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Time Evolution in Quantum Dynamics and Lie Algebras

Alberto Acevedo

Published Jan 1, 2014• 0 citations
OpenAlex

Abstract

In Quantum mechanics solving for the time evolution of a system is a very difficult problem. We use a Lie algebra method to solve the problem rather than typical perturbation methods. We will take a look at this Lie algebra method, the contrast between Quantum and Classical dynamics, and dissipation.

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This chapter aims to develop an appropriate notation and to facilitate the subsequent discussion of the role played by the Lie algebras. It is unlikely that one manages to express a sufficiently realistic model only in terms of the operators that span a finite-dimensional Lie algebra, but it may be possible to simulate most of the relevant features of many physical phenomena by means of such simple models. The chapter discusses well-known approximate methods for the treatment of the time-evolution equation. If the Hamiltonian operator belongs to a finite-dimensional Lie algebra, then the differential equation for the time-evolution operator is easier to solve, and even the calculation of transition probabilities and expectation values of observables may be straightforward. The most relevant feature of this application of the Lie algebraic method is the occurrence of a matrix equation of finite dimension in contrast with the expansion in a basis set of state vectors, which typically leads to matrices of infinite dimension.
Quantum technologies rely on the control of quantum systems at the level of individual quanta. Mathematically, this control is described by Hamiltonian or Liouvillian evolution, requiring the application of various techniques to solve the resulting dynamic equations. Here, we present a tutorial for how the quantum dynamics of systems can be solved using a Lie-algebra decoupling method. The approach involves identifying a Lie algebra that governs the dynamics of the system, enabling the derivation of differential equations to solve the Schrödinger equation. As background, we include an overview of Lie groups and Lie algebras aimed at a general-physicist audience. We then prove the Lie-algebra decoupling theorem and apply it to both closed and open dynamics. The results represent a broad methodology to find the dynamics of quantum systems with applications across many fields of modern quantum research.
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.
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.
A representation of invariants in quantum mechanics for a general time-dependent Hamiltonian is provided. The main properties of these operators and their relevance for solving the Schrödinger equation are discussed in light of this representation. In particular, the well-known Lewis and Riesenfeld’s method is shown to be recovered quite directly by the present approach. A detailed analysis is concerned with Hamiltonians which at each instant are elements of a finite dimensional Lie algebra. Exploiting the Wei and Norman’s theory for such Hamiltonians, it is found that the evolution of a quantum system is always determined, in part or fully, by an invariant which belongs to the same algebra.
A variety of dynamics in this world can be approximately treated by a model: a driven and damped parametric oscillator. An intensive investigate of this time-dependent model from both classical and quantum mechanical points of view provides a dynamical insight to resolve the classical and quantum dynamics in order to understand the more complicated time-dependent evolutions not only happens in the macroscopic classical scale for the synchronized behaviors but also in the coherent microscopic scale for a quantum state control. By using a Lie transformation on a general time-dependent quadratic Hamiltonian, we exactly solve the dynamical behaviors of the driven and damped parametric oscillator to give a universal physical mechanism for the classical parametric control on the quantum state evolution in a dissipation system.