Rubidium 87 D Line Data Daniel A. Steck Theoretical Division (T-8), MS B285 Los Alamos National Laboratory Los Alamos, NM 87545 25 September 2001 (revision 1.6,14 October 2003) 1 Introduction In this reference we present many of the physical and optical properties of 87Rb that are relevant to various quantumopticsexperiments. Inparticular,wegiveparameters thatare usefulintreatingthe mechanicaleffects of light on 87Rb atoms. The measured numbers are given with their originalreferences, and the calculated numbers are presented with an overview of their calculation along with references to more comprehensive discussions of their underlying theory. We also present a detailed discussion of the calculation of fluorescence scattering rates, because this topic is often not treated carefully in the literature. The current version of this document is available at http://steck.us/alkalidata, along with “Cesium D Line Data” and “Sodium D Line Data.”Please send comments and corrections to [email protected]. 2 87Rb Physical and Optical Properties Some useful fundamental physical constants are given in Table 1. The values given are the 1998 CODATA recommended values, as listed in [1]. Some of the overall physical properties of 87Rb are given in Table 2. 87Rb has 37 electrons, only one of which is in the outermost shell. 87Rb is not a stable isotope of rubidium, decaying to β− + 87Sr with a total disintegration energy of 0.283 MeV [2] (the only stable isotope is 85Rb), but has an extremely slowdecay rate, thus makingiteffectively stable. This is the onlyisotope we consider in this reference. The mass is taken from the high-precision measurement of [3], and the density, melting point, boilingpoint, and ◦ heat capacities (for the naturally occurring form of Rb) are taken from [2]. The vapor pressure at 25 C and the vapor pressure curve in Fig. 1 are taken from the vapor-pressure model given by [4], which is 1961.258 log P =−94.04826− −0.037716 87 T +42.57526log T (solid phase) 10 v T 10 (1) 4529.635 log P =15.88253− +0.000586 63 T −2.991 38log T (liquid phase), 10 v T 10 where P is the vapor pressure in torr, and T is the temperature in K. This model should be viewed as a rough v guide rather than a source of precise vapor-pressure values. The ionizationlimit is the minimumenergy required to ionize a 87Rb atom;this value is taken from Ref. [5]. The optical properties of the 87Rb D line are given in Tables 3 and 4. The properties are given separately for each of the two D-line components; the D line (the 52S −→ 52P transition) properties are given in 2 1/2 3/2 Table3,andthe opticalproperties ofthe D line(the 52S −→52P transition) are giveninTable4. Of these 1 1/2 1/2 two components, the D transition is of much more relevance to current quantum and atom optics experiments, 2 2 87RB PHYSICAL AND OPTICAL PROPERTIES 2 because it has a cycling transition that is used for cooling and trapping 87Rb. The frequencies ω of the D and 0 2 D transitions were measured in [6] and [7], respectively (see also [8, 9] for more informationon the D transition 1 1 measurement);thevacuumwavelengthsλandthewavenumbersk arethendeterminedviathefollowingrelations: L 2πc 2π λ= k = . (2) ω L λ 0 The air wavelength λ = λ/n assumes index of refraction of n = 1.000 268 21, corresponding to dry air at a air ◦ pressure of 760torr and a temperature of22 C. The index of refraction is calculated fromthe Edl´en formula[10]: (cid:1)(cid:2) (cid:3) (cid:2) (cid:3) (cid:6) (cid:4) (cid:5) 2 406 030 15 997 0.001388 23 P n =1+ 8342.13+ + × −f 5.722−0.0457κ2 ×10−8 . (3) air 130−κ2 38.9−κ2 1+0.003671 T Here, P is the air pressure intorr, T is the temperature in ◦C,κ is the vacuum wavenumber k /2π inµm−1,and L f is the partial pressure of water vapor in the air, in torr. This