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

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of the two angular momenta can be expressed by the total electron angular momentum J = L+ S. Noting that the fine structure Hamiltonian (2.4) can be rewritten as J 2 = S 2 +L 2 +2 L· S (2.5) Hfs = A 2 (J2 −L 2 −S 2 ). (2.6) The energy-splitting due to the fine structure can be treated as a perturbation to the central potential approximation. The difference in the energy levels is hence given by ∆E = A [J(J +1)−S(S +1)−L(L+1)]. (2.7) 2 The resulting energy level structure and configurations for the cases of 87Rb and 6Li are shown in figure 2.3. The coupling of the angular momenta leads to J = 1/2 for the ground state and J = 1/2 or J = 3/2 for the excited state. Central potential approximation 2 P (L=1) 2 S (L=0) 2 P3/2 (J=3/2) 2 P1/2 (J=1/2) 2 S1/2 (J=1/2) Fine Structure 7.123 THz D line D 1 line D 2 line 87 Rubidium Central potential approximation 2 P (L=1) 2 S (L=0) 2 P3/2 (J=3/2) 2 P1/2 (J=1/2) 2 S1/2 (J=1/2) Fine Structure 10.053 GHz D line D 1 line D 2 line 794.978 nm 780.941 nm 670.992nm 670.977 nm 6 Lithium Figure 2.3: Ground and first excited states of 87 Rb and 6 Li in the central potential and fine structure approximations. Energy splittings are not to scale. The D-line transition is split into two separate lines, called the D1 and D2line. The D2-line transition is the transition relevant to the experiments described in this dissertation. 7
2.3.2 Hyperfine Structure So far the analysis has neglected the interaction with the total angular momentum I of the nucleus. This interaction is significantly smaller than the fine structure interaction; however, it is still possible to observe the hyperfine structure in many atomic transitions, as for example in 87 Rb. For the case of 6 Li however, the hyperfine structure can only be resolved in the ground state, not in the 2 2 P3/2 excited state, where the linewidth of the transition is larger than the hyperfine splitting. The Hamiltonian describing the hyperfine interactions, including the magnetic dipole moment and the electric quadrupole moment of the nucleus, is given by [25] Hhfs = Ahfs I · 3( J +Bhfs I · J) 2 + 3 2 ( I · J)−I(I +1)J(J +1) . (2.8) 2I(2I −1)J(2J −1) Ahfs is the magnetic dipole constant, Bhfs is the electric quadrupole constant. Defining the total atomic angular momentum F = I + J leads to the following solution for the energy splitting ∆Ehfs = 1 2 AhfsK +Bhfs 3 2 K(K +1)−2I(I +1)J(J +1) , (2.9) 4I(2I −1)J(2J −1) where K = F(F +1)−I(I +1)−J(J +1). The resulting energy structure is shown in figures 2.4 and 2.5. The difference in the magnitude of the hyperfine splitting, especially in the 2 P3/2 excited state, has large consequences for the experimental sequence and absorption imaging in particular (see chapter 7.5.1). 2.4 Branching Ratios Transitions between different energy levels are governed by electric dipole transition rules. The relative probabilities for the different decay channels from the excited state are known as branching ratios. Traditional laser cooling techniques, like the mag- neto optical trap, see chapter 2.9, work best on a true two-level atom, and any additional decay channels complicate the experimental setup, effectively limiting the applicability to a small number of elements. For single-photon cooling on the other hand, having at least three energy levels is essential. The branching ratios have a large effect on the efficiency of single-photon cooling, see chapters 3 and 5. 8
Page 1 and 2: Copyright by Kirsten Viering 2012
Page 3 and 4: Experiments to Control Atom Number
Page 5 and 6: Acknowledgments First of all, I wou
Page 7 and 8: Experiments to Control Atom Number
Page 9 and 10: Table of Contents Acknowledgments v
Page 11 and 12: Chapter 7. Lithium Apparatus 92 7.1
Page 13 and 14: List of Tables 2.1 Physical propert
Page 15 and 16: 4.1 Vacuum chamber . . . . . . . .
Page 17 and 18: 7.39 Future vertical axis optics .
