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Quantum Dynamics Using Lie Algebras, with Explorations in the Chaotic Behavior of Oscillators

Ryan Sayer

Published Jan 1, 2012• 0 citations
OpenAlex

Abstract

We study the time evolution of driven quantum systems using analytic, algebraic, and numerical methods. First, we obtain analytic solutions for driven free and oscillator systems by shifting the coordinate and phase of the undriven wave function. We also factorize the quantum evolution operator using the generators of the Lie algebra comprising the Hamiltonian. We obtain coupled ODE's for the time evolution of the Lie algebra parameters. These parameters allow us to find physical properties of oscillator dynamics. In particular we find phase-space trajectories and transition probabilities. We then search for chaotic behavior in the Lie algebra parameters as a signature for dynamical chaos in the quantum system. We plot the trajectories, transition probabilities, and Lyapunov exponents for a wide range of the following physical parameters: strength and duration of the driving force, frequency difference, and anharmonicity of the oscillator. We identify conditions for the appearance of chaos in the system.

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We study the time evolution of driven quantum systems using analytic, algebraic, and numerical methods. First, we obtain analytic solutions for driven free and oscillator systems by shifting the coordinate and phase of the undriven wave function. We also factorize the quantum evolution operator using the generators of the Lie algebra comprising the Hamiltonian. We obtain coupled ODE's for the time evolution of the Lie algebra parameters. These parameters allow us to find physical properties of oscillator dynamics. In particular we find phase-space trajectories and transition probabilities. We then search for chaotic behavior in the Lie algebra parameters as a signature for dynamical chaos in the quantum system. We plot the trajectories, transition probabilities, and Lyapunov exponents for a wide range of the following physical parameters: strength and duration of the driving force, frequency difference, and anharmonicity of the oscillator. We identify conditions for the appearance of chaos in the system.
We propose an anharmonic oscillator driven by two periodic forces of different frequencies as a time-dependent model for investigating quantum dissipative chaos. Our analysis is done in the framework of the statistical ensemble of quantum trajectories in a quantum state diffusion approach. The quantum dynamical manifestations of chaotic behavior, including the emergence of chaos, properties of strange attractors, and quantum entanglement, are studied by numerical simulation of the ensemble averaged Wigner function and von Neumann entropy.
The matrix U' contains the full dynamics of a bilinear oscillator, because any observable of the system is an analytical function of the operators that determine the matrix representation. One cannot solve the time evolution equation in the general case, but there are many exactly solvable problems that simulate and enable one to understand the dynamics of more elaborate and realistic quantum-mechanical models. In the case of the solvable model just discussed it is easy to derive analytical expressions for operators in the Heisenberg picture as well as for invariants or constants of the motion. The mathematical treatment of the driven oscillator just discussed is trivial because the Hamiltonian operator belongs to a solvable Lie algebra. This model is a convenient starting point to illustrate the use of the Lie algebraic method, but it is also an oversimplification of most oscillatory phenomena. Nonlinear Hamiltonians are useful in many physical applications such as the simulation of frictional forces and dissipation effects.
The relationship between chaos and quantum mechanics has been somewhat uneasy--even stormy, in the minds of some people. However, much of the confusion may stem from inappropriate comparisons using formal analyses. In contrast, our starting point here is that a complete dynamical description requires a full understanding of the evolution of measured systems, necessary to explain actual experimental results. This is of course true, both classically and quantum mechanically. Because the evolution of the physical state is now conditioned on measurement results, the dynamics of such systems is intrinsically nonlinear even at the level of distribution functions. Due to this feature, the physically more complete treatment reveals the existence of dynamical regimes--such as chaos--that have no direct counterpart in the linear (unobserved) case. Moreover, this treatment allows for understanding how an effective classical behavior can result from the dynamics of an observed quantum system, both at the level of trajectories as well as distribution functions. Finally, we have the striking prediction that time-series from measured quantum systems can be chaotic far from the classical regime, with Lyapunov exponents differing from their classical values. These predictions can be tested in next-generation experiments.
This paper investigates the application of quantum coherence to the analysis of chaotic systems. The core claim is that leveraging the inherent properties of quantum coherence can reveal hidden patterns within chaotic systems. This work models chaotic systems as quantum systems and utilizes quantum coherence to identify critical parameters and potential control methods. The novelty lies in exploring the intersection of quantum mechanics and chaos theory, presenting a new approach to analyzing complex, nonlinear dynamics. We present a theoretical framework, focusing on the impact of quantum coherence on the behavior of specific chaotic systems, without experimental validation. The analysis provides insights into the potential for quantum-inspired control strategies and a deeper understanding of the underlying mechanisms driving chaotic dynamics. The central mathematical framework involves Hamiltonian operators, wave functions, and coherence measures, emphasizing the shift from classical deterministic descriptions to a quantum probabilistic one.
A method based off of operator consideration for solving the time evolution of a wave function is developed. The method is applied to free space, constant force and harmonic oscillator potentials where general solutions are derived for the wave function time evolutions for these potentials. Solving the Heisenberg equations of motion is discussed. A method to obtain exact solutions to Heisenberg's equations of motion is developed based off of solutions of classical equations of motion. Approximation methods for the time evolution of operators and wave functions are developed. A connection between classical and quantum chaos is made where it is realized that whenever a classical system behaves chaotically, the quantum system will as well.
Studying quantum properties of a system has been quite popular in quantum mechanics. One of the most important systems that are very crucial to the framework of quantum mechanics is the system of harmonic oscillator a system whose classical evolution is known to exhibit peculiar chaotic dynamics. We are motivated to investigate the behavior of quantum properties for a system with position and time dependent perturbed. Starting with Hamiltonian, we determined the equation of motion and obtained the wave function. The energy of the whole system using the operator ordering method was found. We show that the quantum mechanical picture alludes to a chaotic dynamics as expected. This is evidenced through the appearance of energy level crossings. An additional signature to this chaotic dynamics is observed in the transition of Eigen values from real to imaginary. We also show numerically that one can give the behavior of the system is Poincare section. By so doing we confirmed that increasing and decreasing the perturbation amplitude of the system becomes chaotic.