Circular Dichroism of RbHe and RbN 2 Molecules
aa r X i v : . [ phy s i c s . a t o m - ph ] J u l Circular Dichroism of RbHe and RbN Molecules
B. Lancor, E. Babcock, R. Wyllie, and T. G. Walker Department of Physics, University of Wisconsin-Madison, Madison, WI 53706, USA Juelich Centre for Neutron Science, Garching 85747, Germany (Dated: October 21, 2018)We present measurements of the circular dichroism of optically pumped Rb vapor near the D1 resonance line.Collisions with the buffer gases He and N reduce the transparency of the vapor, even when fully polarized.We use two methods to measure this effect, show that the He results can be understood from RbHe potentialcurves, and show how this effect conspires with the spectral profile of the optical pumping light to increase thelaser power demands for optical pumping of very optically thick samples. PACS numbers: 32.70.-n,32.80.Xx,33.55.+b
Optical pumping of alkali-metal atoms at high temperaturesand high buffer gas pressures is a powerful technique for pre-cision spectroscopies (clocks, magnetometers) [1, 2], and forcollisional transfer of angular momentum to noble gas nucleivia spin-exchange collisions [3]. The ability to spin-polarizelarge quantities of nuclei using spin-exchange optical pump-ing (SEOP) normally requires the alkali-metal vapors to be ∼
100 optical depths at line center. The light is only able topenetrate such dense samples if the atoms cease to absorblight once they are fully spin-polarized [4]. It is for this rea-son that optical pumping at high optical thickness is only pos-sible using circularly polarized light resonant with S / → P / (colloquially, “D1”) transitions, where angular momen-tum conservation forbids m S = 1 / atoms to absorb the he-licity h = 1 pumping light. In contrast, optical pumping withS / → P / (“D2”) light tends to partially polarize the atomsin strongly absorbing states [5].This paper presents quantitative measurements of the nor-mally forbidden absorption of h = D1 light by m S = 1 / Rb atoms in atmospheric pressure cells containing He andN buffer gases. We are led to this investigation by the re-peated observations in our lab and elsewhere that the amountof light required for SEOP is substantially greater than mod-els predict [6]. Furthermore, previous studies of SEOP withRb-K mixtures observed that alkali-metal polarizations satu-rated significantly below 100% even for high optical pump-ing rates [7]. This suggested a light-intensity dependent spin-relaxation rate. Finally, when pumping pure Rb with broad-band light sources, it was observed that the Rb polarizationsaturated well below 100%, again suggesting that the pump-ing light does not go dark for polarized atoms [7].Since these experiments are typically done with gas pres-sures of several atmospheres, it is natural to ask to what extentpressure broadening gives a small amount of D2 character tothe D1 resonance. Several previous studies suggest this. First,spectroscopic measurements show that the red wing of the D2line certainly overlaps the D1 resonance [8]. In addition, Hecollisions can collisionally transfer 5P / population to 5P / , Throughout the paper we shall assume the quantization axis is parallel tothe pumping light propagation direction. Thus h = 1 light corresponds tophotons with h J z i = 1 . albeit with a small cross section [9]. The existence of thisexcited-state spin-relaxation process implies that during colli-sions with He atoms the 5P / and 5P / states are somewhatmixed. Finally, from a theoretical perspective it is well knownthat the fine-structure interaction is partially decoupled by theintramolecular fields, thereby mixing P / and P / molecularstates [10]. - H a L V H r LH GH z L P (cid:144) P (cid:144) Σ+ h Ν @ (cid:144) D @ (cid:144) D @ (cid:144) D FIG. 1. Adiabatic energy curves for RbHe molecules adapted fromRef. [11] with the method of Ref. [10]. The projection of the totalelectronic angular momentum along the interatomic axes is given inbrackets. The curve crossings between the photon-dressed 5s σ stateand the two excited-state curves means that photon absorption is al-lowed during a collision. As explained in the text, these absorptionprocesses are not subject to the free-atom dipole selection rules, al-lowing normally angular momentum forbidden transitions to occur. The partially D2-like absorption of light during RbHe col-lisions can be understood by considering the lowest adiabaticpotential energy curves for Rb-He molecules, shown in Fig. 1.It is well known [10] that RbHe collisions are non-adiabatic,but for a qualitative understanding the adiabatic curves shouldbe useful. The curves that correlate with the two 5p fine struc-ture levels are designated 5p [Ω] , where
Ω = 1 / , / is themagnitude of the projection of the electronic angular momen-tum on the interatomic axis. The Ω = 3 / state has pureP / character, while the Ω = 1 / states contain significantmixtures of the two fine-structure components for interatomicseparations r < a . Also shown in Fig. 1 is the groundstate potential sσ shifted up by the energy of a D1 photon.The crossing between the 5p [3 / state and the repulsive wallof the photon-shifted sσ state at r = 8 a means that res-onant D2-like light absorption can occur during RbHe colli-sions. Since kT /h = 9400 GHz at typical 180 ◦ C tempera-tures for SEOP, the curve crossing at an energy of h × buffer gas andHe rich He-N buffer gas mixtures (typical of SEOP cells),near the Rb first resonance lines. When the dichroism mea-surements are combined with known pressure-broadened line-shapes, the results show that the light absorption cross sec-tion for h = 1 D1 light by fully