The Pennsylvania State UniversityThe Graduate School

Eberly College of Science

PROGRESS TOWARDS A MEASUREMENT OF THE ELECTRON

ELECTRIC DIPOLE MOMENT USING COLD CESIUM

A Dissertation inPhysics

byCheng Tang

© 2018 Cheng Tang

Submitted in Partial Fulfillmentof the Requirementsfor the Degree of

Doctor of Philosophy

May 2018

The dissertation of Cheng Tang was reviewed and approved∗ by the following:

David S. WeissProfessor of PhysicsDissertation Advisor, Chair of Committee

Nathan D. GemelkeAssistant Professor of Physics

Stéphane CoutuProfessor of Physics and of Astronomy & Astrophysics

Venkatraman GopalanProfessor of Materials Science and Engineering

Richard W. RobinettProfessor of PhysicsAssociate Head for Undergraduate and Graduate Students

∗Signatures are on file in the Graduate School.

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Abstract

This dissertation describes our continued effort at Penn State to push the sensitivityon the electron electric dipole moment (EDM) down to the level of 2.5×10−30 e ·cmusing trapped cold cesium and rubidium atoms. The sensitivity to the EDMincreases linearly with the electric field. The electric field is generated usingelectrodes made of the transparent conductor, indium tin oxide (ITO), which allowsoptical access to the atoms. The maximum electric field the electrodes can sustain isaround 37 kV/cm, 2 to 3 times lower than the maximum fields attained in mock-ups.While we are continuing to work to increase the strength of the electric field, wehave developed an alternative EDM measurement scheme that works at low field.This alternative measurement scheme begins by the simple process of preparingatoms in the m = 0 magnetic sublevel. By measuring the probability of detectingatoms in the m = 0 state after evolution in electric and magnetic fields, we canobtain a sensitivity [1] close to the original measurement scheme.

We enumerate several notable aspects of our alternative measurement scheme.First, detection of all magnetic sublevels recovers the full sensitivity of precessionfrom the m = 0 state, which is the same for all precession phases. Second, whenthe total population in the even magnetic sublevels is measured, it does not dependon quadratic energy shifts due to electric or magnetic fields that are perpendicularto the measurement axis. Third, we can construct a combination of the populationsof magnetic sublevels to extract a pure sinusoidal signal.

We have also refined many experimental techniques in the process of restoringthe vacuum and the atom signal. In particular, we constructed a stack of Cooketriplets for collecting the fluorescence from the atoms, which provides the essentialdata in the experiment. This newly developed imaging system successfully resolvedthe blurring issue in our previous design.

We did not perform a measurement of the EDM in this dissertation. It is likelythat a measurement can be performed in a year.

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Table of Contents

List of Figures vii

Acknowledgments xvi

Chapter 1Introduction 11.1 EDMs and Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Experimental Searches for EDMs . . . . . . . . . . . . . . . . . . . 3

1.2.1 Electron EDM Experiments . . . . . . . . . . . . . . . . . . 51.2.2 Nuclear EDM Experiments . . . . . . . . . . . . . . . . . . . 6

1.3 An eEDM Search With Trapped Cold Atoms . . . . . . . . . . . . . 7

Chapter 2Overview of Our Experiment 82.1 Atom Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2 State Preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.3 The EDM Measurement . . . . . . . . . . . . . . . . . . . . . . . . 132.4 Challenges to Solve . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

Chapter 3Fluorescence Imaging of the Atoms 173.1 Summary of the problems with a pair of Fresnel lenses . . . . . . . 183.2 Positioning and Mounting the PDAs . . . . . . . . . . . . . . . . . 203.3 Optical Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.3.1 Lens Array of 2 . . . . . . . . . . . . . . . . . . . . . . . . . 233.3.2 Design of a Cooke Triplet . . . . . . . . . . . . . . . . . . . 243.3.3 The Effects of Plane-Parallel Plates . . . . . . . . . . . . . . 253.3.4 Imaging the Atoms and their Mirror Image . . . . . . . . . . 27

3.4 Mechanical Mount of the Cooke Triplets . . . . . . . . . . . . . . . 283.5 Imaging System Tests . . . . . . . . . . . . . . . . . . . . . . . . . . 29

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Chapter 4High Voltage and Field Plates 324.1 Prior Studies on the Creation of High Voltage Using ITO-coated

Glass in Vacuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324.2 ITO-coated Glass Plates After Breakdown . . . . . . . . . . . . . . 344.3 High Voltage Test with a Second Set of ITO-coated Glass Plates . . 364.4 Interferometric Measurements of the Plate Separations . . . . . . . 40

4.4.1 Principle of White Light Interferometry . . . . . . . . . . . . 414.4.2 Measurement of the Mirror Displacement . . . . . . . . . . . 454.4.3 Differential Measurement of Plate Separation . . . . . . . . 464.4.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

Chapter 5Alternative Scheme of an EDM Measurement 495.1 The Scheme Using Audio Frequency Transitions . . . . . . . . . . . 495.2 The Alternative Scheme by Switching Basis . . . . . . . . . . . . . 52

5.2.1 The Quantum Projection Uncertainties . . . . . . . . . . . . 565.2.2 Comparison with Larmor Precession . . . . . . . . . . . . . 595.2.3 Probability in Even Magnetic Sublevels . . . . . . . . . . . . 625.2.4 Single Harmonic from a Linear Combination of Magnetic

Sublevels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 655.2.5 Extension to Higher Integer and Half Integer F . . . . . . . 66

5.3 Effect of a Transverse Field in Measurement of Magnetic Field . . . 685.4 Effect of a Transverse Field in an EDM Measurement . . . . . . . . 705.5 Summary of the EDM Measurement . . . . . . . . . . . . . . . . . 725.6 A Scheme for Measuring the Tensor Polarizability . . . . . . . . . . 72

Chapter 6Restoring the Vacuum and Atom Signals 796.1 Improved Top Vacuum Window with Low Birefringence . . . . . . . 796.2 Baking the Vacuum . . . . . . . . . . . . . . . . . . . . . . . . . . . 836.3 Auxiliary 2-D Imaging System . . . . . . . . . . . . . . . . . . . . . 856.4 Parallel 1-D Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . 87

6.4.1 Mode Matching . . . . . . . . . . . . . . . . . . . . . . . . . 896.4.2 Setup and Alignment . . . . . . . . . . . . . . . . . . . . . . 916.4.3 Cavity Characteristics . . . . . . . . . . . . . . . . . . . . . 956.4.4 Extracting Information from the Cavity Reflectance . . . . . 976.4.5 Cavity Locks . . . . . . . . . . . . . . . . . . . . . . . . . . 101

6.5 Launch by Pushing . . . . . . . . . . . . . . . . . . . . . . . . . . . 1076.6 Stopping and Cooling Atoms between the Plates . . . . . . . . . . . 108

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6.7 Optical Pumping . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

Chapter 7Conclusion 114

Appendix AInsensitivity of the Probability in Even Magnetic Sublevels to

Quadratic Energy Shifts in the Transverse Direction 116A.1 Condensed but Complete Proof . . . . . . . . . . . . . . . . . . . . 116A.2 Symmetry/Antisymmetry of a π/2 Rotation . . . . . . . . . . . . . 118A.3 Explicit Formula of a π Rotation . . . . . . . . . . . . . . . . . . . 119

Appendix BAutomated Calibration of the AOMs 121B.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121B.2 The AOM Calibration system . . . . . . . . . . . . . . . . . . . . . 121B.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

Bibliography 123

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List of Figures

1.1 Transformation of an elementary particle with a nonzero spin (S)and a nonzero EDM (d) under T or P. The spin is odd under timereversal but even under parity transformation. The EDM is evenunder time reversal but odd under parity. An electron with the spinand EDM pointing in the same direction ends up with the spin andEDM pointing in the opposite directions after either one of thesetransformations. But only one of the spin and EDM configuration isfound in experiments. Therefore, the existence of a non-zero EDMviolates both parity symmetry and time reversal symmetry. . . . . . 2

1.2 Progress in experimental searches for the electron EDM and EDMspredicted by theoretical extensions to the standard model. Thedashed vertical lines are the projected shot-noise-limited sensitivitiesof some ongoing EDM experiments. This figure is adapted from [2,3]. 4

2.1 A schematic of our experiment apparatus created. Not shown in thisfigure are field plates threaded by the lattices and µ-metal shieldssurrounding the field plates. . . . . . . . . . . . . . . . . . . . . . . 9

2.2 Cooling and imaging between the plates. AR, anti-reflection coating;HR, high-reflection coating. Atoms are cooled using 3 pairs ofcounter-propagating beams. One pair is perpendicular to this pageand parallel to the plates. The other 2 pairs are formed by 2 beamsincident transversely and their mirror reflection off the high reflectioncoating of the center plate. The fluorescence of the atoms is collectedonto the photodiode arrays (PDAs). Figure adapted from [4]. . . . . 10

3.1 (a) Diagram representing a typical aspheric Fresnel lens with spher-ical aberration corrected. The steps in the actual Fresnel lens aremuch smaller and denser. (b) A pair of Fresnel lenses can forma sharp image on axis at unitary conjugate. These diagrams aremodified from [5]. . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

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3.2 The point spread functions of a pair of Fresnel lenses. (a) Imageon a PDA of a 2 mm section of illuminated atoms at varying fieldposition. The signal from each pixel is plotted as a dot. The dots areconnected by dashed lines just to guide the eyes. (b) The same datapresented in 2-D format. The image data, represented by colors, isplotted versus the field position in the object space. Figure adaptedfrom [4]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.3 (a) Photo of the space within the concentric shields. (b) 2Dschematic of the cylinder, breadboard and the PDA. The aimed ob-ject plane is one of the 2 surfaces of the center plate. Only the PDAon the left-hand side is shown. The other one is mirror-symmetricwith respect to the center plate. The dimensions are in mm. Byplacing the PDA against the cylinder, we gain 29 mm beyond thebreadboard. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.4 (a) Computer design of the PDA mount consisting of 2 parts. Onepart is the PDA holder (shown as transparent) with 3 screws holeto mate with 3 tapped holes on the side of the PDA board. Theother part is the base (shown in grey) with slots for adjusting theposition transverse to the optical axis. The 2 parts are joined byscrews and nuts. The slots at the joint provide adjustability of theaxial position. The atoms to be imaged are on the right. The ridgebelow the base touches the edge of the breadboard and serves asa guide when the base is translated. The part that sticks out ofthe base to the left can touch the wall of the cylinder to preventfurther tipping of the mount. (b) PDA mount is 3D printed fromblack PLA, assembled with screws and attached to the PDA. Thetriangular void in the PDA holder provides the space for the cablesto the photo-diodes to run through. . . . . . . . . . . . . . . . . . . 22

3.5 The layout of optics to image a field of view of 80 mm. We usean array of 2 Cooke triplets, each covering a field of view of 40mm. The images formed by the array elements do not overlap witheach other. To prevent light beyond the designed field of view frominterfering with the image, we insert a horizontal black sheet at aheight between the array elements. . . . . . . . . . . . . . . . . . . 23

3.6 Photo of the glass plates inside the cell. Atoms are trapped betweenthe plates. Fluorescence of the atoms, half of which bounces off thehigh-reflection coating of the center plate, transmits through theanti-reflection coated surfaces of the outer plates and the uncoatedsurfaces of the glass cell. . . . . . . . . . . . . . . . . . . . . . . . . 25

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3.7 A plane-parallel plate displaces the rays emanating from an objectpoint o. By extending backward the refracted ray, we find theimaginary intercept between the refracted ray and the optical axisat o’. The angles of incident and refraction are denoted by i and rrespectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.8 The lens elements of the triplets are truncated on the sides to fit intothe available space. (a) Front view. (b) Side view of the convexlens. (c) Side view of the concave lens. . . . . . . . . . . . . . . . . 28

3.9 (a) Computer design of the triplet mount. The top and bottomhalves are mirror-symmetric. (b) The triplet mount in-situ withlenses in place. The mount is closed and cinched with a nylon thread.The entire structure is placed at 84 mm from the surface of thecenter plate before the optics for laser cooling is set up. . . . . . . . 29

3.10 The point spread functions of 2 Cooke triplets. (a) Image on the−z PDA of a 3 mm section of illuminated atoms at varying fieldposition. The signal from each pixel is plotted as a dot. The dots areconnected by dashed lines just to guide the eyes. (b) The same datapresented in 2-D format. The image data, represented by colors, isplotted versus the field position in the object space. Each columncorresponds to a curve above. Data acquired by Teng Zhang. . . . 31

4.1 Diagram of electrodes for generating high voltage separated by fusedsilica spacers. The exact configuration varies in different trials butthe common feature is one center plate sandwiched between 2 groundplates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4.2 (a) Spidery web on the glass plates. (b) Zoomed-in view of thespidery web. (c) A large crater along the long edge between theplane surface (dark) and curved surface (bright) on a ground plate.(d) Another big crater. . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.3 Current at zero applied voltage. (a) Typical 60 Hz noise of themeasured current in our high voltage circuit (b) Measurements ofcurrent with a sampling rate of 60/s. . . . . . . . . . . . . . . . . . 37

4.4 Temporal evolution of the current through the top and bottombranches of the circuit. (a) Complete trace including charging anddischarging. (b) Zoomed-in view of the data in the time intervalafter charging and before discharging. . . . . . . . . . . . . . . . . . 38

4.5 Power supply set at positive 13.5 kV. (a) Complete trace includingcharging and discharging. (b) Zoomed-in view of the data in thetime interval after charging before discharging. . . . . . . . . . . . . 39

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4.6 Power supply set at positive 13.0 kV. (a) Complete trace includingcharging and discharging. (b) Zoomed-in view of the data in thetime interval after charging before discharging. The data points areconnected by dashed lines to guide the eyes. . . . . . . . . . . . . . 40

4.7 The currents acquired at a rate of 1000 samples/s after the highvoltage has been held on for 3 mins. (a) The currents in the timedomain show that the oscillation does not decay over the 10 s ofmeasurement. (b) Frequency spectrum, obtained by taking the FastFourier transform of the data in (a), shows frequency componentsin addition to the 60 Hz noise. The most dominant frequency otherthan 60 Hz is 18.3± 0.1 Hz. This frequency is the same when theapplied voltage is at negative 14 kV and at negative 15.1 kV. . . . . 41

4.8 (a) The average current over 10 s after the charging current hasdecayed vs the applied high voltage. (b) The vacuum pressure readby an ion gauge located at the top 6-way cross after the high voltagehas been applied for over 3 mins. The pressures at voltages between0 and -12 kV are not recorded but can be assumed to be around1.3× 10−10 torr. Some of the measurements are repeated on anotherday. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.9 Typical setup of a Michelson interferometer . . . . . . . . . . . . . 434.10 The calculated signal of the photo-diode of a Michelson interfer-

ometer assuming the beam splitter is an uncoated piece of glass.(a)Interference fringes of a monochromatic source. (b)Interferencefringes of monochromatic sources of 3 different colors. (c)Interferencefringe of a broadband laser . . . . . . . . . . . . . . . . . . . . . . . 44

4.11 Measurement of plate separations using a Michelson interferometer 454.12 Plate separation measurements on the −z side. (a) The locations

where we sample the plate separations with the center of the platesbeing the origin. (b) Plate separation versus height y. (c) Plateseparation versus x. . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

5.1 Energy levels of the F = 3 hyperfine level of the ground state(62S1/2) of cesium in a strong electric field. . . . . . . . . . . . . . . 50

5.2 Populations versus Larmor frequency. Red curve represents atomsin (|+3〉z + |−3〉z)/

√2 state, prepared and detected using audio

transition, as in the original scheme (Equation (5.7)). Blue curverepresents atoms in |0〉x state, as in the alternative scheme (Equa-tion (5.16)). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

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5.3 Precession from the state |3, 0〉. Both sub-figures are periodic withrespect to the phase with a minimum period of π. (a) The prob-abilities to be in the different magnetic sublevels m as a functionof phase. Because the probability to be in m is identical to theprobability to be in −m, the probabilities to be in m and −m aresummed in this figure. (b) The inverse of the uncertainties obtainedby detecting individual magnetic sublevels as a function of phase.The dashed line is the inverse of the combined uncertainty obtainedby independent measurements of all magnetic sublevels. The dottedline is the inverse of the combined uncertainty obtained in precessionfrom a stretched state. . . . . . . . . . . . . . . . . . . . . . . . . . 56

5.4 Sequential analysis of individual sublevel measurements after pre-cession from the state |3, 0〉. We measure the magnetic levels ina decreasing sequence of |m| starting from |m| = 3. The upperrow shows the probabilities of finding the atom in a given magneticsublevel after it was not found in previous measurements. Thelower row shows the inverse of the corresponding phase uncertain-ties. The dotted line represents the sensitivity of typical Larmorprecession. And the solid flat line represents the combined uncer-tainty of this sequence of measurements. The first column is forthe first measurement, which is the same as the corresponding colorshown in Figure 5.3. The second and third columns, for the secondand third measurements respectively, differ from the correspondingcolors shown in Figure 5.3. Not shown in this figure is the fourthmeasurement, which yields 100% probability and zero sensitivity. . . 60

5.5 Precession from the state |3, 3〉. (a) Probabilities to be in individualmagnetic sublevels m (color coded) as a function of phase. (b)The inverse of the uncertainties obtained with individual magneticsublevels as a function of phase. The dotted line is the inverse of theuncertainty obtained by measuring the projection 〈Fx〉. The upperbound of this figure (1/δφm = 6) is the inverse of the uncertainty ofthe superposition of stretched states. . . . . . . . . . . . . . . . . . 61

5.6 (a) Measurement of the sum of the population in the even magneticsublevels during precession from the state |3, 0〉. (b) The inverseof the uncertainty from this measurement (solid line). The bestsensitivity of the even magnetic sublevels is the same as the bestsensitivity of the m = 0 state shown in Fig. 5.3. The dashed line isthe inverse of the combined uncertainty obtained by independentmeasurements of all magnetic sublevels. The dotted line is theinverse of the uncertainty obtained in typical Larmor precession. . . 63

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5.7 The precession of spherical harmonics. The spherical harmonic of thestate that precesses starting from |3, 0〉x is shown by the solid blackline and its symmetry axis is represented by the grey arrow. Thespherical harmonics of the eigenstates of Fx are shown with dashedlines (blue:|3, 0〉x, red:|3,±1〉x), cyan:|3,±2〉x, magenta:|3,±3〉x. Eachsub-figure shows the unprecessed states |3,m〉x along with a snap-shot of the precessing state at a phase corresponding to an extremumin Fig. 5.6. The extrema correspond to points of relatively goodoverlap with successive |m| values. . . . . . . . . . . . . . . . . . . . 64

5.8 (a) Measurement of the 6th harmonic in precession starting fromthe state |3, 0〉. (b) The inverse of the uncertainty from this mea-surement (solid line). The dashed line is the inverse of the combineduncertainty obtained by independent measurements of all magneticsublevels. The dotted line is the inverse of the uncertainty obtainedin typical Larmor precession. . . . . . . . . . . . . . . . . . . . . . . 65

5.9 Shot-noise-limited phase uncertainty scaling with F . Combineduncertainties obtained by independent measurements of all magneticsublevels in precession from the state |F, 0〉 for integer F (red dots)or |F, 1/2〉 for half integer F (blue dots) are compared with the uncer-tainties obtained with the superposition of stretched states (dashedline) and the uncertainties obtained in typical Larmor precession(dotted line). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

5.10 (a) The probability of measuring the state |3/2, 1/2〉x in precessionfrom |3/2, 1/2〉x. (b) The inverse of the uncertainty from thismeasurement (solid line). The dashed line is the inverse of thecombined uncertainty obtained by independent measurements of allmagnetic sublevels. The dotted line is the inverse of the uncertaintyobtained by measuring the expectation value 〈Fx〉 in precession froma stretched state. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

5.11 Schematic of angular momentum precession. (a) Angular momen-tum initialized in one of the eigenstates of Fx precesses in a magneticfield along z. (b) Angular momentum initialized in one of the eigen-states of Fx precesses in a magnetic field at an angle γ from the zaxis . A magnetic level in the x basis evolves into the correspondinglevel in the x′ basis, where x′ is related to x by a rotation of φaround B. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

5.12 β versus φ for sin(γ) = 0.1. . . . . . . . . . . . . . . . . . . . . . . 695.13 Precession in the presence of a transverse magnetic field assuming

sin(γ) = 0.1 (solid line) and in the absence of a transverse magneticfield (dashed line). The vertical dotted lines mark π − 2γ and π + 2γ. 69

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5.14 (a) Eigenenergies scaled to the tensor Stark shift as a function oftransverse magnetic field in units of linear Zeeman shift scaled tothe tensor Stark shift, for F = 3. (b) Energy difference between thestretched states as a function of the transverse field. Figure adaptedfrom [4] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

5.15 Differences between adjacent peaks and valleys in a scan of the biasmagnetic field. We assume a tensor Stark shift of 5 Hz, a transversemagnetic field of 10 Hz and a precession time of 3 seconds. . . . . . 71

5.16 Evolution of |3, 0〉 in a static electric field only. (a) Probability ofdetecting m = 0 versus phase. There are 4 fringes in a π incrementof ωα2τ . (b) The inverse of the shot-noise-limited phase uncertaintyversus phase. The phase uncertainty diverges as the probability hitsthe minima. The phase uncertainty stays low outside the narrowband around the singularities. . . . . . . . . . . . . . . . . . . . . . 74

5.17 Evolution from |4, 0〉 in a static electric field only. (a) Probabilities ofdetecting individual magnetic sublevels are periodic with a baselineperiod of π/2 of ωα2τ . Only magnetic sublevels with non-zeroprobabilities are shown. (b) Phase uncertainties obtained fromquantum projection into individual magnetic sublevels. . . . . . . . 76

6.1 (a) The top part is the original design of the flange for mountingthe top vacuum window showing the secondary joint. The secondaryindium joint connects the window flange to a standard ConFlat®

flange shown as the bottom part. A standard ConFlat® flange comeswith a knife edge, which is not used in an indium seal and is avoidedby using a copper ring. (b) The modified flange is cut from theoriginal flange. It merges the functionality of the original flange andthe copper ring. It requires only one indium gasket to seal. . . . . . 80

6.2 Setup for measuring the birefringence of the vacuum window usingcrossed polarizers technique. All optical elements are aligned per-pendicular to the incident laser beam. The transmitted power isrecorded while the window is rotated about the laser beam. . . . . . 82

6.3 Transmittance through the crossed polarizers with the vacuumwindow placed in between the polarizers in various directions. . . . 82

6.4 (a) Difference between 2 shots of the background with the GO-5000M-USB camera. (b) Sum over rows vs. column index. (c)Zoom-in of (a). (d) Sum over columns vs. row index. . . . . . . . 87

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6.5 (a) An image of atoms in the MOT with background subtracted.Only pixels enclosed by the white ellipse contribute to the analysisin (b) and (c). (b) Sum over rows vs. column index. (c) Sum overcolumns vs. row index. . . . . . . . . . . . . . . . . . . . . . . . . 88

6.6 Diagram representing our lattices. The lattices are derived froma single fiber amplifier seeded with a Yag laser. The lattices areenhanced by Fabry-Perot cavities. . . . . . . . . . . . . . . . . . . . 89

6.7 (a) Diagram illustrating the technique of positioning the beammidway between the plates. For simplicity, only 2 of the 3 platesand 1 of the 2 lattice beams are shown. (b) 4 spots appear on thewhite card as we tilt the beam towards -z by tilting the folding mirror.The spots from the reflection off the plates are dimmer/smaller thanthe original spots from the transmitted beams. The spots fromreflection are on the +z side of the original spots. (c) When thebeams are tilted towards +z and hitting the plates, the reflectionspots appear on the left of the original spots. . . . . . . . . . . . . . 93

6.8 Transmission (a,b) and the Pound-Drever-Hall error signal (c) simul-taneously captured by a digital oscilloscope. The most prominenttransmission peak in (a) is the fundamental cavity mode, TEM00.This peak is truncated in (b), which zooms into (a) and provides adetailed view of higher modes and sidebands. Visible in this figureare the 2nd excited modes marked by the cyan arrow and the 4thexcited modes marked by the magenta arrow. The 2nd excitedmodes can be TEM11, TEM20 and TEM02. Sidebands of the TEM00are marked by the red arrows. . . . . . . . . . . . . . . . . . . . . 96

6.9 (a) Reflectance and transmittance versus cavity round-trip loss ε.Reflectance is plotted with the left vertical axis and transmittanceis plotted with the right vertical axis. (b) Reflectance divided bythe transmittance (R/T) versus ε. . . . . . . . . . . . . . . . . . . . 100

6.10 Diagram representing the servo for locking a cavity. . . . . . . . . . 1026.11 The loop response when the system was modulated by the laser

frequency and locked with PZT. The amplitude (a) and phase (b) ofthe error signal sent to the PID circuit, Verr, divided by the controlvoltage of the PZT, Vc vs. the modulation frequency. The blue curvewas measured without damping while the red curve was measuredwith damping. Figure from [4]. . . . . . . . . . . . . . . . . . . . . 104

6.12 The loop response when the system is modulated by the PZT andlocked with PZT. The amplitude (a) and phase (b) of the errorsignal sent to the PID circuit, Verr, divided by the control voltageof the PZT, Vc vs. the modulation frequency. . . . . . . . . . . . . 105

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6.13 (a) and (b) show zoomed-in views of Figure 6.12 around 9.5 kHzafter removing the 12db/octave slope. (c) and (d) show the responseof GL deduced from (a) and (b) by inverting Equation (6.21) . . . . 106

6.14 Movement of the center of the atom cloud after applying a push withthe bottom MOT beams. The error bars represent the standarderrors of the centers of the clouds obtained by fitting the atomdistributions to Gaussian functions, and they are about the size ofthe points. Fitting the motion to a parabola is performed with databefore 1 ms. The data falls short of the fit after 1 ms because of thesmearing and uneven distribution of the probe beams as explainedin the text. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

6.15 Atoms launched to the target height versus trap depth . . . . . . . 1096.16 Atoms stopped in the lattice with vertical cooling alone vs. cooling

pulse duration. The number of atoms stopped increases with coolingduration initially but peaks around 0.4 ms, after which the atomsare heated out of the trap. . . . . . . . . . . . . . . . . . . . . . . 110

6.17 Time evolution of the width of the cloud after cooling showingbreathing behavior. The oscillation frequency is about twice thetrap frequency. The oscillation damps out due to dephasing amongthe atoms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

6.18 (a) Atoms detected in the F = 3 hyperfine level after depumpingof varying duration. (b) Atoms detected in F = 3 after 2 ms ofdepumping at varying bias magnetic field in the x direction. . . . . 113

xv

Acknowledgments

I am grateful to my advisor, Professor David Weiss, for letting me work on thisexciting project and providing guidance all along. He is a man with brilliantintuitions as well as scientific rigor. I am always amazed by his wide intellectualhorizon, ranging from precision measurement to quantum computing, from non-equilibrium dynamics to supersolid. He is the leader that I aspire to follow.

