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Theoretical calculation of saturated absorption spectra for multi-level

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Please note that terms and conditions apply. INSTITUTE OF PHYSICS PUBLISHING JOURNAL OF PHYSICS B: ATOMIC, MOLECULAR AND OPTICAL PHYSICS J. Phys. B: At. Mol. Opt. Phys. 39 (2006) 2709–2720 doi:10.1088/0953-4075/39/12/007

Theoretical calculation of saturated absorption spectra for multi-level atoms

L P Maguire1, R M W van Bijnen2,EMese3 and R E Scholten1

1 School of Physics, University of Melbourne, Victoria 3010, Australia 2 Eindhoven University of Technology, PO Box 513, 5600MB Eindhoven, The Netherlands 3 Department of Physics, Dicle University, Diyarbakir, Turkey

E-mail: [email protected]

Received 7 February 2006 Published 24 May 2006 Online at stacks.iop.org/JPhysB/39/2709

Abstract We have developed a model for calculating saturated absorption spectra for dipole transitions in multi-level atoms. Using a semiclassical density matrix formalism, we derive a set of coupled differential equations for the internal state of the in a standing wave light field. The equations are solved using standard integration techniques. The absorption at each detuning is found from an average of the absorption for a number of velocities along the laser field, thermally weighted. The method is relatively efficient computationally yet quantitatively predicts important details of saturated absorption spectra including saturation, crossover , merging of absorption lines at high intensity and optical pumping between hyperfine levels. We have measured saturated absorption and fluorescence spectra of 85Rb, and compare to our computational results for a 36-level model. (Some figures in this article are in colour only in the electronic version)

1. Introduction

Since the demonstration by Bennet [1] and Lamb [2] of spectral hole burning and saturated absorption with , saturated absorption spectroscopy has become a crucial part of many experiments in atomic physics and related areas, for example for the investigation of atomic and molecular structure [3] and as a frequency reference [4]. The nonlinear interaction between an atom (or molecule) and a near-resonant standing wave field lies at the heart of saturated absorption spectroscopy. Such interactions, between atoms and standing waves, are also fundamental to gas lasers [2, 5, 6], and trapping [7–9], and to many experiments in atom optics (e.g., [10]). The prediction of saturated absorption spectra, and more generally of the internal state and external motion of atoms in standing waves, has been difficult due to the rapid variation

0953-4075/06/122709+12$30.00 © 2006 IOP Publishing Ltd Printed in the UK 2709 2710 L P Maguire et al

mF -4 -3 -2 -1 0 1 2 34

28 29 30 31 32 33 34 35 36 F' = 4 120.7 MHz 21 2223 24 25 26 27 F' = 3 2 P3/ 63.4 MHz 2 16 17 18 19 20 F' = 2 29.3 MHz F' = 1 13 14 15 780nm σ+

Rb85

67891011 12 F = 3 2 S1 3.036 GHz 2 12345 F = 2

2 2 85 Figure 1. Energy level diagram for the 36-level 5 S1/2, 5 P3/2 states of Rb. A typical laser- coupled transition is shown for σ + circular polarization, between states α = 12 and β = 36. Energy separations are from [21].

