In Sec. , we saw that any solution to the one-dimensional Mathieu equation, whose canonical form includes a coefficient which is -periodic in , can be expressed as a sum of frequencies as in Eq. :

Most Paul-trapping experiments have been conducted in stability region A (Fig. ), where the motion of a particle is dominated by the lowest-order terms in the above equation (Eq. ), i.e., those terms for which . An additional simplification results from the fact that for , , which implies that . For these ion trajectories in the lowest-order stability region, the term of frequency corresponds to a slowly-oscillating secular motion. The term of frequency represents the -periodic drive of the oscillating voltage. The amplitude of the secular motion is much larger than that of the -periodic motion.

In this region, can be separated into a large-amplitude slow motion and a small-amplitude fast motion. is described by an approximation which assumes radial and axial solutions of the form:[4]

where corresponds to either **x**, **y**, or axial motion; is the slowly-oscillating; large-amplitude secular displacement; and corresponds to the fast, typically small-amplitude ``micromotion.'' Since and --- that is, since the micromotion has small amplitude and large frequency relative to the secular motion--- plugging the approximate solution into the Mathieu equation (Eq. ):

reduces to:

Neglecting the small constant term introduced by non-zero values of **a**, the general solution to the second-order differential equation above is:

This expression for the micromotion shows that: (1) the micromotion is out of phase with the alternating voltage; (2) the amplitude of the micromotion is proportional to the secular displacement, , from the center of the trap, indicating that the micromotion will be most pronounced when an ion's secular displacement is at a maximum.[4,39]

With Eq. as an approximation for the micromotion () in terms of the secular motion (), the instantaneous acceleration of can be obtained through substitution into the Mathieu equation:

The average value of --- and thus the frequency, , of the secular motion--- can be determined by integrating Eq. over a single period () of the micromotion, since the average acceleration of the micromotion will evaluate to zero over this interval.

Interpreting the expression above as the equation of motion for a simple harmonic oscillator, , implies a secular frequency of .

**Figure:** 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`.

**Figure:** **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`.

**Figure:** 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.

This first-order approximation describes the motion of a single Paul-trapped particle as a fast ``micromotion'' superposed on a slower ``secular'' motion:[4]

where the frequency of the micromotion, , is approximately that of the oscillating potential, and the frequency of the secular motion is . This description accounts for the qualitative features of a particle's motion in the axial and radial dimensions (Fig. , ).

However, the Fourier spectrum (Fig. ) of a Paul-trapped particle in stability region A shows that there are more than two frequencies (i.e., the secular frequency and the micromotion freqeuncy) of oscillation. Moreover, there is no Fourier peak corresponding to the * exact* frequency, , of the oscillating potential--- rather, there are two peaks at slightly lower and higher frequencies than .
For a more accurate * quantitative* description of the oscillating particle's motion, we must modify our simple model, which assumes a
single-frequency micromotion superposed on a single-frequency secular oscillation. The Fourier spectrum of Fig. is accurately
described by Eq. , which expresses the solution to the Mathieu equation as an infinite series of harmonic oscillations with frequencies perturbed by from integer multiples of .

Note that the Mathieu equation is reminiscent of many chaotic systems in that it resonates at frequencies perturbed by from harmonics of . Unlike equations describing chaotic systems, the Mathieu equation is a linear equation and cannot, as such, produce chaotic motion. Excitement of superharmonics in systems governed by the Mathieu equation is known as parametric resonance and is due to time-varying coefficients rather than to nonlinearities.[23]

Fri May 12 10:36:01 EDT 1995