Rotating Field

This is defined as for constant field in $z$ direction and an oscillating field in the $x$- $y$ plane,

$$B_z(t)=B_0={\rm const},\quad B_{\Vert}(t) = B_1e^{i\omega t}.$$ (2.10)

Our equation for $\psi_2$ thus becomes

$$i \ddot{\psi}_2 = i\frac{\dot{B_{\Vert}}}{B_{\Vert}} \dot{\psi}_2 + \left[ \frac{\d... ...}}{B_{\Vert}} B_z -i B_z^2 -i \vert B_{\Vert}\vert^2 - \dot{B}_z \right] \psi_2$$

$$= -\omega \dot{\psi}_2 + i[\omega B_0 - B_0^2 - \vert B_1\vert^2 ]\psi_2$$

$$\leadsto \ddot{\psi}_2 - i\omega \dot{\psi}_2 + [B_0^2 + \vert B_1\vert^2 - \omega B_0]\psi_2=0.$$ (2.11)

This can be solved using the exponential ansatz method $\psi_2(t)= ce^{-izt}$ which yields a quadratic equation for $z$,

$$z^2 - \omega z + [\omega B_0 -B_0^2 - \vert B_1\vert^2] =0$$

$$z_{\pm} = \frac{\omega}{2}\pm \frac{1}{2}\sqrt{\omega^2 + 4B_0^2 -4\omega B_0 + 4 \vert B_1\vert^2} = \frac{\omega}{2}\pm \frac{1}{2} \Omega_R$$

$$\Omega_R \equiv \sqrt{ (\omega-2B_0)^2+ 4 \vert B_1\vert^2} .$$ (2.12)

Note that the term $2B_0$ in the Rabi-frequency is determined by the level-splitting $\Delta\equiv 2B_0$ in absence of the time-dependent field $B_{\Vert}(t)$. The solution for $\psi_2(t)$ (from which $\psi_1(t)$ follows immediately) therefore is

$$\psi_2(t) = c_1 e^{i\left( \frac{\omega}{2}+ \frac{\Omega_R}{2}\right)t} + c_2 e^{i\left( \frac{\omega}{2}- \frac{\Omega_R}{2}\right)t}$$

$$= e^{i\frac{\omega}{2}t}\left[ c'_1 \cos \frac{\Omega_R}{2} t + c'_2 \sin \frac{\Omega_R}{2} t\right].$$ (2.13)

We can choose, e.g., the initial condition $\psi_2(0)=1$ from which follows

$$\psi_2(t) = e^{i\frac{\omega}{2}t} \left[\cos \frac{\Omega_R}{2} t + c'_2 \sin \frac{\Omega_R}{2} t\right]$$

$$0=\psi_1(0) = \left.\frac{i\dot{\psi}_2 + B_z \psi_2}{ B_{\Vert}}\right\Vert _{t=0}= \frac{-\frac{\omega}{2} + i\frac{\Omega_R}{2}c'_2 + B_0 }{B_1}$$

$$\leadsto c'_2 = -i \frac{\omega-2B_0}{\Omega_R}$$ (2.14)

This leads to

$$\vert \psi_2(t)\vert^2 = \cos ^2\frac{\Omega_R}{2} t + \frac{(\omega-2B_0)^2}{\Omega_R^2} \sin ^2\frac{\Omega_R}{2} t$$ (2.15)

$$= \frac{(\omega-2B_0)^2}{\Omega_R^2} + \frac{4\vert B_1\vert^2}{\Omega_R^2} \cos ^2\frac{\Omega_R}{2} t .$$

Note that the quantum-mechanical oscillations at constant ${\bf B}$ (e.g., the case ${\bf B}=(T_c,0,0)$ in Eq. (VI.1.18)) occur for a time-independent Hamiltonian. The Rabi-oscillations occur in a time-dependent Hamiltonian containing a time-dependent term (`time-dependent field'). These two often get mixed up in the literature.