of the field as atoms move through the standing wave, the nonlinearity of the atom–laser coupling and the complex internal structure of most atoms of interest. For two-level atoms in a travelling wave, analytic solutions are possible [11]. If the field is composed of a strong pump beam and a weak counter-propagating probe beam, rate equations [12, 13], continued fractions [6] or perturbative methods [14, 15] can be used. With two counter-propagating beams of equal intensity, the strong nonlinearity of the atom–field coupling, even at the low intensities typical of saturated absorption spectroscopy, leads to sensitive dependence on the velocity, laser intensity and atom trajectory, illustrated by complex limit cycle behaviour, period doubling and chaos [16]. Analytic calculations are possible for two-level atoms, by transformation to a reference frame rotating at the rate of change of the atom–laser coupling [17], but most atoms of interest have many more levels coupled to the laser field; for example, 85Rb with N = 36 states (figure 1). It has recently been shown [13] that optical pumping between hyperfine levels, and hence the time dependence of the interaction as the atom traverses the light field, plays a substantial role in determining the structure of saturated absorption spectra, at least in the conventional strong pump–weak probe configuration. Quantum Monte Carlo methods have been successfully applied to multi-level atoms in standing wave fields, for calculating the external motion of the atoms [18] and for calculating the internal state of the atoms in a travelling wave [19]. Extension to standing waves would be straightforward, but computationally demanding. In our model, we begin with the quantum master equation for the atom–laser interaction and derive a set of coupled differential equations for the time development of the atomic internal state density matrix. For a given velocity component along the laser field, the equations are integrated for small time steps along the atomic trajectory and the density matrix is averaged over that trajectory. We then average the density matrix solutions from a Maxwellian-weighted sampling of velocity groups, rather than using Monte Carlo generation of random atomic trajectories as in our earlier work [16]. The model is computationally orders of magnitude more efficient, yet quantitatively predicts saturated absorption spectra for multi-level atoms, including the basic absorption peaks, power broadening, and merging of the hyperfine and crossover resonances. We confirm that optical pumping between hyperfine Calculation of saturated absorption spectra for multi-level atoms 2711

Vapour cell λ λ PBS /4 /4 BS

Photodiode λ/2 Lens Lens Photomultiplier Fibre-coupled laser

Figure 2. Experimental arrangement for saturated absorption spectroscopy with a circularly polarized standing wave. Zero-order anti-reflection-coated λ/2andλ/4 waveplates are used to obtain equal power σ +-polarized counter-propagating beams in the vapour cell. BS, ; PBS, polarizing BS. levels is important in determining the saturated absorption spectra as noted in [13], but find that the pumping process is much slower than suggested. We have also measured saturated absorption and fluorescence spectra for a standing wave; that is, with equal pump and probe intensities, for comparison to the theoretical results.

2. Evolution of the density matrix in a standing wave

We consider multi-level 85Rb (figure 1), which can be described by an N × N Hermitian density matrix ρ [20]. The diagonals are the populations of the respective energy levels and the off-diagonals reflect the coherences between levels. The matrix elements ρβα are labelled by the substates with α, β = 1,...,36 for 85Rb. Each label identifies the angular momentum F and projection MF of the magnetic substate. The atom is subjected to a standing wave, created by two electromagnetic waves of the same angular frequency ω, tuned near dipole , propagating in opposite directions along the z axis (figure 2). The evolution of the atomic density matrix is given by Liouville’s equation [20] i¯hρ˙ = [H,ρ](1) where H is the full system Hamiltonian which should include terms for the atom and the atom–laser interaction, and for the vacuum and laser fields. We explicitly include spontaneous decay phenomenologically and take the laser field to be unchanging, so that the effective Hamiltonian reduces to H = H A + H I (2) where H A is the atomic Hamiltonian and H I is the atom–laser interaction. The evolution of the density matrix is then

i¯hρ˙ = [H,ρ]+Lrelax(ρ). (3)

Lrelax describes the decay of the excited-state populations and off-diagonal coherences for a given atomic state ρ. This evolution equation is often referred to as the quantum master equation, and with appropriate Hamiltonian and relaxation terms, is equivalent to the optical Bloch equations [11]. The matrix elements for the atomic Hamiltonian are zero except for the diagonals: A = Hβα hω¯ βγ δβα (4) 2712 L P Maguire et al where in generalhω ¯ βγ is the energy difference between states β and γ , and in this case γ is the ground state selected by the laser. In the dipole approximation, the interaction Hamiltonian is the scalar product of the atomic dipole µ =−er with the electric field E, i.e. H I =−eE · r, where r is the electron coordinate. The standing wave field typically used in saturated absorption spectroscopy is circular, for example σ +σ +, where the polarizations are with respect to the direction of propagation of each beam separately. That is, i(ωt−kz) i(ωt+kz) E = E0 ˆσ +[e +e +c.c.] iωt −iωt = E ˆσ + cos(kz)(e +e )

≡ E ˆσ +E.ˆ (5)