formula is appropriate for laboratory conditions and has an estimated uncertainty of ≤10−8. The lifetimes are taken from a recent measurement employing beam-gas-laser spectroscopy [11]. Inverting the lifetime gives the spontaneous decay rate Γ (Einstein A coefficient), which is also the natural (homogenous) line width (as an angular frequency) of the emitted radiation. The spontaneous emission rate is a measure of the relative intensity ofa spectral line. Commonly,the relative intensity is reported as an absorption oscillator strength f, which is related to the decay rate by [12] e2ω2 2J +1 Γ= 0 f (4) 2π(cid:6) m c32J(cid:2)+1 0 e for a J −→J(cid:2) fine-structure transition, where m is the electron mass. e The recoil velocityv isthe change inthe87Rbatomicvelocitywhen absorbingoremittingaresonant photon, r and is given by ¯hk v = L . (5) r m The recoil energy ¯hω is defined as the kinetic energy of an atom moving with velocity v =v , which is r r ¯h2k2 ¯hω = L . (6) r 2m The Doppler shift of an incident lightfield of frequency ω due to motionof the atom is L v atom ∆ω = ω (7) d c L forsmallatomicvelocitiesrelativetoc. Foranatomicvelocityv =v ,theDopplershiftissimply2ω . Finally, atom r r if one wishes to create a standing wave that is moving with respect to the lab frame, the two traveling-wave components must have a frequency difference determined by the relation ∆ω λ sw v = , (8) sw 2π 2 because ∆ω /2π is the beat frequency of the two waves, and λ/2 is the spatial periodicity of the standing wave. sw For a standing wave velocity of v , Eq. (8) gives ∆ω = 4ω . Two temperatures that are useful in cooling and r sw r trappingexperiments arealsogivenhere. Therecoiltemperature isthetemperature correspondingtoanensemble with a one-dimensional rms momentum of one photon recoil h¯k : L ¯h2k2 T = L . (9) r mk B 3 HYPERFINE STRUCTURE 3 The Doppler temperature, ¯hΓ T = , (10) D 2k B is the lowest temperature to which one expects to be able to cool two-level atoms in optical molasses, due to a balance of Doppler cooling and recoil heating [13]. Of course, in Zeeman-degenerate atoms, sub-Doppler cooling mechanisms permit temperatures substantially below this limit [14]. 3 Hyperfine Structure 3.1 Energy Level Splittings The 52S −→52P and52S −→52P transitionsare the componentsofafine-structure doublet,andeach 1/2 3/2 1/2 1/2 ofthese transitionsadditionallyhavehyperfine structure. Thefinestructure isaresultofthecouplingbetweenthe orbital angular momentumL of the outer electron and its spin angular momentumS. The total electron angular momentum is then given by J=L+S , (11) and the corresponding quantum number J must lie in the range |L−S|≤J ≤L+S . (12) (cid:7) (Here we use the convention that the magnitude of J is J(J +1)h¯, and the eigenvalue of J is m ¯h.) For the z J ground state in 87Rb, L = 0 and S = 1/2, so J = 1/2; for the first excited state, L = 1, so J = 1/2 or J = 3/2. Theenergy ofanyparticularlevelisshiftedaccordingtothe valueofJ,so theL=0−→L=1(Dline)transition is split into two components, the D line (52S −→52P ) and the D line (52S −→52P ). The meaning 1 1/2 1/2 2 1/2 3/2 ofthe energy levellabels isas follows: the first numberis the principalquantumnumber ofthe outer electron, the superscript is 2S+1, the letter refers to L (i.e., S ↔L = 0, P ↔ L = 1, etc.), and the subscript gives the value of J. The hyperfine structure is a result of the couplingof J with the total nuclear angular momentumI. The total atomic angular momentum F is then given by F=J+I . (13) As before, the magnitude of F can take the values |J−I|≤F ≤J +I . (14) Forthe 87Rbgroundstate, J =1/2and I =3/2,soF =1or F =2. Forthe excited state ofthe D line(52P ), 2 3/2 F can take any of the values 0, 1, 2, or 3, and for the D excited state (52P ), F