Experimental and Theoretical Aspects of Quantum Chaos in Rydberg Atoms in.Strong Fields by Hong Jiao B.S., University of California, Berkeley (1987) M.S., California Institute of Technology (1989) Submitted to the Department of Physics in partial fulfillment of the requirements for the degree of Doctor of Philosophy at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY February 1996 @ Massachusetts Institute of Technology 1996. All rights reserved. Signature of Author. Department of Physics U- ge December 4, 1995 Certified by.. Daniel Kleppner Lester Wolfe Professor of Physics Thesis Supervisor Accepted by. ;AssAGiUS. rrS INSTITU It George F. Koster OF TECHNOLOGY Professor of Physics FEB 1411996 Chairman, Departmental Committee on Graduate Students LIBRARIES 8 Experimental and Theoretical Aspects of Quantum Chaos in Rydberg Atoms in Strong Fields by Hong Jiao Submitted to the Department of Physics on December 4, 1995, in partial fulfillment of the requirements for the degree of Doctor of Philosophy Abstract We describe experimental and theoretical studies of the connection between quantum and classical dynamics centered on the Rydberg atom in strong fields, a disorderly system. Primary emphasis is on systems with three degrees of freedom and also the continuum behavior of systems with two degrees of freedom. Topics include theoret- ical studies of classical chaotic ionization, experimental observation of bifurcations of classical periodic orbits in Rydberg atoms in parallel electric and magnetic fields, analysis of classical ionization and semiclassical recurrence spectra of the diamagnetic Rydberg atom in the positive energy region, and a statistical analysis of quantum manifestation of electric field induced chaos in Rydberg atoms in crossed electric and magnetic fields. Thesis Supervisor: Daniel Kleppner Lester Wolfe Professor of Physics Contents 1 Introduction 1.1 Preface . .. .................... 1.1.1 Rydberg Atom in Strong Fields .... ........ 1.1.2 Theories of Quant um Chaos .............. 1.1.3 The Goal of the T hesistrong .......Fields ... ........... ... 1.2 Background of Atoms in S.esis......... .......... 1.2.1 The Hamiltonian ......trong Fields.............. 1.2.2 Rydberg Atoms 1.2.3 Symmetries . .. .................... 1.3 A Brief History ..... .. ................. 1.3.1 Early Progress in Diamagnetic Rydberg Atoms ... 1.3.2 Closed-Orbit Theory and Scaled-Energy Spectroscopy 1.3.3 Spectroscopy on Lithium ................ 1.3.4 Numerical Advanc es .. ... ...... .. 1.4 Overview of the Experiment .................. 1.5 Outline of the Thesis .. ... .............. 2 Experimental Techniques 33 2.1 Atom ic Beam ............................... 35 5 CONTENTS 2.1.1 Atomic Beam Source . ............... .... 35 2.1.2 Doppler Broadening and Transit Time Linewidth . .. 37 2.2 Lasers and Optics....... ............... ..... 38 2.3 The Magnet .......... ............... .. ... 41 2.3.1 Magnetic Field Profile ............... .. ... 42 2.3.2 Field Monitoring . ............... ..... 42 2.4 Interaction Region ....... ............... ..... 46 2.4.1 Fluorescence Detection .... 46 2.4.2 Electric Field Plates . ............... ..... 48 2.4.3 Stray Electric Field. ..... 49 .... ,........... 2.5 Detection of Rydberg Atoms . ..... 51 2.5.1 Field Ionization .... ............... ..... 51 2.5.2 The Detector ..... ............... ..... 53 2.6 High-Resolution Spectroscopy ............... S . 60 2.6.1 Field Calibration . ... .. 60 N 2.6.2 Conventional Lithium )pectrumI S . 66 2.6.3 Scaled-Energy Spectroscopy 3 Stepwise Excitation Scheme 75 3.1 Two-Photon Transition vs. Stepwise Excitation . .... 76 3.2 Fine and Hyperfine Structure ........ ....... 79 3.2.1 The Hamiltonians... ............... 80 3.2.2 Measured Values .................. 82 3.2.3 Isotope Shift ..................... 88 3.3 Fine and Hyperfine Structure in a Magnetic Field . ... 89 3.3.1 The Hamiltonian .................. 89 3.3.2 2S, 2P, and 3S Energy Levels in a Magnetic Field 90 CONTENTS 3.3.3 Electric Dipole Transitions in the High Field Regime ... .. 94 3.4 Experimental Realization ......................... 97 3.4.1 Laser Operation .......................... 97 3.4.2 Optics ... ... ... ... .. .. ... ... .. .. .. 101 3.4.3 Monitoring the Stepwise Excitation ... ... .... ... .. 106 4 Classical Chaos 111 4.1 Integrable Hamiltonians ......................... 112 4.1.1 Hamilton Equations of Motion . ... ... ... .. ... ... 112 4.1.2 Hydrogen Atom in a Uniform Electric Field ... ... ... 114 4.1.3 Surface of Section ......................... 116 4.2 Canonical Perturbation Theory and Classical Chaos ... .. ... .. 119 4.2.1 One Degree of Freedom ...................... 119 4.2.2 Many Degrees of Freedom .................... 123 4.2.3 KAM Theorem .......................... 