I am thankful to Neal Solmeyer and Kunyan Zhu, who built much of theexperiment apparatus and brought me up to speed in the experiment. They havebeen consistently providing support even after graduating from Penn State. I alsothank Teng Zhang for working with me and carrying on with the endeavor.

I am thankful to Professor Nate Gemelke who encouraged me while I was downand gave me a tutorial every time I met him on a commuter bus. I am thankful toProfessor Kurt Gibble who taught me a great deal of laser physics. I am gratefulto Professor Ken O’Hara who shared expertise with me.

I appreciate Dr. Xianli Zhang who helped me out when I was struggling alone.Thanks to Josh Wilson, Dr. Lin Xia and Yang Wang for giving me a hand whenmounting the cell and installing the vacuum. They are also of great fun to workwith.

I thank Ted Corcovilos and Felipe Giraldo Mejia who gave me interesting ideas.Thanks to Fan Zou for giving us a try. I thank Yang Ge and Wei Xu for manyinteresting discussions.

I am thankful for the support from the National Science Foundation. Theopinions in this dissertation do not necessarily reflect the views of the NationalScience Foundation.

The formatting of this dissertation is based on the LaTex template created byProfessor Gary L. Gray. The optical layouts in this dissertation are created usingComponentLibrary created by Alexander Franzen.

Last but not the least, I am indebted to my parents who support me incompleting this journey.

xvi

Chapter 1 |Introduction

The permanent electric dipole moment of electrons arising from symmetry violationsis very tiny. As we now know, it must be smaller than 10−28 e · cm [6, 7]. Thisdissertation concerns our continued effort to push the sensitivity further downto 10−30 e · cm. The motivation for measurements of the electron EDM to suchgreat precision is to understand the matter and anti-matter asymmetry in theuniverse. In this chapter, I review the motivation for searches for the EDM. Moredetailed explanation of the motivations can be found in the dissertations of mypredecessors [4, 8, 9].

1.1 EDMs and SymmetriesA nonzero permanent electric dipole moment (EDM) of elementary particles breaksboth parity (P) and time-reversal (T) symmetries. To see that, let’s considerthe intrinsic EDM, d, and the spin, S. The EDM must be aligned with the spinaccording to the Wigner-Eckart theorem. Otherwise, the EDM would introduce anadditional quantum number, which is against the absence of a corresponding degreeof freedom in experimental observations. Under parity transformation, the spinstays the same while the EDM flips direction (Figure 1.1). Under time reversal, thespin flips direction while the EDM stays the same. Under either transformation,an electron with the spin and EDM pointing in the same direction ends up withthe spin and EDM pointing in the opposite directions. Experiments on the atomicstructure find just one but not both states before and after the transformations,violating both parity and time reversal symmetries. If nature had followed parityand time reversal symmetries, the EDM would have to be identically 0.

1

s

d

s

d

Time reversal

s

Parity

Transformationd

Figure 1.1. Transformation of an elementary particle with a nonzero spin (S) and anonzero EDM (d) under T or P. The spin is odd under time reversal but even underparity transformation. The EDM is even under time reversal but odd under parity. Anelectron with the spin and EDM pointing in the same direction ends up with the spinand EDM pointing in the opposite directions after either one of these transformations.But only one of the spin and EDM configuration is found in experiments. Therefore, theexistence of a non-zero EDM violates both parity symmetry and time reversal symmetry.

Discrete symmetries including parity, time reversal and charge conjugate, C,were believed to be valid symmetries of the fundamental laws of physics until1957. Parity was found to be maximally violated in the weak interaction in theWu experiment [10] following the proposal of Lee and Yang [11]. In this famousexperiment, Wu observed that the electrons from beta decay of cobalt-60 cameout more in the direction opposite to the nuclear spin whereas the parity oppositeprocess, in which the electrons came out in the same direction as the nuclear spin,happened less often. Parity violation is also observed in pion decay [12], where theemitted antimuon is always left-handed. Pion decay also breaks C because the muondecayed from pion is right-handed. But pion decay obeys the combined operationof charge conjugation and parity transformation (CP). Therefore, CP is a bettersymmetry than parity transformation alone. But CP is found to be violated by afractional amount in neutral kaons. Because of the CPT theorem, which assertsthat the combined operation of all 3 discrete symmetries must be a good symmetryin any quantum field theory even though individual symmetries can be violated,CP violation implies T violation. Direct T violation proves more difficult to observe

2

than CP violation, but it was confirmed in 2012 by the different transformationrates of the forward and backward processes of B0 meson transformations [13].

These symmetry violations are accounted for in the standard model. Thestandard model, based on the phenomenological phase factor in the Cabibbo-Kobayashi-Maskawa (CKM) matrix, predicts tiny EDMs of various elementaryparticles. For instance, the prediction on the electron EDM is de < 10−38 e · cm [14],where e is the electron charge. This prediction is 10 orders of magnitude smallerthan the current experimental sensitivity.

The standard model, though vastly successful in explaining a wide range ofphenomena, is believed to be incomplete. For instance, the standard model does notexplain the asymmetry between matter and anti-matter in the universe. Becausematter and anti-matter can be transformed into each other under CP, the asymmetrybetween matter and anti-matter could be a manifestation of CP violation. Theknown source of CP violation in the standard model is too small to account forthe asymmetry between matter and anti-matter in the universe. Extensions to thestandard model, such as supersymmetry, look for other sources of CP violationand consequently predict larger EDMs in the range between 10−26 and 10−33 e · cm(see Figure. 1.2). These predictions are within the sensitivities of current precisionexperiments or experiments in the near future.

1.2 Experimental Searches for EDMsExperimental searches for EDMs began after Edward Purcell and Norman Ramseypointed out that the existence of an EDM is subject to experimental tests. Sincethen, experimental searches for EDMs have been performed on a variety of systemsincluding neutrons, nucleons of diamagnetic atoms and electrons. EDMs can beprobed through their electric dipole interactions with electric fields. Generally,EDM experiments look for changes in precession frequencies between reversals ofelectric field directions. Because charged particles are accelerated under electricfields, searches for EDMs in protons and electrons have been performed with neutralatoms or molecules. In a non-relativistic model, neutral atoms are insensitive tothe EDMs of their constituent particles because the electrons would rearrangethemselves to cancel out the external electric field. This observation, stated inSchiff’s theorem [15], is evaded however due to relativistic effects when the electrons

3

get close to the nucleus as pointed out by Sandars [16]. Moreover, the EDM ofa neutral atom is enhanced compared with an isolated elementary particle. Thisenhancement is often described by the enhancement factor, R, as in

da = Rde, (1.1)

where da is the EDM of the atom. This enhancement is stronger in atoms withheavy nuclei.

Though no EDM has been found in any system so far, experiments have beennarrowing down the limits on the size of EDMs, constraining proposed theoriesbeyond the standard model. In fact, a large portion of supersymmetry theories hasalready been excluded by experiments. I review below some recent advances inexperimental searches for EDMs using atoms and molecules.

10-25 10-26 10-27 10-28 10-29 10-30 10-31 10-32 10-33

de (e cm)10-40 10-41

Split SUSY

SO(10) GUT

AccidentalCancellation

Naive SUSY

Lepton FlavorChanging

Left-RightSymmetric

Multi-Higgs

Alignment

Seesaw Neutrino Yukawa Couplings

Approx.Universality

Approx.CP

HeavysFermions

ExactUniversality

Standard Model

ExtendedTechnicolor

experimentally excluded

Berkeley 1990Berkeley 1994

Berkeley 2002

Imperial 2010

unconstrainedt

generic modelsSUSY variantsstandard model

ACME 2013

Penn S

tate

JILA

ACM

E

Imperial Y

bF fou

ntain

Figure 1.2. Progress in experimental searches for the electron EDM and EDMs predictedby theoretical extensions to the standard model. The dashed vertical lines are the projectedshot-noise-limited sensitivities of some ongoing EDM experiments. This figure is adaptedfrom [2,3].

4

1.2.1 Electron EDM Experiments

The current best limit on the size of the electron EDM comes from the AdvancedCold Molecule EDM Experiment (ACME), a collaboration between Harvard Uni-versity and Yale University. The 1st generation ACME reported a limit on theelectron EDM [6,17]:

|de| < 9.4× 10−29 e · cm, (1.2)

a 12-fold improvement over the previous best limit set by the ytterbium fluoride(YbF) EDM experiment at Imperial College of London. The ACME experimentmakes use of the carefully chosen molecule thorium monoxide (ThO). Like YbF,ThO features a large effective electric field of Eeff ≈ 77.6 GV/cm [6], muchlarger than the largest electric field created in the laboratory. In molecular EDMexperiments, the effective electric field that polarizes the heavy atom is the strongelectric field between the ions, while the purpose of the laboratory field is to setthe direction of the molecular dipole. The ACME Experiment is insensitive tomagnetic field noise by operating in the (σδ)3∆1 molecular state, a state with1 unit of spin projection, 2 units of orbital angular momentum projection and1 unit of total angular momentum projection. This state has a nearly canceledmagnetic dipole moment of µ = 0.00440(5)µB, where µB is the Bohr magneton.The Ω-doublets of ThO allow flipping the effective electric field direction withoutchanging the laboratory electric field. The H electronic state of ThO has a lifetimeof 1.8 ms. A number of upgrades that will improve the statistical sensitivity arebeing implemented in the 2nd generation ACME experiment.

While the 1st generation ACME result was published in 2014, it was recentlyconfirmed by a slightly weaker result from the molecular ion EDM experiment atJILA. JILA’s limit on electron EDM is [7]

|de| < 1.3× 10−29 e · cm. (1.3)

The molecular ion used in the JILA experiment was HfF+. This independentphysical system provides a strong systematic check of the ACME result. Unlikethe molecular beam in the ACME experiment, the molecular ions were trappedby a rotating electric field. This trap along with the long lifetime of the 3∆1 stateof HfF+ enabled coherent spin precession lifetimes of 700 ms. The experiment at

5

JILA was mainly limited by statistical sensitivity, specifically by the ion counts,which will be improved by an order of magnitude with a larger trap volume in the2nd generation of this experiment.

Before the recent advances in molecular EDM experiments, the best limiton electron EDM was set by thallium EDM experiment at Berkeley [18]. Thisexperiment was performed on 2 pairs of counter-propagating beams of Tl and Naatoms in an electric field of 123 kV/cm. Tl has a larger enhancement of the electronEDM than Na does, and Na served as co-magnetometer. Its sensitivity was limitedby the systematic effect due to motional magnetic fields as atoms move across anelectric field, despite the up/down beam cancellation.

1.2.2 Nuclear EDM Experiments

While paramagnetic atoms with a single valence electron, such as cesium, rubidiumand thallium, are sensitive to the electron EDM, diamagnetic atoms with pairedvalence electrons have the electron EDM nearly canceled but can be sensitiveto EDMs from the nucleus. A nuclear EDM search with a long history is themeasurement of the mercury EDM. Through decades of refinement, the mercuryEDM experiment at the University of Washington has achieved nano-Hertz frequencysensitivity. The latest published limit on the mercury EDM is

|dHg| < 7.4× 10−30 e · cm. (1.4)

This limit on the mercury EDM can be used to set limits on the EDMs of theconstituent nucleons. The inferred limit on the EDMs of the neutron and protonare |dn| < 1.6× 10−26 e · cm and |dp| < 2.0× 10−25 e · cm respectively. The limit onthe neutron EDM deduced from mercury is better than the current neutron EDMlimit obtained using ultracold neutrons. The mercury EDM experiment features a4-cell geometry: 2 central cells with oppositely oriented electric fields and 2 outercells with no electric field. The outer cells have no EDM sensitivity but serve asmagnetometers, which are used in data analysis to perform data cuts.

A relatively young experiment on nuclear EDMs is the radium EDM searchat Argonne National Laboratory. Radium has an enhancement factor 2 to 3orders of magnitude larger than mercury does due to its octupole-deformed nucleus.This experiment achieves seconds of spin precession time by cooling and trapping

6

radium atoms using lasers. Laser cooling and trapping also dramatically reducethe motional magnetic field. The cost of working with radium is that radium isunstable with a half-life of 14.9 days and that it has a very low vapor pressure.In 2015, the group published its proof of principle measurement, yielding a limiton the radium EDM of |dRa| < 5.0 × 10−22 e · cm [19]. A year later, the groupimproved the result by a factor of 36 [20], setting a new limit on the radium EDM:

|dRa| < 1.4× 10−23 e · cm. (1.5)

One major upgrade was extending the coherence time from 2 seconds to 20 secondsafter fixing a vacuum leak and improving the trap geometry. The major limitationof this experiment was statistical sensitivity, which will be improved by increasingthe number of photons each atom contributes and using a new source of radium.

1.3 An eEDM Search With Trapped Cold AtomsOur experiment at Penn State uses cold paramagnetic atoms in far-off-resonancedipole traps to search for the electron EDM. Specifically, the atoms of interest areheavy alkali atoms, cesium and rubidium. Cesium and rubidium are the 2nd and3rd heaviest alkali species, with precisely calculated enhancement factors of 125.9and 24.6 respectively [21] (the heaviest alkali atom, francium, has an enhancementfactor of 910 [22], but it is radioactive). The precision of the enhancement factors ofalkali atoms will allow us to precisely determine the size of the electron EDM fromthe measured atomic EDM. Like the radium EDM experiment, laser-cooled alkaliatoms trapped in optical lattices can be interrogated with a coherence time as longas 5 s with negligible systematic effect from the motional magnetic fields. Unlikeradium, the technologies of laser cooling and trapping of cesium and rubidiumare well developed and cold cesium and rubidium are available in large numbers(108), facilitating good statistical sensitivities. And finally, the electron EDMmeasurements using the 2 species provide a strong systematic check. I give anoverview of our experiment in the following chapter.

7

Chapter 2 |Overview of Our Experiment

Our experiment is built to measure the tiny differential phase shift of the stretchedstates (states with |m| = F , where m is the magnetic quantum number of a generalatomic angular momentum, F .) of the lower hyperfine levels of heavy alkali atomsarising from a non-zero EDM d interacting with a strong electric field. A diagram ofour experimental apparatus is shown in Figure 2.1. The atoms (cesium or rubidium)are vaporized from the oven, slowed down through the Zeeman slower, collectedby a magneto-optical trap (MOT), loaded into one of the lattices and launchedinto the region between the field plates, where the Ramsey spectroscopy will beperformed. I explain the individual components in the following sections. The basicdesign follows from prior dissertations [4, 9] on this experiment. The remainingchallenges when I took over the experiment from my predecessors are presented inSection 2.4.

2.1 Atom ProcessingThe alkali atoms are stored in ovens, one for cesium and one for rubidium. Thereservoirs are heated with heater tape to vaporize the alkali in the vacuum. Thealkali vapor comes out of the oven through a nozzle, which must be heated abovethe temperature of the oven itself to prevent alkali atoms from clogging the nozzle.Atoms coming out of the nozzle have their velocity mostly in the axial direction.The velocity spread in the transverse direction can be cooled in order to improvecapture efficiency in later stages if needed, though this has not been implementedyet. The large axial velocity component is slowed down in the one-meter longZeeman slower. A slowing beam, detuned from the cycling transition and combined

8

MOT

collect and precool

lattice-guided launch

Zeeman slower

Oven

108 atoms/s

Launch ~3x to fill up the lattices

8 cm

capture and cool atoms in 1.064 μm optical lattices

Figure 2.1. A schematic of our experiment apparatus created. Not shown in this figureare field plates threaded by the lattices and µ-metal shields surrounding the field plates.

with a repumping beam, is sent in the opposite direction to that of the atomvelocity. As the atoms are slowed down, the Doppler shift decreases. The Zeemanslower creates a spatially varying magnetic field to account for the change of theDoppler shift. The atoms exit the Zeeman slower with a small velocity and driftinto the MOT.

The magneto-optical trap consists of a pair of Helmholtz coils, 3 pairs of counter-propagating beams detuned from the cycling transition and a repumping beam.We collect atoms in the MOT for about 500 ms of loading time. We then turn offthe magnetic field and cancel the earth field with a set of bias coils. In the nearzero field environment, we cool the atoms through polarization gradient cooling(PGC) using the MOT beams, first with high intensity and close detuning and thenwith lower intensity and further detuning. Before the PGC cooling stages, we shiftthe location of the magnetic field zero of the MOT to load atoms into the paralleloptical lattices.

The optical lattices, enhanced by Fabry-Perot cavities, guide the atoms vertically

9

850 mm up after the atoms are launched by radiation pressure. The atoms arelaunched to a region well shielded from magnetic fields and away from the MOT.The MOT produces large magnetic fields. Our approach to transporting atoms fromthe MOT to the magnetically shielded region is different from the approach adoptedin the experiment measuring the radium EDM [19]. In the Ra EDM experiment,the atoms are trapped to the minimum waist of a focused Gaussian beam andthe atoms move along with the minimum waist of the beam as the focusing lensis translated. Our approach by launching has the advantage of fast speed: theatoms can reach the target positions in 415 ms after launch. The launch processheats up the atoms. Though the atoms are cooled in a moving frame before theyleave the MOT, further cooling is needed after they reach the target height. Thecooling at the top is produced by another 3 pairs of beams (see Figure 2.2) throughpolarization gradient cooling. After being cooled, the atoms stay trapped in thelattices.

PDA PDA

ARAR AR

Transverse cooling

HRFusedsilicaspacer

Fusedsilicachamber

36 cm

Figure 2.2. Cooling and imaging between the plates. AR, anti-reflection coating; HR,high-reflection coating. Atoms are cooled using 3 pairs of counter-propagating beams.One pair is perpendicular to this page and parallel to the plates. The other 2 pairs areformed by 2 beams incident transversely and their mirror reflection off the high reflectioncoating of the center plate. The fluorescence of the atoms is collected onto the photodiodearrays (PDAs). Figure adapted from [4].

The atoms are trapped between 3 parallel plates. The outer plates are alwaysgrounded while the center plate can be connected to either positive or negative highvoltage power supplies. When the high voltage is connected, the 3 plates create 2

10

regions separated by 10 mm where strong electric fields of equal magnitude 1 pointto opposite directions. This enables us to cancel the common mode magnetic fieldnoise in an EDM measurement. Another advantage of this 3-plate configurationis the elimination of any strong electric field between the outer plates and thesurrounding environment, including the vacuum chamber, which is grounded aswell.

The 2 outer plates and the 2 lattices are labeled as “+z” and “-z” respectively,according to the following experiment coordinate system. The z axis is perpendicularto the plates. The x direction is the moving direction of atoms in the Zeemanslower. The y direction is vertically up, parallel to the plate surfaces. These 3axes form a right-handed coordinate system. This coordinate system is also usedthroughout this dissertation.

Between the center plate and either of the outer plates are 2 spacers made offused silica. These spacers have been machined to be 4 mm thick within 1 µm

accuracy to ensure that the plates are parallel to each other. The plates and thespacers are clamped together. When the plates are held vertically, the center plateis held completely by friction. The plates are inside a one-meter long glass cellunder vacuum. The plates are held in the middle of the cell by titanium rodsconnecting to the top and bottom flanges of the cell.

The plates are surrounded by 4 layers of µmetal shields outside the vacuum.These shields are necessitated by our extreme sensitivity to magnetic fields. Themagnetic fields change the phase of the Ramsey fringe much like the EDM does inan electric field but with a much larger magnitude. The Hamiltonian of an atom inboth the electric and magnetic fields is [23]:

H = −(daE + µB)FF, (2.1)

where F is the total atomic angular momentum, E and B are the electric andmagnetic fields in the laboratory, respectively, and µ = 0.9 × 10−27 J/Gauss isthe magnetic dipole moment. Assuming an electron EDM de at our projectedsensitivity of 2.5× 10−30 e · cm, a ground state cesium atom in a laboratory electric

1The regions where atoms are trapped are around the centers of the plates, at least 10 mmaway from the edges of the plates. The uniformity of the fields around the trapped atoms ischaracterized in Section 4.4.

11

field 2 E of 150 kV/cm will experience an energy shift of the order of 17 nHz. Thesame energy shift can be induced by a magnetic field of only 12 femto Gauss. Withour µmetal shields, the residual magnetic field at the center inside the shields isaround several micro-Gauss. Inside the innermost shield, there are a set of coilsdesigned and made by Solmeyer [9] to cancel the bias and first order gradients ofthe residual magnetic fields. The coils are driven by a battery powered currentsource to minimize noise in the magnetic field. The shields keep the MOT out,so the magnetic field from the MOT is reduced by a factor of 5× 104 inside theshields. Outside the shields, currents run through a set of 3 rectangular coils toroughly cancel out the 3 components of the net magnetic field around the platesfrom the earth, ion pumps and the Zeeman slower.

To avoid ferromagnetic materials and Johnson noise from conductors inside theshield, mounts for optics inside the shields are custom made, mostly of plastic. Forexample, the mounts for the triplets for collecting fluorescence from the atoms are3D-printed with nylon and the mounts for the photodiode arrays are 3D-printed withpolylactic acid (PLA) (details covered in Chapter 3). We used mostly plastic screwsfor holding pieces together. My favorite screws are fiberglass screws, which are non-magnetic, non-metallic, and stronger than most other plastic screws. Some smallamount of titanium is used and it is non-magnetic as well. To avoid nickel found inoff-the-shelf photodiodes, Kunyan Zhu custom-made non-magnetic photodiodes [4],which are some of the most critical equipment in our experiment. Where epoxyis needed, we choose the non-magnetic type of epoxy over the magnetic type. Weavoid springs in these mounts as best as we can. So these mounts are harder tocontrol than the off-the-shelf counterparts but work nonetheless.

2.2 State PreparationAfter we cool the atoms loaded from multiple launches, we prepare them into a stateready for EDM measurements. The initial state required for an EDM measurementdepends on the scheme. We can either prepare the atoms in the m = 0 state in they basis, or drive the atoms from m = 0 in the z basis to the superposition of thestretched states using a composite audio frequency pulse, which has been worked

2The designed voltage is 150 kV/cm, which was demonstrated in mock-up experiments(Chapter 4). In general, the EDM sensitivity is better with a stronger field.

12

out by Kunyan Zhu [4]. The audio frequency pulse is an oscillating magnetic fieldtransverse to the electric field. The pulse is produced by the coils inside the shields.

To prepare the atoms in the |F = 3,m = 0〉 state, we start by optically pumpingthe atoms to one of the stretched states in the F = 4 hyperfine level using circularlypolarized light in a magnetic field aligned with the propagation direction of theoptical pumping beam, k, which is slightly off from the y direction. We thentransfer the atoms to the |F = 3,m = 0〉 state using a series of microwave adiabaticfast passage (AFP) pulses when the degeneracy among the Zeeman levels is liftedby a bias magnetic field. The magnetic field needed in this process is again providedby the coils inside the shields.

2.3 The EDM MeasurementBefore the start of an EDM measurement, we measure the residual magnetic fieldsso that we can drive appropriate currents through the coils inside the shields tocancel out the residual static fields. The measurement of the magnetic fields can beperformed while the electric field is off using the same signal fringes for measuringthe EDM by virtue of its strong sensitivity to magnetic fields. The signal fringesfrom atoms trapped between the plates not only provide information on the averagefields inside the magnetic shields but also their spatial gradients. The differentprecession phases between the “+z” and “-z” lattices tell the gradient along z. Thedifferent precession phases among sections of atoms along the vertical direction tellthe gradient along y. The atoms can spread over 8 cm in the vertical direction,and we can divide the atoms into at least 20 sections.

With most static magnetic fields canceled, we can perform EDM measurementson atoms in both lattices using the Ramsey technique. The simultaneous measure-ment at both lattice sites cancels the phase induced by the magnetic field commonto both lattices. A set of EDM measurements alternate the high voltage polaritiesand hence the electric field directions. This will further cancel the spatially stablegradient of the magnetic field.

The shot-noise-limited uncertainty on the electron EDM determined from anEDM measurement with a single atom initialized in the superposition of the

13

stretched states is given by

δde a single atom = ~FgF2REτ , (2.2)

where F is the total atomic angular momentum, gF is the Landé g-factor, E isthe electric field, R is the enhancement factor and τ is the coherence time. Theuncertainty is reduced by performing the measurement with an ensemble of atoms,N , in a single experimental sequence. The uncertainty is further reduced byrepeating the sequence over a period of time T . The net uncertainty from repeatedmeasurements is then:

δde = ~FgF2REN

√τTD

, (2.3)

whereD is the experiment duty cycle, i.e. the fraction of time when the measurementis running. If we perform the measurement using cesium atoms in the F = 3hyperfine level for a period of T = 24 hours, with an electric field of 150 kV/cm, 3second coherence time and half duty cycle, the uncertainty on the electron EDMwill be 2.5× 10−30 e · cm.

Due to instrumental noises, the measurement is most sensitive where thederivative of the signal with respect to the phase is largest. This can be achievedby introducing a tiny bias magnetic field in the z direction, Bz, to shift the phaseto the maximum slope if the magnetic field is stable. The required stability is 0.1micro-Gauss assuming that the coherence time is 3 seconds.

On the other hand, if the magnetic field is not stable enough to keep themeasurement around the maximum slope, the measurement will sample differentphases in the Ramsey fringes. Still, the EDM shift can be deduced from the phasedifference between the 2 lattices. As long as the magnetic field noise is common toboth lattices we do not have to worry about the magnetic field noise of individuallattices. If the fringe is of a single frequency, which is the case for an evolution fromthe superposition of the stretched states, the phase difference can be extracted byplotting the signal from one lattice as the abscissa and the other as the ordinate.The data from the 2 lattices forms an ellipse, whose area tells the phase differencebetween the 2 lattices.

14

2.4 Challenges to SolveMy predecessors [4,8,9] had completed most of the construction of this experiment.In fact, they were about to perform the proof of principle measurement of the EDMbefore they had a breakdown of the electric field plates. Their experimentationrevealed some challenges for an EDM measurement, which I try to address in thisthesis.

The first challenge is the resolution of the image system (Figure 2.2) that collectsEDM data from the atoms. The resolution target is only 3 mm in 1-D, limited bya depth of view of 4 mm when we image the atoms and their mirror image off thecenter plate. But achieving this resolution is not easy when we have a numericalaperture of 0.5 or above and a field of view of 8 cm. Though it appeared to workwell in test setups, a pair of Fresnel lenses with large aperture and short focallength in situ form poor images of the atoms. We realized that this discrepancy wasbecause of the different scattering properties of the atoms and the object in the testsetups. The atoms, probed by circularly polarized light, scatter light in a dipolepattern, covering most directions. The test object is a piece of translucent tape,which, when illuminated by a laser, scatters light mostly in the forward direction.So these test setups failed to test the entire aperture of the Fresnel lenses, but onlya fraction of it close to the optical axis. As it turned out, the aberrations fromthe large numerical aperture caused severe image blur. I address this challenge inChapter 3.

The second challenge is generating a high voltage with ITO-coated glass plates.High voltage is important in our EDM experiment because the sensitivity to theEDM scales linearly with the applied voltage. High voltage also creates the tensorenergy structure necessary for creating the superposition of the stretched states,which has the optimal single-atom EDM sensitivity. Though Fang [8] was ableto demonstrate an electric field of 150 kV/cm in her test setup, Solmeyer [9] wasonly able to reach 25 kV/cm at maximum using the actual experimental setup.In Chapter 3, I will present an explanation of the discrepancy and experimentaleffort in trying to understand how ITO coating behaves at high voltages. Furtherinvestigation on generation of high voltage with ITO-coated glass plates is beingcarried out by Felipe Giraldo and Fan Zou. While it is not clear at this momentwhat will be the highest voltage we will be able to attain with ITO coated glass, I

15

present an alternative EDM measurement scheme that works at a lower voltage inChapter 5.

