\subsection{Time Dependent Perturbation}
Revisited treatment~\ref{ss:revisit}
\begin{eqnarray*}
\Omega(t)+i\Pi(t) &=& -\Omega_0\sin\omega_v t +i\Pi_0\cos\omega_v t 
=\frac{1}{2i}\left(-\Omega_0(e^{i\omega_v t}-e^{-i\omega_v t}) -\Pi_0(e^{i\omega_v t}+e^{-i\omega_v t})\right)\\
&=&\frac{-1}{2i}\left((\Omega_0+\Pi_0)e^{i\omega_v t}+(-\Omega_0 +\Pi_0)e^{-i\omega_v t})\right)\\
&=&\left(X_0e^{i\omega_v t}-Y_0e^{-i\omega_v t})\right)
\end{eqnarray*}
where 
\begin{eqnarray*}
X_0 &=& -\frac{\Omega_0+\Pi_0}{2i}= -\frac{1}{2i}\beta_z^0\omega_a(\frac{\gamma}{\gamma+1} +\gamma\frac{\omega_v}{\omega})\beta_{xy}\\
Y_0&=&-\frac{\Omega_0-\Pi_0}{2i} =-\frac{1}{2i}\beta_z^0\omega_a(\frac{\gamma}{\gamma+1} - \gamma\frac{\omega_v}{\omega})\beta_{xy}\\
\end{eqnarray*}
\begin{eqnarray*}
\omega_0(t) &=& \omega_0+\pi_0\sin^2\omega_v t = \omega_0 +\onehalf\pi_0(1-\cos\omega_v t)
=(\omega_0+\onehalf\pi_0)-\frac{1}{2}\pi_0\cos\omega_v t\\
&=& \omega_0^\prime + \pi^\prime_0\cos\omega_v t\\
\end{eqnarray*}
where $$\omega_0^\prime=\omega_0+\frac{\pi_0}{2}=\frac{eB}{mc}\left(a_\mu+\frac{1}{\gamma}-a_\mu\frac{\gamma}{\gamma+1}\frac{(\beta_z^0)^2}{2}\right)
\ \ {\rm and}\ \ \pi_0^\prime=\frac{eB}{mc}\left(a_\mu\frac{\gamma}{\gamma+1}\frac{(\beta_z^0)^2}{2}\right)$$
Also $$\op = -(\op_0-\omega) = -\omega_a(1-\frac{\gamma}{\gamma+1}\frac{(\beta_z^0)^2}{2})$$
Then we can write, 
\begin{eqnarray*}
{\cal H}&=& {\cal H}_0 + {\cal H}^\prime
\end{eqnarray*}
where in the rotating frame
\begin{eqnarray*}
{\cal H}_0 &=& -\frac{1}{2}\begin{pmatrix} \omega_0^\prime-\omega & 0 \cr 0 & -\omega_0^\prime +\omega\end{pmatrix}
=\begin{pmatrix} \omega^\prime/2 & 0 \cr 0 & -\omega^\prime/2\end{pmatrix}\\\\
{\cal H}^\prime &=& -\frac{1}{2}\begin{pmatrix}\pi_0^\prime\cos\omega_v t & X_0 e^{i\omega_v t}-Y_0 e^{-i\omega_v t}   \cr
X^*_0 e^{-i\omega_v t}-Y^*_0 e^{i\omega_v t} & -\pi_0^\prime\cos\omega_v t\end{pmatrix}   
\end{eqnarray*}
The solutions to the unperturbed hamiltonian are
\begin{eqnarray*}
a(t) &=& a_0 e^{-i\op t/2}\\
b(t) &=& b_0 e^{i\op t/2}
\end{eqnarray*}
and
\begin{eqnarray*}
\dot c_a&=& -i[c_a\pi_0^\prime(t) +c_b (X_0(t)-Y_0(t)) e^{i\op t}]
\end{eqnarray*}
