The Quantum Physics of Atomic Frequency Standards Recent Developments TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk The Quantum Physics of Atomic Frequency Standards Recent Developments Jacques Vanier Université de Montréal, Montréal, Canada Cipriana Tomescu Université de Montréal, Montréal, Canada CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2016 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Version Date: 20150617 International Standard Book Number-13: 978-1-4665-7697-1 (eBook - PDF) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. 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Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com Contents Preface....................................................................................................................xiii Introduction ............................................................................................................xvii Authors ....................................................................................................................xix Chapter 1 Microwave Atomic Frequency Standards: Review and Recent Developments .......................................................................................1 1.1 Classical Atomic Frequency Standards .....................................2 1.1.1 Cs Beam Frequency Standard ......................................2 1.1.1.1 Description of the Approach Using Magnetic State Selection ..............................3 1.1.1.2 Review of Frequency Shifts and Accuracy ....7 1.1.1.3 Frequency Stability of the Cs Beam Standard ......................................................15 1.1.1.4 Recent Accomplishments ...........................16 1.1.2 Hydrogen Maser .........................................................33 1.1.2.1 Active Hydrogen Maser ..............................33 1.1.2.2 Passive Hydrogen Maser .............................48 1.1.2.3 Frequency Stability of the Hydrogen Maser ..........................................................53 1.1.2.4 State of the Art of Recent Developments and Realizations ..................57 1.1.3 Optically Pumped Rb Frequency Standards ..............69 1.1.3.1 General Description ....................................69 1.1.3.2 State-of-the-Art Development ....................71 1.2 Other Atomic Microwave Frequency Standards .....................82 1.2.1 199Hg+ Ion Frequency Standard ..................................83 1.2.1.1 General Description ....................................83 1.2.1.2 Frequency Shifts .........................................85 1.2.1.3 Linear Trap .................................................88 1.2.2 Other Ions in a Paul Trap ...........................................90 1.2.2.1 171Yb+ and 173Yb+ Ion Microwave Frequency Standards ..................................91 1.2.2.2 201Hg+ Ion Microwave Frequency Standard ...................................................92 1.3 On the Limits of Classical Microwave Atomic Frequency Standards .................................................................................93 Appendix 1.A: Formula for Second-Order Doppler Shift ..................94 Appendix 1.B: Phase Shift between the Arms of Ramsey Cavity ......95 v vi Contents Appendix 1.C: Square Wave Frequency Modulation and Frequency Shifts ..........................................................................95 Appendix 1.D: Ring Cavity Phase Shift ............................................97 Appendix 1.E: Magnetron Cavity ......................................................98 Chapter 2 Recent Advances in Atomic Physics That Have Impact on Atomic Frequency Standards Development ................................101 2.1 Solid-State Diode Laser .........................................................102 2.1.1 Basic Principle of Operation of a Laser Diode ..........102 2.1.2 Basic Characteristics of the Semiconductor Laser Diode ..............................................................105 2.1.3 Types of Laser Diodes ..............................................106 2.1.4 Other Types of Lasers Used in Special Situations .....108 2.2 Control of Wavelength and Spectral Width of Laser Diodes ..................................................................109 2.2.1 Line Width Reduction ..............................................109 2.2.1.1 Simple Optical Feedback ..........................109 2.2.1.2 Extended Cavity Approach .......................109 2.2.1.3 Feedback from High-Q Optical Cavities ...112 2.2.1.4 Electrical Feedback ..................................112 2.2.1.5 Other Approaches .....................................112 2.2.1.6 Locking the Laser to an Ultra-Stable Cavity ........................................................113 2.2.2 Laser Frequency Stabilization Using an Atomic Resonance Line ........................................................116 2.2.2.1 Locking the Laser Frequency