2 RUBIDIUM 85 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+ +5)(项b)( 1+10-8(0.601-0.00972T)P 1+0.0036610T -f(037345-0004012)×10-8 Here, P is the air pressure in Pa, T is the temperature in C, K is the vacuum wave number k /2 in um-I,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 5P1/2 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 5P1/2 state and 26 20(9) for the 5-P3/2 state, the third used photoassociation spectroscopy [20(as quoted by [19), with a lifetime of 26.23(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 52P1/2 state and 26.25(8)ns for the 52P3/2 state. Note that at present levels of theoretical accul racy, we do not distinguish between lifetimes of the 5Rb 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 absorption oscillator strength f, which is related to the decay rate by 23 2J+1 for a J-J fine-structure transition, where me is the electron mass. The recoil velocity vr is the change in the srb atomic velocity when absorbing or emitting a resonant photon The recoil energy hwr is defined as the kinetic energy of an atom moving with velocity v= vr, which is h2k 2m The Doppler shift of an incident light field of frequency w due to motion of the atom is △u for small atomic velocities relative to c. For an atomic velocity vatom vr, the Doppler shift is simply 2wr. Finall 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 △sw入 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 I Weighted means were computed according to u=(C, rjwi)(E, wi), where the weights w, were taken to be the invers variances of each measurement, w=1/02. The variance of the weighted mean was estimated according to o2=(>, wi(ej 0)/(n-1)>i wil, and the uncertainty in the weighted the square root of this variance. See Refs. [16, 17] for more details4 2 Rubidium 85 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 spec￾troscopy [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 85Rb 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