from the F = 1 ground state to the excited field system to simultaneously evolve as the atom moves states. These terms become significant only at laser in- through the standing wave. The calculation is terminated tensities such that power broadening causes significant when a trajectory takes the atom out of the beam, or af- amplitude at the .F = 1 to .F = 0,1,2 transitions, or in ter 200 lifetimes. To produce the absorption spectrum, the case of two frequency pumping. Further we invoke the density matrix is calculated for many random trajec- hermiticity (i.e. pap = p*^) so that we are left with a tories at a given laser detuning, and averaged. The laser final set of 46 equations. detuning is then incremented and the process repeated. For a pure laser polarization, using a,fi to denote The absorption is normally calculated from the den- ground and excited states respectively, coupled when sity matrix as Im{Tr(C^/3 ppo,)}, but this is applicable a + v = /3, we have for the F = 1 ground state diago- only for a single travelling wave, or as an approxima- nals: tion in the strong pump - weak probe situation. In fact, 17 the off-diagonal coherences, pap{a ^ ,0), average to zero dpaa 2^ (5) because the atom sees a modulated field intensity as it dt P00 passes through the nodes and anti-nodes of the standing wave [13,10]. We take the total fluorescence emitted from F = 2 ground state diagonals: the atoms as our measure of the effective absorption, cal- 24 culated from: dpa > (a|/i*|0> (9) dt 0=9 24 (6) where JJ, is the atomic dipole operator; (/3\fx\a) — 0=9 {a\(j,*\/3) = /x. Fluorescence arises from spontaneous emission only, and hence depends only on the substate non-linear phenomena that has previously not been con- populations. For a a+ circularly polarized field on res- sidered. Using our approach we have, for the first time, onance, the atoms are optically pumped into the F = demonstrated excellent agreement with experimental sat- 2,m/ = 2 ground and F = 3,m/ = 3 excited states, urated absorption spectra for multi-level atoms, includ- so the population of the latter state is a good measure ing hyperfine and cross-over resonances and saturation of the total fluorescence. Note that stimulated emission broadening. drives oscillations in the off-diagonal coherences, which We gratefully acknowledge the support of the Aus- then contribute to the population inversion (Eq. 8), and tralian Research Council and (T.J.O.) the support of an hence indirectly to the average fluorescence. This is a Australian Postgraduate Award. fundamental difference between the rate equations pre- viously used [7] and our density matrix equation model derived from the optical Bloch equations. For a linear travelling wave, only atoms with the ap- propriate combination of Doppler-shifted fre- quency and laser detuning will be on or near resonance, and therefore have significant interaction with the field. [1] W.E. Lamb, Jr., Phys. Rev 134 A1429 (1964) The introduction of a second counter-propagating field [2] W. Demtroder, Laser spectroscopy, 2e, Springer-Verlag, adds saturation broadening, cross-over resonances and Berlin pp444 (1981) multi-photon processes so that it is no longer possible to [3] C. Cohen-Tannoudji, Les Houches, Fundamental select only on-resonant atoms. In fact, we find atom-field Systems in Quantum Optics, 1-161 (1990) couplings for velocities well above the primary resonance [4] Special issue, J. Opt. Soc. Am. B 6 2020-2226 (1989) that are not predicted in the standard perturbation the- [5] S. Stenholm and W.E. Lamb, Jr., Phys. Rev. 181 618 ory [6], arising from the exact contribution of all higher (1969) order nonlinear terms. Hence we must include atoms [6] S. Haroche and F. Hartmann, Phys. Rev. A 6 1280 with the full range of velocities from our Maxwellian dis- (1972) tribution. Although our density matrix equation model [7] S. Nakayama, Physica Scripta T70 64 (1997) is expensive computationally, it ensures the inclusion of [8] K. Molmer, Y. Castin and J. Dalibard, J. Opt. Soc. Am. B 10 524 (1993) all nonlinear processes that contribute to the observed [9] E.V.Bakalanov and V.P.Chebotaev, Sov.Phys.JETP 33 spectrum. 2 300 (1971); op cit 35 287 (1972) With 70 = lW/cm , saturated absorption spectra [10] K. Molmer, Physica Scripta 45 246 (1992) were obtained for both rubidium and sodium (Figs. 2 [11] P.W. Milonni, M.-L. Shih and J.R. Ackerhalt, Chaos in & 3), with 104 atoms per detuning. In both cases we ob- Laser-Matter interactions, World Scientific, Singapore serve not only Doppler broadening but power broadening (1987) and merging of the hyperfine and cross-over resonances. [12] D.J. Gauthier, M.S. Malcuit and R.W. Boyd, Phys. These effects are in good agreement with the experimen- Rev. Lett 61 1827 (1988) tal results for rubidium (Fig. 2). The hyperfine and cross- [13] D.T. Pegg and W.E. Schulz, J. Phys. B 17 2233 (1984) over resonances should be calculable at lower intensities. [14] V.I. Balykin, Opt. Commun. 33 31 (1981) However, this would require many more atoms to reduce [15] J.J. McClelland and M.H. Kelley, Phys. Rev. A 31 3704 the statistical noise, such that the smaller and narrower (1985) saturation peaks could be distinguished, and to average [16] P.M. Farrell, W.R. MacGillivray and M.C. Standage, out large oscillations in the populations at small laser Phys. Rev. A 37 4240 (1988) detunings for on-resonance atoms. [17] K. Blum, Density Matrix Theory and Applications, 2e, _ Instead, using the_sodium model, we consider the Plenum Press, New York (1996) F = 1, m/ = 1 and F — 2, m/ = 2 ground state pop- [18] T.J. O'Kane, R.E. Scholten, P.M. Farrell and M.R. ulations (Fig. 4). The F — 1, m/ = 1 populations are de- Walkiewicz, "Nonlinear processes in standing wave laser termined solely by spontaneous decay (i.e. fluorescence) interactions with multi-level atoms", submitted. while the F = 2,ra/ = 2 ground state is coupled via [19] L. Allen and J.H. Eberly, Optical Resonance and pumping to the F = 3, m/ = 3 excited state, hence both Two-level Atoms, Wiley, New York (1975) these substates are independent of or only marginally [20] C.W. Oates, K.R. Vogel and J.L. Hall, Phys. Rev. Lett subject to the oscillations discussed above. They show 76 2866 (1996) clearly resolved hyperfine and cross-over resonances for [21] G. Belin, Physica Scripta 4 269 (1971) [22] CM. Bowden and J.P. Dowling, Phys. Rev. A 47 1247 2 x 10* atoms per detuning, at twice saturation intensity (1993) (12mW/cm2). In summary, we have developed a detailed model for atom interactions with a standing wave, including care- ful treatment of non-linear components. The statistical nature of vapour cell experiments hides a depth of such m, -3 -2 -1 0 1 2 3

