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Abstract Exploring Quantum Chaos in a Spin 1/2 Atom ABSTRACT EXPLORING QUANTUM CHAOS IN A SPIN 1/2 ATOM DRIVEN BY A 3D CHAOTIC MAGNETIC FIELD by Jared Goettemoeller Quantum chaos is a field of research that attempts to introduce chaos theory into quantum mechanics. There has already been a significant body of work on classically chaotic systems viewed through a quantum lens, so we looked at a spin 1/2 atom driven by a 3d chaotic magnetic field. We used mathematical tools such as Lyapunov exponents, Bloch spheres, and correlation functions to analyze the system. We found some mixed results and that it is likely more interesting to use a spin 1 atom. EXPLORING QUANTUM CHAOS IN A SPIN 1/2 ATOM DRIVEN BY A 3D CHAOTIC MAGNETIC FIELD A Thesis Submitted to the Faculty of Miami University in partial fulfillment of the requirements for the degree of Master of Science Department of Physics by Jared Christopher Goettemoeller Miami University Oxford, Ohio 2017 Advisor: Perry Rice Reader: Leno Pedrotti Reader: Samir Bali Reader: Jennifer Blue @2017 Jared Christopher Goettemoeller This select thesis titled EXPLORING QUANTUM CHAOS IN A SPIN 1/2 ATOM DRIVEN BY A 3D CHAOTIC MAGNETIC FIELD by Jared Christopher Goettemoeller has been approved for publication by College of Arts and Sciences and Department of Physics Perry Rice Leno Pedrotti Samir Bali Jennifer Blue Contents Acknowledgements xiii Introduction1 1 Quantum Chaos3 1.1 Lie Groups and Algebra.............................3 1.2 Semiclassical Chaos................................5 1.3 Quantum Chaos through a Hamiltonian.................... 10 2 Chaotic Magnetic Fields 11 2.1 Lorenz System................................... 11 2.2 Lindblad Master Equation............................ 16 3 Methodology 19 3.1 Test Parameters.................................. 19 3.2 Analytical Tools.................................. 19 3.2.1 Lyapunov................................. 20 3.2.2 Bloch Sphere............................... 20 3.2.3 Correlations................................ 21 4 Results 23 4.1 Lyapunovs for σi(t)................................ 24 4.1.1 No Collapse Operator.......................... 24 4.1.2 Collapse Operator............................. 25 iii 4.2 Bloch Sphere................................... 27 4.2.1 No Collapse Operator.......................... 27 4.2.2 Collapse Operator............................. 28 4.2.3 Longer Time............................... 30 4.3 Correlations.................................... 31 4.3.1 No Collapse Operator.......................... 31 4.3.2 Collapse Operator............................. 40 4.3.3 More Time Steps............................. 49 4.4 Lyapunovs for σi((t) With a Random Hamiltonian............... 50 4.4.1 No Collapse Operator.......................... 50 4.4.2 Collapse Operator............................. 51 4.5 Bloch Sphere With a Random Hamiltonian................... 52 4.5.1 No Collapse Operator.......................... 52 4.5.2 Collapse Operator............................. 53 4.6 Correlations With a Random Hamiltonian................... 54 4.6.1 No Collapse Operator.......................... 54 4.6.2 Collapse Operator............................. 56 4.6.3 More Time Steps............................. 59 5 Conclusions 60 A Code for Lorenz Plots and Lyapunovs 62 B Code for Correlations 68 C Code for Bloch Sphere 73 D Code for Random Hamiltonian 76 References 78 iv List of Figures 2.1 Lorenz with σ = 10, ρ = 0:5, β = 8=3...................... 12 2.2 Lorenz with σ = 10, ρ = 10, β = 8=3...................... 13 2.3 Lorenz with σ = 10, ρ = 12, β = 8=3...................... 13 2.4 Lorenz with σ = 10, ρ = 16, β = 8=3...................... 14 2.5 Lorenz with σ = 10, ρ = 20, β = 8=3...................... 14 2.6 Lorenz with σ = 10, ρ = 24:4, β = 8=3..................... 15 2.7 Lorenz with σ = 10, ρ = 28, β = 8=3...................... 15 2.8 Lorenz with σ = 10, ρ = 100, β = 8=3...................... 16 3.1 Lyapunov exponents showing chaos....................... 20 3.2 Lyapunov exponents showing no chaos..................... 20 3.3 Bloch sphere showing chaos........................... 21 3.4 Bloch sphere showing no chaos.......................... 21 3.5 Correlation showing chaos............................ 22 3.6 Correlation showing no chaos.......................... 22 4.1 Lyapunov exponents for ρ = 28, σ = 10, and β = 8=3 with γ = 0 and 1 ... 