Rydberg Atoms in Strong Fields: A Testing Ground for Quantum Chaos by Michael Courtney B.S., summa cum laude, Louisiana State University, (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 1995 () Massachusetts Institute of Technology 1995. All rights reserved. Signat.ire of Author - - A - _ _ _ /-1 . Department of Physics November 10, 1994 . I - I ' Certified by. - -%=. f; - - I Daniel Kleppner Lester Wolfe Professor of Physics Thesis Supervisor C Accepted by _ ... , George F. Koster Professor of Physics Chairman, Departmental Committee on Graduate Students Science MtSSCHtU.e;Ti !',,NSTITI i T M.R 02 1995 Rydberg Atoms in Strong Fields: A Testing Ground for Quantum Chaos by Michael Courtney Submitted to the Department of Physics on November 10, 1994, in partial fulfillment of the requirements for the degree of Doctor of Philosophy Abstract Rydberg atoms in strong static electric and magnetic fields provide experimentally accessible systems for studying the connections between classical chaos and quantum mechanics in the semiclassical limit. This experimental accessibility has motivated the development of reliable quantum mechanical solutions. This thesis uses both ex- perimental and computed quantum spectra to test the central approaches to quantum chaos. These central approaches consist mainly of developing methods to compute the spectra of quantum systems in non-perturbative regimes, correlating statistical descriptions of eigenvalues with the classical behavior of the same Hamiltonian, and the development of semiclassical methods such as periodic-orbit theory. Particular emphasis is given to identifying the spectral signature of recurrences-quantum wave packets which follow classical orbits. The new findings include: the breakdown of the connection between energy-level statistics and classical chaos in odd-parity dia- magnetic lithium, the discovery of the signature of very long period orbits in atomic spectra, quantitative evidence for the scattering of recurrences by the alkali-metal core, quantitative description of the behavior of recurrences near bifurcations, and a semiclassical interpretation of the evolution of continuum Stark spectra. Thesis Supervisor: Daniel Kleppner Lester Wolfe Professor of Physics 10 25 5 20 0 15 -5 10 5 -10 0 -15 2 -2 10 5 0 5 .5 0 -10 -15 -5 3 -2 Jl,,,,,,,,,,IIIIIJII_I ___I J_ITrTJf#J .f##~ 1off tttss If f f f ff //////// 1 I11I J,......... 11 70 20 - 60 15 50 10 s 40 5 BN O W , 30 0- iocwwb6iWbsW; 'I 20 -5 10 -10 0 -15 - -10 K _ . 1...A d -- -2 -'--..~~. 0 10 -1 rho C. Figure 0-1: Potential surfaces for Rydberg atoms in strong fields. (a), (c), and (e) are in cylindrical coordinates. (b), (d), and (f) are in "regularized" semiparabolic coordinates. (a) and (b) Diamagnetic hydrogen; (c) and (d) Hydrogen in an electric field; (e) and (f) Hydrogen in parallel electric and magnetic fields. 4 4 strength [arb. units] 4 3 2 1 0 0 1/2 2/3 3/4 4/5 5/6 7/8 3/5 7/9 Figure 0-2: Recurrence spectra of lithium in an electric field. Details are described in Chapter 8. The curves in the horizontal plane represent the scaled action of the parallel orbit and its repetitions as a function of scaled energy. Locations of bifur- cations are marked with small open circles. New orbits created in bifurcations have almost the same action as the corresponding return of the parallel orbit. Measured recurrence strengths are shown in the z direction. Recurrences are especially strong at scaled energies slightly lower than bifurcations. Orbits created by bifurcation of the parallel orbit are shown along the bottom. The 2/4, 4/6 and 6/8 orbits are repetitions of the 1/2, 2/3, 3/4 orbits respectively,so their shapes are identical. 5 6 Contents 1 Introduction 21 1.1 What is Quantum Chaos? .................. 23 1.1.1 Quantum Mechanics in Non-Perturbative Regimes. 23 1.1.2 Statistical Descriptions of Quantum Chaos ..... 25 1.1.3 Semiclassical Methods and Periodic-Orbit Theory . 26 1.1.4 Direct Application of the Correspondence Principle 28 1.2 Why Rydberg Atoms? .................... 28 1.3 Outline of Thesis ....................... 30 2 Quantum Mechanics of Rydberg Atoms 33 2.1 One-Electron Atoms .................. ........ 33 2.2 Atomic Units ...................... ........ 35 2.3 Methods of Computation ................ ........ 35 2.3.1 Spherical Sturmian Basis ............ ........ 36 2.3.2 Basis of Zero-Field Eigenstates ........ ........ 38 2.3.3 Matrix Diagonalization ............ ........ 39 2.4 Atoms in an Electric Field ............... ........ 40 2.4.1 Hydrogen in an Electric Field ......... ........ 40 2.4.2 Alkalis in an Electric Field ........... ........ 42 2.4.3 Ionization Processes in an Electric Field . ........ 