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2.4 Magnetic Trapping The Zeeman effect can be used to spatially confine neutral atoms. This feat was first accomplished in 1985 on a sample of neutral atomic sodium [9]. This can be understood by a quite straightforward extension of the main result of the previous section where it was shown that in the low field limit an external magnetic field shifts the magnetic sublevels of an atom by an amount given by ∆EZE = µBgFmF |B|. (2.34) This shift in energy is correctly viewed as a confining potential for the atoms, the force on which is found from taking the gradient of this expression. FZE = −µBgFmF(∇|B|) (2.35) In the previous two equations I have been explicit that it is the mag- nitude of the magnetic field which is important and not the vector quantity. The reason for this is that under certain conditions, which are met in our trap almost everywhere, the atomic magnetic moments follow the direction of the confining magnetic field. The condition for this adiabatic following can be written as ωL ≫ |dB/dt|/B, where ωL = µB/ is the Larmor precession rate in the applied magnetic field. If this condition is not met, it can result in trap loss via a process known as Majorana spin flips. Examination of Eqs. 2.34 and 2.35 reveals that there are two main classes of atomic states with respect to magnetic trapping. The first, com- monly called high-field seeking states, satisfy gFmF < 0 and therefore mini- 40
mize their energy in high magnetic fields. The second, called low-field seeking states, satisfy gFmF > 0 and minimize their energy in magnetic field min- ima. Earnshaw’s theorem prohibits an electrostatic field from stably trapping a charged particle, however it does not rule out the possibility of trapping a dipole [38]. Clearly, for a trap to be stable atoms must accumulate at a field extrema, but local maxima in the absence of sources are forbidden by Maxwell’s equations [55]. Therefore atoms must be trapped at a local field minimum, so low-field seeking atomic states must be used. Ground state 87 Rb has three low-field seeking states. They are |2, 2〉, |2, 1〉 and |1, −1〉, where the notation |F,mF 〉 is being used. The term “ground state” refers to all of the states contained in the 5 2 S1/2 spectroscopic term, not just the one with the lowest energy. The state |2, 2〉 couples most strongly to external magnetic fields, providing the tightest confinement for a given magnetic field. For this reason we trap 87 Rb atoms in this state. One may worry that this state is not the lowest in energy and may decay into the F = 1 manifold, disrupting the experiment. This concern can be put to rest because this is not an allowed electric dipole transition as ∆L = 0 and so the lifetime of atoms in the F = 2 manifold is much longer than the duration of the experiment. While there are many electric current configurations which can form suitable trapping potentials [56] we use the simplest configuration possible. Our field is produced from a pair of circular coils with counter-propagating currents. This geometry is known as the anti-Helmholtz configuration and it 41
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Copyright by Gabriel Noam Price 200
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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 and 52: Pluging Eq. 2.18 into this Hamilton
Page 53 and 54: Following the same logic leads to a
Page 55: gF = −1/2 using the approximate e
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)
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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
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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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