Quantum dynamics of the parametric oscillator.

Quantum Information is becoming increasingly important in the 21st century. The potential for massive advances in communication, cryptography and even quantum teleportation motivates a need for greater understanding of matter at the quantum scale. Unlike its classical counterpart, matter at the nanoscopic level behaves in a manner far removed our intuition. At present well-defined solutions pertaining to the Hamiltonian-the operator used to define quantum statesexist for both the time-independent Schrödinger equation and the time dependent Schrödinger equation. The solution to the time dependent parametric and driven Hamiltonian operators is less commonly known. It will be the focus of my research in the 10-week REU program. Using the analytical methods of operator theory and linear algebra, numerical methods usingh Mathematica and techniques of Lie algebras developed in the quantum group at Brigham Young University, this project aims to examine particular cases of various time dependent Hamiltonians for the quantum states of light and find the time evolution of these states. By studying various states: