Copyright by Kirsten Viering 2006 - Raizen Lab - The University of ...

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in brackets describes a Wigner-3j-symbol, which includes the selection rules for optical transitions. It is related to the Clebsch-Gordan-coefficients via the following equation Jk 1 Jj −Mk q Mj = (−1) J k−1−Mj (2Jj + 1) 〈Jk − Mk, 1q|Jj − Mj〉. (2.31) Further evaluation of the reduced matrix element is necessary in order to cal- culate the dipole matrix element. The reduced matrix element in the F-basis can be transferred into the J-basis by factoring out the F-dependance, 〈IkJkFk||r||IjJjFj〉 = (−1) 1+J k+Ij+Fj 2Fj + 1 2Fk + 1 The term in curled brackets represents a Wigner-6j-symbol. Jk Ij Fk Fj 1 Jj 〈Jk||r||Jj〉. (2.32) By inserting eq. 2.30 and 2.32 into 2.28 we express the matrix element as µ jk = e〈J kF kM k|rq|JjFjMj〉 Jk 1 Jj = e · (−1) 1+Jk+I+Fj+J k−Mk −Mk q Mj × 2Fj + 1 Jk I Fk 2Fk + 1 〈Jk||r||Jj〉. (2.33) Fj 1 Jj We can connect the reduced matrix element 〈J ′ ||r||J〉 with the Einstein-coefficients Ajk for spontaneous emission, respectively the inverse of the partial lifetime [10], A kj = 1 τ kj = ω 3 kj 3πɛ0c 3 1 2J k + 1 |〈J k||er||Jj〉| 2 . (2.34) Hence the dynamics of the system are included in the Einstein-coefficients, while the geometrical properties are represented by the Wigner-symbols. By applying eq. 2.34 and eq. 2.33 we get the following result for the energy shift [13], 11
|µ kj| ∆Ej = k 2E2 1 1 − ωkj + ωl ωkj − ωl E = k 2 1 1 3πɛ0c − Akj ωkj + ωl ωkj − ωl 3 ω3 (2Jk + 1) kj |e · (−1) 1+J k+I+Fj+J k−Mk 2Fj + 1 Jk 1 Jj Jk I Fk 2Fk + 1 | −Mk q Mj Fj 1 Jj 2 = 3πc 2 Akj(2Jk + 1) I ω k 2 · (ω2 − ω2 kj l kj ) × | 2Fj + 1 Jk 1 Jj Jk I Fk 2Fk + 1 | −Mk q Mj Fj 1 Jj 2 . (2.35) The sum of the Einstein coefficients and the corresponding transition strengths include the contributions from the possible dipole transitions in the atom. The relative transition strengths depend on the involved angular momenta J and F, the projection M and the polarization of the laser beam and are given by the geometry of the system which we have expressed by the Wigner-3j and Wigner-6j-symbols. Nevertheless, the absolute value of the AC-Stark shift is determined by the intensity of the laser beam. 12
Page 1 and 2: Copyright by Kirsten Viering 2006
Page 3 and 4: Spatially resolved single atom dete
Page 5 and 6: Acknowledgments First of all I woul
Page 7 and 8: Spatially resolved single atom dete
Page 9 and 10: Chapter 5 Stimulated Raman transiti
Page 11 and 12: List of Figures 2.1 Schematic of th
Page 13 and 14: 6.1 Schematic of the possible Raman
Page 15 and 16: are therefore useful tools. A tweez
Page 17 and 18: eigenfunctions |n〉 with the corre
Page 19 and 20: and c ′ g(t) = cg(t) (2.12) c ′
Page 21 and 22: 2.3 Interaction of a multi-level sy
Page 23: Figure 2.2: Plot of the AC-Stark sh
Page 27 and 28: on the transition frequency but als
Page 29 and 30: Figure 3.2: Plot of the AC-Stark sh
Page 31 and 32: Chapter 4 Experiments on the magic
Page 33 and 34: I Figure 4.1: Scattering Rate depen
Page 35 and 36: Figure 4.3: Sketch of the Zeeman-Sh
Page 37 and 38: Chapter 5 Stimulated Raman transiti
Page 39 and 40: The scheme for a Resonant Raman tra
Page 41 and 42: variation of the resonance transiti
Page 43 and 44: Figure 5.3: Possible configurations
Page 45 and 46: to adiabatically reduce the three-l
Page 47 and 48: as the final state probability. 5.4
Page 49 and 50: Rabi frequency β given by β f i
Page 51 and 52: larizations stay constant, while th
Page 53 and 54: Chapter 6 Raman transitions of Sodi
Page 55 and 56: e in the ˆz-direction, B = Bˆz. T
Page 57 and 58: With the calculation of the detunin
Page 59 and 60: Figure 6.3: Dependance of the final
Page 61 and 62: In the absence of a magnetic field,
Page 63 and 64: used to determine the necessary mag
Page 65 and 66: The product of the two rotation mat
Page 67 and 68: Figure B.1: Grotrian diagram for So
Page 69 and 70: Figure B.3: Schematic of the cyclin
Page 71 and 72: Figure B.4: Sodium 3 2 S1/2 ground
Page 73 and 74: F’=2 F’=1 F’=0 mF = −1 mF =
Page 75 and 76:
Appendix C Einstein-coefficients fo
Page 77 and 78:
Bibliography [1] J. V. Prodan, W. D
Page 79 and 80:
[24] K. D. Stokes, C. Schnurr, J. R
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