Rubidium 87 d Line data Daniel Adam Steck Oregon Center for Optics and Department of Physics, University of oregon
Rubidium 87 D Line Data Daniel Adam Steck Oregon Center for Optics and Department of Physics, University of Oregon
Copyright 2001, by Daniel Adam Steck. All rights reserved This material may be distributed only subject to the terms and conditions set forth in the Open Publication License, v1.0orlater(thelatestversionispresentlyavailableathttp://www.opencontent.org/openpub/).distribution of substantively modified versions of this document is prohibited without the explicit permission of the copyright holder. Distribution of the work or derivative of the work in any standard(paper) book form is prohibited unless prior permission is obtained from the copyright holder Original revision posted 25 September 2001 This is revision 2.0.1, 2 May 2008 Cite this document as DanielA.Steck,rubIdium87DLineData,availableonlineathttp://steck.us/alkalidata(revision2.0.1 2May2008) Author contact information Daniel Steck Department of Physics 274 University of Oregon Eugene, Oregon 97403-1274 dsteckQuoregon edu
Copyright c 2001, by Daniel Adam Steck. All rights reserved. This material may be distributed only subject to the terms and conditions set forth in the Open Publication License, v1.0 or later (the latest version is presently available at http://www.opencontent.org/openpub/). Distribution of substantively modified versions of this document is prohibited without the explicit permission of the copyright holder. Distribution of the work or derivative of the work in any standard (paper) book form is prohibited unless prior permission is obtained from the copyright holder. Original revision posted 25 September 2001. This is revision 2.0.1, 2 May 2008. Cite this document as: Daniel A. Steck, “Rubidium 87 D Line Data,” available online at http://steck.us/alkalidata (revision 2.0.1, 2 May 2008). Author contact information: Daniel Steck Department of Physics 1274 University of Oregon Eugene, Oregon 97403-1274 dsteck@uoregon.edu
1 INTRODUCTION 1 Introduction In this reference we present many of the physical and optical properties of sRb that are relevant to various quantum optics experiments. In particular, we give parameters that are useful in treating the mechanical effects of light on Rb atoms. The measured numbers are given with their original references, and the calculated numbers are presented with an overview of their calculation along with references to more comprehensive discussions of their underlying theory. At present, this document is not a critical review of experimental data, nor is it even guaranteed to be correct; for any numbers critical to your research, you should consult the original references. We also present a detailed discussion of the calculation of fuorescence scattering rates, because this topic is often not treated clearly in the literature. More details and derivations regarding the theoretical formalism here may be found in Ref. 1 Thecurrentversionofthisdocumentisavailableathttp://steck.us/alkalidata,alongwithcesiumD Line Data, " Sodium D Line Data, " and"Rubidium 85 D Line Data. This is the only permanent URL for this document at present: please do not link to any others. Please send comments, corrections, and suggestions to dsteck@oregon. edu. 2 Rubidium 87 Physical and Optical Properties Some useful fundamental physical constants are given in Table 1. The values given are the 2006 COdaTa recommended values, as listed in [2]. Some of the overall physical properties of Rb are given in Table 2.Rubidium 87 has 37 electrons, only one of which is in the outermost shell. sTRb is not a stable isotope of rubidium, decaying to B-+ 87Sr with a total disintegration energy of 0.283 Mev [3(the only stable isotope is sRb), but has an extremely slow decay rate, thus making it effectively stable. This is the only isotope we consider in this reference. The mass is taken from the high-precision measurement of [4, and the density, melting point, boiling point, and heat capacities(for the naturally occurring form of Rb)are taken from 3. The vapor pressure at 25 C and the por pressure curve in Fig. I are taken from the vapor-pressure model given by 5, which is 4215 log1o Pv=2.881+4.857-T(solid phase) 1og10P=2.881+43124040 (liquid phase) where P is the vapor pressure in torr(for Pv in atmospheres, simply omit the 2.881 term), and T is the temperature in K. This model is specified to have an accuracy better than +5% from 298-550K. Older, and probably less- accurate, sources of vapor-pressure data include Refs. 6 and 7. The ionization limit is the minimum energy required to ionize a SRb atom; this value is taken from Ref [8] for he optical properties of the sTRb D line are given in Tables 3 and 4. The properties are given separately each of the two D-line components; the D2 line(the 52S1/2 52P 3/2 transition o properties are given in Table 3, and the optical properties of the DI line(the 5-S1/2-452P1/2 transition) are given in Table 4. Of these two components, the D2 transition is of much more relevance to current quantum and atom optics experiments because it has a cycling transition that is used for cooling and trapping Rb. The frequency wo of the D2 was measured in 9, while the frequency of the Di transition is an average of values given by [10 and [11; the vacuum wavelengths A and the wave numbers k are then determined via the following relations Due to the different nuclear masses of the two isotopes srB and 87Rb, the transition frequencies of 87Rb are shifted slightly up compared to those of 85Rb. This difference is reported as the isotope shift, and the values are taken from [10.