18 19 20 21 22 23 24

59.6 MHz . 267.2 MHz 13 14 15 16 17

157.1 MHz 10 11 12 F=1 72.3 MHz 9 F = 0 0.01 589nm 780nm -1500 -1000 -500 0 500 1000 Detuning (MHz)

FIG. 3. F — 3, m/ = 3 excited state population versus de- >1/2 j= . = tuning in Na23, again at 1 W/cm2. The lower mass of sodium 1.772 GHz 6.8347 GHz gives a much broader Doppler absorption profile. The hy- F = 1 perfine F = 2 excited state and the F = 2, 3 cross-over are ,223 8; 7 Na Rb not visible due to the large power broadening relative to the splitting between the states (59 MHz). The statistical uncer- 2 FIG. 1. Energy level diagram for the 20-level 3 51/2, tainties are shown for —30 MHz and 0 MHz; calculated points 2 23 2 2 3 P3/2 states of Na and 24-level 5 5i/2, 5 P3/2 states of are connected by straight lines. Rb . The excited state numbering is shown for the full 24 levels used for Rb; the F = 0,1 excited states are omitted in the Na model. 0.136 (a)F = 1,mf =

0.134

0.04

c 0.03 • O i. 0.02 s. 0.01 £

0.00 0.146 -600 -400 -200 0 200 400 600 Detuning (MHz) 0.142 FIG. 2. F = 3, m/ — 3 excited state population versus de- tuning in Rb87. (•) Theoretical calculation for a laser tuned to 2 0.138 the F = 2->F = 3 transition, and intensity Jo = 1 W/cm , -100 -50 0 showing large power and Doppler broadening effects. Each Detuning (MHz) point is an average of 104 atoms with a Maxwellian distri- bution of velocities. Statistical uncertainties for 10 measure- ments are shown at 0 MHz and 400 MHz; calculated points are connected by straight lines. (-) Experimental measure- ment of absorption using configuration in fig. 7.10 of [2], for a 40 mm vapor cell at room temperature, with tuneable diode laser (300 kHz linewidth). The beam was elliptical, 0.6 mm x 0.7 mm (1/e2), and circularly polarised, with-Jo « 1.3 W/cm2. Note that the experimental curve has been displaced for clar- ity. 2 FIG. 4. Ground state (3 5i/2) populations for Na with laser intensity set at twice saturation (12mW/cm2) and 2 x 104 atoms per detuning, (a) F = l,m/ = 1 sate. In the absence of power broadening effects we see the strong reso- nance at -59 MHz due to the F = 2 hyperfine energy level and smaller cross-over resonance. The F = 2 hyperfine resonance is strongest as our model only couples the F = 1, F = 2 levels via relaxation and does not couple the F = 1, F — 3 levels at all. At zero detuning, strong pumping to the F = 3, m/ = 3 state results in little relaxation into the F = 1 ground state. (b) F = 2, rrif — 2 state. The strong resonances now occur at the crossover and F = 3 (zero detuning) hyperfine resonance due to the full coupling of the F = 2, F = 3 transition. The F — 2 hyperfine resonance is weak as most atoms are lost via relaxation to the F = 1 ground state before they can be pumped across to the F = 1,irij = 2 ground state. Statistical uncertainties for 10 measurements are shown at —60 MHz and —30 MHz; calculated points are connected by straight lines.