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 temper rature 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 t temperatures substantially below this limit 25 3 Hyperfine Structure 3.1 Energy Level Splittings The 5 S1/2-5-P3/2 and 5-S1/2-5-P1/2 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 and 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 J 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, so J=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 nto two components, the D, line(52S1/2--52P1/2)and the D2 line(52S1/2-52P3 /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=l, etc. ), and the subscript gives the value of ex The hyperfine structure is a result of the coupling of J with the total nuclear angular momentum I. The total omic angular momentum F is then given by As before, the magnitude of F can take the values J-1≤F≤J+Ⅰ For the Rb ground state, J=1/2 and I=5/2, so F=2 or F=3. For the excited state of the D2 line(52P3/2) F can take any of the values 1, 2, 3, or 4, and for the Di excited state(5P1/2), F is either 2 or 3. Again, the atomic energy levels are shifted according to the value of F. Because the fine structure splitting in oRb 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 H=A3+B131,J)2+3-(+1)J(+1) 2I(2I-1)J(2J-1) +C101·J)+201·J)2+2I·J)(+1)+J(J+1)+31-3(1+1)J(J+1)-5(1+1)J(J+1) I(-1)(2I-1)J(-1)(2J-1)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 85Rb, 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 85Rb ground state, J = 1/2 and I = 5/2, so F = 2 or F = 3. For the excited state of the D2 line (52P3/2), F can take any of the values 1, 2, 3, or 4, and for the D1 excited state (52P1/2), F is either 2 or 3. Again, the atomic energy levels are shifted according to the value of F. Because the fine structure splitting in 85Rb 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)