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produces a quadrupole magnetic field. The form of the magnetic field produced in this configuration can be arrived at by first considering the form of the magnetic field produced by a single circular loop carrying a current I in a region of permittivity µ which is given by [56] where Bz = µI 2π 1 K(k (R + ρ) 2 + (z) 2 2 ) + R2 − ρ2 − z2 Bρ = µI z 2πρ (R + ρ) 2 + (z) 2 k 2 = (R − ρ) 2 + z2E(k2 ) −K(k 2 ) + R2 + ρ2 − z2 (R − ρ) 2 + z2E(k2 ) 4Rρ (R + ρ) 2 + z 2 (2.36a) (2.36b) (2.37) and K(k 2 ) and E(k 2 ) are the complete elliptic integrals of the first and second kind, respectively [57]. The terms in this expression are clarified in Fig. 2.4. Figure 2.4: Geometry of Eq. 2.36. The red loop of current produces a magnetic field at the point (ρ,z). The magnetic field is decomposed into components parallel to ˆz and to ˆρ. 42
If we now consider the field from two such loops separated by a distance d with equal but opposite current we can arrive at the expression for the magnetic field in the anti-Helmholtz configuration. The full mathematical form is not very illuminating, but a plot of the result is shown in Fig. 2.5. This figure shows the magnitude of the magnetic field along the axis of symmetry of the two loops. The central portion of this potential is used for trapping. Note that near the center the potential is nearly linear, deviating only in the vicinity of the current carrying loops, at the surface of which the field magnitude reaches its maximum value. |B| Figure 2.5: Magnitude of the magnetic field along the symmetry axis of the two loops. The central portion of the potential is used for trapping. The potential is approximatly linear near the center and reaches its maximum value at the surface of the circular loops of current. Luckily, we can justify making a huge simplification because the trapped atoms in this experiment only occupy the region near the center of the trap. 43 z
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Copyright by Gabriel Noam Price 200
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Single-Photon Atomic Cooling by Gab
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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 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: mize their energy in high magnetic
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
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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
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[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
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adicals,” Phys. Rev. Lett. 94, 02
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