Before getting into the theory of trapped particles, let's get familiar with the computer simulation TrapApp. TrapApp is a Windows program, and it can be installed such that a Trap ``group'' with an icon for the Trap executable file is created. To install TrapApp, double click on the Install.EXE file enclosed with the distribution package. If you prefer not to install the program, you may run TrapApp.EXE from the Windows File Manager. Important: before running TrapApp, make sure that file BWCC.DLL is in your WINDOWS directory. This file is a Borland dynamically-linked library (DLL) which is distributed free of charge and which is essential for running TrapApp.
The opening screen of the simulation (Fig. ) depicts the radial positions of four singly-ionized magnesium atoms in a Paul trap. The initial positions of the magnesium ions describe the unit cell for a face-centered cubic lattice; the initial velocities were assigned according to Maxwell-Boltzmann statistics at K. Use your mouse to press [Run]. The upper left graph shows the ions oscillating within the trap. On the right side of your screen, there are two data analysis graphs. The top graph shows data pertaining to the ``tagged'' (black) particle; the lower graph displays data pertaining to the whole system.
Figure: Opening screen of program TrapApp after 5 s of simulation.
Press [Stop] to halt the simulation. Moving your mouse within the upper right graph, hold the left mouse button down. A coordinate reader will display the coordinates of your cursor in the lower left corner of the graph. Now click the right mouse button within the graph window. A pop-up menu will appear on the upper left corner of the graph. This pop-up menu allows you to choose different types of analysis graphs or change the scaling of the graph's axes. Click Graph Type and then Phase Space... to view the quasiperiodic trajectory of the tagged ion.
Allow the program to run long enough to collect at least 256 data points (about s of simulation time), then use the right mouse button inside the upper right graph to select the Fast Fourier Transform ( FFT) graph option for a frequency analysis of the tagged ion's axial position or x-position. You will see a well-defined peak corresponding to the secular (slow) oscillations and two peaks representing the (fast) micromotions of the particle. (See Sec. for the theory describing the frequencies of these peaks.)
Use your mouse to select State Data from the menu bar at the top of the TrapApp window. Choose Select Data to Collect from the pop-up menu. You will be presented with a dialog box containing analysis selections pertaining to instantaneous values, running averages, and cumulative averages of energy; mean particle separation; and root-mean square particle distance from the origin. There are also analysis options for the ratio of potential energy of particle interactions to kinetic energy (this ratio is the coupling), heat capacity of the system, pair distribution function, and distribution of the ions' total, radial, and axial displacements from the center of the trap. Sec. elaborates on these data options. Just for fun, you might try selecting (by clicking the mouse inside the appropriate check-boxes) running averages for total energy, kinetic energy, and potential energy. Then click the right mouse button on the lower right graph, click Graph Type vs. Time, select the appropriate running average values, and run the simulation to watch the running-average parameters evolve to their equilibrium values.
It is important to understand how to control which data are collected and which data are displayed on the graphs. When a data item (such as instantaneous kinetic energy) is selected (checked) in the State Data dialog box, program TrapApp creates a data vector to store the data. The State Data and Particle Data dialogs do not, however, affect the display of the analysis graphs. To modify which data are displayed on the graphs, you must click the right mouse button inside the appropriate graph or select Inspect Graph from the pop-up menu which appears when Particle Data or State Data is selected from the menu bar. By separating the tasks of collecting and graphing data, TrapApp allows the user to collect a large amount of data without slowing the simulation or cluttering the screen by plotting data that need not be continuously monitored. If you accidentally attempt to graph an item for which data is not being collected, TrapApp will gracefully ignore your mistake.
Now select the Files option from the menu bar. Click on Open... and load the file entitled intrCrys.trp. Dismiss the comment screen explaining the configuration of 2 Paul-trapped ions and run the simulation for about s. Perform a Fourier transform of the tagged ion's x-position. You may examine Paul trap parameters via the following sequence of mouse clicks: Parameters Trap Type Paul Trap. The specifications of the Paul trap (explained in Sec. ) appear in a dialog box. How does the high-frequency FFT peak compare with the frequency at which the Paul trap's electric potential is oscillating?
Click the right mouse button inside the upper left graph of the ions' positions. A pop-up menu appears with options for changing the viewing projection ( Projection XY, XZ, YZ), inspecting the axes scale (for zooming in or out), or tagging a particle (the black ``tagged'' particle is the one whose position is monitored on the upper right ``particle'' graph). Compare the XY, YZ, and XZ projections. What do you notice about the spatial orientation of the crystal? Display the Running Average of Mean Separation on the State Data Graph. Select the Two-Ion Equilibrium Sep. option from the Parameters menu. How does predicted equilibrium separation distance compare to mean separation displayed on the State Data Graph?
Load file introPen.trp. Run the simulation and use the particle analysis graph to view the ions' axial, radial, and x-motions. Inspect the ion's motion in the XZ (or YZ) plane and in the XY plane. Observe that the axial motion of a single Penning-trapped ion is perfect harmonic oscillation and that the axially-directed magnetic field induces a rotational radial motion (Sec. ).