- Opening screen of program
`TrapApp`after 5 s of simulation. - Radial and axial positions of 32 Paul-trapped particles--- (
*left*): Radial projection of positively charged aluminum particles experimentally observed by Wuerker, et al. (*center, right*): Computer simulation of 32 Mg ions, showing (*center*) instantaneous radial (XY) and (*right*) axial (XZ) positions.*source file*:`paul32cr.trp`. - Electrode structure of quadrupole ion trap. Typical dimensions are m, m.
*source: Winter, H. et al., Am. J. Phys., Vol. 59, No. 9, 1991.* - Stability regions for the one-dimensional Mathieu equation. Note the failure of curves , and , to converge. The published series approximations are inaccurate for ``large'' values of
**q**. - Stability region A as a function of and . This region is defined by the intersection of the lowest-order stability regions of the Mathieu equations for axial and radial motions.
- Axial position and
**x**-position as a function of time for a single particle trapped in stability region A with . Note that: (*a*) both motions consist of a small-amplitude, high-frequency ``micromotion'' superposed on a larger-amplitude, slower secular motion; (*b*) the frequency of axial secular oscillations is twice that of secular oscillations on the**x**-axis; (*c*) the frequencies of axial and**x**-position micromotions are roughly equal.*source file*:`paul1A.trp`. **x**-position vs. axial position for a single Paul-trapped particle in stability region A---*left*: note that one secular oscillation of the**x**4-position corresponds to exactly two secular oscillations of the axial position, due to the fact that the potential in the axial direction is twice as steep as in the**x**-direction for this special case of**a=0**(no static potential).*right*: when allowed to evolve over a long period of time, the system demonstrates its quasiperiodicity (which arises from the fact that the ratios of the secular frequencies to those of the micromotions are irrational) by filling a bounded region without retracing its path.*source file*:`paul1A.trp`.- Fourier transform of a single Paul-trapped particle in Mathieu region A: = 0, , MHz. Fourier amplitudes are expressed in decibels. The theoretical frequencies were calculated from the application of Eq. 2.12 to the approximations (developed in Sec. 2.8) of motion in Region A.
*source file*:`paul1A.trp`.*Note*: since the resolution of the FFT depends on the time step,**dt**, you may need to change the time step of`paul1A.trp`to generate a transform of this resolution. - Failure of series approximations of characteristic values to converge. According to Mathieu theory, . However, the published series approximations lack the precision needed to force convergence of the even and odd characteristic value curves.
- Approximate zones of the two known Paul-trapping stability regions. Note that the exact boundary depicted for region B is inaccurate to the extent that the series approximations for the characteristic value curves (especially and ) depart from their theoretical behavior in the proximity of region B. should converge to and to .
**x**-position vs. axial position for a Paul-trapped single ion in stability region B.*source file*: paul1B.trp.- Fourier transform of
**x**-position of a Paul-trapped single ion in region B. Peak frequencies (MHz): .*source file*: paul1B.trp. - Fourier transform of axial position of a single Paul-trapped ion in region B. Peak frequencies (MHz): .
*source file*: paul1B.trp. - Blümel's four dynamical regimes for a collection of Paul-trapped
particles--- (
*1*): in the high-energy Mathieu regime, interparticle spacing is large, trajectories are virtually uncorrelated, and no heating occurs; (*2a*): in region C of the chaotic regime, non-negligible Coulomb interactions give rise to heating, and higher-density configurations (smaller ) correspond to higher heating rates; (*2b*): in region C of the chaotic regime, higher-density configurations correspond to slower heating rates; (*3*): the quasiperiodic regime is characterized by quasiperiodic trajectories and the absence of heating; (*4*): the crystalline phase. - Running averages of energy and for a melted
**15**-ion crystal which evolves through the heating regime to the Mathieu regime. Note the gain in kinetic energy and increase in for the heating phase. When the Mathieu regime is reached, and**<KE>**oscillate about stable average values.*source file*: 15melt.trp. - Fourier transforms (
**2048**data points) of**x**-position (*left*) and (*right*) axial position of one of**5**Paul-trapped ions in a non-heating cloud configuration (Blümel's ``quasiperiodic'' regime) for region A of the Mathieu stability. m.*source file*:`cloud5A.trp`. - Fourier transform (
**2048**data points) of kinetic energy of an undamped two-ion crystal on the**z**-axis in stability region A: the dominating contribution to kinetic energy is the micromotion. (The transform is*not*scaled to decibels.)*source file*:`2CrA.trp`. - Geometry of an
**8**-Mg ion crystal in stability region A with (*left*) and without (*right*) damping. The left-hand configuration represents a metastable state with slightly higher energy than the configuration on the right.*source file*:`8CrA.trp`. - Equipartition of energy for 10 Paul-trapped magnesium ions in stability region B: the running averages (evaluated over s) of total, potential, and kinetic energies are displayed. The total energy of about eV is partitioned
equally among potential and kinetic components, indicating a temperature of approximately
**1660**K.*source file*:`paul10B.trp`. - Axial position and
**x**-position vs. time for one of**10**ions Paul trapped in region B of Mathieu stability.*source file*:`paul10B.trp`. - Fourier transforms (
**2048**data points) of**x**-position (*left*) and axial position (*right*) of a Paul-trapped ion in stability region B. Peak**x**-frequencies (MHz): , , , , , , . Peak axial frequencies (MHz): , , , , , , , , , .*source file*:`paul10B.trp`. **x**-position vs. axial position for one of**10**Paul-trapped ions in region B of Mathieu stability.*source file*:`paul10B.trp`.- Two Mg ions evolving from a crystalline configuration to
the Mathieu regime upon removal of viscous damping.
*source file*:`2CrB.trp`. - Geometry of a damped
**8**-Mg ion crystal in stability region B.*left*:*XZ*projection shows that the crystal is not planar--- it measures about m in the radial direction, m is the axial direction;*center*:*XY*projection shows radial order;*right*:**x**-position vs. axial position, showing periodic variation in axial placement.*source file*:`8CrB.trp`. - Large micromotion oscillations (m peak-to-peak) for a
**32**-ion ordered structure in stability region B. These figures have the same scale and represent the same ordered configuration at different phases of the micromotion.*source file*:`32CrB.trp`. - Axial, radial, and x-positions of a single Penning-trapped Be ion. Note that the cyclotron motion (represented by fluctuations in the radial position) has the largest frequency and smallest amplitude, while the magnetron motion (represented by the
**x**-oscillations) has the smallest frequency and largest amplitude. The axial oscillations have a frequency between those of the magnetron and cyclotron motions.*source file*:`introPen.trp`. **x**-positions vs. axial position of a single Penning-trapped Be ion.*source file*:`introPen.trp`.

Fri May 12 10:36:01 EDT 1995