is either 1 or 2. Again, the 1 1/2 atomic energy levels are shifted according to the value of F. Because the fine structure splitting in 87Rb is large enough to be resolved by many lasers (∼ 15 nm), the two D-linecomponents are generally treated separately. The hyperfine splittings,however, are much smaller,and it is useful to have some formalism to describe the energy shifts. The Hamiltonian that describes the hyperfine structure for each of the D-line components is [12, 15] 3(I·J)2+ 3I·J−I(I +1)J(J +1) H =A I·J+B 2 , (15) hfs hfs hfs 2I(2I −1)J(2J −1) which leads to a hyperfine energy shift of 1 3K(K +1)−2I(I +1)J(J +1) ∆E = A K+B 2 , (16) hfs 2 hfs hfs 2I(2I −1)2J(2J −1) where K =F(F +1)−I(I +1)−J(J +1) , (17) 3 HYPERFINE STRUCTURE 4 A is the magnetic dipole constant, and B is the electric quadrupole constant (although the term with B hfs hfs hfs applies only to the excited manifoldof the D transition and not to the levels with J =1/2). These constants for 2 the87RbDlinearelistedinTable5. ThevalueforthegroundstateA constantisfromarecent atomic-fountain hfs measurement[16],whiletheconstantslistedforthe52P manifoldweretakenfromarecent,precisemeasurement 3/2 [6]. The A constant for the 52P manifold is taken from another recent measurement [7]. The energy shift hfs 1/2 given by (16) is relative to the unshifted value (the “center of gravity”)listed inTable 3. The hyperfine structure of 87Rb, along with the energy splitting values, is diagrammedin Figs. 2 and 3. 3.2 Interaction with Static External Fields 3.2.1 Magnetic Fields Eachofthe hyperfine (F)energy levelscontains2F+1magneticsublevels thatdeterminethe angulardistribution ofthe electron wavefunction. Inthe absence ofexternal magneticfields, these sublevels are degenerate. However, when an external magnetic field is applied, their degeneracy is broken. The Hamiltonian describing the atomic interaction with the magnetic field is µ H = B(g S+g L+g I)·B B ¯h S L I (18) µ = B(g S +g L +g I )B , ¯h S z L z I z z ifwe takethemagneticfieldtobe alongthez-direction (i.e.,alongtheatomicquantizationaxis). InthisHamilto- nian,the quantitiesg , g , and g are respectively the electron spin,electron orbital,and nuclear “g-factors” that S L I account forvarious modificationsto the corresponding magneticdipole moments. The values for these factors are listed in Table 6, with the sign convention of [15]. The value for g has been measured very precisely, and the S valuegivenisthe CODATArecommended value. Thevalueforg isapproximately1,buttoaccountforthe finite L nuclear mass, the quoted value is given by m g =1− e , (19) L m nuc which is correct to lowest order in m /m , where m is the electron mass and m is the nuclear mass [17]. e nuc e nuc The nuclear factor g accounts for the entire complex structure of the nucleus, and so the quoted value is an I experimental measurement [15]. If the energy shift due to the magnetic field is smallcompared to the fine-structure splitting,then J is a good quantum number and the interaction Hamiltoniancan be written as µ H = B(g J +g I )B . (20) B ¯h J z I z z Here, the Land´e factor g is given by [17] J J(J +1)−S(S +1)+L(L+1) J(J +1)+S(S +1)−L(L+1) g =g +g J L 2J(J +1) S 2J(J +1) (21) J(J +1)+S(S +1)−L(L+1) (cid:5)1+ , 2J(J +1) where the second, approximate expression comes from taking the approximate values g (cid:5) 2 and g (cid:5) 1. The S L expression here does not include corrections due tothe complicatedmultielectron structure of 87Rb[17]and QED effects [18], so the values of g given in Table 6 are experimental measurements [15] (except for the 52P state J 1/2 value, for which there has apparently been no experimental measurement). If the energy shift due to the magnetic field is small compared to the hyperfine splittings, then similarlyF