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S PIN EXCHANGE O PTICAL PUMPING OF N EON AND ITS A PPLICATIONS R AJAT K. G HOSH A D ISSERTATION P RESENTED TO THE FACULTY OF IN P RINCETON U NIVERSITY C ANDIDACY FOR THE D EGREE OF D OCTOR OF P HILOSOPHY R ECOMMENDED FOR A CCEPTANCE BY THE D EPARTMENT OF P HYSICS A DVISER : M ICHAEL V. R OMALIS N OVEMBER 2009 c Copyright by Rajat K. Ghosh, 2009. All Rights Reserved Abstract Hyperpolarized noble gases are used in a variety of applications including medical diagnostic lung imaging, tests of fundamental symmetries, spin filters, atomic gyroscopes, and atomic magnetometers. Typically 3 He is utilized because large 3 He polarizations on the order of 80% can be achieved. This is accomplished by optically pumping an alkali vapour which polarizes a noble gas nucleus via spin exchange optical pumping. One hyperpolarized noble gas application of particular importance is the K3 He co-magnetometer. Here, the alkali atoms optically pump a diamagnetic noble gas. The magnetic holding field for the alkali and noble gas is reduced until both species are brought into hybrid magnetic resonance. The co-magnetometer exhibits many useful attributes which make it ideal for tests of fundamental physics, such as insensitivity to magnetic fields. The co-magnetometer would demonstrate increased sensitivity by replacing 3 He with polarized 21 Ne gas. Tests of CPT violation using co-magnetometers would be greatly improved if one utilizes polarized 21 Ne gas. The sensitivity of the nuclear spin gyroscope is inversely proportional to the gyromagnetic ratio of the noble gas. Switching to neon would instigate an order of magnitude gain in sensitivity over 3 He. In order to realize these applications the interaction parameters of 21 Ne with alkali metals must be measured. The spin-exchange cross section σse , and magnetic field enhancement factor κ0 are unknown, and have only been theoretically calculated. There are no quantitative predictions of the neon-neon quadrupolar relaxation rate Γquad . In this thesis I test the application of a K-3 He co-magnetometer as a navigational gyroscope. I discuss the advantages of switching the buffer gas to 21 Ne. I discuss the feasibility of utilizing polarized 21 Ne for operation in a co-magnetometer, and iii construct a prototype 21 Ne co-magnetometer. I investigate polarizing 21 Ne with optical pumping via spin exchange collisions and measure the spin exchange rate coefficient of K and Rb with Ne to be 2.9 × 10−20 cm3 /s and 0.81 × 10−19 cm3 /s. We measure the magnetic field enhancement factor κ0 to be 30.8 ± 2.7, and 35.7 ± 3.7 for the K-Ne, and the Rb-Ne pair. We measure the quadrupolar relaxation coefficient to be 214 ± 10 Amagat·s. Furthermore the spin destruction cross section of Rb, and K with 21 Ne is measured to be 1.9 × 10−23 cm2 and 1.1 × 10−23 cm2 . iv Acknowledgements I would first like to thank my advisor Michael Romalis for his help in the lab. His passion to study fundamental physics is what drives the entire lab group. Without his assistance and guidance this work would not have been possible. Of special note is his availability to discuss the underlying physics so that one can gain a deeper view of the atomic processes which we study. The projects described herein have greatly benefited from the fruitful collaboration with my colleagues in the Romalis lab. These projects could not have been accomplished by me alone. Although I could never fully express my appreciation and admiration for my fellow colleagues I would still like to take the time to thank them each individually. First I would like to thank Tom Kornack. He built the first generation CPT violation experiment. Without his help I would not have been able to get any gyroscope data. I would also like to thank Γιὼγos Bασιλὰκηs, and Sylvia Smullin for their help with the scalar magnetometer experiment. I would also like to thank Vishal Shah for his help with regards to the spin exchange rate measurements. I really appreciate all the time you have taken to discuss all things atomic physics. This work has benefited from the support of the many staff members in the physics department. I would like to thank Bill Dix for his help in machining the parts of my experiment when I was unable to do so. I would also like to thank Ted Lewis in the metal stockroom for all of his help. I would be remiss if I did not thank Mike Peloso for all his help teaching me how to machine in the student shop and always suggesting alternative designs, or easier ways in which to machine my designs. I will miss our talks, or more accurately shouts, about guitars, and music theory over the commercial lathe and drill press. I would like to thank Mary Santay, Barbara Grunwerg, Kathy Warren, John Washington, from the purchasing and receiving department. You really make it a v pleasure to come visit the A level. I would also like to thank Claude Champagne of the purchasing department for always taking the time to try and make all of us smile. I would also like to thank Regina Savadge, Ellen Webster Synakowski, and Mary Delorenzo as the vanguards of the atomic physics group. I would also like to thank Mike Souza for the glass cells which only he could so expertly craft. I would like to give a special thanks to Laurel Lerner for all her help in all things, especially near the end of my time here. During my tenure here I have spent much of my time with my fellow grad students in the deepest darkest Jadwin. You have been both great colleagues, and even better friends. First I must thank Scott Seltzer for our many tea time conversations about not only atomic physics, but politics, cinema, and the finer points of life in general. I do miss our discussions about british comedy, and the beauty of a proper Jewish Deli and the perfect matza ball soup. I would like to thank the enthusiasm Justin Brown has shown over the years. If only we could all have the energy you do I think the World wouldn’t need coffee. As for my good friend Γιὼγos Bασιλὰκηs I will miss your contagious love of science, and being able to debate the biggest proponent of the Greek culture. My only hope is that they do not one day find the work of my thesis had already been scribed on one of Aristotle’s tablets. I would also like to thank Ranjit Chima. Friends like you are very rare. I think life in Princeton would have not been nearly as fun had you not been here. I would also like to acknowledge all the past members of the atomic physics group from whom I have learnt a great deal over the years including Micah Ledbetter, Hui Xia, Kiwoong Kim, Charles Sule, Andre Baranga, Oleg Polyakov, Parker Meares, Scott Seltzer and Dan Hoffman. I will miss my conversations with Dan about all things, and with Kiwoong about when it is appropriate to acknowledge commoners. vi I would like to give thanks to my readers, and my committee members for taking the time to read my thesis and giving useful suggestions for the improvement of this manuscript. I am also grateful to Scott Seltzer for his careful reading of the first draft of my thesis. Finally and most importantly I would like to thank all of my family. Without you none of this would have been possible. I would like to thank my sister Sheila for all her support. Your dedication to your own work encourages me to always stay positive and persevere. You have always listened to me when I needed comforting, and cheered me when I need cheering. I don’t think I could have gone through this without you. I have to give a special thanks to my parents. Without your guidance none of what I have accomplished would have been possible. I would like to thank my mother for her constant encouragement, her support, and her unwavering belief in me. Without you pushing me when I needed to be pushed, and encouraging me when I needed to be encouraged I would not have made it this far. I must also thank my father for his constant help in both discussing all thing physics, and in his advice. From the time I was little you have always helped me in my education, and taught me a great deal. Your constant encouragement has meant all the difference to me. To all of my family I owe you a debt which I can never acknowledge enough, or ever justly repay. You inspire me. vii Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v Table of Contents x List of Tables xi List of Figures xiv 1 Introduction 1 2 Background 7 2.1 Optical Pumping background . . . . . . . . . . . . . . . . . . . . . . . 8 2.1.1 Atomic Energy Levels . . . . . . . . . . . . . . . . . . . . . . . 8 2.1.2 Optical Pumping . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.1.3 Effects of Pumping Rate and resonance Lineshape on Optical Pumping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.2 2.1.4 Evolution of Alkali polarization due to Optical Pumping . . . 17 2.1.5 Dynamics of polarized alkali at low magnetic fields . . . . . . 20 Spin Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.2.1 Spin Exchange Collisions . . . . . . . . . . . . . . . . . . . . . 24 2.2.2 Spin Destruction Collisions . . . . . . . . . . . . . . . . . . . . 29 2.2.3 Diffusion wall collisions, and Magnetic field Gradients . . . . 30 viii 2.3 2.4 2.5 3 4 Monitoring polarized Alkali . . . . . . . . . . . . . . . . . . . . . . . . 32 2.3.1 Optical Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.3.2 Light Shifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 Coupled Spin Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.4.1 Optical Pumping of Noble Gas . . . . . . . . . . . . . . . . . . 42 2.4.2 Interaction of polarized alkali with polarized noble gas . . . . 44 Manipulation of polarized noble gas spins, and Magnetic shielding . 48 2.5.1 Adiabatic Fast Passage . . . . . . . . . . . . . . . . . . . . . . . 49 2.5.2 Nuclear Magnetic Resonance . . . . . . . . . . . . . . . . . . . 52 2.5.3 Magnetic Shielding . . . . . . . . . . . . . . . . . . . . . . . . . 55 Nuclear Spin Gyroscope 59 3.1 Co-magnetometer Gyroscope Implementation and behaviour . . . . 60 3.2 Effect of Experimental Imperfections on Gyroscope Performance . . 66 3.3 Zeroing the Co-magnetometer Gyroscope . . . . . . . . . . . . . . . . 67 3.4 Co-magnetometer Gyroscope Sensitivity . . . . . . . . . . . . . . . . . 70 Initial tests of an alkali-Neon co-magnetometer 75 4.1 Magnetometer setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 4.2 Neon Polarization Measurements and Preliminary Neon Co-Magnetometer data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 4.3 Influence of Quadrupole collisions in Polarizing neon nuclei . . . . . 77 4.3.1 4.4 T1 measurement of Neon . . . . . . . . . . . . . . . . . . . . . 80 Improving Magnetometer Sensitivity . . . . . . . . . . . . . . . . . . . 84 4.4.1 Removing Birefringence and false Faraday Rotation signals . 84 4.4.2 Controlling and Monitoring the Laser stability . . . . . . . . . 85 4.4.3 Miniaturization of Gyroscope . . . . . . . . . . . . . . . . . . . 86 4.4.4 Alternate methods to heat Cell, and remove Convection noise 87 ix 5 6 Measurement of parameters for Polarizing Ne with K or Rb metal 91 5.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 5.2 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 5.2.1 NMR detection . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 5.2.2 Electron Paramagnetic Resonance Shift . . . . . . . . . . . . . 99 5.2.3 Alkali Polarization . . . . . . . . . . . . . . . . . . . . . . . . . 102 5.2.4 Alkali Polarization Decay Constant measurement . . . . . . . 103 5.2.5 Back Polarization measurement . . . . . . . . . . . . . . . . . . 104 5.2.6 Alkali density measurement . . . . . . . . . . . . . . . . . . . . 105 5.3 Fermi Contact interaction κ0 Results . . . . . . . . . . . . . . . . . . . 107 5.4 Results of neon quadrupolar relaxation Γquad measurement . . . . . . 107 5.5 Spin exchange Rate coefficient Results . . . . . . . . . . . . . . . . . . 108 5.6 Measurement of Spin destruction cross-sections of neon with Rb and K110 5.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 Feasibility of utilizing 21 Ne in a co-magnetometer 6.1 114 Effects of Light Propogation and alkali relaxation on Rb-Ne co-magnetometer simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 6.2 Simulation of Noble gas relaxation . . . . . . . . . . . . . . . . . . . . 117 6.3 Noise mechanisms in a Rb-Ne co-magnetometer . . . . . . . . . . . . 118 6.4 7 6.3.1 Spin Projection Noise . . . . . . . . . . . . . . . . . . . . . . . . 119 6.3.2 Photon Shot Noise . . . . . . . . . . . . . . . . . . . . . . . . . 120 Results of Rb-Ne co-magnetometer simulation . . . . . . . . . . . . . 122 Conclusions and future work 125 A Properties of Ne21 129 x List of Tables 2.1 Line broadening and shift of K in various Gases . . . . . . . . . . . . 16 2.2 Spin Destruction Cross Sections . . . . . . . . . . . . . . . . . . . . . . 30 3.1 Gyroscope performance comparison . . . . . . . . . . . . . . . . . . . 74 5.1 K-Ne spin exchange parameters . . . . . . . . . . . . . . . . . . . . . . 110 5.2 Rb-Ne spin exchange parameters . . . . . . . . . . . . . . . . . . . . . 110 5.3 Fermi Contact Interaction . . . . . . . . . . . . . . . . . . . . . . . . . 110 5.4 Rb-Ne and K-Ne spin destruction cross sections. . . . . . . . . . . . . 112 7.1 K-Ne spin exchange parameters . . . . . . . . . . . . . . . . . . . . . . 125 7.2 Rb-Ne spin exchange parameters . . . . . . . . . . . . . . . . . . . . . 126 7.3 Fermi Contact Interaction . . . . . . . . . . . . . . . . . . . . . . . . . 126 7.4 Rb-Ne and K-Ne spin destruction cross sections. . . . . . . . . . . . . 126 A.1 Properties of Ne21 relevant for optical pumping . . . . . . . . . . . . 130 xi List of Figures 1.1 Basic operation of an Atomic magnetometer . . . . . . . . . . . . . . . 3 2.1 Alkali metal energy level diagram . . . . . . . . . . . . . . . . . . . . 9 2.2 Optical pumping of the electron spin of an alkali atom . . . . . . . . . 12 2.3 Ground-state Zeeman level splitting . . . . . . . . . . . . . . . . . . . 26 2.4 Spin-exchange collisions can cause atoms to switch hyperfine levels . 26 2.5 Spin-temperature distribution . . . . . . . . . . . . . . . . . . . . . . . 28 2.6 Principle of optical rotation . . . . . . . . . . . . . . . . . . . . . . . . 33 2.7 Branching ratios for the D1 and D2 transitions . . . . . . . . . . . . . 35 2.8 Methods for detecting optical rotation . . . . . . . . . . . . . . . . . . 36 2.9 Methods for detecting optical rotation . . . . . . . . . . . . . . . . . . 37 2.10 Polarized Noble gas screens transverse fields . . . . . . . . . . . . . . 48 2.11 Effective magnetic field in the rotating frame . . . . . . . . . . . . . . 51 2.12 NMR tip of atoms with spin ~S . . . . . . . . . . . . . . . . . . . . . . . 54 2.13 Affinity of Magnetic fields line for Magnetic Shields . . . . . . . . . . 57 3.1 Schematic of the co-magnetometer implemented as a gyroscope . . . 61 3.2 Side view of the gyroscope configuration for the co-magnetometer . 62 3.3 In-situ calibration of non-contact displacement sensors. . . . . . . . . 62 3.4 Comparison of co-magnetometer gyroscope signal to displacement sensor signal with no free parameters . . . . . . . . . . . . . . . . . . 63 xii 3.5 noise spectrum of the comagnetometer gyroscope . . . . . . . . . . . 63 3.6 Suppression of an applied magnetic field gradient by the co-magnetometer compared to that of a non-compensating magnetometer . . . . . . . . 64 3.7 The co-magnetometer suppressing magnetic fields . . . . . . . . . . . 65 3.8 Response of co-magnetometer to a magnetic field transient . . . . . . 65 3.9 Long term drift of gyroscope . . . . . . . . . . . . . . . . . . . . . . . 72 4.1 Experimental setup of Neon Magetometer . . . . . . . . . . . . . . . . 76 4.2 T2 time of ≈ 14minutes for Neon polarization when operating away from the compensation point in the co-magnetometer configuration. 4.3 77 Compensation behaviour of the K-Ne comagnetometer to an externally applied magnetic transient field . . . . . . . . . . . . . . . . . . 78 4.4 T1 of 105 minutes for a 1.6atm cell of Ne at 170C◦ . . . . . . . . . . . . 79 4.5 Theoritical simulation of the noble gas spin for positive gain κ . . . . 83 4.6 Theoritical simulation of the noble gas spin for negative gain κ . . . . 83 4.7 Magnetic field homogeneity for a 3cm×3cm region in magetic shields 87 4.8 Silvered oven holding Boron-Nitride housing for Pyrex cell . . . . . . 89 5.1 Experimental Setup for measuring spin exchange parameters of K, and Rb with Ne . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 5.2 Representative NMR signal of polarized 21 Ne gas . . . . . . . . . . . 99 5.3 Representative EPR shifts after 2 hours of polarization . . . . . . . . . 101 5.4 Determination of Alkali polarization via RF sweep over Zeeman levels103 5.5 Potassium Polarization decay as a result of Pump beam being manually chopped . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 5.6 Absolute Potassium Back polarization as Neon is flipped via Adiabatic fast passage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 5.7 Neon Relaxation as a function of cell pressure. . . . . . . . . . . . . . 108 xiii 5.8 Absolute neon polarization as function of time for determination of Spin Exchange Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 5.9 Scatter in Spin exchange rate measurements for K-Ne . . . . . . . . . 111 5.10 Scatter in Spin exchange rate measurements for Rb-Ne . . . . . . . . 111 6.1 Absolute Rb polarization as function of propagation distance through cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 xiv Chapter 1 Introduction Since the time of the first compass Man has been intrigued by the properties of magnetic fields. This fascination only increased with the birth of modern electronics. However the reach of magnetism extends well beyond that of electronics. The duality of the theory of magnetism and electricity proved to be a fundamental realization when special relativity was utilized to unify them. Even light has been shown to be a form of electromagnetic radiation. The properties of magnetic fields have been used for navigational purposes, and as a means to unify seemingly dissimilar forces in physics for over a century. In the modern era one finds even more applications of magnetic fields. A few of these include medical imaging, detection of explosives, generation of electricity, tests of fundamental symmetries, and maglev trains. Therefore, it should be no surprise that the ability to precisely and accurately measure magnetic fields with as much sensitivity as possible is of prime importance. During the past century scientists have continually sought to improve the sensitivity of magnetic field measurements. This began with Hall probes, fluxgates, and ultimately in superconducting quantum interference devices (SQUIDS). For the past few decades these SQUIDS have been the state of the art with respect to 1 measuring magnetic fields. They exhibit sensitivity on the order of 1fT when operating in an environment with superconducting shields (Clarke & Braginski (2004)). However this decade has seen the re-emergence of spin precession based atomic magnetometers with improved sensitivity which now surpass that of SQUIDS. In 1957 Dehmelt (Dehmelt (1957)) proposed observing the precession of alkali spins in order to measure the strength of a magnetic field. This was carried out experimentally the same year by Bell and Bloom (Bell & Bloom (1957)). These works were aided by the contribution of Kastler who created a technique to produce significant population changes from free atoms in the ground state (Kastler (1957)). For this work he was awarded the 1966 Nobel prize in physics. Most modern day spin precession atomic magnetometers operate on the same underlying principles as those used by Bell, Bloom, and Kastler. Spins are polarized by optically pumping with a laser beam, and allowed to precess in a magnetic field according to Larmor precession. See fig.1.1. ~ = γ~B ω (1.1) Here the frequency of precession is directly proportional to the strength of the magnetic field. This precession can be detected via optical rotation of the plane of polarization of a linearly polarized probe laser beam. The plane of polarization of the probe laser becomes rotated as it interacts with a magnetically polarized sample. This leads to a simple way in which to measure the alkali precession frequency, and hence the magnetic field. Normally the atoms one uses to measure precession are alkali atoms. This is for multiple reasons. The most important of which is that because alkali only have one valence electron, they are simple spin systems. That is their total spin can be characterized by the sum of the nuclear spin, and valence electron spin. Furthermore 2 B Pump Beam ω F Probe Beam Figure 1.1: For operation of a magnetometer alkali atoms are polarized via circular polarized pump beam. A probe beam measures the orientation of the alkali polarization via Faraday rotation as the atoms precess due to a magnetic field. The frequency of precession can be used to determine the strength of the magnetic field. 3 the ground state is spherically symmetric. This leads to long lived ground state polarization lifetimes. Additionally the alkali atoms can be easily interrogated with lasers via a strong optical transition. This enables us to pump the majority of the alkali into specific states so that their magnetic moments are all parallel. In order to pump the alkali metal in its gas state, we heat the metal sample and rely on the saturated vapour pressure to supply the vapour to be optically pumped. The sensitivity of the spin precession magnetometer is characterized by the time for which the spins can precess coherently. 1 γT2 ∆B = (1.2) Where T2 is the transverse coherence time of the optically pumped vapour. Quantum mechanically we can interpret this spin precession as the measurement of the splitting of the Zeeman levels of the alkali, in this case in its ground state. The magnetic linewidth can be expressed as ∆B = ∆ω γ (1.3) Thus to optimize the sensitivity of the magnetometer, we maximize the spin precession coherence time, and effectively minimize the linewidth of the Zeeman resonance line. Therefore, we can express the magnetometer sensitivity δB in terms of the Zeeman level linewidth ∆B, and the signal to noise ratio of the Zeeman resonance. δB = ∆B SNR (1.4) When the alkali atoms collide with the cell walls they depolarize. In order to increase the polarization lifetime one normally introduces a buffer gas into the cell. One can coat the cell walls with either parrafin, or silane coatings as these reduce 4 the alkali depolarization rate upon collision. The buffer gas densities are typically 105 − 106 times greater than the alkali density under normal operating conditions. Typically inert noble gases are utilized for this purpose. In fact if the noble gas utilized has a nonzero nuclear spin one can transfer some of the alkali electron spin polarization to the noble gas nuclear spin via an interaction known as spin exchange collisions. This can be used to construct co-magnetometers, which are a type of magnetometer containing a polarized noble gas species. The strong D1 transitions of the alkali can easily be accessed by modern solid state diode lasers. These lasers can easily be tuned to pump the alkali vapour, and are quite stable in frequency. The preferred diode laser is a DFB (distributed feedback) laser. These have reflection gratings etched directly on the diode surface which result in operation with very stable frequency. This frequency stability makes them ideal for measuring the alkali polarization as a linearly polarized detuned probe beam. The sensitivity to polarization of the probe beam due to spin projection noise varies inversely as the number of alkali in the cell. This effect will be discussed later in this text. 1 ∆Sx ∝ √ N (1.5) One can increase this by either increasing the vapour pressure of the alkali, by increasing the temperature, or by increasing the path length of the vapour cell. If one increases the density of the cell the alkali interact with themselves via spin exchange collisions which broaden the Zeeman resonance. For years this had been the limiting mechanism in improving atomic magnetometer sensitivity. Typically cells had been constructed with large volume, with low density, at room temperature in order to minimize the reduction in polarization lifetime due to spin exchange collisions. In 2002 Allred et al. (2002) experimentally demonstrated a magnetometer which 5 operated with small measurement volume, near zero field which did not experience broadening of the Zeeman resonance due to spin exchange collisions. It does so by operating in a spin exchange relaxation free regime (SERF) which was first proposed by Happer & Tam (1977). Many of the magnetometers described in this thesis are based on this effect. In this thesis we describe the potential application of enriched neon as a buffer gas. We also demonstrate the application of a potassium-helium co-magnetometer as a sensitive gyroscope. We show that the sensitivity of this gyroscope can be further improved if we switch the buffer gas from helium to neon. This potassium-helium co-magnetometer was originally used for tests of Lorentz violation, and CPT invariance. CPT (charge parity time reversal) is a discrete symmetry of the universe. Breaking of this symmetry would lead to a violation of Lorentz symmetry. While these symmetries are conserved in the Standard Model they may be broken in a more fundamental underlying theory. There exist theories of quantum gravity which have been shown to violate Lorentz symmetry in some way. Switching to a neon buffer gas will also increase of the sensitivity of the co-magnetometer to these effects. However many of the parameters governing the interaction between potassium, or rubidium with enriched 21 Ne have not been measured. The spin exchange rate between the transfer of polarization of the alkali to the noble gas nuclei has not been measured. Neither has the Fermi contact interaction between these pairs of atoms. In order to design and construct an optimized magnetometer one must have a knowledge of these parameters. We measure them here. We construct the first 21 Ne co-magnetometer and study its behaviour. And finally we discuss what future work must be carried out in order to use tests of fundamental symmetries. 