If $c_a^{(0)}=c_b^{(0)}=\onroot2$, the
\begin{eqnarray*}
 c_a^{(1)}&=& -\frac{i}{\sqrt{2}}\int_0^t[\pi_0^\prime(\tp) + (X_0(\tp)-Y_0(\tp)) e^{i\op \tp}]d\tp\\
 c_a^{(1)}&=& -\frac{i}{\sqrt{2}}\int_0^t[\pi_0^\prime\cos\omega_v \tp + 
(X_0e^{i\omega_v \tp}-Y_0e^{-i\omega_v \tp}) e^{i\op \tp}]d\tp\\
&=& -\frac{i}{\sqrt{2}}[\pi^\prime_0\frac{\sin\omega_v t}{\omega_v} + 
(X_0\frac{e^{i(\omega_v+\op) t} -1}{i(\omega_v+\op)} +Y_0\frac{e^{-i(\omega_v-\op)t}-1}{i(\omega_v-\op)})]\\
c_b^{(1)}&=& -\frac{i}{\sqrt{2}}[-\pi^\prime_0\frac{\sin\omega_v t}{\omega_v} + 
(X^*_0\frac{e^{-i(\omega_v+\op) t} -1}{-i(\omega_v+\op)} +Y^*_0\frac{e^{i(\omega_v-\op)t}-1}{-i(\omega_v-\op)})]
\end{eqnarray*}
We see that there is no frequency shift to first order as all terms average to zero or a constant over time.
\begin{eqnarray*}
c_a^{(2)}&=& -i\int_0^t\left\{\left[\pi_0^\prime\cos\omega_v \tp\frac{-i}{\sqrt{2}}[\pi^\prime_0\frac{\sin\omega_v \tp}{\omega_v} + 
(X_0\frac{e^{i(\omega_v+\op) \tp} -1}{i(\omega_v+\op)} +Y_0\frac{e^{-i(\omega_v-\op)\tp}-1}{i(\omega_v-\op)})]\right]\right.\\
&+&\left. \left[\frac{-i}{\sqrt{2}}[-\pi^\prime_0\frac{\sin\omega_v \tp}{\omega_v} + 
(X^*_0\frac{e^{-i(\omega_v+\op) \tp} -1}{-i(\omega_v+\op)} +Y^*_0\frac{e^{i(\omega_v-\op)\tp}-1}{-i(\omega_v-\op)})]\right]
(X_0e^{i\omega_v \tp}-Y_0e^{-i\omega_v \tp}) e^{i\op \tp}\right\}d\tp\\
\end{eqnarray*}
Let's integrate
\begin{eqnarray*}
c_a^{(2)}&=& -\frac{1}{\sqrt{2}}\left\{\left[[-{\pi^\prime_0}^2\frac{\cos(2\omega_v t)}{4\omega^2_v} + 
\frac{\pi_0^\prime X_0}{i(\omega_v + \omega)}(\frac{e^{i(2\omega_v+\op) t} -1}{2i(2\omega_v+\op)} 
+\frac{e^{i\op t}-1}{2i\op} -\frac{\sin\omega_v t}{\omega_v}) \right.\right.\\
&+&\left.\left. \frac{\pi_0^\prime Y_0}{i(\omega_v-\op)}
(\frac{e^{i(\op)t}-1}{2i(\op)}  +\frac{e^{-i(2\omega_v-\op)t}-1}{-2i(2\omega_v-\op)}
-\frac{\sin(\omega_v t)}{\omega_v})\right]\right.\\
&+&\left. \left[\frac{-\pi^\prime_0X_0}{2i\omega_v}(\frac{e^{i(2\omega_v+\op)t}-1}{i(2\omega_v+\op)}
-\frac{e^{i\op t}-1}{i\op})+\frac{\pi^\prime_0 Y_0}{2i\omega_v}(\frac{e^{i\op t}-1}{i\op}-
\frac{e^{-i(2\omega_v-\op)t}-1}{-i(2\omega_v-\op)})\right. \right.\\
&+&\left. \left. 
