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:
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. ):
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:
From Eq. :
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 x4-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:
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.