$\newcommand{\+}{^{\dagger}}\newcommand{\angles}[1]{\left\langle #1 \right\rangle}\newcommand{\bose}{\,{\rm n}}\newcommand{\bra}[1]{\left\langle #1 \right\vert}\newcommand{\braces}[1]{\left\lbrace #1 \right\rbrace}\newcommand{\bracks}[1]{\left\lbrack #1 \right\rbrack}\newcommand{\braket}[2]{\left\langle #1 \vert #2 \right\rangle}\newcommand{\braketi}[2]{\left.\left\langle #1 \right\vert #2 \right\rangle}\newcommand{\braketd}[2]{\left\langle #1 \left\vert #2 \right\rangle\right.}\newcommand{\ck}{{\rm C_{K}}}\newcommand{\dd}{{\rm d}}\newcommand{\dos}{\,{\cal D}}\newcommand{\ds}[1]{\displaystyle{#1}}\newcommand{\eofm}[3]{\left\langle #1 \left\vert #2 \right\vert #3 \right\rangle}\newcommand{\expo}[1]{{\rm e}^{#1}}\newcommand{\fermi}{\,{\rm f}}\newcommand{\fvec}[1]{\,\vec{\rm #1}}\newcommand{\half}{{1 \over 2}}\newcommand{\ic}{{\rm i}}\newcommand{\iff}{\Leftrightarrow}\newcommand{\imp}{\Longrightarrow}\newcommand{\kb}{{\rm k_{B}}}\newcommand{\kelvin}{\,{\rm K}}\newcommand{\ket}[1]{\left\vert #1 \right\rangle}\newcommand{\ketbra}[2]{\left\vert #1 \right\rangle\left\langle #2 \right\vert}\newcommand{\mat}[1]{{\sf #1}}\newcommand{\ob}[2]{\overbrace{ #1 }^{#2}}\newcommand{\ol}[1]{\overline{#1}}\newcommand{\pars}[1]{\left( #1 \right)}\newcommand{\partiald}[3][]{{\partial^{#1}#2 \over \partial #3^{#1}}}\newcommand{\pp}{{\cal P}}\newcommand{\raiz}[2][]{\,\sqrt[#1]{\, {#2}\, }\,}\newcommand{\sen}{\,{\rm sen}}\newcommand{\sgn}{\,{\rm sgn}}\newcommand{\ss}[1]{\scriptstyle{#1}}\newcommand{\sss}[1]{\scriptscriptstyle{#1}}\newcommand{\tk}{{\rm T_{K}}}\newcommand{\trace}{{\rm Tr}}\newcommand{\totald}[3][]{{{\rm d}^{#1}#2 \over {\rm d}^{#1}#3}}\newcommand{\ub}[2]{\underbrace{ #1 }_{#2}}\newcommand{\ul}[1]{\underline{#1}}\newcommand{\verts}[1]{\left\vert #1 \right\vert}\newcommand{\wt}[1]{\widetilde{#1}}$
Átomo + Radiación: Ecuaciones de
Movimiento
Uno de los propósitos de esta sección es evaluar la
probabilidad $\pp_{+}\pars{t}$ de que el átomo se encuentre en el
estado excitado $\ket{+}$, en el instante $t$, la cual viene dada por
$\angles{n\pars{t}} = \angles{b\+\pars{t}b\pars{t}} = \pp_{+}\pars{t}$.
Ello sugiere la introducción de la función de
Green-Keldysh ${\rm G}\pars{t,t'}$:
$$
{\rm G}\pars{t,t'} \equiv -\ic\angles{\tk b\pars{t}b\+\pars{t'}} \equiv
-\ic\Theta\pars{t,t'}\angles{b\pars{t}b\+\pars{t'}}
-\ic\Theta\pars{t',t}\angles{b\+\pars{t'}b\pars{t}}
$$
\begin{align*}
\mbox{Note que}&
\left\lbrace\begin{array}{rcl}
{\rm G}^{<}\pars{t,t'} & = &
-\ic\angles{b\+\pars{t'}b\pars{t}}
\\[2mm]
{\rm G}^{>}\pars{t,t'} & = & -\ic\angles{b\pars{t}b\+\pars{t'}}
\end{array}\right.