E0,E are the (real) electric field amplitudes for the travelling and standing wave fields, ˆσ + is the unit polarization vector for σ + circularly polarized light and Eˆ = cos(kz)(eiωt +e−iωt ) describes the spatial and time dependence of the field, where k = ω/c and z is distance along the laser field. It is useful to write the interaction in terms of the Rabi frequency where H I = h ¯ , with matrix elements I = | I|  Hβα β H α =−eβ| ˆ · r|αEEˆ

= h ¯ βαEˆ (6) + + where ˆ = ˆσ + for the σ σ field and βα is the Rabi frequency for a given transition between two different substates α and β [22]:   3π 1/2 = C 0 E. (7) βα βα k3τh¯

Cβα is the angular momentum coupling coefficient for the α → β transition and the lifetime 2 τ = 25.8 ns for the rubidium 5 P3/2 state [23]. Writing the matrix multiplication of (1) explicitly, the time evolution of the density matrix elements is i    ρ˙ =−iω ρ − H I ρ − ρ H I (8) βα βα βα h¯ βγ γα βγ γα γ to which we add phenomenological decay terms [11]:

1 − ρ forρ ˙ (β = α) (9) 2τ βα βα 1 − ρ forρ ˙ (10) τ ββ ββ 1  + (C )2ρ forρ ˙ (11) τ βα ββ αα β

where the ground and excited states are explicitly α = 1,...,12,β = 13,...,36 (85Rb). We transform the off-diagonal elements to a reference frame evolving at the rate of change of the atom–laser coupling, defining  iωt ρβα e (β = α) ρβα = (12) ρβα (β = α). Calculation of saturated absorption spectra for multi-level atoms 2713

Rearranging and taking the time derivative, we have   −iωt ρ˙βα = ρ˙βα − iωρβα e . (13) We substitute this into (8) and multiply by exp(iωt). The off-diagonals are then 1 i    ρ˙ =−i(ω − ω)ρ − ρ − H I ρ − ρ H I (14) βα βα βα 2τ βα h¯ βγ γα βγ γα γ and similarly for the excited- and ground-state diagonals. There are N 2 = 1296 coupled differential equations, with coefficients that have a cosinusoidal time dependence as the atom moves through the standing wave field. The problem is simplified by several symmetries and approximations. We first eliminate terms oscillating at twice the optical frequency (the rotating wave approximation [11]). The density = ∗ matrix is Hermitian (i.e., ραβ ρβα), thus reducing the problem to N(N +1)/2 equations. I = The interaction Hβα 0 for all α, β pairs except those coupled by the laser, that is where = ± 2 =  2 = ± F 0, 1 and MF (5 S1/2) + ν MF (5 P3/2), where ν 0, 1 is the polarization of the connecting photon; ν = +1 in our case. We therefore specifically exclude all off-diagonal terms that do not directly couple an excited state to a ground state. This approximation drastically reduces the number of equations for a circularly polarized single-frequency laser field. Elliptical or linear fields induce coherences which cannot be ignored and hence are generally more difficult to solve [24]. We also exclude off-diagonal pumping terms that couple from the ground hyperfine level 2 85 that is not selected by the laser, i.e. 5 S1/2F = 2for Rb. This is a reasonable approximation for rubidium vapour at room temperature, where the Doppler spread is much smaller than the ground-state splitting, and assuming that the saturated laser line width is negligible. The separations between excited-state hyperfine levels are much smaller, and so coupling to multiple excited states is included. For 85Rb and circular polarization, we are left with 48 equations, which can be separated into four distinct groups. In the following, α and β label the ground and excited substates explicitly; for 85Rb, α = 1,...,12 and β = 13,...,36. 2 5 S1/2F = 2 ground-state diagonals 1 27 ρ˙ = (C )2ρ . (15) αα τ βα ββ β=13 2 5 S1/2F = 3 ground-state diagonals 36 1 36 ρ˙ = 2 cos(kz) Im(ρ ) + (C )2ρ . (16) αα βα βα τ βα ββ β=16 β=16 2 5 P3/2 excited-state diagonals 12 1 ρ˙ =−2 cos(kz) Im(ρ ) − ρ . (17) ββ βα βα τ ββ α=6 Off diagonals 1 ρ˙ = i cos(kz)(ρ − ρ ) − i(ω − ω)ρ − ρ . (18) βα βα ββ αα βα βα 2τ βα In a σ +σ + field, the first sums in (16), (17) reduce to (at most) three and one term. The atomic velocity dependence is not obvious from these equations, but of course is manifest through the cos(kz) dependence in the time derivatives. 2714 L P Maguire et al