24 4.2 Lyapunov exponents for ρ = 28, σ = 10, and β = 8=3 with γ = 0 and 2 ... 24 4.3 Lyapunov exponents for ρ = 28, σ = 10, and β = 8=3 with γ = 0 and 3 ... 24 4.4 Lyapunov exponents for ρ = 0:5, σ = 10, and β = 8=3 with γ = 0 and 1 .. 24 4.5 Lyapunov exponents for ρ = 0:5, σ = 10, and β = 8=3 with γ = 0 and 2 .. 25 4.6 Lyapunov exponents for ρ = 0:5, σ = 10, and β = 8=3 with γ = 0 and 3 .. 25 4.7 Lyapunov exponents for ρ = 28, σ = 10, and β = 8=3 with γ = 0:5 and 1 .. 25 4.8 Lyapunov exponents for ρ = 28, σ = 10, and β = 8=3 with γ = 0:5 and 2 .. 25 v 4.9 Lyapunov exponents for ρ = 28, σ = 10, and β = 8=3 with γ = 0:5 and 3 .. 26 4.10 Lyapunov exponents for ρ = 0:5, σ = 10, and β = 8=3 with γ = 0:5 and 1 . 26 4.11 Lyapunov exponents for ρ = 0:5, σ = 10, and β = 8=3 with γ = 0:5 and 2 . 26 4.12 Lyapunov exponents for ρ = 0:5, σ = 10, and β = 8=3 with γ = 0:5 and 3 . 26 4.13 Bloch sphere for ρ = 28, σ = 10, and β = 8=3 with γ = 0 and 1 ....... 27 4.14 Bloch sphere for ρ = 28, σ = 10, and β = 8=3 with γ = 0 and 2 ....... 27 4.15 Bloch sphere for ρ = 28, σ = 10, and β = 8=3 with γ = 0 and 3 ....... 27 4.16 Bloch sphere for ρ = 0:5, σ = 10, and β = 8=3 with γ = 0 and 1 ...... 27 4.17 Bloch sphere for ρ = 0:5, σ = 10, and β = 8=3 with γ = 0 and 2 ...... 28 4.18 Bloch sphere for ρ = 0:5, σ = 10, and β = 8=3 with γ = 0 and 3 ...... 28 4.19 Bloch sphere for ρ = 28, σ = 10, and β = 8=3 with γ = 0:5 and 1 ...... 28 4.20 Bloch sphere for ρ = 28, σ = 10, and β = 8=3 with γ = 0:5 and 2 ...... 28 4.21 Bloch sphere for ρ = 28, σ = 10, and β = 8=3 with γ = 0:5 and 3 ...... 29 4.22 Bloch sphere for ρ = 0:5, σ = 10, and β = 8=3 with γ = 0:5 and 1 ..... 29 4.23 Bloch sphere for ρ = 0:5, σ = 10, and β = 8=3 with γ = 0:5 and 2 ..... 29 4.24 Bloch sphere for ρ = 0:5, σ = 10, and β = 8=3 with γ = 0:5 and 3 ..... 29 4.25 Bloch sphere for ρ = 28, σ = 10, and β = 8=3 with γ = 0 and 1. This is out to t = 10000.................................... 30 4.26 Bloch sphere for ρ = 0:5, σ = 10, and β = 8=3 with γ = 0 and 3. This is out to t = 10000.................................... 30 4.27 Correlation of hσx(t1 + t2)σy(t1)i for ρ = 0:5, σ = 10, and β = 8=3 with γ = 0 and 1 ....................................... 31 4.28 Correlation of hσx(t1 + t2)σz(t1)i for ρ = 0:5, σ = 10, and β = 8=3 with γ = 0 and 1 ....................................... 31 4.29 Correlation of hσy(t1 + t2)σz(t1)i for ρ = 0:5, σ = 10, and β = 8=3 with γ = 0 and 1 ....................................... 31 4.30 Correlation of hσx(t1 + t2)σy(t1)i for ρ = 10, σ = 10, and β = 8=3 with γ = 0 and 1 ....................................... 31 4.31 Correlation of hσx(t1 + t2)σz(t1)i for ρ = 10, σ = 10, and β = 8=3 with γ = 0 and 1 ....................................... 32 vi 4.32 Correlation of hσy(t1 + t2)σz(t1)i for ρ = 10, σ = 10, and β = 8=3 with γ = 0 and 1 ....................................... 32 4.33 Correlation of hσx(t1 +t2)σy(t1)i for ρ = 24:4, σ = 10, and β = 8=3 with γ = 0 and 1 ....................................... 32 4.34 Correlation of hσx(t1 +t2)σz(t1)i for ρ = 24:4, σ = 10, and β = 8=3 with γ = 0 and 1 ....................................... 32 4.35 Correlation of hσy(t1 +t2)σz(t1)i for ρ = 24:4, σ = 10, and β = 8=3 with γ = 0 and 1 ....................................... 33 4.36 Correlation of hσx(t1 + t2)σy(t1)i for ρ = 28, σ = 10, and β = 8=3 with γ = 0 and 1 ....................................... 33 4.37 Correlation of hσx(t1 + t2)σz(t1)i for ρ = 28, σ = 10, and β = 8=3 with γ = 0 and 1 ....................................... 33 4.38 Correlation of hσy(t1 + t2)σz(t1)i for ρ = 28, σ = 10, and β = 8=3 with γ = 0 and 1 ....................................... 33 4.39 Correlation of hσx(t1 + t2)σy(t1)i for ρ = 0:5, σ = 10, and β = 8=3 with γ = 0 and 2 ....................................... 34 4.40 Correlation of hσx(t1 + t2)σz(t1)i for ρ = 0:5, σ = 10, and β = 8=3 with γ = 0 and 2 ....................................... 34 4.41 Correlation of hσy(t1 + t2)σz(t1)i for ρ = 0:5, σ = 10, and β = 8=3 with γ = 0 and 2 ....................................... 34 4.42 Correlation of hσx(t1 + t2)σy(t1)i for ρ = 10, σ = 10, and β = 8=3 with γ = 0 and 2 ....................................... 34 4.43 Correlation of hσx(t1 + t2)σz(t1)i for ρ = 10, σ = 10, and β = 8=3 with γ = 0 and 2 ....................................... 35 4.44 Correlation of hσy(t1 + t2)σz(t1)i for ρ = 10, σ = 10, and β = 8=3 with γ = 0 and 2 ......................................
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