6 21 Ne for Chapter 2 Background In order to understand the experiments involving alkali metal-neon optical pumping it would be prudent to first review the physical processes pertaining to optical pumping. We begin by discussing the basics of polarizing alkali via optical pumping. We discuss the properties of the energy levels relevant to optical pumping as well as the effects of line broadening on optical pumping. We also briefly discuss utilizing polarized alkali to measure magnetic fields. The process of optical pumping does not polarize the alkali to unity. In the next section we discuss the relaxation mechanisms which limit the alkali polarization. Relaxation mechanisms of particular interest include spin relaxation due to spin exchange collisions, spin destruction collisions, and magnetic field gradients. Additionally we discuss the means by which the alkali polarization is interrogated. This includes a number of schemes based on optical rotation utilizing polarimeters. We also investigate spurious signals due to the effects of light shifts. We discuss the consequences of adding diamagnetic noble gas to the vicinity of polarized alkali atoms. We investigate the transfer of polarization from the alkali spin to the noble gas nuclei via spin exchange optical pumping. We describe the response of the coupled system and the dynamics of an atomic co-magnetometer. 7 Finally we discuss useful techniques with which to manipulate the polarized noble gas spins, such as adiabatic fast passage, and nuclear magnetic resonance. 2.1 Optical Pumping background Optical pumping is a technique which can be used to polarize the spins of an atomic species through application of laser light. The most popular atomic species to utilize during optical pumping experiments are the alkali atoms. Typically the transitions utilized to optically pump the alkali are the D1 or D2 doublets. In this section we discuss the basic principles of optical pumping. We discuss the properties which make alkali atoms the preferred species for many optical pumping experiments. We investigate the properties of the alkali atoms energy levels which are relevant for optical pumping and ways to manipulate these levels to produce a long lived ground state alkali polarization. Finally we discuss the dynamics of optical pumping alkali in terms of the optical pumping rate, and the spin relaxation rate. 2.1.1 Atomic Energy Levels Alkali metals are a convenient choice for optical pumping because they possess a single unpaired valence electron. The spectroscopic properties are well approximated by ignoring the interaction of the filled electron sublevels and concentrating on the valence electron, and its interaction with the atomic nucleus. As such the wavefunctions describing the energy levels are well described by total angular momentum quantum numbers of the valence electron spin and the nuclear spin. Consider the ground state S shell of the alkali atom. The spin of the valence electron is S= 1/2, and the orbital angular momentum in this state is L= 0. Thus the total electron angular momentum is J= 1/2. The first excited state is the P 8 F=I+3/2 F=I+1/2 F=I–1/2 F=I–3/2 2 P3/2 p F=I+1/2 F=I–1/2 2 P1/2 D1 s D2 F=I+1/2 F=I–1/2 2 Orbital Structure S1/2 Fine Structure Hyperfine Structure Figure 2.1: Energy level splitting of the ground state and first excited state of an alkali metal atom. The fine structure splits first excited state further into J=1/2 and J=3/2 levels. The hyperfine structure further splits the J energy levels. Not drawn to scale. shell with L= 1. Due to fine structure L·S coupling the P state splits into 2 P3/2 and 2P 1/2 states. Here we use the standard spectroscopic notation 2S+1 L J to describe the energy levels. Historically the 2 S1/2 → 2P1/2 and 2 S1/2 → 2P3/2 transitions are referred to as the D1 and D2 lines. See fig. 2.1. The nuclear spin of the alkali metal I couples to the total electron spin J via the hyperfine interaction to further split the energy levels into states with good quantum number F. Here F is the total angular momentum F=I+J. It follows from the Wigner-Eckart theorem (see Cohen-Tannoudji 1972) that the total electron angular momentum J must be parallel to F. Thus when we probe the hyperfine manifolds the orientation of the total atomic angular momentum vector we also determine the total electron angular momentum vector direction. Implementation of a magnetic field lifts the degeneracy between different Zeeman sublevels of states with the same total angular momentum F, but different projection m f along the quantization axis defined by the magnetic field. The resulting energy splitting between Zeeman levels is proportional to the field for small 9 field strengths, typically less than a Gauss. This splitting gives rise to Larmor precession of the atoms between the energy levels with frequency ω L = ∆EL = γ | B|. Here the gyromagnetic ratio γ for alkali atoms is given by γ ≈ ±2π × (2.8MHZ/G)/(2I+1), where I is the nuclear spin of the atomic nucleus and the sign corresponds to the hyperfine manifold, ie. The F = I + 1/2 manifold yields a + sign in the gyromagnetic ratio. For large magnetic fields however the Zeeman energy level splitting is nonlinear and is given by the Breit-Rabi splitting. We can calculate this by studying the ground state Hamiltonian of the alkali atom. The ground state Hamiltonian is of the form: H = A J I · J + gs µ B S · B − g I µ N I · B (2.1) where µ N is the nuclear magneton, A J = 2h̄ωhf /(2I + 1) is the hyperfine coupling constant, gs ≈2 is the electron g-factor, µ B =9.274×10−24 J/T is the Bohr magneton, and g I the nuclear g-factor. The hyperfine coupling constant is specific to the atom under discussion. The energy spectrum can be calculated from the eigenvalues of the Hamiltonian to be (Corney (1977)): h̄ωhf h̄ωhf E( F = I ± 1/2, m F ) = − − g I µ N Bm F ± 2(2I + 1) 2 r x2 + 4xm F +1 2I + 1 (2.2) Where x≡ 2( gs µ B + g I µ N ) B ( gs µ B + g I µ N ) B = (2I + 1) A J h̄ωh f (2.3) Notice that the energy spacing as a function of the magnetic field is now nonlinear. In the low field adjacent sublevels would be separated by g I µ N Bm F . This 10 clearly scales with the magnetic field. In the high field regime the last term in eq. 2.2 causes the splitting to no longer be directly proportional to the magnetic field. 2.1.2 Optical Pumping The experiments described in this thesis require a large source of polarized spin. One could thermally polarize the sample using brute force by introducing the sample into a large magnetic field: Pther = tanh 1 2 gs µ B B kB T ! (2.4) where gs ≈2 is the electron g-factor and µ B =9.274×10−24 J/T is the Bohr magneton. The polarization can only be raised to 2% at room temperature if one implements large magnetic fields on the order of 10T . For comparison the typical absolute thermal polarization is only 1 × 10−7 at room temperature, in Earth’s field. The more elegant technique of optical pumping can yield alkali polarization on the order of unity. To describe optical pumping we consider a toy model where the atom has no nuclear spin. However, the following scheme is of a general nature and can be extended to the case where I6= 0. Typically one utilizes the ground state D1 transition of an alkali metal for optical pumping for a variety of reasons. First, the alkali doublet have strong oscillator strength which leads to larger absorption by the pump laser. Second, it is theoritically possible to polarize the ground state to unity if the D1 is excited (Franzen & Emslie (1957)). In the case of D2 pumping the maximum achievable polarization is limited to 1/2. This transition is not efficient for optical pumping. It can be utilized under conditions where the gas density is rare, and there is little collisional mixing in the excited P state. The details of collisional mixing will be described shortly. 11 ing + σ mp Pu 2 P1/2 2 S1/2 Quenching Quenching Collisional Mixing Spin Relaxation mJ = +1/2 mJ = -1/2 Figure 2.2: Optical pumping of the electron spin of an alkali. The experiments in this work rely on optically pumping alkali by tuning to the D1 transition. To describe this process first consider the D1 2 S1/2 → 2P1/2 transition of an atom in a magnetic field as shown in fig.2.2. Let us define the quantization axis along the direction of the magnetic field. The ground and excited state sublevel degeneracy is lifted in the presence of a magnetic field and can each be resolved into states with magnetic quantum number m J . The goal of optical pumping is to increase the ground state population of one of the magnetic sublevels, 2 S1/2 (m J = +1/2) in this example. To accomplish this we first utilize σ+ photons to excite the 2 S1/2 (m J = −1/2) →2 P1/2 (m J = +1/2) transition. In the excited state two effects occur. The first is spontaneous emission to both 2 S1/2 magnetic sublevels, and the second is collisional mixing in the excited state (Walker & Happer (1997)). Due to the nonzero orbital angular momentum of the excited state the alkali wavefunction is not spherically symmetric. This causes collisional mixing effects to occur. Collisional mixing transfers the atom from the excited P m J = +1/2 to the m J = −1/2 sublevel. Typically when alkali metal is optically pumped the alkali vapour is in the vicinity of other gases. These can include noble gas, and a 12 molecular quenching gas. The collision between alkali vapour, and noble buffer gas leads to collisional mixing in the excited state. Physically these may be described as the electron orbital angular momenta coupling to the rotational angular momenta of the molecule temporarily created by the pair of colliding atoms (Wu et al. (1985))(Walker & Happer (1997)). In general the Hamiltonian of such an interaction can be expressed as (Wu et al. (1985)): Hcm = γ(r )S · N (2.5) Here the S refers to the alkali spin, N to the rotational angular momentum of the temporarily formed molecule, and γ(r ) as the coupling between the two. This last factor depends on the inter-atomic potential, the separation of the constituents of the temporarily formed molecule, and the corresponding wavefunction density of the pair. During collision the degree to which the alkali wavefunction is perturbed determines the collisional mixing. Atoms with large polarizability have greater collisional mixing cross-sections. The ground state of the alkali is less susceptible to wavefunction deformation than the P state sublevels because it is spherically symmetric (Corney (1977)). If the pumping scheme is iterated the atoms eventually populate the 2 S1/2 m J = +1/2 state, as atoms in this state can no longer absorb the σ+ light. This represents an ideal case. There exist mechanisms to prevent this. Fluorescent light from the spontaneous decay of the excited P state can interact with other alkali in the sample and excite atoms out of the 2 S1/2 m J = +1/2 sublevel. This is referred to in the literature as radiation trapping. This can be prevented by introducing a molecular quenching gas. A typically quenching agent is N2 . N2 molecules collide with the excited alkali atoms and transfer them to the ground state without re-irradiation of the alkali (Happer (1972)). This can occur because the N2 molecule has a large 13 number of closely spaced energy levels, due to its rotational, and vibrational level structure. As the N2 molecules return to their ground state they may either transfer the energy from the alkali atom to its rotational or vibrational modes to be reradiated at a frequency different from the D1 line. The excited alkali atoms become non-radiatively quenched. Typically 100 Torr of Nitrogen is sufficient to prevent radiation trapping in cells with alkali densities near 1014 /cm3 . There are also spinrelaxation mechanisms which can transfer atoms between the two ground state magnetic sublevels to depolarize the sample. These will be discussed in the next section. The alkali nucleus is strongly coupled to the total electron angular momentum via the hyperfine interaction. Thus when we polarize the electrons we also successfully polarize the alkali nuclei. This leads to some interesting effects. The alkali electron spin polarization decay is slower than one would initially estimate. This is due to the electron spins being re-polarized via the hyperfine interaction with the polarized nucleus. This effect is termed the slowing down factor, and will be quantitatively described later in this work. 2.1.3 Effects of Pumping Rate and resonance Lineshape on Optical Pumping In describing optical pumping it is useful to define a quantity named the optical pumping rate. The optical pumping rate is defined as the rate at which an unpolarized alkali atom absorbs photons from the pump laser. Rop = Z I (ν) σ (ν)dν hν (2.6) I (ν) is the light spectral density in units of Watts cm−2 Hz−1 , and σ (ν) is the photon absorption cross section. 14 The photon absorption cross section is defined by the atomic response as a function of incident photon frequency. In general it is influenced by the natural lifetime of the atomic state of interest, pressure broadening of the atomic resonance, and Doppler broadening of the resonance due to thermal velocity distribution. The absorption cross section can be related to the classical electron radius by: Z ∞ 0 σ (ν)dν = πre c f (2.7) This is valid regardless of the spectral line shape. In eq.2.7 the oscillator strength f is the quantum mechanical correction factor to the classical expression for the relation between absorption cross section, and classical electron radius. It is a dimensionless number and depends on which transition we are pumping. For the D1 line it is ≈ 1/3, and for D2 it is ≈ 2/3(Migdalek & Kim (1998)). The absorption cross section is given by: σ (ν) = πre c f Re[V(ν − ν0 )] (2.8) where V(ν − ν0 ) is the atomic lineshape. It is given by the Voight profile (Happer & Mathur (1967)), and includes the effects of pressure broadening, natural lifetime, and Doppler broadening. V(ν − ν0 ) = Where Z ∞ 0 ′ ′ L(ν − ν )G(ν − ν0 )dν ′ ! √ √ 2 ln 2[(ν − ν0 ) + iΓ L /2] 2 ln 2/π V(ν − ν0 ) = w ΓG ΓG (2.9) (2.10) And the function w is given in terms of the complex error function as: 2 w( x ) = e− x (1 − erf(−ix )) 15 (2.11) Gas He He Ne Ne Ar Ar Kr Kr Xe Xe N2 N2 Energy Level P1/2 P3/2 P1/2 P3/2 P1/2 P3/2 P1/2 P3/2 P1/2 P3/2 P1/2 P3/2 Half-halfwidth 1.55 ± 0.03 2.06 ± 0.04 0.85 ± 0.02 1.16 ± 0.04 2.45 ± 0.03 1.98 ± 0.03 2.31 ± 0.05 2.31 ± 0.05 2.75 ± 0.03 2.75 ± 0.03 2.45 ± 0.03 2.45 ± 0.03 Shift +0.45 ± 0.04 +0.24 ± 0.04 −0.41 ± 0.02 −0.62 ± 0.02 −2.31 ± 0.05 −1.52 ± 0.04 −1.65 ± 0.04 −1.16 ± 0.04 −1.79 ± 0.06 −1.79 ± 0.06 −1.83 ± 0.04 −1.32 ± 0.04 Table 2.1: Comparison of the line broadening and shift parameters for K in various Gases. All experimental measurements were made in the 400 − 420K temperature regime. The line broadening parameter and shift parameter have units of 10−9 rad s−1 atom −1 cm3 . Data taken from Lwin & McCartan (1978) er f denotes the standard error function. Here ΓG is the linewidth of the Gaussian contribution to the lineshape due to Doppler broadening for an atom of mass M ! √ −4 ln 2(ν − ν0 )2 2 ln 2/π exp G(ν − ν0 ) = ΓG Γ2G (2.12) r (2.13) ν0 ΓG = 2 c 2k B T ln 2 M And Γ L is the linewidth of the Lorentzian contribution to the Voight profile due to pressure broadening. L(ν − ν0 ) = Γ L /2π (ν − ν0 )2 + (Γ L /2)2 (2.14) The values for Γ L as a function of pressure is listed for K vapour in table 2.1. 16 2.1.4 Evolution of Alkali polarization due to Optical Pumping The evolution of the polarization of the alkali atoms due to optical pumping can calculated. In order to compute the polarization achieved via optical pumping one must clarify whether the target cell contains any buffer gas. For certain applications it becomes advantageous to operate without a buffer gas (see Knappe et al. (2006),Cates et al. (1988),and Pustelny et al. (2006) for more information), and utilize a cell coating instead. However those applications will not be discussed in this work. Once the D1 2 S1/2 → 2P1/2 transition is excited the electron will decay back to both the m J = 1/2, or m J = −1/2 sublevels. For the case of no buffer gas the branching ratios are determined by the Clebsch-Gordon coefficients to be 1/3, and 2/3 respectively (Budker et al. (2004)). However when buffer gas is introduced in the cell, collisional mixing in the excited states alters the branching ratio to be 1/2 for both cases. Under these conditions we can now calculate the time evolution of the alkali polarization. Let us denote the populations in the m J = +1/2 and m J = −1/2 sublevels as N+ , and N− respectively. The pumping rate is defined as the rate at which an unpolarized atom absorbs a photon. The m J = +1/2 state is unable to absorb a photon if one uses σ+ light. Therefore the rate at which the m J = −1/2 state absorbs a photon is at twice the optical pumping rate. This is because the optical pumping rate is defined as per unpolarized atom. Using this and including the 1/2 branching ratio we can describe the pumping process by realizing − − dN dt . Thus: 1 R R dN+ = (2Rop ) N− + rel N− − rel N+ dt 2 2 2 dN+ dt = (2.15) Where Rrel is the relaxation rate. By noting that the polarization can be expressed 17 as: P = 2 < Sz >= N+ − N− N+ + N− (2.16) the time evolution of the polarization can be ascertained. Assuming P(0) = 0: P(t) = Pequil (1 − e( Rop + Rrel )t ) (2.17) During equilibrium: Pequil = Rop Rop + Rrel (2.18) This is only strictly true in the case of an atom with zero nuclear spin. The electron spin and nuclear spin are coupled via the hyperfine interaction. As the electron spin precesses it drags the nuclear spin along with it. This results in a slower precession frequency than that of a free electron in a magnetic field (Budker et al. (2004)). It is modified as: γ= γe 2I + 1 (2.19) Where γe = gs µb /h̄ is the gyromagnetic ratio for a free electron, and has magnitude 2π × 2.8MHz/G. The coupling between the valence electron and the nucleus also alters the rate at which the valence electron is depolarized. Just as the hyperfine interaction can polarize the nucleus when we pump the valence electron, the opposite can occur. When the electron spin is depolarized, the nuclear spin can re-polarize the electron. Thus the actual rate at which the electron is either depolarized by collisions, or polarized by a laser must be modified. To good approximation one can describe this effect by a linear slowing down factor ǫ (Walker & Happer (1997)). For atoms in low magnetic field where γB << Rse the system can be described accurately in terms of a two level system. The equation of motion 18 of the atom can now be described as: dS 1 1 = γe B × S + Rop sb z − S − Rrel S dt ǫ+1 2 (2.20) This is often referred to as the Bloch equation in the literature. Here the first term describes the precession of the spins in a magnetic field. The second term describes the pumping of the spins to their equilibrium value. Here ǫ is given by: ǫ( I, β) = (2I + 1)coth( β/2)coth( β[ I + 1/2]) − coth2 ( β/2) (2.21) (2.22) P = tanh( β/2) (2.23) β = ln 1+P 1+P Here I is the nuclear spin, and β is the spin temperature of the system (Walker & Happer (1997)), which can be related to the polarization P. A more complete discussion on spin temperature will be described later in this work. But for now it is sufficient to note the special limiting cases for ǫ for low and high polarization respectively as: ǫ( I, β << 1) = 4I ( I + 1)/3 (2.24) ǫ( I, β >> 1) = 2I (2.25) Although this alters the time evolution of the electron spin polarization to: P(t) = Pequil (1 − e( Rop + Rrel )t/(ǫ+1) ) The equilibrium polarization reached remains the same eq 2.18. 19 (2.26) 2.1.5 Dynamics of polarized alkali at low magnetic fields Optical pumping is used to polarize the spins of an alkali species. An application of the polarized alkali spin of particular importance is the determination of magnetic fields. This is accomplished by determination of the precession frequency of the alkali by monitoring the spin dynamics and relating it to the magnetic field. The alkali undergo Larmor precession due to the interaction of the spins with the magnetic field: H = γalkali h̄~B · ~S (2.27) The dynamics of the alkali due to this interaction gives a response: d~ i S = [H, ~S] dt h̄ (2.28) Noting that the components of the spin follow the commutation relation: [Sx , Sy ] = iSz (2.29) we can describe the evolution of Sz as d~ Sz = iγalkali (−iBx Sy + iBy Sx ) dt (2.30) One can describe the evolution of the Sx , and Sy components of the spin with similar equations. Inclusion of the effects of optical pumping, and spin relaxation modify the spin evolution given by eq.2.30 to that given by the phenomonological equation eq.2.20. The magnetometer bandwidth can be calculated utilizing a set of simplified Bloch equations. Assume we impose an oscillating field of the form By = B0 exp(−iωt). Let us rewrite eq.2.20 in terms of the components of the alkali polarization Pex , and 20 Pze . Then, for a magnetometer where Bx , and Bz are set to zero imposing a field By gives the following behaviour: 1 dPxe = γe By Pze − Rtot Pxe dt ǫ+1 1 dPze = −γe By Pxe − Rtot Pze + Rop dt ǫ+1 (2.31) (2.32) The first term represents precession about By , the second represents spin relaxation, and the third term represents optical pumping. Solution of the above equations yields: Pxe = Pze γe B0 Rtot − iω (ǫ + 1) S = R( Pxe ) = Pze γe B0 R2tot − ω 2 (ǫ + 1)2 (2.33) (2.34) ω0 is the frequency of the spin precession in the ambient magnetic field. For low frequencies the signal can be simplified to obtain: S= 2.2 Pze γe B0 Rtot (2.35) Spin Relaxation In order to achieve optimal sensitivity and minimize the magnetic linewidth during optical pumping experiments, one must maximize the spin polarization lifetime. Since magnetic linewidth is correlated to the lifetime of the Zeeman states, we are effectively maximizing the spin polarization. In general we must consider two cases, the longitudinal polarization lifetime T1 , and the transverse polarization lifetime T2 . Here T1 is the lifetime of the polarization parallel to the magnetic holding field, and T2 is the lifetime of polarization in a perpendicular direction. 