\frac{X^2_0}{-i(\omega_v+\op)}(t -\frac{e^{i(\omega_v+\op)t}-1}{i(\omega_v+\op)})
-\frac{X_0 Y_0}{-i(\omega_v+\op)}(\frac{e^{-i(2\omega_v)t}-1}{-i(2\omega_v)}
-\frac{e^{-i(\omega_v-\op)t}-1}{-i(\omega_v-\op)})\right.\right.\\
&+&\left.\left.\frac{ Y_0X_0}{-i(\omega_v-\op)}(\frac{e^{i(2\omega_vt)}-1}{2i\omega_v}-\frac{e^{i(\omega_v+\op)t}-1}{i(\omega_v+\op)})
-\frac{Y_0^2}{-i(\omega_v-\op)}(t-\frac{e^{-i(\omega_v-\op)t}-1}{-i(\omega_v-\op)})\right]\right\}\\
\end{eqnarray*}

If we keep only to order $(\beta_z^0)^2$ then we can drop $(\pi_0^\prime)^2$, $\pi_0^\prime  X_0$ and $\pi_0^\prime Y_0$ since $\pi_0^\prime$ is order $\beta_z^0)^2$.
Then 
\begin{eqnarray*}
c_a^{(2)}&\sim& -\frac{1}{\sqrt{2}}\left[
\frac{|X^2_0|}{-i(\omega_v+\op)}(t -\frac{e^{i(\omega_v+\op)t}-1}{i(\omega_v+\op)})
-\frac{X^*_0 Y_0}{-i(\omega_v+\op)}(\frac{e^{-i(2\omega_v)t}-1}{-i(2\omega_v)}
-\frac{e^{-i(\omega_v-\op)t}-1}{-i(\omega_v-\op)})\right.\\
&+&\left.\frac{ Y^*_0X_0}{-i(\omega_v-\op)}(\frac{e^{i(2\omega_vt)}-1}{2i\omega_v}-\frac{e^{i(\omega_v+\op)t}-1}{i(\omega_v+\op)})
-\frac{|Y_0^2|}{-i(\omega_v-\op)}(t-\frac{e^{-i(\omega_v-\op)t}-1}{-i(\omega_v-\op)})\right]\\
&\sim& 
 -\frac{1}{\sqrt{2}}\left[
\frac{X^2_0}{-i(\omega_v+\op)}(t -\frac{e^{i(\omega_v+\op)t}-1}{i(\omega_v+\op)})-\frac{Y_0^2}{-i(\omega_v-\op)}(t-\frac{e^{-i(\omega_v-\op)t}-1}{-i(\omega_v-\op)})\right.\\
&-&\left. X_0 Y_0\left(\frac{e^{-i(2\omega_v)t}-1}{-(2\omega_v)(\omega_v+\op)}-\frac{e^{2i\omega_v t}-1}{(\omega_v-\op)(2\omega_v)}+\frac{(e^{-i(\omega_v-\op)t}-1) + (e^{i(\omega_v+\op)t}-1)}{\omega_v^2-\op^2}\right)
\right]\\
&\sim& 
 -\frac{1}{\sqrt{2}}\left[
\frac{X^2_0}{-i(\omega_v+\op)}(t -\frac{e^{i(\omega_v+\op)t}-1}{i(\omega_v+\op)})-\frac{Y_0^2}{-i(\omega_v-\op)}(t-\frac{e^{-i(\omega_v-\op)t}-1}{-i(\omega_v-\op)})\right.\\