is a good quantum number, so the interaction Hamiltonianbecomes [19] H =µ g F B , (22) B B F z z 3 HYPERFINE STRUCTURE 5 where the hyperfine Land´e g-factor is given by F(F +1)−I(I +1)+J(J +1) F(F +1)+I(I +1)−J(J +1) g =g +g F J 2F(F +1) I 2F(F +1) (23) F(F +1)−I(I +1)+J(J +1) (cid:5)g . J 2F(F +1) The second, approximate expression here neglects the nuclear term, which is a correction at the level of 0.1%, since g is much smaller than g . I J Forweakmagneticfields,the interactionHamiltonianH perturbs thezero-fieldeigenstates ofH . Tolowest B hfs order, the levels split linearly according to [12] ∆E|F mF(cid:3) =µBgFmF Bz . (24) The approximate g factors computed from Eq. (23) and the corresponding splittings between adjacent magnetic F sublevels are given in Figs. 2 and 3. The splitting in this regime is called the anomalous Zeeman effect. For strong fields where the appropriate interaction is described by Eq. (20), the interaction term dominates the hyperfine energies, so that the hyperfine Hamiltonianperturbs the strong-field eigenstates |J m I m (cid:6). The J I energies are then given to lowest order by [20] 3(m m )2+ 3m m −I(I +1)J(J +1) E|JmJ ImI(cid:3) =AhfsmJmI +Bhfs J I 2J(22JJ−1I)I(2I −1) +µB(gJmJ +gImI)Bz . (25) The energy shift in this regime is called the Paschen-Back effect. For intermediate fields, the energy shift is more difficult to calculate, and in general one must numerically diagonalize H +H . A notable exception is the Breit-Rabi formula [12, 19, 21], which applies to the ground- hfs B state manifoldof the D transition: (cid:2) (cid:3) 1/2 ∆E ∆E 4mx E|J=1/2mJ ImI(cid:3) =−2(2I +hfs1) +gIµBmB± 2hfs 1+ 2I +1 +x2 . (26) In this formula,∆E =A (I +1/2) is the hyperfine splitting, m = m ±m = m ±1/2 (where the ± sign is hfs hfs I J I taken to be the same as in (26)), and (g −g )µ B x= J I B . (27) ∆E hfs In order to avoid a sign ambiguityin evaluating (26), the more direct formula I 1 E|J=1/2mJ ImI(cid:3) =∆Ehfs2I+1 ± 2(gJ +2IgI)µBB (28) can be used for the twostates m=±(I+1/2). The Breit-Rabi formulais useful in finding the small-fieldshift of the “clock transition” between the m = 0 sublevels of the two hyperfine ground states, which has no first-order F Zeeman shift. Using m=m for small magnetic fields, we obtain F (g −g )2µ2 ∆ω = J I BB2 (29) clock 2¯h∆E hfs to second order in the field strength. If the magnetic field is sufficiently strong that the hyperfine Hamiltonianis negligible compared to the inter- action Hamiltonian,then the effect istermed the normal Zeeman effect for hyperfine structure. Foreven stronger fields,there arePaschen-Back andnormalZeeman regimesforthe finestructure, where states withdifferent J can mix, and the appropriate form of the interaction energy is Eq. (18). Yet stronger fields induce other behaviors, such as the quadratic Zeeman effect [19], which are beyond the scope of the present discussion. Thelevelstructure of87Rbinthepresence ofamagneticfieldisshowninFigs.4-6intheweak-field(anomalous Zeeman) regime through the hyperfine Paschen-Back regime. 3 HYPERFINE STRUCTURE 6 3.2.2 Electric Fields An analogous effect, the dc Stark effect, occurs in the presence of a static external electric field. The interaction Hamiltonianin this case is [22–24] 1 1 3J2−J(J +1) H =− α E2− α E2 z , (30) E 2 0 z 2 2 z J(2J −1) where we have taken the electric field to be along the z-direction, α and α are respectively termed the scalar 0 2 and tensor polarizabilities, and the second (α ) term is nonvanishing only for the J = 3/2 level. The first term 2 shifts all the sublevels with a given J together, so that the Stark shift for the J = 1/2 states is trivial. The onlymechanism for breaking the degeneracy of the hyperfine sublevels in (30) is the J contribution inthe tensor z