复旦大学：物理实验课程《核相关基础实验（I）》参考资料_铷87精细能级 Rubidium 87 D Line Data

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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)

3. 2 INTERACTION WITH STATIC EXTERNAL FIELDS which leads to a hyperfine energy shift of △Ehs==AhK+Bhf K(压+1)-2(+1)J(J 4I(2-1)J(2J-1) (16) CH 5K2(K/4+1)+K[(I+1)+J(J+1)+3-3(I+1)J(J+1)-5(1+1)J(J+1) I(-1)(2-1)J(J-1)(2J-1) K=F(F+1)-I(I+1)-J(J+1 Ahfs is the magnetic dipole constant, Bhfs is the electric quadrupole constant, and Chfs is the magnetic octupole constant(although the terms with Bnfs and Chfs apply only to the excited manifold of the D2 transition and not to the levels with J =1/2). These constants for the Rb D line are listed in Table 5. The value for the ground state Ahfs constant is from a recent atomic-fountain measurement 29, while the constants listed for the 5-P3/2 manifold were taken from a recent, precise measurement [9]. The Ahfs constant for the 5-P1/2 manifold is the average from the recent measurements of [10 and [ 11; because the discrepancy between the measurements is so large, the older recommended value of 26 is also included in the average. These measurements are not yet sufficiently precise to have provided a nonzero value for Chfs, and thus it is not listed. The energy shift given by(16) is relative to the unshifted value(the"center of gravity")listed in Table 3. The hyperfine structure of sRb, along with the energy splitting values, is diagrammed in Figs. 2 and 3 3.2 Interaction with Static External fields 3.2.1 Magnetic fields Each of the hyperfine(F) energy levels contains 2F +I magnetic sublevels that determine the angular distribution of 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 atomic interaction with the magnetic field is Hi PE(ss+9,L+,I).B h(9sS2+92L2+91l2)B2 if we take the magnetic field to be along the z-direction(i.e, along the atomic quantization axis). In this hamilto- nian, the quantities s, gL, and gr are respectively the electron spin, electron orbital, and nuclear "g-factors"that account for various modifications to the corresponding magnetic dipole moments. The values for these factors are listed in Table 6, with the sign convention of 26. The value for gs has been measured very precisely, and the value given is the CODATA recommended value. The value for g, is approximately 1, but to account for the finite uclear mass, the quoted value is given by which is correct to lowest order in me/ muc, where me is the electron mass and muc is the nuclear mass [30 The nuclear factor gr accounts for the entire complex structure of the nucleus, and so the quoted value is an experimental measurement 26 If the energy shift due to the magnetic field is small compared to the fine-structure splitting, then j is a goo quantum number and the interaction Hamiltonian can be written as HB=(gJ2+91I2)B2 20)

6 3.2 Interaction with Static External Fields which leads to a hyperfine energy shift of ∆Ehfs = 1 2 AhfsK + Bhfs 3 2K(K + 1) − 2I(I + 1)J(J + 1) 4I(2I − 1)J(2J − 1) + Chfs 5K2 (K/4 + 1) + K[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) , (16) where K = F(F + 1) − I(I + 1) − J(J + 1), (17) Ahfs is the magnetic dipole constant, Bhfs is the electric quadrupole constant, and Chfs is the magnetic octupole constant (although the terms with Bhfs and Chfs apply only to the excited manifold of the D2 transition and not to the levels with J = 1/2). These constants for the 87Rb D line are listed in Table 5. The value for the ground state Ahfs constant is from a recent atomic-fountain measurement [29], while the constants listed for the 52P3/2 manifold were taken from a recent, precise measurement [9]. The Ahfs constant for the 52P1/2 manifold is the average from the recent measurements of [10] and [11]; because the discrepancy between the measurements is so large, the older recommended value of [26] is also included in the average. These measurements are not yet sufficiently precise to have provided a nonzero value for Chfs, and thus it is not listed. The energy shift given by (16) is relative to the unshifted value (the “center of gravity”) listed in Table 3. The hyperfine structure of 87Rb, along with the energy splitting values, is diagrammed in Figs. 2 and 3. 3.2 Interaction with Static External Fields 3.2.1 Magnetic Fields Each of the hyperfine (F) energy levels contains 2F + 1 magnetic sublevels that determine the angular distribution of 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 atomic interaction with the magnetic field is HB = µB ~ (gSS + gLL + gI I) · B = µB ~ (gSSz + gLLz + gI Iz)Bz, (18) if we take the magnetic field to be along the z-direction (i.e., along the atomic quantization axis). In this Hamiltonian, the quantities gS, gL, and gI are respectively the electron spin, electron orbital, and nuclear “g-factors” that account for various modifications to the corresponding magnetic dipole moments. The values for these factors are listed in Table 6, with the sign convention of [26]. The value for gS has been measured very precisely, and the value given is the CODATA recommended value. The value for gL is approximately 1, but to account for the finite nuclear mass, the quoted value is given by gL = 1 − me mnuc , (19) which is correct to lowest order in me/mnuc, where me is the electron mass and mnuc is the nuclear mass [30]. The nuclear factor gI accounts for the entire complex structure of the nucleus, and so the quoted value is an experimental measurement [26]. If the energy shift due to the magnetic field is small compared to the fine-structure splitting, then J is a good quantum number and the interaction Hamiltonian can be written as HB = µB ~ (gJ Jz + gI Iz)Bz. (20)