&-&\left. X_0 Y_0\left(\frac{e^{-i(2\omega_v)t}-1}{-(2\omega_v)(\omega_v+\op)}-\frac{e^{2i\omega_v t}-1}{(\omega_v-\op)(2\omega_v)}+\frac{e^{-i(\omega_v-\op)t} + e^{i(\omega_v+\op)t}-2}{\omega_v^2-\op^2}\right)
\right]\\
&\sim& 
 -\frac{1}{\sqrt{2}}\left[
\frac{X^2_0}{-i(\omega_v+\op)}t-\frac{Y_0^2}{-i(\omega_v-\op)}t +
\frac{X^2_0}{i(\omega_v+\op)}( \frac{e^{i(\omega_v+\op)t}-1}{i(\omega_v+\op)})-\frac{Y_0^2}{i(\omega_v-\op)}(\frac{e^{-i(\omega_v-\op)t}-1}{-i(\omega_v-\op)})\right.\\
&-&\left. X_0 Y_0\left(\frac{e^{-i(2\omega_v)t}-1}{-(2\omega_v)(\omega_v+\op)}-\frac{e^{2i\omega_v t}-1}{(\omega_v-\op)(2\omega_v)}+\frac{e^{-i(\omega_v-\op)t} + e^{i(\omega_v+\op)t}-2}{\omega_v^2-\op^2}\right)
\right]\\
c_a^{(2)}&\sim& 
 -\frac{1}{\sqrt{2}}\left[
\frac{X^2_0}{-i(\omega_v+\op)}t-\frac{Y_0^2}{-i(\omega_v-\op)}t +
\frac{X^2_0}{i(\omega_v+\op)}( \frac{e^{i(\omega_v+\op)t}-1}{i(\omega_v+\op)})-\frac{Y_0^2}{i(\omega_v-\op)}(\frac{e^{-i(\omega_v-\op)t}-1}{-i(\omega_v-\op)})\right.\\
&+&\left. \frac{X_0 Y_0}{\omega_v^2-\op^2}\left(\frac{2i\op\sin 2\omega_v t}{(2\omega_v)}+\frac{2\omega_v(\cos 2\omega_v t-1)}{(2\omega_v)}+2( e^{i\op t}\cos\omega_vt-1) \right)
\right]\\
c_b^{(2)}&\sim&-{c_a^{(2)}}^* 
\end{eqnarray*}

The terms that will yield a time dependence that does not average to zero are
\begin{eqnarray*} 
\vev{c_a^{(2)}}&\rightarrow& -i\int_0^t\left\{ -\left[\frac{i}{\sqrt{2}}[ 
(X_0\frac{e^{-i(\omega_v+\op) \tp} -1}{-i(\omega_v+\op)} +Y_0\frac{e^{i(\omega_v-\op)\tp}-1}{-i(\omega_v-\op)})]\right]
(X_0e^{i\omega_v \tp}-Y_0e^{-i\omega_v \tp}) e^{i\op \tp}\right\}d\tp\\
&\rightarrow& -i\int_0^t\left\{ -\left[\frac{i}{\sqrt{2}}[ 
(X^2_0\frac{1}{-i(\omega_v+\op)} -Y^2_0\frac{1}{-i(\omega_v-\op)})]\right]\right\}
d\tp\\
&=&-\frac{i}{\sqrt{2}}\left(\frac{X_0^2}{\omega_v + \op}t -\frac{Y_0^2}{\omega_v-\op}t\right)\\
&=&-\frac{i}{\sqrt{2}}\left(\frac{X_0^2(\omega_v-\op) - Y_0^2(\omega_v +\op)}{\omega^2_v-\op^2}t\right)\\
