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

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the polarizations as any perpendicular combination of fields, i.e. { ˆy, ˆz}, which therefore allows transitions with ∆m = ±1. The number of atoms making the transition from the initial to the final state remains independant of the local orientation of the atoms, because lab frame and atom frame are equivalent in the absence of a magnetic field. In the presence of a magnetic field the local field orientation defines a quantization axis. In fig. 5.6 we have given some examples of how the magnetic field determines the quantization axis. The unprimed axis define the lab frame, the atom frame is marked by primed coordinates. In figure (a) the magnetic field points along Figure 5.6: Definition of the quantization axis by an external magnetic field. The direction of the magnetic field is represented by the red arrow. Unprimed coordinates belong to the lab frame, coordinates in the atom frame are primed. the ˆz-axis of the lab frame. The quantization axis ˆz ′ corresponds to the ˆz-axis in the lab frame. This causes the atoms to view the { ˆx, ˆy} polarization of the Raman beams the same as in the lab frame. Hence the coupling to the state | f 〉 is governed by ∆m = 0 transitions. If the magnetic field points along one of the polarization axis, for example along the ˆy-axis as in (b), the polarization in the atom frame is { ˆy ′ , ˆz ′ }. This configu- ration therefore leads to ∆m = ±1 transitions. Generally the magnetic field can point in an arbitrary direction (c). This corresponds to a linear combination of the two cases discussed before and transitions with ∆m = 0 and ∆m = ±1 happen. All our calculations will be done in the lab frame, where the Raman beam po- 37
larizations stay constant, while the quantization axis rotates according to the magnetic field orientation. The rotation angles, as they are defined in fig. 5.6 (c), are given by sin θ(r) = cos θ(r) = sin φ(r) = cos φ(r) = B2 x(r) + B2 y(r) B2 x(r) + B2 y(r) + B2 z(r) Bz(r) B2 x(r) + B2 y(r) + B2 z(r) Bx(r) B2 x(r) + B2 y(r) (5.35) (5.36) (5.37) By(r) B2 x(r) + B2 y(r) . (5.38) Thus we are able to determine the effective Raman-Rabi frequencies βi,0 in the atom frame by rotating the Raman transition operator. For further calculations it is useful to go back to the spherical tensor notation, T(k,q). k describes the rank of the operator, q the ˆz component of the operators angular momentum. We can define β0,0 = T(1, 0). In the absence of a magnetic field, the transition operator T(1,0) corresponds to ∆m = 0 transitions. In the rotated frame this tensor becomes T(1, 0) = D 1 q0 (r) T′ (1, q), (5.39) q where D1 (r) are the Wigner rotation matrix elements which can be expressed using the q0 spherical harmonics Y1q(θ(r), φ(r)), D 1 q0 (r) = (−1)q 4π 3 Y1q(θ(r).φ(r)), (5.40) The angles θ and φ describe the rotation of the quantization axis. The relation between the spherical tensor operator in the lab and the rotated 38
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 and 24: Figure 2.2: Plot of the AC-Stark sh
Page 25 and 26: |µ kj| ∆Ej = k 2E2 1 1 − ω
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: Rabi frequency β given by β f i
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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