Rydberg Atoms in an Oscillating Field: Extracting Classical Motion from Quantum Spectra by Neal W
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Rydberg Atoms in an Oscillating Field: Extracting Classical Motion from Quantum Spectra by Neal W. Spellmeyer B.S., University of Wisconsin, Madison (1992) 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 1998 © Massachusetts Institute of Technology 1998. All rights reserved. Signature of Author Department ofPhysics December 5, 1997 Certified by Aaniel Kleppner Lester Wolfe Professor of Physics Thesis Supervisor Accepted by George F. Koster Professor of Physics Chairman, Departmental Committee on Graduate Students Rydberg Atoms in an Oscillating Field: Extracting Classical Motion from Quantum Spectra by Neal W. Spellmeyer Submitted to the Department of Physics on December 5, 1997, in partial fulfillment of the requirements for the degree of Doctor of Philosophy Abstract We present an experimental and theoretical investigation of the connections between classical and quantum descriptions of Rydberg atoms in external electric fields. The technique of recurrence spectroscopy, in which quantum spectra are measured in a manner that maintains constant classical scaling laws, reveals the actions of the closed orbits in the corresponding classical system. We have extended this technique with the addition of a weak oscillating electric field. The effect of this perturbing field is to systematically weaken recurrences in a manner that reveals the ac dipole moments of the unperturbed orbits, at the frequency of the applied field. We outline a version of closed orbit theory developed to describe these experiments, and show that it is in good agreement with the measurements. The experiments also show good agreement with semiquantal Floquet computations. We describe a computational method that diagonalizes a Floquet Hamiltonian to compute spectra of both hydrogen and lithium in oscillating fields. We use this new type of recurrence spectroscopy to measure classical trajectories of two closed orbits in the lithium Stark system. We extract the classical trajectories by evaluating the inverse Fourier transform of the ac dipole moment measured over some range of frequencies. Within the resolution of the experiment, the measured trajectories show excellent agreement with computed classical trajectories. Thesis Supervisor: Daniel Kleppner Lester Wolfe Professor of Physics To my parents, Robert and Marjorie. Contents 1 Introduction 17 1.1 The Semiclassical Limit ......................... 17 1.1.1 WKB and EBK Quantization .................. 19 1.1.2 Periodic Orbit Theory ...................... 20 1.2 Rydberg Atoms .... ............... .......... 22 1.2.1 Wave-Packet Experiments . .................. 23 1.3 Other Time-Dependent Systems ................... .. 24 1.4 Outline of the Thesis ........................... 25 2 Semiclassical Mechanics of Rydberg Atoms in Oscillating Fields 27 2.1 Hamiltonian and Classical Motion . .................. 27 2.2 Closed Orbit Theory for a Time-Independent System . ........ 30 2.3 Closed Orbit Theory for a Time-Dependent System . ......... 34 2.3.1 Behavior at Small Oscillating Field Amplitude ........ 37 2.4 The Parallel Orbit ............................ 40 2.5 Bifurcations ................................ 44 2.6 Chaos .......... .. .. 48 3 Quantum Mechanics of Rydberg Atoms in Oscillating Fields 51 3.1 Stark Effect ...... ........ ....... ....... .... 51 3.2 Ionization . .. .. 52 3.2.1 Static Field Ionization ...................... 53 3.2.2 Microwave Ionization ....................... 53 3.3 Stark Structure in an Oscillating Field . .......... 55 3.4 The Floquet Method ...................... 57 3.4.1 Numerical Results ................... 60 3.4.2 Stark Structure in a Perpendicular Oscillating Field . 62 3.4.3 Recurrence Spectra in Oscillating Fields ....... 66 4 Experimental Method 69 4.1 Lasers ............... ....... .. 71 4.1.1 Red Laser ......... ....... ... 72 4.1.2 Diode Laser ........ .......... 72 4.1.3 Yellow Laser ........ .......... 73 4.2 Locking .............. ... ...... 73 4.3 Atomic Beam Source ....... ....... ... 76 4.4 Stepwise Excitation Efficiency .. S . 76 4.5 Interaction Region ........ .......... 81 4.6 Excitation of Rydberg States ... S . 82 4.7 RF System ............. .......... 84 4.8 RF Field Amplitude Calibration. S . 86 4.8.1 Error Analysis....... .......... 92 4.9 Static Field Calibration ...... S. .... 93 4.10 Recurrence Spectroscopy ..... ......... 95 4.10.1 Error Analysis....... .......... 96 5 Recurrence Spectroscopy in a Static Electric Field 99 5.1 Recurrence Spectra, E = -0.4 ...................... 100 5.1.1 Experimental Reproducibility ... ....... ....... 101 5.1.2 Comparison with Computation and Theory ...... .... 