polarized atoms is σ =1 . ± . × − cm at a N density of 1 amg [12], and σ = 1 . ± . × − cm at a He density of 1 amg. Mea-surements at two He pressures confirm the RbHe molecules asthe source of the absorption. Despite the small size of thesecross sections in comparison to on-resonant cross sections of × − cm , they are sufficient to significantly lower thealkali-metal polarization and increase the power required tooptically pump optically thick vapors, especially for broad-band light sources.This paper, a full account of Ref. [13], is organized as fol-lows. In Section I we discuss the circular dichroism of spin-1/2 atoms, and discuss its consequences for optical pump-ing of optically thick vapors. In Section II we present ourmeasurements of circular dichroism, using two quite differentmethods. We then present in Section III a theoretical estimateof the effect and show that the expected size is in accord withthe experimental results. Section IV is a more detailed dis-cussion of the consequences of the measurements for opticalpumping. Section V discusses complications from hyperfineinteractions that appear for a small portion of the experimentaldata. I. EFFECTS OF REDUCED CIRCULAR DICHROISM ONOPTICAL PUMPING OF THICK VAPORS
To facilitate some of the discussions later in the paper, weinclude here a brief discussion of the relationship betweencircular dichroism and optical pumping. We will assumemonochromatic light for this section. A more complete anal-ysis, including key spectral averaging effects, will be given inSec. IV.Circular dichroism is the differential absorption of light ofopposite helicities. To avoid the difficult problem of deter-mining absolute atom densities, it is convenient to measure anormalized circular dichroism, defined as C = (cid:10) σ − (cid:11) − (cid:10) σ (cid:11) h σ − i + h σ i (1)where the average is over the various populations ρ m of themagnetic sublevels of the atom, (cid:10) σ h (cid:11) = X m ρ m σ h m , (2) and σ h m is the absorption cross section for light of helicity h = ± by an atom in sublevel m . The atoms are assumed tobe purely longitudinally polarized parallel to a bias magneticfield.The general form of the light absorption cross section forspin-1/2 atoms is [7] σ h P = (cid:10) σ h (cid:11) = σ (1 − P ∞ h P ) (3)where the atomic polarization is P = ρ / − ρ − / and thelight is assumed to propagate parallel to the magnetic field.The absorption cross section for unpolarized light is σ . Fornow, we neglect hyperfine structure. Since the absorptioncross section depends only on the product of the helicity andthe spin-polarization, the dichroism can be measured by re-versing the direction of either the helicity or the polarization.The parameter P ∞ is limited to the range − ≤ P ∞ ≤ .Using Eq. 3 in Eq. 1, it follows that C = P P ∞ . (4)Thus P ∞ is the normalized circular dichroism of a fully spin-polarized atom. For alkali-metal atoms P ∞ ( ν ) ≈ for lightnear the center of the D1 line, and P ∞ ( ν ) ≈ − / at the D2line[5]. Therefore C ≈ P for an alkali-metal vapor probed byresonant D1 light. We shall refer to the condition C = P , orequivalently P ∞ = 1 , as “full dichroism”.The value of P ∞ also affects the optical pumping process.Under conditions of large pressure broadening, full electronspin-randomization in the excited state, and nuclear spin con-servation in the excited state, the optical pumping process formonochromatic h = 1 light obeys [7] d h F z i dt = R P ∞ − P ) − Γ P (5)where h F z i is the total angular momentum of the atom (elec-tronic plus nuclear), R = σ I/hν is the optical pumpingrate from light of intensity I , and Γ is the ground-state spin-relaxation rate which may be due to a variety of collision pro-cesses. Thus we see that the steady-state polarization is P = P ∞ R Γ + R (6)so that the maximum polarization that is attainable at infinitepumping rate is P = P ∞ .The steady-state absorption rate is A = R (1 − P ∞ P ) = Γ R Γ + R (cid:20) R Γ (cid:0) − P ∞ (cid:1)(cid:21) (7) ≈ Γ R Γ + R (cid:20) R Γ (1 − P ∞ ) (cid:21) (8)where the approximation holds for − P ∞ ≪ . Forfull dichroism, the atoms absorb at the rate Γ P , the amountrequired to repolarize the atoms due to ground-state spin-relaxation. For reduced dichroism, the scattering rate in-creases by the factor Υ ≈ (cid:20) R Γ (1 − P ∞ ) (cid:21) (9) R ( ) / Γ z FIG. 2. Optical pumping rate as a function of propagation distance z , for a monochromatic pumping beam. In the case of full dichro-ism, the light is attenuated only by the need to repolarize atoms thatundergo spin-relaxation collisions. When polarized atoms continueto absorb light due to imperfect dichroism, the light attenuation ismuch more rapid. The cell length l was chosen so that the vapor is100 optical depths thick. which can be much greater than 1 when the spin-polarizationis high ( R ≫ Γ ). This condition is necessary for pumping ofoptically thick vapors, where the rule of thumb is that in orderto fully polarize the cell volume, the pumping rate at the cellentrance must be at least the optical depth times the relaxationrate.To illustrate the effects of reduced dichroism, Fig. 2 com-pares monochromatic light propagation for full dichroism tothat with a modest P ∞ = 0 . , for an optical thickness of100. For the ideal case, the intensity steadily decreases at therate governed by spin-relaxation. The reduced dichroism casesees an initial intensity decrease that is five times larger, re-sulting in the light being able to polarize only about 40% ofthe atoms. II. DICHROISM MEASUREMENTS
In this section we present the two methods used to mea-sure the normalized circular dichroism of Rb–buffer gasmolecules.
