Rubidium 87 D Line Data - Steck

1 Rubidium 87 D Line DataDaniel A. SteckTheoretical Division (T-8), MS B285 Los Alamos National LaboratoryLos Alamos, NM 8754525 September 2001(revision , 14 October 2003)1 IntroductionIn this reference we present many of the physical and optical properties of87Rb that are relevant to variousquantum optics experiments. In particular, we give parameters that are useful in treating the mechanical effects oflight on87Rb atoms. The measured numbers are given with their original references, and the calculated numbersare presented with an overview of their calculation along with references to more comprehensive discussions oftheir 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 current version of this document is available , along with Cesium DLine Data and Sodium D Line Data.

2 Please send comments and corrections Physical and Optical PropertiesSome 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 of87Rb are given in Table 37 electrons, only one of which is in the outermost is not a stable isotope of Rubidium , decayingto +87Sr with a total disintegration energy of MeV [2] (the only stable isotope is85Rb), but has anextremely slow decay rate, thus making it effectively stable. This is the only isotope we consider in this mass is taken from the high-precision measurement of [3], and the density, melting point, boiling point, andheat capacities (for the naturally occurring form of Rb) are taken from [2]. The vapor pressure at 25 Candthevapor pressure curve in Fig. 1 are taken from the vapor-pressure model given by [4], which islog10Pv= 26 716 87T+ 26 log10T(solid phase)log10Pv= 53 + 586 63T 38 log10T(liquid phase),(1)wherePvis the vapor pressure in torr, andTis the temperature in K.

3 This model should be viewed as a roughguide rather than a source of precise vapor-pressure values. The ionization limit is the minimum energy requiredto ionize a87Rb atom; this value is taken from Ref. [5].The optical properties of the87Rb D line are given in Tables 3 and 4. The properties are given separatelyfor each of the two D-line components; the D2line (the 52S1/2 52P3/2transition) properties are given inTable 3, and the optical properties of the D1line (the 52S1/2 52P1/2transition) are given in Table 4. Of thesetwo components, the D2transition is of much more relevance to current quantum and atom optics experiments,287RB PHYSICAL AND OPTICAL PROPERTIES2because it has a cycling transition that is used for cooling and trapping87Rb. The frequencies 0of the D2andD1transitions were measured in [6] and [7], respectively (see also [8, 9] for more information on the D1transitionmeasurement); the vacuum wavelengths and the wave numberskLare then determined via the following relations: =2 c 0kL=2.

4 (2)The air wavelength air= /nassumes index of refraction ofn= 268 21, corresponding to dry air at apressure of 760 torr and a temperature of 22 C. The index of refraction is calculated from the Edl en formula [10]:nair=1+ +2 406 030130 2+15 2 388 23P1+ 671T f 2 10 8.(3)Here,Pis the air pressure in torr,Tis the temperature in C, is the vacuum wave numberkL/2 in m 1,andfis the partial pressure of water vapor in the air, in torr. This formula is appropriate for laboratory conditionsand has an estimated uncertainty of 10 lifetimes are taken from a recent measurement employing beam-gas-laser spectroscopy [11]. Invertingthe lifetime gives the spontaneous decay rate (EinsteinAcoefficient), which is also the natural (homogenous)line width (as an angular frequency) of the emitted spontaneous emission rate is a measure of the relative intensity of a spectral line. Commonly, the relativeintensity is reported as an absorption oscillator strengthf, which is related to the decay rate by [12] =e2 202 0mec32J+12J +1f(4)for aJ J fine-structure transition, wheremeis the electron recoil velocityvris the change in the87Rb atomic velocity when absorbing or emitting a resonant photon,and is given byvr= hkLm.

5 (5)The recoil energy h ris defined as the kinetic energy of an atom moving with velocityv=vr,whichis h r= h2k2L2m.(6)The Doppler shift of an incident light field of frequency Ldue to motion of the atom is d=vatomc L(7)for small atomic velocities relative toc. For an atomic velocityvatom=vr, the Doppler shift is simply 2 r. Finally,if one wishes to create a standing wave that is moving with respect to the lab frame, the two traveling-wavecomponents must have a frequency difference determined by the relationvsw= sw2 2,(8)because sw/2 is the beat frequency of the two waves, and /2 is the spatial periodicity of the standing a standing wave velocity ofvr,Eq.(8)gives sw=4 r. Two temperatures that are useful in cooling andtrapping experiments are also given here. The recoil temperature is the temperature corresponding to an ensemblewith a one-dimensional rms momentum of one photon recoil hkL:Tr= h2k2 LmkB.