21 One can express the longitudinal lifetime T1 as: 1 1 = ( R + Rop + R pr ) + Rwall + Rinh T1 ǫ + 1 sd (2.36) Here the first term is due to spin destruction collisions. These collision may be of several types, including collisions between alkali atoms, collisions between the alkali atoms and the buffer gas, or collisions between the alkali atoms and the quenching gas. sel f bu f f er Rsd = Rsd + Rsd quench + Rsd (2.37) Each of the collisional spin destruction rate mechanisms can be described by: Rsd = nσv (2.38) where n is the density of the gas species in question which is colliding with the alkali, and σ is the spin destruction cross section. Calculation of the spin destruction crpss-sections can be performed from a knowledge of the atomic wavefunction, interaction potential and the interaction Hamiltonian. These calculations are normally accurate to 50% with the measured value, due to imprecise knowledge of the specifics of the collisional interaction (Walker & Happer (1997)). The last term v is the relative velocity of the colliding pair and is given by: v= r 8κ B T πM (2.39) and M is the reduced mass of the alkali and its colliding partner: 1 1 1 = + ′ M m m (2.40) The second term in eq (2.36) is the optical pumping rate. This affects T1 be22 cause the absorption of the pump beam alters the angular momentum state of the alkali. The next term is similar, but due to absorption of the probe beam. These relaxation mechanisms depolarize the valence electrons, while leaving the nucleus unaffected. Thus the sum of the first three terms must be divided by the slowing down factor because of the previously mentioned re-pumping of the electron due to the polarized nucleus. The next term is due to collisions with the wall. In uncoated cells collisions with the cell wall completely depolarize the alkali atoms. The last term is due to magnetic field inhomogeneity. The pumped alkali align with the net magnetic holding field. If a gradient is present it can locally alter the direction of the net magnetic field and change T1 . Consider the mean free path λ in the gas. Since the Larmor frequency is much faster than the transit rate v/λ between atomic collisions the atomic polarization follows the total magnetic field. This leads to relaxation when the atom changes direction after a collision (Budker et al. (2004)). The atom experiences a small magnetic field, which varies slowly compared to the Larmor frequency. It is transverse to the holding field and corresponds to a rotation of the total magnetic field vector with a frequency ω ≈ δBv Bholding λ . It can be treated as the flipping probability of an atom with spin oriented along a magnetic holding field, by the presence of a fluctuating transverse magnetic field. Magnetic field gradients are normally not the dominant alkali relaxation mechanism. Typical gradients found in the low field comagnetometer experiment are 10µG/cm, with a holding field of 1mG. This gives a negligible alkali relaxation rate due to gradient relaxation. Additionally magnetic field gradients cause the atoms to precess due to local variation in the magnetic field and de-phase. This causes relaxation of the transverse component of the spins. Additionally the mechanisms discussed earlier pertaining to the T1 relaxation similarly affect the transverse polarization because they 23 also randomly orient the spin polarization. The T2 lifetime can thus be given as: 1 1 1 = + Rse + R grad T2 T1 qse At high field where γB >> Rse Happer & Tam (1977) give (2.41) 1 qse as: 1 2I (2I − 1) = qse 3(2I + 1)2 Whereas at low magnetic field where γB << Rse they show that (2.42) 1 qse → 0. Rse is the spin exchange rate between alkali atoms, and Rgrad is the dephasing due to nonuniform magnetic field across the sample cell. The significance of eq. (2.42) and its physical description will be described in more detail in the next section. In order to maximize the sensitivity of most polarization experiments one attempts to maximize the T2 because it is the relevant parameter in the determination of the precession frequency. One also attempts to maximize T1 in a sense because it limits the maximum achievable value of T2 . It would be prudent to review each of the aforementioned pumping and relaxation mechanisms in more detail. In the next section we do so. 2.2.1 Spin Exchange Collisions When alkali atoms collide there is the possibility of their exchanging spin states (Purcell & Field (1956)). This can be shown as: |+i A |−i B → |−i A |+i B (2.43) At large alkali density this has been shown to be the dominant relaxation mechanism. The following treatment on spin exchange collisions follows the treatment by Budker et al. (2004). 24 One can describe the spin exchange mechanism by realizing that the interatomic potential during collision has a spin dependent contribution: V (r ) = V0 (r ) + S A · SB V1 (r ) (2.44) Where V0 is a spin independent interaction term, and S A and SB are the respective spins of the colliding alkali. The wavefunction of a free atom before collision can be expressed in the |S, MS i basis in terms of the singlet |0, 0i, and triplet |1, 0i states as: 1 |ψ(0)i = |+i A |−i B = √ (|1, 0i + |0, 0i) 2 (2.45) During collision the singlet and triplet states acquire a relative phase 1 |ψ(0)i = √ (|1, 0i + ei∆φ(t) |0, 0i) 2 (2.46) where the relative phase acquired is: 2π ∆φ(t) = h Z t 0 V1 [r (t)]dt (2.47) The atoms have a probability of undergoing a spin exchange collision when ∆Φ is an odd multiple of π. In this case |ψ(t)i → |−i A |+i B (2.48) Though spin exchange collisions conserve total mF quantum number during collision they can alter the F state of the alkali. These collisions are fast with respect to the nuclear hyperfine interaction, thus they do not affect the nuclear spin state. These collisions also cause decoherence of the transverse spins, and dephasing because the two hyperfine manifold actually have gyromagnetic ratios which 25 F=2 -2 +ωL F=1 -1 -1 +ωL −ωL 0 0 +ωL −ωL +1 +ωL +2 +1 Figure 2.3: Ground-state Zeeman sublevels for the case I=3/2. Sublevels are labeled by their m F azimuthal quantum number. Note that the energy level splitting changes sign for the different F manifolds. This causes both F manifold to have gyromagnetic ratios with opposite sign Figure 2.4: During spin-exchange collisions the total angular momentum F 1 + F 2 is conserved, the atoms may switch between m F hyperfine sublevels. The hyperfine levels F = I ± 1/2 are represented here by the colors red and blue. The atoms is different hyperfine levels have gyromagnetic ratios with different sign,t equal magnitude. They will precess in opposite directions and decohere. while having the same magnitude differ in sign. See fig.2.3 and fig.2.4. This can readily be seen by solving the ground state alkali Hamiltonian eq. (2.1)(Walker & Happer (1997)): and solving for the eigenvalues in the | F, m F i basis. ω F= I +1/2 = −ω F= I −1/2 = 2π gs µ B (2I + 1)h (2.49) This decoherence causes a broadening of the Zeeman sublevels which at high 26 field and low polarization can be expressed as (Happer & Tam (1977)): 1 2I (2I − 1) = qse 2(2I + 1)2 (2.50) This expression is accurate in the high field regime where ω Larmor >> Rse . This effect also contributes to the reduction of the T2 of the alkali atom precession. See eq. (2.41). In the low field limit where ω Larmor << Rse we find interesting behaviour as spin exchange processes no longer contribute to the transverse decoherence of the spins. This can be explained if we think in terms of the relative populations of the different hyperfine states. When the spin exchange rate is much higher than both the optical pumping and relaxation rates the atoms mix the m f sublevel populations. This condition applies at high alkali density when spin exchange collisions occur frequently. Here the steady state m f sublevel population can be described by a spin temperature distribution (Anderson et al. (1959),Anderson et al. (1960)). See fig.2.5. ρ( F, m F ) = 1 βm F e ZF (2.51) where ZF = Σe βm F = sinh[ β( F + 1/2)] sinh( β/2 (2.52) Here the sublevel population possess a Boltzmann like distribution but with a spin temperature β instead of the normal kinetic thermal temperature. We find that because of the high spin exchange rate the atoms spend time in both hyperfine manifolds. However they do so with a probability corresponding to the spin temperature distribution. Since the spin exchange rate is much higher than the Larmor precession rate the atoms motion can be thought of as a slower averaged precession due to the fact that the gyromagnetic ratios in each hyperfine manifold has opposite sign but have unequal time spent in each manifold. This results in a net 27 F=2 e-2β F=1 mF=-2 e-β 1 e+β e-β 1 e+β mF=-1 mF=0 mF=+1 e+2β mF=+2 Figure 2.5: When the rate of spin-exchange collisions is high, the Zeeman sublevel populations are given by a Boltzmann distribution which is characterized by a spin-temperature. The sublevel population in this case scales as eβm F . The case of nuclear spin I= 3/2 is shown. rotation of the spins in one direction. The alkali experience spin exchange collisions and hop between the two hyperfine manifolds. The two hyperfine manifolds possess gyromagnetic ratios with opposite sign. The atoms precess a small amount between collisions. However the atoms spend a larger amount of time in the hyperfine manifold with larger F value. This is determined by the spin temperature distribution. Thus even though the atoms hops back and forth between the hyperfine manifolds it preferentially spends a larger fraction of time in the hyperfine manifold with the larger F value. This causes the atom to have a net precession in the orientation consistent with the sign of the gyromagnetic ratio of the larger F valued hyperfine manifold. Since this precession is coherent, spin exchange is effectively removed as a source of relaxation to first order. The spin exchange relaxation rate becomes (Allred et al. (2002)): Rse = ( Q) 2 ωser f Tse 2 − (2I + 1)2 2 (2.53) Q is described in eq. (2.55). In the equation above Tse is the time between spin exchange collisions, and depends on the specifics of the cell temperature and buffer 28 gas pressure. However under typical conditions of the experiments in this thesis it is ≈ 15µS. Optical pumping schemes operating in this regime are called spin exchange relaxation free (SERF). The rate of precession is modified from the traditional Larmor rate by (Allred et al. (2002)): ωSERF = 2πgs µ B B ( Q)h (2.54) Q = 1+ I ( I + 1) S ( S + 1) (2.55) Where for high polarization. One finds the dependence of ǫ on polarization as (Savukov & Romalis (2005)): Q( P) = 2I + 1 2 − 32I++P12 (2.56) 2.2.2 Spin Destruction Collisions Spin destruction collisions are the next leading mechanism for spin relaxation. Pictorially they may be represented as: |↑i A + |↓i B → |↓i A + |↓i B (2.57) They occur through when the spin angular momentum becomes coupled to the orbital angular momentum of the atom-atom collision. This may be between the alkali atoms and any of the other alkali atoms, buffer gas atoms, or quenching gas molecules. This angular momentum coupling is hypothesized to be due to a spin axis interaction (Bhaskar et al. (1980)): VSA = 2 bR b − 1) · S λS · (3 R 3 29 (2.58) Alkali Metal K Rb Cs σsel f 1 × 10−18 cm2 9 × 10−18 cm2 2 × 10−16 cm2 σHe 8 × 10−25 cm2 9 × 10−24 cm2 3 × 10−23 cm2 σNe 1 × 10−23 cm2 −− −− σN2 −− 1 × 10−22 cm2 6 × 10−22 cm2 Table 2.2: Spin destruction cross sections of alkali atoms with various gases. Adapted from Allred et al. (2002) A list of spin destruction cross sections of alkali gases with typical buffer, and quenching gases is listed in table 2.2. 2.2.3 Diffusion wall collisions, and Magnetic field Gradients When alkali interact with the cell walls they normally become completely depolarized. Physically this process involves the alkali becoming adsorbed on the glass surface. Though the adsorption time is typically between 10µ S and 100nS, the atoms experience very large magnetic and electric field emanating from the glass surface during this time. This is sufficient to depolarize both the spin of the electron and the nucleus. In order to minimize this effect one can either treat the cell wall with an anti-relaxation coating or fill the cell with a high pressure of buffer gas to decrease the alkali diffusion to the cell wall. In this work we consider the second option. If one assumes that the alkali becomes fully depolarized when it becomes adsorbed on the wall then one can model the polarization in the cell by the diffusion equation. d P = D ∇2 P dt (2.59) Here D is the diffusion constant, where λ is the mean free path, and v the average thermal velocity. D= 1 λv 3 30 (2.60) The diffusion constant for K in neon is given by(Franz & Volk (1982)): √ DK − Ne = 0.19cm2 /s 1 + T/273.15K p amagat ! (2.61) Where the rate of K relaxation for the case of a spherical cell is found by solving eq. (2.59): Rdi f f = DK − Ne π 2 (2.62) a Here the temperature T is given in Celsius, and p is the buffer gas pressure in amagat, and a is the cell radius. This is only strictly valid for a spherical cell. Franzen (1959) gives the diffusion constant of Rb in Ne as: √ DRb− Ne = 0.31cm2 /s 1 + T/273.15K p amagat ! (2.63) The alkali may also experience relaxation due to traveling through a magnetic field gradient. The specifics of this depend on cell geometry, diffusion constant, and magnetic gradient orientation. The gradient relaxation rate is defined as: R grad = D ∇B B 2 (2.64) Where δB is the magnetic field variation over the cell, and D is the diffusion constant. The variation in the magnetic field also causes broadening of the alkali precession frequency ≈ γ∇ B. This local variation of the field also causes dephasing of the spin precession and places a limit on T2 , the transverse relaxation time. For a formal treatment of this effect see Cates et al. (1988). We take a more simplified approach to describe this effect. We can compute this by imagining the cell partitioned into two halves each differing in field by δB. In the regime where the atoms are free to move about the 31 cell unimpeded they dephase according to: 2γδBT2 ≈ 1 (2.65) However in actuality the atoms travel across the cell and experience a δB given by the time averaged field. This movement can be treated as a random walk across the cell. Thus the difference in averaged field experienced by two alkali atoms is: δB δBavg ≈ √ N (2.66) Where N is the number of collisions before dephasing. N can be expressed in terms of T2 as N≈ v T2 R (2.67) Here R is the cell dimension. Noting that one can generalize this argument from a cell of dimension R to a buffer gas included cell of mean free path λ one can solve this set of equations to find: λ 1 > (2γδB)2 T2 v 2.3 (2.68) Monitoring polarized Alkali 2.3.1 Optical Rotation In order to determine the orientation of the alkali polarization we utilize the technique of optical rotation. A linearly polarized beam, which is slightly detuned from the optical transition frequency utilized for pumping, is employed as a probe beam. One uses a weak probe beam so that the alkali vapour is not significantly pumped along the probe direction. As the probe beam interacts with the magnetized sample in the ground state the plane of rotation on the probe beam rotates. 32 θ Figure 2.6: When linearly polarized light passes through a medium which is polarized the axis of polarization of the light rotates. This rotation angle is proportional to the projection of atomic spin along the propagation direction. See fig.2.6. To understand this recall that the linear polarized beam is equivalent to a composition of two equal beams which are circularly polarized with opposite helicity. The rotation occurs when the index of refraction of the probing transition differs for the transitions of different helicity. To describe this effect we follow the approach of Mort et al. (1965), and Erickson (2000). To describe optical rotation mathematically let us first define the probe beam in terms of electromagnetic waves. The electric field of such waves propagating along the xb direction can be described as: E (0) = E (0) = E0 yb + c.c. 2 E0 E0 (yb + ib z) + (yb − ib z) + c.c. 4 4 (2.69) (2.70) Where c.c is the complex conjugate. After propagating the length of the cell l = tc/n(ν) the electric field along the yb becomes: E(l ) = E0 iωn+ (ν)l/c E0 e (yb + ib z) + eiωn− (ν)l/c (yb − ib z) + c.c. 4 4 (2.71) Where n+ , and n− are the indices of refraction for σ+ and σ− polarized light re- 33 spectively. It becomes convenient to define the quantities: n(ν) = [n+ (ν) + n− (ν)]/2 (2.72) ∆n(ν) = [n+ (ν) − n− (ν)]/2 (2.73) We can then substitute this into eq. (2.71) to obtain: E(l ) = E0 iωn(ν)l/c iω∆n(ν)l/c E0 e e (yb + ib z) + eiωn(ν)l/c e−iω∆n(ν)l/c (yb − ib z) + c.c. (2.74) 4 4 One can ignore the common phase factor exp(iωn(ν)l/c when determining the optical rotation angle. The rotation angle is defined to be: θ= πνl Re[n+ (ν) − n− (ν)] c (2.75) We can substitute this into eq. (2.74) to find the electric field vector of the emerging probe beam as: E(l ) = E0 (cosθ yb − sinθb z) (2.76) One can show that an atomic vapour is a medium where through appropriate choice of laser frequency n+ (ν) 6= n− (ν). To validate this assertion let us consider the following. First the index of refraction of light for the D1 transition is given by: n(ν) = 1 + nre c2 f Im[V(ν − ν0 )] 4ν (2.77) Where f is the familiar oscillator strength, and V(ν − ν0 ) is the Voight profile of the transition. We notice that when the probe beam interacts with the ground state, only transitions which obey the quantum selection rules ∆m j = ±1 will occur depending on the helicity of the light. Each σ± component of the linearly polarized probe beam then couples to the ground state depending on the population of the 34 D1 Transition D2 Transition 2 2 P3/2 1 1 3/4 2 mJ = -1/2 P3/2 mJ = +1/2 1/4 1/4 3/4 mJ = +1/2 mJ = +3/2 2 S1/2 S1/2 mJ = -3/2 mJ = -1/2 Figure 2.7: Branching ratios for the D1 and D2 transitions. ground state sublevel. Thus we can effectively write: nre c2 f D1 Im[V(ν − νD1 )] 4ν (2.78) nre c2 f D1 n− (ν) = 1 + 2ρ(−1/2) Im[V(ν − νD1 )] 4ν (2.79) n+ (ν) = 1 + 2ρ(+1/2) For an unpolarized vapour ρ(+1/2) = ρ(−1/2). However for a polarized vapour, Px = ρ(+1/2)−ρ(−1/2) , ρ(+1/2)+ρ(−1/2) we see that n+ 6= n− . Substituting this into equation 2.75, and recalling that ρ(+1/2) + ρ(−1/2) = 1 we find: θ= πlnre cPx (− f D1 Im[V(ν − νD1 )]) 2 (2.80) One can generalize this result taking into account the effect of the D2 line on the probe beam to obtain: πlnre cPx θ= 2 1 − f D1 Im[V(ν − νD1 )] + Im[V(ν − νD2 )] 2 (2.81) The negative sign and factor of 1/2 can be attributed to the different branching ratios for the D2 transition. See fig.2.7. To detect the rotation of the polarization angle one uses a polarizing beam splitter to split the probe beam into two orthogonally polarized beams. These are fed into two photo-diodes whose output is fed through a subtracting photodiode am35 Polarizing Beamsplitter Linear Polarizer Cell Probe Laser Photodiode Photodiode Figure 2.8: Optical detection with a linearly polarized probe beam passing through a polarized cell, then being split by a polarizing beam splitter into two separate photodiode detectors. The resultant signals from each photo-detector is fed through a photodiode amplifier and then into a subtraction circuit, giving the rotation. plifier. See fig.2.8. The plane of polarization of the probe beam is normally adjusted so that one obtains equal signal in each photodiode in the case where there are no polarized alkali. This corresponds to the point where the beam splitter is at 45◦ to the probe beam polarization. Then the difference signal is directly proportional to the polarization. In this case the signal on each photodiode is given by: I1 = I0 sin2 (θ − π ) 4 (2.82) I2 = I0 cos2 (θ − π ) 4 (2.83) where I0 is the intensity of the probe beam, and I1 and I2 are the intensities on the two photo-detectors. Solving for the rotation angle in radians we get: 1 Θ = sin−1 2 I1 − I2 I1 + I2 (2.84) Which is often useful to write as: 2 6 20 Θ = Φ 1 + Φ3 + Φ5 + Φ7 + · · · · · · · · · 3 5 7 36 (2.85) Linear Polarizer Photodiode Input Faraday Modulator Cell Lock-In Amplifier Linear Polarizer Probe Laser Reference Figure 2.9: Optical detection with a linearly polarized probe beam passing through a Faraday rotator, then a polarized cell, a second linear polarizer at 90degrees to the first, and finally a photodiode detector. The photo-detector signal is fed through a photodiode amplifier and using a Lockin amplifier referenced to the Faraday modulator frequency. The in phase component of the Lockin amplifier gives the rotation where Φ= I1 − I2 2( I1 + I2 ) For the case where Φ << 1 we have Θ = I1 − I2 . 2( I1 + I2 ) (2.86) For sensitive polarimetry one often alters the detection scheme to increase the sensitivity or decrease the 1/ f noise of this measurement. An example of this is to send the probe beam through a Faraday modulator which modulates the plane of polarization of the probe beam by a small angle via application of strong internal magnetic fields. See fig.2.9. In this arrangement one first passes the probe beam through a plane polarizer, then through the Faraday modulator, then the cell, and finally through a second polarizer set at 90◦ to the first polarizer before finally terminating on a photodiode. The resultant signal on the photodiode is: I = I0 sin2 [θ + αsin(ωmod t)] (2.87) Here α denotes the amplitude of modulation of the probe polarization axis, I0 denotes the light intensity transmitted through the cell and ωmod denotes the frequency of modulation of the Faraday modulator. To low order one can Taylor 37 expand the eq. (2.87) to obtain: I ≈ I0 [θ 2 + 2θαsin(ωmod t) + α2 sin2 (ωmod t)] (2.88) One then references the detected signal to a lockin amplifier to detect only the Fourier component of the signal at the Faraday modulation frequency. This method greatly reduces 1/ f noise and gives a signal: ILockin ≈ 2I0 θα (2.89) Another commonly used detection scheme is to employ a dichroic plate before the polarizing beam splitter in the standard detection arrangement. The dichroic plate is a type of poor polarizer which greatly attenuates but does not extinguish one particular orientation of polarized light. These often have extinguish or attenuation ratios on the order of 100 : 1. The attenuating axis of the dichroic plate is aligned with the polarization axis of the probe beam. This increases the magnitude of the signal in one photo-detector relative to the other, and effectively increase the rotation angle. This however does not alter the maximum sensitivity as the relative noise on both channels remains unaffected. This system also requires a careful calibration by comparing the output signal as the probe polarization axis is varied because the signal to angle conversions factor is a function of angle. 