The third challenge is reducing the amount of vector light shift induced by thetrapping light. The vector light shift varies with light intensity and is unstableover time. It will in general also be different for our two lattices. It is thereforehard to cancel using magnetic fields. Vector light shifts arise from residual circularpolarization in the otherwise linearly polarized light. The trapping light used in ourexperiment is enhanced by two parallel Fabry-Perot cavities. Though we send intothe cavities linearly polarized light with residual circular polarization around 10−6

in fractional intensity and the circular polarization from the input can be effectivelyfiltered by the Brewster plates in the cavity, the polarization within the cavities ismostly limited by birefringent elements inside the cavities, such as vacuum windows.The vacuum windows are not meant to be birefringent by design. But mechanicalstress and thermal gradient can give rise to birefringence at the level we care about.Though the Brewster plates filter out part of the circular polarization created bythe vacuum windows in every cavity trip, the vacuum windows add more circularpolarization in every cavity trip as well. So the best way to reduce vector lightshift is reducing birefringence of the windows. I talk about the improved vacuumwindow in Chapter 6.

16

Chapter 3 |Fluorescence Imaging of the Atoms

Fluorescence from the atoms provides all the signals in our measurement of theEDM. The signals of interest are usually the probabilities for the atoms to bein some particular states, and the probabilities can be obtained by detecting thepopulation in these states out of an ensemble of atoms. The population of atoms ismeasured by collecting the fluorescence of the atoms.

Spatial resolution will give an important handle on systematic errors in anEDM measurement. In addition, an EDM measurement requires magnetic fieldcancellation. In order to measure the field gradient using angular momentumprecession of atoms (see Chapter 5), spatial resolution is needed. We target theresolution of the imaging system to 3 mm, limited by the 4 mm depth of the field.Our imaging sensors are the custom-made nonmagnetic photodiodes [4]. Eachphotodiode is 3.2 mm by 4.7 mm in size. The photodiodes have been arranged intoarrays that match the shape of the atoms in the lattices. The atoms are spreadover 8 cm inside the lattice beams. So the imaging system is supposed to cover afield of view of 8 cm with unitary magnification.

To reduce photon shot noise, we would like to collect the fluorescence as efficientlyas possible. We collect the mirror reflection from the high reflection coating surfaceof the center plate in addition to the light directly from the atoms. We also use alarge aperture for our imaging lenses.

Finally, the imaging system, including both the lenses and photodiode arrays(PDAs), has to physically fit into the region inside the magnetic shields, subject totight spatial constraints.

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3.1 Summary of the problems with a pair of FresnellensesThe first attempt at imaging the atoms was using Fresnel lenses shown in Figure 3.1.There are a few advantages of Fresnel lenses at a glance. First, Fresnel lenses arethin by construction, so they can easily fit into the available space. Second, Fresnellenses are available in large diameters, so the numerical aperture and thus lightcollection efficiency can be very large. To add to the list, large plastic asphericFresnel lenses with a variety of focal lengths are available off the shelf at low prices.Even though the off-the-shelf aspheric lenses are corrected for on-axis aberrationonly at an infinite conjugate ratio (Figure 3.1a), by using 2 identical lenses with theiraspheric surfaces facing each other, the on-axis aberration can be well correctedat unitary conjugate (Figure 3.1b) as well. Fresnel lenses also come with a coupleof disadvantages. First, off-the-shelf plastic Fresnel lenses cannot be coated withanti-reflection coating so the light collection efficiency is reduced. But with largeapertures, the Fresnel lenses still have decent light collection efficiency. Second,Fresnel lenses come with steps by design (Figure 3.1) and these steps scatter light.But this scattered light presents no problem at the designed level of resolution.

a b

Figure 3.1. (a) Diagram representing a typical aspheric Fresnel lens with sphericalaberration corrected. The steps in the actual Fresnel lens are much smaller and denser.(b) A pair of Fresnel lenses can form a sharp image on axis at unitary conjugate. Thesediagrams are modified from [5].

Test images of atoms with a pair of identical Fresnel lenses are shown inFigure 3.2. A 2 mm section of illuminated atoms was imaged into 20 mm on thePDA around unitary conjugate, whereas an aberration-free imaging system should

18

0 1 2 3 4 5 6 7 8 9 10111213141516Pixel

0.00

0.05

0.10

0.15

0.20

0.25

Light

(a)

0 10 20 30 40Object (mm)

0

10

20

30

40

50

60

Imag

e (m

m)

(b)

0.00

0.05

0.10

0.15

0.20

0.25

Figure 3.2. The point spread functions of a pair of Fresnel lenses. (a) Image on a PDAof a 2 mm section of illuminated atoms at varying field position. The signal from eachpixel is plotted as a dot. The dots are connected by dashed lines just to guide the eyes.(b) The same data presented in 2-D format. The image data, represented by colors, isplotted versus the field position in the object space. Figure adapted from [4].

map a 2 mm section of the object into a 2 mm section in the image. The blur isparticularly severe for off-axis positions in the field, from where the point spreadfunction has a long tail. This blurred imaging system prevented us from measuring

19

and canceling the field gradient near the atoms.There are 2 major reasons for this failure. First, the imaging system was out

of focus due to spatial constraints. The pair of Fresnel lenses has individual focallengths of 70 mm and thus a collective focal length of 35 mm. The minimumaxial length from the object to the image, which occurs at unitary conjugate, is4× 35 = 140 mm. The breadboard for mounting the lenses and PDAs is 304 mmwide. With atoms to be imaged roughly centered, the length of each side is 152mm. The PDA was out of focus due to bulky parts in the mount of the PDA anddeviation from the unitary conjugate.

Second, though the aspheric surfaces of the Fresnel lenses correct well thespherical aberration on axis at the focal plane, they do not address the off-axisaberrations such as astigmatism. One result of astigmatism is the Petzval fieldcurvature which describes a curved image surface of a straight object. For instance,a converging lens forms an image of an off-axis point in the field closer to the lensthan it does for an on-axis point. In general, the axial displacement of the actualimage from the paraxial image plane is given by [24]

Petzval sum = h2

2∑i

1fini

, (3.1)

where h is the field height and the sum is taken over all lens elements in the system.Since both Fresnel lenses in our system are converging lenses, the Petzval sum ispositive. The Petzval field curvature is not a problem if the imaging sensor has amatching curvature like the one in the Kepler space observatory. But our imagingsensors, the PDAs, are straight and cannot be bent.

We overhauled our imaging system to correct the above problems. We addressthe aberration issue using Cooke triplets as described in Section 3.3, and thedefocusing issue by careful design of the mounts of the PDAs to fully utilize theavailable space inside the shields as described in Section 3.2

3.2 Positioning and Mounting the PDAsIt is preferred to place the PDAs and the lenses as far away from the atoms aspossible. That way, the angular field of view is reduced and so is the challengeof handling the aberrations. The previous design placed the PDA within the

20

breadboard, which sits inside the acrylic cylinder within the shields (see Figure 3.3).But there is more space inside the cylinder beyond the box region defined by thebreadboard. Our PDAs, together with the board that the photodiodes are solderedto, is 106 mm high and can fit into the arc area as shown in Figure 3.3b. Using thecylinder instead of the breadboard as the constraint, the maximum object to imagedistance available is 180 mm. That is 29 mm more than what the breadboard canoffer. Accounting for the displacement of the image due to the glass cell and plates(Section 3.3.3) and leaving tolerance, we set our designed object to image distanceto be 160 mm, which is 20 mm more than what the pair of stock Fresnel lensesrequires.

Figure 3.3. (a) Photo of the space within the concentric shields. (b) 2D schematic ofthe cylinder, breadboard and the PDA. The aimed object plane is one of the 2 surfacesof the center plate. Only the PDA on the left-hand side is shown. The other one ismirror-symmetric with respect to the center plate. The dimensions are in mm. By placingthe PDA against the cylinder, we gain 29 mm beyond the breadboard.

Having worked out the range of available space for the PDAs, we have to designthe appropriate mounts to put the PDAs at the desired locations. Because themount eventually has to be secured to the breadboard, it is clear from a glanceat Figure 3.3b that the mounting structure cannot be vertically straight. As aconsequence, the mount needs some structure to prevent it from tipping beforeit is secured by screws. Additionally, we need the adjustability of the axial andtransverse positions relative to the optical axis so that we can move the PDA tofind the focus of the image. Last but not least, the mount should not introduce

21

extra pieces that further limit the available space. Our design that meets all of theabove requirements is shown in Figure 3.4.

Figure 3.4. (a) Computer design of the PDA mount consisting of 2 parts. One part isthe PDA holder (shown as transparent) with 3 screws hole to mate with 3 tapped holeson the side of the PDA board. The other part is the base (shown in grey) with slots foradjusting the position transverse to the optical axis. The 2 parts are joined by screwsand nuts. The slots at the joint provide adjustability of the axial position. The atoms tobe imaged are on the right. The ridge below the base touches the edge of the breadboardand serves as a guide when the base is translated. The part that sticks out of the base tothe left can touch the wall of the cylinder to prevent further tipping of the mount. (b)PDA mount is 3D printed from black PLA, assembled with screws and attached to thePDA. The triangular void in the PDA holder provides the space for the cables to thephoto-diodes to run through.

3.3 Optical DesignHaving set the object to image distance to 160 mm, the angular field of viewbecomes ±26.6. If we use an aperture that matches the size of the field, thenumerical aperture is then 0.45. Simultaneously achieving such a wide field of viewand a large numerical aperture has proved difficult. Taking a step back, we reducethe aperture by half but use 2 sets of lenses to cover the entire field.

22

3.3.1 Lens Array of 2

There are several reasons why we have chosen to use an array of 2 but not more.First, while there is no constraint to the vertical size of the lenses we can putin place, the horizontal dimension is limited to about 40 mm by the mirrors forlaser-cooling of the atoms. And 40 mm is half of the targeted field of view. Second,with an array of 2, the images formed by the 2 sets of lenses do not overlap atunitary conjugate when the field of view is twice the array spacing (see Figure 3.5),whereas images formed by an array of more than 2 overlap with each other and arehard to separate. Lastly, a lens array of more than 2 will further limit the aperturesize.

Figure 3.5. The layout of optics to image a field of view of 80 mm. We use an arrayof 2 Cooke triplets, each covering a field of view of 40 mm. The images formed by thearray elements do not overlap with each other. To prevent light beyond the designed fieldof view from interfering with the image, we insert a horizontal black sheet at a heightbetween the array elements.

By using an array of 2, each with aperture radius of 20 mm, the numericalaperture becomes 0.24. The image is separated into 2 sections. Reconstruction ofthe image of the atoms requires rearranging signals acquired by the PDAs usingthe mapping results presented in Section 3.5. In the next subsection, we deal withthe aberrations in each array element.

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3.3.2 Design of a Cooke Triplet

A Cooke triplet consists of 3 elements, with one diverging lens sandwiched between2 converging lenses. It is the simplest lens system that deals with all Seidelaberrations. The triplet is symmetric about the center element (see Figure 3.5) sothat coma and distortion are eliminated. The curvatures of the elements can beadjusted to correct astigmatism.

In particular, the center lens is negative so that the Petzval sum is reduced. Thespacing between the lenses helps to maintain a focusing power when the Petzvalsum is small or zero. The focusing power is given by

1f

= 1y1

∑i

yifi, (3.2)

where yi are the ray heights at individual lens elements, fi are the focal lengths ofindividual lenses and the sum is taken over all lens elements. The focusing powerdepends on yi, whereas the Petzval sum does not (see Equation 3.1). Because ofthe focusing power of the lenses on the sides and the spacing to the center lens, theray height at the center lens is lower than that at the outer lenses. This reduces theamount of focusing power that the negative lens element takes out of the overallfocusing power so that the compound lens system still has a positive focusing power.

Still the presence of a negative lens demands a strong focusing power from thepositive lenses in order to keep the overall focal length below 40 mm. Positivelenses with strong focusing power need high index of refraction and short radiiof curvature. Short radii of curvature introduce higher order aberrations that arehard to correct. So we have to trade off between higher order aberrations andthe Petzval sum. We end up with a small but nonzero Petzval sum. We dealwith the residual Petzval sum by defocusing from the paraxial focus. Defocusingcompensates astigmatism off-axis at the price of on-axis resolution. By pickingan appropriate defocusing position, we are able to achieve more or less uniformresolutions of 3 mm across the field.

Chromatic aberration could have be corrected by using crown glass for thepositive lenses and flint glass for the negative lens in the Cooke triplet. But crownglass does not support very strong focusing power which is what we need for ahigh numerical aperture. So our design of the triplet does a bad job in chromatic

24

aberrations, which are not a serious concern in our experiment. Our imaging systemis designed for 2 wavelengths at the cycling transitions of cesium (852 nm) andrubidium (780 nm). The focal lengths at these two wavelengths differ by less thanthe depth of focus as discussed in Section 3.3.4.

We could have added more lenses into the Cooke triplet to further correct theaberration. But more lenses complicate the assembly. Since the triplet meets ourdesign target, we adopt the triplet design.

3.3.3 The Effects of Plane-Parallel Plates

Figure 3.6. Photo of the glass plates inside the cell. Atoms are trapped between theplates. Fluorescence of the atoms, half of which bounces off the high-reflection coating ofthe center plate, transmits through the anti-reflection coated surfaces of the outer platesand the uncoated surfaces of the glass cell.

Our imaging system does not image the atoms directly, but through a glassplate and a glass cell (Figure 3.6). Both the glass plate and the cell wall can betreated as plane-parallel plates. Light rays passing through the plane-parallel platesare refracted twice, emerging out in the same direction but displaced as shown inFigure 3.7. The displacement at a fixed location down the optical axis is invariantwith respect to translation of the plane-parallel plates. By translating one of the

25

plane-parallel plates to the position where the two touch each other, we convert theoptical system with 2 plane-parallel plates into the equivalent optical system withone plate whose thickness is the sum of the thicknesses of the individual plates.The plate is 6.1 mm thick and the wall of the cell is 5 mm thick. Therefore, thetotal thickness of glass is 11.1 mm.

ir

d

i

o o'

Figure 3.7. A plane-parallel plate displaces the rays emanating from an object point o.By extending backward the refracted ray, we find the imaginary intercept between therefracted ray and the optical axis at o’. The angles of incident and refraction are denotedby i and r respectively.

We calculate the displacement due to the glass plates as follows. From Figure 3.7,we deduce that the axial displacement of the intercept with the optical axis is

oo′(i) = d(1− tan(r)/tan(i)), (3.3)

where d is the thickness of the plate and i and r are the angles of incidence andrefraction respectively. In the paraxial limit, Equation (3.3) reduces to:

limi→0

oo′(i) = d(1− 1/n), (3.4)

where n is the index of refraction of the glass and the index of refraction of theair is approximately 1. Therefore the paraxial virtual image of the object formedby the plates is displaced by d(1 − 1/n) towards the plates. Given a total glassthickness of 11.1 mm, the object is brought closer towards the lenses or the PDA

26

by 3.7 mm. The insertion of the plane-parallel glass plates reduces the effectiveobject to image distance, even though it increases the optical path length due tothe index of refraction of glass.

To the extent that the angle of incidence deviates from the paraxial limit, theintercept (o′) between the extended refracted ray and the optical axis is dependenton the angle of incidence, as given by Equation (3.3). So the rays refracted by theplane-parallel plate cannot be extended backwards to the same virtual image point,resulting in aberration. We can estimate the size of this aberration by comparingthe ray heights of the marginal ray and the paraxial ray at the paraxial imageplane.

hm − hp = [oo′(im)− oo′(o)]tan(im), (3.5)

where the subscript m denotes the quantities for the marginal ray and the subscriptp denotes the quantities for the paraxial ray. For the on-axis marginal ray, theaberration is 0.072 mm. For the oblique marginal ray, the largest aberration is0.515 mm. These are small compared with the resolution of the Cooke triplets orthe pixel size of the PDAs. Hence the plane-parallel plates in our system do notpresent an issue. Otherwise, we would have to further tune the curvatures of thesurfaces of the triplet to correct the aberrations due to the plates.

3.3.4 Imaging the Atoms and their Mirror Image

The atoms and their mirror image by the high reflection coating of the center plateare displaced by 4 mm. In order to prevent blurring from this displacement, thedepth of focus of the optical system has to be larger than 4 mm. The depth offocus is related to the numerical aperture (NA)

∆z = ∆ytan(arcsin(NA)) = 9 mm, (3.6)

where ∆y = 3 mm is the transverse resolution. Therefore the mirror image of theatoms does not compromise the resolution of our optical system.

27

R20

36

a b c

Figure 3.8. The lens elements of the triplets are truncated on the sides to fit into theavailable space. (a) Front view. (b) Side view of the convex lens. (c) Side view of theconcave lens.

3.4 Mechanical Mount of the Cooke TripletsThere is no off-the-shelf mount for the set of Cooke triplets shown in Figure 3.5.We want the spacing between the 2 sets of Cooke triplets to be around 1 mm, andwe do not want the mount to introduce more than 1 mm to the width of the lensassembly due to the spatial constraints in the horizontal transverse direction. Infact, we have trimmed the lenses with 20 mm radius to 36 mm in width as shownin Figure 3.8. By truncating 2 mm from each side of the lenses, we lose only 3.7%of the aperture of the lenses.

The traditional technique for mounting compound lenses uses a tube for hostingthe lenses and sets the spacing between them using ring spacers. The tube andring spacers occupy the space to the sides of the lenses and is therefore unsuitablein our system.

Our design of the triplet mount is shown in Figure 3.9. Like other optics weplace close to the atoms in this experiment setup, this mount is made of plastic, ormore specifically, 3D-printed from nylon with selective laser sintering. Nylon itselfis white. To minimize scattering from the nylon surfaces, we dye it black usingJacquard iDye Poly. The void in this mount is designed to match the shape of thetrimmed lenses and the only extra width occupied by the mount is the thickness ofthe wall, which is 0.7 mm. The spacing between the lenses is set by the thicknessof the ridges printed on the mount. The ridges work like partial rings. They setthe spacing while minimizing the blocked aperture. With the 4 sets of ridges in

28

our design, we have 98.7% of clear aperture. The lenses are inserted in place fromthe side after the opening on the side is widened by bending the thin wall of themount. The thin wall is restored to flat and the opening closes in after the lensesare in place. This one-piece design minimizes the hassle of assembly and the use ofepoxy.

Figure 3.9. (a) Computer design of the triplet mount. The top and bottom halvesare mirror-symmetric. (b) The triplet mount in-situ with lenses in place. The mount isclosed and cinched with a nylon thread. The entire structure is placed at 84 mm fromthe surface of the center plate before the optics for laser cooling is set up.

3.5 Imaging System TestsWe tested the imaging system before it was put in-situ. Special care was taken tochoose test objects that imitate the atoms. The atoms emit incoherent fluorescencein all directions and we have adopted large apertures in order to collect as muchlight from the atoms as possible. So the test object has to emit incoherent lightin all directions as well. Direct laser light is a coherent directional beam, andtherefore does not meet the requirement. But we can diffuse the laser beam to

29

make incoherent scattered light at the wavelength of the fluorescence. We find thatthick paper diffuses light more uniformly than translucent tape does, though itstrongly attenuates the light.

We also have to pick an imaging sensor when we don’t have the actual PDAsavailable for testing on the side. Typical imaging sensors are too small to coverthe extent of the image. Our first attempt to resolve the size issue of the sensorsis forming an image of the test object on a piece of paper and demagnifying theimage on the paper using another lens system with a low numerical aperture sothat the secondary image fits into a typical CCD camera. This worked poorly forreasons we did not resolve. Instead, we formed the image on a typical CCD sensorand translated the CCD sensor as we scanned the object.

We found that the imaging system works to our specification in the test setupwith the imitating object. But that is no substitute for an in-situ test with theactual atoms and the actual PDAs. We carried out the in-situ test of the imagingsystem using a method similar to that used by [4]. First, we depump the atomstrapped in the lattices to the lower hyperfine level of F = 3 using prolonged (100ms) polarization gradient cooling without re-pumping. The atoms in F = 3 donot scatter the light on the cycling transition, F = 4 to F ′ = 5. We then pulse onthe beams resonant with the cycling transition and a separate repumping beamintersecting the lattice from a transverse direction. This transverse repumpingbeam has a beam waist of 0.5 mm. Even though the beams on the cycling transitionilluminate all atoms in the lattice, only the atoms illuminated by the repumpingbeam emit fluorescence. The resultant fluorescing atoms have a width of 3 mm asmeasured by an auxiliary imaging system. This small section of fluorescing atomsis detected by the imaging system, and we measure the response of the systemusing a PDA. The results are presented as point spread functions in Figure 3.10.The image of a 3 mm section of the atoms has most of the energy concentrated ona single pixel, achieving our design goal. The mapping from the object to the imageis linear, which is a result of a negligible distortion. The results are separated into2 sections, which is a consequence of the array of 2 sets of lenses. Atoms beyondthe field of ±45 mm do not have significant signal on the PDA. They are blockedby the black screen separating the top and bottom halves. Atoms at the field of±(40, 45) mm can be imaged to the pixel at the midpoint.

30

0 5 10 15 20 25Pixel

0.0

0.2

0.4

0.6

0.8

1.0

Light

(a)

40 20 0 20 40Object (mm)

40

20

0

20

40

Imag

e (m

m)

(b)

0.0

0.2

0.4

0.6

0.8

1.0

Figure 3.10. The point spread functions of 2 Cooke triplets. (a) Image on the −z PDAof a 3 mm section of illuminated atoms at varying field position. The signal from eachpixel is plotted as a dot. The dots are connected by dashed lines just to guide the eyes.(b) The same data presented in 2-D format. The image data, represented by colors, isplotted versus the field position in the object space. Each column corresponds to a curveabove. Data acquired by Teng Zhang.

31

Chapter 4 |High Voltage and Field Plates

High voltage is important to the measurement of EDMs because the sensitivity islinearly proportional to the electric field. Other experiments studying EDMs [19,25]have developed mature techniques using metal electrodes for generating high electricfield. Our lab at Penn State has pioneered studying the creation of high electricfields using ITO-coated glass plates in vacuum. The ACME experiment studyingthe EDM of ThO [17] also uses ITO-coated glass plates as the electrodes to generatethe electric field but they need only a very small electric field (142 V/cm [26]) in thelaboratory. We have chosen ITO-coated glass plates as our electrodes because wewant to have optical access from 3 dimensions to cool the atoms trapped betweenthe plates and we want to minimize Johnson noise by avoiding bulk conductors. Webegin by reviewing our prior studies on the high voltage properties of ITO-coatedglass plates.

4.1 Prior Studies on the Creation of High VoltageUsing ITO-coated Glass in VacuumThe basic configuration of our high voltage systems consists of 3 plates with roundededges and corners as shown in Figure 4.1. The center plate can be connected tothe positive or negative high voltage supplier or the ground, while the 2 outerplates are grounded at all times so that the field between the outer plates and thesurrounding vacuum chamber is nearly zero. The spacing between the plates is setby cylindrical spacers made of fused silica. In the early trials of this experiment,the spacers were 3 mm thick. In the later trials, the spacers were increased to 4

32

high voltage

grou

nd p

late

grou

nd p

late

center p

late

spacers

Figure 4.1. Diagram of electrodes for generating high voltage separated by fused silicaspacers. The exact configuration varies in different trials but the common feature is onecenter plate sandwiched between 2 ground plates.

mm thick to improve optical access. We can trap atoms in the 2 gaps created bythe 3 plates and study EDM physics there. The electric field is enhanced at theedges and corners. The peak field around the corners of the center plate is 1.7times the field between the centers of the plates. This peak field at the cornerscan be avoided by making the electrodes in the Rogowski profile [27] instead of around shape. But the Rogowski profile is difficult to make with glass, and moreimportantly, it requires a too thick center plate.

Fang Fang, the first Ph.D. student in our lab, carried out high voltage tests in amock-up setup consisting of an ITO-coated glass plate sandwiched between 2 argonconditioned copper plates [8]. The plate spacing she used was 3 mm. She was ableto apply a positive voltage of 45 kV to the center plate with 5 pA current throughthe vacuum. But when she applied a negative voltage of 18 kV to the center plate,the current through the vacuum was 500 pA, 2 orders of magnitudes higher.

The asymmetry between the positive high voltage test and the negative highvoltage test demonstrates the different field-emission properties of the ITO-coatedglass plate and argon-conditioned copper plates. When a positive voltage wasapplied to the center plate, the 2 argon-conditioned copper plates were cathodes

33

and the low current through the vacuum showed that the argon-conditioned copperelectrodes had low field emission. On the other hand, when a negative voltage wasapplied to the center plate, the ITO-coated glass plate was the cathode, and thehigher current through the vacuum could be due to the stronger field emission fromthe ITO coating surface.

Neal Solmeyer, the second Ph.D. student in our lab, in the actual experimentalsetup for an EDM measurement used a set of 3 ITO-coated glass plates spaced by4 mm [9]. He measured the average current flowing through the vacuum after thecharging current decayed down to a negligible value. He found the average currentincreased linearly with the voltage applied to the center plate between negative 9kV and positive 9 kV. But he observed a breakdown at 10 kV, where the averagecurrent increased sharply beyond the configured measurement range (2 nA) of thepicoammmeter and a faint blue glow was spotted on the plates. Small patches ofdiscoloration were found on the rounded edges of the center plate and the thin ITOcoating was identified as damaged.

A bluish glow is often observed in corona arcing, which is an electrical dischargein gas or liquid. The breakdown electric field in a gas depends on only the pressureas given by Paschen’s law. In contrast, the breakdown in ultra-high vacuum (about1× 10−10 torr in our setup) is caused not by the residual pressure but the surfacequality of the electrodes. I discuss the electrodes’ surfaces in the next section.

4.2 ITO-coated Glass Plates After BreakdownThe ITO coated glass plates after breakdown were examined with an opticalmicroscope after we took them out of the vacuum chamber. The most ubiquitousfeature on the plates is the spidery web like the one shown in Figure 4.2a and b.The spidery web is not everywhere on the plates but it spreads across a large area.Since no image had been taken at comparable resolution before the breakdown, wecannot assert for certain that this spidery web is a consequence of the breakdown,but it seems very likely.