Page 19 and 20: The development of techniques to la
Page 21 and 22: Ptorr 10 5 10 6 10 7 10 8 10 9 260
Page 23: Property Symbol Value Reference Ato
Page 27 and 28: The Wigner-Eckhart theorem can be u
Page 29 and 30: The total magnetic moment of an ato
Page 31 and 32: interaction. The shift in the energ
Page 33 and 34: (a) (b) scattering length a 0 Energ
Page 35 and 36: a laser field with frequency ω. Th
Page 37 and 38: Here ω0 is the resonance frequency
Page 39 and 40: Optical molasses do not provide a s
Page 41 and 42: J=1 J=0 -1 Δ σ + σ - σ - 0 +1 -
Page 43 and 44: phase-space density is defined as
Page 45 and 46: which maintains a constant η = U/k
Page 47 and 48: 2.12.1 Saturated Absorption Spectro
Page 49 and 50: Figure 2.22: Saturation absorption
Page 51 and 52: Signal 0 Signal (a) (b) Signal Freq
Page 53 and 54: (a) (b) (c) (d) Figure 2.24: Absorp
Page 55 and 56: y more complex energy level structu
Page 57 and 58: ight side of the box. An irreversib
Page 59 and 60: time (a) (b) (c) Figure 3.3: Single
Page 61 and 62: them to move from B to A, but close
Page 63 and 64: Figure 4.1: Vacuum chamber. Photogr
Page 65 and 66: ( ( Figure 4.3: Helicoflex seal. (a
Page 67 and 68: 88 mm 4.2.3 Auxiliary Coils 62 mm Q
Page 69 and 70: 5 2 P 3/2 5 2 S 1/2 384. 230 484 46
Page 71 and 72: distribution of the MOT master lase
Page 73 and 74: 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
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of atoms in the two hyperfine groun
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difference in the lifetimes is dete
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Chapter 7 Lithium Apparatus Creatin
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angle valve rotary feedthrough ion
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an ion gauge controller (Granville
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viewports (MDC, 450002) are attache
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not trapped by the MOT impinge on t
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Should contamination of the vacuum
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Figure 7.12: The assembled Zeeman s
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7.2.2 MOT Coils A second set of coi
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BGauss 250 200 150 100 50 30 20 10
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are required to create the high mag
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The coils are then checked for shor
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+ B A Figure 7.23: Feshbach coil H-
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ensure that no particulates build u
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118 Figure 7.26: Optical layout of
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The error signal shows three lines,
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into the CO2 laser. The optical lay
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Thorlabs PDA10A Coupler ZDC-10-1 to
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Imaging ber Imaging ber from Imagin
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Table 7.4 summarizes the beam power
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CO2 laser 40 MHz AOM f=2 in f=2 in
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132 Figure 7.38: Optical layout aro
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Feshbach coil objective lenses atom
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MOT beams are no longer traveling a
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The second card (NI6733) offers eig
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Averaging over the magnetic subleve
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2 2 P 3/2 D 2=670.977nm 2 2 S 1/2 m
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The right hand side of equation 7.1
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= E0 1 −n2 √2 exp σ0 ± iE0
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esidual potentials besides gravitat
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Figure 8.1: MOT loading. Typical lo
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A linear ramp typically changes the
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A typical MOT is density-limited at
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ubidium [113-115] and cesium[116].
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that most of the noise is due to th
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(a) (c) y . x z X=0.0mm Z=2.5mm 1)
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Figure 8.11: Different shapes of MO
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ZnSe lens f=5 in knife edge ZnSe wi
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shown in figure 8.13. By matching t
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Figure 8.15: Atom number and temper
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275 Hz. Using these values the beam
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system. Maybe the easiest way to ac
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Appendix A Magic Wavelength of Hydr
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transitions for the excited state p
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Bibliography [1] J.J. Thomson. Cath
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[22] C.J. Foot. Atomic Physics. Oxf
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[46] E.D. Black. An introduction to
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[67] J.L. Hanssen. Controlling Atom
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[88] D.S. Naik, C.G. Peterson, A.G.
Page 205 and 206:
[110] J.W. Goodman. Introduction to
Page 207:
[132] C. Degenhardt, H. Stoehr, U.
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