3. 2 INTERACTION WITH STATIC EXTERNAL FIELDS Here, the Lande factor g, is given by [30 J(J+1)-S(S+1)+L(L+1),J(J+1)+S(S+1)-L(L+1) 2J(J+1) 2J(J+1) J(J+1)+S(S+1)-L(L+1) where the second, approximate expression comes from taking the approximate values gs a 2 and g a 1. The expression here does not include corrections due to the complicated multielectron structure of Rb [30 and QED effects[31], so the values of g, given in Table 6 are experimental measurements [26](except for the 52P1/2 state 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 similarly F a good quantum number, so the interaction Hamiltonian becomes s HB=HBgF F: B2. where the hyperfine Lande g-factor is given by F(F+1)-I(I+1)+J(J+1 2F(F+1) 2F(F+1) (23) F(F+1)-I(I+1)+J(J+1) 2F(F+1) The second, approximate expression here neglects the nuclear term, which is a correction at the level of 0. 1% since gr is much smaller than g For weak magnetic fields, the interaction Hamiltonian HB perturbs the zero-field eigenstates of Hhfs. To lowest order, the levels split linearly according to 23 △E AB g The approximate gp factors computed from Eq(23)and the corresponding splittings between adjacent magnetic 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 Hamiltonian perturbs the strong-field eigenstates m, I mi. The energies are then given to lowest order by 3 EyJmJ I mi)=Ahfsmjm,+ Bhfs 3(m;m)2+m;m-I(I+1)J(J+1) 2J(2J-1)I(21-1) +p(9m+9nm)B2.(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 Hhfs HB. A notable exception is the Breit- Rabi formula 23, 32, 34, which applies to the ground- state manifold of the d transition E1=1/2 mIMi= 2(27+1)+9AmB±AE In this formula, A Ehfs =Ahfs(I +1/2)is the hyperfine splitting, m=mtmy=mrt1/2(where the t sign is aken to be the same as in(26), and △Ehfs