The second prominent feature on the plates is craters, like the one shown inFigure 4.2b and c. I find about 20 craters, big or small, on the ground plates butnone on the center plate. Most of these craters are along the long edge betweenthe outer planar surface and the curved surface of the ground plates. The biggest

34

Figure 4.2. (a) Spidery web on the glass plates. (b) Zoomed-in view of the spideryweb. (c) A large crater along the long edge between the plane surface (dark) and curvedsurface (bright) on a ground plate. (d) Another big crater.

of these craters are shown in Figure 4.2b and c. They are visible to naked eyesand would have been noticed had they existed before the breakdown. Therefore,these craters are certainly the aftermaths of the breakdown. They are created byionized particles bombarding the ITO-coated glass. I specifically searched aroundthe location on the center plate corresponding to these craters and I found nothingother than discoloration along the long edge of the center plate. These observationsare consistent with the fact that the breakdown occurred at a positive voltageapplied to the center plate when the ground plates served as the cathodes. Electronsemitted by the cathodes bombarded the center plate, created ionized particles andpotentially deoxidized the ITO. The positive ions, accelerated by the electric fieldtowards the ground plates, created the craters.

After the first set of ITO-coated glass plates suffered a breakdown, we brokethe vacuum and replaced the plates with a second set of ITO-coated glass plates. Ipresent the results of the high voltage tests of this set of ITO-coated glass plates in

35

the following section.

4.3 High Voltage Test with a Second Set of ITO-coated Glass PlatesWhen this round of high voltage tests was carried out, we wanted to find out moreabout the current flowing through the vacuum than just its average value. Solmeyeronly recorded the average of the current largely because of limitations related tothe 60 Hz noise in the measurement of the current [9]. A typical cycle of the 60 Hznoise is shown in Figure 4.3. We can reduce the noise by 2 orders of magnitudes byacquiring data at a rate of 60 samples/s (See Figure 4.3b). The phase with respectto the 60 Hz noise at which the data is sampled changes the offset of the data. Forinstance, if the data is sampled at a π/2 phase, the data is always at the peaks ofthe 60 Hz noise. To avoid this unnecessary offset, we trigger the start of acquisitionat the zero phase with respect to the power line.

Using the above technique, we can observe the temporal evolution of the currentwith much better precision. Figure 4.4a shows the temporal evolution of thecurrent through the top and bottom branches of the circuit when the center plateis connected to positive 5 kV. At a time around 1.5 s, we switch on the relayconnecting the center plate to the high voltage power supply. The dip immediatelyafter is the charging current, with a charging time constant of about 26 ms. Thepicoammeters we use to measure the current saturate at 4 nA in the selectedmeasurement range. At around 2.5 s, we disconnect the center plate to the powersupply and connect the center plate to the ground. The spike immediately afteris the discharging current. The high voltage is applied to the center plate in thetime interval between charging and discharging. This interval with high voltageapplied is magnified in Figure 4.4b. While we do not observe anything special whenpositive 5 kV is applied, we do observe 2 types of behaviors at higher voltages.

The first type of behavior is additional spikes or dips after the charging currenthas decayed to a negligible amount, as shown in Figure 4.5. There are 2 identifiablediscontinuities during the 1 second when the center plate is held at positive 13.5kV. Additionally, the current is offset from zero. When the high voltage is kept onfor a longer time at +12 kV, we notice that these extra dips occur once every 10 s.

36

0 5 10 15 20Time (ms)

2.01.51.00.50.00.51.01.52.02.5

Curr

ent

(nA

)

a)

0 1 2 3 4 5Time (s)

0.0080.0100.0120.0140.0160.0180.0200.0220.0240.026

Curr

ent

(nA

)

b)

Figure 4.3. Current at zero applied voltage. (a) Typical 60 Hz noise of the measuredcurrent in our high voltage circuit (b) Measurements of current with a sampling rate of60/s.

The frequency of these spikes or dips increases when the applied voltage increases.The second type of behavior is the oscillation at 18.3 Hz as shown in Figure 4.6.

The oscillation persists for several minutes after charging when we keep the highvoltage on, as shown in Figure 4.7.

Neither the additional spikes or dips nor the oscillation at 18.3 Hz can beattributed to the ohmic conductance of the spacers. If the spacers have ohmicconductance, the currents will have a constant offset. But the offset in our mea-surements of the currents is very tiny compared with the amplitudes of spikes, dipsand oscillation.

Nevertheless, we take the averages of the currents when a high voltage isapplied after the charging current has decayed down to an insignificant amountand plot them in Figure 4.8a. Along with the currents, the vacuum pressures

37

0 2 4 6 8 10Time (s)

6

4

2

0

2

4

6

curr

ent

(nA

)

a)+05kv

topbottom

2.0 2.5 3.0 3.5 4.0Time (s)

0.6

0.5

0.4

0.3

0.2

0.1

0.0

0.1

curr

ent

(nA

)

b)

Figure 4.4. Temporal evolution of the current through the top and bottom branches ofthe circuit. (a) Complete trace including charging and discharging. (b) Zoomed-in viewof the data in the time interval after charging and before discharging.

at the corresponding voltages are plotted in Figure 4.8b. The average currentsincrease at an approximately linear rate as the applied voltage increases until thevoltage reaches 15.6 kV on the negative side or 13.7 kV on the positive side, wherethe average currents as well as the vacuum pressure increase sharply. We have aweb camera focused on the glass plates and use it to monitor the plates in thevisible spectrum when a voltage is applied to the plates. But no light in the visiblespectrum was ever detected throughout this high voltage test. Because the rise invacuum pressure indicates that gas particles have been liberated from the electrodesand that we want to prevent the ITO coating from damage, we do not increase thevoltage any further.

The necessary condition for a current through the vacuum is some source of fieldemission, that is electrons tunneling from a conductor into vacuum. It is unclearwhether the source of field emission is on the metal parts or on the ITO coated glass.Our metal parts have been electropolished but not conditioned. Electropolishing

38

0 2 4 6 8 10Time (s)

6

4

2

0

2

4

6

curr

ent

(nA

)

a)+13.5 kv

topbottom

2.0 2.5 3.0 3.5 4.0Time (s)

0.6

0.5

0.4

0.3

0.2

0.1

0.0

0.1

curr

ent

(nA

)

b)

Figure 4.5. Power supply set at positive 13.5 kV. (a) Complete trace including chargingand discharging. (b) Zoomed-in view of the data in the time interval after chargingbefore discharging.

is different from conditioning. Conditioning is the standard practice to preparemetal surfaces for high voltages. It uses breakdowns to remove surface defectscausing the breakdowns. Though the goal is to sustain high voltage in vacuum,conditioning can be done using breakdowns either in vacuum or in dilute buffergases such as argon or krypton. In either case, conditioning targets defects onthe surface causing breakdowns. Electropolishing, on the other hand, relies on anelectrical-field-induced chemical reaction. The rate of reaction is approximatelylinearly dependent on the strength of the field. Electropolishing removes materialacross a large surface. Though electropolishing removes material faster where thefields are stronger, the standard commercial electropolishing services reduce thesurface roughness by half. So electropolishing is a much less efficient method toprepare metal surfaces for high voltage. We have not attempted to condition ourITO coated plates, for fear of ablating too much of the coating. However, in aseparate test chamber, we have started to do just that, to see what types of surface

39

0 2 4 6 8 10Time (s)

6

4

2

0

2

4

6

curr

ent

(nA

)

a)+13.0 kv

topbottom

2.0 2.5 3.0 3.5 4.0Time (s)

0.6

0.5

0.4

0.3

0.2

0.1

0.0

0.1

curr

ent

(nA

)

b)

Figure 4.6. Power supply set at positive 13.0 kV. (a) Complete trace including chargingand discharging. (b) Zoomed-in view of the data in the time interval after chargingbefore discharging. The data points are connected by dashed lines to guide the eyes.

preparation might be consistent with a mixture of metal parts and ITO coatedglass.

4.4 Interferometric Measurements of the Plate Sep-arationsPrecision measurements of the plate separations are needed in order to preciselydetermine the electric fields between the plates from an applied voltage and tounderstand the forces exerted by the electric field gradients on the atoms. Theprecise knowledge of the electric fields is essential to the precision measurement ofthe tensor polarizability of the atoms and the force on the atoms is an importantpart of the analysis on the systematic errors in the measurement of the EDM. The

40

0 2000 4000 6000 8000 10000Time (ms)

0.20

0.15

0.10

0.05

0.00

0.05

0.10

0.15

curr

ent

(nA

)

a)

0 10 20 30 40 50 60 70frequency (Hz)

0.00

0.01

0.02

0.03

0.04

0.05

0.06

am

plit

ude (

nA

)

b)

Figure 4.7. The currents acquired at a rate of 1000 samples/s after the high voltage hasbeen held on for 3 mins. (a) The currents in the time domain show that the oscillationdoes not decay over the 10 s of measurement. (b) Frequency spectrum, obtained bytaking the Fast Fourier transform of the data in (a), shows frequency components inaddition to the 60 Hz noise. The most dominant frequency other than 60 Hz is 18.3± 0.1Hz. This frequency is the same when the applied voltage is at negative 14 kV and atnegative 15.1 kV.

plates separations are measured using white light interferometry.

4.4.1 Principle of White Light Interferometry

The core of our setup for white light interferometry is a Michelson interferometer(Figure 4.9). A beam of light is split into two by a beam splitter. Each of them isretro-reflected by a mirror. They recombine after the beam splitter. The intensityof the combined beam is detected using a photo-diode.

Figure 4.10 illustrates the signals detected by a photo-diode as one scans theposition of one of the 2 mirrors, say M2, for both a monochromatic light source anda broadband light source. When the light is monochromatic, the interference fringe

41

200

150

100

50

0

50

Aver

age

curre

nt (p

A)

(a)

topbottom

15 10 5 0 5 10 15Voltage (kV)

1.3

1.4

1.5

1.6

Pres

sure

(×10

10 to

rr)

(b)

Figure 4.8. (a) The average current over 10 s after the charging current has decayed vsthe applied high voltage. (b) The vacuum pressure read by an ion gauge located at thetop 6-way cross after the high voltage has been applied for over 3 mins. The pressuresat voltages between 0 and -12 kV are not recorded but can be assumed to be around1.3× 10−10 torr. Some of the measurements are repeated on another day.

is observable over a long scan range (Figure 4.10a). The fringe spacing is half of thewavelength of the light. When the 2 arms of the interferometer have the same lightpath (l1− l2 = 0), the signal detected by the photodiode depends on the differentialphase shift induced by the beam splitter on the reflected and transmitted beams. Ifthe beam splitter is a piece of uncoated glass, the reflected beam incurs a π phaseshift when it is incident from the air while the transmitted beam incurs no phaseshift. Because of this differential phase shift, the light from the 2 paths, l1 and l2,completely destructively interferes at the equal-arm condition. As one varies thewavelength of the light (Figure 4.10b), the fringe spacing varies but the fringe isalways dark when l1 − l2 = 0. If the light source emits a spectrum of wavelengths(Figure 4.10c), the dark fringes of individual wavelengths constructively form a darkfringe at the equal-arm condition. As the path lengths deviate from the equal-arm

42

Figure 4.9. Typical setup of a Michelson interferometer

condition, the fringes of different wavelengths lose coherence with respect to eachother, resulting in reduced contrast. When the light path difference is more thanthe coherence length of the light source, the contrast is lost and no interferencecan be observed. If the frequency spectrum of the light source follows a Gaussiandistribution, the interference signal is modulated by a Gaussian envelope, centeredat the equal-arm location. In our setup, we use a broadband laser at 830 nm with50 nm line-width. That implies that the width of the Gaussian envelope is about14 µm.

Our discussion so far has assumed that the beam splitter is an uncoated pieceof glass, which splits the light unevenly. The beam splitter we actually use in thelab is coated with 50% reflection coating on one surface and anti-reflection coatingon the other. These coated surfaces alter the differential phase shift, so at theequal-arm condition, the fringe does not have to be dark. Nonetheless, the fringecontrast is still the best at the equal-arm condition.

The equal-arm condition, however, may not be satisfied at the same mirrorpositions for all wavelengths if there is a differential dispersion in the paths of l1and l2. This is the case in the setup of Figure 4.9: the path that undertakes l2passes through the beam splitter twice, whereas the path that undertakes l1 doesnot pass through the beam splitter at all. The light that passes through the beamsplitter experiences dispersion. To compensate the light path through the beamsplitter, a glass plate made of the material (BK7) of the beam splitter is added tothe path of l1 so that l1 − l2 = 0 can hold true for all wavelengths.

To use the Michelson interferometer to measure a plate separation involvesapplying the equal-arm condition 3 times or less. We take the Michelson inter-ferometer shown in Figure 4.9, place it close to the plates and direct the beam

43

a)

b)

3 2 1 0 1 2 3Path difference l1−l2 (µm)

c)

Figure 4.10. The calculated signal of the photo-diode of a Michelson interferometerassuming the beam splitter is an uncoated piece of glass. (a)Interference fringes of amonochromatic source. (b)Interference fringes of monochromatic sources of 3 differentcolors. (c)Interference fringe of a broadband laser

from the laser perpendicular to the plate surfaces, as shown in Figure 4.11, usinganother beam splitter and some auxiliary mirrors. The laser light is retro-reflectedby the center plate to the Michelson interferometer. We begin by measuring theseparations on the right (dAB) and repeat the measurement for the other side.When the light path from surface A to M1 (lA1) is the same as the path fromsurface A to M2 (lA2), two equal-arm conditions are satisfied (lA1 − lA2 = 0 andalso lB1 − lB2 = 0) and we obtain an envelope of interference fringes as we scanthe location of M2 within the coherence range around the equal-arm location. Werecord the location of M2 that yields the maximum contrast and label it with“d0”. When M2 is displaced from d0 to the right by the plate separation (about4mm), another equal-arm condition is satisfied. That is lB1 − lA2 = 0. We labelthis location of M2 with “d4”. Thereby, the separation between the plates (dAB)is mapped to the displacement of M2 (d4− d0), which can be measured withoutcontacting the plates. Similarly, there is a 3rd equal-arm condition when M2 is

44

displaced from d0 to the left by the plate separation. We label this location of M2with “d4′”. We can deduce dAB from either d4− d0 or d4′ − d0. We have chosen touse d4− d0.

Figure 4.11. Measurement of plate separations using a Michelson interferometer

One caveat is that the phase of the “d4” fringe may differ from that of “d0”.In the case of “d0”, The 2 arms of the interferometer do not incur additionaldifferential phase shifts other than those induced by the beam splitter. On theother hand, in the case of “d4”, one of the 2 arms is reflected by the high-reflectioncoating on the center plate while the other arm is reflected by the anti-reflectioncoating on the outer plate (see Figure 4.11). These surfaces induce different phaseshifts in addition to those induced by the beam splitter. Because the “d4” and“d0” interference fringes have different phases, the separation between the locationsof maximum contrasts is not the same as the separation between the locations ofeither the brightest or the darkest fringes.

4.4.2 Measurement of the Mirror Displacement

The mirror displacement (d4− d0) in white light interferometry can be measuredusing a variety of methods. One method is counting the number of fringes ofthe Michelson interferometer with a monochromatic source while the mirror isbeing scanned. This method requires combining a broadband light source with amonochromatic light source at a distinct wavelength, sending them both to theMichelson interferometer and separating them for detection using a splitter andfilters.

Alternatively, the mirror displacement can be measured using the encoder clicks

45

sent out by the translation stage (Physik Instrumente M-110.1DG) on which themirror (M2) is mounted. Measurement of the mirror displacement using the encoderclicks proceeds as follows. We start the translation stage 10 microns before “d0”and move it at 0.04 mm/s towards “d4”. During the movement, the translationstage sends out encoder clicks every 103 nm. The encoder clicks are counted by adigital counter. Both the interference fringes and encoder clicks are recorded by anoscilloscope during the beginning 1 second and the ending 1 second of the scan. Theencoder clicks are broadened by a digital delay generator before being sent to anoscilloscope because of the oscilloscope’s limited bandwidth. From the oscilloscopetraces, we find out the maximum contrast of the white light interference fringes andthe corresponding encoder click. We subtract from the total count of encoder clicksthe number of clicks before the “d0” maximum contrast and the number of clicksafter the “d4” maximum contrast. This way, we obtain the mirror displacementin terms of the number of encoder clicks between “d0” and “d4”. This method formeasuring of the mirror displacement requires calibrating the spacing between theencoder clicks and making sure that the measurement is reproducible.

We carry out calibration on the encoder clicks to account for the error inthem and the angle between the translation axis and light propagation direction.The calibration uses the same Michelson interferometer, with the broadband laserreplaced by a narrow-band laser (1 MHz line-width at 852.347 nm). We scan thetranslation stage through the same range from “d0” to “d4”. The interferencefringes and the encoder clicks are counted by separate digital counters and theyare both recorded by an oscilloscope during the beginning 1 second and the ending1 second of the scan. From the oscilloscope traces, we find out the encoder clickscorresponding to the first and the last interference fringe and obtain the numberof fringes in terms of the number of encoder clicks. Since the wavelength andhence the fringe spacing are well known, we know the precise distance between theencoder clicks.

4.4.3 Differential Measurement of Plate Separation

Our goal is to map the plate separations across an area of 10 cm by 1 cm nearwhere the atoms are trapped. Having completed the measurement of one separation,we translate the interferometer transverse to the plates so that the beam hits on

46

a different location on the plates. We could repeat the measurement describedin section 4.4.2. But to the extent that the interferometer is stable and that theencoder clicks are reproducible, the “d0” fringe stays the same with respect tothe encoder clicks even though the separation may change. Therefore, the onlymeasurement worth repeating is the “d4” measurement. The “d4” measurement ata shifted location can be simplified into the comparison of the phase of the “d4”interference fringes with that of a reference location, for which the plate separationhas been measured using the method described in section 4.4.2. The comparison ofphases, which constitutes the differential measurement, is ambiguous to an integermultiple of 2π. But because the gradient of the plate separation is small, as longas we translate the interferometer in small steps, we can assume the phase shiftbetween adjacent locations is smaller than π and thereby remove the 2π phaseambiguity.

Specifically, we translate the interferometer in steps of 2 mm. In each step, wescan the mirror near the maximum contrast of the “d4” interference fringe over therange of 10 µm (Because of the backlash of the translation stage, the usable range isnarrower but wide enough to yield several interference fringes). We remove the DCoffset of the light intensity incident on the photo-diode by acquiring data using theAC mode of the oscilloscope. We record the positions of the first 2 zero-crossingsin the oscilloscope traces. We typically use only the first zero-crossing but we needthe second one in case that the first one shifts out of the scan range. We recordthe positions in nm converted from the number of encoder clicks.

4.4.4 Results

Setting the origin of our coordinate system to be the center of the plates. Thelocation of our absolute measurement has x and y coordinates of (+5 mm, +50mm). Measurements at all other locations are referenced to this location. Thelocations where we sample the plate separations are mapped in Figure 4.12a. Andthe measurement results are plotted in Figure 4.12b and c. These measurementshave an uncertainty of ±0.4 µm limited by our ability to locate the maximumcontrasts of the interference fringes. Across a hundred millimeter on the plate, theplate separation differs by about 4 µm. The plates are at an angle of 40 µ-radianfrom being perfectly parallel.

47

5 0 5

40

20

0

20

40

x (mm)

y (mm)(a)

40 20 0 20 40y (mm)

4005

4006

4007

4008Se

para

tion

(m

)(b) x=+5 mm

x= 0 mmx=-5 mm

4 2 0 2 4x (mm)

4005

4006

4007

4008

Sepa

ratio

n (

m)

(c)

Figure 4.12. Plate separation measurements on the −z side. (a) The locations wherewe sample the plate separations with the center of the plates being the origin. (b) Plateseparation versus height y. (c) Plate separation versus x.

48

Chapter 5 |Alternative Scheme of an EDMMeasurement

Because of the leakage currents through the vacuum described in the previouschapter, the strongest electric field we can create with ITO-coated glass plates is 37.5kV/cm. This not only reduces the EDM sensitivity but also makes it a challengeto prepare the superposition of stretched states mostly sensitive to the EDM.In this chapter, we propose an alternative measurement scheme that overcomesthe challenge of state preparation. The proposed scheme measures the angularmomentum precession from the m = 0 magnetic level. The m = 0 level is easy toprepare and it is sensitive to the EDM and magnetic fields. Though we developedthis scheme independently, the narrow line shape and good sensitivity obtainedby detecting m = 0 were demonstrated by Xu and Heinzen [1]. This scheme canalso be adapted to making precision measurements of the tensor polarizabilities.We begin by reviewing the original scheme, which prepares the superposition ofthe stretched states (states with |m| = F ) by driving atoms from the m = 0 stateusing a transverse magnetic field oscillating at audio frequencies.

5.1 The Scheme Using Audio Frequency TransitionsThis scheme is most elegant as long as the audio frequency transitions betweenthe magnetic levels are not limited by the energy separations. This scheme hasalso been described in Fang’s dissertation [8], Solmeyer’s dissertation [9] and Zhu’sdissertation [4].

Consider an atom in electric and magnetic fields. The tensor Stark shift induced

49

by the electric field on the magnetic levels of the F = 3 hyperfine level of theground state (62S1/2) of cesium is given by

EE(m) = −12α2

m2

5 E2, (5.1)

where m is the magnetic quantum number, E is the electric field, α2 is the tensorpolarizability of the F = 3 ground state of cesium and the zero of the energy hasbeen shifted such that E(m = 0) = 0. The Zeeman shift induced by a magneticfield, B, is given by

EB(m) = µBgFBm. (5.2)

The tensor Stark shift is quadratic in m whereas the Zeeman shift is linear in m.The vector light shift induced by the trapping light is also linear in m and henceis considered a fictitious magnetic field. With a large enough electric field andsmall magnetic fields, the same magnitude m levels are nearly degenerate, whilethe degeneracy among |m| pairs is lifted as in Figure 5.1. Atoms in the m = 0 state

Figure 5.1. Energy levels of the F = 3 hyperfine level of the ground state (62S1/2) ofcesium in a strong electric field.

can be driven coherently to the superposition of stretched states:

(|m = +3〉+ |m = −3〉)/√

2 ≡ |+〉 , (5.3)

using an oscillating transverse magnetic field (called a π pulse) that containsfrequency components corresponding to all the ∆m = 1 transitions (see Chapter 6of [4]). The degeneracy between |m = +3〉 and |m = −3〉 states can be lifted bythe interaction of the EDM, da, with the electric field. The energy splitting dueto the EDM is 23

4daE. After an evolution time τ , the state becomes (omitting anoverall phase factor):

|+〉 e−iHτ/~ = (|m = +3〉 e−i 34daEτ/~ + |m = −3〉 e+i 3

4daEτ/~)/√

2 (5.4)

50

The above state can be rewritten in the basis of |+〉 ≡ (|m = +3〉+ |m = −3〉)/√

2and |−〉 ≡ (|m = +3〉 − |m = −3〉)/

√2:

|+〉 e−iHτ/~ = |+〉 cos(34daEτ/~) + i |−〉 sin(3

4daEτ/~). (5.5)

A second π pulse will then take |+〉 back to |m = 0〉. The probability of an atomreturning to |m = 0〉 is

Pm=0 = cos2(34daEτ/~) ≡ cos2(3ωτ) (5.6)

≡ cos2(3φ) = 12[cos(6φ) + 1], (5.7)

where ω = 14daE/~ is the precession frequency induced by the EDM interaction

with the electric field and φ = ωτ . By measuring the fraction of atoms returningto the |m = 0〉 state, we can determine the EDM da.

Because the EDM, da, is linearly present in the phase φ, the EDM uncertaintyδd is linearly related to the phase uncertainty δφ:

δd = 4~Eτ

δφ. (5.8)

The shot-noise-limited phase uncertainty is the uncertainty of the probabilitydivided by the slope of the probability with respect to the precession phase, that is

δφ =√pm=0 − p2

m=0/

∣∣∣∣∣dpm=0

dφ

∣∣∣∣∣ = 1/6. (5.9)

Because both the probability uncertainty and the slope are sinusoidal with thesame frequency, the ratio between them is constant. The sensitivity is linearlyproportional to the energy separation of the basis states. The energy separation isgreatest when the angular momentum is in the superposition of stretched states, asin Equation (5.3).

We have so far only considered the sensitivity limited by quantum projection.In an experiment, we also have to deal with instrumental noise, which is the sameat all phases. When the quantum projection noise is small, the instrumental noisecan become the dominant source of noise. It is, therefore, best to operate the

51

measurement where the slope is largest. The slope of Pm=0 is

dPm=0

dφ= −3sin(6φ), (5.10)

which has a maximum of 3 occurring at φ = π/12 + nπ/6, where n is any integer.The probability of detecting an atom in |m = 0〉 when the slope is maximized is1/2.

The challenge of this measurement scheme is generating a strong electric fieldand controlling the unwanted vector light shift such that the tensor Stark shiftdominates the energy structure. Given vector light shifts as large as 13 Hz [28] anda tensor Stark shift of EE(1) = 5.6 Hz at 37 kV/cm, it is difficult to prepare thesuperposition of stretched states using an audio frequency transition.

5.2 The Alternative Scheme by Switching BasisMeasurement of the EDM requires a coherent superposition of magnetic sublevels.Coherent superpositions can be generated using a transition to degenerate levels.Coherent superpositions can also be obtained by choosing another basis. Aneigenstate in one basis is naturally a coherent superposition state on a differentbasis. Specifically, |m = 0〉x in the x basis can be represented in the z basis byapplying a passive Wigner rotation to the coordinate system. The representationin the z basis is:

|m = 0〉x =√

54 |m = +3〉z −

√3

4 |m = +1〉z

+√

34 |m = −1〉z −

√5

4 |m = −3〉z .(5.11)

The |m = 0〉x state, in the z basis, not only has amplitudes in |m = +3〉z and|m = −3〉z, but also amplitudes in |m = +1〉z and |m = −1〉z. All these states havesensitivity to the EDM as well as to magnetic fields. But the states |m = +3〉zand |m = −3〉z have 3 times the sensitivities of |m = +1〉z and |m = −1〉z. It is,therefore, desirable to have larger amplitudes in |m = +3〉z and |m = −3〉z. Ofall the 7 eigenstates in the x basis, |m = 0〉x has the strongest amplitudes in|m = +3〉z and |m = −3〉z. Therefore, |m = 0〉x is the most suitable eigenstate for

52

a measurement sensitive to the EDM or magnetic fields.To prepare atoms in the |m = 0〉x state, we can optically pump the atoms to a

stretched state of the F = 4 hyperfine level of the ground state and then transferthe atoms to |m = 0〉x in the F = 3 hyperfine level through a series of adiabaticfast passage (AFP) microwave pulses with high fidelity.

Measurement of the EDM starts by initializing atoms in |m = 0〉x. In thepresence of electric field and magnetic field in the z direction, the system evolvesby the Hamiltonian written in the z basis:

Hz = (µB + dE)F

m− 12α2

E2

5 m2 ≡ ~(ωLm− ωα2m2), (5.12)

where ωL captures the linear shifts of the magnetic levels arising from the magneticfield or the EDM and ωα2 describes the quadratic shifts due to the tensor Starkshift. After some evolution time of τ , the state becomes:

|m = 0〉xe−iHτ/~ = e−i(3ωL−9ωα2 )τ

√5

4 |m = 3〉z − e−i(ωL−ωα2 )τ

√3

4 |m = 1〉z

+ e−i(−ωL−ωα2 )τ√

34 |m = −1〉z − e

−i(−3ωL−9ωα2 )τ√

54 |m = −3〉z .

