Experiments to Control Atom Number and Phase-Space Density in ...

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7.5.1 Absorption Imaging Determining the absorption scattering cross section in the case of lithium, espe- cially in the presence of external magnetic fields, is not as straight forward as it is for rubidium. The excited states are not resolved, thus making the two-level assumption used to calculate the scattering cross-section invalid. In addition the existence of the second hyperfine ground state cannot be neglected. 7.5.1.1 Absorption Scattering Cross Section at Zero Magnetic Field For imaging the atoms at zero magnetic field, for example for measuring the temperature of the MOT, both MOT and repump light are required. Without repump light, the atoms will only scatter a few photons and then fall into the lower state, where they are transparent to the MOT light. The absorption imaging beam used at zero magnetic field is thus derived from the tapered amplifier, which has both MOT and Repump frequencies. In a multi-level atom, the on-resonance scattering cross section will differ from the two-level approximation (σ0 = 3λ2), and all possible transitions have to be taken 2π into account. The transition matrix element is given by µ = 〈(J2I)F2mF2|ˆµ(1,q)|(J1I)F1mF1〉, (7.1) where the subscript 1 corresponds to the ground state, the subscript 2 to the excited state respectively. In lithium the excited state energy levels are degenerate to within the transition linewidth, so all possible values for F2 have to be taken into account. The transition matrix element will be shown to be independent of the polarization of the laser light, and a random mixture of polarizations is therefore assumed. In addition, the magnetic sublevels of the ground state will be mixed. An even mixture of these states is assumed in the following calculation. ê = a(q)êq q (7.2) |a(q)| 2 = 1. (7.3) q 139
Averaging over the magnetic sublevels requires each component of the transition 1 matrix element to be weighed with the degeneracy factor . The possible values for 2F1+1 the quantum number are I = 1,F1 = 3/2, F1 = 1/2, F2 = 1/2, F2 = 3/2, F2 = 5/2 J1 = 1/2 and J2 = 3/2. The transition matrix element is then given by µ 2 = 1 2F1 +1 q 2 |a(q)| F2,mF 2 ,mF 1 |〈(J2I)F2mF2|ˆµ(1,q)|(J1I)F1mF1〉| 2 . (7.4) This matrix element can be rewritten using the Wigner-Eckhart-Theorem [26], µ 2 1 = |a(q)| 2F1 +1 2 (−1) 2(F2−mF ) F2 2 −mF2 1 F1 q mF1 2 q F2 mF 2 ,mF 1 where the term in parenthesis is known as Wigner-3j-symbol. ×|〈(J2I)F2|ˆµ(1)|(J1I)F1〉| 2 , (7.5) The sum over the magnetic sublevels is independent of the polarization and the final level. The equation thus reduces to µ 2 = 1 |〈(J2I)F2|ˆµ(1)|(J1I)F1〉| 3(2F1 +1) 2 . (7.6) F2 Using further angular momentum coupling, equation 7.6 can be simplified using the reduced dipole matrix element of the D2 line, 〈(L2S)J2||ˆµ(1)||(L1S)J1〉: µ 2 = 1 δI2I1(−1) 3(2F1 +1) F2 2(J2+I1+F1+1) (2F1 +1)(2F2 +1) 2 J2 × I F2 |〈(L2S)J2||ˆµ(1)||(L1S)J1〉| 2 . (7.7) F1 1 J1 The transition matrix element for the MOT light (F1 = 3/2) and the repump light (F1 = 1/2) are therefore µ 2 = 1 2 |〈(L2S)J2||ˆµ(1)||(L1S)J1〉| 6 = 2 3 µ20, (7.8) The on-resonance absorption scattering cross section is thus given by where σ0 = 3λ2 2π . σMOT = σRepump = 2 3 σ0 = λ2 π 140 (7.9)
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Copyright by Kirsten Viering 2012
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Experiments to Control Atom Number
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Acknowledgments First of all, I wou
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Experiments to Control Atom Number
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Table of Contents Acknowledgments v
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Chapter 7. Lithium Apparatus 92 7.1
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List of Tables 2.1 Physical propert
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4.1 Vacuum chamber . . . . . . . .
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7.39 Future vertical axis optics .