43 2.5 Rydberg Atom Diamagnetism ............. ........ 45 2.5.1 Symmetries in Rydberg Atom Diamagnetism. ........ 45 2.5.2 Regions of Interest in Rydberg Atom Diamagnetism. 48 7 2.5.3 Diamagnetism in Alkalis .............. 48 2.5.4 Simple Features of Rydberg Atom Diamagnetism 50 2.6 Parallel Electric and Magnetic Fields 52 2.6.1 Symmetries........ ................... 54 2.6.2 Regions of Interest ..... ................... 56 2.6.3 Hydrogen in Parallel Fields ................... 58 2.6.4 Lithium in Parallel Fields ................... 60 2.6.5 Summary of Parallel Fields. 64 3 The Onset of Classical Chaos in Rydberg Atoms 67 3.1 What is Chaos? ................... ........... ....69 3.2 Symmetries and Constants of Motion ....... ........... ....71 3.3 Scaling Rules .................... ........... ....72 3.4 Core Potential for Alkalis .............. ........... ....73 3.4.1 A Simple Model ............... ........... ....73 3.4.2 A More Accurate Core Model ....... ........... ....76 3.4.3 Estimates of Core Effects .......... ........... ....76 3.5 Semiparabolic Coordinates ............ ........... ....78 3.6 Computational Results .............. 79 3.6.1 Hydrogen .................. ........... ....81 3.6.2 Lithium ................... ........... ....85 3.6.3 Heavier Alkalis .............. ........... ....95 3.7 Summary ...................... ........... ....97 4 Experimental Method 99 4.1 Atomic Beam Source ................ ........... 100 4.2 Magnetic Field .................. ........... 102 4.3 Electric Field ................... ........... 103 4.4 Excitation Scheme and Metrology ........ ........... 104 4.5 Excitation and Detection ............. ........... 106 4.6 Data Acquisition and Computer Control ..... ........... 112 8 4.7 Spectroscopy at Constant Scaled Energy ............... 112 5 Energy Level Statistics and Quantum Chaos 115 5.1 Claims and Application of Nearest-Neighbor Distribution 116 5.2 Diamagnetism. .............. 118 ... 5.2.1 Hydrogen in a Magnetic Field . 118 ... 5.2.2 Lithium in a Magnetic Field . 118 ... 5.3 Electric Field .............. 123 ... 5.3.1 Hydrogen in an Electric Field . 123 ... 5.3.2 Lithium in an Electric Field . 123 ... 5.4 Parallel Electric and Magnetic Fields . 127 ... 5.4.1 Hydrogen in Parallel Fields . 127 ... 5.4.2 Lithium in Parallel Fields .... 129 ... 5.5 Summary and Discussion ......... 131 ... 6 Periodic Orbit Spectroscopy 133 6.1 Periodic-Orbit Theory .......... 134 ... 6.2 Closed-Orbit Theory .......... 138 ... 6.2.1 Bifurcations ............ 140 ... 6.2.2 Core-Scattered Recurrences . 143 ... 6.3 Computing Recurrence Spectra .... ............ 144 6.4 Classical Scaling Rule and Alkali-Metals . 146 ... 6.5 Paradox .................. 148 ... 6.6 Effects of Finite h ............. 149 ... 7 Recurrence Spectroscopy of Diamagnetic Rydberg Atoms 151 7.1 Hydrogen ............................ .... 154 7.1.1 Onset of Chaos ..................... .... 155 7.1.2 Perpendicular Orbit in Odd-Parity Spectrum ..... .... 160 7.1.3 Nearly Circular Orbit .................. .... 163 7.1.4 Pre-Bifurcation Recurrences ............. .... 165 9 7.1.5 Very Long Period Orbits ............................ 167 7.1.6 Ionizing Orbits of Diamagnetic Hydrogen ........... 177 7.2 Lithium ................................. ...... 179 7.2.1 Core Scattering .......................... 180 7.2.2 Recurrence Proliferation as a Measure of Chaos ....... 186 8 Recurrence Spectroscopy in an Electric Field 191 8.1 Hydrogenic Closed Orbits ........................ 193 8.1.1 Hamiltonian and Separation of Variables ........... 193 8.1.2 Period Ratios and Their Relation to Closed Orbits ..... 194 8.1.3 An Illustration: The Closed Orbit With Period Ratio 9/10 . 197 8.2 Recurrence Spectra in the Continuum Regime ............ ..... 198 8.2.1 Continuum Recurrence Spectra for Small Action ....... 199 8.2.2 Behavior at a Bifurcation .................... 201 8.2.3 Comparison with Closed-Orbit Theory ............ ..... 203 8.2.4 Continuum Recurrence Spectra for Large Action ....... 205 8.3 Recurrence Spectra in the Quasidiscrete Regime . 207 8.3.1 Recurrence Spectrum at High Action .............. 207 8.3.2 Recurrence Spectrum at Low Action .............. 209 8.3.3 Evidence for Core Scattering . 211 8.3.4 Core Scattering and Spectral Evolution . 214 8.3.5 Recurrence Spectrum at Low Scaled Energy ......... 216 8.4 Recurrence Spectra for m = 1 ...................... 218 8.5 Which Scaled Action? . ........................ 221 8.6 Summary and Discussion .................................. 224 9 Parallel Field Recurrence Spectra 227 9.1 Hydrogen in Parallel Fields ................................ 228 9.2 Lithium in Parallel Fields ........................ 237 10 Recurrence Statistics 243