2.3.2 Light Shifts When operating with an off resonant, or nearly resonant laser beam one must take special precautions to avoid the effects of light shifts. Light shifts mimic the behaviour of a magnetic field and will alter both the precession frequency, and orientation of free spins. Light shifts can arise from two different mechanisms. Our 38 description will follow that by Appelt et al. (1998). The mechanism for light shifts typically encountered in precision frequency measurements is due to the AC Stark effect of the electric field of the light beam interacting with the atomic vapour. This interaction can be described by: δH = ∆Eν − ih̄ < Γ >= − E∗ · α(ν) E 2 (2.90) Here the energy shift is given by ∆Eν , α(ν) describes the complex atomic polarizability, and < Γ > is the average photon absorption rate. It can be related to the photon flux Φ, and the absorption cross-section σ (ν) by: < Γ >= σ(ν)Φ(1 − s · S) (2.91) where S is the atomic spin, and s is the photon spin vector. This is given by: s = ib ǫ×b ǫ∗ (2.92) where b ǫ is the unit Jones polarization vector. Jones vectors are a convenient way to describe polarization. In Jones terminology a σ+ , σ− , and π polarized beams are given by: ǫσ + ǫσ − 1 1 =√ i 2 0 1 1 =√ −i 2 0 39 (2.93) (2.94) 1 ǫπ = 0 0 (2.95) Because the Hamiltonian is an analytic function, its real and imaginary parts are related. One can apply the Kramers-Kronig relations to eq. (2.90) to find the energy shift to be: ∆Eν = h̄ πre c f Φ(1 − 2s · S) Im[V(ν − ν0 )] 2 (2.96) One can readily see that this has the same form as the Zeeman interaction: ∆Eν = h̄γe B LightShi f t · S (2.97) where the effective magnetic field the atom experiences is: B LightShi f t = −πre c f Φ Im[V(ν − ν0 )]s γe (2.98) Here we ignore the common offset given by eq. (2.96) and only retain the relevant spin dependent energy shift. One can generalize this and include the effect of the D2 line by replacing s in the above equations by s/2. Since the light shift interaction appears as a fictitious magnetic field the atoms respond by precessing around the total effective magnetic field B = B0 + B LightShi f t . This effect can be quite large when B0 ≈ B LightShi f t . Thus we try to minimize this effect whenever possible. Note that the eq. (2.98) becomes zero either when Im[V(ν − ν0 )] or s become zero. The first case corresponds to a laser beam exactly on resonance, as Im[V(ν − ν0 )] has a dispersive shape. One can easily verify using the Jones vectors that the second case where s vanishes corresponds to the case of linearly polarized light. To minimize the effects of light shifts we ensure that the 40 pump beam is tuned to resonance. We must also ensure that beams which are off resonance, such as the probe, are linearly polarized. There exists a second mechanism by which light shifts can occur in the optical pumping system. During optical pumping atoms spend a small amount of time out of the ground state and in the excited state. The gyromagnetic ratio of the alkali is different in the excited state. Thus atoms which were not pumped into the excited state will acquire a phase relative to atoms which remained in the ground state the entire time. This effect is generally much smaller than the precession frequency itself and is negligible in our systems. 2.4 Coupled Spin Dynamics In a vast number of fundamental physics experiments such as electric dipole moment search (Romalis et al. (2001)), CPT violation (Kornack et al. (2008)), and pulmonary imaging (Oros & Shah (2004)), it would be useful to polarize noble gas nuclei. Due to spin dependent interactions the electron spin of the alkali can be used to polarize the nuclear spin of a noble gas. This is called spin exchange optical pumping (SEOP). This is the typical way in which noble gases are polarized, with the exception of helium. In some cases it is advantageous to polarize helium gas via a technique called metastable exchange optical pumping. That technique will not be discussed here (Schearer (1968)). In the following sections we discuss the mecahnisms for SEOP and the dynamics of the interaction between the polarized alkali, and the polarized noble gas. 41 2.4.1 Optical Pumping of Noble Gas The spin dependent interaction between the alkali and the noble gas during collision is: V1 (r ) = γ(r ) N · S + Ab (r ) Ib · S (2.99) Here the noble gas spin is denoted as Ib , the alkali spin by S, and the rotational angular momentum by N. The spin-rotation interaction arises from the magnetic fields created by the motion of the charges of the colliding atoms. The second term arises from the hyperfine interaction of the nucleus of the noble gas, and the alkali. Here the coupling coefficients are strong functions of the inter-atomic separation r. The subscripts a refer to the alkali atom, while the subscript b to the noble gas atoms. The spin-rotation term in the interaction potential leads to alkali spin relaxation. The second term leads to spin exchange between the alkali electron spin and the noble gas nuclear spin. In the spin temperature regime we can express the rate equations governing the polarization of the spins by (Walker & Happer (1997)): d h Fz i = − Γ a ( γ ) h Sz i − dt Γ a ( Ab )[ǫ( Ib , β b ) hSz i − h Ibz i] (2.100) d h Ibz i = Γb ( Ab )[ǫ( Ib , β b ) hSz i − h Ibz i] dt (2.101) Due to the principle of detailed balance we can say that: nb Γb ( Ab ) = n a Γ a ( Ab ) where the density of the vapour species is given as ni . For the case of 42 (2.102) 21 Ne, Ib is 3/2. Thus eq. (2.101) simplifies to: d h Ibz i = Γb ( Ab )[3 hSz i − h Ibz i] dt (2.103) Since ǫ is a function of spin temperature this is valid uner conditions of high spin temperature where Pa ≈ 1. We can readily see that in steady state, if all relaxation mechanism are suppressed, then the noble gas nuclear spin expectation value is one third of the alkali spin expectation value. However the nuclear spin of 21 Ne is 3/2, while the electron spin is 1/2. Taking this into account we can re-write eq. (2.103) in terms of the polarization of each species respectively as: 3 1 Pb = 3 Pa 2 2 (2.104) Or, simply Pa = Pb , under conditions of high spin temperature. We see that in steady state and under the absence of relaxation mechanisms the noble gas polarization equilibrates with the alkali vapour polarization. This argument is valid for an atom with any nuclear spin value. However in practice the noble gas polarization does not reach the same value as the alkali polarization due to strong relaxation mechanisms. These include spin destruction collisions, gradient relaxation, and long range magnetic dipolar and quadrupolar field interaction. Typical polarizations are on the order of 40 − 50% under conditions of high pumping rate. The equilibrium nuclear polarization can be written as: Pb = P a ab ǫ Rse b ab + 1/T I /Sz Rse 1 bz ab = Where the spin exchange rate Rse dh Ibz i dt rate for neon polarization. 43 (2.105) = Γb ( Ab ), and 1/T1b is the relaxation 2.4.2 Interaction of polarized alkali with polarized noble gas The interaction of the polarized alkali and polarized noble gas yields interesting behaviour. The magnetic field experienced by the alkali is due to both the classical magnetic holding field in which the atoms are present, and the magnetic field created by the polarized noble gas. The field experienced by the alkali due to the noble gas is not given by the classical expression of a magnetic dipole field. This is because collisions between the alkali and noble gas deform the wavefunction of the alkali and also overlap. The dominant interaction between the alkali electron spin and the noble gas nuclear spin is described by eq. (2.99). The hyperfine interaction coefficient arises from the Fermi-contact magnetic fields of the two atoms (Herman (1965)): A b (r ) = 8πgs µb |Ψb (0)|2 3Ib (2.106) Due to the deformation of the wavefunction during collision it becomes: A b (r ) = 8πgs µb |ηΦ( R)|2 3Ib (2.107) The enhancement factor η is the ratio of the wavefunction at the noble gas nucleus during collision to the unperturbed wavefunction in the absence of noble gas. This isotropic hyperfine interaction also leads to the frequency shift of the precession frequencies of both the alkali and the noble gas. The shift is described by the parameter κ0 , where κ0 is the ratio of the magnetic field experienced by the alkali due to the collision with the noble gas, and that which would arise from a Fermi contact interaction with no η wavefunction enhancement factor (Schaefer et al. (1989a)). κ0 = Z 4πR2 |ηΦ0 ( R)|2 e−V0 ( R)/k B T 44 (2.108) Where V0 is the spin independent interaction between the Noble gas and the alkali. Normally this is fit to pseudo-potentials using data from scattering experiments. The dominant interaction between the alkali nuclear spin and the alkali electron spin is: Hhyp = A a Ia · S (2.109) The A a term in the isotropic interaction can produce a pressure shift in the hyperfine splitting. This is because the valence electron density is perturbed by the noble gas resulting in a shift in the energy levels. For the most part optical pumping experiments measure magnetic resonance of Zeeman levels, and are thus not very sensitive to shifts in the A a parameter. This is an important effect in the operation of atomic clocks. The effective magnetic field the alkali experience due to the polarized noble gas for a spherical volume is: B= 8πκ0 MP 3 (2.110) Where M is the magnetization density of a fully polarized noble gas, and P is the noble nuclei polarization. The alkali atoms precess in the resultant field caused by both the static magnetic holding field, and the magnetic field produced by the noble gas nuclei. The motion of the alkali polarization can be described by the phenomenological Bloch equations (Kornack & Romalis (2002)): ∂Pe γe 8πκ0 = B+ Mnoble Pnoble + L × Pe + Ω × Pe e ∂t Q( P ) 3 ab n e + Rop s pump + + R probe s probe + Rse P − Rrel P /Q( Pe ) ∂Pn = γn ∂t 8πκ0 B+ Malkali Pe 3 (2.111) ab × Pn + Ω × Pn + Rse (Pe − Pn ) − Rrel Pn (2.112) γe is the gyromagnetic ratio of a free electron, Rrel is the alkali relaxation rate, Pn ab is the alkali-noble gas spin exchange rate. R is the nuclear polarization, and Rse op 45 and R probe are the pump and probe beam pumping rates. One can write this system of equations in complex form by multiplying the y component by the imaginary number i, adding it to the x component, and solving the resulting complex differential equation. For small non-equilibrium excitation of the spins when there is no transverse magnetic field present we can solve this set of equations. The linear approximations to the solutions is: 8πκ0 8πκ0 z z Mnoble Pnoble ) P⊥e − Rtot P⊥e + i Mnoble Pnoble )/Q( Pe ) 3 3 (2.113) ′ 8πκ 8πκ 0 0 n z z P⊥ (t) = −iγn ( Bn + Malkali Palkali ) P⊥n − Rtot P⊥n + i Malkali Palkali ) P⊥n 3 3 (2.114) ′ e P⊥ (t) = (−iγe ( Bn + Here the xb, and yb components correspond to the real and imaginary components of the solution. One can see that the magnetic fields experienced by the alkali and Noble gas species differ. This is because in addition to the holding field each species experiences the magnetic field produced by the magnetization of the other sample. Bze = Bn + 8πκ0 z Mnoble Pnoble 3 (2.115) Bzn = Bn + 8πκ0 z Malkali Palkali 3 (2.116) The co-magnetometer experiments in this thesis operate near a noble-alkali hybrid resonance described by the magnetic field compensation point: n Bcomp =− 8πκ0 8πκ0 z z Malkali Palkali − M Noble PNoble 3 3 (2.117) Although the alkali is not at zero field, the field is low enough that the alkali remains in the SERF regime. Operation at the compensation point leads to screening 46 of transverse magnetic fields. This will be discussed shortly. One application of particular interest is the signal dependence of the co-magnetometer on rotation. Solving the Bloch equations one finds the signal to be (Kornack et al. (2005b)): Srot Pze γe Ωy = γn Rtot γn γe2 2 Bz e 1− Q( P ) − C − 2 Bz + n − D + · · · γe B Rtot (2.118) where C= γe Pze Rnse ≈ 10−3 n γn Pz Rtot (2.119) Me Rese ≈ 10−5 Mn Rtot (2.120) D= where Ωy is the angular rotation rate of the apparatus. Pze is the electron polarization, Rnse is the noble gas spin exchange rate, Pzn is the noble gas polarization, Rtot is the total alkali spin relaxation rate, and Rese is the alkali spin exchange rate. The signal is defined to be S = R( Pxe ). For a holding field set to the compensation point: Pze γe Ωy S(Ωy ) = γn Rtot γn 1− Q( Pe ) − C − D + O(10−6 ) γe (2.121) One can see that the system is sensitive to any rotations about the yb axis. The experimental implications of this will be discussed later, suffice it to say that this allows one to operate the coupled system as a gyroscope. The signals from rotation about the other two axes is much smaller. This is by a factor of approximately 100 for the xb and 105 for the b z axes. The coupled system also exhibits interesting shielding behaviour when set to the compensation point. The signal becomes insensitive to small applied transverse magnetic fields. When the system is tuned to the compensation point the noble gas spins are aligned with the holding field. If a small transverse field in47 (a) 21Ne cancels the external field Bn (b) K feels no field SK MK I 21 M 21 Ne 21Ne compensates for Bx K feels no change SK MK II 21 Ne Bn n B M 21 Ne Ne B B Bn Bn Bx Figure 2.10: When set to the compensation point the polarized noble gas screens transverse magnetic fields. teracts with the system the noble gas spins will adjust to align itself with the total magnetic field. To first order the noble gas cancels out any transverse field. This suppresses fields transverse to the holding field. Magnetic fields parallel to the holding field do not tip the alkali since they will not adjust the coupled system. 2.5 Manipulation of polarized noble gas spins, and Magnetic shielding In experiments where noble gas is polarized it is often necessary to manipulate the noble gas orientation. Examples of this include performing adiabatic fast passage for determination of alkali noble gas spin exchange cross section (Chann & Walker (2002)), and nuclear magnetic resonance for determination of κ0 (Stoner & Walsworth (2002)), and pulmonary imaging (Oros & Shah (2004)). Many optical pumping experiments benefit from operation in a low magnetic field environment (Kornack et al. (2008)). This can be achieved by utilizing magnetic shielding. This suppresses the magnetic field the atomic species experience by several orders of 48 magnitude. In this section we discuss each of these techniques for manipulation of both the polarized noble gas, and shielding in greater detail. 2.5.1 Adiabatic Fast Passage In optical pumping experiments it is often useful to flip the noble gas polarization by 180◦ . The most common technique for achieving this with relatively low loss of polarization is Adiabatic Fast Passage (AFP). To achieve this one applies a magnetic field perpendicular to the magnetic holding field of the alkali. This field is then swept in frequency through the magnetic resonance frequency of the alkali due to the holding field. Alternatively the frequency of the perpendicular applied field can be held constant, and the strength of the field can be varied. Under appropriate conditions this process can result in the inverting of the alkali spins. To explain this effect it is necessary to review some dynamics of operators viewed in a rotating frame. ~ in an initially inertial frame. If one transfers to a frame Consider the vector A ~ then A ~ transforms as: which is rotating with frequency Ω ~ dA dt ! ~ dA dt ! = rotating = inertial ~ dA dt ~ dA dt ! ! inertial rotating ~ − ~Ω × A (2.122) ~ + ~Ω × A (2.123) Now let us consider a spin ~S held with a field B0 in the inertial lab frame. It undergoes Larmor precession according to: ~ dA dt ! inertial 49 = γ~S × ~B0 (2.124) In the rotating frame this becomes: d~S dt ! d~S dt ! or, rotating rotating = γ~S × ~B0 − ~Ω × ~S (2.125) ! (2.126) = γ~S × ~ ~B0 + Ω γ The system behaves as if it experiences an effective magnetic field ~Be f f = ~B0 + Ω γ (2.127) in the rotating frame. The spins rotate about this effective field in the rotating frame. Let us consider the special case where the applied field ~B1 is rotating with ~ Ω is equal to the precession frequency ω0 = −γB0 of the alkali. Here we find: ~Be f f = ~B0 + ~B1 + ω b z γ or since ω0 = −γB0 , ω ~Be f f = B0 b z + B1 xb + b z γ ~Be f f = B1 xb (2.128) (2.129) (2.130) When the frequency of the applied transverse field is exactly on resonance with the precession frequency due to the holding field in the rotating frame, the spins experience a static transverse field perpendicular to the holding field. This causes the spins to rotate about the xb and flip. In the more general case where Ω 6= ω0 one does not completely cancel out the effect of the holding field as described in the transition from equation 2.129 to eq. (2.130). The atoms still experience a static field B1 xb but also experience a 50 B0 ω γ Beff θ B1 Figure 2.11: Here one can see that if a transverse field is applied at a frequency other than the resonant frequency it appears as having both transverse and axial components in the rotating frame. The relative strength of these components is a function of the detuning from resonance. The atoms precess around the effective magnetic field which lies at an angle θ to the holding field. As one sweeps the frequency of the transverse field B1 from far below resonance to above it the angle θ goes from 0 to π. Thus we can flip the orientation of the spins in this manner. field Ω − ω0 z along γ b the holding field axis. See fig.2.11. The alkali now rotate around the general field ~Be f f in the rotating frame. As the frequency Ω is swept the field along b z changes. The angle ~Be f f makes with the holding field is also changed. If Ω is varied slowly then the spins will follow and continue to precess around the field Be f f . Using this knowledge we can sweep the field from far below the resonant frequency to far above the resonant frequency to flip the spins by π radians. To ensure that the alkali spins follow Be f f one must change the frequency of the transverse field slowly. Physically this condition is satisfied when the precession rate is very large compared to the rate at which the direction θ (t) of Be f f is changing (Powles (1958)). This condition is most stringent when the applied frequency is equal to ω0 or geometrically when θ = π 2. So, dBe f f (t) θ̇ = 1 B1 dt 51 (2.131) or, 1 1 dΩ << γB1 B1 γ dt (2.132) Substituting the condition ω = γB1 which occurs at θ = π/2 we find: dΩ << ω02 dt (2.133) to ensure the spins adiabatically follow the Be f f . One cannot sweep the magnetic field frequency arbitrarily slowly. When one has a Be f f which is not parallel to the holding field the spins will dephase and lose polarization after a time T1 . We must sweep the spins faster than a time T1 to retain spin polarization. Mathematically this corresponds to: γB1 dΩ << T1 dt (2.134) Thus to ensure one can flip the spin by sweeping the frequency of the applied field one must satisfy: γB1 dΩ << << ω02 T1 dt (2.135) 2.5.2 Nuclear Magnetic Resonance In order to determine the κ0 magnetic enhancement factor one must determine noble gas polarization. To do this we utilize the techniques of nuclear magnetic resonance. As this is quite a broad field only the fundamentals which were applied in the experiments in this thesis will be reviewed. Nuclear magnetic resonance (NMR) is the technique by which polarized spins precessing in a magnetic field are manipulated and detected. These are tipped or excited so that they possess a transverse polarization. As the transverse magnetization rotates they produces an oscillating magnetic field which is typically detected 52 by inductive pickup coils. To tip the spins one excites them with a magnetic field pulse which is at the same frequency as the precession frequency of the atoms. If the excitation tipping field, along the xb direction, is of strength B1 then the field experienced in the frame rotating at the Larmor frequency of the nuclei is: ~Be f f = ~B0 + ~B1 + ω b z γ or since ω0 = −γB0 , ω ~Be f f = B0 b z + B1 xb + b z γ ~Be f f = B1 xb (2.136) (2.137) (2.138) This argument follows directly from that in the previous section, where we assume the holding field along the b z direction. We can see that the nuclei will now precess along the xb direction. Typically the applied field however is of the form: ~B1 = B1 cos(ωt) xb (2.139) which is not equivalent to a field rotating at the precession frequency. However it can be decomposed into two counter-rotating fields at the precession frequency: 1 ~B1 = 1 B1 (cos(ωt) − sin(ωt)) b σ+ + B1 (cos(ωt) + sin(ωt)) b σ− 2 2 (2.140) The σ− component is located far enough off resonance that its effect is negligible in most cases. We will not discuss it but it can be of interest in certain comagnetometers operating in the SERF regime. If the duration of the magnetic field pulse is for a time t then we see that the angle the spins rotate through an angle given by: Θ= 1 γBt 2 53 (2.141) B0 ω S Β1cos (ω t) θ Figure 2.12: If an oscillating magnetic field B1 is applied at the atom precession frequency orthogonal to the holding field B0 then the atoms will become tipped off axis. See fig.2.12. In order to detect the field produced by the precession of the tipped spin one constructs an inductive pickup coil. To ensure that the signal is larger than the various instrument noise one utilizes the pickup coil to create a resonant LC circuit. Here a capicator is placed is parallel with the pickup coil. For an ideal coil of inductance L, and resistance r the impedance of parallel RLC circuit is given by (Fukushima & Roeder (1981)): Ztank = r − iωL(1 − ω 2 CL − r2 C/L)−1 r 2 + ω 2 L2 (2.142) The resonant condition occurs when: 1 − ω 2 CL − r2 C/L = 0 (2.143) However for most systems the resistance of the pickup is quite small. Here we 54 get the resonant frequency to be: 1 ω= √ LC (2.144) The impedance at the resonant condition is given by: Zresonant r 2 + ω 2 L2 = r (2.145) Typically the resonant frequency of the pickup is tuned by placing a variable capicator in the RLC circuit, and adjusting it. It is often convenient to define the quantity Q = ωL/r. This is referred to as the quality factor of the pickup circuit. For an inductive pickup coil the Q factor is the factor by which the pickup voltage is magnified at the resonance condition as compared to the case of detecting the precessing spins directly with the pickup coil, and not using a RLC circuit. 2.5.3 Magnetic Shielding For sensitive magnetometry experiments it is often advantageous to operate inside magnetic shields. These reduce or eliminate magnetic field contributions due to the Earth’s field, and laboratory power line 60Hz magnetic noise. This ensures that the field in which the atoms are held is well known and uniform. These shields typically are made from ferromagnetic materials, have high permeability, and are easy to both magnetize and demagnetize. Typical commercial brands are made from Mu-Metal. Other materials often used in magnetic shield manufacture are Conetic alloys, and Moly Permalloy. Recently work has been done to observe the effectiveness of utilizing ceramic ferrite as a shield (Kornack et al. (2007)) which have lower thermal noise than the Johnson noise of Mu-Metal shields at room tem- 55 perature. The standard arrangement for magnetic shielding is to make multiple concentric cylinders which are nested within each other. If one performs the calculation they find that the shielding factor in the transverse and axial direction differ for a cylindrical shield. The transverse shielding factor is given by (Jackson (1999)): ST = Bi µt = B0 2R (2.146) where µ is the magnetic permeability of the shield, t is its thickness, and R its radius. Here B0 is the uniform field outside the shields, and Bi the field inside the shield. The axial shielding factor is given by (Khriplovich & Lamoreaux (1997)): SA = 2µtR1/2 Bi ≈ B0 L2/3 (2.147) where L is the length of the cylinder. See fig.2.13. This approximation is only valid in the region where 4 < L/R < 80. Normally one must place holes in magnetic shields for various electronic feedthroughs, power cables, optical beam paths, etc. This decreases the shielding factor slightly. For a hole of radius r the field perpendicular to the shield surface falls off as (Khriplovich & Lamoreaux (1997)): B(l ) = Be−1.5l/r (2.148) where l is the distance from the hole. The shielding from nested cylinders is not simply the product of the individual shielding factors of each layer. This is because the internal shields affect the boundary conditions used in the previous calculation of the shielding factors. The 56 Magnetic Shields Air Baxial Figure 2.13: The magnetic fields lines have an affinity for magnetic shields. The above figure shows how magnetic fields lines from a previously homogeneous field warp in the presence of magnetic shields. The solution of the Laplace equation in cylindrical polar coordinates gives a magnetic scalar potential Φ = (Ck ρ + Dρk )cosθ. The relevant boundary conditions for the magnetic field compok nents normal and tangential the surface are µk dΦ dρ |rk = µk+1 1 dΦk+1 rk dθ |rk dΦk+1 1 dΦk dρ |rk , and rk dθ |rk = The magnetic field pattern can be constructed by solution of the Laplace equation, subject to application of the boundary conditions described above. 