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 details4 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 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 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