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about F, their projections along B (the quantization axis) changes 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. Therefore the Zeeman Hamiltonian (Eq. 2.20) can be written in a more useful form, in this case, by taking the projection of J and I along F 〈 HZE = µB gJ F · J〉F · B 〈 + gI F(F + 1) F · I〉F · B , (2.21) F(F + 1) where the terms in the angled brackets refer to an expectation value. This can be written more compactly as where B has been taken along ˆz and gF is given by gF = gJ HZE = µBgFFzB, (2.22) 〈 F · J〉 〈 + gI F(F + 1) F · I〉 . (2.23) F(F + 1) The expectation values can be written in a more illuminating manner; for example 〈 F · J〉 can be rewritten using the following relation Squaring both sides of this equation and using reveals F + J = I + 2 J. (2.24) I · J = 1 2 (F 2 − I 2 − J 2 ) (2.25) 〈 F · J〉 = F 2 + J2 − I2 . (2.26) 2 36
Following the same logic leads to a similar equation for the second expectation value one finds 〈 F · I〉 = F 2 + I2 − J2 . (2.27) 2 If these terms are collected and evaluated in the coupled basis |I,J,F,Fz〉 F(F + 1) − I(I + 1) + J(J + 1) gF = gJ +gI 2F(F + 1) F(F + 1) + I(I + 1) − J(J + 1) . 2F(F + 1) (2.28) This still leaves the problem of determining gJ and gI. The latter accounts for the complex structure of the nucleus and so its value is found experimentally. The value of gJ, however, can be found in a manner very similar to that used to find gF, the difference being that one must consider the coupling between S and L and take the projection of these vectors along J. The result is gJ = gL J(J + 1) − S(S + 1) + L(L + 1) +gS 2J(J + 1) J(J + 1) + S(S + 1) − L(L + 1) , 2J(J + 1) (2.29) where gS and gL are the electron spin and electron orbital “g-factors.” The value of each of the “g-factors” considered above are listed in Ta- ble 2.3. The value of gS is known extraordinarily well and its measurement [52] has served as an important test of QED theory [53]. The value of gL quoted is derived from gL ≈ 1 − me 37 mnuc (2.30)
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 and 50: Figure 2.1: 87 Rb D2 Transition Hyp
Page 51: Pluging Eq. 2.18 into this Hamilton
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 .
Page 101 and 102: Chapter 3 Experimental Apparatus Th
Page 103 and 104:
Figure 3.1: The vacuum chamber duri
Page 105 and 106:
Nor-Cal Products Inc. (AMV-1502-CF)
Page 107 and 108:
and monitor the necessary vacuum le
Page 109 and 110:
3.1.3 Lower Chamber The lower chamb
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a) top ridge 1 cm b) metal jacket h
Page 113 and 114:
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
Page 123 and 124:
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.
Page 133 and 134:
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
Page 139 and 140:
λ λ λ λ
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process. The two coils are arranged
Page 143 and 144:
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
Page 151 and 152:
(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
Page 167 and 168:
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
Page 175 and 176:
used to construct it into the remai
Page 177 and 178:
μ μ μ Figure 4.18: Side view of
Page 179 and 180:
Figure 4.20: Geometry of the optica
Page 181 and 182:
could potentially have undergone th
Page 183 and 184:
detunings. The result of scattering
Page 185 and 186:
are switched off causing non-optica
Page 187 and 188:
Figure 4.25: Number (■) and tempe
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traps. If we model the ensembles in
Page 191 and 192:
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
Page 197 and 198:
the short lived 2p state. Atoms whi
Page 199 and 200:
Bibliography [1] R. Frisch, “Expe
Page 201 and 202:
[15] C. C. Bradley, C. A. Sackett,
Page 203 and 204:
[31] E.T. Jaynes, “Information th
Page 205 and 206:
[49] J. Ye, S. Swartz, P. Jungner,
Page 207 and 208:
[66] J.P. Gordon and A. Ashkin, “
Page 209 and 210:
[85] C.-S. Chuu, Direct Study of Qu
Page 211 and 212:
adicals,” Phys. Rev. Lett. 94, 02
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