The Quantum Physics of Atomic Frequency Standards Recent Developments This page intentionally left blank 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 The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint Except as permitted under U.S Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers For permission to photocopy or use material electronically from this work, please access www.copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400 CCC is a not-for-profit organization that provides licenses and registration for a variety of users For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe 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 Microwave Atomic Frequency Standards: Review and Recent Developments .1 1.1 Classical Atomic Frequency Standards .2 1.1.1 Cs Beam Frequency Standard 1.1.1.1 Description of the Approach Using Magnetic State Selection 1.1.1.2 Review of Frequency Shifts and Accuracy 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 Recent Advances in Atomic Physics That Have Impact on Atomic Frequency Standards Development 101 2.1 2.2 2.3 2.4 2.5 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 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 Laser Optical Pumping 119 2.3.1 Rate Equations 120 2.3.2 Field Equation and Coherence 122 Coherent Population Trapping 127 2.4.1 Physics of the CPT Phenomenon 129 2.4.2 Basic Equations 131 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 vii Contents 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 Microwave Frequency Standards Using New Physics 191 3.1 3.2 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 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 3.2.4.4 3.2.4.5 3.2.4.6 3.2.4.7 3.2.4.8 3.3 3.4 Magnetic Field 221 Temperature Control 221 Capture and Selection Zone 221 Detection Zone 221 Supporting Systems 221 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 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 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 ix Contents 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 References 443 Sagna N., Mandache C., and Thomann P 1992 Noise measurement in single-mode GaAl As diode lasers In Proceedings of the European Forum on Time and Frequency 521 Santarelli G., Audoin C., Makdissi A., Laurent P., Dick G.J., and Clairon C 1998 Frequency stability degradation of an oscillator slaved to a periodically interrogated atomic resonator IEEE Trans Ultrason Ferroelectr Freq Control 45: 887 Santarelli G., Governatori G., Chambon D., Lours M., Rosenbusch P., Guena J., Chapelet F., Bize S., Tobar M., Laurent Ph., Potier T., and Clairon A 2009 Switching atomic fountain clock microwave interrogation signal and high-resolution phase measurements IEEE Trans Ultrason Ferroelectr Freq Control 56 (7): 1319 Santarelli G., Laurent Ph., Lemonde P., Clairon A., Mann A.G., Chang S., Luiten A.N., and Salomon C 1999 Quantum projection noise in an atomic fountain: A high stability cesium frequency standard Phys Rev Lett 82: 4619 Sarosy E.B., Johnson W.A., Karuza S.K., and Voit F.J 1992 Measuring frequency changes due to microwave power variations as a function of C-field setting in a rubidium frequency standard In Proceedings of the Annual Precise Time and Time Interval Applications and Planning Meeting, ION Publication, 229 Schwindt P.D.D., Olsson R., Wojciechowski K., Serkland D., Statom T., Partner H., Biedermann G., Fang L., Casias A., and Manginell R 2009 Micro ion frequency standard In Proceedings of the Annual Precise Time and Time Interval Applications and Planning Meeting, ION Publication, 509 Shah V., Gerginov V., Schwindt P.D.D., Knappe S., Hollberg L., and Kitching J 2006 Continuous light-shift correction in modulated coherent population trapping clocks Appl Phys Lett 89: 151124 Shahriar M.S and Hemmer P.R 1990 Direct excitation of microwave–spin dressed state using a laser-excited resonance