
\sect{Spin resonance}

%Baruch's  D43.

Spin Resonance: Consider a spin $\half$ particle with magnetic
moment   in a constant magnetic field ${B_{0}}$ in the ${z}$
direction and a perpendicular rotating magnetic field with
frequency $\omega$ and amplitude ${B_{1}}$; the Hamiltonian is
\[\ \hat{H} =\hat{H} _{0} + \half\hbar \omega_{ 1}
\left[\sigma_{ x}cos\left(\omega t\right) +\sigma_{
y}sin\left(\omega
t\right)\right]\]\\
where ${\hat{H}_{0}=\half\hbar\omega_{ 0}\sigma_{ z},
\half\hbar\omega_{0}=\mu B_{0}, \half\hbar\omega_{1}= \mu B_{1}}$
and ${\sigma _{x}, \sigma _{y}, \sigma _{z}}$ are the Pauli
matrices. The equilibrium density matrix is ${\hat{\rho}_{
eq}=exp\left( -\beta \hat{H}_{ 0}\right)/Tr \left[exp\left(-\beta
\hat{ H} _{0}\right)\right]}$, so that the heat bath drives the
system towards equilibrium with ${\hat{H} _{0}}$ while the weak
field ${B_{1}}$ opposes this tendency. Assume that the time
evolution of the density matrix ${\hat{\rho}\left(t\right)}$  is
determined by
\[\    d \hat{\rho} /dt  =  -\frac{i}{h}  \left[\hat{H} , \hat{\rho}\right ]
-\frac{\hat{\rho}-\hat{\rho}_{eq}}{\tau}\]\\
\begin{itemize}
\item [(a)]
Show that this equation has a stationary solution of the form
${\delta \rho _{11}=-\delta\rho   _{22}=a, \,\,\,\delta\rho_{
12}=\delta\rho_{  21}^{*}= be ^{-i\omega t}}$ where
${\delta\hat{\rho} =\hat{\rho}-\hat{\rho}_{eq}}$.
\item [(b)]
The term   ${-\left[\hat{\rho}-\hat{\rho}_{ eq}\right]/\tau}$
represents ${\left(- i/\hbar\right ) \left[(\hat{H}_{bath})
\hat{\rho} \right ]}$ where ${\hat{H}_{bath}}$ is the interaction
Hamiltonian with a heat bath. Show that the power absorption is
\[ \frac{ d}{dt}  Tr\left[(\hat{H}  + \hat{H}_{ bath})\hat{\rho}
\right ] = Tr\left[\frac{d\hat{H}}{dt} \hat{\rho}\right]\]
\item [(c)]
Determine ${b}$ to first order in ${B_{1}}$ (for which ${a=0}$ can
be assumed), derive the power absorption and show that it has a
maximum at ${\omega =\omega_{ 0}}$, i.e. a resonance phenomena.
Show that ${\left(d/dt\right) Tr\left( \hat{\rho}\hat{ H}
\right)=0}$, i.e. the absorption is dissipation into the heat
bath.\\

\end{itemize}


