Gaussian Wigner Function Research Articles

Photon-number distributions for quantum fields generated in nonlinear optical processes.

We show that in a large class of nonlinear optical processes which are used to generate the squeezed states of the radiation field, the quantum state of the generated radiation is given by a Gaussian Wigner function. This is so even when losses in the medium are included. The photon-number distributions for such fields are evaluated. The number distributions exhibit oscillatory character for the range of parameters for which the field is squeezed.

Nonnegative mixed states in Weyl-Wigner-Moyal theory

We classify the gaussian Wigner functions corresponding to mixed states and show that, unlike the case of pure states, not all nonnegative mixed states are gaussian.

A comparison of local and global single Gaussian approximations to time dynamics: One-dimensional systems

The detailed calculation of the dynamics of a chemical system is usually not considered due to the size and cost of the computation. It is thus useful to examine various approximation methods. Such methods first need to be tried out on simple systems, like one-dimensional motion. Here two approaches to approximating the solutions of the Schrödinger and von Neumann equations by single time-dependent Gaussians are explored and contrasted, explicitly for one-dimensional barrier penetration. The first approach, in which no tunneling occurs, is local in nature and characterized by an expansion of the equations of motion to second order about the average position of the Gaussian wave packet or about the average position and momentum of the Gaussian Wigner function. This approach was first introduced by Heller [E. J. Heller, J. Chem. Phys. 62, 1544 (1975)]. Here both Heller’s approach and a more general truncation method are considered. Indeed tunneling can be incorporated if second-order terms in the quantal von Neumann equation are included. However, the resulting dynamics is unstable for kinetic energies where the exact wavepacket (and Wigner function) splits into nonnegligible parts that represent reflection and transmission. In contrast, the second approach is a global method which is obtained by applying appropriate closure approximations to the equations of motion for the first- and second-order position and momentum expectation values. This method allows tunneling and is stable at all kinetic energies. It is also possible to approximate the global equations in such a way that the local equations are obtained.