A. Direct Optical Method
We deduced the normalized circular dichroism of fully po-larized atoms P ∞ by comparing the absorption of positive he-licity light by atoms with polarizations ± P using the appara-tus and methods described here.A schematic of the apparatus is shown in Fig. 3. A Rb va-por cell, contained in a flowing hot-air oven, was opticallypumped by a circularly polarized frequency narrowed diodearray bar providing 35 Watts of power at 795nm, with a spec-tral width of of ∼
125 GHz [20]. A holding field of 50 G wasapplied in the pump propagation direction. A probe beam, oneof two single-frequency external cavity diode lasers tunable
Pump laser Probe B z l/4l/4 Lock-InLock-In
FIG. 3. Apparatus for measuring the circular dichroism of Rb-Hevapor. The pump laser, propagating parallel to a magnetic field, spin-polarizes Rb atoms by optical pumping. The polarization is set eitherparallel or antiparallel to the field, as determined by the orientationof a quarter-wave plate. The fractional transmission of a weak, cir-cularly polarized, tunable probe laser is determined by the ratio ofphotodiode voltages before and after traversal of the cell. The cir-cular dichroism is then determined from the transmissions for bothdirections of Rb polarization. The absolute Rb polarization is mea-sured with the apparatus of Fig. 4. around 780 nm and 795 nm, was attenuated to P < µ W, sentthrough a chopper operating at 485 Hz, and linearly polarizedwith a polarizing beam splitter cube. Directly in front of theoven, the beam went through a non-polarizing beam splitterplate to provide a reference signal proportional to the incidentintensity, and a quarter wave plate to produce circular polar-ization. The reference and transmitted intensities were mea-sured on silicon photo-diodes and sent to lock-in amplifiersreferenced to the chopper frequency. To change the directionof the atomic spin polarization relative to the probe helicity,the pump λ/ plate was rotated 90 ◦ , thus reversing the pumplaser helicity. To obtain the relation between the incident andtransmitted intensities in the absence of Rb (thus accountingfor loss in the windows of the oven and cell), a measurementwas taken at room temperature.Three natural abundance Rb cells were used in this exper-iment. The pure N cell is a 4.5 cm diameter blown Pyrexsphere, with 2.80 amg N [12], the low pressure SEOP cell is aclosed 4.9 cm long cylinder (Corning 1720 body with GE180windows), containing 0.80 amg He and 0.07 amg of N . Thehigh pressure SEOP cell is a blown GE180 sphere of diameter3.5 cm, filled with 3.27 amg of He and 0.13 amg N . Thetemperature of the cell was monitored by an infrared thermo-couple, and controlled by an analog controller. Temperaturesranging from ∼ ◦ C to ∼ ◦ C, corresponding to [Rb]= − × cm − , were used to produce appropriate op-tical thickness for transmission measurements at a range offrequencies. At the lower temperatures, the pump laser powerwas reduced to lower background noise.For the pure N cell, Rb liquid droplets on the walls of thecell produced an uncontrolled temperature dependence anddrift to the probe transmission as the droplets slowly movedon the face of the cell. To account for this, the probe laser and Pump laser Probe λ/4 B z Lock-in
Polarizer refsig B RF FIG. 4. Measurement of polarization via transverse EPR spec-troscopy. As the DC magnetic field is swept to match the Rb ZeemanEPR frequencies, the index of refraction of the Rb atoms is mod-ulated at the 26.4 MHz RF driving frequency, thus modulating thepolarization of the probe laser. This modulation is detected and de-modulated by the lock-in amplifier. The Rb polarization is deducedfrom the ratio of areas of the EPR resonances. an additional 856 nm laser were coupled into a fiber, with the856 nm laser transmission serving to measure small variationsof the transmission of the cell walls with time.The Rb spin polarization was measured with transverseelectron paramagnetic resonance (EPR) spectroscopy [14,15], as shown in Fig. 4. A 26.4 MHz RF field was appliedperpendicular to the holding field by driving a pair of 9 cmdiameter coils, separated by 7 cm, with a synthesized functiongenerator. The amplitude was controlled with a voltage con-trolled attenuator. As the holding field was swept through theEPR resonances of either isotope, the transverse componentof the probe beam, now linearly polarized, acquired a polar-ization modulation at the RF frequency, which was convertedto an intensity modulation with a polarizer oriented at 45 ◦ tothe original probe polarization direction. The probe laser fre-quency was detuned from the D2 resonance 1-2 nm so that ithad negligible effect on the optical pumping. The probe inten-sity was measured by a fast photodetector, the output of whichwas amplified by two RF amplifiers. This signal was demod-ulated to 100 KHz by mixing it with a 26.5 MHz signal froma second synthesized function generator. This 100 KHz signalwas then sent to a lock-in amplifier referenced to a signal gen-erated by mixing the outputs of the two function generators.The phase of the lock-in was chosen to produce LorentzianEPR spectra, which were recorded in the lock-in and sent to acomputer for analysis. The spin-polarization was deduced us-ing the area ratio method of Ref. [14]. This method is insen-sitive to the broadening mechanism of the EPR lines, and inparticular can be applied for both low and high spin-exchangerates, i.e. over a wide range of temperatures. All the polariza-tion measurements were done with Rb, with correspondingmeasurements made on Rb at low temperatures where thetwo species were not in spin-temperature equilibrium.The probe beam propagates through the cell at an angle θ = 17 . ± . ◦ . The modulated component of the Fara- day rotation is proportional to P x sin θ and is therefore muchsmaller than would be detected by a transverse probe. How-ever, we wish to know the polarization averaged along theprobe beam direction, so there was a tradeoff between theEPR signal, proportional to sin θ and the DC circular dichro-ism, which is degraded by a factor cos θ . We chose θ to be assmall as possible and still observe high quality EPR signals.At high densities, this is not a problem, but for line-centermeasurements where the temperatures (as low as 70 ◦ C) arenot sufficient to produce spin-temperature equilibrium [16], itwas essential to observe many EPR peaks in order to back outthe polarization.We wish to find the normalized circular dichroism, fromwhich we can deduce P ∞ . The basic parameters observed bythe experiment are the transmissions I ± for circularly polar-ized light of helicity h propagating at angle θ to the atomicpolarization ± P . Then the absorption cross section for theprobe light becomes [17] σ h P ( θ ) = σ (1 − P ∞ h P cos θ ) (10)Since the absorption cross section is proportional to h P , re-versing the direction of the atomic spins is equivalent to re-versing the helicity of the probe. We found it advantageousto reverse the pump polarization and leave the probe helic-ity fixed, as the light transmission of the oven and cell win-dows depended slightly on the probe helicity. When the pumppolarization was reversed, the atomic polarization satisfied (1 − | P ′ Rb | ) = (1 − P Rb ) to within 5% at worst, and typi-cally to within much less. The imperfect atomic polarizationreversal was measured and corrected for in the data. We alsocorrected each intensity measurement to account for an ob-served residual transmission of the probe beam even at veryhigh optical depths ( ∼ . ). This probably results from asmall amount of light at the free-running probe laser wave-length. Data points with transmission < ∼ were not useddue to the uncertainty in this correction. A correction was alsomade for typical probe helicity h = 0 . .We extract P ∞ from the corrected transmitted intensities I ± = I o exp( − [Rb] σ ± P ( θ ) l ) by finding C = − ln (cid:16) I − I o (cid:17) + ln (cid:16) I + I o (cid:17) − ln (cid:16) I − I o (cid:17) − ln (cid:16) I + I o (cid:17) = P P ∞ cos θ. (11)Note that in forming this ratio the optical thickness [Rb] l andinstrumental gains cancel. Combining with measurements of θ and P gives P ∞ = C P cos θ . (12)For regions of the data where P ∞ ≈ , it is particularly cru-cial to make an accurate measurement of − P . This wascomplicated by the fact that we did not achieve high atomicpolarizations at low [Rb], where the on resonance data mustbe taken. At [Rb] ∼ . × cm − we observed P from0.9 to 0.95 in all three cells, while at [Rb] > × cm − we observed P ≥ . in the He-N cells, and P ∼ . inthe N cell. The typical uncertainty of the EPR measurementswas ∼ or less in (1 − P ) .For the near resonance data taken in the spherical cells,the uncertainty in P was the largest uncertainty. Etalon in-terference effects in the walls of the cylindrical cell (0.75 % to 1.0 % ) were comparable to the uncertainty in P near res-onance, and were the dominant contribution to the errors offresonance. Etalon effects were smaller in the spherical cells( < . ), where drifts in I o (0.25 % to 0.5 %) dominatedoff resonance uncertainties for the spherical SEOP cell, andrelative intensity drift between the 795 nm and 856 nm lasers(0.5 % ) was the primary source of off resonant error in the pureN cell. For most of the frequency range covered, the data wastaken at high density where P was close to 1 and the uncer-tainty in P was a minor component of the total uncertainty.The uncertainties in the probe propagation angle and the mea-sured probe intensities were small under all conditions.An important check on the reliability of the measurementswas a detailed study of the dichroism very near the D1 linecenter, where σ − ≫ σ giving P ∞ ≈ . In this region, hy-perfine interactions are quite important, especially for the lowpressure cell, and a complex structure is observed. This studywill be presented in Sec. V. The results are quite sensitiveto the accuracy of the measurement of (1 − P ) , and the suc-cessful understanding of the experiment in this region givesadditional confidence in the off-resonant results.The measured P ∞ near resonance, from transmission data,for both SEOP cells is shown in Fig. 5. The circular dichroismdeviates significantly from one within the typical 1000 GHzbandwidth of broadband sources such as diode array bars of-ten used for SEOP. The implications of this finding will bediscussed in Sec. IV. A very important result from Fig. 5 isthe agreement between the two cells, despite their very differ-ent He pressures. At detunings outside the atomic linewidthof 15-60 GHz [18], the absorption of negative helicity lightis proportional to the buffer gas pressure. Thus, only if thenormally forbidden absorption of positive helicity light is alsoproportional to the buffer gas pressure will the dichroism bepressure independent. The agreement between the two cellsat different pressures confirms that the source of the impuredichroism in these cells is Rb–buffer-gas collisions.For the high [ He] cell, we were able to measure P ∞ acrossthe entire frequency range from the D1 to the D2 resonance,and this is plotted in Fig. 6. (The gap in the data from 784-6nm is due to lack of coverage of that range by our two probelasers.) As expected, the dichroism goes from +1 to − . asthe light is tuned between the two resonances. However, thesign of the dichroism flips at . ± . nm, a significantshift from the . nm zero crossing that would result froma naive model that takes the cross-section at each frequency tobe the sum of Lorentzian pressure-broadened D1 ( P ∞ = 1 )and D2 ( P ∞ = − / ) lineshapes.To allow us to isolate the contributions of Rb-N collisions,we measured P ∞ in the pure N cell (Fig. 7). Pure He cellscannot be used for these measurements because some N isnecessary to eliminate radiation trapping [3], so these pure N results allow us to subract out the small N contribution in theSEOP cells.Given the importance of these measurements for D1 opticalpumping, it is convenient to extract the absorption cross sec- P ¥ -600 -400 -200 0 200 400 600 Detuning from D1(GHz) [He]=3.27 Amg [He]=0.80 Amg Naive Dichroism Fit w/Mol. Cross Section FIG. 5. Normalized circular dichroism results near the D1 resonanceline. The agreement between cells of different He pressure verifiesthat the effects originates from absorption in RbHe collisions. Thedashed line denotes the frequency dependence of the dichroism mak-ing the naive assumption of purely Lorentzian line broadening, andthe solid line is the result of using the molecular absorption crosssection from Eq. 14. P ¥ Detuning from D1(GHz)
Wavelength (nm) [He]=3.27 Amg [He]=0.80 Amg [He]=3.27 Amg OP Method [He]=0.80 Amg OP Method Naive Dichroism
FIG. 6. Measured circular dichroism of RbHe molecules in the re-gion between the first resonance lines of Rb. The solid curve showsthe expected dichroism from the very naive assumption of purelyLorentzian broadened lines. tion for positive helicity light by fully polarized atoms. To dothis, we derive the product [Rb] lσ = − ln ( I + /I o ) − ln ( I − /I o )2 (13)from the transmission data, and fit it near resonance to the ex-pected lineshape from [18] to find [Rb] l , l being the probepropagation distance through the cell. Our measured line-shapes agree well with Ref. [18] in the near-resonant region.The N data deviates from this lineshape at detunings morethan ∼
100 GHz. Given σ and P ∞ , we then deduce σ ± us-ing Eq. 3. The three cross-sections for Rb-N are shown inFig. 8.Assuming the N cross-section is proportional to [N ], we P ¥ Detuning from D1(GHz)
