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

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2.5.2.1 Optical Bloch Equations The optical Bloch equations are used to describe an ensemble of two- level atoms interacting with a monochromatic optical field [69] as shown in Fig. 2.6. To describe the situation in the figure we must find the expectation ω Figure 2.6: Ensemble of N two-level atoms per unit volume interacting with a monochromatic optical field. value 〈µ(t)〉 = Ψ ∗ µ Ψd 3 r (2.55) of the microscopic polarization operator µ = −er by solving Schrödinger’s equation, i ∂Ψ ∂t = HΨ. (2.56) Here H = H0+Hcoh+Hdamp is the sum of the unperturbed atomic Hamiltonian H0, the Hamiltonian describing the coherent evolution of the atom driven by the optical field Hcoh = −µ · E(t) and the Hamiltonian due to inter-atomic damping processes such as collisions. The inclusion of the third term means that we must consider the system as a quantum ensemble, not as individual atoms. Because the system we are describing comprises a large number of 50
atoms its wavefunction Ψ(t) is typically intractable. Instead we use the density matrix formalism which provides a convenient way of describing the evolution of the ensemble average expectation value of the dipole operator 〈µ(t)〉 even when Ψ(t) is unknown. The density matrix can be defined in two steps. In the first step, we consider the expectation value of the dipole operator for a single atom 〈µ(t)〉 = 〈Ψ(r,t)|µ|Ψ(r,t)〉. (2.57) We then expand Ψ(r,t) in a complete set of orthonormal eigenfunctions of the unperturbed atomic Hamiltonian H0 Ψ(r,t) = cn(t)un(r), (2.58) n where all of the time dependence has been placed in the coefficients cn(t). We then evaluate the expectation value of the dipole operator in this basis 〈µ(t)〉 = c ∗ m(t)〈um(r)|µ|un(r)〉cn(t). (2.59) m,n By defining µmn ≡ 〈um(r)|µ|un(r)〉 and Rnm ≡ cnc ∗ m this can be written much more compactly as 〈µ(t)〉 = Rnmµmn = (Rµ)nn = Tr(Rµ). (2.60) m,n n Next we apply this approach to an atomic ensemble. To do so we define ρnm(t) ≡ Rnm(t), (2.61) 51
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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
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Rubidium group when I first joined.
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Single-Photon Atomic Cooling Public
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2.5.2.4 Magneto-Optical Trap . . .
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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 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: maxima. Our experiment makes extens
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
Page 111 and 112: a) top ridge 1 cm b) metal jacket h
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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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