3.2 Interaction with Static External Fields 7 Here, the Land´e factor gJ is given by [30] gJ = gL J(J + 1) − S(S + 1) + L(L + 1) 2J(J + 1) + gS J(J + 1) + S(S + 1) − L(L + 1) 2J(J + 1) ≃ 1 + J(J + 1) + S(S + 1) − L(L + 1) 2J(J + 1) , (21) where the second, approximate expression comes from taking the approximate values gS ≃ 2 and gL ≃ 1. The expression here does not include corrections due to the complicated multielectron structure of 87Rb [30] and QED effects [31], so the values of gJ given in Table 6 are experimental measurements [26] (except for the 52P1/2 state 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 similarly F is a good quantum number, so the interaction Hamiltonian becomes [32] HB = µB gF Fz Bz, (22) where the hyperfine Land´e g-factor is given by gF = gJ F(F + 1) − I(I + 1) + J(J + 1) 2F(F + 1) + gI F(F + 1) + I(I + 1) − J(J + 1) 2F(F + 1) ≃ gJ F(F + 1) − I(I + 1) + J(J + 1) 2F(F + 1) . (23) The second, approximate expression here neglects the nuclear term, which is a correction at the level of 0.1%, since gI is much smaller than gJ . For weak magnetic fields, the interaction Hamiltonian HB perturbs the zero-field eigenstates of Hhfs. To lowest order, the levels split linearly according to [23] ∆E|F mF i = µB gF mF Bz. (24) The approximate gF factors computed from Eq. (23) and the corresponding splittings between adjacent magnetic 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 Hamiltonian perturbs the strong-field eigenstates |J mJ I mI i. The energies are then given to lowest order by [33] E|J mJ I mI i = AhfsmJmI + Bhfs 3(mJmI ) 2 + 3 2mJmI − I(I + 1)J(J + 1) 2J(2J − 1)I(2I − 1) + µB(gJ mJ + gI mI )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 Hhfs + HB. A notable exception is the Breit-Rabi formula [23, 32, 34], which applies to the groundstate manifold of the D transition: E|J=1/2 mJ I mI i = − ∆Ehfs 2(2I + 1) + gI µB m B ± ∆Ehfs 2 1 + 4mx 2I + 1 + x 2 1/2 . (26) In this formula, ∆Ehfs = Ahfs(I + 1/2) is the hyperfine splitting, m = mI ± mJ = mI ± 1/2 (where the ± sign is taken to be the same as in (26)), and x = (gJ − gI )µB B ∆Ehfs . (27)

3. 2 INTERACTION WITH STATIC EXTERNAL FIELDS In order to avoid a sign ambiguity in evaluating(26), the more direct formula E=1/2m1m)=△B2+1+2(+29)nB can be used for the two states m=+I+1/2). The Breit-Rabi formula is useful in finding the small-field shift of the"clock transition"between the mp=0 sublevels of the two hyperfine ground states, which has no first-order Zeeman shift. Using m= mp for small magnetic fields, we obtain c=(-0)1B2 to second order in the field strength. If the magnetic field is sufficiently strong that the hyperfine Hamiltonian is negligible compared to the inter- action Hamiltonian, then the effect is termed the normal Zeeman effect for hyperfine structure. For even stronger fields, there are Paschen-Back and normal Zeeman regimes for the fine structure, where states with different 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 32, which are beyond the scope of the present discussion The level structure of 87Rb in the presence of a magnetic field is shown in Figs. 4-6 in the weak-field(anomalous Zeeman) regime through the hyperfine Paschen-Back regime 3.2.2 Electric Fields An analogous effect, the dc stark effect, occurs in the presence of a static external electric field. The interaction Hamiltonian in this case is 27, 35, 36 1m232-J(J+1) J(2J-1) where we have taken the electric field to be along the z-direction, ao and a are respectively termed the scalar nd tensor polarizabilities, and the second (a) term is nonvanishing only for the J= 3/2 level. The first term shifts all the sublevels with a given J together, so that the Stark shift for the J=1/2 states is trivial. The only mechanism for breaking the degeneracy of the hyperfine sublevels in(30) is the J2 contribution in the tensor erm. This interaction splits the sublevels such that sublevels with the same value of mFI remain degenerate An expression for the hyperfine Stark shift, assuming a weak enough field that the shift is small compared to the △E 1Fmp)=-2 E E F+3(2F+2)F(2F-1((+ F(F+1)3X(X-1)-4F(F+1) 〓a2 1) X=F(F+1)+J(J+1)-I(I+1) 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 analog to the Paschen-Back effect for magnetic fields. to many nm away from resonance, where the rotating-wave approximation is invalid), since the optical potentia.? The static polarizability is also useful in the context of optical traps that are very far off resonance(i.e, seve given in terms of the ground-state polarizability as V=-1/2aoE, where E is the amplitude of the optical field. a slightly more accurate expression for the far-off resonant potential arises by replacing the static polarizability with the frequency-dependent polarizability 37