3. Solutions

These equations were solved by Runge–Kutta integration with the initial populations evenly distributed among the ground-state diagonals and all other elements of the density matrix set to zero. The observed spectrum in a typical saturated absorption experiment is a statistical average of the spectra of many individual, mutually noninteracting atoms [25]. The atoms have different trajectories through the light field, including a Maxwellian distribution of velocities along the field:    m mv2 f(v)dv = exp − dv (19) 2πkBT 2kBT where m is the atomic mass, kB Boltzmann’s constant and T the vapour temperature. The atoms also experience time-dependent field intensity and varying interaction time due to different transverse trajectories across the laser beam profile. To generate the average spectrum, we calculate an average atomic density matrixρ( ) ¯ for a range of detunings . In previous work [16], we have used a Monte Carlo process to calculate atomic trajectories distributed over all possible transverse and longitudinal paths. While ensuring proper weighting, the method was computationally inefficient. Here, we discretize the sampling space into specific longitudinal velocity groups and neglect intensity variation across the laser field. That is, at each detuning, the density matrix is determined for a sampling of velocities v along the standing wave, weighted by the Maxwellian distribution: nv ρ( )¯ = ρ¯v(vi )f (vi ) (20) i=1 whereρ ¯v(vi ) is the time-averaged density matrix calculated for a given velocity vi and f(vi ) = = is the Maxwellian probability normalized such that i f(vi ) 1. We typically use nv 200 velocity groups equally distributed over three Doppler widths, i.e.    1 kBT 1 vi = 3i + i = 0,...,nv − 1. (21) nv m 2 The distribution is offset from zero because stationary atoms do not traverse the nodes and antinodes of the standing wave, producing erroneous results for small velocity groups. For a linear travelling wave, only atoms with the appropriate combination of Doppler- shifted resonance frequency and laser detuning will be on or near a resonance, and therefore have significant interaction with the field. The introduction of a second counter-propagating field adds saturation broadening, crossover resonances and multi-photon processes so that it is no longer possible to select only on-resonant atoms. In fact, we find atom–field couplings for velocities well above the primary resonance that are not predicted in the standard perturbation theory [15]. Hence, we must include atoms with the full range of velocities from our Maxwellian distribution. Our model includes all nonlinear processes that contribute to the observed spectrum while remaining computationally tractable. Eachρ ¯v(vi ) is an average over all the possible transverse atomic trajectories for a given vi . Rather than sampling such trajectories, we find that assuming a fixed uniform field amplitude, corresponding to the central intensity of a Gaussian laser profile, and fixed interaction time, are reasonable approximations (validated by appropriate tests; see later). Thus, 1 nt ρ¯ (v ) ≈ ρ(v , ,t ) (22) v i n i j t j=0 Calculation of saturated absorption spectra for multi-level atoms 2715 where the time average is over nt time steps (j = 1,...,nt ), typically 50 lifetimes, and the atoms are initially at z = 0. For each velocity group and detuning, the coupled differential equations (15)–(18)forthe atomic density matrices ρ(vi , ,tj ) were integrated, calculating a new position and hence a new field amplitude and driving rate βα cos(kz) for each time tj . The time step dt was limited by the shortest time interval involved in the atomic evolution, i.e. by the lifetime of the excited states, the Rabi frequencies and the detunings, such that dt τ, −1, −1 where for the largest detuning ωβα − ω must be taken. A time step of 0.2 ns was found to be the maximum which led to stable solutions. We use the total fluorescence emitted from the atoms as our measure of the effective = absorption [26, 27], calculated from F β Fβ where Fβ is the fluorescence due to decay from a given excited state β,  2 Fβ = (Cαβ ) ρββ . (23) α Fluorescence arises from spontaneous emission only, and hence can be calculated from the substate populations; that is, the diagonal elements of the density matrix. Since all atoms in an excited state decay at the same rate, the absorption is given by the total excited-state population β ρββ .