&\sim& 1 -\exp\left( -\frac{i}{\sqrt{2}}\left(\frac{X_0^2(\omega_v-\op) - Y_0^2(\omega_v +\op)}{\omega^2_v-\op^2}t\right)\right)\\
&\sim& 1 -\exp\left( -\frac{i}{\sqrt{2}}\left(\frac{(X_0^2- Y_0^2)\omega_v-(X_0^2+Y_0^2)\op}{\omega^2_v-\op^2}t\right)\right)
=1-e^{-i\lambda t}\\
\end{eqnarray*}
\begin{eqnarray*}
X_0^2+Y_0^2&=& -\onehalf(\Omega_0^2+\Pi_0^2)=-\onehalf\left(z_0\frac{eB}{mc}\right)^2\left(\left(a_\mu\frac{\gamma}{\gamma+1}\right)^2 +\frac{\omega_v^2}{\omega^2}\left(a_\mu +\frac{1}{\gamma+1}\right)^2
\right)\left(\frac{\omega_v}{c}\right)^2\beta_{xy}^2(t)\\
&=&-\onehalf(\beta_z^0)^2\omega_a^2\left(\left(\frac{\gamma}{\gamma+1}\right)^2 +\frac{\omega_v^2}{\omega^2}\left(1 +\frac{1}{a_\mu(\gamma+1)}\right)^2
\right)\beta_{xy}^2(t)\\
X_0^2-Y_0^2&=& -\Omega_0\Pi_0=-(\beta_z^0\omega_a)^2\frac{\gamma}{\gamma+1}(1+\frac{1}{a_\mu(\gamma+1)})\frac{\omega_v}{\omega}\beta_{xy}^2(t)
\end{eqnarray*}
and
\begin{eqnarray*}
\lambda&=& -(\beta_z^0\omega_a)^2\beta^2_{xy}(t)\left(\frac{1}{\gamma+1}\right)^2\left[\left(\gamma(\gamma+1)+\frac{\gamma}{a_\mu}\right)\frac{\omega^2_v}{\omega}
-\onehalf\left(\gamma^2+\frac{\omega^2_v}{\omega^2}((\gamma+1) + \frac{1}{a_\mu})^2\right)\op\right]\frac{1}{\omega_v^2-\op^2} 
\end{eqnarray*}
In the limit where $\omega_v\ll \omega$,
\begin{eqnarray*}
\lambda&\sim& \onehalf\frac{(\beta_z^0\omega_a)^2}{\op}\beta^2_{xy}(t)\left(\frac{\gamma}{\gamma+1}\right)^2\\
\opp^2&=&\op^2+\lambda_0^2\sim\omega_a^2\left[\left(1-\frac{\gamma}{\gamma+1}\frac{(\beta_z^0)^2}{2}\right)^2 + \frac{(\beta_{xy}\beta_z^0)^2}{2}\left(\frac{\gamma}{\gamma+1}\right)^2\right]\\
&=& \omega_a^2\left[1 -(\beta_z^0)^2(\frac{\gamma}{\gamma+1}-\frac{1}{2}\frac{\gamma-1}{\gamma+1})\right]\\
&\sim& \omega_a^2\left[1 -\onehalf(\beta_z^0)^2\right]^{1/2}
\end{eqnarray*}
to order $(\beta_z^0)^2$. That is $\omega_a$ is shifted by average of $\beta_z^2$ which is $\onehalf(\beta_z^0)^2$ as long as $\beta_z$ is sinusoidal.