( See 11, 12 for less accurate measurements. The air wavelength Aair A/n assumes an index of refraction of n= 1.000 266 501(30) for the D2 line and n= 1.000 266 408 (30) for the D1 line, corresponding
1 Introduction 3 1 Introduction In this reference we present many of the physical and optical properties of 87Rb that are relevant to various quantum optics experiments. In particular, we give parameters that are useful in treating the mechanical effects of light on 87Rb atoms. The measured numbers are given with their original references, and the calculated numbers are presented with an overview of their calculation along with references to more comprehensive discussions of their underlying theory. At present, this document is not a critical review of experimental data, nor is it even guaranteed to be correct; for any numbers critical to your research, you should consult the original references. We also present a detailed discussion of the calculation of fluorescence scattering rates, because this topic is often not treated clearly in the literature. More details and derivations regarding the theoretical formalism here may be found in Ref. [1]. The current version of this document is available at http://steck.us/alkalidata, along with “Cesium D Line Data,” “Sodium D Line Data,” and “Rubidium 85 D Line Data.” This is the only permanent URL for this document at present; please do not link to any others. Please send comments, corrections, and suggestions to dsteck@uoregon.edu. 2 Rubidium 87 Physical and Optical Properties Some useful fundamental physical constants are given in Table 1. The values given are the 2006 CODATA recommended values, as listed in [2]. Some of the overall physical properties of 87Rb are given in Table 2. Rubidium 87 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 [3] (the only stable isotope is 85Rb), but has an extremely slow decay rate, thus making it effectively stable. This is the only isotope we consider in this reference. The mass is taken from the high-precision measurement of [4], and the density, melting point, boiling point, and heat capacities (for the naturally occurring form of Rb) are taken from [3]. The vapor pressure at 25◦C and the vapor pressure curve in Fig. 1 are taken from the vapor-pressure model given by [5], which is log10 Pv = 2.881 + 4.857 − 4215 T (solid phase) log10 Pv = 2.881 + 4.312 − 4040 T (liquid phase), (1) where Pv is the vapor pressure in torr (for Pv in atmospheres, simply omit the 2.881 term), and T is the temperature in K. This model is specified to have an accuracy better than ±5% from 298–550K. Older, and probably lessaccurate, sources of vapor-pressure data include Refs. [6] and [7]. The ionization limit is the minimum energy required to ionize a 87Rb atom; this value is taken from Ref. [8]. 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 D2 line (the 52S1/2 −→ 5 2P3/2 transition) properties are given in Table 3, and the optical properties of the D1 line (the 52S1/2 −→ 5 2P1/2 transition) are given in Table 4. Of these two components, the D2 transition is of much more relevance to current quantum and atom optics experiments, because it has a cycling transition that is used for cooling and trapping 87Rb. The frequency ω0 of the D2 was measured in [9], while the frequency of the D1 transition is an average of values given by [10] and [11]; the vacuum wavelengths λ and the wave numbers kL are then determined via the following relations: λ = 2πc ω0 kL = 2π λ . (2) Due to the different nuclear masses of the two isotopes 85Rb and 87Rb, the transition frequencies of 87Rb are shifted slightly up compared to those of 85Rb. This difference is reported as the isotope shift, and the values are taken from [10]. (See [11, 12] for less accurate measurements.) The air wavelength λair = λ/n assumes an index of refraction of n = 1.000 266 501(30) for the D2 line and n = 1.000 266 408(30) for the D1 line, corresponding