term. This interaction splits the sublevels such that sublevels with the same value of |m | remain degenerate. F An expression for the hyperfine Stark shift, assuming a weak enough field that the shift is small compared to the hyperfine splittings, is [22] 1 1 [3m2 −F(F +1)][3X(X−1)−4F(F +1)J(J +1)] ∆E|JIF mF(cid:3) =−2α0Ez2− 2α2Ez2 F (2F +3)(2F +2)F(2F −1)J(2J −1) , (31) where X =F(F +1)+J(J +1)−I(I +1) . (32) For stronger fields, when the Stark interaction Hamiltonian dominates the hyperfine splittings, the levels split according to the value of |m |, leading to an electric-field analogto the Paschen-Back effect for magnetic fields. J The staticpolarizabilityisalsousefulinthe contextofopticaltraps thatareveryfaroffresonance (i.e.,several tomanynmawayfromresonance, where the rotating-waveapproximationisinvalid),since theopticalpotentialis given in terms of the ground-state polarizabilityas V =−1/2α E2, where E is the amplitude of the opticalfield. 0 A more accurate expression for the far-off resonant potential arises by replacing the static polarizabilitywith the frequency-dependent polarizability[25] ω2α α (ω) = 0 0 , (33) 0 ω2−ω2 0 where ω is the resonant frequency of the lowest-energy transition (i.e., the D resonance); this approximate 0 1 expression is validfor lighttuned far to the red of the D line. 1 The 87Rb polarizabilitiesare tabulated inTable 6. Notice that the differences inthe excited state and ground state scalar polarizabilities are given, rather than the excited state polarizabilities, since these are the quantities that were actually measured experimentally. The polarizabilities given here are in SI units, although they are often given in cgs units (units of cm3) or atomicunits (units of a3, where the Bohr radius a is given in Table 1). 0 0 The SI values can be converted to cgs units via α[cm3] = 5.95531×10−22α[Hz/(V/cm)2] [25], and subsequently the conversion to atomic units is straightforward. The level structure of 87Rb in the presence of an external dc electric field is shown in Fig. 7 in the weak-field regime through the electric hyperfine Paschen-Back regime. 3.3 Reduction of the Dipole Operator The strength of the interaction between 87Rb and nearly-resonant optical radiationis characterized by the dipole matrix elements. Specifically, (cid:7)F m |er|F(cid:2) m(cid:2) (cid:6) denotes the matrix element that couples the two hyperfine sublevels |F m (cid:6) and |F(cid:2) m(cid:2) (cid:6) (whereFthe primedFvariables refer to the excited states and the unprimed variables F F refer to the ground states). To calculate these matrix elements, it is useful to factor out the angular dependence and write the matrix element as a product of a Clebsch-Gordan coefficient and a reduced matrix element, using the Wigner-Eckart theorem [26]: (cid:7)F m |er |F(cid:2)m(cid:2) (cid:6)=(cid:7)F(cid:8)er(cid:8)F(cid:2)(cid:6)(cid:7)F m |F(cid:2)1m(cid:2) q(cid:6) . (34) F q F F F 4 RESONANCE FLUORESCENCE 7 Here, q is an index labeling the component of r in the spherical basis, and the doubled bars indicate that the matrix element is reduced. We can also write (34) in terms of a Wigner 3-j symbol as (cid:2) (cid:3) √ (cid:2) (cid:7)F mF|erq|F(cid:2)m(cid:2)F(cid:6)=(cid:7)F(cid:8)er(cid:8)F(cid:2)(cid:6)(−1)F(cid:1)−1+mF 2F +1 mF(cid:2) 1q −Fm . (35) F F Notice that the 3-j symbol (or, equivalently,the Clebsch-Gordan coefficient) vanishes unless the sublevels satisfy (cid:2) (cid:2) m = m +q. This reduced matrix element can be further simplified by factoring out the F and F dependence F F intoaWigner6-j symbol,leavingafurther reduced matrixelementthatdepends onlyontheL,S,andJ quantum numbers [26]: (cid:7)F(cid:8)er(cid:8)F(cid:2)(cid:6)≡(cid:7)J I