Raman interaction Phys Rev Lett 65: 1865 Shahriar M.S., Hemmer P.R., Kalz D.P., Lee A., and Prentiss M.G 1997 Dark-state-based three-element vector model for the stimulated Raman interaction Phys Rev A 55: 2272 Shirley J.H., Lee W.D., and Drullinger R.E 2001 Accuracy evaluation of the primary frequency standard NIST-7 Metrologia 38: 427 Simon E., Laurent Ph., and Clairon A 1998 Measurement of the Stark shift of the Cs hyperfine splitting in an atomic fountain Phys Rev A 57: 436 Singh G., DiLavore P., and Alley C.D 1971 GaAs-laser-induced population inversion in the ground-state hyperfine levels of Cs133 IEEE J Quantum Elect QE7: 196 Sortais Y 2001 Construction d’une Fontaine Double Atomes Froids de 87Rb et 133Cs; étude des Effets Dépendant du nombre d’atomes dans une Fontaine PhD Thesis, Université ParisVI, France Steane A 1997 The ion trap quantum information processor Appl Phys B 64: 623 Szekely C., Walls F., Lowe J.P., Drullinger R.E., and Novick A 1994 Reducing local oscillator phase noise limitations on the frequency stability of passive frequency standards: tests of a new concept IEEE Trans Ultrason Ferroelectr Freq Control 41: 518 Szymaniec K., Chałupczak W., Tiesinga E., Williams C.J., Weyers S., and Wynands R 2007 Cancellation of the collisional frequency shift in caesium fountain clocks Phys Rev Lett 98: 153002 Szymaniec K., Chałupczak W., Whibberley P.B., Lea S.N., and Henderson D 2005 Evaluation of the primary frequency standard NPL-CsF1 Metrologia 42: 49 Szymaniec K., Lea S.N., and Liu K 2014 An evaluation of the frequency shift caused by collisions with background gas in the primary frequency standard NPL-CsF2 IEEE Trans Ultrason Feroelectr Freq Control 61 (1): 203 Taghavi-Larigani S., Burt E.A., Lea S.N., Prestage J.D., and Tjoelker R.L 2009 A new trapped ion clock based on 201Hg+ In Proceedings of the IEEE International Frequency Control Symposium, IEEE, 774 444 References Taichenachev A.V., Tumaikin A.M., Yudin V.I., Stähler M., Wynands R., Kitching J., and Hollberg L 2004 Nonlinear-resonance line shapes: dependence on the transverse intensity distribution of a light beam Phys Rev A 69: 024501 Taichenachev A.V., Yudin V.I., Velichansky V.L., Kargapoltsev S.V., Wynands R., Kitching J., and Hollberg L 2004 High-contrast dark resonances on the D1 line of alkali metals in the field of counterpropagating waves JEPT Lett 80: 236 Têtu M., Busca G., and Vanier J 1973 Short-term frequency stability of the Rb87 maser IEEE Trans Instrum Meas 22: 250 Thomas J.E., Hemmer P.R., Ezekiel S., Leiby J.C.C., Picard H., and Willis R 1982 Observation of Ramsey fringes using a stimulated, resonance Raman transition in a sodium atomic beam Phys Rev Lett 48: 867 Tiesinga E., Verhaar B.J., Stoof H.T.C., and van Bragt D 1992 Spin-exchange frequency shift in a caesium atomic fountain Phys Rev A 45: R2671 Tsuchida H and Tako T 1983 Relation between frequency and intensity stabilities in AlGaAs semiconductor lasers Jpn J Appl Phys 22: 1152 Vanier J 1968 Relaxation in rubidium-87 and the rubidium maser Phys Rev 168: 129 Vanier J 1969 Optical pumping as a relaxation process Can J Phys 47: 1461 Vanier J 2002 Coherent population trapping for the realization of a small, stable, atomic clock In Proceedings of the IEEE International Frequency Control Symposium, IEEE, 424 Vanier J 2005 Atomic clocks based on coherent population trapping: a review Appl Phys B 81: 421 Vanier J and Audoin C 1989 The Quantum Physics of Atomic Frequency Standards (Adam Hilger, Bristol) Vanier J and Audoin C 2005 The classical caesium beam frequency standard: fifty years later Metrologia 42 (3): S31 Vanier J and Bernier L.G 1981 On the signal-to-noise ratio and short-term stability of passive rubidium frequency standards IEEE Trans Instrum Meas IM-30: 277 Vanier J., Godone A., and Levi F 1998 Coherent population trapping in cesium: dark lines and coherent microwave emission Phys Rev A 58: 