57 transverse shielding factor for nested cylinders is given by (Sumner et al. (1987)): ST = SnT n −1 ∏ SiT i " 1− Ri R i =1 2 # where the script i refers to each individual layer. The 1 − (2.149) Ri R i =1 2 compression of internal flux (volume loss) in the region between layers. The axial shielding is given by: SA = SnA n −1 ∏ SiA i L 1− i L i +1 (2.150) Here the 1 − LLi term reflects the reduction in volume or compression of flux along i +1 the cylinder length. Before use the magnetic shields must be de-gaussed. This necessitates reduction of the magnetization of the shields to zero. To accomplish this one passes a high alternating current, normally 50 amp-turns, through the shields to saturate them. The current in the shields is then slowly reduced to zero. The idea is that when one passes a high current through the shields the saturation is symmetric on current reversal. If one then reduces the current slowly, much slower than the alternating current frequency, one can eliminate the magnetization when the current becomes zero. 58 Chapter 3 Nuclear Spin Gyroscope Sensitive gyroscopes are utilized in applications ranging from inertial navigation, and studies of the Earth’s rotation, to tests of general relativity (Stedman (1997)). A variety of physical principles have been employed for rotation sensing. These include mechanical gyroscopes, the Sagnac effect for photons (Stedman (1997))(Andronova & Malykin (Andronova & Malykin)) and atoms (Gustavson et al. (1997))(YverLeduc (Yver-Leduc)), the Josephson effect in superfluid 4 He, and 3 He (Avenel et al. (2004)), and nuclear spin precession (Woodman, Franks & Richards (Woodman et al.)). While mechanical gyroscopes operating in low gravity environments remain so far unchallenged (Buchman et al. (2000)) there is much competition in the field of compact gyroscopes operating in Earth’s field. We have developed a nuclear spin gyroscope (Kornack et al. (2005a)) based on the co-magnetometer arrangement described earlier in this work (Allred et al. (2002)). As mentioned earlier the signal from the atom co-magnetometer has a dependence on the rotation of the apparatus, thus it can be utilized as a gyroscope. Though the entire dynamics of the coupled alkali-noble gas system is complicated it is useful to describe the system physically before delving into mathematics. The work in this chapter can also be found in Kornack et al. (2005a). 59 3.1 Co-magnetometer Gyroscope Implementation and behaviour In the co-magnetometer arrangement the K alkali spins are polarized with a pump laser. Through spin exchange collisions the nuclear spins of the 3 He noble gas are polarized parallel to this. Let us imagine the cell which contains noble gas to be in an inertial reference frame. If we rotate the apparatus the alkali spins will quickly be re-pumped along the new pump laser orientation since the T1 of the alkali is on the order of 30ms. The noble gas then precess around the net magnetic field and becomes aligned with the holding field in a time T2 of ≈ 100s. We will focus on the interesting dynamics on a short time scale. Initially if the co-magnetometer is properly zero-ed, and tuned to the compensation point then in the steady state arrangement the noble gas nuclei experience a net magnetic field parallel to the nuclear spin due to the magnetic compensation field. When the apparatus is rotated the compensation field is now at an angle to the nuclear spins causing them to precess about it. As the nuclear spins precess the orientation of the net magnetic field which the alkali experience now changes. This causes the potassium to rotate about the new orientation of the net magnetic field. Consider the component of the magnetic field perpendicular to the pump and probe beam optical axis which is produced by the noble gas. This field causes the alkali to precess in the plane of the pump and probe beam. It is this alkali precession that we monitor with an off resonantly tuned probe beam. See eq. (2.121). Using Green’s functions for the linearized Bloch equations one can show that the integral of the co-magnetometer signal is proportional to the total rotation angle about the yb axis. This is independent of the time behaviour of Ωy . The angular 60 Position sensors P u m p B ea m H i gh P ower D i ode L aser ?/4 P i ezoel ect r i c S t ack I mmobi l e B l ock Floating Optical Table Lock-in Amplifier Photodiode Analyzing Polarizer Magnetic Shields Field Coils Hot Air Cell SK MK Probe Beam Single Freq. Diode Laser M 3He Polarizer I 3He Bz Faraday Pockel Cell Modulator x y z Figure 3.1: A schematic of the co-magnetometer being implemented as a gyroscope. Note the non-contact position sensors used to detect the rotation, and the piezo stack used to force the apparatus to oscillate frequency of the alkali can be related to the co-magnetometer signal by: Ωy = γg S (3.1) where S is the signal from the co-magnetometer in units of magnetic field and γg is given by: γg ≈ Q( Pe ) 1 − γn γe −1 (3.2) The apparatus was driven to rotate by a piezo stack placed between the optical table upon which the co-magnetometer sits and a heavy immovable concrete block. This is depicted in fig.3.1, and fig.3.2. Six non-contact displacement sensors were used to monitor the table orientation. They were mounted to the posts at the base of the optical table. They operate by measuring the capacitance between the sensor and a target metallic plate. The target plates were mounted to the floating portion of the optical table. The sensors was calibrated and found to give a linear response over displacement am61 Cell Spin polarization Pump Beam Probe Beam Photodiode Table rotation Piezo Stack y Position Sensors z Immobile Block y x x z Figure 3.2: Alternate side view of the gyroscope configuration for the comagnetometer 5.0 Signal (volts) 4.5 4.0 3.5 3.0 2.5 2.0 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 Horizontal displacement from target (cm) Figure 3.3: In-situ calibration of the non-contact displacement sensors for determination of absolute rotation. plitudes of a few centimeters. For comparison during operation the amplitude of displacement was on the order of millimeters. The sensors were each individually calibrated in-situ. See fig.3.3. In order to convert the co-magnetometer signal to a gyroscope we must integrate the co-magnetometer signal. Using eq. (3.1) we convert to angular units. The resulting fit of the co-magnetometer gyroscope signal to the orientation signal from the non-contact displacement sensors in shown in figure 3.4 The gyroscope signal agrees with the position measurement to within the 3% calibration accuracy. The gyroscopic signal has also been shown to be insensitive 62 50 0 0 − 50 − 20 E ff ecti v e F i el d ( f T ) R otati on ( µ r ad / sec) 100 20 − 100 0 2.5 5 7.5 10 12.5 Time (s) 0.02 25 0.015 20 15 0.01 10 0.005 F i el d ( fT / H z1/ 2 ) A ngl e R andom W al k ( degr ees/ hour 1/ 2 ) Figure 3.4: Comparison of co-magnetometer gyroscope signal to displacement sensor signal with no free parameters. The solid line depicts the co-magnetometer signal, and the dashed line the signal from the position sensors. 5 0 0 200 400 600 800 0 1000 Frequency (hour − 1 ) Figure 3.5: Fourier transform of the noise spectrum of the comagnetometer gyroscope. The discrete noise peaks are an artifact caused by the periodic zeroing routines for the co-magnetometer to the other two components of the angular velocity, and only depends on the angular velocity vector perpendicular to the plane containing the pump and probe beams. Using the relation between the magnetic field measurement of the co-magnetometer and the rotation measurement of the gyroscope one can calculate the noise spectrum of the gyroscope from previous magnetic noise measurements of the comagnetometer. As one can see from fig 3.5 the noise spectrum of the gyroscope √ √ is nearly flat at 1.0ft/ Hz, or translated to angular units 1.4 × 10−5 rad/ hour for frequencies above 400 hour−1 . At frequencies below this one sees a clear 1/ f noise 63 F i el d S u p p r essi on F actor 100 dBy/dx dBy/dy 10− 1 dBy/dz dBx/dz −2 10 dBz/dz 10− 3 10− 4 0.1 0.2 0.5 1 2 5 10 Frequency (Hz) Figure 3.6: Suppression of an applied magnetic field gradient by the comagnetometer compared to that of a non-compensating magnetometer. Colored points refer to measurements made with square wave modulation instead of sinusoidal modulation. dependence with a 1/ f noise knee at 0.05Hz. The noise of the gyroscope in magnetic units is much less than the magnetic Johnson contribution from the magnetic shielding. This is because the co-magnetometer acts as a magnetic field suppressor. This is discussed in more detail in section 2.6. The magnetic field is suppressed even though the alkali metal and Noble gas have different spatial distributions. (see fig 3.6). The reason for this behaviour is because the noble gas diffusion rate is much lower than the nuclear spin precession rate. That is γn Bn >> R D >> Rnsd , where Rnsd is the nuclear spin destruction rate. The magnitude of the polarization of the noble gas is constant in value, and orients itself parallel to the local magnetic field. Thus the noble gas is able to cancel a non-uniform external field on a point by point basis. An oscillating magnetic field was applied and effectively shielded by the comagnetometer. This is shown in fig 3.7. In addition, using the linearized Bloch equations one can show that the rotation angle created by a magnetic field transient is zero as long as the spin polarizations are not rotated by a large amount during the transient. Fig 3.8 shows the response of the gyroscope to a magnetic field transient. The co-magnetometer demonstrates a reduction of the spin rotation angle by a factor of 400 as compared to that of a standard potassium co-magnetometer. 64 Bx By 10 Bz 0.1 1 0.01 M easu r ed F i el d ( p T ) F i el d S u p p r essi on F actor 1 0.1 0.001 0.1 0.2 0.5 1 2 5 10 Frequency (Hz) 60 0.3 40 0.2 20 0.1 0 A ngl e ( r ad ) M agneti c F i el d ( p T ) Figure 3.7: The co-magnetometer suppresses magnetic fields. Thus the contribution from the Johnson noise of the magnetic shields is greatly reduced. The field suppression in the xb,b y, and b z directions are measured relative to that of a K magnetometer. The solid line represents the theoretical prediction. 0 − 20 − 0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 Time (s) Figure 3.8: Response of co-magnetometer(dashed red line) to a magnetic field transient(solid line). The total rotation angle(blue dashed line) is proportional to the integral of the co-magnetometer signal. It is also much smaller than the rotation angle from a K magnetometer(dashed black line). 65 3.2 Effect of Experimental Imperfections on Gyroscope Performance Gyroscope performance can be affected by a number of experimental imperfections of the system. One such imperfection is that arising from not properly zeroing the magnetic field on the co-magnetometer. The only component of the magnetic field or light shift which contributes to the signal in first order is the Bx field. It causes a false signal of (Kornack et al. (2005a)): e n S( Bx ) = Bx Pze (Cse + Cse )/Bn (3.3) e Cse = ( Rese Pzn )/( Rtot Pze ) (3.4) where is the electron spin exchange correction, and n = (γe Pze Rnse )/(γn Pzn Rtot ) Cse (3.5) is the nuclear spin exchange correction factor which arise from solving eq. (2.111). e ≈ 10−2 Substituting the measured values for Pze ,Pzn , Rese , Rnse , and Rtot we find Cse n ≈ 10−3 . Since both C e and C n are small this field dependence is heavily and Cse se se suppressed by a factor of ≈ 105 . Pze is the alkali polarization,Pzn the noble gas polarization, Rese is the alkali metal-noble gas spin exchange rate for an alkali atom and, Rnse is the same for a noble gas atom. Rtot = R pump + Rese + Resd + R probe Misalignment of the pump and probe beam so that they are not orthogonal also causes a systematic rotation signal. If they are misaligned an angle α away from 90◦ we measure a signal: S = αR pump /Rtot 66 (3.6) Under typical operating conditions of the gyroscope a misalignment of 1µrad gives a false signal of 10−8 rad/s. However this can be corrected for because a true rotation signal has no dependence on the pumping rate. One could correct for this by varying the intensity of the pump beam, and aligning the probe beam until there is no longer a gyroscope signal at the frequency at which the pump intensity is being varied. One can also measure a false signal if the probe beam polarization is not fully linearly polarized, but has a small circular polarization component. This gives a signal of: S = sm Rm /Rtot (3.7) where the circular polarization of the probe beam is given by sm , and Rm gives the pumping rate of the probe beam. This pumping by the probe beam can be eliminated by zeroing the probe beam light shift. This will be described further in the next section. Other experimental imperfections only contribute to the gyroscope signal in second order. For comparison the signal from an improperly zeroed field By is given by: S= γe By Pze e n ( Bz − ( Bz + Lz )Cse − (2Bz + Lz )Cse ) Bn Rtot (3.8) where Bz is the amount by which the b z field is tuned away from the compensation point. The Effects of imperfections in a Rb-Ne co-magnetometer are discussed in Sec.6.4. 3.3 Zeroing the Co-magnetometer Gyroscope In order for the co-magnetometer signal to be properly calibrated it must be zeroed. That is the net magnetic field along both the xb, and yb directions must be 67 zero. The light shifts must be removed, and the b z field must be tuned to the compensation point. To do this we employ a quasi-static modulation technique. That is we modulate certain components of the field at very low frequency. That is a frequency much lower than the resonance of the noble gas. In the 3 He comagnetometer this corresponds to a frequency of roughly 7 Hz during normal operation conditions. This frequency is determined by the field experienced by each atom species when the co-magnetometer is brought to the compensation point. The steady state solution to the coupled Bloch equation which govern the comagnetometer gives the co-magnetometer signal in the steady state regime in units of magnetic field as: Ωy sm Rm + αR p S = Ly + + + Bz γn γe Pze By γe − Lx Bn Rtot γe + Rtot Bx Bz ( Bz + Lz ) − L x Lz Bn (3.9) We shall consider the zero-ing of each component of the magnetic field or light shift separately and in greater detail. If one were to calculate the dependence of 3.9 on By they would find: δS ∝ Bz δBy (3.10) We see that the dependence of the signal is directly proportional to the Bz component of the field. So if we modulate the By component of the field we observe a modulation of the signal at the same frequency. Thus we can zero the Bz component by modulating the By field, monitoring the signal, and varying Bz until the modulation in the signal disappears. This corresponds to the point where the compensation field cancels out the field contribution from the magnetization of the atoms, and any residual field due to the magnetic shielding. Once the field has been tuned to this compensation point one can use a similar 68 method to zero the By field. One sees that the signal dependence on Bz is: By δS γe ∝ n + Lx δBz B Rtot (3.11) We ignore the contribution of the Bx Bz term here as it is the product of two small numbers and is suppressed compared to the other terms in the above expression. We see that the modulation of Bz and readjusting the By field will now zero a linear combination of the By field and the lightshift Lx . We independently zero Lx as well. This will be discussed later. Thus to zero By we iterate the modulation of Bz followed by independent zeroing of Lx . To zero Bx we take a slightly different approach. One can write the dependence of the signal on Bz as: δ2 S ∝ Bx δBz2 (3.12) Thus we can eliminate Bx if we modulate the second derivative of Bz . To do this we asymmetrically modulate Bz between a zero, and non-zero value. If one were to perform this modulation symmetrically one would simply be repeating the earlier procedure used to zero By . Although this response would be different since the Bx Bz term would now be of significant size compared to By since the latter component has been zeroed. This function would also be dependent on Lx and By . Once the magnetic field components have been zero-ed we zero the contribution of lightshifts. We consider the Lx Lz term of eq. (3.9). We see: δS ∝ Lx δLz (3.13) δS ∝ Lz δL x (3.14) Thus we can modulate the appropriate component of the light shift to zero the other. The light shift of a laser goes as the frequency of detuning, modified by the 69 Voight profile, and the degree of circular polarization. To modulate the light shift of the pump beam Lz one modulates the frequency of the laser. This is because tuning the pump to be directly on resonance maximizes pumping efficiency. To modulate the lightshift of the probe beam Lx one varies the degree of circular polarization. To do this we attempt to cancel any birefringence in the beam before it strikes the cell. There are a few methods to do this including placing a Pockel cell in the beam path or a plate of stressed glass. The stress of the glass can be varied by using a piezo to compress it. This compression can be modulated. One must also ensure that the pump and probe beam must be properly orthogonalized. The co-magnetometer signal does depend to first order on the product of the angle misalignment α and the pumping rate R pump . However in practice one cannot make the pumping rate high enough to make it the only term which contributes to the signal. The signal dependence on the pumping rate goes as: R pump Ωy R pump δS ∝ αK + γe δR pump Rtot γn R2tot (3.15) where K is a factor for calibrating the misalignment angle to signal units. To vary the pumping rate we feed the pump beam on maximum intensity through two crossed polarizers. In between these polarizers we place a liquid crystal waveplate. By varying the retardance of the waveplate we are able to change the polarization of the beam between the polarizers, and adjust the intensity of the pump upon exiting the polarizers. Thus we can vary the angle α until the dependence on the signal vanishes. 3.4 Co-magnetometer Gyroscope Sensitivity The fundamental sensitivity of the gyroscope is accurately described by the spin projection noise of the co-magnetometer. The measurement uncertainty of the co70 magnetometer is dominated by the noise from the alkali metal spins. In terms of the angular frequency this can be expressed as: γn δΩy = γe r Q( Pe ) Rtot nV (3.16) where n is the alkali metal density, and V is the measurement volume. Currently we operate the gyroscope with a K-3 He co-magnetometer. The fun√ damental sensitivity for this is δΩy ≈ 1.2 × 10−8 rad/s/ Hz. This is roughly 50 √ times lower than the realized sensitivity of δΩy ≈ 5.0 × 10−7 rad/s/ Hz. This discrepancy can be attributed to the large amounts of angular noise contributed from the probe beam laser. By looking at eq. (3.16) we see that the fundamental sensitivity is a function of the ratio of the noble gas and alkali gyromagnetic ratios. By switching to a K-21 Ne co-magnetometer should instigate an immediate improvement in the sensitivity by a factor of roughly 10, since 21 Ne has a gyromagnetic ratio roughly an order of magnitude smaller than that of 3 He. For a cell with 10cm3 volume, K density of 1014 cm−3 , and density of 6 × 1019 cm−3 this would imply a fundamental √ sensitivity of δΩy ≈ 2.0 × 10−10 rad/s/ Hz. 21 Ne To be useful for applications such as navigation, a gyroscope must be small and portable. The most widely used gyroscope for navigational purposes is the fiber-optic gyroscope. After nearly two decades of improvement the sensitivity of these gyroscopes have approached their fundamental sensitivity. Compact state √ of the art fiber-optic gyroscopes have sensitivity of δΩy ≈ 2.0 × 10−8 rad/s/ Hz (Sanders et al. (2000)). New compact interferometer gyroscopes using cold atoms √ with a shot noise sensitivity of δΩy ≈ 1.4 × 10−7 rad/s/ Hz (Canuel et al. (2006))and √ δΩy ≈ 5.0 × 10−9 rad/s/ Hz (Müller et al. (2007)). Another newly proposed gyroscope is one which operates with MEMS technology. These hold promise if 71 0.6 A ngl e ( d egr ees) 0.4 0.2 0 − 0.2 − 0.4 Constant drift of 0.1 deg/h − 0.6 0 1 2 3 4 5 Time (hours) Figure 3.9: Long term drift of gyroscope they continue to be developed in the future. Current designs have recently overcome the extreme sensitivity to temperature drift past MEMS groscopes exhibited.(Trusov et al. (2008)). However the current sensitivity of temperature insen√ sitive MEMS gyroscopes are δΩy ≈ 5.0 × 10−4 rad/s/ Hz (Trusov et al. (2008)). This renders them unusable for navigational applications in the near future. The compact co-magnetometer gyroscope is competitive with the methods mentioned above. The measurement volume is roughly 10 cm3 , though the present implementation occupies a square approximately 2m to a side. This setup can be miniaturized. In fact many of the techniques found in the miniaturization of atomic clocks can be used to minituarize the gyroscope (Knappe et al. (2006), and Knappe (2004)). The lasers can be made more compact quite easily. In fact miniaturizing the magnetic shields improves their performance. The dominant source of long term drift in the co-magnetometer is due to temperature drift in the system. Fig3.9 shows the long term drift of the gyroscope. This should also improve dramatically upon minutuarization. The co-magnetometer gyroscope sensitivity is competitive with larger commercial gyroscopes. For comparison gyroscopes based on the Sagnac effect (Stedman (1997)) have achieved sensitivities of √ δΩy ≈ 2.0 × 10−10 rad/s/ Hz using a ring laser with an enclosed area of 1m2 , and 72 √ δΩy ≈ 6.0 × 10−10 rad/s/ Hz using an atom interferometer with path length of 2m (Gustavson et al. (2000)). One gyroscope which has a much greater sensitivity is Gravity Probe B. However the wonderfully low drift which it demonstrates, significantly decreases when operated under Earth’s gravity. This makes it much less suitable for terrestrial based application. 73 Type 74 Large Scale (∼2 m) Ring Laser Gyro (CII) Atom Interferometer (Yale) Intermediate Scale (∼50 cm) Mechanical (Gravity Probe B) Superfluid 3 He (Orsay) Atomic Interferometer (HYPER) Atomic Fountain (Paris) Atomic Spin ‘NMRG’ (Litton) Small Scale (∼10 cm) Fiber-optic Gyro (Honeywell) Atomic Spin (This work) Miniature Scale (< 1 cm) MEMS (CMU) Realized Sensitivity √ rad/s/ Hz Projected Sensitivity √ rad/s/ Hz Drift Citation 2.2 × 10−10 6.0 × 10−10 — 2.0 × 10−10 — 1.3 × 10−4 Stedman (1997) Gustavson et al. (2000) — 1.4 × 10−7 — — 2.9 × 10−6 — 3.0 × 10−10 2.0 × 10−9 3.0 × 10−8 — 3.0 × 10−14 2.1 × 10−5 — — 9.0 × 10−4 Buchman et al. (2000) Avenel et al. (2004) Jentsch et al. (2004) Yver-Leduc (Yver-Leduc) Woodman, Franks & Richards (Woodman et al.) 2.3 × 10−8 5.0 × 10−7 — 2.0 × 10−10 1.7 × 10−6 7.0 × 10−4 Sanders et al. (2000) 3.5 × 10−4 1.8 × 10−4 0.5 Xie & Fedder (2003) rad/hour Table 3.1: A survey of gyroscope performance. Chapter 4 Initial tests of an alkali-Neon co-magnetometer The main objective of these experiments on 21 Ne is to ultimately create a neon co-magnetometer which can be utilized for experiments on tests of fundamental symmetries, and for deployment as a sensitive gyroscope. We describe construction of a 21 Ne co-magnetometer and investigate the practical problems limiting its performance. 4.1 Magnetometer setup The co-magnetometer operates with orthogonal pump and probe beams. These are both distributed feedback lasers (DFB). These lasers have a reflection grating etched on the diodes themselves providing a much more stable single frequency emission. The pump beam has a power of 20 − 40mW, passes through the typical λ/4 plate and is expanded to a cross section of 3cm2 . It is tuned to the D1 resonance of K. The probe beam is linearly polarized and passes through the cell and is split by a polarizing beam splitter into two photodiode detectors. It has a power of 75 Figure 4.1: Experimental setup of Ne Magnetometer 10mW and is tuned 0.2nm away from the D1 resonance of K. The probe beam cross section is 1.25 × 1.25cm2 defined by a mask. See fig. 4.1. The cell is filled with a K, 1.6 atm 21 Ne, and 60torr nitrogen mixture in a Pyrex cell of volume 8.0cm3 . It is heated to 180C◦ . The cell is encased in a glass oven. The cell is heated via a hot air line which feeds into the glass oven. The glass housing is placed inside a set of 4 concentric Mu-metal magnetic shields, which sits upon a standard optical table. Multiple turns of wire run inside the magnetic shields for de-gaussing purposes. These are only connected and run when the magnetometer in not under operation. Inside the magnetic shields are sets of Helmholtz coils, and cosine windings wound around a G−7 frame for magnetic field generation. The internal current used to drive these internal magnetic fields is created by a custom current source. It is based on a mercury battery voltage reference with a FET input stage followed by an op-amp and transistor output stage (Baracchino et al. (1997)). 