For reference we note that 
\begin{equation}
\opp = \op\left(1+\frac{\lambda_0^2}{\op^2}\right)^{1/2}\sim \op+\frac{\lambda_0^2}{2\op}\rightarrow \frac{\lambda_0^2}{2\op} = \lambda\label{eq:omegapp}
\end{equation}
At the magic momentum
\begin{eqnarray*}
X_0^2+Y_0^2
&\sim&-\onehalf(\beta_z^0)^2\omega_a^2\left(\left(\frac{\gamma}{\gamma+1}\right)^2 +\frac{\omega_v^2}{\omega^2}\gamma^2
\right)\frac{\gamma^2-1}{\gamma^2}\\
X_0^2-Y_0^2&\sim&-(\beta_z^0\omega_a)^2\frac{\gamma}{\gamma+1}\gamma\frac{\omega_v}{\omega}\frac{\gamma^2-1}{\gamma^2}\\
\lambda_0&=&\frac{(\beta_z^0\omega_a)^2}{\omega_v^2-\op^2}\left[\frac{\gamma^2}{\gamma+1}\frac{\omega_v^2}{\op}-
\onehalf\left(\left(\frac{\gamma}{\gamma+1}\right)^2 +\frac{\omega_v^2}{\op^2}\gamma^2 \right)\op\right]\frac{\gamma^2-1}{\gamma^2}\\
&=&\frac{(\beta_z^0\omega_a)^2}{\omega_v^2-\op^2}\left[\frac{\gamma^2-\onehalf\gamma^2(\gamma+1)}{\gamma+1}\frac{\omega_v^2}{\op}
-\frac{1}{2}\left(\left(\frac{\gamma}{\gamma+1}\right)^2  \right)\op\right]\frac{\gamma^2-1}{\gamma^2}\\
&=&\frac{(\beta_z^0\omega_a)^2}{\omega_v^2-\op^2}\left[\frac{\onehalf\gamma^2(1-\gamma)}{\gamma+1}\frac{\omega_v^2}{\op}
-\onehalf\left(\left(\frac{\gamma}{\gamma+1}\right)^2  \right)\op\right]\frac{\gamma^2-1}{\gamma^2}\\
&=&\frac{(\beta_z^0\omega_a)^2}{\omega_v^2-\op^2}\left[\frac{\onehalf\gamma^2(1-\gamma)(\gamma+1)}{(\gamma+1)^2}\frac{\omega_v^2}{\op}
-\onehalf\left(\left(\frac{\gamma}{\gamma+1}\right)^2  \right)\op\right]\frac{\gamma^2-1}{\gamma^2}\\
&=&\frac{(\beta_z^0\omega_a)^2}{\omega_v^2-\op^2}\frac{\gamma^2}{2(\gamma+1)^2}\left[\frac{\omega_v^2-\op^2}{\op}
-\gamma^2\frac{\omega_v^2}{\op}\right]\frac{\gamma^2-1}{\gamma^2}\\
\opp^2&=&\op^2+\lambda_0^2\sim\omega_a^2\left[\left(1-\frac{\gamma}{\gamma+1}\frac{(\beta_z^0)^2}{2}\right)^2
+\frac{(\beta_z^0)^2}{2}\frac{\gamma-1}{\gamma+1}\left(1-\gamma^2\frac{\omega_v^2}{\omega_v^2-\op^2}\right)\right]\\
&\sim&\op^2+\lambda_0^2\sim\omega_a^2\left[1-\frac{(\beta_z^0)^2}{2}
-\frac{(\beta_z^0)^2}{2}\frac{\gamma-1}{\gamma+1}\left(\gamma^2\frac{\omega_v^2}{\omega_v^2-\op^2}\right)\right]\\
\end{eqnarray*}
%Then
%\begin{eqnarray*}
%\opp&=&\sqrt{\op^2+\lambda^2}\\
% &\sim &\omega_a\left(1- (\beta_z^0)^2(1 + \frac{\omega_a}{2\op}\left[\gamma^2\frac{\omega_v^2}{(\omega_v^2-\op^2)}\right]\frac{\gamma-1}{\gamma+1})\right)^{1/2}\\
%&\sim& \omega_a\left(1- (\beta_z^0)^2(1 - \frac{\omega_a}{2\op}\left[1-\gamma^2\frac{\omega_v^2}{(\omega_v^2-\op^2)}\right]\frac{\gamma-1}{\gamma+1})\right)^{1/2}\\
%\end{eqnarray*}
%where we use the fact that $\omega_a \sim \op$ and to second order in $\beta_z^0$.