\\[5mm]\mbox{tal que}\
{\rm G}^{\rm\pars{r}}\pars{t,t'}
&= \Theta\pars{t - t'}
\bracks{{\rm G}^{>}\pars{t,t'} - {\rm G}^{<}\pars{t,t'}}
=-\ic\Theta\pars{t - t'}\angles{\bracks{b\pars{t},b\+\pars{t'}}}
\\[3mm]\mbox{y}\
{\rm G}^{\rm\pars{r}}\pars{t,t^{-}} & \equiv
\lim_{t' \to t^{-}}{\rm G}^{\rm\pars{r}}\pars{t,t'}
=\ic\angles{\sigma_{z}\pars{t}}
\\[3mm]\imp\quad&
\begin{array}{|c|}\hline\\ \mbox{}\\
\quad\pp_{+}\pars{t} =
\half - \half\,\ic\,{\rm G}^{\rm\pars{r}}\pars{t,t^{-}}\quad
\\ \mbox{}\\ \hline
\end{array}
\end{align*}
${\rm G}\pars{t,t'}$ satisface la ecuación de movimiento
\begin{equation}
\ic\,\partiald{{\rm G}\pars{t,t'}}{t}\ =\
\ob{\delta\pars{t,t'}\angles{\bracks{b\pars{t},b\+\pars{t'}}}}
{\ds{-\delta\pars{t,t'}\angles{\sigma_{z}\pars{t}}}}\ -\
\ic\angles{\tk\bracks{b\pars{t},H}b\+\pars{t'}}
\label{GbbAR}
\end{equation}
donde
$$
H = E_{-} + \Delta\,b\+b
+ \sum_{k}\pars{a_{k}\+a_{k} + \half}\omega_{k}
-{\cal V}
\sum_{k}\omega_{k}^{1/2}\sen\pars{kZ}\pars{a_{k}\+b + b\+a_{k}}\,,\quad
{\cal V} = \raiz{4\pi \over \Omega}\,\wp
$$
es el hamiltoniano del Átomo + Radiación ( ver
sección anterior ).
Note que
$\bracks{b,H} = \Delta b
+{\cal V}\sigma_{z}\sum_{k}\omega_{k}^{1/2}\sen\pars{kZ}a_{k}\,,\qquad
\sigma_{z} = 2n - 1 = 2b\+b - 1$
La Ec. \eqref{GbbAR} se reduce a
\begin{align}
\pars{\ic\,\partiald{}{t} - \Delta}{\rm G}\pars{t,t'} &=
-\delta\pars{t,t'}\angles{\sigma_{z}\pars{t}}
+ {\cal V}\sum_{k}\omega_{k}^{1/2}\sen\pars{kZ}{\cal F}_{k}\pars{t,t'}
\label{GcompletaAR}
\\[3mm]
{\cal F}_{k}\pars{t,t'} & =
-\ic\angles{\tk\sigma_{z}\pars{t}a_{k}\pars{t}b\+\pars{t'}}
\end{align}
${\cal F}_{k}\pars{t,t'}$ satisface
\begin{equation}
\ic\,\partiald{{\cal F}_{k}\pars{t,t'}}{t} =
\delta\pars{t,t'}
\angles{\bracks{\sigma_{z}\pars{t}a_{k}\pars{t},b\+\pars{t'}}}
-\ic\angles{\tk\bracks{\sigma_{z}\pars{t}a_{k}\pars{t},H}b\+\pars{t'}}
\label{ecmoffkttp}
\end{equation}
con
$$
\bracks{\sigma_{z}a_{k},b\+} = \bracks{\sigma_{z},b\+}a_{k} = 2b\+a_{k}
$$
y
\begin{align}
\bracks{\sigma_{z}a_{k},H} & =
\sum_{q}\omega_{q}\ \ob{\bracks{\sigma_{z}a_{k},a_{q}\+a_{q}}}
{\ds{\delta_{kq}\sigma_{z}a_{k}}}