8 3.2 Interaction with Static External Fields In order to avoid a sign ambiguity in evaluating (26), the more direct formula E|J=1/2 mJ I mI i = ∆Ehfs I 2I + 1 ± 1 2 (gJ + 2IgI )µB B (28) can be used for the two states m = ±(I + 1/2). The Breit-Rabi formula is useful in finding the small-field shift of the “clock transition” between the mF = 0 sublevels of the two hyperfine ground states, which has no first-order Zeeman shift. Using m = mF for small magnetic fields, we obtain ∆ωclock = (gJ − gI ) 2µ 2 B 2~∆Ehfs B 2 (29) to second order in the field strength. If the magnetic field is sufficiently strong that the hyperfine Hamiltonian is negligible compared to the interaction Hamiltonian, then the effect is termed the normal Zeeman effect for hyperfine structure. For even stronger fields, there are Paschen-Back and normal Zeeman regimes for the fine structure, where states with different 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 [32], which are beyond the scope of the present discussion. The level structure of 87Rb in the presence of a magnetic field is shown in Figs. 4-6 in the weak-field (anomalous Zeeman) regime through the hyperfine Paschen-Back regime. 3.2.2 Electric Fields An analogous effect, the dc Stark effect, occurs in the presence of a static external electric field. The interaction Hamiltonian in this case is [27, 35, 36] HE = − 1 2 α0E 2 z − 1 2 α2E 2 z 3J 2 z − J(J + 1) J(2J − 1) , (30) where we have taken the electric field to be along the z-direction, α0 and α2 are respectively termed the scalar and tensor polarizabilities, and the second (α2) term is nonvanishing only for the J = 3/2 level. The first term shifts all the sublevels with a given J together, so that the Stark shift for the J = 1/2 states is trivial. The only mechanism for breaking the degeneracy of the hyperfine sublevels in (30) is the Jz contribution in the tensor term. This interaction splits the sublevels such that sublevels with the same value of |mF | remain degenerate. An expression for the hyperfine Stark shift, assuming a weak enough field that the shift is small compared to the hyperfine splittings, is [27] ∆E|J I F mF i = − 1 2 α0E 2 z − 1 2 α2E 2 z [3m2 F − F(F + 1)][3X(X − 1) − 4F(F + 1)J(J + 1)] (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 |mJ |, leading to an electric-field analog to the Paschen-Back effect for magnetic fields. The static polarizability is also useful in the context of optical traps that are very far off resonance (i.e., several to many nm away from resonance, where the rotating-wave approximation is invalid), since the optical potential is given in terms of the ground-state polarizability as V = −1/2α0E2 , where E is the amplitude of the optical field. A slightly more accurate expression for the far-off resonant potential arises by replacing the static polarizability with the frequency-dependent polarizability [37] α0(ω) = ω 2 0 α0 ω 2 0 − ω2 , (33)

3.3 REDUCTION OF THE DIPOLE OPERATOR where wo is the resonant frequency of the lowest-energy transition (i. e, the DI resonance); this approximat expression is valid for light tuned far to the red of the Di line. The STRb polarizabilities are tabulated in Table 6. Notice that the differences in the 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 cm )or atomic units(units of a3, where the Bohr radius ao is given in Table 1) The SI values can be converted to cgs units via a[cm]=(100h/4TEo (a/)[Hz/(V/cm)]=5.955 213 79(30)x 10-22(a/h)[Hz/(V/cm)2](see [37] for discussion of units), and subsequently the conversion to atomic units is straightforward The level structure of 7Rb in the presence of an external dc electric field is shown in Fig. 7 in the weak-field egime through the electric hyperfine Paschen-Back regime 3.3 Reduction of the Dipole Operator The strength of the interaction between STRb and nearly-resonant optical radiation is characterized by the dipole matrix elements. Specifically, (F mpler)denotes the matrix element that couples the two hyperfine sublevels F mp)and Fmp(where the primed variables refer to the excited states and the unprimed variables refer to the ground states). To calculate these matrix elements, it is useful to factor out the angular dependence nd write the matrix element as a product of a Clebsch-Gordan coefficient and a reduced matrix element, using the Wigner-Eckart theorem 38 (F mplerglf'mp)=(FllerllF)(F mFlF'I mp q) Here, q is an index labeling the component of r in the spherical basis, and the doubled bars indicate that the atrix element is reduced. We can also write(34) in terms of a Wigner 3-3 symbol FmlFm)=(Fp-)-1+mVF+(F1F e q-p Notice that the 3-j symbol (or, equivalently, the Clebsch-Gordan coefficient) vanishes unless the sublevels satisfy mp=mp+g. This reduced matrix element can be further simplified by factoring out the F and F dependence to a Wigner 6-j symbol, leaving a further reduced matrix element that depends only on the L, S, and quantum numbers 38 (F|erF)≡{ J I FerN T'F (Jer)(-1)F++1+V(2F+1)(2J+1) I. 1 FF I Again, this new matrix element can be further factored into another 6-3 symbol and a reduced matrix element nvolving only the L quantum number (‖er‖J)≡{ L SerL' SJ) L|er2(-1)y+L+1+S②2+1(2+1{L The numerical value of the (J=1/2erlJ'=3/2)(D2)and the(J=1/2 erlJ=1/2)(D1)matrix elements are given in Table 7. These values were calculated from the lifetime via the expression 39 3丌60bc32)+/(Jerl)2 Note that all the equations we have presented here assume the normalization convention ∑ J Mer J"M)H2=∑ J MleralJ'M)2=1(Jlrl) M