Next the polarization to second order
\begin{eqnarray*}
a&=& \frac{1}{\sqrt{2}}[1- i\left(\pi_o^\prime \frac{\sin\omega_v t}{\omega_v} + X_0\frac{e^{i(\omega_v +\op)t}-1}{i(\omega_v + \op)} 
+ Y_0\frac{e^{-i(\omega_v-\op)t}-1}{i(\omega_v-\op)}\right)+c_a^{(2)}]e^{-i(\op/2+\omega/2+\lambda)t}\\
a^*&=& \frac{1}{\sqrt{2}}\left(1 +i[\pi^\prime_0\frac{\sin\omega_v t}{\omega_v} + 
(X_0^*\frac{e^{-i(\omega_v+\op) t} -1}{-i(\omega_v+\op)} +Y_0^*\frac{e^{i(\omega_v-\op)t}-1}{-i(\omega_v-\op)})]+{c^*_a}^{(2)}\right)e^{i(\op/2+\omega/2+\lambda) t}\\
b&=&\frac{1}{\sqrt{2}}\left(1 -i[-\pi^\prime_0\frac{\sin\omega_v t}{\omega_v} + 
(X_0^*\frac{e^{-i(\omega_v+\op) t} -1}{-i(\omega_v+\op)} +Y_0^*\frac{e^{i(\omega_v-\op)t}-1}{-i(\omega_v-\op)})]+c_b^{(2)}\right)e^{i(\op/2+\omega/2+\lambda) t}
\end{eqnarray*}
\begin{eqnarray*}
%\vev{s_x}&=& \left(\cos(\op-\omega)t -2\pi_0^\prime\frac{\sin\omega_v t}{\omega_v}\sin(\op-\omega)t\right)+\ldots\\
%\vev{s_y}&=& \left(\sin(\op-\omega)t -2\pi_0^\prime\frac{\sin\omega_v t}{\omega_v}\cos(\op-\omega)t\right)+\ldots\\
|a|^2&\sim& \onehalf\left[1-2|X_0|\frac{\sin(\omega_v+\op)}{\omega_v+\op}-2|Y_0|\frac{\sin(\omega_v-\op)t}{\omega_v-\op}
+|X_0|^2\frac{2(1-\cos(\omega_v+\op)t}{(\omega_v+\op)^2}+|Y_0|^2\frac{2(1+\cos(\omega_v-\op)t}{(\omega_v-\op)^2}\right.\\
&+&\left. 2 \Re\left[X_0Y_0^*\frac{e^{-2i\omega_v t} + 1 -2e^{-i\omega_v t}\cos(\op t) + e^{2i\omega_v t}+1-2e^{i\omega_v t}\cos\op t}{\omega_v^2-\op^2}\right]+c_a^{(2)}+{c_a^{(2)}}^*\right)\\
|b|^2&\sim& \onehalf\left[1+2|X_0|\frac{\sin(\omega_v+\op)}{\omega_v+\op}+2|Y_0|\frac{\sin(\omega_v-\op)t}{\omega_v-\op}
+|X_0|^2\frac{2(1-\cos(\omega_v+\op)t}{(\omega_v+\op)^2}+|Y_0|^2\frac{2(1+\cos(\omega_v-\op)t}{(\omega_v-\op)^2}\right.\\
&+&\left. 2\Re\left[X_0Y_0^*\frac{e^{-2i\omega_v t} + 1 -2e^{-i\omega_v t}\cos(\op t) + e^{2i\omega_v t}+1-2e^{i\omega_v t}\cos\op t}{\omega_v^2-\op^2}\right]+c_b^{(2)}+{c_b^{(2)}}^*\right)\\
|a|^2-|b|^2&\sim&-2|X_0|\frac{\sin(\omega_v+\op)t}{\omega_v+\op}-2|Y_0|\frac{\sin(\omega_v-\op)t }{\omega_v-\op} + 2 \Re c_a^{(2)}\\
a^*b&\sim& \onehalf\left(1 +2i\pi_0^\prime\frac{\sin\omega_v t}{\omega_v}
+ 4 |X_0|^2 \frac{e^{-i(\omega_v+\op)t}\sin^2(\omega_v+\op)t/2)}{(\omega_v+\op)^2} 