\nonumber
\\[3mm] & -{\cal V}\sum_{q}\omega_{q}^{1/2}\sen\pars{qZ}\braces{%
\ob{\bracks{\sigma_{z}a_{k},a_{q}\+b}}
{\ds{-\delta_{kq}b - 2a_{q}\+ba_{k}}}\ +\
\ob{\bracks{\sigma_{z}a_{k},b\+a_{q}}}{\ds{2b\+a_{q}a_{k}}}}
\nonumber
\\[5mm] & =
\omega_{k}\sigma_{z}a_{k} + {\cal V}\omega_{k}^{1/2}\sen\pars{kZ}b
\nonumber
\\[3mm]&
\mbox{} + 2{\cal V}\sum_{q}\omega_{q}^{1/2}\sen\pars{qZ}a_{q}\+ba_{k}
- 2{\cal V}\sum_{q}\omega_{q}^{1/2}\sen\pars{qZ}b\+a_{q}a_{k}
\label{conmumaspro}
\end{align}
Con este resultado, Ec. \eqref{ecmoffkttp} satisface
\begin{equation}
\pars{\ic\,\partiald{}{t} - \omega_{k}}{\cal F}_{k}\pars{t,t'} =
2\delta\pars{t,t'}\angles{b\+\pars{t}a_{k}\pars{t}} +
{\cal V}\omega_{k}^{1/2}\sen\pars{kZ}{\rm G}\pars{t,t'}
\label{ecFreduc}
\end{equation}
donde hemos despreciado los dos últimos términos en
la identidad \eqref{conmumaspro}.
$\fermi_{k}\pars{t,t'} \equiv {\cal F}_{k}\pars{t,t'}_{{\cal V}\ =\ 0}$
satisface la ecuación de movimiento
\begin{equation}
\pars{\ic\,\partiald{}{t} - \omega_{k}}\fermi_{k}\pars{t,t'} = 0
\label{ecfreduc}
\end{equation}
puesto que en el estado inicial el átomo se encuentra en el estado
excitado $\ket{+}$ en ausencia de fotones:
$$
\angles{b\+\pars{t}a_{k}\pars{t}}_{{\cal V}\ =\ 0}
=
\expo{-\ic\pars{\omega_{k} - \Delta}t}\
\ob{\angles{b\+a_{k}}_{{\cal V}\ =\ 0}}{\ds{=\ 0}}\phantom{AA} =\ 0
$$
\eqref{ecFreduc} y \eqref{ecfreduc} conducen a:
$$
\pars{\ic\,\partiald{}{t} - \omega_{k}}
\bracks{{\cal F}_{k}\pars{t,t'} - \fermi_{k}\pars{t,t'}} =
2\delta\pars{t,t'}\angles{b\+\pars{t}a_{k}\pars{t}} +
{\cal V}\omega_{k}^{1/2}\sen\pars{kZ}{\rm G}\pars{t,t'}
$$
la cual puede reescribirse como la ecuación integral
\begin{align*}
{\cal F}_{k}\pars{t,t'} & = \fermi_{k}\pars{t,t'} +
\oint_{\ck}{\rm g}_{k}\pars{t,t''}\bracks{%
2\delta\pars{t'',t'}\angles{b\+\pars{t''}a_{k}\pars{t''}} +
{\cal V}\omega_{k}^{1/2}\sen\pars{kZ}{\rm G}\pars{t'',t'}}\,\dd t''
\\[3mm] & =
\fermi_{k}\pars{t,t'} +
{\rm g}_{k}\pars{t,t'}\angles{b\+\pars{t'}a_{k}\pars{t'}} +
2{\cal V}\omega_{k}^{1/2}\sen\pars{kZ}
\oint_{\ck}{\rm g}_{k}\pars{t,t''}{\rm G}\pars{t'',t'}\,\dd t''
\end{align*}