A solution with Mathieu's Equation

For a more formal analysis refer to [6]. Starting with the equation,

$$\frac{d^2z}{d\eta^2}+\left(a_z-2q_z \cos(2\eta)\right)z=0, \eta... ...ga t}{2}, a_z=\frac{-16eU}{md^2_0\omega^2}, q_z=\frac{-8eV}{md_0^2\omega^2}$$

We apply Floquet's theorem and the subsequent corollary to suppose solutions of the form,

$$u_1(\eta)=e^{\mu \eta}\phi_1(\eta), u_2(\eta)=e^{-\mu \eta}\phi_2(\eta)$$

The conditions for stability require that $\mu$ be purely imaginary. It is typical to write $\mu = \alpha + i \beta$ , and so we can take a fourier expansion of the $\phi$ , and recalling that the original equation contains $\cos(2\eta)$ , we assume a general solution,

$$z(\eta)=A\sum_{n=-\infty}^{\infty}C_{2n}e^{i(2n+\beta)}\eta + B \sum_{n=-\infty}^{\infty} C_{2n}e^{-i(2n+\beta)\eta}$$

$$z(\eta)=A'\sum_{n=-\infty}^{\infty}C_{2n}\cos((2n+\beta)\eta)$$

As before, we can find a useful recurssion relation. Define:

$$D_{2n} \equiv \frac{a_z -(2n+\beta)^2}{q_z} \longrightarrow D_{2n}C_{2n}-C_{2n-2}-C_{2n+2}=0$$

When $n=0$ we have

$$D_0=\frac{a_z-\beta^2}{q_z}=\frac{C_{-2}}{C_0}+\frac{C_{2}}{C_0}$$

With this recurssion relationship we may solve for $\beta$ with increasing levels of accuracy, for example,

$$C_{2n} = \frac{C_{2n-2}}{D_{2n}}+\frac{C_{2n+2}}{D_{2n}}$$

$$= \frac{ \frac{C_{2n-4}}{D_{2n-2}}+ \frac{C_{2n}}{D_{2n-2}}}{D_{2n}}+\frac{ \frac{C_{2n}}{D_{2n+2}}+ \frac{C_{2n+4}}{D_{2n+2}}}{D_{2n}}$$

As a first approximation, we set $C_{\pm 4}=0$ and obtain

$$D_0 = \frac{1}{D_{-2}} + \frac{1}{D_2}$$

$$\frac{a-\beta^2}{q}=q\left(\frac{1}{a-(-2+\beta)^2} + \frac{1}{a -(2+\beta)^2}\right)$$

When we assume $4 \gg \beta^2, \beta, a$ we obtain the approximation,

$$\beta=\sqrt{a+\frac{q^2}{2}}$$

If we take $U=0 \ni a=0$ then we find we have recovered the approximation of the previous section,

$$\omega_z=\frac{2\sqrt{2}eV}{md_0^2\omega}$$

From such references as [8] we know that the next approxmation is

$$\beta=\sqrt{\frac{a_z + q_z^2\left(\frac{1}{2} + \frac{a}{8}\right) + \frac{q^4}{128}}{1-q^2\left(\frac{3}{8} + \frac{5a}{16}\right)}}$$

The motion will have frequencies of $(2n + \beta)$ , of which the lowest and second to the lowest correspond roughly with the secular approximation secular and micromotion. We could carry this process on ad infinitum ad nauseum. This is but one method to solve for $\beta$ . The other method is the numerical method we used to find the stable points of the iso-$\mu$ . A third method is to use the more technical solutions to the Mathieu equation devoloped by the mathematicians. We will close this report with a brief review of one such solution.