+ 4|Y_0|^2\frac{e^{i(\omega_v-\op)t}\sin^2(\omega_v-\op)t/2}{(\omega_v-\op)^2}\right.\\
&-&\left. 4|X_0Y_0| \frac{e^{-i\op t}(\cos\op t-\cos\omega_v t)}{(\omega_v^2-\op^2)} 
\right)e^{i(\op+\omega)t}e^{2i\lambda t}
\end{eqnarray*}
We want the expectation value of something like
\begin{eqnarray*}
\bm\beta\cdot\bm\sigma&=&
\begin{pmatrix}-\beta_z^0\sin\omega_v t & -\beta_{xy}^0\cos(\beta_z^0\sin\omega_v t) e^{-i\omega t}\cr
-\beta_{xy}^0\cos(\beta_z^0\sin\omega_v t)e^{i\omega t} & \beta_z^0\sin\omega_v t\end{pmatrix}
\end{eqnarray*}
and the
\begin{eqnarray*}
\vev{\bm\beta\cdot\bm\sigma}&=&-a^*\beta_z^0\sin\omega_v t)a -a^*\beta_{xy}^0\cos(\beta_z^0\sin\omega_v t)e^{-i\omega t}b
-b^*\beta_{xy}^0\cos(\beta_z^0\sin\omega_v t)e^{i\omega t}a + b^*\beta_z^0\sin\omega_v t b\\
&=&(|b|^2-|a|^2)\beta_z^0\sin\omega_v t -(a^*be^{-i\omega t}+ab^*e^{i\omega t})\beta_{xy}^0\cos(\beta_z^0\sin\omega_v t)\\
&=&-2\left(|X_0|\frac{\sin(\omega_v+\op)t }{(\omega_v + \op)^2}+|Y_0|\frac{\sin(\omega_v-\op)t}{(\omega_v-\op)^2}\right)\beta_z^0\sin\omega_v t\\
&-&\left(\cos(\op+2\lambda) t   +2\pi_0^\prime\frac{\sin\omega_v t}{\omega_v}\sin(\op+2\lambda) t
 +4|X_0|^2\cos(\omega_v-2\lambda)t \frac{\sin^2(\omega_v+\op)t/2}{(\omega_v+\op)^2}\right.\\
&+&\left.4|Y_0|^2\cos(\omega_v+2\lambda)t\frac{\sin^2(\omega_v-\op)t/2}{(\omega_v-\op)^2}
+4|X_0Y_0|\cos(2\lambda t) \frac{\cos\op t-\cos\omega_v t)}{\omega_v^2-\op^2}
\right) \beta_{xy}^0\cos(\beta_z^0\sin\omega_v t)
\end{eqnarray*}
where we keep to order $(\beta_z^0)^2$.

What happens in the limit where $\omega_v << \op$. 

\begin{eqnarray*}
\vev{\bm\beta\cdot\bm\sigma}
&=&
-\left(\cos(\op+2\lambda) t +2\pi_0^\prime t \sin(\op+2\lambda) t
 +4|X_0|^2\cos(2\lambda)t \frac{\sin^2(\op)t/2}{\op^2}\right.\\
&+&\left.2|Y_0|^2\cos(2\lambda)t\frac{\sin^2(-\op)t/2}{\op^2}
+4X_0Y_0\cos(2\lambda t) \frac{\cos\op t)}{-\op^2}
\right) \beta_{xy}^0\\
&=&
-\left(\cos(\op+2\lambda) t +2\pi_0^\prime t \sin(\op+2\lambda) t
 +4\frac{\cos(2\lambda t)}{\op^2}((X_0^2+Y_0^2) \sin^2(\op t/2)+2X_0Y_0 \cos\op t)\right) \beta_{xy}^0\\
&=&
-\left(\cos(\op+2\lambda) t +2\pi_0^\prime t \sin(\op+2\lambda) t
 +\frac{\cos(2\lambda t)}{\op^2}((2X_0^2+2Y_0^2)(1-\cos(\op t))+4X_0Y_0 \cos\op t)\right) \beta_{xy}^0
\end{eqnarray*}