F(cid:8)er(cid:8)J(cid:2)I(cid:2) F(cid:2)(cid:6) (cid:8) (cid:9) (cid:7) (cid:2) =(cid:7)J(cid:8)er(cid:8)J(cid:2)(cid:6)(−1)F(cid:1)+J+1+I (2F(cid:2)+1)(2J +1) J(cid:2) J 1 . (36) F F I Again, this new matrix element can be further factored into another 6-j symbol and a reduced matrix element involvingonly the L quantum number: (cid:7)J(cid:8)er(cid:8)J(cid:2)(cid:6)≡(cid:7)LS J(cid:8)er(cid:8)L(cid:2) S(cid:2) J(cid:2)(cid:6) (cid:8) (cid:9) (cid:7) (cid:2) =(cid:7)L(cid:8)er(cid:8)L(cid:2)(cid:6)(−1)J(cid:1)+L+1+S (2J(cid:2)+1)(2L+1) L(cid:2) L 1 . (37) J J S The numerical valueof the (cid:7)J =1/2(cid:8)er(cid:8)J(cid:2)=3/2(cid:6) (D ) and the (cid:7)J =1/2(cid:8)er(cid:8)J(cid:2)=1/2(cid:6) (D ) matrixelements are 2 1 given in Table 7. These values were calculated from the lifetime via the expression [27] 1 ω3 2J +1 = 0 |(cid:7)J(cid:8)er(cid:8)J(cid:2)(cid:6)|2 . (38) τ 3π(cid:6) ¯hc32J(cid:2)+1 0 Note that all the equations we have presented here assume the normalizationconvention (cid:10) (cid:10) |(cid:7)J M|er|J(cid:2)M(cid:2)(cid:6)|2 = |(cid:7)J M|er |J(cid:2) M(cid:2)(cid:6)|2 =|(cid:7)J(cid:8)er(cid:8)J(cid:2)(cid:6)|2 . (39) q M(cid:1) M(cid:1)q There is, howeve√r, another common convention (used in Ref. [28]) that is related to the convention used here by (J(cid:8)er(cid:8)J(cid:2)) = 2J +1(cid:7)J(cid:8)er(cid:8)J(cid:2)(cid:6). Also, we have used the standard phase convention for the Clebsch-Gordan coefficients as given in Ref. [26], where formulae for the computation of the Wigner 3-j (equivalently, Clebsch- Gordan) and 6-j (equivalently,Racah) coefficients mayalso be found. The dipolematrixelements forspecific |F m (cid:6)−→|F(cid:2)m(cid:2) (cid:6) transitionsarelistedinTables9-20asmultiplesof (cid:7)J(cid:8)er(cid:8)J(cid:2)(cid:6). The tables are separated by the grouFnd-state F nFumber and the polarizationof the transition (where σ+-polarized light couples m −→ m(cid:2) = m +1, π-polarized light couples m −→ m(cid:2) = m , and σ−-polarized light couples m −→m(cid:2) =mF −1).F F F F F F F F 4 Resonance Fluorescence 4.1 Symmetries of the Dipole Operator Althoughthehyperfine structure of87Rbisquitecomplicated,itispossibletotakeadvantageofsomesymmetries of the dipole operator in order to obtain relatively simple expressions for the photon scattering rates due to resonance fluorescence. In the spirit of treating the D and D lines separately, we willdiscuss the symmetries in 1 2 this section implicitly assuming that the light is interacting with only one of the fine-structure components at a time. First, notice that the matrix elements that couple to any single excited state sublevel |F(cid:2) m(cid:2) (cid:6) add up to a F factor that is independent of the particular sublevel chosen, (cid:10) 2J +1 |(cid:7)F (m(cid:2) +q)|er |F(cid:2)m(cid:2) (cid:6)|2 = |(cid:7)J(cid:8)er(cid:8)J(cid:2)(cid:6)|2 , (40) F q F 2J(cid:2)+1 qF 4 RESONANCE FLUORESCENCE 8 (cid:2) as can be verified fromthe dipolematrixelement tables. The degeneracy-ratio factor of(2J+1)/(2J +1) (which is 1 for the D line or 1/2for the D line) is the same factor that appears in Eq. (38),and is a consequence ofthe 1 2 normalization convention (39). The interpretation of this symmetry is simply that all the excited state sublevels decay at the same rate Γ, and the decaying population “branches” into various ground state sublevels. Another symmetry arises from summing the matrix elements from a single ground-state sublevel to the levels (cid:2) in a particular F energy level: (cid:8) (cid:9) (cid:10) (cid:2) 2 SFF(cid:1) := (2F(cid:2)+1)(2J +1) FJ(cid:2) JF I1 |(cid:7)F mF|F(cid:2) 1(mF −q)q(cid:6)|2 q (cid:8) (cid:9) (41) (cid:2) 2 (cid:2) J J 1 = (2F +1)(2J +1) (cid:2) . F F I This sum SFF(cid:1) is independent of the particular ground state sublevel chosen, and also obeys the sum rule (cid:10) SFF(cid:1) =1. (42) F(cid:1) Theinterpretationofthissymmetryisthatforanisotropicpumpfield(i.e.,apumpingfieldwithequalcomponents in all three possible polarizations), the coupling to the atom is independent of how the population is distributed among the sublevels. These factors SFF(cid:1) (which are listed in Table 8) provide a measure of the relative strength of each of the F −→ F(cid:2) transitions. In the case where the incident light is isotropic and couples two of the F levels, the atom can be treated as a two-level atom, with an effective dipole moment given by 1 |diso,eff(F −→F(cid:2))|2 = SFF(cid:1)|(cid:7)J||er||J(cid:2)(cid:6)|2 . (43) 3 The factorof1/3inthisexpression comesfromthe factthatanygivenpolarizationofthe field onlyinteracts with one (of three) components of the dipole moment, so that it is appropriate to average over the couplings rather than sum over the couplings as in (41). When the light is detuned far from the atomic resonance (∆ (cid:11) Γ), the light interacts with several hyperfine levels. If the detuning is large compared to the excited-state frequency splittings, then the appropriate dipole strength comes from choosing any ground state sublevel |F m (cid:6) and summing over its couplings to the excited F states. In the case of π-polarized light,the sum is independent of the particular sublevel chosen: (cid:8) (cid:9) (cid:10)(2F(cid:2)+1)(2J +1) FJ(cid:2) JF(cid:2) I1 2|(cid:7)F mF|F(cid:2)1mF 0(cid:6)|2 = 13 . (44) F(cid:1) This sum leads to an effective dipole moment for far detuned radiation given by 1 |d |2 = |(cid:7)J||er||J(cid:2)(cid:6)|2 . (45) det,eff 3 The interpretation of this factor is also straightforward. Because the radiation is far detuned, it interacts with the full J −→J(cid:2) transition; however, because the lightis linearly polarized, it interacts with only one component of the dipole operator. Then, because of spherical symmetry, |dˆ|2 ≡ |erˆ|2 = e2(|xˆ|2+|yˆ|2+|zˆ|2) = 3e2|zˆ|2. Note ± that this factor of 1/3 also appears for σ light, but only when the sublevels are uniformly populated (which, of course, is not the equilibrium configuration for these polarizations). The effective dipole moments for this case and the case of isotropic pumping are given in Table 7. 4.2 Resonance Fluorescence in a Two-Level Atom 4 RESONANCE FLUORESCENCE 9 In these two cases, where we have an effective dipole moment, the atoms behave like simple two-level atoms. A two-level atom interacting with a monochromatic field is described by the optical Bloch equations [27], iΩ ρ˙ = (ρ˜ −ρ˜ )+Γρ gg ge eg ee 2 iΩ ρ˙ =− (ρ˜ −ρ˜ )−Γρ (46) ee ge eg ee 2 iΩ ρ˜˙ =−(γ+i∆)ρ˜ − (ρ −ρ ) , ge ge 2 ee gg where the ρ are the matrix elements of the density operator ρ := |ψ(cid:6)(cid:7)ψ|, Ω := −d·E /¯h is the resonant Rabi ij 0 frequency, d is the dipole operator, E is the electric field amplitude (E = E cosω t), ∆ := ω − ω is the 0 0 L L 0 detuning of the laser field from the atomic resonance, Γ = 1/τ is the natural decay rate of the excited state, γ := Γ/2+γ is the “transverse” decay rate (where γ is a phenomenological decay rate that models collisions), c c ρ˜ := ρ exp(−i∆t) is a “slowly varying coherence,” and ρ˜ = ρ˜∗ . In writing down these equations, we have ge ge ge eg made the rotating-wave approximation and used a master-equation approach to model spontaneous emission. Additionally,wehaveignoredanyeffects due tothemotionofthe atomanddecays orcouplingstoother auxiliary states. In the case of purely radiative damping(γ =Γ/2), the excited state populationsettles to the steady state solution (Ω/Γ)2 ρ (t→∞)= . (47) ee 1+4(∆/Γ)2+2(Ω/Γ)2 The (steady state) total photon scattering rate (integrated over all directions and frequencies) is then given by Γρee(t→∞): (cid:2) (cid:3) Γ (I/I ) sat R = . (48) sc 