Wavelength (nm) [N ]=2.80 Amg Absorption Method Naive Dichroism FIG. 7. Measured circular dichroism of RbN molecules in the re-gion between the first resonance lines of Rb. The solid curve showsthe expected dichroism from the very naive assumption of purelyLorentzian broadened lines. -18 -17 -16 -15 C r o ss S e c t i on ( c m ) h=0 h=1 h=-1 FIG. 8. Cross sections, normalized to 1 amg N density, for absorp-tion of σ , σ − , and σ light by fully polarized Rb atoms. The solidline is the linear fit to the σ data. fit σ to a line in the region near the D1 line, giving σ ( ν ) = σ ( ν ) + ( ν − ν ) dσ dν (cid:12)(cid:12)(cid:12)(cid:12) ν (14) σ ( ν )[N ] = 1 . ± . × − cm amg , (15) ] dσ dν (cid:12)(cid:12)(cid:12)(cid:12) ν = − ± × − cm amg GHz . (16)We then use this result to extract the normalized cross sectionat the D1 line for He-Rb from the SEOP cells, finding σ ( ν )[He] = 1 . ± . × − cm amg , (17) dσ dν (cid:12)(cid:12)(cid:12)(cid:12) ν = 6 ± × − cm amg GHz . (18)The Rb-N cross sections from Fig. 8 were also used alongwith the data from the SEOP cells to get the Rb-He cross -18 -17 -16 -15 -14 C r o ss S e c t i on ( c m ) FIG. 9. Cross sections, normalized to 1 amg He density, for absorp-tion of σ , σ − , and σ light by fully polarized Rb atoms. The solidline is a linear fit to σ in the vicinity of the D1 resonance. sections across the full frequency range. These are shown inFig. 9.The σ measurements in Fig. 9 are to our knowledge thefirst far-wing absorption measurements of the RbHe molecule.Previously, Gallagher and coworkers [8, 19] measured near-and far-wing emission spectra, and pointed out the non-quasi-static nature of the RbHe emission spectrum. If we assumethat the D2 red-wing emission spectrum roughly correspondsto the contribution to σ from the p [3 / state, we would de-duce × − cm from Ref. [8]. As will be discussedin Sec. III, calculations suggest that the p [3 / contributesabout 1/3 of σ . This is in excellent agreement with our mea-surements, though this should be considered somewhat fortu-itous given that a number of assumptions have been made incomparing the two experiments. B. Optical Pumping Method
As a second, quite different, method for measuring P ∞ atfrequencies off the D1 resonance, we used a 30 W frequencynarrowed (50 GHz linewidth) external cavity laser [20] to op-tically pump the atoms at different frequencies ν and mea-sured the resulting polarization P ( ν ) , pumping rate R ( ν ) , andspin-relaxation rate Γ . This method was used only on theSEOP cells. For off-resonant pumping, we assume P ∞ is es-sentially constant for the small range of frequencies present inthe pumping beam, so Eq. 6 becomes P ∞ ( ν ) = (cid:18) R ( ν ) (cid:19) P ( ν ) . (19)For most of the data, Γ ≫ R ( ν ) .We chopped the pumping laser with a mechanical shutterand measured the spin-polarization as a function of time, as il-lustrated in Fig. 10, using Faraday rotation [16, 21]. For smallpolarizations, the rising transient builds up polarization to thesteady state value (19) with a rate constant ( R ( ν ) + Γ) /η , andthe falling transient decays with a rate constant Γ /η , wherethe slowing-down factor η = 10 . for natural abundance Rbtakes into account the spin inertia due to the alkali-metal nu-clei at low polarization in spin-temperature equilibrium [15].The Faraday rotation was calibrated by pumping on the D1resonance, where longitudinal EPR spectroscopy [21] verifiedthat the polarization was very close to 100%.We took data using two separate frequency narrowed diodearray bars, one with a free running wavelength of 794 nmat room temperature and the other with a free running wave-length of 800 nm at room temperature. With these two bars,narrowed and frequency shifted by our external cavity design[20], we were able to access wavelengths from 782 nm to 810nm by also changing the temperature from -20 ◦ C to 45 ◦ C us-ing a water-cooled thermo-electric mount. P (G+R)/h=48.4/s G/h=45.6/s(G+R)/h=59.3/s G/h=45.0/s
785 nm793 nm
FIG. 10. Pumping/decay transients used to measure P ∞ . Pumpinglight is turned on at t = 0 , and the polarization builds up to thesteady-state value of Eq. 19. The exponential build-up and decayconstants allow the pumping rate R and the relaxation rate Γ to bemeasured. Note that the sign of the polarization reverses for 785 nmpumping as compared to 793 nm pumping, due to the dichroism ofthe vapor being negative for the former case. The measured values of P ∞ using this optical pumpingmethod are shown in Fig. 6. They agree quite well with thedirect circular dichroism measurements described above. Thetwo SEOP cells were used for the measurements, and againwe find that the dichroism is the same within errors for thetwo cells, consistent with the RbHe molecular absorption hy-pothesis.With this method, the zero crossing of the dichroism is par-ticularly dramatic as the signal of Fig. 10 reverses sign near790 nm. III. ESTIMATES
In this section we will present estimates of the dichroism ofRbHe molecules in the vicinity of the D1 resonance line. Toour knowledge, the circular dichroism of these molecules hasnever been studied before, nor have calculations been pub-lished. A full calculation is beyond the scope of this work,but the considerations given here will attempt to explain ourobservations. We consider the light absorption process as a collision,where the ground-state potential curve is shifted up by onephoton energy. As seen from Fig. 1, 795 nm light can beresonantly absorbed to the 5p[3/2] state during a collision.In the absence of fine-structure mixing by the atom-atom in-teractions, the 5p[1/2] state that correlates at large r to theP / state would not absorb the σ light from a spin-polarizedRb atom. However, the fine-structure interaction causes the5p[1/2] states to be mixtures of P / and P / character, withthe P / part giving rise to an allowed light absorption. Inthis case, the Rabi coupling will depend on the interatomicseparation r , going to zero at large r .To simplify the calculations, we will assume that we canneglect the variation in the light-atom coupling with collisionangle, replacing the angle-dependent Rabi coupling with itsangular average. We will also assume that we can neglectrotational coupling of the adiabatic curves [10]. With theseassumptions, the light-absorption process is isotropic and wecan consider the problem as a simple collision in the radial co-ordinate, with excitation to each of the excited-state potentialcurves considered separately.Assuming a classical trajectory with distance of closest ap-proach r (assumed to occur at time t = 0 ), it is convenient todefine α ( r ) = Z rr drv ( r ) ∆ V ( r ) (20) v ( r ) = v ∞ s − b r − V sσ ( r ) mv ∞ (21)where K = mv ∞ / is the initial kinetic energy, correspond-ing to initial speed v ∞ , and b is the impact