University of Arkansas

*With support from the National Science Foundation*

This is the gateway to a collection of applets developed in order to help visualize the evolution of some quantum mechanical systems of interest in quantum optics, atomic physics, and quantum information theory. The basic tools used are the Bloch Sphere, for two-level systems (such as a two-level atom, a spin 1/2 system or a Cooper-pair box), and the Q-function (sometimes called the Husimi function) for a quantum harmonic oscillator, such as a single mode of the electromagnetic field, or a nanoresonator.

The applets are highly interactive: they can be resized and the viewpoint can be rotated in real time, and a wide range of initial conditions and parameters may be chosen.

The applets currently available are:

**The Bloch Equations:**This applet integrates the Bloch equations for a two-level system interacting with a classical, but otherwise completely arbitrary, electromagnetic field. The user can specify any function for the field's complex amplitude and watch the resulting evolution as a trajectory on the Bloch sphere.**The Jaynes-Cummings Model:**This applet shows the evolution of a two-level system coupled to a quantum harmonic oscillator in the rotating-wave approximation (Jaynes-Cummings model). The evolution of the two-level system is plotted on the Bloch sphere, and simultaneously a 3-D plot of the evolving Q function of the field is displayed. The initial state of the atom is arbitrary; the initial state of the oscillator is restricted to be a coherent state.**The Spin-Boson Hamiltonian:**This applet shows the evolution of a two-level system coupled to a quantum harmonic oscillator*without*the rotating-wave approximation. This is practically a research problem, since the equations do not admit of a closed-form analytical solution and may exhibit a bewildering range of behaviors. A physical example (among many) of such a system is a Cooper-pair box coupled to a nanoresonator (see, e.g., E. K. Irish*et al*., Phys. Rev. B**72**, 195410 (2005)).

An interesting feature of these applets is a "math reader" java class that allows the user to enter any mathematical expression in a text field. A separate demo with links to the code is available here.

The applets were developed by Jared Ostmeyer (jostmey AT uark DOT edu) under the supervision of Julio Gea-Banacloche (jgeabana AT uark DOT edu) during the summer of 2006 as part of an NSF-sponsored REU program in the Physics Department at the University of Arkansas. The code is freely available; please contact any of us for it.

In order to run these applets properly it is generally best to have the most recent version of the Java virtual machine installed in your computer. This seems to be especially true for Mac OS computers.

This page has been accessed times since September 12, 2006.