2 1+4(∆/Γ)2+(I/I ) sat In writing down this expression, we have defined the saturation intensity I such that sat (cid:2) (cid:3) 2 I Ω =2 , (49) I Γ sat which gives (with I =(1/2)c(cid:6) E2) 0 0 c(cid:6) Γ2¯h2 0 I = , (50) sat 4|ˆ(cid:6)·d|2 where ˆ(cid:6) is the unit polarization vector of the light field, and d is the atomic dipole moment. With I defined sat in this way, the on-resonance scattering cross section σ, which is proportional to R (∆ = 0)/I, drops to 1/2 of sc its weakly pumped value σ when I = I . More precisely, we can define the scattering cross section σ as the 0 sat power radiated by the atom divided by the incident energy flux (i.e., so that the scattered power is σI), which from Eq. (48) becomes σ 0 σ = , (51) 1+4(∆/Γ)2+(I/I ) sat where the on-resonance cross section is defined by ¯hωΓ σ = . (52) 0 2I sat Additionally, the saturation intensity (and thus the scattering cross section) depends on the polarization of the pumping light as well as the atomic alignment, although the smallest saturation intensity (Isat(mF=±2→m(cid:1)F=±3), discussed below)isoftenquotedasarepresentative value. Somesaturationintensitiesandscatteringcrosssections corresponding to the discussions in Section 4.1 are given in Table 7. A more detailed discussion of the resonance fluorescence from a two-level atom, including the spectral distribution of the emitted radiation, can be found in Ref. [27]. 4 RESONANCE FLUORESCENCE 10 4.3 Optical Pumping If none of the special situations in Section 4.1 applies to the fluorescence problem of interest, then the effects of optical pumping must be accounted for. A discussion of the effects of optical pumpingin an atomic vapor on the saturation intensity usinga rate-equation approach can be found inRef. [29]. Here, however, we willcarry out an analysis based on the generalization of the optical Bloch equations (46) to the degenerate level structure of alkali atoms. The appropriate master equation for the density matrix of a F →F hyperfine transition is [30–33] g e ⎡ ⎫ (cid:10) (cid:10) ⎪⎪ ∂∂tρ˜αmα,βmβ = −2i ⎣δαe Ω(mα,mg)ρ˜gmg,βmβ −δgβ Ω(me,mβ)ρ˜αmα,eme ⎪⎪⎪⎪⎬ mg me ⎤ (pumpfield) (cid:10) (cid:10) ⎪⎪⎪ +δαg Ω∗(me,mα)ρ˜eme,βmβ −δeβ Ω∗(mβ,mg)ρ˜αmα,gmg⎦⎪⎪⎪⎭ me mg ⎫ ⎪⎪ − δαeδeβ Γρ˜αmα,βmβ ⎪⎪⎪⎪⎪⎪ Γ ⎪⎪ − δαeδgβ 2 ρ˜αmα,βmβ ⎪⎪⎪⎪⎪ ⎬ Γ − δαgδeβ 2 ρ(cid:10)˜α1 m(cid:19)α,βmβ ⎪⎪⎪⎪⎪(dissipation) + δαgδgβ Γq=−1 ρ˜e(mα+q),e(mβ+q) (cid:20) ⎪⎪⎪⎪⎪⎪⎪⎪⎪ (cid:7)F (m +q)|F 1m q(cid:6)(cid:7)F (m +q)|F 1m q(cid:6) ⎪⎭ e α g α e β g β (cid:21) + i(δ δ −δ δ )∆ρ˜ (free evolution) αe gβ αg eβ αmα,βmβ (53) where Ω(me,mg)=(cid:7)Fg mg|Fe1me −(cid:22)(me−mg)(cid:6)Ω−(me−mg) (54) 2F +1 =(−1)Fe−Fg+me−mg 2Fg+1 (cid:7)Fe me|Fg 1mg (me−mg)(cid:6)Ω−(me−mg) e is the Rabi frequency between two magnetic sublevels, 2(cid:7)F ||er||F (cid:6)E(+) e g q Ω = (55) q ¯h (+) is the overallRabifrequency with polarizationq (E is the field amplitudeassociated withthe positive-rotating q component, withpolarizationq inthe spherical basis),and δ isthe Kronecker deltasymbol. Thismaster equation ignores coupling to F levels other than the ground (g) and excited (e) levels; hence, this equation is appropriate for a cycling transition such as F = 2 −→ F(cid:2) = 3. Additionally, this master equation assumes purely radiative damping and, as before, does not describe the motionof the atom. To calculate the scattering rate from a Zeeman-degenerate atom, it is necessary to solve the master equation for the steady-state populations. Then, the total scattering rate is given by (cid:10) R =ΓP =Γ ρ , (56) sc e eme,eme me where P is the total population in the excited state. In addition, by including the branching ratios of the e spontaneous decay, it is possible to account for the polarization of the emitted radiation. Defining the scattering