Symbolic Computation Approach to Nonlinear Dynamics

The description and demonstration of a short-time approximation method are presented for solution of the Heisenberg equation using a computer for operator algebra symbolic computation to obtain an operator series. Multi-mode nonlinear processes can be studied by this method in a fully quantum way; the symbolic operator or standard numerical results as well as the graphical dependences can be computed. Although a use for the method can be seen in its ability to improve numerical predictions rather than to discover new states or effects, we have obtained squeezing parameters and variances of photon number for initially squeezed states (and some other cases) in degenerate parametric processes.