76 N eon P ol ar i zat i on ( ar b. ) 0 100 200 300 400 500 600 Time (s) Figure 4.2: T2 time of ≈ 14minutes for Neon polarization when operating away from the compensation point in the co-magnetometer configuration. 4.2 Neon Polarization Measurements and Preliminary Neon Co-Magnetometer data Neon nuclei were polarized by optically pumping K vapour. The compensation point for neon was approximately 250µG, which corresponds to a neon magnetization of 8µG, or 0.08% polarization. We have also tipped the neon spins with a uniform tipping field which was fed into cosine windings. This yielded a T2 time of approximately 14minutes at low field. See fig.4.2. We have also been able to show that the K-Ne co-magnetometer shares the same transient response, that is to compensate for external magnetic transient fields, as the K-He comagnetometer. See fig.4.2. In short we have demonstrated operation of a K-Ne co-magnetometer, albeit one with low Ne polarization. 4.3 Influence of Quadrupole collisions in Polarizing neon nuclei We were also able to measure the T1 time of the spin exchange optically pumped neon, as both a function of cell pressure and temperature. It seems that the dom- 77 1 500 0 0 -1 -500 -2 0 50 100 150 200 T ransverse M agneti c F i el d ( pT ) 1000 21 N e-K gyroscope si gnal ( V ) 2 -1000 Time (sec) Figure 4.3: Compensation behaviour of the K-Ne comagnetometer to an externally applied magnetic transient field. inate limiting factor in achieving large neon polarization is due to quadrupolar relaxation. These occur during neon-neon collisions when large electric field gradients are created and couple to the quadrupole moments of the respective nuclei causing depolarization. We believe this is the main cause of nuclear relaxation due to the following reasons. First, the T1 time of the longitudinal neon polarization is of the form: T1 = 1 Rse + Rrelax (4.1) So, by knowing the Rse and Rrelax one should be able to reproduce the T1 . Where, Ne− Ne Rrelax = R gradient + σsd [ Ne]ν and R gradient = D Ne where 2 D Ne = 0.79cm /s 78 √ ∇B B 2 1 + T/273.15 Pneon /1amagat (4.2) (4.3) (4.4) P o l a r i z a t i o n ( a r b . u n i t s ) 12 10 8 6 4 2 0 0 100 200 300 400 500 Time (minutes) Figure 4.4: T1 of 105 minutes for a 1.6atm cell of Ne at 170C◦ . For a 6.5atm cell of K-Ne we have a T1 time of 34minutes. Using the theoritical value of the spin exchange rate and estimating the gradients to be 50µG/cm we expect a T1 of approximately 140minutes. The gradient calculation was estimated by modeling the cell as a uniformly polarized cube of magnetization 320µG. Combining Rse , Rsd and the contribution due to magnetic gradients we still cannot reproduce the observed T1 times. We attribute the discrepancy to a quadrupolar relaxation mechanism. Furthermore we varied the spin exchange rate by measuring T1 at various temperatures without observing a significant change in the T1 time. In fact by increasing the temperature of the 1.6atm cell from 170C◦ to 190C◦ we only observe the T1 change from 105 to 97 minutes. The K density varies by a factor of over 2 in this temperature range, which would have had a more dramatic effect on the T1 if this were the leading contribution to T1 . It was difficult to raise the temperature above 190◦ because the optically thick cell was not uniformly polarized by the pump beam and caused a lower neon polarization. We are further supported by the fact that cells with higher neon pressures have significantly lower T1 times. For instance at 170C◦ the 6.5atm cell has a T1 of 35.1 ± 1.9minutes 79 whereas the 1.6atm cell is 104.7 ± 0.7minutes. See fig.4.4. The contribution from neon quadrupolar relaxation can be quantified by measuring the T1 times of cells with different pressure. See fig 5.7. The slope of this fit is 214 ± 10min.atm. This is discussed in more detail in sec.5.4. Grover (1983) has also polarized neon and suggests that neon quadrupolar relaxation is the dominant source of neon relaxation. His data suggests a relaxation of 240min.atm which is consistent with our data. Although he performed his experiments with Rb instead of K most relevant exchange cross sections do not vary dramatically between the two alkali species. We can still attribute the dominant relaxation to be dependent on the neon properties rather than that of the alkali. Finally we investigated the effect of filling the cells with both 21 Ne and 4 He. Originally this was done to broaden the optical resonance of the K line and decrease the optical thickness of the cell to ensure uniform polarization of the cell. This was to ensure that the low T1 of the cell was not due to relaxation and inefficient pumping caused by a non-uniformly polarized cell. We filled the cell with 1.73atm of neon and 3.24atm of helium. We measured a T1 of 40.7 ± 1.4 minutes. This leads us to believe that the effect of Ne-Ne collisions may be comparable to that of Ne-He collisions. 4.3.1 T1 measurement of Neon The inference that quadrupolar relaxation is the dominant relaxation mechanism for neon is dependent on the accurate measurement of the T1 time. To put this on a firm footing it would be prudent to discuss the measurment of T1 in more detail. Determining the spin dynamics of the co-magnetometer and determining the T1 data is not trivial. A problem arises because the T2 time is comparable to the T1 time for the measurements made inside magnetic shields. The large T2 makes measuring the T1 time difficult. To measure the T1 we first 80 tipped the neon spins by a small angle using field coils wound in a cosine winding arrangement within the magnetic shields. As a result of tipping the spins the Faraday rotation measured by the probe beam became modulated at the precessional frequency. The signal from the photodiode subtracting amplifier was fed into a Labview program which fit the data and calculated the precessional frequency, amplitude, and decay constant. These can be utilized to calculate the T2 time, and polarization. However when the T2 time is large and comparable to T1 the cell would significantly increase in polarization during the decay of the transverse polarization. This would not make it possible to accurately determine the T1 time. Instead we quenched the transverse oscillations to eliminate this potential problem. To quench the oscillations we first partially polarize the sample. Subsequently the Faraday rotation signal from the detection system was inverted and used to generate an oscillating magnetic field in a direction orthogonal to both the probe and pump directions. This caused a nonlinear response which flipped the spins to make them anti-aligned with the pumping direction. Subsequently a tipping pulse was applied, and data was recorded for 60 seconds. Then the signal from the detection system was again fed into the field coils. This then served to quench the transverse oscillations. Let us describe this process in more detail theoretically: ~S = a sin(φ(t)) [ xb cos(ωt) − yb sin(ωt)] + a cos(φ(t))b z (4.5) where a is equal to the magnetization. The quenching field can be expressed as: h i ~Bquench = κ ~S · xb yb (4.6) Here κ is a parameter with units of s−1 which is dependent on factors such as the 81 gain of the coils, and magnetic moments of the spins. So the total magnetic field acting on the spins in the laboratory frame is: h i ~ ~B = ~B0 b z + κ S · xb yb (4.7) ~B = ~B0 b z + κa sin(φ(t)) cos(ωt)yb (4.8) ~Brotating = κa sin(φ(t)) cos(ωt)yb (4.9) If we transfer to the frame rotating at the neon precessional frequency this transforms to: Similarly the behaviour of our spins can be described as: d~S = γ~S × ~Binertial dt inertial (4.10) d~S = γ~S × ~Brotating dt rotating (4.11) or, d~S = γ~S × κa sin(φ(t)) cos(ωt)yb dt rotating (4.12) We are interested in the behaviour of the b z component of the spin. This becomes: d~Sz dφ = − a sin(φ(t)) = γκa2 sin2 (φ(t)) cos2 (ωt)yb dt dt (4.13) Solving this we find the behaviour of the spins as a function of time: φ(t) = 2 tan −1 tan(φ0 /2)exp −κ t sin(2ωt) + 2 4ω (4.14) where φ0 is the initial tip angle. One can see that in the case of positive κ the quenching field aligns the spins with the magnetic holding field during equilib- 82 Angle (degrees) 1 0.8 0.6 0.4 0.2 0 0 2 4 6 8 10 Time (s) Angle (degrees) Figure 4.5: Theoritical simulation of the noble gas spin. The quenching field eliminates transverse oscillations and aligns the spins with the holding field for positive κ. 175 150 125 100 75 50 25 0 0 2 4 6 8 10 12 14 Time (s) Figure 4.6: Theoritical simulation of the noble gas spin. The quenching field flips the spins so that the are anti-aligned for negative κ. rium. see fig 4.5. For negative values of κ we see that the spins align anti-parallel to the magnetic holding field. That is the equilibrium value of the azimuthal angle is π radians, regardless of the initial tip angle φ0 . See fig 4.6 As time goes on and the spins are re-pumped along their original direction the polarity of the quenching pulses must be flipped so as not to re-flip the spins to their anti-aligned state again. A Labview program was utilized to control this entire data taking process and monitor the spin orientation relative to the magnetic holding field. 83 4.4 Improving Magnetometer Sensitivity A number of technical schemes for improving the magnetometer noise have been investigated before the final alkali-neon comagnetometer prototype is built. The first we discuss is eliminating spurious signal from the gyroscopic signal. 4.4.1 Removing Birefringence and false Faraday Rotation signals Birefringence in the probe beam can cause a false signal. If there is any degree of circular polarization in the probe beam the system will additionally pump along the probe beam axis and introduce light shifts into the system. To reduce this effect we have utilized a cubic cell shape rather than the typical spherical cell. For a spherical cell there is no dichroic effect produced for a diametrically traversing beam. For a spherical cell the light experiences some dichroism as varying amount of light are reflected off the cell wall for light polarized in the plane of reflection, and perpendicular to the plane of reflection. Random walk of the beam off centre, possibly due to convection currents of air, would induce some degree of circular polarization. This is not an issue for a cubic cell since the surfaces are planar. In a spherical cell different segments of the probe beam travel varying distances through the cell. Therefore segments of the probe beam experience different amount of Faraday rotation. This effect is also eliminated for a cubic cell to first order. Circular birefringence can also be induced from reflections with mirrors, and via transmission through various optical elements. This can particularly be the case for stressed lenses, and mirrors. To eliminate this problem the probe beam has been forced to travel through a variable waveplate. The variable waveplate consists of a nematic liquid crystal with polar molecules. As an electric field is applied the polar crystals orient themselves along the electric field axis. The popu- 84 lation which is oriented along the field is a function of the electric field magnitude. When the probe beam enters the liquid it will experience different indices of refraction along the direction of the electric field and perpendicular to it, inducing a tunable phase lag. A square wave function generator to control this retardance and to eliminate or cancel the circular polarization of the beam has been built. It seems though that utilizing a stressed plate of glass to alter the birefringence gives a better noise performance, and will most likely be used in the future neon co-magnetometer. 4.4.2 Controlling and Monitoring the Laser stability We were unable to obtain DFB lasers which operated at the Potassium D1 frequency. Thus we had to cool commercially available Eagle Yard laser diodes to −30◦ C in order to tune it to the required frequency. To achieve this we attached two thermoelectric coolers (TEC’s) in series to the laser diode baseplate. These were in turn connected to a large heat sink which was cooled with a continuously run chilled water supply cold plate. Originally the setup was cooled via a mechanical fan, but the level of measurement noise induced by vibration of the air against the heat sink was unacceptable. There is the additional complication that the dew point in the lab is approximately 12◦ C. Thus electronic shorting do to condensation of the electrical circuitry was a major concern. To circumvent this problem we encased the laser and dual TEC setup in an aluminum housing. Holes were drilled in this housing and air was continuously flowed through the setup at positive pressure. The water vapour was first removed from the air by flowing the air through a commercial dessicant called Drierite. This setup was utilized for both the pump and probe laser arrangements. With recent advances in DFB lasers one can now purchase lasers which are evacuated around the actual diode. These can be cooled without danger below the dew 85 point of the laboratory. Another concern in the setup was due to the laser frequency shifting with time. To monitor this we constructed a constant pressure Fabry Perot etalon. It consists of an Invar tube, which has low thermal expansivity, surrounded by Nichrome wire, along with a thermocouple to measure temperature. Two confocal mirrors are fitted to both ends of the hollow Invar tube, and attached to a piezo. The Nichrome wire is utilized as a heater and used via negative feedback to keep the temperature of the Invar constant. The entire setup enclosed inside three stainless steel tubes and evacuated. The idea was to utilize this to look for correlations between pressure, and temperature change of the lasing frequency and stabilize them. This is currently being tested on a different setup. 4.4.3 Miniaturization of Gyroscope In an effort to miniaturize the current setup one runs into difficulties creating homogeneous magnetic fields. With the standard Helmholtz arrangement the active region, where the non linearity in the field is less than 1%, is small compared to the radius of the coils. In order to miniaturize the system a dual pair of coils were modeled analytically. The current ratio in the two pairs was taken to be a rational integer. Using a numerical simulation we solved for the inter coil separation as a function of the integer current ratio while imposing elimination of 4th order nonlinearity in the field dependence at the symmetric centre of the coils. Using this process we found that the optimal compact arrangement occurred for a turn ratio of 1 : 1. The coils are a distance 1.2cm and 2.2cm from the location of the cell along the field axis, and have a diameter of 3.6cm. The analytical solution is not strictly valid in the vicinity of magnetic shielding. We used the simulation as a starting point to estimate the optimal placement of theses coils within Mu metal shields. This was modeled using the commercial 86 Figure 4.7: Magnetic field homogeneity for a 3cm×3cm region. Here the coils are carrying a 10mA current. computer program Maxwell. See fig.4.7. An optimized magnetic field homogeneity was realized by simultaneously varying both the radii of the coils and their positions independent of each other. This solution is independent to first order to small asymmetric misplacing of the wires. We were able to achieve a magnetic field homogeneity of 0.1% in a region 2cm by 2cm which was produced by coils of diameter 4cm. 4.4.4 Alternate methods to heat Cell, and remove Convection noise A source of error in the magnetometer signal is due to random walk of the laser beam caused by the air convection used to heat the cell. In order to reduce this we have enclosed the cell in an evacuated glass chamber. The cell is surrounded by a 6 boron-nitride sheets. These are finally attached to a boron-nitride rod around which is wrapped a coaxial mineral insulated heating cable. The entire enclosure is cemented together utilizing aluminum nitride. A current is sent through the coaxial cable, which is shorted at one end. The power dissipated through the cable is used to heat the boron nitride walls, which heat the cell by conduction. The 87 entire enclosure is enclosed in a evacuated glass chamber which has been silvered to reduce radiative loss of heat. The silver coating is scratched to prevent Johnson noise from currents in the coating. The co-axial coils carry a 10mA current and heat the cell via direct conduction. Because of the coaxial arrangement the magnetic field due to the current is eliminated to first order. Boron-Nitride was chosen because it is has a large thermal conductivity. (120 W/mK). Since it is not a metal it has limited magnetic properties. The fusing cement has a nominal conductivity of 110 W/mK. However experiments were performed and indicate the conductivity of the cement form is 30 W/mK. An arrangement utilizing Kapton heaters was tested to heat the cells. This system was theoretically modeled. A temperature gradient exist across the boronnitride oven which can be explained by heat loss due to radiation. To eliminate this problem we are attempting to redesign the boron nitride oven, to keep the temperature gradient to a minimum. We also theoretically studied the radiation exchange problem between the oven and magnetic shielding. We operate such that the magnetic shielding is not raised above its Curie temperature. It was also shown that one could simply use a twisted pair of heater coils to heat the cell. This is the scheme which is currently being used. The advantage over Kapton heaters is that the twisted pair can operate at higher temperature, whereas the Kapton heaters melt at 200C◦ . This is because the twisted wire contains a high temperature glass sheath. To reduce the noise further we oscillate the current in the twisted wire at a frequency higher than the bandwidth of co-magnetometer. The bandwidth of the co-magnetometer is ≈ 100Hz, and the current is oscillated at ≈ 25KHz. This also ensures that the spin precess through a small angle during the period of current oscillation. This coaxial wire has an outer sheath of copper, and an inner sheath made 88 89 Figure 4.8: Silvered oven holding Boron-Nitride housing for Pyrex cell. These are all enclosed in a 4layer concentric Mumetal magnetic shield system. of Everdur 655. This is a higher resistance non magnetic metal. These were chosen to reduce the inhomogeneous DC field produced from the wires themselves. Other wires such as Nichrome have been experimented with. The oven is placed inside a glass encasing which is then evacuated. This is then silvered to reduce heat loss due to radiation leakage. The silver surface is scratched so as to reduce the effect of eddy currents. A version of the gyroscope where the Mu-metal shields have been replaced with a smaller ceramic ferrite shield has been implemented for use as a new test of √ CPT violation. It has a realized noise of 1fT/ Hz, using a K-He co-magnetometer. This is roughly 2.5 times more sensitive than the previous incarnation of the co√ magnetometer gyroscope which had a magnetic field sensitivity of 2.5fT/ Hz, √ and 5 × 10−7 rad/s/ Hz. In the current setup if one switched to a K-Ne co√ magnetometer cell the expected sensitivity would be 5 × 10−8 rad/s/ Hz due to the 21 Ne gyromagnetic ratio being a factor of ≈ 10 smaller than that of 3 He. See eq. (2.121). 90 Chapter 5 Measurement of parameters for Polarizing Ne with K or Rb metal Many of the applications currently using 3 He, including the co-magnetometer, would benefit, and realize increased sensitivity by substituting 3 He with polarized 21 Ne gas. In fact, tests of CPT violation using co-magnetometers would be greatly improved if one utilizes polarized 21 Ne gas (Kornack et al. (2008)). Additionally the nuclear spin co-magnetometer gyroscope would realize an order of magnitude gain in sensitivity (Kornack et al. (2005a)). Very little is known about parameters which govern the spin-exchange polarization of 21 Ne 21 Ne. In order to realize these applications, and test the feasibility of a co-magnetometer the interaction parameters of 21 Ne with alkali metals must be measured. The spin-exchange cross section σse , and magnetic field enhancement factor κ0 have only been theoretically calculated (Walker (1989a)). Furthermore there are no quantitative predictions of the neon-neon quadrupole relaxation rate Γquad . In this work we investigate polarizing 21 Ne with optical pumping via spin exchange collisions and measure the relevant spin exchange rate coefficient, magnetic field enhancement factor, and quadrupolar relaxation coefficient. Fur- 91 thermore we measure the spin destruction cross section of Rb, and K with 21 Ne, and find agreement with the values found in the literature. Finally we discuss the feasibility of utilizing polarized 21 Ne for operation in a co-magnetometer. 5.1 Theory In this work we refer to the spin exchange rate measurements of the K-21 Ne system. However the measurement technique is identical to that utilized for the Rb-21 Ne system. 21 Ne becomes polarized via spin-exchange collisions with polarized alkali atoms. During spin exchange collisions the electron wavefunction of the alkali atom overlaps with the noble gas nuclei. They interact via a hyperfine Fermi contact interaction of the form Hse = α~In · S~a (5.1) where ~In refers to the noble gas nuclear spin operator, and S~a the alkali electron spin operator. During collision this interaction leads to an exchange of angular momentum from the alkali valence electron to the noble gas nuclei. The collision also brings both the alkali valence electron, and the noble gas nuclei into close proximity. This causes them to both experience strong magnetic fields due to the the interaction of their magnetic moments. This results in a change in the Larmor precession frequency of both species. This frequency shift is described in terms of a magnetic field enhancement factor κ0 (Walker & Happer (1997)). It is defined as the ratio of the Larmor frequency shift of each species caused by the contact interaction and to the shift caused by the classical magnetic field generated by each other gas species. The value of κ0 is often much greater than unity. It can be measured by comparing the precessional frequency shift of an alkali atom the noble gas is in contact with, to the actual magnitude of the magnetic field produced by the noble 92 gas nuclei. For light alkali atoms the dominant spin exchange mechanism for polarizing noble gas involves binary collisions. Here the spin-exchange rate constant can be expressed in terms of the familiar spin-exchange cross-section as Grover (1983): κ a = σse vK − Ne (5.2) Where vK − Ne is the relative velocity between the potassium, and neon atoms. The polarization of neon atoms via spin exchange collisions with potassium atoms of number density [K] can be described by (Walker & Happer (1997),Appelt et al. (1998)): 3 ∂PNe 1 3 = κ a [K ](ǫ(W, β) PK − PNe ) − PNe Γquad 2 ∂t 2 2 (5.3) Here the coefficient κ a represents the spin-exchange rate constant. The terms PK and PNe are related to the longitudinal spin polarization of the alkali atom of spin S, and neon atom with nuclear spin K by PNe = hWz i /W, and PK = 2 hSz i. The neon relaxation is dominated by the quadrupolar relaxation rate Γquad (Adrian (1965)). One method to measure κ a , suggested by Gentile (Gentile & McKeown (1993)), is to simply measure the rate of rise of neon polarization at PNe = 0. This transforms eq. (5.3) to ∂PNe 5 = κ a [K ] PK ∂t 3 (5.4) We measure the buildup of neon polarization as a function of time extrapolated to the slope at zero neon polarization to determine the alkali-neon spin exchange rate κ a . These measurements are made for times where the buildup of neon polarization remains linear. In this work we additionally use a method based on re-polarization in the dark 93 to measure κ a (Chann & Walker (2002),Kadlecek et al. (1998)). Here the polarized neon spins repolarize the K spins in the absence of optical pumping. In this case the total K spin, F = I + S, evolves as ∂Fz = D ∇2 Fz − ΓK Sz + κ a [ Ne](Wz − ǫ(W, β)Sz ) ∂t (5.5) Here the first term represents the contribution due to diffusion of K through the cell. The second term represents depolarization via spin destruction collisions. We expect the contributions from diffusion to be small compared to the other terms. The diffusion term is ≈ 2840 times smaller than the spin destruction rates. When the neon has reached steady state polarization one can simplify eq. (5.5) as: κa = 1 2 Γ K PK0 ( 32 PNe0 − 12 ǫPK0 )[ Ne] (5.6) Each of these quantities can be measured independently. ΓK can be measured by chopping the pump beam, and analyzing the resulting decay in the alkali polarization. [ Ne] is directly measured as the vapour cells are filled. P Ne0 can be computed directly by performing NMR on the polarized sample. PK0 can be measured by scaling the polarization when the pump beam illuminates the cell according to the optical rotation signal in the light, and the dark. The individual measurements for determining the spin exchange rate, spin destruction cross-sections, and κ0 values are described in greater detail in the following sections. The dominant source of relaxation in neon arises from the quadrupolar relaxation rate Γquad . According to Adrian (1965) the relaxation rate is proportional to the neon filling pressure. One can measure this contribution by measuring the T1 of neon cells with different filling pressure. The dominant source of alkali relaxation is due to spin destruction collisions. 