234 Vanier J., Godone A., and Levi F 1999 Coherent microwave emission in coherent population trapping: origin of the energy and of the quadratic light shift In Proceedings of the Joint Meeting of the European Forum on Time and Frequency/IEEE International Frequency Control Symposium, IEEE, 96 Vanier J., Kunski R., Brisson A., and Paulin P 1981 Progress and prospects in rubidium frequency standards J Phys Coll 42 (C8), Suppl 12: 139 Vanier J., Kunski R., Cyr N., Savard J.Y., and Têtu M 1982a On hyperfine frequency shifts caused by buffer gases: application to the optically pumped passive rubidium frequency standard J Appl Phys 53: 5387 Vanier J., Kunski R., Paulin P., Têtu M., and Cyr N 1982b On the light shift in optical pumping of rubidium 87: the techniques of “separated” and “integrated” hyperfine filtering Can J Phys 60: 1396 Vanier J., Levine M., Janssen D., and Delaney M 2001 Coherent population trapping and intensity optical pumping: on their use in atomic frequency standards In Proceedings of the 6th Symposium on Frequency Standards and Metrology, P Gil Ed (World Scientific, London) 155 Vanier J., Levine M., Janssen D., and Delaney M 2003a Contrast and linewidth of the coherent population trapping transmission hyperfine resonance line in 87Rb: effect of optical pumping Phys Rev A 67: 065801 Vanier J., Levine M., Janssen D., and Delaney M 2003b On the use of intensity optical pumping and coherent population trapping techniques in the implementation of atomic frequency standards IEEE Trans Instrum Meas 52: 822 References 445 Vanier J., Levine M., Kendig S., Janssen D., Everson C., and Delaney M 2004 Practical realization of a passive coherent population trapping frequency standard In Proceedings of the IEEE International Ultrasonics, Ferroelectrics, and Frequency Control Joint 50th Anniversary Conference, IEEE, 92 Vanier J., Levine M., Kendig S., Janssen D., Everson C., and Delaney M 2005 Practical realization of a passive coherent population trapping frequency standard IEEE Trans Instrum Meas 54 (6): 2531 Vanier J., Simard J.F., and Boulanger J.S 1974 Relaxation and frequency shifts in the ground state of Rb85 Phys Rev A 9: 1031 Vanier J and Strumia F 1976 Theory of the optically pumped cesium maser Can J Phys 54: 2355 Vanier J., Têtu M., and Bernier L.G 1979 Transfer of frequency stability from an atomic frequency reference to a quartz-crystal oscillator IEEE Trans Instrum Meas 28: 188 Vian C., Rosenbusch P., Marion H., Bize S., Cacciapuoti L., Zhang S., Abgrall M., Chambon D., Maksimovic I., Laurent Ph., Santarelli G., Clairon A., Luiten A., Tobar M., and Salomon C 2005 BNM-SYRTE fountains: recent results IEEE Trans Instrum Meas 54 (2): 833 Warrington R.B., Fisk P.T.H., Wouters M.J., and Lawn M.A 2002 A microwave frequency standard based on laser-cooled 171Yb+ ions In Proceedings of the 6th Symposium Frequency Standards and Metrology, P Gil Ed (World Scientific, London) 297 Weyers R.S.S and Wynands R 2006 Effects of microwave leakage in caesium clocks: theoretical and experimental results In Proceedings of the European Forum on Time and Frequency 173 Weyers S., Gerginov V., Nemitz N., Li R., and Gibble K 2012 Distributed cavity phase frequency shifts of the caesium fountain PTB-CSF2 Metrologia 49: 82 Weyers S., Lipphardt B., and Schnatz H 2009 Reaching the quantum limit in a fountain clock using a microwave oscillator phase locked to an ultra stable laser Phys Rev A 79: 031803(R) Wineland D.J and Itano W.M 1981 Spectroscopy of a single Mg+ ion Phys Lett A 82: 75 Wolf P., Bize S., Clairon A., Landragin A., Laurent Ph., Lemonde P., and Borde Ch.J 2001 Recoil effects in microwave atomic frequency standards: an update In Proceedings of the 6th Symposium on Frequency Standards and Metrology, P Gil Ed (World Scientific, London) 593 Wynands R and Weyers S 2005 Atomic fountain clocks Metrologia 42: S64 Xiao Y., Novikova I., Phillips D.F., and