parameter forthe collision. The difference potential is ¯ h ∆( r ) = V e ( r ) − V sσ ( r ) − ¯ hω . Then, in the weak intensity limit, first orderperturbation theory gives the probability of finding the atomin the excited state e after a single collision is | c e | = (cid:12)(cid:12)(cid:12)(cid:12)Z ∞ r drv ǫ ( r ) cos α ( r ) (cid:12)(cid:12)(cid:12)(cid:12) . (22)The molecular Rabi frequency is given in terms of the atomicRabi frequency as ǫ ( r ) = ǫ β ( r ) / (23)where the factor of 2 accounts for the angular average overcollision orientations, and β is the P / mixing angle to bediscussed below. From this we get the excitation rate to a sin-gle adiabatic potential by averaging over the collision impactparameters: R = [He] v ∞ Z πdb | c e | = σ I/ ¯ hω (24)The atomic Rabi frequency ǫ is related to the light intensityand the atomic lifetime τ by ǫ = 3 λ I/ (2 πhcτ ) , giving fi-nally σ ( v ∞ ) = 3 λ v ∞ πτ [He] Z πdb | c e | ǫ . (25)This expression is then averaged over a Maxwellian velocitydistribution.We begin by considering absorption to the 5p[3/2] curve inFig. 1. This state is of pure P / character, so that β = π .D1 light is sufficiently far detuned that non-adiabatic effectsshould not be too important, and we consider the process as aLandau-Zener transition from the ground state 5s σ + hν at thecrossing point. The probability of excitation is P LZ = π ¯ hǫ v r | d ∆ /dr | , (26)with all quantities evaluated at the crossing point. Performingthe impact parameter integration, the thermal average, and ac-counting for the two-fold | Ω | degeneracy then gives σ opt = [He] 3 π / λ r c τ | d ∆ /dr | e − V sσ /T . (27)For 1 amg of He, this evaluates to . × − cm at the D1resonance, about a factor of 2 smaller than a direct numericalintegration of Eq. 22 that gives . × − cm .For the 5p[1/2] curve that correlates to the P / state at r = ∞ , the Landau-Zener approximation is invalid becausethe phase α does not vary rapidly enough with r . The r − dependent wavefunction, written in terms of the atomicfine-structure states | JM i , is | p [1 / i = cos( β/ (cid:12)(cid:12)
12 12 (cid:11) +sin( β/ (cid:12)(cid:12)
32 12 (cid:11) . The calculated mixing angle β ( r ) is shownin Fig. 11. Numerical integration of Eqs. 22 and 25 gives . × − cm at 1 amg. H a L Β H r L FIG. 11. Fine-structure mixing angle for the 5p[1/2] states as a func-tion of interatomic separation.
Thus our estimate of the absorption cross section σ =9 . × − cm at 1 amg is very close to the measured value.Given that we have ignored rotational coupling of the adia-batic curves and the anisotropic nature of the light-moleculeRabi coupling, this excellent agreement may be consideredsomewhat fortuitous. Nevertheless, it confirms that it is quitereasonable for the circular dichroism of RbHe molecules to beat the level we observe in the experiment. IV. CONSEQUENCES OF REDUCED CIRCULARDICHROISM FOR SPIN-EXCHANGE OPTICAL PUMPING
We now turn to the implications of these results for spin-exchange optical pumping of He. With the advent of high power diode array bars having 10s to 100s of Watts of low-cost power available at 795 nm, it has become standard prac-tice to use these lasers for SEOP [22]. One drawback of theselasers is that they have relatively broad spectral widths, typ-ically several nm (1 nm=475 GHz at 795 nm). This wasshown to be mitigated by the practice of running at multi-atmosphere He pressures, where the atomic line could bebroadened and directly interact with a more substantial frac-tion of the laser linewidth [22]. However, the highest He po-larizations, around 80%, have only been obtained with spec-trally narrowed lasers [6, 23, 24]. We shall see that the re-duced dichroism effect explains much of this behavior, sincethe off-resonant light of an un-narrowed laser accentuates thedecrease in dichroism. In addition, the frequency spectrum ofthe light changes as the light propagates through the cell, asthe resonant portions of the spectrum are preferentially atten-uated, while the off-resonant portions experience little attenu-ation. This causes a further decrease in the effective dichroismas the light propagates.We begin by generalizing the results of Sec. I to pumpingsources with non-zero bandwidth. The absorption rate of h =1 light by spin-1/2 atoms of electron spin-polarization P is e A = e R − g RP ∞ P (28)where e f = R dνf ( ν ) . The resulting optical pumping evolu-tion is given by d h F z i dt = 12 (cid:16) g RP ∞ − e RP − Γ P (cid:17) (29)In steady-state, the equilibrium polarization reached by theatoms is P = g RP ∞ Γ + e R . (30)We see from Eq. 30 that in the limit of large pumping rates,the alkali-metal polarization saturates at the pumping-rate-weighted mean value of P ∞ .The steady-state absorption rate is found by substitutingEq. 30 in Eq. 28 to get e A = Γ e R Γ + e R e R Γ − " g RP ∞ e R (31) ≈ Γ e R Γ + e R (cid:20) g Rδ ∞ (cid:21) ( δ ∞ ≪ (32)where δ ∞ = 1 − P ∞ measures the deviation of P ∞ from itsmaximum value of 1. If δ ∞ = 0 , the ideal case, the absorp-tion rate becomes Γ P , and the light dissipation is simply theamount required to compensate for the collisional losses. Athigh pumping rates, as required for optical pumping of opti-cally thick vapors, the factor g Rδ ∞ / Γ can be large and thelight is absorbed by the atoms at a much greater rate.To explore the consequences of reduced dichroism for op-tical pumping at high densities, we have simulated the opticalpumping and light propagation effects described above. Ateach point in the optical pumping cell, the pumping rate is cal-culated from the spectral profile of the light at that position,and the atomic polarization is calculated from the pumpingrate, P ∞ , and the ground-state spin-relaxation rate (calculatedas described in Ref. [6]) according to Eq. 30. Then the spec-tral profile of the light is propagated to the next place in thecell using dI ( ν ) dz = − [Rb] σ ( ν ) I ( ν ) [1 − P P ∞ ( ν )] . (33)The diffusion layer at the front of the cell is accounted for inan approximate manner using the model of Ref. [3]. Heatingeffects [25] have been neglected. The electron spin is assumedto completely relax in the excited state, and the nuclear spin isassumed to be conserved in the excited state. P o w e r ( W ) R b P o l a r i z a t i on FIG. 12. Light propagation results with full (solid red) and measured(dashed green) circular dichroism, for pumping with a laser whosespectral profile is shown in Fig. 13. The upper graph shows the poweras a function of position. The reduced dichroism produces a fasterattenuation of the light. The lower graph shows the correspondingRb polarizations, which are limited to values less than 100% even atthe cell entrance, but then decay further as the spectral profile of thelight is increasingly off-resonant (see Fig. 13).
