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

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where m is the atomic mass. By setting ∆xsp = ∆xacc one finds the optimal pulse duration Tacc = 2m F 2 1/3 and the acceleration limited resolution thus becomes ∆xacc = 2 2mF (5.6) 1/3 . (5.7) Let us assume that only the final state is shifted by the potential. The transition between the initial state |i〉 and the final state | f 〉 can occur at any time during the pulse time T, and so the uncertainty in the final momentum of the atoms is ∆p = FT. Simple multiplication with the spectral resolution limit (eq. 5.2) leads to the fundamental limit given by the uncertainty relation between space and momentum µm 5.3 Raman transitions in a three level atom ∆x∆p = . (5.8) The total electro-magnetic field driving a stimulated Raman transition can be described by the sum of the two individual light beams with frequencies ω1 and ω2, E(r, t) = 1 2 · E1 · e i(k·r−ω1t) + E1 · e −i(k·r−ω1t) + E2 · e i(k·r−ω2t) + E2 · e −i( k·r−ω2t) . (5.9) There exist three possible configurations of a three-level system with the initial state |i〉, an intermediate state |I〉 and a final state | f 〉, that is driven by a two-photon process, usually called the ”ladder”, ”V” or ”lambda” configurations, which are schemat- ically shown in fig. 5.3 [26]. A stimulated Raman transition between two hyperfine levels of the ground state corresponds to the Λ-configuration. 29
Figure 5.3: Possible configurations of a three-level system driven by a two-photon process, from left to right: ladder-configuration, V-configuration, Λ-configuration 5.3.1 Three-level amplitude equations In chapter 2 we developed the coherent amplitude formalism for the interaction of an atom with an electric field. The amplitude equations in a three level system and with two monochromatic light sources become [27] and ˙ci = iΩ∗ 1 2 cIe i∆1t , (5.10) ˙c f = iΩ∗ 2 2 cIe i∆2t where the Rabi frequencies Ω1 and Ω2 are defined by and (5.11) ˙cI = iΩ1 2 cie −i∆1t + iΩ2 2 c f e −i∆2t , (5.12) Ω1 = −E1 〈I|µ|i〉 (5.13) Ω2 = −E2 〈I|µ| f 〉; (5.14) 30
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: variation of the resonance transiti
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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