124 4.3 The Diamagnetic Hydrogen Atom . .... ... ... ... ... ... 126 4.3.1 Surface of Section ......................... 128 4.3.2 An Approximate Constant of Motion . .... ... ... ... 135 4.3.3 A Brief Remark .......................... 136 4.4 Chaos in Open Systems ......................... 136 5 A Semiclassical Method 143 5.1 Semiclassical Quantization ........................ 144 5.1.1 WKB Expansion and Bohr-Sommerfeld Quantization .. ... 144 5.1.2 EBK Quantization ........................ 146 5.2 Periodic-Orbit Theory .......................... 147 5.2.1 Background ............................ 147 5.2.2 The Trace Formula ........................ 149 CONTENTS 5.3 Closed-Orbit Theory ........................... 150 5.3.1 Basic Formulation............................. 151 5.3.2 Technique of Scaled-Energy Spectroscopy .. ..... ..... 152 6 Rydberg Atoms in Parallel Fields 155 6.1 Qualitative Features ........................... 156 6.2 Classical Ionization .. .. .. ...... ....... ... ...... .157 6.2.1 Classical Ionization Time ... ...... ...... ...... 160 6.2.2 Relationship to Closed Orbits ...... ...... ...... 161 6.3 Experimental Results ........................... 165 6.3.1 Recurrence Spectroscopy . ........ ........ .... 165 6.3.2 Bifurcations ............................ 166 6.3.3 Numerical Results ........................ 171 6.4 Sum m ary ................................. 171 7 Diamagnetic Rydberg Atoms 175 7.1 Classical Description ........................... 176 7.2 Semiclassical Recurrence Spectra ............ ........ 181 7.3 Summary and Discussion ......................... 183 8 Rydberg Atoms in Crossed Fields 187 8.1 The Hamiltonian ............................. 189 8.2 Classical Dynamics ............................ 190 8.2.1 Surface of Section ......................... 191 8.2.2 Lyapunov Exponents ....................... 195 8.2.3 Arnold Diffusion ......................... 197 8.3 Nearest-Neighbor Spacings Distribution ................. 199 8.3.1 Regular Region .......................... 200 CO0NTE~NTS 8.3.2 Chaotic Region .......................... 200 8.3.3 Transition Region ......................... 201 8.4 Quantum Computations ......................... 202 8.5 R esults .. .. ... .. ... .. ... .. ... .. ... .. ... .. 203 8.6 Sum m ary ................................. 205 9 Conclusion 207 APPENDICES A Magnet Power Supply 209 B Cryogenic Considerations 213 C Magnetic Field of a Finite Solenoid 217 D Other Excitation Schemes for Lithium 219 D.1 2S - 3D - Rydberg ........................... 219 D.2 2S 2P - 3D -+ Rydberg ....................... 221 D.3 2S -4 2P - Rydberg ........................... 223 D.4 2S -+ 3P - Rydberg ........................... 224 E Clebsch-Gordon Coefficients for the 2P States 225 F An Alternative Pumping Scheme 227 G Hamilton Equations of Motion 231 H Lyapunov Exponent 235 Bibliography 10 CONTENTS List of Figures 1-1 Excitation scheme . .. .. .. .. 2-1 Schematic diagram of the apparatus . .. .. .. .. .. .. 34 2-2 Field of a split-coil magnet ........................ 43 2-3 Calculation and measurement of the total field . .. .. .. .. 44 2-4 Hall probe calibration against magnetic field . .. .. .. .. 45 2-5 Interaction region ............................. 47 2-6 Stray electric field measurement of lithium at n=76 . .. .. .. .. 50 2-7 Gain of the MCP in a magnetic field . .. .. .. .. .. .. .. 55 2-8 Lithium ion in the finging magnetic field . .. .. .. .. .. .. 56 2-9 Biasing circuit for electron detection .. .. .. .. .. .. .. .. 58 2-10 Detector setup ............................... 59 2-11 Energy levels of lithium at n = 21 .. .. .. .. .. .. .. .. .. 61 2-12 Experimental measurement of the magnetic field . .. .. .. .. .. 62 2-13 Energy levels of m = -1 lithium at n = 31 . .. .. .. .. .. .. 64 2-14 Energy levels of lowest lying states of m = -1 lithium at n = 31 . 65 2-15 Electric field to voltage ratio calibration . .. .. .. .. .. .. 66 2-16 Experimental recurrence spectrum of diamagnetic lithium at E= -0.2 2-17 Experimental recurrence spectrum of diamagnetic lithium at f = -0.1 3-1 New excitation scheme .......................... LIST OF FIGURES 3-2 Principal transitions of 7Li and 6Li . .. ... ... ... 84 3-3 Experimental scan of the 2S --+ 2P transition . ... .. ... 85 3-4 2S - 2P 3/2 transition ................. .. 85 3-5 2S --+ 2P1/ 2 transition ................ .. 86 3-6 Fine and hyperfine splittings of 7Li . ... .. ... ... 87 3-7 Fine and hyperfine states of 2P in a magnetic field .. ... 90 3-8 Hyperfine states of 2P1/ 2 in a magnetic field . .. .. ... 92 3-9 Hyperfine states of 2P 3/2 in a magnetic field . .. .. ... 92 3-10 Hyperfine states of 2S in a magnetic field . ... .. ... 93 3-11 Hyperfine states of 3S in a magnetic field . ... .. ... 93 3-12 Optical layout ....................