(5.13)

Transforming the above state back to the x basis through the inverse Wignerrotation, we obtain the state after evolution in the spinor representation in the xbasis:

0001000

x

e−iHτ/~ =

− 116i√

5 (3 sin (τωL)− e−8iτωα2 sin (3τωL))−1

8

√152 (cos (τωL)− e−8iτωα2 cos (3τωL))

116i√

3 (sin (τωL) + 5e−8iτωα2 sin (3τωL))18 (3 cos (τωL) + 5e−8iτωα2 cos (3τωL))

116i√

3 (sin (τωL) + 5e−8iτωα2 sin (3τωL))−1

8

√152 (cos (τωL)− e−8iτωα2 cos (3τωL))

− 116i√

5 (3 sin (τωL)− e−8iτωα2 sin (3τωL))

x

. (5.14)

Taking the modulus square of the above expression, we find the probabilities of

53

detecting an atom in individual magnetic levels:

p(E,B) =

45256 sin2 (τωL)− 15

128 cos (8τωα2) sin (3τωL) sin (τωL) + 5256 sin2 (3τωL)

15128 cos2 (τωL)− 15

64 cos (3τωL) cos (8τωα2) cos (τωL) + 15128 cos2 (3τωL)

3256 sin2 (τωL) + 15

128 cos (8τωα2) sin (3τωL) sin (τωL) + 75256 sin2 (3τωL)

964 cos2 (τωL) + 15

32 cos (3τωL) cos (8τωα2) cos (τωL) + 2564 cos2 (3τωL)

3256 sin2 (τωL) + 15

128 cos (8τωα2) sin (3τωL) sin (τωL) + 75256 sin2 (3τωL)

15128 cos2 (τωL)− 15

64 cos (3τωL) cos (8τωα2) cos (τωL) + 15128 cos2 (3τωL)

45256 sin2 (τωL)− 15

128 cos (8τωα2) sin (3τωL) sin (τωL) + 5256 sin2 (3τωL)

.

(5.15)The evolution in electric and magnetic fields depends on both the tensor Stark shiftand the Zeeman shift or the shift induced by an EDM interacting with the electricfield. We would like to separate the tensor shift from the linear shift. We will keepthe tensor shift and discard the linear shift in Section 5.6, where we discuss theapplication of the above result to the measurements of the tensor polarizabilities.In this section, our purpose is to use the above result to measure the EDM ormagnetic fields, so we keep the linear shift and discard the tensor shift. We can getrid of the tensor Stark shift in Equation (5.15) using 2 strategies:

1. We can fix the electric field E and the precession time τ . Thereby, the tensorStark shift is a constant and the measurement varies with only the magneticfield B and the direction of the electric field. In particular, we can pick Eand τ such that phase accumulated due to the tensor Stark shift τωα2 is aninteger multiple of 2π. Then the system behaves as if there was no tensorStark shift at all.

2. The probability in the even (or the odd) magnetic sublevels is independent ofthe tensor Stark shift induced by an electric field in the orthogonal direction.This can be verified for F = 3 using Equation (5.15) and is generalized toany F as shown in Appendix A.

We consider the first strategy first. The probabilities of detecting individualmagnetic levels after setting τωα2 to integer multiple of 2π are plotted in Figure 5.3.All these probabilities are sensitive to the EDM through their dependence on ωL.Of particular interest is the probability of detecting an atom in the m = 0 state:

Pm=0x = 164[3cos(ωLτ) + 5cos(3ωLτ)]2, (5.16)

54

which is plotted in Figure 5.2 along with the fringe obtained by using the superposi-tion of stretched states. Compared with the fringes of the superposition of stretched

0.0 0.2 0.4 0.6 0.8 1.0ω (2π/τ)

0.0

0.2

0.4

0.6

0.8

1.0Pro

babili

ty

(|+3〉z + |−3〉z )/√

2

|m=0〉x

Figure 5.2. Populations versus Larmor frequency. Red curve represents atoms in(|+3〉z + |−3〉z)/

√2 state, prepared and detected using audio transition, as in the original

scheme (Equation (5.7)). Blue curve represents atoms in |0〉x state, as in the alternativescheme (Equation (5.16)).

states (Equation (5.7)), the fringes of m = 0 (Equation (5.16)) depend on not onlythe 6th harmonic of the Larmor frequency but also its 2nd and 4th harmonics. Itsnarrow line shape and sensitivity close to that obtained using the superpositionof stretched states was demonstrated by Xu and Heinzen [1] through the activerotation of the state. I have reached the same conclusion using the passive rotationof the axes, which is suitable for our experimental setup. Like any magnetic levels,the m = 0 state can be probed in our experimental setup using the optical cyclingtransition after the population in |m = 0〉 is transferred to the F = 4 hyperfinelevel of the ground state. The slope of the probability with respect to the phaseφ = τωL is:

dPm=0

dφ= −3

64 (13 sin(2φ) + 20 sin(4φ) + 25 sin(6φ)), (5.17)

which has maxima of 2.36 at φ = ±0.31 + nπ, where n is any integer.Xu and Heinzen [1] considered the sensitivity obtained by measuring the m = 0

state only. In the following section, we discuss the sensitivities obtained by detectingindividual magnetic levels

55

5.2.1 The Quantum Projection Uncertainties

The uncertainties of φ determined from the quantum projection into individualmagnetic levels are

δφm =√〈P 2

m〉 − 〈Pm〉2/

∣∣∣∣∣d 〈Pm〉dφ

∣∣∣∣∣ , (5.18)

where Pm = |m〉 〈m| are the projection operators for the individual magnetic levelsand 〈Pm〉 are the probabilities of detecting atoms in |m〉. Using Equation (5.15)for 〈Pm〉, the inverses of phase uncertainties, 1/δφm, are plotted in Figure 5.3.

0.0

0.2

0.4

0.6

0.8

1.0

p m

(a)

0 14

12

34

(rad)

0

2

4

6

1/m

(b)

±3±2±10

Figure 5.3. Precession from the state |3, 0〉. Both sub-figures are periodic with respectto the phase with a minimum period of π. (a) The probabilities to be in the differentmagnetic sublevels m as a function of phase. Because the probability to be in m isidentical to the probability to be in −m, the probabilities to be in m and −m are summedin this figure. (b) The inverse of the uncertainties obtained by detecting individualmagnetic sublevels as a function of phase. The dashed line is the inverse of the combineduncertainty obtained by independent measurements of all magnetic sublevels. The dottedline is the inverse of the combined uncertainty obtained in precession from a stretchedstate.

Out of all the eigenstates of Fx, the |m = 0〉 level is most sensitive. In particular,The smallest phase uncertainty obtained by detecting the probability to be in thestate |3, 0〉 is half of what is obtained by Larmor precession and 22.5% more thanwhat is obtained with the superposition of stretched states. A similar result for

56

F = 4 was first shown in [1]. This sensitivity is available only around a phase of 0or π, where the slope dp0/dφ is close to 0. To make a precision measurement ofthe magnetic field or the EDM, it is often desirable to scan the phase (either byscanning the bias magnetic field or the precession time) over a larger range. Thisgoal can be realized by combining measurements of all magnetic sublevels. Weshow next that the combined sensitivity of measurement of all magnetic sublevelsat all precession phase is the same as the best sensitivity obtained by detecting|m = 0〉.

The sensitivities obtained by detecting individual sublevels can be combined togive a net sensitivity better than that obtained from any single magnetic sublevel.In the absence of correlations, sensitivities from independent measurements, δφi,can be combined as follows,

1δφ2

c

=∑i

1δφ2

i

, (5.19)

where δφc is the combined uncertainty. However, the uncertainties δφm as givenby Eq. (5.18) and plotted in Fig 5.3 are correlated with each other. We can takeinto account the correlations by analyzing the measurement of all the magneticsublevels as a sequence of measurements. If the atom is measured to be in a givensublevel, the measurement is done. If it is not, the probabilities are shared amongthe remaining sublevels, and the next measurement is independent of the priorones. The sensitivities of a sequence of measurements can then be combined usingEq. (5.19). Of course, one expects the same result regardless of the order in whichthis imagined sequence of measurements is made.

We calculate the full sensitivity obtained by detecting all magnetic sublevels bycalculating the result for a sequence of 2F + 1 measurements as follows. Considera normalized state after precession of phase φ:

|ψ〉 =F∑

m=−Fam(φ) |m〉 , (5.20)

where am(φ) are the amplitudes of the state |m〉. Suppose we detect the atom in|m = F 〉 as the first in the sequence of measurements. The phase uncertainty of

57

the first measurement follows directly from Eq. (5.18) and can be written as:

δφ1 =

√pm=F (φ)− pm=F (φ)2

|pm=F |, (5.21)

where pm=F (φ) = |am=F (φ)|2 is the probability to be in m = F and pm=F =dpm=F (φ)/dφ is the slope. The remaining probability, 1/(1− pm=F (φ)), is in theundetected states. The new state is:

|ψ′〉 = 1√1− pm=F (φ)

F−1∑m=−F

am(φ) |m〉 (5.22)

≡ a′m(φ) |m〉 , (5.23)

where the prefactor normalizes the collapsed state and the primed amplitudesa′m(φ) = am(φ)/

√1− pm=F (φ) are the amplitudes after normalization. Then we

carry out the next measurement, say on |m = F − 1〉. The phase uncertaintyobtained by measuring |m = F − 1〉 from the collapsed state is:

δφ2 = 1√1− pm=F (φ)

√p′F−1(φ)− p′F−1(φ)2

|p′F−1|, (5.24)

where√

1− pm=F (φ) accounts for the probability of any of these outcomes occurring,p′F−1 = |aF−1(φ)|2/(1− pF (φ)) is the probability of measuring m = F − 1 from thecollapsed state |ψ′〉 and p′F−1 = dp′m=F−1(φ)/dφ is the slope. In this expressionfor δφ2, the previous measurement has altered both the slope and the projectionuncertainty. We repeat the above procedure until all the remaining probability isin a single magnetic sublevel, which will at last be detected with 100% probability,and therefore offers no measurement sensitivity. The sensitivities of the sequenceof measurements can be combined using Eq. (5.19).

We apply the above method to analyzing the full sensitivity obtainable inprecession from |3, 0〉. We can separate the measurement into a sequence of7 measurements. But we might as well condense the sequence into 4 separatemeasurements by combining the measurement of |+m〉 with that of |−m〉 becausethe state precessed from |m = 0〉x has the same amplitude in |+m〉 and |−m〉.Suppose we first measure |±3〉, followed by |±2〉, then |±1〉 and at last |0〉. The

58

probability of finding the atoms in a given magnetic level after previous nullmeasurements and the inverse of the associated phase uncertainty are plottedin Fig 5.4. Compared with the results in Fig 5.3, the state collapses alter allmeasurements except the first. The combined uncertainty δφc from this sequenceof measurements is 1/

√6 at all precession phases, the same value as the best

uncertainty obtained by measuring |m = 0〉 alone. Even though we have picked aparticular sequence for illustration, the combined sensitivity is the same for anysequence. In the case of precession from |3, 0〉x, the combined uncertainty froma sequence of 4 measurements as specified above is the same as the combineduncertainty from a sequence of 7 measurements.

The above sequential analysis, however, does not invalidate the earlier resultspresented in Figure 5.3, which shows the probability of detecting an individualmagnetic level out of the state immediately after precession, |ψ(φ)〉, and theassociated phase sensitivity. The same results are produced in the sequentialmeasurements when we multiply the probability of detecting a magnetic level froma collapsed state by the probability for |ψ(φ)〉 to be projected into the collapsedstate.

5.2.2 Comparison with Larmor Precession

Though we have argued that the superposition of stretched states is most sensitiveand the sensitivity obtained by precession from m = 0 is close to that obtainedby precession from the superposition of stretched states, we show in this sectionexplicitly that the typical Larmor precession is less sensitive on a single atom level.Larmor precession can be used to measure the EDM in much the same way it isused to measure magnetic fields [29]. A typical Larmor precession measurementdetects the expectation value of some component of the angular momentum 〈F〉. Ithas maximum contrast when atoms are initialized in a stretched state. We considerprecession from a stretched state, say |ψ〉 = |F,m = F 〉x, for an atom with angularmomentum F > 1. The expectation value of the angular momentum precesses in amagnetic field, Bz, at the Larmor frequency, ωL = gFµBB/~. ωL is proportionalto the separation between adjacent magnetic sublevels, which for large F is muchsmaller than the largest energy difference in the problem, which is the separationbetween stretched states. The shot-noise-limited phase or frequency uncertainty

59

0.0

0.2

0.4

0.6

0.8

1.0Pr

obab

ility

(a)1st 2nd 3rd

0 14

12

34

0

1

2

3

4

5

6

1/

(b)

0 14

12

34

Precession angle (rad)0 1

412

34

Figure 5.4. Sequential analysis of individual sublevel measurements after precessionfrom the state |3, 0〉. We measure the magnetic levels in a decreasing sequence of |m|starting from |m| = 3. The upper row shows the probabilities of finding the atomin a given magnetic sublevel after it was not found in previous measurements. Thelower row shows the inverse of the corresponding phase uncertainties. The dotted linerepresents the sensitivity of typical Larmor precession. And the solid flat line representsthe combined uncertainty of this sequence of measurements. The first column is for thefirst measurement, which is the same as the corresponding color shown in Figure 5.3.The second and third columns, for the second and third measurements respectively, differfrom the corresponding colors shown in Figure 5.3. Not shown in this figure is the fourthmeasurement, which yields 100% probability and zero sensitivity.

obtained by detecting 〈Fx〉 scales as 1/√

2F . The question naturally arises whetherone can improve the sensitivity of a Larmor precession measurement by using theability to measure the populations of individual sublevels. The probabilities ofdetecting atoms in individual magnetic levels are pm = 〈ψ′|m〉x 〈m|x |ψ′〉, where|ψ′〉 = e−iHt/~ |ψ〉 is the state after precession and |m〉x are the eigenstates of Fx. Toevaluate pm, we write the initial state |ψ〉 in the z basis, in which the HamiltonianH = ~ωFz is diagonal, using the Wigner rotation matrix, |m〉x = Dy(π/2) |m〉z,

60

the pm are given by:

pm =∣∣∣∣∣ ∑m′m′′

d†mm′(π/2)dm′m′′(π/2) 〈m′′|x |ψ〉 e−im′φ

∣∣∣∣∣2

, (5.25)

where dm′m′′(π/2) are the Wigner rotation matrix elements, and φ = ωτ is thephase accumulated over the precession time of τ . We will use F = 3 as an example,where the initial state |ψ〉 is |3, 3〉x. The probabilities of detecting each magneticsublevel after precession are plotted in Fig. 5.5a. The phase uncertainties obtained

0.0

0.2

0.4

0.6

0.8

1.0

p m

(a)

0 12

32

2 (rad)

0

2

4

6

1/m

(b)

3210-1-2-3

Figure 5.5. Precession from the state |3, 3〉. (a) Probabilities to be in individualmagnetic sublevels m (color coded) as a function of phase. (b) The inverse of theuncertainties obtained with individual magnetic sublevels as a function of phase. Thedotted line is the inverse of the uncertainty obtained by measuring the projection 〈Fx〉.The upper bound of this figure (1/δφm = 6) is the inverse of the uncertainty of thesuperposition of stretched states.

by these measurements are given by the quantum projection noises divided by theslopes of the signals with respect to the phase:

δφm =√〈P 2

m〉 − 〈Pm〉2/

∣∣∣∣∣dpmdφ∣∣∣∣∣ , (5.26)

61

where Pm = |m〉 〈m| are the projection operators for the individual magneticsublevels. The inverse of phase uncertainties, 1/δφm, are plotted in Fig. 5.5b.For comparison, we also calculate the phase uncertainty obtained by measuring〈Fx〉 = ∑

m pmm. The phase uncertainty obtained by detecting 〈Fx〉 is:

δφ〈Fx〉 =√〈F 2

x 〉 − 〈Fx〉2/

∣∣∣∣∣d 〈Fx〉dφ

∣∣∣∣∣ = 1/√

2F , (5.27)

as indicated by the dotted line in Fig. 5.5b. It is evident from Fig. 5.5b that thesensitivity obtained by detecting any single magnetic sublevel is never any betterthan the sensitivity obtained by detecting 〈Fx〉.

Even when combining the sensitivities using the method introduced in theprevious section, the full sensitivity from all magnetic levels is identical to thesensitivity obtained by measuring the expectation value, i.e. 1/

√2F . So the

measurement of the expectation value 〈Fx〉 is the most sensitive measurement intypical Larmor precession, but less sensitive compared with precession from them = 0 state or the superposition of stretched states.

5.2.3 Probability in Even Magnetic Sublevels

We have shown that by combining measurements of individual magnetic sublevelswe can extract the full sensitivity of precession. In this subsection and the next, weintroduce two ways of combining individual magnetic sublevel measurements toyield a single potentially useful fringe.

The sensitivity obtained by taking the sum of the probabilities to be in the evenmagnetic sublevels is shown in Fig. 5.6. The best sensitivity is comparable to theresult for |3, 0〉. Using this combination of individual measurements seems simplerthan keeping separate track of the individual sublevel evolutions, and though thefringe shape is not as simple as a sinusoid, it is not very complicated, containingjust two Fourier components. More importantly, these fringes are unaffected bythe tensor Stark shift, the second order Zeeman shift and any interaction thatis even with respect to m in the orthogonal direction, even though the evolutionof any individual magnetic sublevel does depend on these shifts. Proof of thisinsensitivity is given in Appendix A. The insensitivity to quadratic shifts requiresthe measurement axis to be orthogonal to the fields that give rise to the quadratic

62

0.0

0.2

0.4

0.6

0.8

1.0

Prob

abilit

y

(a)

0 14

12

34

(rad)

0

2

4

6

1/

(b)

Figure 5.6. (a) Measurement of the sum of the population in the even magneticsublevels during precession from the state |3, 0〉. (b) The inverse of the uncertainty fromthis measurement (solid line). The best sensitivity of the even magnetic sublevels is thesame as the best sensitivity of the m = 0 state shown in Fig. 5.3. The dashed line isthe inverse of the combined uncertainty obtained by independent measurements of allmagnetic sublevels. The dotted line is the inverse of the uncertainty obtained in typicalLarmor precession.

energy shift. Of course, since the probabilities to be in even and odd magneticsublevels sum up to 1, we could as well have used the odd magnetic sublevels forthis discussion.

In order to get a physical sense for why keeping track of the even (or odd)populations gives heightened sensitivity, we can visualize the precession of them = 0 state using spherical harmonics. In Fig. 5.7, we show the precession of aninitial |3, 0〉x state at the precession phases corresponding to the first 4 extrema inFig. 5.6, and compare them to the other spherical harmonics in the x-basis. As thestate precesses, the plane of maximum amplitude in the rotating state overlaps inturn with the lobes of the various basis states. Of course, the axis of symmetryof the precessing state rotates, so the precessing state never quite looks like thecomparison states. Still, the results of the rigorous calculation presented in Fig. 5.6

63

0°

45°

90°

135°

180°

0123

0°

45°

90°

135°

180°

0°

45°

90°

135°

180°

0°

45°

90°

135°

180°

Figure 5.7. The precession of spherical harmonics. The spherical harmonic of the statethat precesses starting from |3, 0〉x is shown by the solid black line and its symmetry axisis represented by the grey arrow. The spherical harmonics of the eigenstates of Fx areshown with dashed lines (blue:|3, 0〉x, red:|3,±1〉x), cyan:|3,±2〉x, magenta:|3,±3〉x. Eachsub-figure shows the unprecessed states |3,m〉x along with a snapshot of the precessingstate at a phase corresponding to an extremum in Fig. 5.6. The extrema correspond topoints of relatively good overlap with successive |m| values.

are made graphically clear in these pictures. The precessed state has considerableoverlap in turn with the |3,±1〉 states, the |3,±2〉 states, and then the |3,±3〉states, accounting for the high-frequency component of the fringe in Fig. 5.6.

64

5.2.4 Single Harmonic from a Linear Combination of MagneticSublevels

0.5

0.0

0.5

1.0Pr

obab

ility

(a)

0 14

12

34

(rad)

0

2

4

6

1/

(b)

Figure 5.8. (a) Measurement of the 6th harmonic in precession starting from the state|3, 0〉. (b) The inverse of the uncertainty from this measurement (solid line). The dashedline is the inverse of the combined uncertainty obtained by independent measurementsof all magnetic sublevels. The dotted line is the inverse of the uncertainty obtained intypical Larmor precession.

Another notable combination of individual magnetic sublevel evolution is theone that yields the 2F th order polarization moment. The probability curves in thevarious m levels are generally composed of the sum of the 0th to the 2F th harmonicsof the Larmor frequency. We can single out the highest order harmonic usinga linear combination of the probabilities to be in various m levels with differentweights, αm. In the case of F = 3, the measurement operator that yields thehexacontatetrapole [30] is

P2F =∑m

αm |m〉 〈m| = |0〉 〈0| −87(|1〉 〈1|+ |−1〉 〈−1|) + 11

7 (|2〉 〈2|+ |−2〉 〈−2|)

(5.28)

65

5.2.5 Extension to Higher Integer and Half Integer F

The above results can be extended to higher integer angular momenta in a straight-forward manner. When precessing from the |F, 0〉 state, the shot-noise-limited phaseuncertainty scales as 1/

√2F (F + 1). This is 1/

√F + 1 times what is obtained in

typical Larmor precession starting from a stretched state and√

2F/(F + 1) timeswhat is obtained with an optimal measurement, as illustrated in Fig. 5.9.

1 2 3 4 5 6 7 8 9 10Atomic angular momentum F

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Unce

rtain

ty

|F, 0 x

|F, 1/2 x

(2F) 1

Larmor

Figure 5.9. Shot-noise-limited phase uncertainty scaling with F . Combined uncertaintiesobtained by independent measurements of all magnetic sublevels in precession from thestate |F, 0〉 for integer F (red dots) or |F, 1/2〉 for half integer F (blue dots) are comparedwith the uncertainties obtained with the superposition of stretched states (dashed line)and the uncertainties obtained in typical Larmor precession (dotted line).

The reason that precession from the state |F, 0〉x is more sensitive than precessionfrom |F, F 〉x lies in the fact that the state |F, 0〉x in the z basis has the strongestamplitudes in the stretched states among all eigenstates of Fx. The most sensitivepossible measurement involves creating a superposition of the m = ±F levels,(|+F 〉+ |−F 〉)/

√2, for which the shot-noise-limited uncertainty scales as (2F )−1.

We will discuss preparation of this superposition below. In general, it is lessstraightforward than preparing the eigenstate |F, 0〉x. As can be seen from Fig. 5.9,there is not much inherent sensitivity loss associated with using the more simplyprepared state.

To extend these calculations to half integer angular momenta we prepare atomsin |F, 1/2〉 (or equivalently |F,−1/2〉). The precession of the state |F, 1/2〉 is

66

somewhat qualitatively similar to that of |2F, 0〉, as illustrated by the precessionof |3/2, 1/2〉 shown in Fig. 5.10 compared to the |3, 0〉 curve in Fig. 5.3. Thebest shot-noise-limited uncertainty obtained by using the state |F, 1/2〉 scales as1/√

2F (F + 1− 1/(4F )). Interestingly, these sensitivities neatly interleave theresults for |2F, 0〉 evolution in integer F systems (see Fig. 5.9).

0.0

0.2

0.4

0.6

0.8

1.0

Prob

abilit

y

(a)

0 12

32

2 (rad)

0

1

2

3

1/

(b)

Figure 5.10. (a) The probability of measuring the state |3/2, 1/2〉x in precessionfrom |3/2, 1/2〉x. (b) The inverse of the uncertainty from this measurement (solid line).The dashed line is the inverse of the combined uncertainty obtained by independentmeasurements of all magnetic sublevels. The dotted line is the inverse of the uncertaintyobtained by measuring the expectation value 〈Fx〉 in precession from a stretched state.

Atoms with large magnetic moments are inherently sensitive to magnetic fields,but Larmor precession does not take full advantage. For example, the shot-noise-limited uncertainty from Larmor precession of Dy, where F = 21/2 in the groundstate, is 4.58 times what can be obtained with the superposition of stretched states.If Dy is prepared in the m = 1/2 state instead, the smallest uncertainty obtainedby measuring m = 1/2 alone or by measuring the evolution of all sublevels is onlya factor of 1.35 away from the optimal uncertainty.

67

5.3 Effect of a Transverse Field in Measurement ofMagnetic Field

B

φ

β γ

(b)(a)

x'

x

x'

x

Bz z

γφ

Figure 5.11. Schematic of angular momentum precession. (a) Angular momentuminitialized in one of the eigenstates of Fx precesses in a magnetic field along z. (b)Angular momentum initialized in one of the eigenstates of Fx precesses in a magneticfield at an angle γ from the z axis . A magnetic level in the x basis evolves into thecorresponding level in the x′ basis, where x′ is related to x by a rotation of φ around B.

The discussion so far has been limited to the ideal scenario (Fig. 5.11a) whereatoms are initialized in one of the magnetic levels in the x basis and the magneticfield B is along a direction perpendicular to x, say z. In general, the magneticfield will not be perfectly aligned with the z axis but at angle γ from the z axisas shown in Fig. 5.11b. The transverse component of the magnetic field is givenby B⊥ = Bsin(γ). A magnetic level in the x basis evolves into the correspondinglevel in the x′ basis, where x′ is related to x by a rotation of φ around B. Themeasurement on the evolved state in the original basis of x is solely determinedby the angle β between x and x′. The angle β differs from the precession phase φwhen γ is nonzero and is related to φ through the following equation:

sin(β/2) = sin(φ/2)cos(γ). (5.29)

Although φ ranges from 0 to 2π, β does not have values between π− 2γ and π+ 2γ.To illustrate how this affects fringe shapes, we will compare precession with

sin(γ) = 0.1 to precession without B⊥. Fig. 5.12 shows the relationship between βand φ for sin(γ) = 0.1. The two angle variables are essentially equivalent exceptnear the narrow range that β does not reach. In Fig. 5.13 we plot, for both

68

0 12

32

20

12

Figure 5.12. β versus φ for sin(γ) = 0.1.

sin(γ) = 0.1 and B⊥ = 0, the evolution from the state |3, 0〉x in the x basis of them = 0 state, the even magnetic sublevels and the 6th harmonic. The two evolutionsare nearly the same except in the narrow range where β diverges from φ.

0.0

0.5

1.0|3, 0

0.0

0.5

1.0

Prob

abilit

ies Even

0 12

32

2 (rad)

0.0

0.5

1.0

1.56th Harmonic

Figure 5.13. Precession in the presence of a transverse magnetic field assuming sin(γ) =0.1 (solid line) and in the absence of a transverse magnetic field (dashed line). Thevertical dotted lines mark π − 2γ and π + 2γ.