94 In the absence of optical pumping the spin destruction rate can be expressed as: Rsd = nK σK −K V K −K + n Ne σK − Ne V K −K + n N2 σK − N2 V K − N2 (5.7) In order to measure σK − Ne , one measures the decay of alkali polarization to determine the total spin destruction rate and fit to eq. (5.7). One must modify the measured relaxation rate by accounting for the paramagnetic slowing factor. This is described in further detail in the section regarding measurement of the alkali polarization decay. 5.2 The Experimental Setup 21 Ne gas sample is contained in a cubic pyrex cell of side length 20mm. It is fully illuminated by a high power Sacher littrow diode laser which is tuned to the K D1 line and outputs 100mW of power. This beam passes through a λ/4 waveplate before illuminating and polarizing the K atoms. A Toptica Dl-100 diode laser is utilized as a probe beam. It passes through a linear polarizer and propagates parallel to the pump beam. The pyrex cell contains a K droplet, ≈ 100 torr of nitrogen for quenching and 6.2 atm of 90% isotopically enriched21 Ne gas. It is heated to 180C◦ via a hot air line which heats a glass oven. The oven is placed in the middle of two large 34 inch diameter Helmholtz coils which create a holding field of approximately 16.7 gauss. The Helmholtz coils are powered by a 100V Bipolar amplifier. Perpendicular to these coils are a pair of NMR tip coils, also placed in a Helmholtz configuration, and a pair of RF modulation coils. Along the y axis, out of the plane of the optical table, (see fig 5.1) a NMR pickup coil is placed. It has a Q of ≈ 14, and is tuned to the neon resonant frequency. For the Rb measurements the pump laser is a 2W Coherent 19 diode laser array, and the probe is a 10mW Nanoplus DFB diode. 95 Compensating Coil z x Feedback y Fluxgate Polarizing Beamsplitter NMR Pick-up Oven Probe Laser Cell Photodiodes Subtraction Linear Polarizer NMR tip Coils Current Source λ/4 Lock-In Amplifier (Circular Polarizer) Pump Laser Modulation Reference (Larmor Frequency) Feedback AFP frequency Modulation AFP Amplitude Modulation Magnetometer Signal Figure 5.1: Experimental Setup κ0 is measured with a sequence of EPR-AFP flips, followed by NMR tips. κ0 is calculated by computing the ratio of the magnetic field shift experienced by the K (EPR-AFP) and the magnetic field produced by the neon, as determined from the NMR. To determine the spin exchange constant using the repolarization method a series of EPR-AFP flips was used to determine the K frequency shift. PNe0 was calculated using the measured value for κ0 and the neon pressure in the cell which was measured during filling. [ Ne] is known from the cell filling pressure. To determine PK0 a two step process was used. First the RF source was swept over the K Zeeman levels to determine the K polarization with the pump on. The pump beam was then blocked and the repolarization of K by Ne was measured by monitoring the optical rotation. PK0 was calculated by calibrating the optical rotation obtained during back polarization with the optical rotation measured while the cell was illuminated. The zero optical rotation point when the pump beam is blocked was found by flipping the neon spins with AFP and monitoring the subsequent change in the K optical rotation in the dark. ΓK is determined by manually 96 chopping the pump beam, and measuring the resulting decay in the optical rotation. For the rate of rise measurements the average alkali polarization in the cell PK must be measured while the cell is illuminated. To account for the variation in K polarization across the cell, the probe beam was swept across the cell and the optical rotation was measured. At the centre of the cell RF modulation was used to measure the K polarization. By scaling the polarization as compared to the optical rotation across the cell an estimate of the average polarization of the cell was obtained. By comparing the T1 data from the manufactured cells, and those constructed by others the contribution of neon-neon quadrupolar collisions to overall neon spin relaxation can be estimated. The T1 times were calculated by utilizing EPR lock-AFP flips to calculate the neon polarization. A measurement of the spin destruction rate of rubidium in the rubidium-neon cell and subsequent calculation of the Rb-Ne spin destruction cross section was carried out. The spin destruction rate was measured via the rubidium relaxation in the dark measurement which was previously described. The spin destruction rate was extrapolated to zero probe beam intensity by making multiple spin destruction rate measurements at different probe beam intensities. The variation in probe beam intensity was achieved by placing neutral density filters of different values directly in the beam path before the probe entered the cell. In the following subsections we describe each of the individual measurements in greater detail. 5.2.1 NMR detection In order to determine the spin-exchange rate constant, several different quantities for each rate constant must be measured. P Ne0 is measured using NMR. To 97 accomplish this the neon NMR was obtained by tipping the 21 Ne using two cal- ibrated 9 inch diameter coils in the Helmholtz configuration. Typical tip angles are approximately 30◦ . The resulting neon NMR is detected with a 4 coil pickup with 780 turns per coil. This consists of two pairs of pickup coils with opposite winding orientation so that the combined pickup has zero dipole moment. This is done to minimize coupling of the pickup coil to external fields which are not due to the 21 Ne gas. Each pair of coils are located symmetrically about the cell, and are wound on one Teflon rod. Each coil has 780 turns of enameled copper magnet wire. A numerical model was written to determine optimum coil placement in order to maximize pickup sensitivity. In order to minimize systematic errors the tip coils were calibrated using two independent methods. First the pyrex cell is replaced with a small pickup coil of similiar dimension to the pyrex cell, while the tip coils are operated. The resulting signal is fed into a lockin amplifier and used to calibrate the magnetic field strength. Care is taken to properly align the axis of the pickup coil with that of the tipping coils. Second, the current entering the tipping coils is measured with a clamp on ammeter. The magnetic field produced from the tipping coils is calculated from a knowledge of this current and the coil separation, and diameter. The two calibration schemes agree to 2%, which is within the compound precision of the coil geometry measurements, alignment of pickup coils, and ammeter precision. For NMR detection the pickup signal is fed into a low noise pre-amp, and monitored via computer. A typical neon NMR signal is shown in fig.5.2. The pickup coil is also calibrated using two methods. The first requires a dummy source of known size to create a magnetic field. The magnetic field strength is calculated theoretically based on the dummy coil geometry. The resulting pickup is measured to produce a calibration. The pickup coils are also operated in reverse to produce a 98 0.60.6 0.4 0.4 0.2 0.2 NMR Pickup 0.0 gain 1e4 (volts) 0. -0.2 0.2 -0.4 0.4 -0.6 0.6 -0.8 0.8 0 0.05 0.10 0.15 0.20 Time (s) Figure 5.2: Representative NMR signal of polarized 21 Ne gas magnetic field at the cell location. A small pickup coil is placed here. Using reciprocity arguments (Jackson (1999)) the two signals are compared and agree to 6%. The uncertainty of the Q of the resonant coils is 4%. 5.2.2 Electron Paramagnetic Resonance Shift In order to calculate the κ0 value, the effective magnetic field the K atoms experience must be measured simultaneously with the NMR signal. This effective magnetic field is measured by monitoring the shift in the frequency in the electron paramagnetic resonance(EPR)upon reversal of the neon spins. (Romalis & Cates (1998),Schaefer et al. (1989b),Newbury et al. (1993),Barton et al. (1994)). A double feedback scheme is utilized to accomplish this. The first feedback loop locks the output of the holding field to minimize magnetic field fluctuations from external sources. To accomplish this a fluxgate magnetometer is placed approximately 5 inches from the pyrex cell outside the glass oven. In order to prevent the fluxgate from saturating, from the 16.7gauss field produced from the large 34 inch diameter Helmholtz coils, a solenoid was wound around the fluxgate. This compensating solenoid produces a field which cancels the holding field and results in zero field at the fluxgate. The compensating solenoid is powered by a custom built voltage 99 controlled current supply. The output of the fluxgate is fed through PID feedback electronics to the input of the voltage controlled power supply which powers the large Helmholtz coil holding field. The second feedback loop locks the magnetic holding field to the K resonance. Thus when the K resonance is shifted the holding field for the atoms is also shifted so that the K resonance frequency remains constant. In short the two feedback loops are summed. The first feedback controls the current which supply the holding field for the atoms, while the second feedback give an offset to this field in order to maintain a constant K precession frequency. In order to operate the second feedback loop a small RF coil is used to produce a magnetic field at 12.4957MHz, with a sweep width of 15KHz, and a sweep rate of 340Hz. The optical rotation of the probe beam is measured as the RF coils sweep in frequency across the K magnetic resonance. The derivative of this signal is produced taking the out of phase component after feeding the signal into the lockin amplifier which has been referenced to the RF sweep rate. The out of phase component is fed into an integral feedback box and used to control the voltage controlled current source which powers the compensating solenoid. In this manner we can lock to the K resonance. The output of the integral feedback box is monitored and calibrated by introducing known frequency shifts in the base RF field. That is the RF field frequency is shifted by 100Hz, and the resulting feedback box voltage is monitored. This relation is linear, and is used to calibrate the EPR frequency shifts. This technique is utilized to lock to the end resonance of the F = 2 manifold of the K spectrum. Operating at high field causes the K resonance to be non-linearly spaced due to the Breit-Rabi splitting. The neon NMR frequency is approximately 5500Hz while the system is locked to the K end state. A numerical program is used to calculate the effective gyromagnetic ratio of the end state due to the Breit-Rabi splitting 100 400 Frequency Shift (Hz) 200 0 -200 -400 -600 0 5 10 15 20 25 Time (s) Figure 5.3: Representative EPR shifts after 2 hours of polarization. The spikes in the data are due to temporary loss of lock while the AFP coils are operated. (Woodgate (1989)). This coupled with the NMR data allows the determination of κ0 (Romalis & Cates (1998)). ∆ν = 8π dν( F, M ) κ0 µK [ Ne] PNe 3 dB (5.8) In order to measure the magnetic field enhancement factor κ0 the technique of Adiabatic fast passage (Abragam (1961)) is used to flip the orientation of the polarized neon spins by 180◦ , while keeping locked to the EPR frequency and monitoring the frequency shift. See fig5.3. The peak to peak amplitude of the frequency shift seen using EPR is twice the value of the frequency shift induced by the polarized neon. In order to flip the spin orientation a magnetic field is swept in frequency across the neon NMR resonance. This is accomplished by producing a field with the small set of Helmholtz coils while satisfying the adiabatic fast passage (AFP)conditions (Abragam (1961)): k∇ Bz k2 γω̇ << << γB1 2 B1 B1 101 (5.9) To sweep the frequency of the AFP flipping field a voltage controlled oscillator is fed through a multiplier circuit, and a ramping voltage from a NI-Daq analog output. The resulting signal is then amplified through a 100V Bipolar amplifier to supply the AFP field. The neon spins are flipped every 5 seconds. The frequency is swept from 2500Hz to 7500Hz in 4s. The AFP field strength is ≈ 4Gauss. The flip efficiency is 99.7% In order to optimize flip efficiency it was necessary to slowly ramp up the AFP field strength before the AFP field frequency was swept. The AFP field was raised from 0 to 4 Gauss in 0.7s. 5.2.3 Alkali Polarization To determine the alkali polarization when the cell is fully illuminated, a RF field is swept across the K Zeeman levels, while the optical rotation of the longitudinal probe beam is measured (Chann & Walker (2002)). The transverse RF field causes a depopulation of the end state, and a lower polarization. This in turn reduces the optical rotation of the probe laser. By comparing the area under the peaks of the different transitions of the F = 2 manifold and using eq. (5.11) one is able to determine the K polarization. See fig5.4. The area under a transition peak ( F, m) → ( F, m − 1) depends on the state population ρ Fm , and the RF field Br f as (Chann & Walker (2002)): A Fm ∝ Br2f [ F ( F + 1) − m(m − 1)](ρ Fm − ρ Fm−1 ) (5.10) The spin exchange rate measurement experiment operates in the regime where the magnetic sublevel populations can be described by a spin temperature distribution (Walker & Happer (1997)). Substituting the spin temperature condition ρ Fm ∝ exp( βm), and noting that the spin polarization is given by PK = tanh( β/2) results in 102 Optical Rotation (arb. units) 4 3 2 1 0 -1 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 Time (s) Figure 5.4: Sweep over the A22 , A21 and A20 transitions at reduced Pump beam power. PK = 7r − 3 7r + 3 (5.11) where r= A22 A21 + A11 (5.12) where the state is designated with the indices AFm . Under normal operating conditions the spin polarization is very close to 1. Only the end state transition A22 is visible under these conditions. However one can view the entire spectrum if the pump beam power is sufficiently attenuated. 5.2.4 Alkali Polarization Decay Constant measurement In order to measure the K decay constant the optical rotation due to the K is monitored, as the pump beam is manually chopped. See fig5.5. The decay constant is then determined by fitting the decay curve on a log plot. One must account for the fact that to obtain the true time constant the measured constant must be multiplied by the paramagnetic coefficient or ’slowing down factor’ (Walker & Happer (1997),Chann & Walker (2002),Appelt et al. (1998)). 103 1.0 Absolute K Polarization 0.8 0.6 0.4 0.2 0.0 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 Time (s) Figure 5.5: Potassium Polarization decays as a result of Pump beam being manually chopped. For spin K, and spin temperature β the slowing down factor is given by eq. (2.21): For β << 1 this reduces to 4I ( I + 1)/3 see eq. (2.24)(Walker & Happer (1997)). Only decay data from the low polarization portion of the data set were fit to the β << 1 simplified expression. 5.2.5 Back Polarization measurement In order to measure the K polarization in the dark, the optical rotation due to Ne-K back polarization was measured. The optical rotation signal is routed through a low noise Pre-amp and fed into a NI-DAQ for acquisition. To measure the zero polarization level the AFP Helmholtz coils are run to flip the neon polarization. See fig5.6. To determine the back polarization in the dark the gain adjusted optical rotation to the optical rotation signal with the pump beam is illuminating the cell is compared to that when the pump beam is blocked. Since the alkali polarization was measured when the pump beam was unblocked one can scale the optical rotation signals to the polarization. During operation of the pump beam large (≈ 1) radian optical rotations were observed. In this regime one can no longer use the 104 6x10 -4 K Back Polarization 4 2 0 -2 -4 -6 0 5 10 15 20 25 Time (s) Figure 5.6: Absolute Potassium Back polarization as Neon is flipped via Adiabatic fast passage standard small angle optical rotation formula Φ= I1 − I2 2( I1 + I2 ) (5.13) Where I1 and I2 are the voltages on the individual channels of the balanced polarimeter. Instead one can use: 1 Φ = sin−1 2 I1 − I2 I1 + I2 (5.14) 5.2.6 Alkali density measurement In order to calculate κ a using the rate of rise method, the alkali density must be independently measured. This is accomplished by measuring both the optical rotation of the longitudinal probe beam and its detuning. This enables calculation of the alkali density, while accounting for the fact that the alkali is not polarized to 1. The detuning is measured by monitoring the optical rotation as the probe beam frequency is swept. The probe beam frequency is tuned by varying the temperature, so as to minimize variation in the probe beam intensity. The probe beam is 105 detuned ≈ 0.5nm from the centre of the absorption peak. The pump beam is tuned to resonance by varying the current until the optical rotation experienced by the probe beam is a maximum. There is sufficient pump beam power, 100mW so that the cell is uniformly polarized. To determine the alkali density the optical rotation is measured and fit to: π 1 1 Θ = lnre Px − Im [ L(ν − νD1 )] + Im [ L(ν − νD2 )] 2 3 3 (5.15) where the Lorentzian line shape is given by: L(ν − ν0 ) = Γl /2π + i (ν − ν0 )/π (ν − ν0 )2 + (Γl /2)2 (5.16) Here n is the density of K in the cell, l is the path length which the probe beam propagates through the cell, and re is the classical electron radius. The contribution to the optical rotation from the D2 transition is on the order of a few percent. The density is 4.7 × 1013 /cm3 which is a factor of 4 less than that predicted by the empirical formula which relates the vapour pressure to the cell temperature. However there is often a discrepancy in the value calculated from the saturated vapour pressure and the actual vapour density by at least a factor of 2 (Chann & Walker (2002)). This comes from the fact that the cell may not be of uniform temperature, or from substantial heating due to the pump beam. This effect can also be caused by absorption, or a reduced alkali vapour pressure due to interactions with the glass walls of the cell. This has been observed for pyrex glass cells, which is the same glass employed in these measurements. 106 5.3 Fermi Contact interaction κ0 Results κ0 for the interaction of K with 21 Ne is 30.8 ± 2.7. This is approximately 10% lower than the value of 34 as predicted by Walker Walker (1989b). Although the values do not agree with the predictions it should be noted that the general trend among the predicted values, and experimentally measured κ0 for other noble gas-alkali pairs is for the predicted values to be 10 − 20% higher than those obtained experimentally. When the experimentally measured values are arranged by magnitude the gas mixtures show the same order as those of the theoretically predicted values. In a Rb-Ne system Walsworth has measured κ0 to be 32.0 ± 2.9 (Stoner & Walsworth (2002)), while the value predicted by Walker is 38 (Walker (1989b)). The Fermi-contact interaction was also measured for the rubidium-neon pair. It was found to be 35.7 ± 3.7 which is in agreement with the value found by Stoner & Walsworth (2002) of 32.0 ± 2.9. Both of these values are below that predicted by Walker (1989a) of 38. However both of the experimental measurements follow the expected trend of being larger than the Fermi-contact interaction for the potassium-neon pair. See Table 5.3 5.4 Results of neon quadrupolar relaxation Γquad measurement By comparing the T1 data from the manufactured cells, and those constructed by others the contribution of neon-neon quadrupolar collisions to overall neon spin relaxation can be estimated. In general one expects that for noble gases with a quadrupole moment, the dominant form of relaxation is due to nuclear electric quadrupole interaction (Adrian (1965)). As a consequence of this one expects the spin-lattice relaxation time to vary inversely with the density of the buffer gas. In 107 Quadrupolar Relaxation due to Neon Collisions 0.035 1/Lifetime (1/min) 0.03 0.025 0.02 0.015 0.01 0.005 0 0 1 2 3 4 5 6 7 Cell Pressure (Amagat ) Figure 5.7: Neon relaxation as a function of cell pressure. Here the data satisfies the relationship Pressure×T1 = 214 ± 10Amagat×min. The fact that the T1 is inversely proportional to the cell filling pressure seems to indicate that the dominant relaxation mechanism is due to nuclear electric quadrupole collisions. This is consistent with what one would expect from a buffer gas with spin greater than 1/2. fig 5.7 we see that the cells follow this linear relationship. The data from the cell with the lowest pressure is from Grover (Grover (1983)) whereas the data from the other cells is from this work. Both Grover’s cell and 3.34 atm cell is filled with a Rb-Ne mixture, whereas the others are filled with a K-Ne mixture. The pressure in the Rb-Ne cell is determined by measuring the broadening of the optical D1 transition in Rb. The reason for the large uncertainty in the 3.34 atm cell’s pressure is due to the uncertainty in the literature of the broadening parameters for the D1 transition of Rb in neon gas (Ottinger et al. (1975)). 5.5 Spin exchange Rate coefficient Results The K-21 Ne spin exchange rate, measured with the repolarization method, is 3.36 ± 0.67 × 10−20 cm3 /s. The value obtained by using the theoretically predicted spin exchange cross section by Walker is 1.2 × 10−19 cm3 /s. See Table5.1. Data for the rate of rise method was taken in the limit of low neon polarization. 