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the sum of the two coupled angular momentum vectors J and I, in this case giving the total atomic angular momentum F F = J + I, (2.10) which is similarly constrained by the triangle inequality to have a magnitude with the range |J − I| ≤ F ≤ J + I. (2.11) To get a feeling for how this works, consider the ground state of 87 Rb which has the single term 5 2 S1/2. Because I = 3/2 and J = 1/2, Eq. 2.11 indicates that F can take on two values: 1 or 2. In the excited state of the D2 transition the term is 5 2 P3/2 so F can take on the values 0, 1, 2, or 3. The Hamiltonian describing the interaction between the total electronic angular momentum J and the total nuclear angular momentum I is given by [44–47] Hhfs = Ahfs I · 3( J + Bhfs I · J) 2 + 3 2 ( I · J) − I(I + 1)J(J + 1) , (2.12) 2I(2I − 1)J(2J − 1) where the interaction between J and the magnetic dipole moment and electric quadrupole moment of the nucleus has been included. Higher order terms resulting from interactions with higher order nuclear moments have been ne- glected in this Hamiltonian because experimental measurements are not sufficiently accurate to assign a non-zero contribution to them. In Eq. 2.12, Ahfs is the magnetic dipole constant and Bhfs is the electric quadrupole constant, 30
the latter of which only applies to the excited manifold of the D2 transition and not to levels with J = 1/2. This Hamiltonian can be solved in a manner analogous to that used in the case of fine structure splitting. We note that by squaring both sides of Eq. 2.10 one can write, J · I = 1 2 (F 2 − J 2 − I 2 ). (2.13) This can be used in Eq. 2.12 to solve for the shifts in energy levels due to this interaction. Again we note that the coupling of I and J causes the uncoupled basis |I,Iz,J,Jz〉 to no longer be an eigenvector of the Hamiltonian describing the atom. But as before, simple inspection revels that |I,J,F,mF 〉 is an eigenvector of Eq. 2.12. Taking this as the basis in evaluating the perturbation due to the hyperfine Hamiltonian results in energy splittings given by where ∆Ehfs = 1 2 AhfsK + Bhfs 3 2 K(K + 1) − 2I(I + 1)J(J + 1) , (2.14) 4I(2I − 1)J(2J − 1) K = F(F + 1) − I(I + 1) − J(J + 1) (2.15) is introduced for notational convenience. The nuclear moment constants for the 87 Rb D2 line are given in Ta- ble 2.2. The ground state value was taken from a precise atomic fountain measurement [48], while the excited state values were measured using a het- erodyne technique between two ultra stable lasers referenced to atomic 87 Rb [49]. 31
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: only one value of J is possible fro
Page 49 and 50: Figure 2.1: 87 Rb D2 Transition Hyp
Page 51 and 52: Pluging Eq. 2.18 into this Hamilton
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
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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
Page 179 and 180:
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
Page 185 and 186:
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
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
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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,
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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