The effects of reduced dichroism are most readily seen forbroad-band pumping light at high [He], shown in Fig. 12. Weassume 100 W of broadband pumping light (spectral profileshown in Fig. 13) enters a 10 cm diameter, 7 cm long cellwith 8 amg of He and 50 Torr of N , with [Rb]= × cm − . Under these conditions we estimate a spin-relaxationrate of 630/s [6]. In the full dichroism case the light is at-tenuated only due to ground-state spin-relaxation and the 35W dissipated in the cell is consistent with this. The polar-ization is maintained at a very high level, averaging 97%, asthe pumping rate has not been sufficiently reduced to cause asubstantial Rb polarization drop at the back end of the cell.The on-resonant portion of the laser spectral profile is not yetcompletely attenuated, as shown in Fig. 13, again consistentwith maintenance of a high pumping rate. When the reduceddichroism is taken into account, several changes occur. Thepower dissipation per unit length is substantially increased, asseen in Fig. 12, even at the entrance to the cell before the spec-tral hole is burned. The total power dissipation is much greater than in the ideal case, now 65 W. The polarization drop isnow quite substantial, reducing to 75% at the back of the cell.This is due to two effects: 1) the pumping rate is lower dueto the greater power dissipation and the production of a com-plete hole in the spectral profile (shown in Fig. 13), and 2) theremaining light is in the spectral region with low dichroism,further reducing the maximum attainable polarization. S pe c t r a l F l u x ( pho t on s / s / G H z ) -2000 -1000 0 1000 2000Frequency (GHz) reduced dichroismfull dichroismcell entrance FIG. 13. Measured spectra before and modeled spectra after propa-gation through the cell of Fig 12, modeled as described in the text.Light out in the wings of the spectrum disproportionately contributesto the reduced dichroism, while contributing comparatively less tothe pumping rate. The net effect is significantly greater absorptionthan expected.
From the above observations, some of the advantages ofnarrowband pumping are explained, as illustrated in Fig. 14for a 100 GHz laser with half the power of the broadbandlaser of Figs. 12 and 13. A source with spectral width of100 GHz or less will experience nearly full dichroism, reduc-ing the excess light absorption and increasing the maximumattainable polarization. Furthermore, as the light is attenu-ated, the spectral profile does not change significantly and thepolarization remains very high. These observations are qual-itatively consistent with experimental observations and com-parisons of broad and narrow-band sources [6, 24].
V. MEASUREMENTS NEAR THE D1 LINE CENTER
As an important check on the reliability of the direct opti-cal method, we studied the dichroism near line center wherehyperfine interactions play a role, especially in the low pres-sure SEOP cell. This region is particularly good for testingthe accuracy of our polarimetry as our laser, despite its highintensity, was unable to fully spin-polarize the Rb atoms atthe low temperatures ( ∼ ◦ C) needed for moderate opticalthickness at the line center.Although the hyperfine splittings (3.0 GHz for Rb, 6.8GHz for Rb) are smaller than the pressure-broadened atomiclinewidth, the hyperfine effect on the measured normalizedcircular dichroism is striking, as seen in Fig. 15. The low po-larization of this data also makes an accurate measurement of (1 − P ) critical. Since the atoms are not in spin temperature0 P o w e r ( W ) Position (cm) R b P o l a r i z a t i on FIG. 14. Light propagation results with full (solid red) and mea-sured (dashed green) circular dichroism, for a 100 GHz bandwidthpumping source. The upper graph shows the power as a function ofposition. The reduced dichroism produces a slightly faster attenua-tion of the light, but high polarization is maintained throughout thecell, as shown in the lower graph. equilibrium for this region of the data, we found it necessaryto be able to resolve EPR peaks that were < / the sizeof the primary peak in order to accurately deduce the polar-ization from the EPR peak areas. It was also necessary tomeasure the polarization of the two isotopes independently atlow temperatures.The shape of C near resonance is primarily due to a fre-quency shift between the absorption cross sections of light ofopposite helicity. An unpolarized vapor has the same crosssection for absorption of all helicities of light. In a highly po-larized vapor, most of the atoms are pushed into a stretchedstate in the upper hyperfine manifold, | F = 2 , m F = 2 i for Rb, which absorbs only light of helicity h = − (for thecase of full dichroism). The rest of the atoms are in otherstates of high m f in both the upper and lower hyperfine mani-folds. The high m F states in the upper manifold are preferen-tially spin up, while those in the lower manifold are preferen-tially spin down. Thus, in a spin-temperature-like distribution,most of the atoms that absorb light of helicity h = 1 are in