4. Experiment

The experimental arrangement is shown in figure 2. A tuneable external cavity diode laser [28] was coupled into a single-mode optical fibre. The emerging light was collimated to produce a 2 mm diameter (1/e2) Gaussian profile beam, verified with a commercial beam profiler. The light was split into the two beams forming the standing wave using a polarizing beam splitter cube and the power balanced using a zero-order anti-reflection-coated λ/2 retarder. Intensity noise was measured at less than 1%. The two counter-propagating components were separately circularly (σ +) polarized with two zero-order anti-reflection-coated λ/4 waveplates. The beams overlapped through a standard uncoated borosilicate glass Rb vapour cell of length 8 cm, with natural isotopic abundance (72% 85Rb and 28% Rb87), at room temperature. Nulling the ambient magnetic field (about 60 µT) was found to have minimal effect on the spectra obtained with σ +σ + polarization. The saturated absorption signal was recorded by measuring the transmission of one of the laser beams using a biased silicon PIN photodiode. The laser detuning was simultaneously determined by frequency counting the rf beatnote relative to a second laser stabilized to the F  = 3, 4 crossover peak of 85Rb. The saturated absorption spectra were measured by scanning the laser over a frequency range of up to 1 GHz, in a time of approximately 1 min. Each spectrum was normalized to unity peak absorption.

5. Results

85Rb transmission spectra were measured and calculated for central peak intensities of 1, 4 and 8ISAT (figure 3). The intensity (more formally, the irradiance) is that for one −2 single travelling wave component. ISAT = 1.7mWcm is the saturation intensity for the 2 = = 2  =  = 5 S1/2F 3,MF 3to5P3/2F 4,MF 4 transition. The fluorescence was measured simultaneously for I = 4ISAT to check the validity of using the fluorescence operator (23)as a measure of absorption. 2716 L P Maguire et al

1.0 ISAT

1.0

4 ISAT

1.0

8 ISAT Transmission

0.0

-400 -200 0 200 400 Frequency (MHz)

Figure 3. Relative transmission spectra for 85Rb, measured (black, solid) and calculated (red, dashed), and at I = 4ISAT, the fluorescence (blue dots). The measured data are normalized to the same absorption depth. The calculated results for each intensity are normalized such that the integrated absorption is equal to that of the measured spectrum. The frequency is relative to the  F = 3 → F = 4 resonance. Spectra calculated with nv = 200, integration time 50τ in steps of 0.2 ns.

The calculated spectra were each scaled to normalize the integrated absorption to that of the corresponding measured spectrum, but are otherwise absolute. The calculations show general agreement with the overall Doppler background absorption, and with the location and saturation-broadened peak widths of the hyperfine and crossover resonances. However, several discrepancies are apparent, in particular the height of the crossover peaks. We now investigate the effects of the approximations in our model, seeking possible origins of the disparities. Figure 4 shows the effects of several variations to probe the importance of different approximations made in the model, compared to our standard result given in figure 3, for an intensity of 4ISAT. The results of figure 3 are based on a time-averaged density matrix. If the spectrum is instead calculated from the final integrated density matrix, the results are markedly altered (curve B). The time averaging is defined by (22); for the end-of-trajectory density matrix we set ≈ ρ¯v(vi ) ρ(vi , ,tnt ) (24) such that the density matrix approximates the steady-state solution. Similar effects can be seen by extending the integration time (curve C). The effect on the spectrum is twofold. Firstly, the Doppler-broadened (travelling wave) absorption is significantly less for detunings below resonance. Optical pumping causes loss of atoms to the F = 2 hyperfine ground state, reducing the absorption. The thermal atoms have a mean speed of around 250 m s−1, traversing the light field in 4 µs, comparable to optical pumping times. Figure 5 shows the time dependence of the F = 2 ground-state population, Calculation of saturated absorption spectra for multi-level atoms 2717