6 (9)3 HYPERFINE STRUCTURE3 The Doppler temperature,TD= h 2kB,(10)is the lowest temperature to which one expects to be able to cool two-level atoms in optical molasses, due to abalance of Doppler cooling and recoil heating [13]. Of course, in Zeeman-degenerate atoms, sub-Doppler coolingmechanisms permit temperatures substantially below this limit [14].3 Hyperfine Energy Level SplittingsThe 52S1/2 52P3/2and 52S1/2 52P1/2transitions are the components of a fine-structure doublet, and eachof these transitions additionally have hyperfine structure. The fine structure is a result of the coupling between theorbital angular momentumLof the outer electron and its spin angular momentumS. The total electron angularmomentum is then given byJ=L+S,(11)and the corresponding quantum numberJmust lie in the range|L S| J L+S.(12)(Here we use the convention that the magnitude ofJis J(J+1) h, and the eigenvalue ofJzismJ h.)

7 For theground state in87Rb,L=0andS=1/2, soJ=1/2; for the first excited state,L=1,soJ=1/2orJ=3 energy of any particular level is shifted according to the value ofJ,sotheL=0 L= 1 (D line) transitionis split into two components, the D1line (52S1/2 52P1/2)andtheD2line (52S1/2 52P3/2). The meaningof the energy level labels is as follows: the first number is the principal quantum number of the outer electron, thesuperscript is 2S+ 1, the letter refers toL( , S L=0,P L= 1, etc.), and the subscript gives the hyperfine structure is a result of the coupling ofJwith the total nuclear angular momentumI. The totalatomic angular momentumFis then given byF=J+I.(13)As before, the magnitude ofFcan take the values|J I| F J+I.(14)For the87Rb ground state,J=1/2andI=3/2, soF=1orF= 2. For the excited state of the D2line (52P3/2),Fcan take any of the values 0, 1, 2, or 3, and for the D1excited state (52P1/2),Fis either 1 or 2.

8 Again, theatomic energy levels are shifted according to the value the fine structure splitting in87Rb is large enough to be resolved by many lasers ( 15 nm), thetwo D-line components are generally treated separately. The hyperfine splittings, however, are much smaller, andit is useful to have some formalism to describe the energy shifts. The Hamiltonian that describes the hyperfinestructure for each of the D-line components is [12, 15]Hhfs=AhfsI J+Bhfs3(I J)2+32I J I(I+1)J(J+1)2I(2I 1)J(2J 1),(15)which leads to a hyperfine energy shift of Ehfs=12 AhfsK+Bhfs32K(K+1) 2I(I+1)J(J+1)2I(2I 1)2J(2J 1),(16)whereK=F(F+1) I(I+1) J(J+1),(17)3 HYPERFINE STRUCTURE4 Ahfsis the magnetic dipole constant, andBhfsis the electric quadrupole constant (although the term withBhfsapplies only to the excited manifold of the D2transition and not to the levels withJ=1/2). These constants forthe87Rb D line are listed in Table 5.

9 The value for the ground stateAhfsconstant is from a recent atomic-fountainmeasurement [16], while the constants listed for the 52P3/2manifold were taken from a recent, precise measurement[6]. TheAhfsconstant for the 52P1/2manifold is taken from another recent measurement [7]. The energy shiftgiven by (16) is relative to the unshifted value (the center of gravity ) listed in Table 3. The hyperfine structureof87Rb, along with the energy splitting values, is diagrammed in Figs. 2 and Interaction with Static External Magnetic FieldsEach of the hyperfine (F) energy levels contains 2F+ 1 magnetic sublevels that determine the angular distributionof the electron wave function. In the absence of external magnetic fields, these sublevels are degenerate. However,when an external magnetic field is applied, their degeneracy is broken. The Hamiltonian describing the atomicinteraction with the magnetic field isHB= B h(gSS+gLL+gII) B= B h(gSSz+gLLz+gIIz)Bz,(18)if we take the magnetic field to be along thez-direction ( , along the atomic quantization axis).

10 In this Hamilto-nian, the quantitiesgS,gL,andgIare respectively the electron spin, electron orbital, and nuclear g-factors thataccount for various modifications to the corresponding magnetic dipole moments. The values for these factors arelisted in Table 6, with the sign convention of [15]. The value forgShas been measured very precisely, and thevalue given is the CODATA recommended value. The value forgLis approximately 1, but to account for the finitenuclear mass, the quoted value is given bygL=1 memnuc,(19)which is correct to lowest order inme/mnuc,wheremeis the electron mass andmnucis the nuclear mass [17].The nuclear factorgIaccounts for the entire complex structure of the nucleus, and so the quoted value is anexperimental measurement [15].If the energy shift due to the magnetic field is small compared to the fine-structure splitting, thenJis a goodquantum number and the interaction Hamiltonian can be written asHB= B h(gJJz+gIIz)Bz.