69

5.4 Effect of a Transverse Field in an EDM Measure-mentThe original scheme of an EDM measurement (Section 5.1) relies on the energystructure of the tensor Stark shift. Any transverse magnetic field, EB⊥ , or fictitiousfield due to vector light shifts [4], deforms the energy level structure, as differentmagnetic sublevels in the original basis become coupled. The deformation affectsthe stretched states least, since lifting their degeneracy involves 2F magnetic dipolecouplings. The larger F is, the more resistant the degeneracy is to being lifted, asillustrated in Fig. 5.14a for F = 3. For EB⊥(1) << EE(1), the energy differencebetween the stretched states varies as the sixth power of B (see Fig. 5.14b). Thedegeneracy of the stretched states, and thus the ease with which the superpositionof stretched states can be prepared, is compromised once EB⊥(1) is on the orderof EE(1). In contrast, it is easy to prepare the m = 0 state, so if the transversemagnetic fields cannot be controlled well enough, it may be advantageous to acceptthe modest loss in precision described earlier in this paper, and measure the EDMwith a precession measurement from the m = 0 state.

10

5

0

5

10

(m)/

E(1)

(a)

0.0 0.5 1.0 1.5 2.0 2.5 3.0Transverse field B (1)/ E(1)

0.0

0.1

0.2

0.3

0.4

0.5

|(+

3)(

3)|/

E(1)

(b)

Figure 5.14. (a) Eigenenergies scaled to the tensor Stark shift as a function of transversemagnetic field in units of linear Zeeman shift scaled to the tensor Stark shift, for F = 3.(b) Energy difference between the stretched states as a function of the transverse field.Figure adapted from [4]

70

We introduced 2 strategies in Section 5.2 for decoupling the tensor Stark shiftfrom the linear shift of the magnetic sublevels for the purposes of measuring theEDM. Both of these strategies are compromised when B⊥ 6= 0. In contrast, abias field Bz adds an offset to the precession (and might add noise), but does nototherwise change the precession signal. It is, therefore, possible to increase Bz

to mitigate the detrimental effect of B⊥. To investigate the minimum bias fieldrequired to maintain useful signal fringes, we have carried out a numerical study forthe concrete scenario where the tensor Stark shift is 5 Hz and the transverse fieldis 10 Hz. Taking partial advantage of the above 2 strategies, we set the precessiontime to be 3 seconds and measure the probability to be in the even magneticsublevels. Fig. 5.15 shows the differences between the adjacent peaks and valleysas we scan the bias field from 10 Hz to 200 Hz. Below 60 Hz, the points are erratic,illustrating that there is no stable fringe pattern in that region. Above 60 Hz, thepoints start to form lines. These lines converge to 3 values around a bias field of200 Hz. These 3 values correspond to the differences between adjacent peaks andvalleys of the stable probability curve in Fig. 5.6. It takes a bias field of about 10times the transverse field for there to be a robust precession signal.

25 50 75 100 125 150 175 200Bias B field (Hz)

0.0

0.2

0.4

0.6

0.8

Adja

cent

(pea

k-di

p)

Figure 5.15. Differences between adjacent peaks and valleys in a scan of the biasmagnetic field. We assume a tensor Stark shift of 5 Hz, a transverse magnetic field of 10Hz and a precession time of 3 seconds.

71

5.5 Summary of the EDM MeasurementWe have shown a scheme for measuring the EDM using precession from the m = 0state. The m = 0 state can be easily prepared through optical pumping when thesuperposition of stretched states is hard to create. Precession from the m = 0state is more sensitive than typical Larmor precession and is close in sensitivityto the original scheme based on the superposition of stretched states. In addition,individual magnetic sublevels enhance the sensitivity when the measurement ofjust the m = 0 state is worse than its best sensitivity. We find the total probabilityin even (or odd) magnetic sublevels is insensitive to the tensor Stark shift and thesecond order Zeeman shift. We can also combine the measurements of individualsublevels to yield the highest rank polarization moments. While the superpositionof stretched states is immune to a small transverse magnetic field, the precessionfrom the m = 0 state works in all magnetic field by strengthening the quantizationaxis with a bias field. Strengthening the quantization axis with a bias field maycompromise the magnetic field stability because magnetic fields are usually lessstable and less uniform than the electric field in our experiment.

5.6 A Scheme for Measuring the Tensor PolarizabilityMeasurement of the tensor polarizability complements the measurement of themagnetic field or the EDM using precession of angular momentum. We need toknow the tensor polarizability in one of the 2 methods that decouple the tensorpolarizability from the linear Zeeman shift because the phase induced by the tensorStark shift affects the contrast and sensitivity of the precession signal with respectto the magnetic field or the EDM. Besides playing an important role in the EDMmeasurement, precision measurement of the tensor polarizability provides a testof atomic structure calculations involving short-range interactions [31] and affectfrequency shift due to black body radiation in atomic clocks. The most recentmeasurement of the tensor polarizability of free cesium atoms using an atomicbeam found [32]

α2(F = 4)/h = 3.34(2)(25)× 10−2 Hz

(kV/cm)2 . (5.30)

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We believe we can perform a more precise measurement of the tensor polarizabilitiesof both F = 3 and F = 4 hyperfine levels using our experimental setup designedfor an EDM measurement.

A precision measurement of the tensor polarizability can be accomplished bymeasuring the evolution of the atomic angular momentum in the electric andmagnetic fields, similar to the measurement of the EDM. When measuring theEDM using the precession of angular momentum, we keep the linear Zeeman shift,ωL, but try to discard the tensor Stark shift, ωα2 (Section 5.2). When measuringthe tensor polarizability, we turn it the other way around: keeping the tensor Starkshift while trying to discard the linear Zeeman shift.

When measuring the EDM, we can discard the tensor Stark shift and keepthe linear Zeeman shift by combining the measurement of individual magneticlevels. We do not have the counterpart for measuring the tensor polarizability,unfortunately. There is no combination of populations in magnetic levels thatcleanly removes the linear Zeeman shift but keeps the tensor Stark shift. Thisis because the tensor Stark shift shows up in terms that also contain the linearZeeman shift (Equation 5.15). If we somehow cancel ωL, we will necessarily get ridof ωα2 as well.

Still, we can set ωL = 0 by canceling out the residual magnetic fields by drivingcurrents through the coils. Consider a measurement of the tensor polarizabilityof F = 3. Setting ωL = 0 in Equation 5.15, we obtain the probabilities to be inindividual magnetic levels after an evolution time τ in an electric field only:

p(E) =

0164(−15) (cos (8τωα2)− 1)

0132 (15 cos (8τωα2) + 17)

0164(−15) (cos (8τωα2)− 1)

0

, (5.31)

which depend on only a single frequency 8ωα2 , the frequency splitting between them = ±1 and m = ±3 levels. Having a single frequency is a result of choosing |3, 0〉xas the initial state. Choosing any other magnetic level will necessarily introducemultiple harmonics of ωα2 and have smaller contrast and worse sensitivity. There

73

0.0

0.2

0.4

0.6

0.8

1.0

Prob

abilit

ies

(a)

0 14

12

34

2 (rad)

0

2

4

6

8

1/(

2)

(b)

Figure 5.16. Evolution of |3, 0〉 in a static electric field only. (a) Probability of detectingm = 0 versus phase. There are 4 fringes in a π increment of ωα2τ . (b) The inverse ofthe shot-noise-limited phase uncertainty versus phase. The phase uncertainty divergesas the probability hits the minima. The phase uncertainty stays low outside the narrowband around the singularities.

are only 3 states with non-zero amplitudes or probabilities. 2 of them are identical(p−1 = p+1), and p±1 offer the same information as p0, because of the normalizationcondition. So p0 contains all the sensitivity. Its evolution is plotted in Figure 5.16.

We will consider first the statistical uncertainty attainable with this measurementscheme followed by the systematic uncertainties. As it turns out, the systematicuncertainties dominate over the statistical uncertainty.

Recalling that ωα2(F=3) = 110α2(F = 3)E2/~, we can measure α2 by scanning E

or τ . The electric field is not a convenient parameter to scan in our experimentbecause the electric field can only be stabilized around some discrete values anda continuous scan may compromise the stability of the electric field. But theevolution time is a convenient and reliable scan parameter. Suppose we measurethe probability of the m = 0 state from τ = 0 to 5 seconds. The frequencycan be determined crudely by counting the number of fringes. The frequency

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uncertainty thus obtained is 8δωα2 = 2π 0.2 Hz. Using an attainable electricfield of 37.5 kV/cm, the corresponding uncertainty in the tensor polarizability is0.017× 10−2 Hz/(kV/cm)2.

We can do better by splitting the fringe. The inverse of the shot-noise-limitedphase uncertainty, the quantum projection noise of p0 divided by the slope of p0,is shown in Figure 5.16b. Though the phase uncertainty diverges when p0 hitsminima, the smallest uncertainty is 1/7.745 and the averaged phase uncertainty is1/6.92. The corresponding frequency uncertainty is

δωα2 = δφ√Nτ

. (5.32)

Using a conservative coherence time of τ = 3 s, a cesium atom number of N = 106

attainable in a single launch and the averaged phase uncertainty, the frequencyuncertainty is δωα2 = 2π×7.65×10−6 Hz, which is more than 3 orders of magnitudebetter than the uncertainty obtained by counting the fringes. The shot-noise-limiteduncertainty of the tensor polarizability of F = 3 is thereby 5.5×10−8 Hz/(kV/cm)2.

We can carry out a similar measurement on the tensor polarizability of theF = 4 hyperfine level. Setting the magnetic field to zero, the evolution of themagnetic sublevels in an electric field starting from the m = 0 state is given by

p(E)F=4 =

−35(12 cos(4τωα2)+4 cos(12τωα2)−3 cos(16τωα2)−13)4096

05(12 cos(4τωα2)−28 cos(12τωα2)−21 cos(16τωα2)+37)

1024

0180 cos(4τωα2)+700 cos(12τωα2)+315 cos(16τωα2)+853

2048

05(12 cos(4τωα2)−28 cos(12τωα2)−21 cos(16τωα2)+37)

1024

0

−35(12 cos(4τωα2)+4 cos(12τωα2)−3 cos(16τωα2)−13)4096

, (5.33)

which depends on three frequencies. We cannot, unfortunately, single out a partic-ular frequency from measurements of individual magnetic levels because there arenot enough independent quantities to cancel out any 2 of the 3 frequencies. Wecombine the probabilities to be in ±m magnetic sublevels and plot the evolution of

75

0.0

0.2

0.4

0.6

0.8

1.0

Prob

abilit

y

(a)

0 14

12

34

2 (rad)

0

5

10

15

1/(

2)

(b)±4±20

Figure 5.17. Evolution from |4, 0〉 in a static electric field only. (a) Probabilities ofdetecting individual magnetic sublevels are periodic with a baseline period of π/2 of ωα2τ .Only magnetic sublevels with non-zero probabilities are shown. (b) Phase uncertaintiesobtained from quantum projection into individual magnetic sublevels.

three nonzero magnetic sublevels individually in Figure 5.17aA crude measurement of α2(F = 4) can be obtained by counting the largest

peaks in the evolution of the probability to be in m = 0. These peaks occur fourtimes every 2π increment of ωα2τ . The frequency uncertainty obtained from 5seconds of evolution is 4δωα2 = 2π 0.2 Hz. The corresponding uncertainty in thetensor polarizability in an electric field of 37.5 kV/cm is 0.066× 10−2 Hz/(kV/cm)2.

The inverses of the shot-noise-limited uncertainties of ωα2τ obtained by measure-ments of individual magnetic levels are shown in Figure 5.17b. The smallest phaseuncertainty is 1/(6

√5), and the average phase uncertainty obtained by detecting

m = 0 is 1/6.85. The corresponding frequency uncertainty over a coherence time ofτ = 3 s with N = 106 atoms is δωα2 = 2π × 7.8× 10−6 Hz, The shot-noise-limiteduncertainty of the tensor polarizability of F = 4 is 1× 10−7 Hz/(kV/cm)2.

The measurements of the tensor polarizabilities of both hyperfine levels aresubject to the precision of the electric field, which in our setup is limited by theprecision of the plate separation. The uncertainty in the plate separation is halfa wavelength out of 4 mm. That results in 2× 10−4 fractional uncertainty of the

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tensor polarizabilities.A potential complication in the measurement of the tensor polarizabilities may

come from the vector light shift which is hard to cancel using magnetic fields.Like in an EDM measurement, the effect of the vector light shift can be mitigatedby applying a strong bias field in the z direction. Now the angular momentumevolves in the electric field and a parallel strong magnetic field. The effect of thetensor Stark shift is modulating the angular momentum precession induced by themagnetic field. To extract the tensor Stark shift from the modulation, we canmeasure the evolution at precession time when the magnetic-field-induced phase is

τωL = nπ, (5.34)

where n is any integer. With this condition, the evolution in electric and magneticfields (Equation (5.15)) reduces to the evolution in an electric field only (Equa-tion (5.31) and Figure 5.16) except that the fringe are now sampled at discreteτ that satisfy Equation 5.34. The number of samples available in a fringe cycleis given by the relative size of the linear Zeeman shift and the tensor Stark shift.Assuming a vector light shift of 13 Hz as found out by Zhu [4], a tensor Stark shiftof 5.6 Hz at 37 kV/cm and a bias magnetic field 10 times the vector light shift thatwe concluded in Section 5.4 is needed to beat the vector light shift, there are about5.6 values for τ to construct a fringe cycle. If we increase the magnetic field further,we can obtain more points with which to construct the fringes.

The largest uncertainty of a measurement of the tensor polarizability is likelydue to the tensor light shifts. The tensor light shifts induced by linearly polarizedtrapping light are [4]:

∆vT = vT (F )U(3cos2(α)− 1)m2, (5.35)

where α = 0 is the angle between the linear polarization and the quantization axisand vT (F = 3) = −vT (F = 4) = 0.0035 Hz/µK. Assuming that we can reduce thetrap depth to 20 µK after cooling the atoms in the volume between the plates, themaximum tensor light shifts experienced by the atoms are ∆vT (m = 1) = 0.14Hz,which is an offset of 0.186 × 10−2 Hz/(kV/cm)2 to the measurements of tensorpolarizabilities in an electric field of 37.5 kV/cm. An offset caused by the tensorlight shifts can be determined from the trap depth. The problem caused by tensor

77

light shifts is frequency noise due to light intensity fluctuations and inhomogeneousfrequency shifts. The intensity fluctuation of our trap is below than 5%. If we cancontrol the inhomogeneous frequency shifts to 10%, the uncertainty caused by thetensor light shifts is 0.02× 10−2 Hz/(kV/cm)2.

In contrast to an EDM measurement, measurements of the tensor polarizabilityusing angular momentum precession do not have noise cancellation using the signalsfrom the 2 lattices.

Cancellation of noises from the bias magnetic field can come from simultaneousmeasurements of the transition frequency between the m = 2 state and the m = 3state and the transition frequency between m = −2 and m = 3. When the biasmagnetic field dominates the energy structure, the frequency difference betweenthe 2 transitions is sensitive to the tensor Stark shift but insensitive to the biasmagnetic field.

Summarizing the above considerations, our measurement of the tensor polariz-abilities is most likely limited by the inhomogeneous frequency shifts caused by thetensor light shifts of the trapping light. Nevertheless, we can use our experimentalsetup to improve the precision of the tensor polarizabilities by an amount thatdepends on how well we can manage the inhomogeneous tensor light shifts.

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Chapter 6 |Restoring the Vacuum and AtomSignals

The vacuum provides an isolated environment for the atoms. Atoms will have along lifetime in a good ultra-high vacuum. Atoms are heated out of the oven, slowedby a Zeeman slower, collected by a MOT, loaded into the lattices, and launchedinto the region between the electric field plates well shielded from magnetic fields.

We repeated the process of building the vacuum and processing the atoms. Wecarried out this work by learning from our predecessors [4, 8, 9]. But during thisprocess, we refined many techniques involved. I, therefore, focus on the techniquesthat we developed.

6.1 Improved Top Vacuum Window with Low Bire-fringenceThe vacuum chamber is inside the Fabry-Perot cavities that provide the opticallattice traps for the atoms. The vacuum windows that the cavity light passesthrough alter the light every time it passes through. In particular, the birefringenceof the vacuum windows changes the polarization of light. When the birefringencewas severe, we could not even lock the cavities to the laser [8]. When the birefrin-gence was reduced, we obtained a stable cavity lock but the residual birefringencestill caused significant vector light shifts and limited our ability to perform ameasurement of the EDM [4]. Therefore, further reduction in the birefringence ofthe vacuum windows is important to the success of this experiment.

79

We made several upgrades to the top vacuum window. First, we replaced HeraeusInfrasil® 302 with Suprasil® 3001 as the quartz glass grade of our vacuum window.Suprasil® 3001 offers an absorption 2.5×10−3 that of Infrasil® 302 at the wavelengthof 1064 nm and is expected to reduce the thermal-induced birefringence [4].

indium ring

copper ring

knife edge

indium ring

knife edge

a

b

Figure 6.1. (a) The top part is the original design of the flange for mounting the topvacuum window showing the secondary joint. The secondary indium joint connects thewindow flange to a standard ConFlat® flange shown as the bottom part. A standardConFlat® flange comes with a knife edge, which is not used in an indium seal and isavoided by using a copper ring. (b) The modified flange is cut from the original flange.It merges the functionality of the original flange and the copper ring. It requires onlyone indium gasket to seal.

Second, we modified the flange hosting the vacuum window. In order to reducestress-induced birefringence, we used indium seals which require less minimum

80

torque to seal than the standard ConFlat® seals. For each window, we use indiumgaskets at 2 flange joints: the primary joint that seals the window to a stainless steelflange and the secondary joint that connects the flange to the vacuum chamber [33].We used indium seals at the secondary joint in order to avoid the stress propagatingfrom a ConFlat® seal. The implementation of the secondary joint we had usedrequires a custom-made copper ring and 2 indium gaskets as shown in Figure 6.1a.We modified the flange by cutting away some material and turned it into the designin Figure 6.1b. The modified flange needs only one indium gasket at the secondaryjoint in order to seal. This eliminates the possibility of uneven deformation of the2 indium gaskets.

Third, we managed to reduce the minimum torque on the bolts in order toseal the primary joint to 10 inch-pounds from 12 inch-pounds in prior trials [9].The reduced torque on the bolts means less stress and hence less stress-inducedbirefringence on the window. I describe our method for sealing the primary joint asfollows. We sealed the primary joint separately from the rest of our vacuum systemby mounting the top window flange on a leak detector. We tighten the bolts ina sequence that maximizes the distance to the last bolt tightened and graduallyincrease the torque in steps of 1 inch-pound until no leak was detected. Henceforth,we have never found a leak from the primary joint and the bolts for the primaryjoint have never been tightened since the top vacuum window was mounted on thevacuum chamber and baked together with the entire vacuum chamber. After thewindow was detached from the leak detector by breaking the secondary joint andbefore it was mounted to the vacuum chamber, we measured its birefringence.

We measured the birefringence of the window by placing the window between 2crossed polarizers as shown in Figure 6.2. We sent the Yag laser beam at 1064 nmthrough the crossed Glan-Taylor polarizers with a measured extinction ratio of 106.The first polarizer was aligned with the polarization of the input beam to purify theincident beam. The second polarizer was set at an orientation that minimized thetransmission before anything was placed between the polarizers. With the vacuumwindow placed between the polarizers, we measured the transmission through thecrossed polarizers while scanning the transverse angle of the window. For thismeasurement, we placed the setup in the beam path of the cavity input beam butbefore the cavity. So the laser intensity experienced by the vacuum window in thismeasurement was about 100 times less intense than the light intensity experienced

81

Derived from a Yag laser amplified by a fiber amplifier

Beam block

Window mounted on flange

Crossed Glan-Taylor Calcite Polarizers

Figure 6.2. Setup for measuring the birefringence of the vacuum window using crossedpolarizers technique. All optical elements are aligned perpendicular to the incident laserbeam. The transmitted power is recorded while the window is rotated about the laserbeam.

inside the cavity.The birefringence of this upgraded top window is around or below our measure-

ment sensitivity. Figure 6.3 shows the measurement results when the laser beamis centered on the window. The measured transmittance is around 2× 10−6, one

0°

45°

90°

135°

180°

225°

270°

315°

.51.0

1.52.0×10−6

Transmittance

Figure 6.3. Transmittance through the crossed polarizers with the vacuum windowplaced in between the polarizers in various directions.

order of magnitude improvement over the previous implementations of the indiumseal [9]. With this low level of birefringence, it is hard to identify the birefringenceaxes of the window.

The above results were measured with incident laser power below 10 W and a

82

Gaussian beam waist of 1 mm. By varying the laser power between 1 W to 10 W,we did not observe any power dependence of the transmittance through the crossedpolarizers with the window in between. The observed slowly drifting vector lightshift [4] could be attributed to thermal induced birefringence. This measurementusing the crossed polarizers does not exclude the possibility of thermal inducedbirefringence at a laser power of 100 W, which is the typical intra-cavity powerpassing through the window when it is mounted on the vacuum chamber.

Further reduction of stress-induced birefringence may be possible by removingthe bolts compressing the indium seal after the leak test. The indium forms asecure adhesion with the stainless steel and glass window after the compression. Iattempted to separate a window from a previously sealed flange, and there was noneat way to separate them even after all bolts were removed. In the end, I hadto knock the window through a cushion with a hammer. I eventually managed toseparate the glass window from the steel but the glass broke into 2 pieces. Thisincident qualitatively demonstrated the strong adhesion of indium after compression.But it remains to be seen if the adhesion from indium alone is enough to sustain avacuum-tight seal.

6.2 Baking the VacuumAfter the entire vacuum chamber is sealed and pumped down with a turbo pump, weneed to bake the vacuum to accelerate gas desorption in order to achieve ultra-highvacuum. Most of our baking techniques are inherited from our predecessors. Wewrap heater tapes uniformly around the metal part of the vacuum chamber andaround the aluminum tube set up around the glass cell. Heater tapes are wrappednot directly onto the glass cell but rather on the aluminum tube in order to avoidnon-uniform heating that can potentially crack the cell. We place thermocouplesat various locations on the vacuum chamber including the glass cell to monitor thetemperatures during baking. We then wrap 3 layers of aluminum foil around themetal part of the vacuum chamber and the aluminum tube on top of the heatertapes and thermocouples. The first layer is wrapped tightly against the chamberto spread the heat generated by the heater tapes around. The second and thirdlayers are wrinkled and wrapped loosely around the prior layers to trap heat.

We set up automated acquisition of the temperature readings of the thermo-

83

couples in order to reduce the acquisition time and to monitor the temperaturesat nights. The core part of our automated acquisition is a set of multiplexedthermocouple circuits borrowed from the Gemelke lab, replacing our handheldthermocouple module (Fluke 80TK). Each multiplexed thermocouple circuit cantake in up to 8 independent thermocouples and connect them through an analogmultiplexer ADG507A to a single thermocouple amplifier AD594, which convertsthe voltage across the terminals of thermocouples into temperatures in Celsius oneat a time.

Automated acquisition of temperatures makes use of a USB-6008 input andoutput device from National Instrument, which has both multiple digital outputchannels and multiple analog input channels. We use 3 digital output channelsto simultaneously address the multiplexers on 4 circuits and use 4 analog inputchannels to read the temperatures from the 4 circuits. The temperatures of all 32thermocouples can be read in 4 seconds. This 4 seconds of acquisition time is farfrom the limit of the acquisition devices and can be reduced when desired.

We use care when raising and lowering the temperatures of the vacuum chamberto avoid large thermal gradients, particularly on the glass cell. Specifically, westrive to limit the thermal gradient on the cell below 10C between any 2 of the 6thermocouples placed on the cell. While raising the temperature, we find that thetop of the cell is hotter than the bottom of the cell because hot air rises and leaksfrom the top of the aluminum tube and that cool air leaks in from the bottom ofthe tube. To reduce that temperature gradient, we have to completely turn off theheater tapes on the middle and top part of the aluminum tube surrounding the celland keep only the bottom heater tape on. That way, the cell top is still hotter thanthe cell bottom. To further reduce the temperature gradient on the cell, we have toraise the temperature on the steel part below the cell 10C above the temperatureon the cell. It takes us 3 days to raise the temperature on the chamber to 110C.

While the chamber is held at 110C, we degas the various pumps and theion gauge in the vacuum by turning them on. Of special attention are the ionpumps. The ion pumps accumulated lots of water and gas molecules after beingexposed to the atmosphere for two years. After the ion pumps are brought back tovacuum with a turbo pump, turning on the ion pumps leads to large currents to theion pump power supplies (PS-100). The protection circuits of the power suppliesautomatically shut themselves off when they see a large current. So initially the ion

84

pump power supplies immediately turned themselves off after they were manuallyswitched on. Degassing the ion pumps requires switching them on and letting themautomatically shut themselves off many times until they stay on.

We kept the vacuum chamber at 110C for a week. Towards the end of theweek, the vacuum pressure approached the asymptotic value of 2.4× 10−8 torr. Bythat time, we gradually lowered the temperatures of the vacuum chamber back toroom temperature. When the temperatures settled back to room temperature, thevacuum pressure was 9× 10−11 torr.

The temperature gradient caused by the air flow in the aluminum tube sur-rounding the glass cell unnerved us. In future iterations of baking the vacuum, onecan block or seal the top or the bottom of the aluminum tube to stop the air flow.That will allow the air within the tube to achieve thermal equilibrium and make thetemperature across the glass cell more uniform. More uniform temperatures willallow raising the minimum temperature across the entire vacuum system higher,and that will eventually lead to a better vacuum pressure after baking since theresults of baking is essentially limited by the coldest point in the vacuum system.

6.3 Auxiliary 2-D Imaging SystemI discussed the fluorescence imaging system with non-magnetic PDAs and tripletlenses in Chapter 3, which is the primary imaging system for collecting the EDMdata. But this imaging system has resolution only along one dimension (y) andhas been designed to fit inside the shields. Oftentimes, we like to know about theatom cloud in 2 dimensions and the atoms in the MOT, with less emphasis on thelight collection efficiency. This demand can be met with an off-the-shelf camera.For this purpose, our predecessors used CCD cameras by PULNIX (JAI TM-7AS,and a relatively newer version CV-A55IR), which were digitized by a frame grabber(Data Translation DT-3155) [4]. The system based on the PULNIX camera alongwith the frame grabber was plagued by missing triggers when working with thetiming system (Supertime) triggered by a 60 Hz phase. The reason for that hasbeen unclear. We bypassed this problem by replacing the PULNIX camera with aUSB CMOS camera from JAI (GO-5000M-USB).

Along with this new camera, we have different acquisition software. Thissoftware is based on the sample software named "ImageConversionSample" supplied

85

by JAI, the manufacturer of the camera. But this software has been modified sothat it takes images whenever it receives a trigger and automatically saves thenext image with a different filename without overwriting the existing images. Thefilename always has an extension “.bin” representing the raw binary format of theimage. The first image taken after launching the software is named "1.bin", thesecond image is named “2.bin” and subsequent images are named by incrementingthe numerical part of the filename by 1.

With the above modification, it is possible to acquire multiple images withina single timing sequence. For instance, both the atoms held in a MOT and thebackground of the MOT without the atoms or multiple images of atoms loaded intothe ±z lattices during multiple launches can be taken within a timing sequence.The maximum number of images that can be taken within a single sequence isonly limited by the data transfer rate. The GO-5000M-USB camera connects to acomputer through USB 3.0, which enables a maximum transfer rate of 60 framesper second.