to Linear Optical Absorption ...................................116 2.2.2.2 Locking the Laser Frequency to Saturated Absorption ............................117 2.3 Laser Optical Pumping ..........................................................119 2.3.1 Rate Equations .........................................................120 2.3.2 Field Equation and Coherence .................................122 2.4 Coherent Population Trapping ...............................................127 2.4.1 Physics of the CPT Phenomenon .............................129 2.4.2 Basic Equations ........................................................131 2.5 Laser Cooling of Atoms ........................................................136 2.5.1 Atom–Radiation Interaction .....................................138 2.5.1.1 Effect of a Photon on Atom External Properties: Semi-Classical Approach .......138 2.5.1.2 Quantum Mechanical Approach ...............143 2.5.2 Effect of Fluctuations in Laser Cooling and Its Limit .............................................................158 2.5.3 Cooling below Doppler Limit: Sisyphus Cooling ....160 2.5.3.1 Physics of Sisyphus Cooling .....................160 2.5.3.2 Capture Velocity .......................................164 Contents vii 2.5.3.3 Friction Coefficient ...................................165 2.5.3.4 Cooling Limit Temperature ......................166 2.5.3.5 Recoil Limit ..............................................166 2.5.3.6 Sub-Recoil Cooling ..................................167 2.5.4 Magneto-Optical Trap ..............................................167 2.5.5 Other Experimental Techniques in Laser Cooling and Trapping ...............................................170 2.5.5.1 Laser Atom-Slowing Using a Frequency Swept Laser System: Chirp Laser Slowing .................................171 2.5.5.2 Laser Atom-Slowing Using Zeeman Effect: Zeeman Slower .............................173 2.5.5.3 2D Magneto-Optical Trap ........................177 2.5.5.4 Isotropic Cooling ......................................180 2.5.5.5 Optical Lattice Approach .........................183 Appendix 2.A: Laser Cooling—Energy Considerations ..................189 Chapter 3 Microwave Frequency Standards Using New Physics .....................191 3.1 Cs Beam Frequency Standard ...............................................192 3.1.1 Optically Pumped Cs Beam Frequency Standard ....192 3.1.1.1 General Description ..................................192 3.1.1.2 Frequency Shifts and Accuracy ................194 3.1.1.3 Experimental Determination of Those Shifts ..........................................197 3.1.1.4 Frequency Stability ...................................198 3.1.1.5 Field Application ......................................200 3.1.2 CPT Approach in a Beam ......................................200 3.1.2.1 General Description ..................................200 3.1.2.2 Analysis ....................................................201 3.1.2.3 Experimental Results ................................206 3.1.3 Classical Cs Beam Standard Using Beam Cooling ..............................................................208 3.2 Atomic Fountain Approach ...................................................210 3.2.1 In Search of a Solution .............................................210 3.2.2 General Description of the Cs Fountain ...................211 3.2.3 Functioning of the Cs Fountain ................................213 3.2.3.1 Formation of the Cooled Atomic Cloud: Zone A ..........................................213 3.2.3.2 Preparation of the Atoms: Zone B ............217 3.2.3.3 Interrogation Region: Zone C ...................218 3.2.3.4 Free Motion: Zone D ................................218 3.2.3.5 Detection Region: Zone E.........................218 3.2.4 Physical Construction of the Cs Fountain ................219 3.2.4.1 Vacuum Chamber .....................................219 3.2.4.2 Microwave Cavity .....................................220 viii Contents 3.2.4.3 Magnetic Field ..........................................221 3.2.4.4 Temperature Control .................................221 3.2.4.5 Capture and Selection Zone ......................221 3.2.4.6 Detection Zone .........................................221 3.2.4.7 Supporting Systems ..................................221 3.2.4.8 Advantages and Disadvantages of a Pulsed Fountain .................................222 3.2.5 Frequency Stability of the Cs Fountain ....................223 3.2.5.1 Photon Shot Noise .....................................224 3.2.5.2 Quantum Projection Noise........................225 3.2.5.3 Electronic Noise ........................................225 3.2.5.4 Reference Oscillator Noise: Dicke Effect ...225 3.2.6 Rubidium and Dual Species Fountain Clock ...........226 3.2.7 Frequency Shifts and Biases Present in the Fountain ....................................................................229 3.2.7.1 Second-Order Zeeman Shift .....................230 3.2.7.2 Black Body Radiation Shift ......................232 3.2.7.3 Collision Shift ...........................................237 3.2.7.4 Cavity Phase Shift ....................................240 3.2.7.5 Cavity Pulling ...........................................242 3.2.7.6 Microwave Spectral Purity .......................247 3.2.7.7 Microwave Leakage ..................................247 3.2.7.8 Relativistic Effects ....................................248 3.2.7.9 Other Shifts ...............................................249 3.2.7.10 Conclusion on Frequency Shifts and Accuracy ............................................250 3.2.8 An Alternative Cold Caesium Frequency Standard: The Continuous Fountain ........................251 3.2.8.1 Light Trap .................................................252 3.2.8.2 Interrogation Zone, Microwave Cavity .....253 3.2.8.3 Preliminary Results ..................................255 3.2.9 Cold Atom PHARAO Cs Space Clock ....................257 3.3 Isotropic Cooling Approach ..................................................258 3.3.1 External Cavity Approach: CHARLI ......................258 3.3.2 Approach Integrating Reflecting Sphere and Microwave Cavity: HORACE ...........................260 3.3.3 Different HORACE Approach .................................261 3.4 Room Temperature Rb Standard Approach Using Laser Optical Pumping ....................................................................262 3.4.1 Contrast, Line Width, and Light Shift .....................263 3.4.2 Effect of Laser Radiation Beam Shape ....................272 3.4.3 Expectations Relative to Short-Term Frequency Stability ....................................................................273 3.4.4 Review of Experimental Results on Signal Size, Line Width, and Frequency Stability .......................273 Contents ix 3.4.5 Frequency Shifts .......................................................278 3.4.5.1 Buffer Gas Shift ........................................278 3.4.5.2 Magnetic Field Shift .................................279 3.4.5.3 Light Shift .................................................279 3.4.5.4 Spin-Exchange Frequency Shift ...............284 3.4.5.5 Microwave Power Shift .............................285 3.4.5.6 Cavity Pulling ...........................................286 3.4.6 Impact of Laser Noise and Instability on Clock Frequency Stability ..................................................287 3.4.6.1 Spectral Width, Phase Noise, and Intensity Noise of Laser Diodes ...............288 3.4.6.2 Impact of Laser Noise on Clock Short-Term Frequency Stability ................290 3.4.6.3 Medium- and Long-Term Frequency Stability .....................................................295 3.4.7 Other Approaches Using Laser Optical Pumping with a Sealed Cell .....................................................297 3.4.7.1 Maser Approach ........................................297 3.4.7.2 Laser Pulsing Approach............................297 3.4.7.3 Wall-Coated Cell Approach .....................299 3.5 CPT Approach .......................................................................300 3.5.1 Sealed Cell with a Buffer Gas in Continuous Mode: Passive Frequency Standard ..........................300 3.5.1.1 Signal Amplitude and Line Width ............302 3.5.1.2 Practical Implementation and Its Characteristics ..........................................307 3.5.2 Active Approach in a Cell: The CPT Maser ............315 3.5.2.1 Basic CPT Maser Theory .........................315 3.5.2.2 Frequency Stability ...................................318 3.5.2.3 Frequency Shifts .......................................320 3.5.3 Techniques for Improving S/N Ratio in the Passive IOP and CPT Clock Approach ....................322 3.5.4 CPT in Laser-Cooled Ensemble for Realizing a Frequency Standard ..................................................323 3.6 Laser-Cooled Microwave Ion Clocks ....................................324 3.6.1 9Be+ 303 MHz Radio-Frequency Standard ..............325 3.6.2 113Cd+ and 111Cd+ Ion Trap ........................................327 3.6.3 171Yb+ Laser-Cooled Microwave Frequency Standard ...................................................................328 Appendix 3.A: Frequency Stability of an Atomic Fountain ............329 3.A.1 Shot Noise ................................................................333 3.A.2 Quantum Projection Noise .......................................334 Appendix 3.B: Cold Collisions and Scattering Length ....................337 Appendix 3.C: Optical Absorption of Polarized Laser Radiation Including Optical Pumping ..............................................................338 Appendix 3.D: Basic CPT Maser Theory ........................................341