3.3 Reduction of the Dipole Operator 9 where ω0 is the resonant frequency of the lowest-energy transition (i.e., the D1 resonance); this approximate expression is valid for light tuned far to the red of the D1 line. The 87Rb polarizabilities are tabulated in Table 6. Notice that the differences in the 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 atomic units (units of a 3 0 , where the Bohr radius a0 is given in Table 1). The SI values can be converted to cgs units via α[cm3 ] = (100 · h/4πǫ0)(α/h)[Hz/(V/cm)2 ] = 5.955 213 79(30) × 10−22 (α/h)[Hz/(V/cm)2 ] (see [37] for discussion of units), 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 radiation is characterized by the dipole matrix elements. Specifically, hF mF |er|F ′ m′ F i denotes the matrix element that couples the two hyperfine sublevels |F mF i and |F ′ m′ F i (where the primed variables refer to the excited states and the unprimed variables 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 [38]: hF mF |erq|F ′ m′ F i = hFkerkF ′ ihF mF |F ′ 1 m′ F qi. (34) 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 hF mF |erq|F ′ m′ F i = hFkerkF ′ i(−1)F ′−1+mF √ 2F + 1 F ′ 1 F m′ F q −mF . (35) Notice that the 3-j symbol (or, equivalently, the Clebsch-Gordan coefficient) vanishes unless the sublevels satisfy mF = m′ F + q. This reduced matrix element can be further simplified by factoring out the F and F ′ dependence into a Wigner 6-j symbol, leaving a further reduced matrix element that depends only on the L, S, and J quantum numbers [38]: hFkerkF ′ i ≡ hJ I FkerkJ ′ I ′ F ′ i = hJkerkJ ′ i(−1)F ′+J+1+Ip (2F′ + 1)(2J + 1) J J′ 1 F ′ F I . (36) Again, this new matrix element can be further factored into another 6-j symbol and a reduced matrix element involving only the L quantum number: hJkerkJ ′ i ≡ hL S JkerkL ′ S ′ J ′ i = hLkerkL ′ i(−1)J ′+L+1+Sp (2J ′ + 1)(2L + 1) L L′ 1 J ′ J S . (37) The numerical value of the hJ = 1/2kerkJ ′ = 3/2i (D2) and the hJ = 1/2kerkJ ′ = 1/2i (D1) matrix elements are given in Table 7. These values were calculated from the lifetime via the expression [39] 1 τ = ω 3 0 3πǫ0~c 3 2J + 1 2J ′ + 1 |hJkerkJ ′ i|2 . (38) Note that all the equations we have presented here assume the normalization convention X M′ |hJ M|er|J ′ M′ i|2 = X M′q |hJ M|erq|J ′ M′ i|2 = |hJkerkJ ′ i|2 . (39)

