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\heading{Quantum dynamical echoes} 
\auname{Nitzan Maman}


{\bf Introduction:}


I develop this question from articles witch describes polarization
echoes in NMR. It is demonstrate that it is possible to revers the
time evolution of polarization in a
dipole-coupled nuclear spin system.\\



%%%%%%%%%%%%%%%%%%
{\bf Question:}

Consider a static solid that containing two nuclear spins species,
\emph{I} and \emph{S}. The \emph{I} spins are abundant and the
\emph{S} spins rare. The system is simultaneously irradiated with
two rf fields $B_{1I}$ and $B_{2I}$ of frequencies $\omega_{1I}$
and $\omega_{2S}$ that close to Larmor frequencies and The
resonant offsets are small.\

\textbf{A}. Write the hamiltonian of the system and explain its
terms.\

\textbf{B}. What are the condition for time reversal.

\textbf{C}. Give quality explanation of the phenomena and give
simple classical example that help to understanding this nature.


%%%%%%%%%%%%%
{\bf Answer:}

\textbf{A}. The \emph{S} spins are rare so we can ignore the
$\emph{S-S}$ interactions, and we can treat the system as an
ensemble of subsystems, each consisting of a single \emph{S} spin
and a large number of \emph{I} spins. after the high-field
truncations the hamiltonian compose from 4 terms: 
%
\be{0}
H=H_{I}+H_{S}+H_{IS}+H_{II}
\ee
%
while:
\be{0}
&H_{I}&=\sum_{k}\Omega_{k}I_{kz}+ \omega_{1I}\sum_{k} I_{kz}\\
&H_{S}&=\Omega_{s}S_{z}+ \omega_{1s} S_{kz}
\ee
is the interaction of the spins \emph{I} and spin \emph{S} with
the rf-fields. $\Omega_{k}$ and $\Omega_{s}$ are the resonance
offsets and $\omega_{1I}$ and $\omega_{1s}$ are the rf-field
strengths. According to the question the $\Omega_{k}$ and
$\Omega_{s}$ are negligibly small and may be ignored.

The other two terms are the hetronuclear dipolar coupling between
the spin \emph{S} and the \emph{I} spins.
\be{0}
&H_{IS}&=\sum_{k}b_{k}2I_{kz}S_{z}\ee


with the hetronuclear dipolar constant:
\be{0}
&b_{k}&=-\frac{\mu_{0}\gamma_{I}\gamma_{S}\hbar^{2}}{4\pi
r_{k}^{3}}\frac{1}{2}(3\cos^{2}\theta_{k} -1)\ee



and the homonuclear dipolar coupling of the \emph{I} spins.
\be{0}
&H_{II}&=\sum_{J}\sum_{K}\emph{d}_{jk}[2I_{jz}I_{kz}-\frac{1}{2}({I_{j}^{+}}{I_{k}^{-}}+I_{j}^{-}I_{k}^{+})]
\ee
with the homonuclear \emph{I}\emph{I} dipolar coupling constant:
\be{0}
&\emph{d}_{jk}& =-\frac{\mu_{0}\gamma_{I}^{2}\hbar^{2}}{4\pi
r_{jk}^{2}}\frac{1}{2}(3\cos^{2}\theta_{jk}-1)
\ee
where the $r_{k}$ and $r_{jk}$ are the internuclear distances and
$ \theta_{k}$ and $\theta_{jk}$ are the angels between the
internuclear vectors and the static magnetic field.

\textbf{B}.In the case of strong irradiation on both channels,

$ |\omega_{1I}| \gg | d_{jk} | $ 
and
$ |\omega_{1s} | \gg | b_{k} | $

we will write the hamiltonian in a frame rotated about the y axis
which defined as:
\be{0}
A^{T}=exp[i\frac{\pi}{2}(\sum_{k}I_{ky}+S_{y})]\cdot A
\cdot exp[-i\frac{\pi}{2}(\sum_{k}I_{ky}+S_{y})]
\ee
so that:
\be{0}
&H^{T}&=H_{I}^{T}+H_{S}^{T}+H_{IS}^{T}+H_{II}^{T}
\ee
\be{0}
&H_{I}^{T}&= \omega_{1I}\sum_{k} I_{kz}\\
&H_{S}^{T}&= \omega_{1s} S_{kz}\\
&H_{IS}^{T}&=\sum_{k}b_{k}2I_{kx}S_{x}
\ee
The homonuclear dipolar interaction hamiltonian in the tilted
frame $H_{II}^{T}$ divide into two terms $H_{II}^{T'}$ and
$H_{II}^{T''}$ secular and nonsecular with respect to the large
hamiltonian $H_{I}^{T}$.
\be{0}
&H_{II}^{T'}&=-\frac{1}{2}\sum_{J}\sum_{Kj}\emph{d}_{jk}[2I_{jz}I_{kz}-\frac{1}{2}({I_{j}^{+}}{I_{k}^{-}}+I_{j}^{-}I_{k}^{+})]\\
&H_{II}^{T''}&=\frac{3}{2}\sum_{J}\sum_{Kj}\emph{d}_{jk}\frac{1}{2}[({I_{j}^{+}}{I_{k}^{+}}+I_{j}^{-}I_{k}^{-})]\\
\ee
For large \emph{I} spin fields $ (|\omega_{1I}|\gg|d_{jk}|) $
the nonsecular terms $ H_{II}^{T''} $ may usually be ignored.
After we neglected the nonsecular terms we have the factor
$-\frac{1}{2}$ that multiply the homonuclear dipolar interaction
hamiltonian and experimentally switching the
evolution from one situation to the other, which leads to a
polarization echoes.

\textbf{C}. The magnetic dipolar interaction is anisotropic. Any
network homologous spins can be describe, in the quantization
axis of the external field, by the hamiltonian [6]. Now if a
$\pi/2$ pulse is applied and then irradiation is maintained in
the direction of the polarization (spin lock) with reasonable rf
power, the quantization axis should be taken along the rf field.
During the time, the spin dynamics is described (after we neglect
the non-secular terms) by:
\be{0}
&H_{II}^{'}&=-[\frac{1}{2}]H_{II}
\ee
simple classical system that can demonstrate this phenomena is
of two magnetic moments that repel or attract each other
depending on the angle between their orientation and the
internuclear vector.


\ \\

\includegraphics[height=2in]{nitzan_fig.eps} 

{\footnotesize two magnets constrained to a plane: in
the left side the magnets are repel each other and the energy is
positive. when the magnets rotate in a $\pi/2$ angle (right side)
they are aligned along their internuclear vector and they attract
each other (negative energy).} \label{complex}



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