Walsworth R.L 2006 Diffusion-induced Ramsey narrowing Phys Rev Lett 96: 043601 Yamagushi S., Matsuda I., and Suzuki M 1992 Dependence of the frequency shift of the optical-microwave double resonance signal in the Cs D2 line on the pumping frequency and power of a GaAs semiconductor laser IEEE J Quantum Electron 28: 2551 Zanon T., Guerandel S., de Clercq E., Dimarcq N., and Clairon A 2003 Coherent population trapping with cold atoms In Proceedings of the Joint IEEE International Frequency Control Symposium/European Forum on Time and Frequency, IEEE, 49 Zanon T., Guerandel S., de Clercq E., Holleville D., Dimarcq N., and Clairon A 2005a High contrast Ramsey fringes with coherent-population-trapping pulses in a double lambdaatomic system Phys Rev Lett 94: 193002–1 Zanon T., Tremine S., Guerandel S., de Clercq E., Holleville D., Dimarcq N., and Clairon A 2005b Observation of Raman-Ramsey fringes with optical CPT pulses IEEE Trans Instrum Meas 54: 776 Zhang J.W., Wang S.G., Miao K., Wang Z.B., and Wang L.J 2014 Toward a transportable microwave frequency standard based on laser cooled 113Cd+ ions Appl Phys B 114 (1–2): 183 Zhang J.W., Wang Z.B., Wang S.G., Miao K., Wang B., and Wang L.J 2012 High-resolution laser microwave double-resonance spectroscopy of hyperfine splitting of trapped 113Cd+ and 111Cd+ ions Phys Rev A 86 (2): 022523 446 References Zhang S 2004 Déplacement de Fréquence du au Rayonnement du corps noir dans une Fontaine Atomique a Césium et Amélioration des Performances de l’horloge, PhD dissertation, Department of Physics, Universite Pierre et Marie Curie, Paris VI, France Zhu, M and Cutler L 2000 Theoretical and experimental study of light shift in a CPT-based Rb vapor cell frequency standard In Proceedings of the Annual Precise Time and Time Interval Systems and Applications Meeting, ION Publication, 311 Zhu M., Cutler L.S., Berberian J.E., DeNatale J.F., Stupar P.A., and Tsai C 2004 Narrow linewidth CPT signal in small vapor cells for chip scale atomic clocks In Proceedings of the IEEE International Ultrasonics, Ferroelectrics, and Frequency Control Joint 50th Anniversary Conference, IEEE, 100 CHAPTER Akatsuka T., Takamoto M., and Katori H 2010 Three-dimensional optical lattice clock with bosonic 88Sr atoms Phys Rev A 81: 023402 Angel J.R.P and Sandars P.G.H 1968 The hyperfine structure Stark effect I Theory Proc Royal Soc A 305: 125 Arbes F., Gudjons T., Kurth F., Werth G., Marin F., and Inguscion M 1993 Lifetime measurements of the 3D3/2 and 3D5/2 metastable states in Ca II Z Phys D: Atoms Mol Clusters 25: 295 Arora B., Safronova M.S., and Clark Ch.W 2007 Blackbody radiation shift in a 43Ca+ ion optical frequency standard Phys Rev A 76: 064501 Baillard X., Fouch M., Le Targat R., Westergaard P.G., Lecallier A., Le Coq Y., Rovera G.D., Bize S., and Lemonde P 2007 Accuracy evaluation of an optical lattice clock with bosonic atoms Opt Lett 32: 1812 Barber Z.W., Hoyt C.W., Oates C.W., Hollberg L., Taichenachev A.V., and Yudin V.I 2006 Direct excitation of the forbidden clock transition in neutral 174Yb atoms confined to an optical lattice Phys Rev Lett 96: 083002 Barber Z.W., Stalnaker J.E., Lemke N.D., Poli N., Oates C.W., Fortier T.M., Diddams S.A., Hollberg L., and Hoyt C.W 2008 Optical lattice induced light shifts in an Yb atomic clock Phys Rev Lett 100: 103002 Barwood G.P., Gao K., Gill P., Huang G., and Klein H.A 2001 Development of optical frequency standards based upon the 2S1/2 – 2D5/2 Transition in 88Sr+ and 87Sr+ IEEE Trans Instrum Meas 50: 543 Barwood G.P., Gao K., Gill P., Huang G., and Klein H.A 2003 Observation of the hyperfine structure of the 2S1/2 – 2D5/2 transition in 87Sr+ Phys Rev A 67: 013402 Barwood G.P., Gill P., Huang G., and Klein H.A 2012 Characterization of a 88Sr+ optical frequency standard at 445 THz by two-trap comparison In Proceedings of the Conference Precision Electromagnetic Measurements, IEEE, 270 Bergquist J.C., Itano W.M., and Wineland D.J 1987 Recoilless optical absorption and Doppler sidebands of a single trapped ion Phys Rev A 36: 428 Bergquist J.C., Wineland D.J., Itano W.M., Hemmati H., Daniel H.