We noticed that this GO-5000M-USB CMOS camera has noise correlated alongthe rows more than the columns. Figure 6.4a shows the difference between 2 shotsof the background. Figure 6.4b shows the sum of rows versus the column indexand Figure 6.4c shows the sum of columns versus the row index. It is surprisingthat the sum of rows is a lot noisier than the sum of columns even though they arederived from the same 2-D image. This correlation of noise along the rows is alsoevident from close examination of the 2-D image (Figure 6.4c), which shows fainthorizontal lines.

This correlation of noise along the rows may be a consequence of non-uniformgains of individual pixels on the CMOS sensor. If that is the case, this noise can beremoved by calibrating the CMOS sensor with uniform illumination. This noise canalso be reduced by discarding pixels not contributing to the signal. Take for examplean image of the atoms in the MOT with background subtracted (Figure 6.5a). Theatoms are localized in a small area on the frame. We use pixel data only withinthe ellipse. The noise outside the ellipse does not contribute to the analysis andthe sum of rows (Figure 6.5b) is significantly quieter. The process of selectingpixels only around the atoms is automated in a function called gausfit2D withinthe AtomicPhysics.py module. This python function guesses the centroid and thesize of the atom cloud, discards pixels a certain radius away from the centroid and

86

0 500 1000 1500 2000 2500

0

500

1000

1500

2000

(a)

0 200 400 600 800 10001200

0

200

400

600

800

1000

(c)

0 500 1000 1500 2000 2500Column index

250

200

150

100

50

0

50

100

150

200

Sum

over

row

s

(b)

0 500 1000 1500 2000Row index

1500

1000

500

0

500

1000

1500

Sum

over

colu

mns

(d)

Figure 6.4. (a) Difference between 2 shots of the background with the GO-5000M-USBcamera. (b) Sum over rows vs. column index. (c) Zoom-in of (a). (d) Sum over columnsvs. row index.

from the remaining pixels determines the center and waists by fitting the data toGaussian functions.

6.4 Parallel 1-D LatticesWe used a pair of far-off-resonance parallel one dimensional lattices for trapping theatoms. The lattices also guide the atoms into the region shielded from magneticfields after they are launched from a MOT from below. Far-off-resonance latticesreduce spontaneous emissions but they also require a significant amount of powerto guide the atoms well. Atoms after launch experience spatially varying optical

87

600 800 1000 1200 1400

600

800

1000

1200

1400

(a)MOT with

background subtracted

900 950 1000 1050 1100 1150

Column index [pixels]

0

1000

2000

3000

4000

5000

6000(b)

900 950 1000 1050 1100

Row index [pixels]

01000200030004000500060007000

(c)

Figure 6.5. (a) An image of atoms in the MOT with background subtracted. Onlypixels enclosed by the white ellipse contribute to the analysis in (b) and (c). (b) Sumover rows vs. column index. (c) Sum over columns vs. row index.

potential, which on average is 1/2 of the deepest trap depth. Our atoms havea temperature of 30 µK in free space after polarization gradient cooling in theMOT chamber located one meter below the field plates. Trapping atoms withthat temperature requires a minimum laser power of 15 W per lattice and guidingthe atoms 1 m up to the top requires much more power [34]. A fiber amplifier(IPG Photonics YAR-10-1064-SF-PM) seeded with a Yag laser can output a totalpower of 10 W. To bridge the gap in laser power, we use a pair of Fabry-Perotcavities (Figure 6.6) which enhance the laser power by bouncing the light between2 mirrors about 300 times. The intra-cavity power can be 20 times the input power.I describe how we configure the cavities in the following sections.

88

Vacu

um

Split into 2 beams

High reflection mirrors

BS

PDrefl

PDt

Figure 6.6. Diagram representing our lattices. The lattices are derived from a singlefiber amplifier seeded with a Yag laser. The lattices are enhanced by Fabry-Perot cavities.

6.4.1 Mode Matching

The cavities support modes (TEMij) given by their geometry. The fundamentalmode TEM00 of a symmetric cavity has a minimum waist given by:

ω0 =

√√√√Lλ

2π

√1 + g

1− g = 0.577 mm, (6.1)

where L = 2.36 m is the current cavity length, λ = 1.064 µm is the wavelength,g = 1 − L/R and R = 2 m is the radius of curvature of both mirrors. In orderto find out the required input beam that reaches the minimum waist given byEquation (6.1) after propagating to midway between the 2 mirror, we reverse the

89

beam propagation direction and calculate what the beam supported by the cavitybecomes after leaking out from the input mirror. We first calculate the beam waistat the input mirror of the supported fundamental mode:

ωi =

√√√√Lλ

π

√1

1− g2 = 0.901 mm. (6.2)

The supported mode at the input mirror has the same radius of curvature as theinput mirror. So the q of the beam at the input cavity mirror is:

qi = 1/(

1/R + iλ

πω2i

)= 1.18 − 0.984i m. (6.3)

The qi and ωi above are beam parameters inside the cavity at the input cavitymirror. The input cavity mirror is a plano-convex lens, with a focal length of:

f = −R/(n− 1), (6.4)

where n is the index of refraction of the glass substrate of the mirror and is assumedto be 1.5 in this calculation. This cavity mirror imparts a change of the radius ofcurvature of the beam after the supported mode leaks out of the input mirror. Sothe q outside the cavity at the input mirror is:

qo = 1/(−1/f + 1/qi) = 1.019 − 0.566i m, (6.5)

from which we can deduce the distance to the minimum waist, the Rayleigh lengthand hence the minimum beam waist of the beam outside the cavity:

zto the minimum waist = Re(qo) = 1.019 m, (6.6)

z0o = −Im(qo) = 0.566 m, (6.7)

ω0o =√z0oλ

π= 0.438 mm. (6.8)

So the beam that we need to send in to match the cavity mode has a smaller beamwaist and the minimum waist is 19 mm farther away from the midpoint of thecavity. We use stock lenses to convert the beam from the fiber amplifier to thebeam given by Equation (6.5). It is often convenient to have the beam collimated

90

before the last lens before the cavity. Then the radius of curvature of the beamright after this lens should be the same as the focal length fc, that is

R(l) = l + z20o/l = fc, (6.9)

where R(l) is the radius of curvature at a distance l from the minimum waist ofthe beam specified by qo. Since we have to place fc before the cavity, l must begreater than zto the minimum waist and hence greater than 1 m. Then R(l) is at least1.56 m. Stocked lenses with fc > 1.56 m are not common, so we have a lens with a2 m focal length. Setting fc = 2 m in Equation (6.9), we find the distance can be0.176 m or 1.824 m. 0.176 m is inside the cavity and should therefore be discarded.Setting l = 1.824 m, we find the waist of the collimated beam before fc:

ωfc =√

(l2 + z202)λ

z02π= 0.148 mm, (6.10)

which is close to the 1.15 mm beam waist out of the fiber amplifier after passingthrough the telescope for the pair of optical isolators. Therefore, the lens withfc = 2 m works for us and it should be placed at a distance before the input cavitymirror,

l − zto the minimum waist = 0.805 m. (6.11)

The above calculation depends on the cavity length L, which was forced tochange after the introduction of the 3-axis control of the Brewster plates [4]. We,therefore, change the placement of fc accordingly.

6.4.2 Setup and Alignment

Besides mode matching, there are several additional requirements that we need tomeet when we set up the cavities. First of all, the linear polarization of the trappinglight has to be parallel to the static electric field created by the electrodes [4], that is,the linear polarization perpendicular to the glass plates. Since the normal of glassplates has been aligned with the 45 holes on the optical table, this requirementbecomes aligning the polarization of the trap with the 45 holes on the opticaltable. The trapping light is reflected by a folding mirror with a dielectric coating,which can induce a differential phase shift to the S and P polarizations and mess

91

up the input linear polarization created by the Glan-Taylor calcite polarizer unlessthe linear polarization is completely P polarized. In order to preserve the linearpolarization, the incident beam must hit the folding mirror along the 45 holes aswell.

Second, for 2 mirrors (top and bottom) to form a Fabry Perot cavity, the curvedsurfaces of these two mirrors must retro-reflect the beams. This can be achieved bymoving the beam such that the beam passes through the center of the sphericalsurfaces of the mirrors. This is the widely used method for aligning Fabry Perotcavities. But this method assumes that the mirrors’ orientation is only looselyconstrained or not constrained at all. In our experiment, the mirrors must havethe line between the centers of the spherical mirror surfaces passing through thegap between the plates, which sets a tight constraint. Our method of aligning thecavity turns it the other way around: aligning the mirrors to the beam. We firstsend the beam through the gap between the plates and make it parallel to the platesurface. Then we put in the mirrors and have them retro-reflect the beam.

Third, we want to avoid refraction of the beam by the first cavity mirror(bottom). That is, we want to position the bottom cavity mirror, which is plano-concave, such that its minimum thickness is hit by the beam. This requirement ismore of a simplification to the setup procedure than a necessity of the operationof the cavity. Keeping the beam transmitting through the mirror straight makesit easier to keep track of the direction of the beam. The cavity would work evenif the beam is refracted by the first mirror though at the cost of mode-matchingefficiency due to aberrations at refraction. We do not have a similar requirementon the exit cavity mirror (top), though it is also plano-concave, because there is nodemand on the direction of the exit beam and less still do we care about the modequality of the exit beam.

Our procedure for setting up the cavity with the above requirement is differentfrom the procedure Solmeyer and Zhu used [4, 9]. Our procedure is straightforwardand requires the minimal trials and errors.

The first step of our procedure is aligning the horizontal incident beams perpen-dicular to the plate surfaces. Since the normal of plate surfaces has been alignedwith the 45 holes on the optical table, all we have to do is to align the beam withthe 45 holes.

Second, we place the mirror under the bottom vacuum window to turn the

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beams vertical. This folding mirror can take a range of positions (±10 mm) forthe beam to traverse through the gaps between the plates. But we have to makesure the beams are in the middle of 2 plates and parallel to the plates. In orderto achieve that, we place a white card above the top vacuum window to view thetransmitted beams (Figure 6.7a). We scan the vertical knob of the folding mirror

Incident beam

White Card S1 S2

2 of the 3Plates

a b

c

Figure 6.7. (a) Diagram illustrating the technique of positioning the beam midwaybetween the plates. For simplicity, only 2 of the 3 plates and 1 of the 2 lattice beams areshown. (b) 4 spots appear on the white card as we tilt the beam towards -z by tiltingthe folding mirror. The spots from the reflection off the plates are dimmer/smaller thanthe original spots from the transmitted beams. The spots from reflection are on the +zside of the original spots. (c) When the beams are tilted towards +z and hitting theplates, the reflection spots appear on the left of the original spots.

until the beams hit the upper edges of the plates. We know the beams hit the upperedges of the plates when 2 more beams appear on the white card in addition to theoriginal transmitted beams as shown in Figure 6.7b. These additional 2 beams are

93

the reflections off the plates. We record the separation between the transmittedbeams and the reflected beams on the white card (S1). Next, we reverse the scan ofthe folding mirror until the beams hit the other side of the plates. Now the reflectedbeams should appear on the other side of the transmitted beams (Figure 6.7) sothe separation between them (S2) should flip sign. We carry out the above scan fora range of positions of the folding mirror. The beam position at the folding mirroris midway between the plates when S1 + S2 = 0. We next position the verticalknob of the folding mirror such that the transmitted beams are midway betweenthe extreme positions of the transmitted beams. Now the beams are centered andparallel to the plates.

Third, we place the top cavity mirrors. The spacing between the 2 beams isexactly 10 mm and the mirrors are bigger than 10 mm. So the 2 mirrors needto have an offset from each other. (By convention, the −z top cavity mirror isabove that of +z. But the other way around works equally well in principle).There are holes on the mirror plates to let the beam pass through so that thetransmitted beams can be detected. We position the holes such that the beamscan pass through. Next, we adjust the orientation of the mirror to retro-reflect thebeams. Retro-reflecting the beams might seem trivial at first glance. But muchof the space is under vacuum and we cannot put a card into the vacuum to checkthe beam overlap. After coming off the bottom vacuum windows the beams passthrough several optical elements including a Glan laser prism and split into severalbeams. It is important to identify the primary beam and overlap the primary beamwith the incident beam so that the beams are retro-reflected properly.

Fourth, we mark the positions of the retro-reflected beams using the beampaths for collecting the cavity error signals. The error signals are derived from thereflections of the cavities using the Pound-Drever-Hall technique. The reflectionsof the cavities, like the reflections off the top cavity mirrors, should be the retro-reflections of the incident beams. We use a partial beam splitter placed at theretro-reflection paths before the first step of our procedure to pick off the retro-reflections. And we use mirrors to direct the beams picked off to the fast photodiodesfor collecting the error signal. Now the error signal beam paths are set and we useirises to mark the position of the retro-reflected beams.

Fifth, we mark the positions of the incident beams at the top cavity mirrors.We do this by mounting a camera (a beam profiler) just below the top cavity

94

mirrors. We position the camera such that the beams are centered in the frame ofthe camera and record the centers of the beam profiles on the camera frame.

The fourth and fifth steps are really the preparation for the sixth step whichis to place the bottom cavity mirrors. We adjust the angles and positions of thebottom cavity mirrors such that the reflections from these mirrors pass through theirises we placed in the fourth step and that the transmitted beams do not deflectusing the camera we placed in the fifth step. The angles and the positions of themirrors are in general coupled. But we find that the angles mostly change thereflections and the positions mostly change the transmission. After removing thecamera placed at the beams, there are spikes in the transmission and dips in thereflections. These are the resonant modes of the cavities.

Seventh, we finely tweak the angles of the cavity mirrors to maximize thetransmission of the TEM00 mode.

Lastly, we place the pairs of Brewster plates into the beams. We are careful tokeep the beams relatively centered on the Brewster plates while keeping the −zBrewster plates at a safe distance from the +z beam and vice versa. We adjust theangles of the Brewster plates to minimize the reflections from them and orient theaxes of the galvos perpendicular to the planes of reflection.

With that, the setup and alignment of the cavities are complete. Even thoughthe above procedure is described for both cavities, many steps can be completedfor one cavity at a time.

6.4.3 Cavity Characteristics

After we complete the alignment procedures outlined in the previous section, wecan obtain the cavity transmission and error signal as shown in Figure 6.8. ThePound-Drever-Hall error signal is obtained by modulating the phase of the cavityinput beam by driving an electro-optic modulator (EOM) at frequency Ω = 10.14MHz and demodulating the cavity reflection at Ω. The phase modulation generatessidebands, which are visible in Figure 6.8b. Each sideband carries about 1% of theinput power, corresponding to a modulation depth of β ≈ 0.2 and an RF of Vpp = 1V driving the EOM. The up-and-down asymmetry of the dispersive shapes on thesides in the error signal is a consequence of phase mismatch between the cavityreflection and the local oscillator. This can be corrected by adjusting the phase

95

0

200

400

600

Tran

smiss

ion

(mV) TEM00

2nd4th

(a)

0

10

20

30

Tran

smiss

ion

(mV) Sidebands

(b)

0 1 2 3 4 5 6Time (ms)

0.2

0.1

0.0

0.1

Erro

r sig

nal (

mV)

(c)

Figure 6.8. Transmission (a,b) and the Pound-Drever-Hall error signal (c) simultaneouslycaptured by a digital oscilloscope. The most prominent transmission peak in (a) is thefundamental cavity mode, TEM00. This peak is truncated in (b), which zooms into (a)and provides a detailed view of higher modes and sidebands. Visible in this figure are the2nd excited modes marked by the cyan arrow and the 4th excited modes marked by themagenta arrow. The 2nd excited modes can be TEM11, TEM20 and TEM02. Sidebandsof the TEM00 are marked by the red arrows.

delay between the cavity reflection and the local oscillator. Having symmetricdispersive shapes on the side is an indication of the steepest slope in the centerdispersive shape.

96

The traces in Figure 6.8 are recorded over a period of time during which thecavity length is changed. Even though the frequency of the incident light is keptconstant, we typically characterize features in Figure 6.8, such as the cavity line-width, in units of frequencies This frequency unit differs from the frequency of thelight and it is defined as the mode frequencies of the cavity of length L:

ν = nc

2L, (6.12)

where n is any integer and c is the speed of light. To account for changes incavity length (δL) caused by noise or induced by actuators, we differentiate Equa-tion (6.12),:

δν = −δLL

nc

2L = − νLδL. (6.13)

The mode frequencies are locally linearly proportional to the cavity length whenthe change in cavity length is small, even though the mode frequencies are inverselyproportional to the cavity length as shown by Equation (6.12). The change inour cavity length is of the order of half the wavelength (λ/2), so the linearity ofEquation (6.13) holds valid to an accuracy of λ/(2L) = 1064nm/4.72m = 2.3×10−7.

We can measure the cavity line-width, δv, by comparing it to a known frequencyin a linear scan. One known frequency is the cavity free spectral range (FSR),which can be calculated from the cavity length and is about 64 MHz. The otheris the phase modulation frequency ω = 10.14 MHz, smaller than the FSR. It iseasier to produce a linear scan over a smaller range because our cavities are quitenoisy. So we use the phase modulation frequency to measure the cavity line-width.The modulation frequency manifests itself as the frequency spacing between thezero-crossings on the side and zero-crossing in the center in the Pound-Drever-Hallerror signal as shown in Figure 6.8. A cavity linewidth is about the frequencyspacing between the peak and the dip next to the center zero-crossing of the errorsignal. The cavity linewidth δv determined from the error signal is 212 kHz andthe cavity finesse is about 306.

6.4.4 Extracting Information from the Cavity Reflectance

The single most useful and most convenient source of information of a cavity istypically the cavity transmission. The transverse profile of the transmitted light

97

tells about the mode within the cavity, the transmittance tells about the finesseand loss of the cavity and the transmitted power tells about the trap depth.

However, the usefulness of cavity transmission in our experimental setup ishindered by 2 details of the mounts that hold the top cavity mirrors. First, wecannot directly measure the transmission through the top cavity mirrors but have tomeasure the transmission through the cavity mirrors and another piece of un-coatedfrosted glass. This additional piece of glass was introduced to reduce the mechanicalresonance of the mirror mounts in order to achieve stable lock of the cavities. [4, 9].Because of this additional piece of frosted glass, the transmitted light is diffusedand no information can be learned about the mode of the light before the mirror.Second, the spacing between the 2 lattices is exactly 10 mm, but the distancebetween the edge of the mirror mount and the hole for the light to transit throughis too wide to let the light of both lattices pass through the mirror mount withoutclipping. The compromise that we have taken is setting up the −z lattice first andthen moving the +z top mirror close to the −z lattice before degrading the cavityfinesse of −z. By doing so, we do not introduce unnecessary loss to either cavitybut the consequence is that we can detect only a fraction of the transmission of +z.These 2 flaws of the cavity mirror mounts can be avoided by some modificationsto the mounts. But because the mirrors have been epoxied to the mounts and wedo not want to risk damaging the high reflection coating of the cavity mirrors, wehave decided to live with them and exploit the cavity reflection for informationwhen we need it.

The cavity reflection provides complementary information about the cavity.In the case of a loss-less cavity, the cavity transmittance (Tc) and reflectance(Rc) necessarily add up to 1 and so they carry exactly the same information. Inthe case of a cavity with loss, the cavity transmittance and reflectance do notadd up to 1 in general. But the minimum of reflectance, just like the maximumof transmittance, still corresponds to the optimal mode-matching condition. Inaddition, we can deduce the cavity loss from the minimum reflectance and themaximum transmittance from the cavity loss, as we will show next by calculatingthe transmittance and reflectance from the cavity loss.

In our experimental setup, the cavities are asymmetric with the bottom mirrortransmission of T1 = 0.7% and the top mirror transmission of T2 = 0.035% andthey have the round-trip fractional intensity loss of ε. We define the loss parameter

98

α by α2 = 1− ε. Then the cavity transmission coefficient (Etra/Einc) and reflectioncoefficient (Eref/Einc), in the case of perfect transverse mode matching, are givenby [35]:

EtraEinc

= t1t2α

1− r1r2α2ei2δ, (6.14)

ErefEinc

= r2α2ei2δ − r1

1− r1r2α2ei2δ, (6.15)

where δ ≡ 2πL/λ is the phase shift after the light travels from the bottom mirrorto the top mirror, t1 ≡

√T1 and t2 ≡

√T2 are the transmission coefficients of the

bottom and top cavity mirrors and r21 ≡ 1− t21 and r2

2 ≡ 1− t22 may differ from thereflection coefficients of the mirrors by the loss at the mirror surfaces. The loss atthe mirror surfaces, together with the loss at vacuum windows and Brewster plates,is described by α or ε. We lock the cavity to δ = 0, when the cavity transmission ismaximized and reflection is minimized. The maximum of cavity transmittance andthe minimum of reflectance, which are the modulus square of Equation (6.14) and(6.15), are plotted in Figure 6.9a for a range of cavity loss ε.

Notice that the reflectance is not a monotonic function of loss. In fact, thereflectance is null when the cavity loss is ε0 = 1− r1/r2 = 0.0033. The existenceof such a minimum with respect to cavity loss is because of r1 < r2 in ourexperimental setup. (In all other cases (r1 > r2), the reflectance monotonicallyincreases with cavity loss.) This minimum reflectance with respect to the cavity lossdoes not correspond to the maximum transmittance and the maximum trap depth.The maximum transmittance and the maximum trap depth always monotonicallydecrease with cavity loss. Therefore, we always strive to reduce the cavity loss.

There are 2 possible values for the cavity loss corresponding to a measuredcavity reflectance because the reflectance is not monotonic with respect to thecavity loss. We need other information, either direct or indirect, to help determineif ε < ε0 or ε > ε0. The direct information that can help resolve this ambiguity isthe transmittance or the ratio of transmittance and reflectance (R/T ), either ofwhich is monotonic with respect to cavity loss as shown in Figure 6.9(a)(b). Thisinformation is available for our −z cavity. The −z cavity loss determined from thereflectance is consistent with that determined from the transmittance, suggestingthat the mode matching is close to unity and the transmittance can be inferred

99

0.00 0.01 0.02 0.03 0.04 0.05ε0.0

0.2

0.4

0.6

0.8

Reflect

ance

(a)

0.00 0.01 0.02 0.03 0.04 0.05ε

0

200

400

600

800

R/T

(b)

0.00

0.05

0.10

0.15

0.20

Tra

nsm

itta

nce

Figure 6.9. (a) Reflectance and transmittance versus cavity round-trip loss ε. Re-flectance is plotted with the left vertical axis and transmittance is plotted with the rightvertical axis. (b) Reflectance divided by the transmittance (R/T) versus ε.

from the reflectance when the transmittance cannot be measured in the experiment.The cavity loss of −z has always been above 0.012 (> ε0). The transmittance of +zcannot be measured but we can indirectly resolve the ambiguity of loss. First, weinfer that the trap depth of +z is less than that of −z from the oscillation frequencyof atoms in the trap and therefore the cavity loss of +z is higher than that of −z.Second, dust settles on the cavity components outside the vacuum and the lossincreases with time before the cavity components are cleaned. The observation thatthe cavity reflectance degrades monotonically with time before cleaning suggeststhat the cavity loss of +z is greater than ε0. From the above 2 pieces of evidence,we can conclude that the reflectances of both cavities lie within the right side of

100

the monotonic region of the reflectance curve in Figure 6.9, which can be used todetermine the transmittance from the reflectance without ambiguity.

6.4.5 Cavity Locks

In order to have intra-cavity power larger than the input power, it is necessary tomeet the following resonance condition:

2L = mλ, (6.16)

where L is the cavity length, λ is the wavelength of the light and m is any integer.Because we use cavities to enhance the lattices to guide the atoms during thelaunches, the cavity length is at least the launch height. As it turns out, the cavitiesin our setup are 2.36 m long. Cavities of this length are subjected to a tremendousamount of length noise caused by mechanical vibrations and thermal expansionand contraction. In fact, without any active scan of the cavity length, the noisesalone routinely change the cavity length over a free spectrum range (λ/2). In orderto keep this cavity in resonance, we must actively change the cavity length, L, orthe wavelength, λ to compensate for the noises, so that the cavity is locked to theresonance. We may keep the resonance condition in check by changing either L orλ. Since the 2 cavities share the same light source, varying λ changes both cavitiesby the same amount. But the 2 cavities do not share the same noises. To correctthe noises in 2 cavities independently, we have decided to use 2 sets of actuators tochange the cavity lengths. Each set of actuators consists of a PZT glued to theback of the top cavity mirror and a Brewster plate mounted on a galvanometer.

We use the Pound-Drever-Hall method to lock the cavity to the laser (Fig-ure 6.10). A home-made electro-optic modulator (EOM) is placed along the inputbeam path before the input beam is split into two. We drive the EOM with amodulation frequency, Ω = 10.14 MHz. This generates two frequency sidebands inthe light sent into the cavities. Both the carrier and the sidebands are reflectedby the cavities with reflection coefficients that depend on the frequency detuningfrom the cavity resonance. The reflections of the cavities are detected using fastphotodiodes (Figure 6.6). The signals from the photodiodes are mixed with thelocal oscillator to extract the beat signals between the sidebands and the carrier,which yields the error signal in Figure 6.8c. The error signals are sent to two

101

Cavity excited with EOM Modulated laser

Yag frequency Gf

PZT GPZT

Brewster plates GB

Actuators

Sensor Gs

Photodiode

PID GPID

δL

ε

GLε

Amplifier

Mixer

local oscillatorΩ=10.14 MHz

Vc

Figure 6.10. Diagram representing the servo for locking a cavity.

independent PID circuits, which drive the actuators to compensate for the noises.In order to reduce the noise with the lock, we need to increase the loop gain and

increase the bandwidth. Our bandwidths are limited by the mechanical resonanceof the PZTs at 60 kHz. Mounting the PZTs to the mirror mounts introducesseveral additional resonances at lower frequencies. Across a resonance, the phase ofthe response sweeps through a large range. If the loop gain (GL), which includesthe gain of the actuator (Ga), is in phase with the noise and its magnitude isgreater than one, the servo adds noise to the system and the lock is unstable. Inorder to push the bandwidth beyond the resonances, we must reduce the rangesof phase sweep across the resonances. We do that by using passive damping withSorbothane.

The responses of the actuators, including the resonances of the mounted PZTs,are most directly characterized using transfer functions, the open loop responseof the actuators driven at a range of frequencies. The response of a PZT can bemeasured using a Michelson interferometer. However, we do not have a Michelsoninterferometer immediately available in our setup (Figure 6.10). What we havefor sensing the movement of the PZTs are the Pound-Dever-Hall signals from thecavities. Unlike the signal from a Michelson interferometer, the Pound-Dever-Hallmethod can sense a movement only when the cavity is within a linewidth aroundthe resonances. We must, therefore, keep the cavity at least loosely locked around

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a resonance by closing the servo loop shown in Figure 6.10 in order to detect theresponse of a PZT using the Pound-Drever-Hall method. What we measure withthe loop closed is no longer the transfer function per se. Therefore, we must becareful when interpreting the results.

To measure the closed-loop response of the system, we can drive the systemby modulating one of the actuators in Figure 6.10. Because the bandwidth ofgalvanometer (for rotating the Brewster plate) is limited to the low-frequency range,we have essentially two choices of what to modulate: the frequency of the laser orthe PZT.