1.0

0.8

0.6

0.4

Transmission 0.2

0.0 0.15

0.10 (C) 200 lifetimes (A) Standard (B) Steady-state 0.05 (D) 500 velocities (E) Starting position 0.00 Difference

-0.05 (F) Gaussian intensity profile -0.10 -400 -200 0 200 400 Frequency (MHz)

Figure 4. The effect of different approximations in the calculations. Intensity 4ISAT (6.8mWcm−2) compared to measured spectrum (black points). (A) Standard result as in figure 3; (B) without time averaging; (C) integrated to 200 lifetimes; (D) 500 velocity groups; (E) average over ten starting positions; (F) average for Gaussian intensity profile. Curves A, D and E are almost indistinguishable.

0.6 5S F = 2 0.5 5S F = 3 0.4

0.3

Population 5P F' = 4 (x10) 0.2 5P F' = 3 (x10) 0.1 5P F' = 2 (x10) 0.0 0 1020304050 Time (lifetimes)

Figure 5. Calculated populations of specific hyperfine ground states, as a function of time in the laser field, averaged over velocity groups. Optical pumping to the dark (F = 2) ground state is −2 slow. Note that the model does not include collisional relaxation. I = 4ISAT (6.8mWcm ), =−200 MHz.

illustrating the slow optical pumping loss, over a duration of 50 lifetimes (about 1.5 µs). The difference in optical pumping loss for the time-averaged and end-of-trajectory calculations is shown in figure 6. Taking the final density matrix exaggerates the effects of this loss, compared to averaging each time step as in (22). The latter allows for the contribution from atoms entering the light field, which have not been optically pumped. The loss will be more significant at lower frequencies because the laser couples more strongly to the lower excited states which can decay to the dark F = 2 ground state. 2718 L P Maguire et al

0.6 F = 3 0.5 F = 2 0.4

0.3 5P F' = 1,2,3,4 (x10)

Transmission 0.2

0.1 Time-averaged Steady-state 0.0 -400 -200 0 200 400 Frequency (MHz)

2 Figure 6. Calculated populations of the 5 P3/2 state (lower curves) and the F = 2 (blue) and F = 3 (red) hyperfine ground states, for time-averaged (solid) and end-of-trajectory ‘steady-state’ −2 (broken) models, against laser frequency. I = 4ISAT (6.8mWcm ).