103 5.2 Bifurcation Structure and the Measurement of Orbital Periods . .. 105 6 Recurrence Spectroscopy in an Oscillating Field 115 6.1 Confrontation Between Experiment and Theory at E = -0.4 ..... 115 6.1.1 Semiquantal Floquet Computations ... ... ... .. ... 120 6.1.2 A Test of Closed Orbit Theory . .... ... .... ... 122 6.1.3 Detailed Closed Orbit Theory ... .... ..... .. ... 125 6.1.4 Untransformed Spectra .................. ... 132 6.1.5 High-Frequency Behavior of AC Dipole Moment . ... ... 135 6.2 Recurrence Spectra at E = -0.5 ..... ..... ..... .. ... 137 6.3 Computational Results, E = -6 ......... ........ ... 138 6.3.1 Static Field Recurrence Spectra .... .... ..... ... 139 6.3.2 Oscillating Field Recurrence Spectra ... ...... ... 140 6.3.3 Interaction of Core-Scattered Orbits with an Oscillating Field 144 6.4 Computational Results, c - -4 ................. .... 145 6.4.1 Static Field Recurrence Spectra .... ......... .... 145 6.4.2 Oscillating Field Recurrence Spectra ..... ..... .... 148 7 Quantum Measurement of a Classical Trajectory 151 7.1 Experimental Measurements .. ... 151 7.2 Determining the Phase . ... ... 156 7.3 Extracting the Trajectory . ... .. 157 7.4 Resolution Limit of the Measurement 162 7.5 Discussion ..... .... .... .. 167 8 Conclusion 169 A Scaling Properties of Assorted Systems 173 A.1 Hydrogen in a Static Electric Field . ................. 173 A.2 Harmonic Oscillator ........................... 174 A.3 Charged Particle Bound by a Coulomb Potential ........... 176 A.4 Periodic Perturbations to Simple Scaling Systems . ......... 177 B Some Results for a Perpendicular Field 179 C Atomic Units 183 D Paper: Recurrence Spectroscopy of a Time-Dependent System: A Rydberg Atom in an Oscillating Field 185 Bibliography 191 Acknowledgments 201 List of Figures 2-1 Scaled action of the parallel orbits U and D .... ...... ... 2-2 Period of the parallel orbits U1 and D ...... ..... ...... 2-3 Classical action of the uphill orbit U2, E = -0.4, C = 0.32, f = 0.001 2-4 Action of the parallel orbit U5 as a function of f, phase, and & at E = - 0.4 . ................................ 2-5 Actions of orbits vs. scaled energy ..... ........ ....... 2-6 Allowed winding numbers of closed orbits . ....... ........ 2-7 Surfaces of section in an oscillating field, E = -0.4, CD = 0.32 .... 3-1 Computed spectra: Li, n = 31, F = 23.14 V/cm, v = 580 MHz . .. 3-2 Computed spectrum: H, n = 10, F = 3000 V/cm, v = 15 GHz . .. 3-3 Computed spectra: H, n = 10, F = 3000 V/cm, v = 15 GHz .... 3-4 Computed spectra: H, n = 10, F = 3000 V/cm, v = 15 GHz ..... 4-1 Overview of the experiment ... .... .... ... ....... 70 4-2 Lithium level structure ................. ........ 71 4-3 Schematic of the locking region .. ...... .... ....... 75 4-4 Three-level system ................... ........ 77 4-5 Computed 3s population ................ ........ 79 4-6 Experimental measurement of relative 3s population. ....... 80 4-7 Schematic of the interaction region .. ........ ....... 81 4-8 Experimental spectrum near n = 100 ... ...... ....... 82 4-9 Experimental spectrum near n = 170 ......... ....... 83 4-10 Experimental spectrum near n = 220 ........ ....... 84 4-11 Schematic of the rf system ............... ....... .. 85 4-12 Computed energy levels, Li n = 31 Stark manifold . .......... 87 4-13 Dependence of the rf sideband amplitude on x = kFi/hw ....... 88 4-14 Quantum Floquet computation and experiment for n = 31 Stark man- ifold in an oscillating field, F = 23.6 V/cm, v = 300 MHz ...... 89 4-15 Calibration of rf field amplitude ........... .. 91 4-16 Amplitude of rf field vs. input power . ............ .... 92 4-17 n = 31 Stark manifold, F = 9.409 V/cm . .......... .... 94 4-18 Fit to F vs. V calibration data ..................... 94 4-19 Frequency scaling law .......................... 98 4-20 Energy scaling law ................. ........... 98 5-1 Untransformed experimental scaled spectrum, E = -0.4 ....... 100 5-2 Experimental recurrence spectrum, E = -0.4 . ............. 101 5-3 Averaged recurrence spectrum, E = -0.4 . ............... 102 5-4 Averaged recurrence spectrum and semiquantal computation, E = -0.4 103 5-5 Averaged recurrence spectrum and closed orbit theory, E = -0.4 .. 104 5-6 Experimental recurrence spectra near E = -2 .......... ... 106 5-7 Computed recurrence spectra near E = -2 . .............. 108 5-8 Experimental and computed spectra near E = -2 . .......... 109 5-9 Action of the 2/3 orbit: measurement and theory . .......... 111 5-10 Action of the 3/4 orbit: measurement and theory . .......... 112 6-1 Experimental recurrence spectra, E = -0.4, f = 0.005 . ........ 117 6-2 Experimental recurrence spectra, e = -0.4, f = 0.008 . ........ 118 6-3 Experimental recurrence spectra, E= -0.4, o = 0.32 . ........ 119 6-4 Semiquantal Floquet computation,