2 RUBIDIUM 87 PHYSICAL AND OPTICAL PrOPerties to typical laboratory conditions(100 kPa pressure, 20 C temperature, and 50% relative humidity). The index of refraction is calculated from the 1993 revision [13 of the Edlen formula [14 nair=1+8342.54+ +a1)(项B)(+10n -f(003735-0012)×10-8 Here, P is the air pressure in Pa, T is the temperature inC, K is the vacuum wave number k /2 in um-,and f is the partial pressure of water vapor in the air, in Pa(which can be computed from the relative humidity via the Goff-Gratch equation [15 ). This formula is appropriate for laboratory conditions and has an estimated (30) uncertainty of 3 x 10-8 from 350-650 nm The lifetimes are weighted averages! from four recent measurements; the first employed beam-gas-laser spec troscopy [18], with lifetimes of 27.70(4)ns for the 5-P1/2 state and 26. 24(4)ns for the 5-P3/2 state, the second used time-correlated single-photon counting [19, with lifetimes of 27.64(4)ns for the 5-P1/2 state and 26.20(9) Po for the 52P3/2 state, the third used photoassociation spectroscopy [20](as quoted by [191), with a lifetime of 23(6)ns for the 5-P3/2 state only, and the fourth also used photoassociation spectroscopy 21], with lifetimes of 27.75(8)ns for the 5-P1/2 state and 26.25(8)ns for the 5-P3/2 state. Note that at present levels of theoretical [22]and experimental accuracy, we do not distinguish between lifetimes of the sRb and 87Rb isotopes. Inverting the lifetime gives the spontaneous decay rate r(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 of a spectral line. Commonly, the relative tensity is reported as an absorption oscillator strength f, which is related to the decay rate by[23 or a -J' fine-structure transition, where me is the electron The recoil velocity vr is the change in the Rb atomic velocity when absorbing or emitting a resonant photon and is given by The recoil energy hwr is defined as the kinetic energy of an atom moving with velocity U=vr, which h2k2 2m The Doppler shift of an incident light field of frequency w due to motion of the atom is for small atomic velocities relative to c. For an atomic velocity vatom U, the Doppler shift is simply 2wr. Finally, 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 because Awsw /2 is the beat frequency of the two waves, and A/2 is the spatial periodicity of the standing wave For a standing wave velocity of vr, Eq(8)gives Awsw 4wr. Two temperatures that are useful in cooling and ans were computed according to u=(2i Jui)/i,), where the weights w, were taken to be the inverse variances of each measurement, wi=1/a,. The variance of the weighted mean was estimated according to af=(2; wj(rj u)2)/[(n-1)>, wil, and the uncertainty in the weighted mean is the square root of this variance. See Refs. [16, 17] for more details
4 2 Rubidium 87 Physical and Optical Properties to typical laboratory conditions (100 kPa pressure, 20◦C temperature, and 50% relative humidity). The index of refraction is calculated from the 1993 revision [13] of the Edl´en formula [14]: nair = 1 + " 8 342.54 + 2 406 147 130 − κ 2 + 15 998 38.9 − κ 2 P 96 095.43 1 + 10−8 (0.601 − 0.009 72 T )P 1 + 0.003 6610 T −f 0.037 345 − 0.000 401 κ 2 # × 10−8 . (3) Here, P is the air pressure in Pa, T is the temperature in ◦C, κ is the vacuum wave number kL/2π in µm−1 , and f is the partial pressure of water vapor in the air, in Pa (which can be computed from the relative humidity via the Goff-Gratch equation [15]). This formula is appropriate for laboratory conditions and has an estimated (3σ) uncertainty of 3 × 10−8 from 350-650 nm. The lifetimes are weighted averages1 from four recent measurements; the first employed beam-gas-laser spectroscopy [18], with lifetimes of 27.70(4) ns for the 52P1/2 state and 26.24(4) ns for the 52P3/2 state, the second used time-correlated single-photon counting [19], with lifetimes of 27.64(4) ns for the 52P1/2 state and 26.20(9) ns for the 52P3/2 state, the third used photoassociation spectroscopy [20] (as quoted by [19]), with a lifetime of 26.23(6) ns for the 52P3/2 state only, and the fourth also used photoassociation spectroscopy [21], with lifetimes of 27.75(8) ns for the 52P1/2 state and 26.25(8) ns for the 52P3/2 state. Note that at present levels of theoretical [22] and experimental accuracy, we do not distinguish between lifetimes of the 85Rb and 87Rb isotopes. 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 of a spectral line. Commonly, the relative intensity is reported as an absorption oscillator strength f, which is related to the decay rate by [23] Γ = e 2ω 2 0 2πǫ0mec 3 2J + 1 2J ′ + 1 f (4) for a J −→ J ′ fine-structure transition, where me is the electron mass. The recoil velocity vr is the change in the 87Rb atomic velocity when absorbing or emitting a resonant photon, and is given by vr = ~kL m . (5) The recoil energy ~ωr is defined as the kinetic energy of an atom moving with velocity v = vr, which is ~ωr = ~ 2k 2 L 2m . (6) The Doppler shift of an incident light field of frequency ωL due to motion of the atom is ∆ωd = vatom c ωL (7) for small atomic velocities relative to c. For an atomic velocity vatom = vr, the Doppler shift is simply 2ωr. Finally, 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 vsw = ∆ωsw 2π λ 2 , (8) because ∆ωsw/2π is the beat frequency of the two waves, and λ/2 is the spatial periodicity of the standing wave. For a standing wave velocity of vr, Eq. (8) gives ∆ωsw = 4ωr. Two temperatures that are useful in cooling and 1Weighted means were computed according to µ = (P j xjwj )/( P j wj ), where the weights wj were taken to be the inverse variances of each measurement, wj = 1/σ2 j . The variance of the weighted mean was estimated according to σ 2 µ = (P j wj (xj − µ) 2 )/[(n − 1) P j wj ], and the uncertainty in the weighted mean is the square root of this variance. See Refs. [16, 17] for more details
