Transforming to a rotating frame in the $x$-basis I was reading this paper on analytically Solvable driven time-dependent two level quantum systems. The Hamiltonian considered in the paper is the following: $$H=\sigma_z\cdot J(t)/2)+\sigma_x\cdot h/2$$ 
$U$ is the unitary evolution operator with elements $u_{ij}$ (Equation (2) in the paper). It is given that after transforming to a rotating in the x-basis the following equation is obtained (equation (3) in the paper): 
$$D_+=e^{+i\cdot h\cdot t/2}(u_{11}+u_{21})\cdot 1/\sqrt(2)$$ and $$D_-=e^{-i\cdot h\cdot t/2}(u_{11}-u_{21})\cdot 1/\sqrt(2)$$
What are these $D_+$ and $D_-$? I know that the general form of the rotation operator(which is also often denoted by $D$) for the spin system is given by:
$$
D(\hat n,\theta)=\mathrm e^{-i\theta\ \hat{n}\cdot\vec S}
$$
and when I computed it for rotation about the x-axis I got the rotation operator to be:$$D(\hat n,\theta)=\left[ {\begin{array}{cc}
   \cos(\theta/2) & -i\sin(\theta/2) \\ -i\sin(\theta/2) & \cos(\theta/2) \      \end{array} } \right]$$
What are the $D$s obtained while transforming to a rotating frame in the attached picture? Am I doing anything wrong here? 
The terminology is also quit confusing for me. Why is it called a "rotating" frame rather than the frame rotated by $\theta$? But, then that also does't make sense because in the original paper there is no $\theta$? So, then, what do they mean by "rotating"?
 A: If $|\pm\rangle$ are the eigenvectors of ${\hat \sigma}_x$, ${\hat \sigma}_x |\pm\rangle = \pm |\pm\rangle$, then a rotating $x$-basis is defined as
$$
|+\rangle(t) = \exp\left(-i\frac{\omega t}{2} {\hat \sigma}_x\right)|+\rangle = e^{-i\omega t/2 } |+\rangle
$$
$$
|-\rangle(t) = \exp\left(-i\frac{\omega t}{2} {\hat \sigma}_x\right)|-\rangle = e^{i\omega t/2 }|-\rangle
$$
Notice that it is still an eigenbasis of ${\hat \sigma}_x$, ${\hat \sigma}_x |\pm\rangle(t) = \pm |\pm\rangle(t)$. Check why it is referred to as a rotating basis by looking at the transformation between $|\pm\rangle(t)$ and $|\uparrow\rangle$, $|\downarrow\rangle$. 
In the present case take $\omega = h$ and let $d_\pm(t)$ be the expansion coefficients of the state vector in the rotating basis, that is,
$$
|\psi(t)\rangle = d_+(t)  |+\rangle(t) + d_-(t)|-\rangle(t)
$$
Since ${\hat \sigma}_z |\pm\rangle(t) = e^{\mp i ht/2} |\mp \rangle$, the Schroedinger equation yields
$$
i \frac{d|\psi\rangle}{dt} = i {\dot d}_+  |+\rangle(t) +  i d_+(t)  \frac{d}{dt} |+\rangle(t) + i {\dot d}_-(t) |-\rangle(t) + i d_-(t) \frac{d}{dt} |-\rangle(t) =
$$
$$
= i {\dot d}_+  |+\rangle(t) + \frac{h}{2} d_+(t)  |+\rangle(t) + i {\dot d}_-(t) |-\rangle(t) - \frac{h}{2} d_-(t) |-\rangle(t) =
$$
$$
\left[ \frac{J(t)}{2} d_-(t)e^{iht} + \frac{h}{2}d_+(t)\right] |+\rangle(t) + \left[ \frac{J(t)}{2} d_+(t)e^{-iht} - \frac{h}{2}d_-(t)\right] |-\rangle(t)
$$
wherefrom after simplification and identification follows
$$
{\dot d}_\pm(t) = - i \frac{J(t)}{2}e^{\pm iht} d_\mp(t)
$$
This looks very much like eq.(4) in the paper, but coefficients $d_\pm(t)$ are not yet related to the matrix elements $u_{11}$ and $u_{21}$ of the evolution operator ${\hat U}$, as required by eq.(3). To obtain the latter, and with it the meaning of the functions $D_\pm(t)$, let us look at the alternative expression for $|\psi(t)\rangle$ obtained by applying ${\hat U}(t)$ to the initial state vector in the z-basis, $|\psi(0) \rangle = c_1(0)|\uparrow\rangle + c_2(0)|\downarrow\rangle$:
$$
|\psi(t)\rangle = {\hat U}(t)|\psi(0) \rangle = c_1(0) {\hat U}(t)|\uparrow\rangle + c_2(0) {\hat U}(t)|\downarrow\rangle =
$$
$$
= \left[ u_{11}(t)c_1(0) - u^*_{21}(t) c_2(0)\right] |\uparrow\rangle + \left[ u_{21}(t) c_1(0) + u^*_{11}(t)c_2(0)\right]|\downarrow\rangle
$$
If we switch now to the x-basis and then to the rotating x-basis this reads
$$
|\psi(t)\rangle = \left[ u_{11}(t)c_1(0) - u^*_{21}(t) c_2(0)\right] \frac{1}{\sqrt{2}}\left(|+\rangle + |-\rangle\right) + \left[ u_{21}(t) c_1(0) + u^*_{11}(t)c_2(0)\right]\frac{1}{\sqrt{2}}\left(|+\rangle - |-\rangle\right) =
$$
$$
= \left[  \left[\frac{1}{\sqrt{2}} \left(u_{11} + u_{21} \right) e^{iht/2}\right] c_1(0) + \left[ \frac{1}{\sqrt{2}} \left(u_{11} - u_{21} \right) e^{-iht/2}\right]^* c_2(0)\right] |+\rangle(t) + 
$$
$$
+ \left[  \left[\frac{1}{\sqrt{2}} \left(u_{11} - u_{21} \right) e^{-iht/2}\right] c_1(0) - \left[ \frac{1}{\sqrt{2}} \left(u_{11} + u_{21} \right) e^{iht/2}\right]^* c_2(0)\right] |-\rangle(t)  =
$$
$$
=  \left[  D_+(t) c_1(0) + D^*_-(t) c_2(0)\right] |+\rangle(t) +  \left[  D_-(t) c_1(0) - D^*_+(t) c_2(0)\right] |-\rangle(t) 
$$
In other words, the functions $D_\pm(t)$ simply provide a convenient reparametrization of the evolution in the rotating x-basis. 
I leave it as an exercise to derive eq.(4) from the identification
$$
d_\pm(t) = D_\pm(t) c_1(0) \pm D^*_\mp(t) c_2(0)
$$
(Hint: coefficients $c_1(0)$, $c_2(0)$ are arbitrary).