thelower manifold, while the overwhelming majority of atomsabsorbing light of h = − are in the upper manifold. As thetwo manifolds have slightly different center frequencies, thisleads to the structure seen in Fig. 15.To model C for near resonance conditions, we can write thespin dependence of the cross sections σ h F m f in terms of thecross sections in the fine structure basis σ h m s = σ o (1 − m s P ∞ h ) (34)Using Clebsch-Gordan coefficients, and neglecting small cor-rections due to excited-state hyperfine structure, σ h F m F ( ν ) = σ o ( ν − ν F ) (cid:16) ∓ m F F h P ∞ (cid:17) (35)(36) for F = I ± / , and the center frequencies of the hyperfinelines are ν F . Then the cross sections σ h can be built fromthe populations ρ F m F that are extracted from the EPR data,giving σ h ( ν ) = X F m F σ h F m F ( ν ) ρ F m F (37)Modeling that used these cross sections, with P ∞ = 1 , is ingood agreement with the near resonance data (Fig. 15). N o r m a li z ed C i r c u l a r D i c h r o i s m Detuning (GHz)
FIG. 15. Normalized Circular Dichroism near resonance, taken atlow density, and low polarization, P = 0 . . Modelling that omitsthe hyperfine (solid blue line) splitting predicts a nearly flat C at P cos θ . Modelling with P ∞ = 1 , but including the ground statehyperfine structure (dashed green line), is in good agreement withthe data. VI. CONCLUSIONS
We have presented here measurements of the spin-dependence of line broadening of alkali-metal atoms in Heand N buffer gases. In particular we observe the breakdownof atomic selection rules due to buffer gas collisions [13]. Wepresented two very distinct methods for making the measure-ments that nonetheless gave very similar results. While line-broadening of alkali-metal atoms by noble gas atoms has beenextensively studied [10], the spin-dependence has not beenpreviously studied. We have pointed out in this paper thatthe unique conditions of spin-exchange optical pumping arequite sensitive to the spin-dependence of far-wing absorption,since such absorption becomes effectively a spin-relaxationmechanism that is proportional to the light intensity. Such amechanism implies not only a limit on the maximum polar-ization that can be obtained, but can greatly increase the laserpower requirements. This effect at least partially accounts forthe substantial improvements observed when using frequencynarrowed lasers [24].In this paper, we have limited our focus to the effects ofreduced dichroism from RbHe and RbN collisions. We havenot attempted to compare modeling predictions with SEOPexperiments because two other effects, non-conservation ofnuclear spin in the excited state and radiation trapping, arealso important and have yet to be included in our models. In1upcoming papers, we will discuss these effects and we intendto present an updated experimental comparison to a full modelthat includes reduced dichroism and these other effects. ACKNOWLEDGMENTS
Discussions with T. R. Gentile were very helpful. Thiswork was supported by the Department of Energy. [1] Y.-Y. Jau, N. N. Kuzma, and W. Happer, Phys. Rev. A, ,061401 (2004).[2] I. K. Kominis, T. W. Kornack, J. C. Allred, and M. V. Romalis,Nature, , 596 (2003).[3] T. Walker and W. Happer, Rev. Mod. Phys., , 629 (1997).[4] N. D. Bhaskar, M. Hou, B. Suleman, and W. Happer, Phys.Rev. Lett., , 519 (1979).[5] W. Happer and W. A. van Wijngaarden, Hyperfine Interact., ,435 (1987).[6] W. C. Chen, T. R. Gentile, T. G. Walker, and E. Babcock, Phys.Rev. A, , 013416 (2007).[7] E. Babcock, I. Nelson, S. Kadlecek, B. Driehuys, L. W. Ander-son, F. W. Hersman, and T. G. Walker, Phys. Rev. Lett., ,123003 (2003).[8] D. L. Drummond and A. Gallagher, J. Chem. Phys., , 3426(1974).[9] M. D. Rotondaro and G. P. Perram, Phys. Rev. A, , 2023(1998).[10] N. Allard and J. Kielkopf, Rev. Mod. Phys., , 1103 (1982).[11] J. Pascale, Phys. Rev. A, , 632 (1983).[12] 1 amg = 2 . × cm − .[13] B. Lancor, E. Babcock, R. Wyllie, and T. G. Walker,arXiv:1003.4292v1 (2010).[14] A. Ben-Amar Baranga, S. Appelt, C. Erickson, A. Young, and W. Happer, Phys. Rev. A, , 2282 (1998).[15] S. Appelt, A. Ben-Amar Baranga, C. J. Erickson, M. V. Ro-malis, A. R. Young, and W. Happer, Phys. Rev. A, , 1412(1998).[16] W. Happer, Rev. Mod. Phys., , 169 (1972).[17] B. Chann, E. Babcock, L. W. Anderson, and T. G. Walker,Phys. Rev. A, , 33406 (2002).[18] M. V. Romalis, E. Miron, and G. D. Cates, Phys. Rev. A, ,4569 (1997).[19] C. Ottinger, R. Scheps, G. W. York, and A. Gallagher, Phys.Rev. A, , 1815 (1975).[20] E. Babcock, B. Chann, I. Nelson, and T. Walker, Appl. Opt., , 3098 (2005).[21] B. Chann, E. Babcock, L. W. Anderson, and T. G. Walker,Phys. Rev. A, , 032703 (2002).[22] B. Driehuys, G. Cates, E. Miron, K. Sauer, D. Walter, andW. Happer, Appl. Phys. Lett., , 1668 (1996).[23] B. Chann, I. Nelson, and T. G. Walker, Opt. Lett., , 1352(2000).[24] B. Chann, E. Babcock, W. Chen, T. Smith, A. Thompson, L. W.Anderson, T. G. Walker, and T. R. Gentile, J. Appl. Phys., ,6908 (2003).[25] D. K. Walter, W. M. Griffith, and W. Happer, Phys. Rev. Lett.,86