4 RESONANCe FLUORESCENCE There is, however, another common convention(used in Ref. [ 40) that is related to the convention used here by(JerllJ"=v2J+1(JlerIJ'"). Also, we have used the standard phase convention for the Clebsch-Gordan coefficients as given in Ref. 38, where formulae for the computation of the Wigner 3-3(equivalently, Clebsch- Gordan) and 6-3(equivalently, Racah) coefficients may also be found The dipole matrix elements for specific F me)-F'mp) transitions are listed in Tables 9-20 as multiples of Jer ). The tables are separated by the ground-state F number(1 or 2)and the polarization of the transition (where a+-polarized light couples mF mp= mp + 1, T-polarized light couples mp mp=mp, and -polarized light couples mp -mp=mp-1) 4 Resonance fluorescence 4.1 Symmetries of the Dipole Operator Although the hyperfine structure of 87Rb is quite complicated, it is possible to take advantage of some symmetries 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 D1 and D2 lines separately, we will discuss the symmetries in 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 Fmp) add up to a factor that is independent of the particular sublevel chosen ∑F(m+q) lerglf'm)2 2J+1 F as can be verified from the dipole matrix element tables. The degeneracy-ratio factor of (2J +1)/(2 +1)(which is 1 for the Di line or 1/2 for the D2 line) is the same factor that appears in Eq. ( 38), and is a consequence of the ormalization convention (39). The interpretation of this symmetry is simply that all the excited state sublevels decay at the same rate T, 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 leve in a particular F energy leve Sr>F+12+{F1 IF mpIF1(mp-9)q)2 (2F+1)(2J+1) This sum SFF, is independent of the particular ground state sublevel chosen, and also obeys the sum rule The interpretation of this symmetry is that for an isotropic pump field (i.e, a pumping field with equal components in all three possible polarizations), the coupling to the atom is independent of how the population is distributed mong the sublevels. These factors SFp(which are listed in Table 8)provide a measure of the relative strength of each of the F- F 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 diso.eff(F→→F)P2=SF、er|J/^)P2 The factor of 1/3 in this expression comes from the fact that any given polarization of the field only interacts 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)

10 4 Resonance Fluorescence There is, however, another common convention (used in Ref. [40]) that is related to the convention used here by (JkerkJ ′ ) = √ 2J + 1 hJkerkJ ′ i. Also, we have used the standard phase convention for the Clebsch-Gordan coefficients as given in Ref. [38], where formulae for the computation of the Wigner 3-j (equivalently, ClebschGordan) and 6-j (equivalently, Racah) coefficients may also be found. The dipole matrix elements for specific |F mF i −→ |F ′ m′ F i transitions are listed in Tables 9-20 as multiples of hJkerkJ ′ i. The tables are separated by the ground-state F number (1 or 2) and the polarization of the transition (where σ +-polarized light couples mF −→ m′ F = mF + 1, π-polarized light couples mF −→ m′ F = mF , and σ −-polarized light couples mF −→ m′ F = mF − 1). 4 Resonance Fluorescence 4.1 Symmetries of the Dipole Operator Although the hyperfine structure of 87Rb is quite complicated, it is possible to take advantage of some symmetries 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 D1 and D2 lines separately, we will discuss the symmetries in 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 ′ m′ F i add up to a factor that is independent of the particular sublevel chosen, X q F |hF (m′ F + q)|erq|F ′ m′ F i|2 = 2J + 1 2J ′ + 1 |hJkerkJ ′ i|2 , (40) as can be verified from the dipole matrix element tables. The degeneracy-ratio factor of (2J + 1)/(2J ′ + 1) (which is 1 for the D1 line or 1/2 for the D2 line) is the same factor that appears in Eq. (38), and is a consequence of the 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 in a particular F ′ energy level: SF F ′ := X q (2F ′ + 1)(2J + 1) J J′ 1 F ′ F I 2 |hF mF |F ′ 1 (mF − q) qi|2 = (2F ′ + 1)(2J + 1) J J′ 1 F ′ F I 2 . (41) This sum SF F ′ is independent of the particular ground state sublevel chosen, and also obeys the sum rule X F ′ SF F ′ = 1. (42) The interpretation of this symmetry is that for an isotropic pump field (i.e., a pumping field with equal components in all three possible polarizations), the coupling to the atom is independent of how the population is distributed among the sublevels. These factors SF F ′ (which are listed in Table 8) provide a measure of the relative strength of each of the F −→ F ′ 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 |diso,eff(F −→ F ′ )| 2 = 1 3 SF F ′ |hJ||er||J ′ i|2 . (43) The factor of 1/3 in this expression comes from the fact that any given polarization of the field only interacts 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)

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复旦大学：物理实验课程《核相关基础实验（I）》参考资料_铷87精细能级 Rubidium 87 D Line Data