The frequency of the laser was what our predecessors modulated when measuringthe response of the cavity system (a speculation in retrospect subject to verification).A control voltage, Vc, was sent to a PZT within the laser to modulate the laserfrequency. That created an effective change in cavity length, VcGf . The cavity wasalso subjected to length change induced by the environment, δL. If the residuallength change after the feedback from the servo was ε, the length change imposedby the servo was −GLε, where −GL is the loop gain of the servo and the minussign is due to the convention that sets negative feedback as the default. The lengthchanges induced by all actuators as well as those induced by the environment addup to the residual length change:

VcGf + δL−GLε = ε. (6.17)

From the above equation, we find that the residual length change with feedback is:

ε = VcGf + δL

1 +GL

. (6.18)

The error signal sent to the PID circuit is then:

Verr = GsVcGf + δL

1 +GL

. (6.19)

Assuming VcGf >> δL, the ratio between the error signal and the control voltageis:

VerrVc

= GsGf

1 +GL

. (6.20)

The above expression was what our predecessors measured [4, 9]. Their data is

103

reproduced in Figure 6.11. The amplitude of Verr/Vc dips down at frequenciesaround 2.2 kHz and 2.5 kHz. These dips are probably caused by spikes of the loopgain, GL, at the corresponding frequencies. The spikes in GL are the manifestationof the mechanical resonances of the PZT. These spikes decreased in size after thePZT mount was damped [4,9].

Figure 6.11. The loop response when the system was modulated by the laser frequencyand locked with PZT. The amplitude (a) and phase (b) of the error signal sent to the PIDcircuit, Verr, divided by the control voltage of the PZT, Vc vs. the modulation frequency.The blue curve was measured without damping while the red curve was measured withdamping. Figure from [4].

The response of the system is qualitatively different when we modulate thesystem through the PZT. The signal that we measure is now:

VerrVc

= GsGPZT

1 +GL

. (6.21)

The loop gain, GL, is the product of individual gains in the loop. When GL ismuch greater than 1, Equation (6.21) becomes:

limGL>>1

VerrVc

= 1GPID

, (6.22)

independent of GPZT . Therefore, this signal is insensitive to the PZT responsewhen the loop gain is large. The measurement results are shown in Figure 6.12.Even with the dense sampling of frequencies, we do not observe the kind of responsethat our predecessors observed at 2.2 kHz or 2.5 kHz. The only visible featurethat resembles a resonance in this measurement is at 9.5 kHz. It is blown up in

104

100

101

V err/V

c

(a)

103 104

frequency (Hz)

150

100

50

0

50

100

phas

e (

)

(b)

Figure 6.12. The loop response when the system is modulated by the PZT and lockedwith PZT. The amplitude (a) and phase (b) of the error signal sent to the PID circuit,Verr, divided by the control voltage of the PZT, Vc vs. the modulation frequency.

Figure 6.13a(b) (after removing the 12db/octave slope). Its amplitude responseis like a dispersive shape and the phase response is like a Lorentzian, whereas anormal resonant amplitude response is like a Lorentzian and a normal resonantphase response is like a dispersive shape. Assuming that the unity loop gain occurswhere the amplitude deviates from the 12db/octave, the loop gain at 9.5 kHz is onlyabout 2. The maximum amplitude deviation there is only of a factor of 2 and themaximum phase deviation is only 80. These abnormal features at 9.5 kHz can beattributed to the 1 in the denominator of Equation (6.21). This 1 reveals the PZTresonance that would otherwise be hidden by the feedback. But it also significantlyalters the response as given by Equation (6.21). By inverting Equation (6.21), wecan deduce the loop gain, GL, from the measured response Verr/Vc, as shown inFigure 6.13cd, which recover the typical response of a resonance.

Even though we can infer the loop gain by the inverting process, the response of

105

0.8

1.0

1.2

1.4

1.6

1.8Am

plitu

de(a)

1.00

1.25

1.50

1.75

2.00

2.25

2.50 (c)

8000 9000 10000 11000frequency (Hz)

10

20

30

40

50

60

70

phas

e (

)

(b)

8000 9000 10000 11000frequency (Hz)

120

130

140

150

160

170(d)

Figure 6.13. (a) and (b) show zoomed-in views of Figure 6.12 around 9.5 kHz afterremoving the 12db/octave slope. (c) and (d) show the response of GL deduced from (a)and (b) by inverting Equation (6.21)

the system when modulated by the PZT is far less sensitive to the resonances thanthe response when modulated by the frequency of the laser. We have decided tomodulate the system by the PZT for convenience reasons. We achieve a lock-servobandwidth of 50 kHz after passive damping. But if we need to study the mechanicalresonances around 2 or 2.5 kHz, it is certainly more effective to measure the responsewhen the frequency of the laser is modulated.

The dominant noise in the cavity system occurs around 40 Hz, which can bedetected using a model 728A accelerometer (manufactured by Wilcoxon) mountedon the breadboard holding the top optics. The vibration noises in the x and ydirection are below the sensitivity of the accelerometer. But the accelerometerdetects an acceleration of 2 mV amplitude along the z. Using the calibration of the

106

accelerometer of 500 mV/g, we deduce that the intrinsic noise in the cavity is ofamplitude 615 nm. Using a lock-servo of 50 kHz bandwidth, the residual noise inthe cavity length is about half a nanometer.

6.5 Launch by PushingThe lattices are far red-detuned from the atomic resonances of cesium and theycan trap atoms in the intensity maxima. After the atoms are loaded into oneof the lattices from the MOT, we launch the atoms to the region shielded frommagnetic fields. In contrast to acceleration by moving molasses described by ourpredecessors [4,8,9], we accelerate the atoms using the pushing force of laser beamsfrom the bottom, before finally cooling them in moving molasses before releasingthem. We begin by turning on the bottom MOT beams (that is the pair of MOTbeams pointing 45 tiled upward) while keeping the top MOT beams off. Thenet force from absorbing photons from the bottom MOT beams is vertically up,while the net force from spontaneous emission is zero. As a result, the atoms areaccelerated upward as shown in Figure 6.14. Though the atoms are farther awayfrom resonance when they are moving due to the Doppler shift, the Doppler shiftat 4 mm/s is small compared to the baseline detuning of the bottom MOT beam.Hence the acceleration should be constant. However, the measured data deviatesfrom constant acceleration. The measured position after 1 ms falls short of theposition expected from constant acceleration. This deviation is likely caused by 2details of the measurement. First, the atoms have moved by about 1 mm duringthe 0.5 ms exposure time of the camera. So the measurement of position has abouta 1 mm of uncertainty when the atoms are moving. This effect can be reducedwith shorter exposure time at the cost of a weaker signal. Second, the image ofatoms is probed with the MOT beams, which have a Gaussian profile centered at 1mm in height. As the atoms move away from the center of the MOT beams, theyexperience a weaker probe. The image of the atoms, after being elongated dueto heating from acceleration, has a Gaussian center lower than that of the atomsthemselves.

Fitting the data before 1 ms to a parabola, we find the acceleration is 6.1mm/ms2. So the atoms reach 4.067 mm/ms, the speed required to be launched tothe region between the plates, in 0.67 ms.

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0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4Time since the start of launch (ms)

0

2

4

6Ve

rtica

l pos

ition

(mm

)

Fit to y = 12at2

Data

Figure 6.14. Movement of the center of the atom cloud after applying a push with thebottom MOT beams. The error bars represent the standard errors of the centers of theclouds obtained by fitting the atom distributions to Gaussian functions, and they areabout the size of the points. Fitting the motion to a parabola is performed with databefore 1 ms. The data falls short of the fit after 1 ms because of the smearing and unevendistribution of the probe beams as explained in the text.

After they reach the target speed, we cool the atoms with moving molasses in 2stages. In the first stage, we cool with high intensity at 17 MHz detuning for 0.2ms. In the second stage, we cool with low intensity at 13 MHz detuning for 1 ms.In order to make sure the atoms are still within the 6 molasses beams of 10 mmsize during the cooling stages, we use the maximum pushing force available duringthe acceleration stage.

The number of atoms launched to the target height varies with the trap depths(Figure 6.15). More atoms reach the target height with deeper lattices as expected.But the gain in atoms launched gradually decreases as the trap depth increases.

6.6 Stopping and Cooling Atoms between the PlatesWe stop and cool atoms using polarization gradient cooling after they reach thetarget height. The cooling beams consist of 3 pairs of counter-propagating beamswith mixed polarization. One pair of them is nearly vertical and the beams are

108

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00

20000

40000

60000

80000

100000

120000

-Z+ZN

um

ber

of

atom

s (a

rbitra

ry u

nits)

Trap depth

Figure 6.15. Atoms launched to the target height versus trap depth

at only 5 mrad angle to the lattice beams. This small angle provides room forplacing optics without degrading the cavity finesses. The beam coming from thetop is linearly polarized while the beam from the bottom is circularly polarized.The beam from the bottom can be shifted to being resonant with the cyclingtransition to provide the probe beam when imaging the atoms. The other 2 pairsare horizontal. They emerge out of 2 fiber cables fed inside the shields, are expandedin the vertical direction and shine upon the plates at roughly ±30 to the normal.The reflection from the high reflection coating surfaces of the center plate providesthe counter-propagating beams. The beams coming out of the 2 fiber cables arepolarized differently. One is S polarized and the other is P polarized.

The alignment of the vertical cooling beams can be easily set by referencing thelattice beams. Though we cannot directly check the overlap between the coolingbeam and the lattice beam inside the vacuum, we can reflect both beams off thevacuum using a pellicle beam splitter, and make sure the reflections overlap ata distance equal to the distance from the plates. By pulsing on only the pair ofbalanced vertical cooling for 0.4 ms, we are able to stop in the lattices half of theatoms that reach the volume between the plates. With this 1-D cooling, keepingthe vertical cooling on for a longer time heats the atoms transversely and eventuallyleads to loss of atoms from the trap as shown in Figure 6.16.

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0.0 0.2 0.4 0.6 0.8 1.0 1.2

Vertical stopping duration (ms)

46

48

50

52

54

56

58

60

62

64

Ato

ms

stopped (

%)

Figure 6.16. Atoms stopped in the lattice with vertical cooling alone vs. cooling pulseduration. The number of atoms stopped increases with cooling duration initially butpeaks around 0.4 ms, after which the atoms are heated out of the trap.

To cool the atoms further, we need transverse cooling in addition to the verticalcooling. In order to align the horizontal beam to the atoms, we could use oneof the horizontal beams as the probe and detect the fluorescence from the atoms.However, this method is hindered by horizontal light scattered into the auxiliaryimaging system and the fluorescence cannot be easily separated from the scatteredprobe light. Luckily, being able to stop atoms with just the vertical cooling beamsprovides another signal for aligning the horizontal beams. The signal is the lossof atoms after shining one horizontal beam onto the atoms stopped with verticalcooling alone. The atoms are probed using one of the vertical beams. This does notinterfere with fluorescence imaging because the vertical cooling beams are small (1mm) and have been aligned to avoid scattering. After one horizontal beam has beenaligned to push away atoms most effectively, the counter-propagating beam can bealigned to overlap with it. The alignment of 3-D cooling is thereby completed.

After 2 ms of 3-D polarization gradient cooling, we observe the breathing modeof the transverse size of the atom cloud as shown in Figure 6.17. The breathingmode was also observed by our predecessors [9]. But we do not observe the sloshingmode as reported by our predecessors [9], presumably because we have better pointto point intensity balance for the cooling beams. The transverse center of the atomcloud in the z direction stays the same within our 20 µm resolution. Averaging

110

0 5 10 15 20 25 30Time after stopping pulse (ms)

0.36

0.38

0.40

0.42

0.44

0.46

0.48

0.50

Full

wid

th a

t half m

axim

um

(m

m)

Figure 6.17. Time evolution of the width of the cloud after cooling showing breathingbehavior. The oscillation frequency is about twice the trap frequency. The oscillationdamps out due to dephasing among the atoms.

along the vertical direction could disguise a sloshing mode of individual sections ofthe atom cloud. To check against this averaging effect, we divide the entire atomclouds into 1.6 mm sections in the vertical direction and find that the z coordinateof the center of each section stays the same as well. These observations, however,do not exclude a sloshing mode in the x direction, which is the viewing direction ofthe auxiliary imaging system.

Both the breathing mode and sloshing mode are the results of conversion betweenpotential energy and kinetic energy after the atoms are cooled at some potentialin the trap. The breathing mode has twice the trap frequency while the sloshingmode has the same frequency of the trap. In order to cool the atoms further, wepulse on the cooling beams again when the kinetic energy peaks, that is when thepotential energy hits its minimum, to further remove the kinetic energy. After that,the atom cloud continues to breathe but with reduced amplitude. After 2 to 3rounds of cooling, the breathing can no longer be observed and the transverse sizestays around 0.47 mm (FWHM). With a trap depth of 140 µK, that correspondsto an atom temperature of 30 µK.

In order to have the transverse size of the atom cloud represent the temperatureof atoms, we choose a short probing time to avoid heating from the probe. Asthe transverse size of the atom cloud becomes smaller, the atoms become colder,

111

and we have to progressively reduce the probing time so that the heating from theprobe does not significantly alter the transverse size of the atoms.

6.7 Optical PumpingAfter the atoms are cooled in the lattices between the plates, we need to preparethe atoms either in the superposition of the stretched states or the state with m=0depending on the measurement scheme. In any measurement scheme, it is necessaryto have most atoms in the same angular momentum state. We do that by opticallypumping atoms to one of the stretched states. In particular, we use σ− polarizedlight resonant at F = 4 to F ′ = 4 transition to pump all the atoms to F = 4,m = 4 (abbreviated as |4, 4〉). Once the atoms reach |4, 4〉, they no longer scatterthe optical pumping light and become dark, provided that the optical pumpinglight is purely circularly polarized and that its propagation direction is alignedwith the magnetic field. If either condition is not satisfied, atoms in |4, 4〉 precessaway from m=4 or are scattered into other Zeeman levels or the other hyperfinelevel. When both conditions are satisfied, optical pumping to the stretched statecauses minimum atom heating. To achieve these conditions, we scan the transversecomponents of the magnetic field and the waveplates for generating the circularlypolarized light to minimize the population driven away from |4, 4〉 state by theoptical pumping light alone. We begin by turning on both the optical pumpinglight and repumping light for 1 ms. Then we keep the optical pumping light onand turn off the repumping light for a time ranging from 2 ms to 1 s. This processis called depumping. After depumping, the atoms are either in |4, 4〉 or F = 3hyperfine level. We clear all the atoms in |4, 4〉 by turning on a traveling beam atthe cycling transition from F = 4 to F ′ = 5. After atoms in F = 4 are clearedaway, atoms in the F = 3 hyperfine level can be detected by turning on both therepumping light as well as the light at the cycling transition. The optimal opticalpumping is achieved when the atoms in the F = 3 hyperfine level is minimum.

Sample data for optimizing the optical pumping using the depumping techniqueis shown in Figure 6.18. With longer depumping time, more atoms are depumpedinto the F = 3 hyperfine level. When we scan a magnetic field or the polarization,we pick a depumping time with the maximum sensitivity (roughly at half of themaximum signal in Figure 6.18a). We pick the magnetic fields (see Figure 6.18b)

112

0 5 10 15 20

Depumping duration (ms)

102030405060708090

100

ato

ms

in F

=3 (

%)

(a)

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8

bz (V)

10

15

20

25

30

35

40

45

50

ato

ms

in F

=3 (

%)

(b)

Figure 6.18. (a) Atoms detected in the F = 3 hyperfine level after depumping ofvarying duration. (b) Atoms detected in F = 3 after 2 ms of depumping at varying biasmagnetic field in the x direction.

or polarization that minimizes the depumping rate. As we continue to decreasethe depumping rate, the slope of the curve as in Figure 6.18a becomes more gentle.And we can improve the sensitivity by appropriately increasing the depumpingtime.

113

Chapter 7 |Conclusion

This dissertation describes my contributions to the eEDM measurement using coldatoms trapped in the optical lattices. We upgraded the imaging system from pairsof Fresnel lenses to stacks of Cooke triplets. The Cooke triplets have resolutionsmatched to the pixel size of our imaging sensors, thereby completely resolving theblurring issue present in the previous implementation of the imaging system. Wehave gained more understanding of the HV leakage current through the vacuum,that has stopped us from turning up the voltage. While the team will continue todecrease the leakage current and to increase the high-voltage, an EDM measurementwith a smaller electric field can be performed using the precession from the m = 0magnetic sublevel. We have restored the vacuum and the ability to cool and trapatoms in the volume between the plates.

At the time of writing this dissertation, Teng Zhang has upgraded the lasersystem with a tapered amplifier and improved the number of atoms trapped betweenthe plates by a factor of 2. He also eliminated a major source of noise to our cavitylock system after replacing the fan in the HVAC system. With the magnetic shieldssoon to be closed, the residual vector light shift with the upgraded vacuum windowwill be found out. With proper field cancellation, a measurement of the tensorpolarizability and a measurement of the EDM can be performed in a year.

While the first measurement of the electron EDM using cold atoms may notbe immediately competitive, future upgrades are possible. The electric field canbe increased by conditioning with a thick layer of ITO. The number of atoms canbe increased by adding transverse cooling after the nozzle. Measurements of theelectron EDM using cold atoms will provide strong systematic checks of the electronEDM obtained using molecular EDMs.

114

Finally, I’m proud to be part of the ambitious community that strives to answerthe universe’s biggest questions using tabletop experiments.

115

Appendix A|Insensitivity of the Probabilityin Even Magnetic Sublevels toQuadratic Energy Shifts in theTransverse Direction

This is a convoluted proof. So I present the condensed proof in the first sectionand fill in the details in the later sections.

A.1 Condensed but Complete ProofWe defineme andmo for the even and odd magnetic quantum numberm respectivelyfor integer F ; or me and mo for the even+1/2 and odd+1/2 magnetic quantumnumber m respectively for half integer F . The completeness relation reads:

1 =∑m

|m〉 〈m| =∑me

|me〉 〈me|+∑mo

|mo〉 〈mo| (A.1)

The operator for a measurement of probability to be in the even magnetic sublevelsin the x basis is Pex = ∑

me |me〉x 〈me|x. Since the Hamiltonian is more convenientlywritten in the z basis, we also write Pex in the z basis using the passive rotation|m〉x = Dy(π/2) |m〉z, where Dy(θ) is the Wigner rotation around y:

Pex =∑me

Dy(π/2) |me〉z 〈me|z D−1y (π/2). (A.2)

116

Because we will work exclusively in the z basis henceforth, we will drop the zsubscripts for the kets and bras. We will rewrite Eq. (A.2) using the followingequation:

∑me

|me〉 〈me|D−1y (π/2) =1

2∑me

Dy(π/2) + D−1y (π/2) |me〉 〈me|

+12∑mo

D−1y (π/2)−Dy(π/2) |mo〉 〈mo| . (A.3)

To prove Eq. (A.3), we expand the rotation operators D using the completenessrelation (A.1) and write 〈m′|Dy(π/2) |m〉 as dm′m(π/2) and 〈m′|D−1

y (π/2) |m〉 asd†m′m(π/2). d†m′m(π/2) is the same as dmm′(π/2) because Dy is unitary and real.The sum over me in the right-hand side of Eq. (A.3) becomes:

12∑me

Dy(π/2) + D−1y (π/2) |me〉 〈me| =

12∑m′me

dm′me(π/2) + dmem′(π/2) |m′〉 〈me|

(A.4)

=∑m′eme

dm′eme(π/2) |m′e〉 〈me| . (A.5)

From (A.4) to (A.5), we separatem′ intom′e andm′o:∑m′ |m′〉 〈me| =

∑m′e|m′e〉 〈me|+∑

m′o|m′o〉 〈me|. The |m′e〉 〈me| terms double and the |m′o〉 〈me| terms drop out be-

cause of dmm′(π/2) = (−1)m′−mdm′m(π/2), which can be shown using the explicitWigner’s formula found in many standard quantum mechanics textbooks such asRef. [36]. Similarly, the sum over mo in the right-hand side of Eq. (A.3) can bewritten as:

12∑mo

D−1y (π/2)−Dy(π/2) |mo〉 〈mo| =

∑m′emo

d†m′emo

(π/2) |m′e〉 〈mo| . (A.6)

The sum of Eq. (A.5) and Eq. (A.6) yields the left-hand side of Eq. (A.3).Using Eq. (A.3), Eq. (A.2) becomes:

Pex = 12 + 1

2D2y (π/2)(

∑me

|me〉 〈me| −∑mo

|mo〉 〈mo|). (A.7)

D2y (π/2) is the same as Dy(π), which can be written explicitly for both integer and

117

half integer F as:Dy(π) =

∑m

(−1)F−m |−m〉 〈m| . (A.8)

It is noted that Dy(π) contains only anti-diagonal components. Using Eq. (A.8),Pex can be written explicitly in the z basis as:

Pex = 12 + 1

2(−1)bF c∑m

|−m〉 〈m| , (A.9)

where bF c is the floor of F . bF c = F for integer F and bF c = F − 1/2 for halfinteger F .

Suppose the Hamiltonian can be expanded in powers of Fz (or m) in the z basis:

H = ωFz + α2F2z + α3F

3z + ..., (A.10)

where αn are the coefficients of the nth order interactions. The measurementoperator in the Heisenberg picture is:

Pex(t) = eiHt/~Pexe−iHt/~ = 1

2 + 12(−1)bF c

∑m

|−m〉 〈m| e−2ωm−2α3~2m3.... (A.11)

The diagonal components in Pex do not interact with the Hamiltonian. The anti-diagonal components interact with only the parts of Hamiltonian that are oddwith respect to m. The parts of the Hamiltonian that are even with respect tom, including quadratic interactions, drop out. Therefore, the probability to be inthe even magnetic sublevels for an integer angular momentum or in the even+1/2magnetic sublevels for a half integer angular momentum is insensitive to quadraticenergy shifts with respect to the magnetic quantum number m.

A.2 Symmetry/Antisymmetry of a π/2 RotationThis section proves the relation:

dmm′(π/2) = (−1)m′−mdm′m(π/2), (A.12)

118

using Wigner’s formula. Wigner’s formula for d(j)m′m(β) is the following [36]:

d(j)m′m(β) =

∑k

(−1)k−m+m′

√(j +m)!(j −m)!(j +m′)!(j −m′)!

(j +m− k)!k!(j −m′ − k)!(k −m+m′)!

×(cos

β

2

)2j−2k+m−m′ (sin

β

2

)2k−m+m′

. (A.13)

Setting β = π/2 in cos and sin, we obtain

d(j)m′m(π/2) =

∑k

(−1)k−m+m′

√(j +m)!(j −m)!(j +m′)!(j −m′)!

(j +m− k)!k!(j −m′ − k)!(k −m+m′)!

× 2−j. (A.14)

I introduce a dummy variable k′ = k −m+m′. Substituting k = k′ +m−m′ intothe above equation and I obtain

d(j)m′m(π/2) =

∑k′

(−1)k′

√(j +m)!(j −m)!(j +m′)!(j −m′)!

(j +m′ − k′)!(k′ −m′ +m)!(j −m− k′)!(k)!

× 2−j. (A.15)

Since k′ is a dummy variable, I might as well replace it with k in the summation.The right hand side of equation A.15 is recognized as d(j)

mm′(π/2) up to a sign thatdepends on the difference between m and m′:

d(j)m′m(π/2) = (−1)m−m′

d(j)mm′(π/2). (A.16)

Q.E.D.

A.3 Explicit Formula of a π RotationThis section shows Eq. (A.8), the β = π case of Wigner’s formula (equation (A.13)).The cosine term in equation (A.13) becomes zero or singular when β = π unlessthe power of the cosine term is zero. So the nonzero elements of d(j)

m′m(π) must have

119

the following:

0 = 2j − 2k +m−m′ = (j +m− k) + (j − k −m′). (A.17)

The two terms in parentheses are the arguments of factorials in the denominatorand must be non-negative. For all the above conditions to hold, we must have:

j +m− k = j − k −m′ = 0. (A.18)

That implies m = −m′ and k = j + m. The former implies only anti-diagonalelements of d(j)

m′m(π) are non-zero, and the latter reduces the summation over k toa single term. Wigner’s formula becomes:

d(j)m′m(π) = δmm′(−1)j−m, (A.19)

which proves Eq. (A.8).

120

Appendix B|Automated Calibration of theAOMs

B.1 IntroductionOne of the routines working in an AMO lab is the calibration of acousto-opticmodulators in order to ensure the right laser power. Laser power affects the resultof laser cooling. The power of the lasers is often controlled by the power of RFsent to the AOMs. The power of the RF is again controlled by a voltage set by thetiming Program. The relationship between the control voltage and the power ofthe laser after the AOM can change over time. Therefore it’s important to updatethe control voltage in the timing sequence on a regular basis in order to maintainthe best cooling configuration. The calibration of the control voltage and the laserpower was carried out by hand previously. Depending on the number of samples inthe calibration, this manual acquisition process could take a long time. Therefore Iimplemented an automated AOM calibration system to speed up this routine job.

B.2 The AOM Calibration systemThe automated calibration system is very simple. It consists of one or more digitaloutput channels controlling the RF power driving the AOMs and one or more analoginput channel to read the power of the laser from a power meter or a photodiode.The program named AOMcalibration takes in the range of the voltage scan and thenumber of samples from the user. It sets the control voltage as one of the voltages

121

in the scan range, waits for a time t for the reading of the laser power to stabilizeand reads the laser power from the analog input channel. Immediately after thevoltage is acquired, it moves on to the next voltage and repeats the process until thelast voltage in the range. After all samples are acquired, the program immediatelyplots the laser power versus voltage curve on the screen. The wait time t I use is 1seconds. If 60 samples are required, the calibration process can be completed inone minute. If more samples are required, the program can be set to run by itselfwithout human intervention.

The automated calibration system does not alter the existing timing sequenceprogram Supertime but rather is built upon the existing Supertime hardware. Ituses the same digital output channels from Supertime. But because the Supertimedoes not have any analog input, it requires a separate analog input card (for instanceNI USB-6008). The software, however, is completely separate from the Supertime.The AOMcalibration uses the updated DAQmx application programming interface,whereas the Supertime uses the obsolete DAQmx application programming interfacefrom National Instruments. The AOMcalibration uses the same channel specificationas the Supertime. To specify a digital channel in AOMcalibration, the user canlook up the corresponding channel number and card number in Supertime.

B.3 ConclusionUnlike Supertime, automated AOM calibration is not necessary for running theexperiment, because the same job can be accomplished by hand. Nonetheless, theautomated AOM calibration liberates the experimenters from the routine job sothat they can focus on something more interesting and they will not hesitate toexecute it when a calibration is in demand. This program also verifies at least thatpart of the existing hardware of Supertime is compatible with DAQmx.

122

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VitaCheng Tang

Cheng was born in China. He moved to Singapore in 2005 and obtained hisBachelor degree in physics from National University of Singapore in 2010. Hestarted graduate study in physics at Penn State in 2011. Since then, he has beenworking on the search for the permanent electric dipole moment of the electron.