3 HYPERFINE STRUCTURE trapping experiments are also given here. The recoil temperature is the temperature corresponding to an ensemble with a one-dimensional rms momentum of one photon recoil hkL h2k2 mk The Doppler temperature 2KB 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 24]. Of course, in Zeeman-degenerate atoms, sub-Doppler cooling echanisms permit temperatures substantially below this limit 25 Hyperfine Structure 3.1 Energy Level Splittings The 52S1/2-5P32 and 52S1/2-52P1/2 transitions are the components of a fine-structure doublet, and each of these transitions additionally have hyperfine structure. The fine structure is a result of the coupling between the orbital angular momentum L of the outer electron and its spin angular momentum S. The total electron angular omentum is then given by J=L+s (11) nd the corresponding quantum number J must lie in the range L-S≤J≤L+S. (Here we use the convention that the magnitude of J is VJ(+1)h, and the eigenvalue of 2 is m, h )For the ground state in Rb, L=0 and S=1/2, so J=1/2: for the first excited state, L= l, soJ=1 /2 or J=3/2 The energy of any particular level is shifted according to the value of so the L=0-L=l(D line) transition is split into two components, the DI line(52S1/2-5 P1/2)and the D2 line(52S1/2-52Pa/2).The meaning of the energy level labels is as follows: the first number is the principal quantum number of the outer electron, the superscript is 2S+l, the letter refers to L (i.e, S+L=0, P+L=l, etc. ) and the subscript gives the value of The hyperfine structure is a result of the coupling of J with the total nuclear angular momentum I. The total tomic angular momentum F is then given by J+I eore tude of f can take the values J-≤F≤J+Ⅰ 14) For the Rb ground state, J=1/2 and I=3/2, so F=l or F=2. For the excited state of the D2 line(5 P3/2) F can take any of the values 0, 1, 2, or 3, and for the Di excited state(5-P1/2), F is either 1 or 2. Again,the atomic energy levels are shifted according to the value of F. Because the fine structure splitting in 7Rb is large enough to be resolved by many lasers(15 nm), the two D-line components 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 23, 26-28 Hhfs=AhsI·J+Bh IJ)2+2(·J)-I(I+1)J(J+1) 2I(2I-1)J(2J-1) +c10J+200J)2+21.J)(+1)+J(+1)+3-3(+1)J(+1)-51(+1)J(+1 I(I-1)(2-1)J(J-1)(2J-1) (15)
3 Hyperfine Structure 5 trapping experiments are also given here. The recoil temperature is the temperature corresponding to an ensemble with a one-dimensional rms momentum of one photon recoil ~kL: Tr = ~ 2k 2 L mkB . (9) The Doppler temperature, TD = ~Γ 2kB , (10) 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 [24]. Of course, in Zeeman-degenerate atoms, sub-Doppler cooling mechanisms permit temperatures substantially below this limit [25]. 3 Hyperfine Structure 3.1 Energy Level Splittings The 52S1/2 −→ 5 2P3/2 and 52S1/2 −→ 5 2P1/2 transitions are the components of a fine-structure doublet, and each of these transitions additionally have hyperfine structure. The fine structure is a result of the coupling between the orbital angular momentum L of the outer electron and its spin angular momentum S. 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) (Here we use the convention that the magnitude of J is p J(J + 1)~, and the eigenvalue of Jz is mJ ~.) For the 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. The energy of any particular level is shifted according to the value of J, so the L = 0 −→ L = 1 (D line) transition is split into two components, the D1 line (52S1/2 −→ 5 2P1/2) and the D2 line (52S1/2 −→ 5 2P3/2). The meaning of the energy level labels is as follows: the first number is the principal quantum number of the 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 coupling of J with the total nuclear angular momentum I. 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) For the 87Rb ground state, J = 1/2 and I = 3/2, so F = 1 or F = 2. For the excited state of the D2 line (52P3/2), F can take any of the values 0, 1, 2, or 3, and for the D1 excited state (52P1/2), F is either 1 or 2. Again, the 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-line components 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 [23, 26–28] Hhfs = AhfsI · J + Bhfs 3(I · J) 2 + 3 2 (I · J) − I(I + 1)J(J + 1) 2I(2I − 1)J(2J − 1) + Chfs 10(I · J) 3 + 20(I · J) 2 + 2(I · J)[I(I + 1) + J(J + 1) + 3] − 3I(I + 1)J(J + 1) − 5I(I + 1)J(J + 1) I(I − 1)(2I − 1)J(J − 1)(2J − 1) , (15)