108 7x10 -4 Neon Polarization 6 5 4 3 2 1 0 0 1 2 3 4 5 Time (minutes) Figure 5.8: Absolute neon polarization as function of time to determine spin exchange rate constant by rate of rise method. Here the rate of rise of neon polarization remained linear. See fig5.8. The rate of rise method gives a value of 2.34 ± 0.52 × 10−20 cm3 /s for the K-21 Ne spin exchange rate. Additionally neon was polarized using spin exchange optical pumping with Rb metal. The Rubidium-Neon spin exchange rate was also measured and found to be 0.80 ± 0.16 × 10−19 cm3 /s. Walker (1989a) predicts a value of 1.66 × 10−19 cm3 /s. See Table 5.2. However this value is not in agreement with the previously measured value by Chupp & Coulter (1985) of 4.66 × 10−19 cm3 /s. However Chupp & Coulter (1985) determined their rubidium density by directly applying the saturated vapour pressure calculated by the empirical formula given by Alcock et al. (1984). This formula can be in disagreement with the actual alkali density by as much as a factor of two (Chann & Walker (2002)). Chupp fits his data directly through the origin implying that the contribution of neon quadrupolar relaxation collisions to the T1 is negligible. We have shown in this work that this is incorrect, and that the dominant contribution to the neon T1 is in fact due to quadrupolar collisions. This coupled with the fact that the more precise measurements carried out by Chann & Walker (2002) indicate a statistical variation in the rate constants of 109 Spin Exchange Rate Method Repolarization Theoretical Prediction Rate of Rise cm3 /s 3.36 ± 0.67 × 10−20 1.2 × 10−19 2.34 ± 0.52 × 10−20 cm2 Table 5.1: Spin exchange parameter of a potassium-neon system Spin Exchange Rate Method Repolarization Theoretical Prediction Rate of Rise cm3 /s 0.80 ± 0.16 × 10−19 1.66 × 10−19 0.82 ± 0.18 × 10−19 Table 5.2: Spin exchange parameter of a rubidium-neon system 25% upon subsequent measurements implies that this disagreement is not entirely unexpected. 5.6 Measurement of Spin destruction cross-sections of neon with Rb and K Utilizing the data from the potassium relaxation in the dark measurement one can compare the measured spin destruction rate to that predicted by using the known Species K-Ne (this work) K-Ne (prediction) Walker (1989a) Rb-Ne (this work) Rb-Ne (Walsworth) Stoner & Walsworth (2002) Rb-Ne (prediction) Walker (1989a) Fermi Contact Interaction 30.8 ± 2.7 34 35.7 ± 3.7 32.0 ± 2.9 38 Table 5.3: Fermi contact interaction measurements for the K-Ne, and Rb-Ne systems in this work, and compared to both predictions ans measurements by other groups. 110 4.0x10 -20 Spin Exchange Rate (cm3/s) 3.8 3.6 3.4 3.2 3.0 2.8 2.6 0 1 2 3 4 5 6 Figure 5.9: Scatter in Spin exchange rate measurements for K-Ne 1.0x10 -19 Spin Exchange Rate (cm3/s) 0.9 0.8 0.7 0.6 0.5 0 1 2 3 4 5 Figure 5.10: Scatter in Spin exchange rate measurements for Rb-Ne 111 Species K-Ne K-Ne K-Ne Rb-Ne Rb-Ne Rb-Ne Rb-Ne Group This work Franz Franz & Volk (1982) Walker (prediction) Walker (1989a) This work Franz and Volk Franz & Volk (1976) Franzen Franzen (1959) Walker (prediction) Walker (1989a) 10−23 cm2 1.1 ± 0.1 1.41 ± 0.14 1.6 1.9 ± 0.2 1.9 5.2 1.8 Table 5.4: Spin destruction cross sections for K-Ne, and Rb-Ne as compared to both theory and other measurements spin destruction cross sections, and gas densities in the cells using eq. (5.7). For the case of potassium with neon the predicted spin destruction rate is 8% smaller in the 1.6atm cell, and 8%greater in the 6.2atm cell than the measured value. This is reasonable agreement since the relevant spin destruction cross sections have only been measured to one significant figure. The spin destruction rates are approximately 2840 times larger than the contribution due to gradient relaxation. It is larger than the rate of diffusion to the cell wall by a factor of 40. A measurement of the spin destruction rate of rubidium in the rubidium-neon cell and subsequent calculation of the Rb-Ne spin destruction cross section was carried out. The spin destruction rate was measured via the rubidium relaxation in the dark measurement which was previously described. The spin destruction rate was extrapolated to zero probe beam intensity by making multiple spin destruction rate measurements at different probe beam intensities. The variation in probe beam intensity was achieved by placing neutral density filters of different values directly in the beam path before the probe entered the cell. The alkali-Ne spin destruction cross sections are listed in Table 5.4. 112 One can accurately predict the polarization of the neon gas by using eq. (2.105). PNe = Palkali ǫ Rse W/S Rse + 1/T1 (5.17) Here the measured spin exchange rate for the Rse was substituted into eq. (5.17) and T1 was calculated assuming that the quadrupolar relaxation mechanism is the dominant relaxation mechanism. Making these assumptions, while taking into account the non-uniform alkali polarization profile across the cell,a polarization of 0.55% at 140◦ C was expected. A value of 0.8 ± 0.13% was measured using EPR. 5.7 Conclusion The spin exchange rate coefficient for both the Rb-21 Ne, and K-21 Ne systems have been measured using two different techniques. The values from the rate of rise and repolarization techniques are consistent. The spin exchange rate coefficient for Rb-21 Ne does not agree with the previous value in the literature. We claim this discrepency is caused by the exclusion of relaxation due to the neon quadrupolar relaxation from previous measurements. The Fermi contact interaction for both the Rb-21 Ne, and K-21 Ne pairs have been measured. There is agreement with the previously measured value for the Rb-21 Ne system. The neon quadrupolar relaxation has been measured. Also the various alkali spins destruction cross-sections with 21 Ne have been measured. These agree with the previous values quoted in the literature. We have modeled the neon polarization as a function of alkali density and have show that the polarization dynamics are well described by assuming quadrupolar relaxation is the dominant form of relaxation, and spin exchange the dominant polarizing interaction. 113 Chapter 6 Feasibility of utilizing 21Ne in a co-magnetometer The main objective of these experiments on 21 Ne is to ultimately create a neon comagnetometer which can be used for experiments on tests of fundamental symmetries, and for deployment as a sensitive gyroscope. However application of 21 Ne in a co-magnetometer requires polarization higher than that observed for the set of spin-exchange rate measurements at 140◦ C. One can increase the neon polarization by increasing the alkali density. However this requires additional laser power to ensure the optically thick cell remains uniformly polarized. Experiments were carried out at higher density at 180◦ C and resulted in ≈ 8% neon polarization. A Mathematica model was used to calculate the sensitivity of the co-magnetometer at normal operating temperatures by optimizing the laser power. Here the sources of noise included were the effects of spin projection noise, and photon shot noise. The first effect is due to the Heisenberg uncertainty principle as applied to the transverse components of the spins and the fact that they do not commute. The second source of noise is the noise of the rotation signal of the probe beam for a 114 balanced polarimeter setup. We ignore the effects of light shift noise because we presume to operate with the pump and probe beams orthogonal to each other. In this orientation the magnetometer signal is sensitive to the axis perpendicular to the plane which contains both laser beams. Thus it is in-sensitive to light shift noise caused by fluctuations in the ellipticity of the probe beam to first order. Let us describe the Rb-Ne comagnetometer simulation in greater depth. 6.1 Effects of Light Propogation and alkali relaxation on Rb-Ne co-magnetometer simulation The propagation of the pump and probe beams must be modeled as the beams can be strongly absorbed and result in non-uniform polarization through the cell. The propagation of light through the cell is a function of the alkali polarization in the cell which is given by eq. (2.18). The propagation dynamics can be described by: dRop = −nσ(ν0 )(1 − Pequil ) Rop dx As one can see the attenuation dRop dx (6.1) vanishes if the alkali is fully polarized. How- ever this is never achieved in practice due to spin relaxation due to spin destruction collisions. These can be described by: sd sd se Rrel = n Rb σRb − Rb V Rb + n Ne σRb− Ne V Ne + R pr + (ǫ + 1) Rwall + R Rb− Ne n Rb (6.2) Rwall is given by eq. (2.62), V i− j is given by eq. (2.39), ni refers to the density per cm−3 of species i, and ǫ is given by eq. (2.56). R pr is the pumping rate due to the 115 probe beam and is given by: λ probe σRb− xs (λ probe ) × Pprobe × wlhc 1 Exp −σRb− xs (λ probe )n Rb 2 R pr = (6.3) w refers to the width of the cell, and l its length. h is Planck’s constant, c is the speed of light, and P probe is the probe beam power. The photon absorption cross section σRb− xs is given by: σRb− xs (ν) = re c ( f D1 V(ν − νD1 ) + f D2 V(ν − νD2 )) (6.4) Solution of eq. (6.1) gives a pumping rate profile across the cell as: Rop ( x ) = Rrel W Rop−int Exp[ Rop−int Rrel − σRb−xs (λ D1 )n Rb x ] Rrel (6.5) Here P pump is the pump beam power, and x is the propagation distance through the cell. The function W is the principal value of the Lambert W-function. This is defined as the inverse of the function f (W ) = WeW . It is also refer Rop−int = σRb− xs (λ D1 ) Ppump λ D1 1 Ahc (6.6) where A is the cross sectional area of the cell, and λ D1 is the wavelength of the D1 line. An alkali polarization profile for a pancake cell of dimensions 6 × 15mm under condition of high pumping rate (≈ 7800s−1 ) is shown in fig.6.1 116 Absolute Polarization 0.98 0.96 0.94 0.92 0.1 0.2 0.3 0.4 0.5 0.6 Distance Across cell (cm) Figure 6.1: Absolute Rb polarization as function of propagation distance through cell, for pancake cell of dimensions 6 × 15 × 15mm and a pumping rate of ≈ 7800s−1 . 6.2 Simulation of Noble gas relaxation We are able to relate the noble gas polarization in the steady state to the alkali polarization using eq. (2.105). However eq. (2.105) indicates that the steady state polarization of the noble gas not equilibrate with the alkali polarization due to strong noble gas relaxation mechanisms. In this work we model the effects of both quadrupolar relaxation, and relaxation due to diffusion of the noble gas in a magnetic field gradient. The quadrupolar relaxation rate of neon is taken from the measured relaxation rate as a function of cell pressure. This is depicted in fig.5.7. The magnetic field gradient in the cell can be evaluated utilizing the technique of magnetic vector potential(Jackson (1999)). The field inside the cell can be modeled as a uniformly polarized mass. The magnetic field at an arbitrary point inside the field can be evaluated by replacing the polarized mass with a surface current of density equal to the cell’s magnetization (Jackson (1999)). This enables calculation of the relaxation rate given by eq. (2.64) on a point by point basis, which is averaged over the cell. This is calculated numerically on a lattice. The noble gas T1 117 then becomes: 1 Pneon = + T1 12840s Z volume D Ne− Ne ∇ B( Pneon ) B( Pneon ) 2 dV (6.7) In the above equation Pneon represents the pressure of neon in the cell in amagat. The denominator of the first term describes the relaxation of neon due to quadrupolar relaxation. The Pneon 12840 is the quadrupolar relaxation rate given in s−1 . It is equal to 214Amagat·mins. D Ne− Ne is the self diffusion coefficient of neon, and ∇ B is the magnetic field gradient taken at every point in the cell. The eq. (6.7) gives 1 T1 in units of s−1 . For a cell with pressure of 3 amagat the relaxation rate due to diffusion is ≈ 30% as large as the quadrupolar relaxation rate. 6.3 Noise mechanisms in a Rb-Ne co-magnetometer We describe the noise contributions to the magnetometer sensitivity in more detail. The total noise in the magnetic field measurement has two major contributions. The first is the spin projection noise due to the quantum uncertainty in the spin’s orientation. The second contribution is due to the photon shot noise. To obtain the total noise in the magnetic field measurement these contributions must be added in quadrature. δB = q 2 + δB2 δBspn psn (6.8) The descriptions of the uncertainty mechanisms below follow that of Savukov et al. (2005). 118 6.3.1 Spin Projection Noise To determine the spin projection noise of the co-magnetometer let us first consider the commutation relation between the transverse components of the alkali spin. [ Fx , Fy ] = iFz (6.9) This leads to the uncertainty relation δFx δFy ≥ | Fz | 2 (6.10) Let us operate under conditions of full polarization since this minimizes the uncertainty relation. Due to symmetry δFx = δFy for spins which are not in a squeezed state. We can describe the signal from N alkali atoms as making N uncorrelated measurements. In such a case the uncertainty in the spin of the transverse components becomes: δFx = r h Fz i 2N (6.11) It is important to note that only uncorrelated measurements improve the sensitivity of the measurement. In the atomic magnetometer setup the probe beam continuously measures the spin projection. Correlated measurements from the same atoms do not improve sensitivity. To take this into consideration let us define a quantity χ(τ ) as the degree of loss of spin coherence from a measurement at time t = 0, and the time t = τ. χ(τ ) = e−τ/T2 (6.12) Gardner (1990) gives the total uncertainty in a continuous measurement as: Z t 1/2 2 τ δ h Fx i = δFx 1− χ(τ )dτ τ 0 t 119 (6.13) " δ h Fx i = δFx 2T2 2T22 (e−t/T2 − 1) + τ t2 #1/2 (6.14) Since typical measurement time for the magnetometer system satisfies t >> T2 we can combine eq. (6.11), and eq. (6.14) to obtain the total uncertainty: δ h Fx i = r 2Fz T2 BW N (6.15) where BW= 1/2t is the bandwidth of the measurement. Here N describes the total number of alkali spin that the probe beam interacts with. In units of root mean square noise per root Hz we can rewrite the total uncertainty as: δ h Fx irms = r 2Fz T2 N (6.16) This can be written in terms of magnetic noise by using eq. (2.35) and by setting S → Fx /2 and Pze → Fz /2 to find: δ h Birms = δ h Fx i Rtot γe Fz (6.17) 6.3.2 Photon Shot Noise We will describe the photon shot noise associated with detection of a probe beam with a balanced polarimeter setup. It is convenient to define the total photon flux as: Φ′ = Z ΦdA A (6.18) where Φ is the photon flux per unit area, and Φ is the total photon flux taken over the area of the probe beam. In terms of the photon flux in each photodiode we can write the optical rotation as: θ= Φ1′ − Φ2′ 2(Φ1′ + Φ2′ ) 120 (6.19) If we assume the rotation angle θ is small we can state Φ1′ ≈ Φ2′ . In this case we can write the fluctuation in the photon flux as: δΦ1′ = δΦ2′ = r Φ′ 2 (6.20) This leads to a fluctuation in the optical rotation angle given by: v " u 2 2 # u δθ δθ δ hθ i = t2BW δΦ1′ + δΦ1′ δΦ1′ δΦ1′ δ hθ i = r BW 2Φ′ (6.21) (6.22) In units of root-mean square noise per root Hz this becomes: δ hθ i = r 1 2Φ′ (6.23) For the magnetometer it is more useful to relate the noise in terms of the rotation angle to the atomic polarization. Consider a probe beam detuned from the D1 resonance. We can calculate the optical rotation due to the D1 resonance if we combine eq. (6.23) and eq. (2.81) to obtain: δ h Px irms = πlnre c f √ 2 Φ′ Im[V(ν − ν0 )] (6.24) Where Φ′ is given by: Φ′ = ηPprobe λ probe Exp −nrb lσRb− xs (λ probe ) hc 121 (6.25) Here η is the quantum efficiency η= ( Resp)hc λ probe e (6.26) where Resp is the responsivity of the photodiode in A/W, and e is the charge of the electron. The noise in angular units can be converted to magnetic units by using eq. (2.35) and noting that S = Re( Pxe ) to obtain: Px γe B0 = Pz Rtot (6.27) or, δ h Birms = 6.4 δ h Px irms Rtot γe Pze (6.28) Results of Rb-Ne co-magnetometer simulation For a cell which is cubic with an edge of 5mm at 180◦ C with 15mW pump beam √ one finds the sensitivity averaged over the cell to be 0.77fT/ Hz and compensation point of 4.7Hz. At 200◦ C with 15mW pump beam one finds the sensitivity √ averaged over the cell to be 0.31fT/ Hzand compensation point of 4.7Hz. These √ sensitivities are both below the 1fT/ Hz noise floor caused by technical sources. Thus we could operate a co-magnetometer with these settings. In both of these cases the probe beam was optimized to 10mW and detuned 0.39nm lower than the resonant wavelength. For the situation where we operate at 240◦ C we find that pumping with 75mW √ gives a sensitivity of 0.67fT/ Hz. This is the sensitivity averaged over the entire cell. That is with a probe beam expanded over the entire cell. √ If instead we go to a pancake geometry one can reach 7aT/ Hz for a cell of dimensions 6 × 15 × 15mm with 3atm of neon. This is a reasonable since the pyrex 122 glass we have available has wall thickness of 1mm, which corresponds to being safely able to encase 3atm nominally. Also the cell temperature is 180◦ C which is near the upper temperature a pyrex cell be operated at without discoloration. Thus it should be safe to operate at this temperature for a pyrex cell. The neon frequency at the compensation point is 0.8Hz. This is not high enough that the comagnetometer zero-ing routines can be operated in a reasonable time. The pump laser power is 40mW, and the probe beam is 10mW and detuned 0.4nm below the D1 resonance wavelength. Typically one utilizes ultra stable single frequency DFB laser diodes for precision experiments. These DFB diodes are normally available with power rating up to 100mW. Thus it should be easy to obtain laser for pumping and probing with the required power rating. The angular noise corresponding √ to the atomic shot noise is 5nRad/ Hz in this scheme. Thus it seems feasible to create a rubidium-neon co-magnetometer with fundamental noise approaching comparable to the state of the art atomic co-magnetometers. We can use the experimentally measured values of the alkali-neon spin exchange rate to predict the feasibility of a alkali-neon co-magnetometer. One can accurately predict the polarization of the neon gas by using eq. (2.105). Here we ab and calculate the T assumsubstitute the measured spin exchange rate for the Rse 1 ing that the quadrupolar relaxation mechanism is the dominant relaxation mechanism. Making these assumptions while taking into account the non-uniform alkali polarization profile across the cell we expect a polarization of 0.55% at 140◦ C. A value of 0.8% which was measured using EPR. Ultimately the quantity of interest when constructing a co-magnetometer is the frequency of the noble gas when operating at the compensation point. This is because the frequency of the noble gas sets the time scale for the co-magnetometer. When processes are performed adiabatically for the co-magnetometer we really mean that they are slow compared to the noble gas precession frequency at the 123 compensation point. Thus by increasing this frequency we are able to perform adiabatic tasks, such as zeroing, faster. For practical purposes this is of great use when running the co-magnetometer, and gyroscope experiments. By predicting the noble gas polarization and utilizing the noble gas gyromagnetic ratio we can predict the noble gas precession frequency at the compensation point. Typically for operation of the co-magnetometer the noble gas frequency must be set to at least 7Hz. Again utilizing eq. (2.105) we calculate the frequency at the compensation point to be 2.30Hz at 180◦ C, 3.3Hz at 190◦ C, and 4.7Hz at 200◦ C for a 3atm. At temperatures higher than 190◦ C pyrex cells can often brown because the alkali metal reacts with the glass in the cell wall. We can eliminate this effect by employing aluminosilicate glass instead. We have also investigated the influence of imperfections in the gyroscope for e , and C n from a Rb-Ne co-magnetometer. The spin exchange correction factors Cse se eq. (3.4), and eq. (3.5) can be estimated. For operation with the pancake cell e ≈ 8 × 10−5 , and C n ≈ 3 × 10−4 . We predict a false signal due to misalignment Cse se of the pump-probe orthogonality by 1µrad of ≈ 9 × 10−7 rad/s. The rotational √ sensitivity of the gyroscope is 1 × 10−9 rad/s/ Hz. 124 Chapter 7 Conclusions and future work We have been able to measure many of the parameters necessary to construct an optimized enriched neon co-magnetometer. The neon-neon quadrupolar relaxation rate was found to be 214 ± 10min×amagat. The neon-potassium spin exchange coefficient has measured and is listed tables 7.1. The neon-rubidium spin exchange rate coefficient has been measured and is listed in Table 7.2. The Fermi contact interaction κ0 for neon with both species is listed in table 7.3. The spin destruction cross-sections for each alkali with neon is listed in Table 7.4. We have demonstrated operation of an co-magnetometer gyroscope with mea√ sured sensitivity of ∆Ω ≈ 5.0 × 10−7 rad/s/ Hz. The fundamental sensitivity of √ such a gyroscope using the current K-He mixture is ∆Ω ≈ 1.0 × 10−8 rad/s/ Hz and is limited by angular noise of the probe beam. We have shown that switching √ to 21 Ne would increase the sensitivity of the detector to ∆Ω ≈ 1.0 × 10−9 rad/s/ Hz. Spin Exchange Rate Method Repolarization Theoretical Prediction Rate of Rise cm3 /s 3.36 ± 0.67 × 10−20 1.2 × 10−19 2.34 ± 0.52 × 10−20 cm2 Table 7.1: Spin exchange parameter of a potassium-neon system 125 Spin Exchange Rate Method Repolarization Theoretical Prediction Rate of Rise cm3 /s 0.80 ± 0.16 × 10−19 1.66 × 10−19 0.82 ± 0.18 × 10−19 Table 7.2: Spin exchange parameter of a rubidium-neon system Species K-Ne K-Ne (prediction) Rb-Ne Rb-Ne (Walsworth) Rb-Ne (prediction) Fermi Contact Interaction 30.8 ± 2.7 34 35.7 ± 3.7 32.0 ± 2.9 38 Table 7.3: Fermi contact interaction measurements for the K-Ne, and Rb-Ne systems in this work, and compared to both predictions ans measurements by other groups. Species K-Ne K-Ne K-Ne Rb-Ne Rb-Ne Rb-Ne Rb-Ne Group This work Franz Walker (prediction) This work Franz and Volk Franzen Walker (prediction) 10−23 cm2 1.1 ± 0.1 1.41 ± 0.14 1.6 1.9 ± 0.2 1.9 5.2 1.8 Table 7.4: Spin destruction cross sections for K-Ne, and Rb-Ne as compared to both theory and other measurements 126 Switching to 21 Ne will also enable stricter limits on tests of Lorentz and CPT violation. In future when using 21 Ne to construct an improved co-magnetometer one must further study the effects of quadrupole interactions and cell geometry which differ from those of 3 He. Since the dominant source of relaxation in a neon cell is due to quadrupolar relaxation, and not due to long range dipolar fields one could use a cubic geometry for the measurement cell. This will reduce effects of cell birefringence and beam distortion of the lasers on the magnetometer noise. However in this case the Bloch equations which describe the magnetometer interaction must be modified to include a neon self interacting term due to the non-uniform magnetic field it would experience. Also the effect of quadrupolar splitting shifts the Zeeman levels for the neon gas. This splitting in not uniform for each of the Zeeman levels. To counteract this one needs to with operate with a cell which has zero quadrupole moment. We have found that in order to generate high neon polarization one requires a uniform alkali polarization. One could also imagine creating a thin pancake shaped cell so that the cell will be uniformly polarized. The addition of a quadrupole moment allows one to test more parameters of the standard model than helium. One could test Lorentz invariance by searching for a quadrupole splitting in the Zeeman levels of the 21 Ne which varies at a frequency of twice a sidereal day. One could imagine constructing a vapour cell with both 21 Ne and 3 He. The 21 Ne could be used as a magnetometer and used to correlate any systematic effects experienced by the 3 He . This would give a test for the local Lorentz invariance by measuring the mass anisotropy of 21 Ne (Chupp et al. (1989)). One could also measure the coherence relaxation between the Zeeman levels of 21 Ne. This has implications, and gives a test of the linearity of quantum mechanics (Chupp & Hoare (1991)). The quadrupole moment of 21 Ne interacts with electric field gradients. Thus it could have many potential applica- 127 tions in pulmonary imaging. When the 21 Ne interacts with pulmonary cell walls it could be used as a probe of the surface dynamics. In fact another Noble gas with a quadrupole moment 83 Kr has been shown to successfully detect the difference between hydrophilic and hydrophobic surfaces (Raftery (2006)). Since many lung disease are dependent on the surface to volume ratio of lung tissue this could be useful in diagnosis of pulmonary disease. 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