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

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take into account the velocity of the atoms and their accelerations due to the differential force exerted by the potential for the initial and final state. 5.2.1 Spectral resolution limit During the Raman process one photon is absorbed and another photon of almost the same frequency is emitted into a co-propagating field. Thus the net momentum change of the atom is negligible. The use of a co-propagating configuration avoids a velocity- selection due to Doppler-Shifts which has been observed by Kasevich et al. using a counter-propagating beam configuration [23]. Raman-induced resonance imaging using co-propagating laser fields has been used by Thomas et al. for precision position measurements of moving atoms [24], [25]. If the detuning of both Raman pulses from resonance is large enough to avoid a population of an excited state, incoherent transitions between the two ground states can be avoided. Therefore, the life-time of the ground states determines the bandwidth of the Raman transition. Since the lifetime of the ground states is long, we are able to neglect the bandwidth of the Raman transition. The difference in the Raman beams can be created by passing one beam through an acousto-optic modulator (AOM). By equalizing the optical path length it is possible to minimize the frequency jitter in the difference frequency; in this limit the stability of the difference frequency is then given by the AOM and is usually small and negligible. In the limit of a very stable difference frequency the Fourier-Theorem gives the frequency resolution of the Raman process, ∆ω ≈ 1 . (5.1) T For simplicity we assume the initial state |i〉 to be independant of the magnetic field. If the frequency shift of the final state | f 〉 varies linearly with the position, the force F exerted by the magnetic field on the final state is uniform and the frequency shifts as ∆ω = ∆E F F = ∆x. The quantity is the spatial tuning rate, which determines the spatial 27
variation of the resonance transition frequency. Combining this with eq. 5.1, we yield the spectral resolution limit 5.2.2 Velocity-limited resolution ∆xsp = FT . (5.2) The spatial resolution is not only influenced by the spectral resolution but also by atomic motion. Atoms with a certain velocity vx along the measurement axis will leave the resonant region after having made the transition from the initial state |i〉 to the final state | f 〉. Hence the velocity-limited resolution is ∆xvel = vxT. (5.3) The velocity-limited resolution increases with a decreasing pulse duration, and so a short pulse duration is desirable. However, the spectral resolution limit will decrease with a decreasing pulse time. The optimal pulse duration can be found using eq. 5.2 and 5.3. The velocity-limited resolution is given by ∆x vel = vx . (5.4) F 5.2.3 Acceleration-limited resolution and the Heisenberg uncertainty prin- ciple Atoms do not only move because of their initial velocity; they will also be accelerated by the force due to the energy gradient. This should not be confused with the fact that the net momentum due to absorption and emission of photons is negligible. The acceleration-limitied resolution is ∆xacc = F 2m T2 , (5.5) 28
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: The scheme for a Resonant Raman tra
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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