The crossover peaks are also much more pronounced when taking the final (maximally pumped) density matrix, almost matching the measured depths. The difference can again be attributed to optical pumping. The crossovers arise from strong coupling to two excited- state hyperfine levels. As optical pumping progresses, the population of the upper F  = 3 hyperfine level becomes comparatively small (figure 5). The F  = 3, 4 crossover at −60 MHz therefore does not couple as strongly to the laser, and the transmission is relatively high for the end-of-trajectory calculation rather than when time averaged. The importance of optical pumping in saturated absorption spectroscopy has recently been described, for the conventional strong pump–weak probe case [13]. Both the crossover peak enhancement and the broader depletion at frequencies below the primary resonance were observed. Indeed, optical pumping was found to be of much greater significance than the traditional ‘saturated absorption’ effect which lends its name to this velocity-selective spectroscopy technique. Note, however, that we find the rate of optical pumping to the F = 2 dark state is comparable to the traversal time of the atom across the light field (figure 5). The slow optical pumping rate is consistent with the results of other optical pumping calculations and measurements [29, 30], but in contrast to the suggestion [13] that the equilibrium time is of the order of tens of nanoseconds. The difference between our calculations and the measured crossover heights (figure 3) becomes more pronounced as the intensity increases (although the peak widths remain in agreement). Our model assumes a constant fixed light field; that is, variation of the field as it propagates along the cell is neglected. Absorption will affect the overall power, the intensity profile and particularly the field polarization [31, 32], and because the absorption is detuning dependent, these effects will influence the shape of the spectrum. The variations will be more significant for longer cells, and in our case, the total absorption is ≈25% at room temperature. The measurements show crossover enhancement relative to our fixed field model. Saturation and optical pumping to the dark state will reduce the single-pass Doppler absorption and increase the intensity of the propagated field. The counter-propagating field will be similarly increased due to lower relative absorption, and the effects combine at the resonances, to produce larger peaks relative to the single-pass Doppler background. The exact heights of the features are also sensitive to many technical parameters, including the polarization and transverse spatial profile of the laser beams, the crossing angle in the cell and the magnetic field [13]. While we have taken care to control these experimentally, Calculation of saturated absorption spectra for multi-level atoms 2719 empirically we find that small changes to the polarization of either of the counter-propagating beams can cause large changes to the crossover peak heights. Indeed, optical pumping calculations for elliptically polarized fields [24, 33] show that small ellipticity can have dramatic effects on the atomic density matrix. We next consider the effects of velocity discretization. At low laser intensity, the resonance peak widths are very narrow compared to the atomic velocity distribution, implying that the velocity steps must be small for adequate resolution in the calculation. This is particularly true near zero detunings, where including stationary atoms leads to instabilities because the atoms do not traverse the nodes and antinodes of the standing wave. Longer integration times (upto200τ) and many more velocity groups (up to 500) were able to stabilize the spectra (see figure 4) but similar results were obtained much more rapidly by offsetting the velocity bins as in (21). Another simplification made in the calculations is starting all atoms from a standing wave antinode. Because the atoms have a velocity component along the standing wave field, the effect of this simplification is negligible. We computed an average of ten calculations in which the only change was the initial starting position of the atoms, equally separated over one standing wave period. The spectrum was almost indistinguishable from our standard result. Finally, using a uniform intensity profile for the light field might seem overly simplistic, but averaging over a Gaussian variation of the intensity does not improve the agreement. Curve F of figure 4 shows the average of ten spectra calculated for different intensity values, corresponding to evenly spaced radial displacements out to r = 1.0 mm, the 1/e2 beam radius in our experiments. The average was weighted by the circumference of a circle at each radius, corresponding to the expected number of atoms at that radius contributing to the overall absorption. At larger radius, although the number of atoms is greater, the beam intensity is smaller, therefore exciting the atoms less and contributing less to the absorption. The two effects do not exactly cancel; the net result is quite similar to the uniform intensity result calculated at half the actual intensity. In summary, we have shown that taking the absorption as the long-time approximately steady-state value, or the time average over the total trajectory, has the greatest effect on our calculated spectra. The absorption depends critically on the slow development of the atomic density matrix through optical pumping processes, on time scales comparable to typical traversal times through a laser beam of 1 mm radius. Thus while the results from our simplified approach are encouraging, we can also see from our model that complex optical pumping effects for the varied trajectories of atoms in a thermal vapour have significant effects on the spectra which cannot be neglected.

6. Future

Our model for multi-level atom interactions with a standing wave provides an effective and computationally tractable method for calculating saturated absorption spectra, showing reasonable agreement with experimental spectral features including hyperfine and crossover resonances and saturation broadening. Compared to relatively simple analytic approaches, the model includes greater insight arising from the multi-level nature of the atoms, yet can be calculated quickly to investigate details of the underlying physical processes, for example optical pumping between hyperfine levels or magnetic sublevels. The model was simplified dramatically by considering only the specific case of σ +σ + polarization and a fixed field. Extension to arbitrary polarization [24, 33] and to a Maxwell– Bloch model which includes absorption effects on the field [34] may enable improvements to the prediction of spectral details and would allow investigation of more complex applications 2720 L P Maguire et al such as optical filters [31], and with further work, magnetic-field-dependent phenomena such as magnetic field sensors [35] and the widely used DAVLL [36] laser locking scheme.

Acknowledgments

This work has been supported by the Australian Research Council and the Australian International Science Linkages programme. EM would like to thank the ICSC World Laboratory for financial support. RES would like to thank M Kennedy-Jones and M Moore for their donation of rf equipment.

References

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