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The development of techniques to la
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Ptorr 10 5 10 6 10 7 10 8 10 9 260
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Property Symbol Value Reference Ato
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2.3.2 Hyperfine Structure So far th
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The Wigner-Eckhart theorem can be u
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The total magnetic moment of an ato
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interaction. The shift in the energ
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(a) (b) scattering length a 0 Energ
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a laser field with frequency ω. Th
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Here ω0 is the resonance frequency
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Optical molasses do not provide a s
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J=1 J=0 -1 Δ σ + σ - σ - 0 +1 -
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phase-space density is defined as
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which maintains a constant η = U/k
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2.12.1 Saturated Absorption Spectro
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Figure 2.22: Saturation absorption
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Signal 0 Signal (a) (b) Signal Freq
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(a) (b) (c) (d) Figure 2.24: Absorp
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y more complex energy level structu
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ight side of the box. An irreversib
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time (a) (b) (c) Figure 3.3: Single
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them to move from B to A, but close
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Figure 4.1: Vacuum chamber. Photogr
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( ( Figure 4.3: Helicoflex seal. (a
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88 mm 4.2.3 Auxiliary Coils 62 mm Q
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5 2 P 3/2 5 2 S 1/2 384. 230 484 46
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distribution of the MOT master lase
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shifts. First a frequency shift of
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transition. However, the laser freq
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4.3.1.4 Upper MOT Horizontal Slave
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provides the beam for the upper MOT
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demon beam λ λ horizontal MOT bea
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scholar, Florian Schreck. It is wri
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Atoms are then optically pumped to
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ferred into the |F = 1,mF = 0〉 or
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5.2 Branching Ratios and Population
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effect of the reflected beam at zer
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y the repulsive trough beams, gaini
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Figure 5.9: Atom number as a functi
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Figure 5.10: Radius of the magnetic
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on the collision rate of atoms insi
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ferred via the single-photon coolin
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By comparison an atomic Fock state
Page 105 and 106: of atoms in the two hyperfine groun
Page 107 and 108: difference in the lifetimes is dete
Page 109 and 110: Chapter 7 Lithium Apparatus Creatin
Page 111 and 112: angle valve rotary feedthrough ion
Page 113 and 114: an ion gauge controller (Granville
Page 115 and 116: viewports (MDC, 450002) are attache
Page 117 and 118: not trapped by the MOT impinge on t
Page 119 and 120: Should contamination of the vacuum
Page 121 and 122: Figure 7.12: The assembled Zeeman s
Page 123 and 124: 7.2.2 MOT Coils A second set of coi
Page 125 and 126: BGauss 250 200 150 100 50 30 20 10
Page 127 and 128: are required to create the high mag
Page 129 and 130: The coils are then checked for shor
Page 131 and 132: + B A Figure 7.23: Feshbach coil H-
Page 133 and 134: ensure that no particulates build u
Page 135 and 136: 118 Figure 7.26: Optical layout of
Page 137 and 138: The error signal shows three lines,
Page 139 and 140: into the CO2 laser. The optical lay
Page 141 and 142: Thorlabs PDA10A Coupler ZDC-10-1 to
Page 143 and 144: Imaging ber Imaging ber from Imagin
Page 145 and 146: Table 7.4 summarizes the beam power
Page 147 and 148: CO2 laser 40 MHz AOM f=2 in f=2 in
Page 149 and 150: 132 Figure 7.38: Optical layout aro
Page 151 and 152: Feshbach coil objective lenses atom
Page 153 and 154: MOT beams are no longer traveling a
Page 155: The second card (NI6733) offers eig
Page 159 and 160: 2 2 P 3/2 D 2=670.977nm 2 2 S 1/2 m
Page 161 and 162: The right hand side of equation 7.1
Page 163 and 164: = E0 1 −n2 √2 exp σ0 ± iE0
Page 165 and 166: esidual potentials besides gravitat
Page 167 and 168: Figure 8.1: MOT loading. Typical lo
Page 169 and 170: A linear ramp typically changes the
Page 171 and 172: A typical MOT is density-limited at
Page 173 and 174: ubidium [113-115] and cesium[116].
Page 175 and 176: that most of the noise is due to th
Page 177 and 178: (a) (c) y . x z X=0.0mm Z=2.5mm 1)
Page 179 and 180: Figure 8.11: Different shapes of MO
Page 181 and 182: ZnSe lens f=5 in knife edge ZnSe wi
Page 183 and 184: shown in figure 8.13. By matching t
Page 185 and 186: Figure 8.15: Atom number and temper
Page 187 and 188: 275 Hz. Using these values the beam
Page 189 and 190: system. Maybe the easiest way to ac
Page 191 and 192: Appendix A Magic Wavelength of Hydr
Page 193 and 194: transitions for the excited state p
Page 195 and 196: Bibliography [1] J.J. Thomson. Cath
Page 197 and 198: [22] C.J. Foot. Atomic Physics. Oxf
Page 199 and 200: [46] E.D. Black. An introduction to
Page 201 and 202: [67] J.L. Hanssen. Controlling Atom
Page 203 and 204: [88] D.S. Naik, C.G. Peterson, A.G.
Page 205 and 206: [110] J.W. Goodman. Introduction to
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[132] C. Degenhardt, H. Stoehr, U.
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