-U., and Leuchs G 1985 Energy and radiative lifetime of the 6d9 6s2 2D5/2 state in Hg II by Doppelr-free two photon laser spectroscopy Phys Rev Lett 55 (15): 1567 Berkeland D.J., Cruz F., and Bergquist J C 1997 Sum-frequency generation of continuouswave light at 194 nm Appl Opt 36: 4159 Berkeland D.J., Miller J.D., Bergquist J.C., Itano W.M., and Wineland D.J 1998 Minimization of ion micromotion in a Paul trap J Appl Phys 83 (10): 5025 Bernard J.E., Marmet L., and Madej A.A 1998 A laser frequency lock referenced to a single trapped ion Opt Commun 150: 170 Bigeon M 1967 Probabilite de transition de la raie 1S0 – 6 3P0 du mercure J Phys 28: 51 References 447 Binnewies T., Wilpers G., Sterr U., Riehle F., Helmcke J., Mehlstäubler T.E., Rasel E.M., and Ertmer W 2001 Doppler cooling and trapping on forbidden transitions Phys Rev Lett 87: 123002 Blythe P.J 2004 Optical Frequency Measurement and Ground State Cooling of Single Trapped Yb+ Ions PhD Thesis, University of London Blythe P.J., Webster S.A., Hosaka K., and Gill P 2003 Systematic frequency shifts of the 467 nm electric octupole transition in 171Yb+ J Phys B: At Mol Opt Phys 36: 981 Bloom B.J., Nicholson T.L., Williams J.R., Campbell S.L., Bishof M., Zhang X., Zhang W., Bromley S.L., and Ye J 2014 An optical lattice clock with accuracy and stability at the 10–18 level Nature 506: 71 Boshier M.G., Barwood G.P., Huang G., and Klein H.A 2000 Polarisation-dependent optical pumping for interrogation of magnetic-field-independent “clock” transition in laser cooled trapped 87Sr+ Appl Phys B: Lasers Opt 71: 51 Boyd M.M., Zelevinsky T., Ludlow A.D., Blatt S., Zanon-Willette T., Foreman S.M., and Ye J 2007 Nuclear spin effects in optical lattice clocks Phys Rev A 76: 022510 Boyd M.M., Zelevinsky T., Ludlow A.D., Foreman S.M., Blatt S., Ido T., and Ye J 2006 Optical atomic coherence at the 1-second time scale Science 314: 1430 Castin Y., Wallis H., and Dalibard J 1989 Limit of Doppler cooling J Opt Soc Am B (11): 2047 Champenois C., Houssin M., Lisowski C., Knoop M., Hagel G., Vedel M., and Vedel F 2004 Evaluation of the ultimate performances of a Ca+ single-ion frequency standard Phys Rev A 33: 298 Champenois C., Knoop M., Herbane M., Houssin M., Kaing T., Vedel M., and Vedel F 2001 Characterization of a miniature Paul-Straubel trap Eur Phys J D 15: 105 Chou C.W., Hume D.B., Koelemeij J.C.J., Wineland D.J., and Rosenband T 2010 Frequency comparison of two high-accuracy Al+ optical clocks Phys Rev Lett 104: 070802 Chwalla M 2009 Precision spectroscopy with 40Ca+ ions in a Paul trap PhD Thesis, LeopoldFranzens-Universitat, Innsbruck, Austria Chwalla M., Benhelm J., and Kim K 2009 Absolute frequency measurement of the 40Ca+ 4s2S1/2 – 3d2D5/2 clock transition Phys Rev Lett 102: 023002 Chwalla M., Benhelm J., Kim K., Kirchmair G., Monz T., Riebe M., Schindler P., Villar A.S., Hansel W., Roos C.F., Blatt R., Abgrall M., Santarelli G., Rovera G.D., and Laurent Ph 2008 Absolute frequency measurement of the 40Ca+ 4s 2S1/2 – 3d 2D5/2 clock transition arXiv:0806.1414v1(physics.atom-ph) Courtillot I., Quessada A., Kovacich R.P., Brusch A., Kolker D., Zondy J.J., Rovera G.D., and Lemonde P 2003 A clock transition for a future optical frequency standard with trapped atoms arXiv:physiscs/0303023 Curtis E.A., Oates C.W., and Hollberg L 2001 Quenched narrow-line laser cooling of 40Ca to near the photon recoil limit Phys Rev A 64: 031403(R) Degenhardt C., Stoehr H., Lisdat C., Wilpers G., Schnatz H., Lipphardt B., Nazarova T., Pottie P.E., Sterr U., Helmcke J., and Riehle F 2005 Calcium optical frequency standard with ultracold atoms: approaching 10−15 relative uncertainty Phys Rev A 72: 062111 Dehmelt H 1973 Proposed 1014 Dν