Single-Photon Atomic Cooling - Raizen Lab - The University of ...

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2.3 The Zeeman Effect The Zeeman effect refers to a shift in an atomic energy level due to the presence of an external magnetic field. Each of the hyperfine levels F contains 2F + 1 magnetic sublevels, labeled by mF, which correspond to different projections of the total atomic angular momentum along the quantization axis, here taken to be along ˆz. These states are represented by the vectors |F,mF 〉 and are eigenvectors of the operators F 2 and Fz. F 2 |F,mF 〉 = 2 F(F + 1)|F,mF 〉 (2.16) Fz|F,mF 〉 = mF |F,mF 〉 (2.17) In the absence of an external magnetic field, sublevels of a common hyperfine state F are degenerate. Application of a magnetic field lifts this degeneracy. The total atomic magnetic moment of the atom is the sum of the electronic and nuclear moments µatom = −µB(gJ J + gI I), (2.18) where µB = h · 1.399 624 604(35) MHz/G is the Bohr magneton and gJ and gI are the total electronic angular momentum and nuclear “g-factors” which arise from projections of J and I along F. The Hamiltonian for the interaction of µatom with an external magnetic field is [50, 51] HZE = −µatom · B. (2.19) 34
Pluging Eq. 2.18 into this Hamiltonian yields HZE = µB(gJ J + gI I) · B. (2.20) If the interaction of the atom with an external field described by this equation is small compared to the hyperfine splitting then F is a good quantum number. A useful method of visualizing this situation is provided by the vector model of angular momentum. This particular situation is depicted in Fig. 2.2, which illustrates that the coupling of J and I can be interpreted as the sum of these two vectors precessing about the total atomic angular momentum vector F. In turn F precesses about the external magnetic field. As J and I precess B Figure 2.2: Vector model of the hyperfine interaction. The coupling of J and I can be interpreted as the sum of these two vectors precessing about the total atomic angular momentum vector F. Similarly, but at a much slower rate, F precesses about the external magnetic field. As J and I precess about F, their projections along B (the quantization axis) change in time. Therefore the quantum numbers associated with these projections Jz and Iz (assuming that B is along ˆz) are not good quantum numbers. In contrast, the projection of F along B is constant in time and so Fz is a good quantum number. 35 F J I
Page 1 and 2: Copyright by Gabriel Noam Price 200
Page 3 and 4: Single-Photon Atomic Cooling by Gab
Page 5 and 6: Acknowledgments First, I would like
Page 7 and 8: Rubidium group when I first joined.
Page 9 and 10: Single-Photon Atomic Cooling Public
Page 11 and 12: 2.5.2.4 Magneto-Optical Trap . . .
Page 13 and 14: List of Tables 2.1 87 Rb Physical P
Page 15 and 16: 3.1 Vacuum Chamber . . . . . . . .
Page 17 and 18: Chapter 1 Introduction This chapter
Page 19 and 20: the momentum kicks due to absorptio
Page 21 and 22: samples by allowing for very long i
Page 23 and 24: Figure 1.2: The energy level struct
Page 25 and 26: quency regime [19]. This then led t
Page 27 and 28: Figure 1.4: Depiction of a simple,
Page 29 and 30: demonstrates the cooling power of a
Page 31 and 32: a) b) c) external potential one-way
Page 33 and 34: worked on by Brillouin [27-29], ide
Page 35 and 36: the atomic ensemble. Not surprising
Page 37 and 38: are untrappable. The SI unit for en
Page 39 and 40: the atomic transition and is called
Page 41 and 42: 2.1 Rubidium Rubidium (Rb) is an al
Page 43 and 44: constant which is given by α = e2
Page 45 and 46: only one value of J is possible fro
Page 47 and 48: the latter of which only applies to
Page 49: Figure 2.1: 87 Rb D2 Transition Hyp
Page 53 and 54: Following the same logic leads to a
Page 55 and 56: gF = −1/2 using the approximate e
Page 57 and 58: mize their energy in high magnetic
Page 59 and 60: If we now consider the field from t
Page 61 and 62: where P is in torr. The lifetime of
Page 63 and 64: component of the dipole oscillation
Page 65 and 66: maxima. Our experiment makes extens
Page 67 and 68: atoms its wavefunction Ψ(t) is typ
Page 69 and 70: In this equation we let H = H0 + Hc
Page 71 and 72: The significance of Isat is that at
Page 73 and 74: Δ ω =ω−Δ ω =ω−Δ Δ
Page 75 and 76: where I/Isat ≪ 1 has been assumed
Page 77 and 78: λ λ Figure 2.9: The Sisyphus cool
Page 79 and 80: process repeats. If however, it dec
Page 81 and 82: σ σ σ Figure 2.10: Geomet
Page 83 and 84: to vary linearly with position alon
Page 85 and 86: 0, ±1 and ∆mF = 0, ±1, allow ex
Page 87 and 88: this figure only the relevant hyper
Page 89 and 90: D2 transition in 87 Rb has a natura
Page 91 and 92: 2.7.2 Saturation Absorption Spectro
Page 93 and 94: ground state depletion due to the p
Page 95 and 96: oadened background can be removed f
Page 97 and 98: where n is the number density of at
Page 99 and 100: where σ 2 c = σ 2 2 kBTt n + m .
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Chapter 3 Experimental Apparatus Th
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Figure 3.1: The vacuum chamber duri
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Nor-Cal Products Inc. (AMV-1502-CF)
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and monitor the necessary vacuum le
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3.1.3 Lower Chamber The lower chamb
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a) top ridge 1 cm b) metal jacket h
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to the appropriate frequency throug
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plexiglass cover piezo U stack grat
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λ Figure 3
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Voltage a b c d e f Frequency Figur
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eams. For example the MOT and optic
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λ Figure 3.12: The slave lasers
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λ λ λ
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λ λ λ λ
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needed when forming a MOT or optica
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master laser is dithered at 20 kHz.
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λ λ
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Output Power > 10 W Wavelength 532
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labeled 1, 2 and 3 in the Fig. 3.21
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λ λ λ λ
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process. The two coils are arranged
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19) connected in parallel. This arr
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3.4.2 Horizontal Imaging The horizo
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Chapter 4 Single-Photon Atomic Cool
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1 s, after which time ∼ 10 8 87 R
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(a) x z magnetically trapped atoms
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h x z magnetically trapped atoms y
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z y x magnetically trapped atoms cr
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Figure 4.5: The effective potential
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depopulation present and it indicat
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numbers sufficiently large to quant
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μ μ μ
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the potential landscape due to the
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Figure 4.12: The number of atoms lo
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Figure 4.13: Incremental atom captu
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netic trap and transfer into the op
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temperature TB of atoms loaded into
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used to construct it into the remai
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μ μ μ Figure 4.18: Side view of
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Figure 4.20: Geometry of the optica
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could potentially have undergone th
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detunings. The result of scattering
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are switched off causing non-optica
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Figure 4.25: Number (■) and tempe
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traps. If we model the ensembles in
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Figure 4.26: Radius of the atomic c
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atoms were removed from the magneti
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trappable |F = 1,mF = −1〉 state
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the short lived 2p state. Atoms whi
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Bibliography [1] R. Frisch, “Expe
Page 201 and 202:
[15] C. C. Bradley, C. A. Sackett,
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[31] E.T. Jaynes, “Information th
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[49] J. Ye, S. Swartz, P. Jungner,
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[66] J.P. Gordon and A. Ashkin, “
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[85] C.-S. Chuu, Direct Study of Qu
Page 211 and 212:
adicals,” Phys. Rev. Lett. 94, 02
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