2.5: Rate Equations
- Page ID
- 48283
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)With the wave equation Eq.(2.2.2) and the expression for the polarization induced by the electric field of the wave, we end up with the complete Maxwell-Bloch equations describing an electromagnetic field interacting with a statistical ensemble of atoms that are located at postions \(z_i\)
\[\left ( \Delta - \dfrac{1}{c_0^2} \dfrac{\partial^2}{\partial t^2}\right ) \vec{E}^{(+)} (z, t) = \mu_0 \dfrac{\partial^2}{\partial t^2} \vec{P}^{(+)} (z, t),\label{eq2.5.1} \]
\[\vec{P}^{(+)} (z, t) = -2N \vec{M} ^* d(z, t) \nonumber \]
\[\dot{d} (z, t) = -(\dfrac{1}{T_2} - j \omega_{eg} )d + \dfrac{1}{2j \hbar} \vec{M} \vec{E}^{(+)} w, \nonumber \]
\[\dot{w} (z, t) = -\dfrac{w - w_0}{T_1} + \dfrac{1}{j\hbar} (\vec{M}^*\vec{E}^{(-)} d - \vec{M} \vec{E}^{(+)} d^*) \nonumber \]
In the following we consider a electromagnetic wave with polarization vector \(\vec{e}\), frequency \(\omega_{eg}\) and wave number \(k_0 = \omega_{eg}/c_0\) with a slowly varying envelope propagating to the right
\[\vec{E} (z, t)^{(+)} = \sqrt{2 Z_{F_0}} A(z, t) e^{j(\omega_{eg} t - k_0 z)} \vec{e}, \nonumber \]
with
\[\left |\dfrac{\partial A(z,t)}{\partial t} \right |, \left |c\dfrac{\partial A(z,t)}{\partial z} \right |\ll |\omega_{eg} A(z, t)| \nonumber \]
Note, we normalized the complex amplitude \(A(t)\) such that its magnitude square is proportional to the intensity of the wave. This will also excite a wave of dipole moments in the atomic medium according to
\[d(z, t) = \hat{d} (z, t) e^{j(\omega_{eg} t - k_0 z)}, \nonumber \]
that is also slowly varying. In that case, we obtain from Eq.(\(\ref{eq2.5.1}\)) in leading order
\[\left (\dfrac{\partial}{\partial z} + \dfrac{1}{c_0} \dfrac{\partial}{\partial t} \right ) A(z, t) = jN \vec{e}^T \vec{M}^* \sqrt{\dfrac{Z_{F_0}}{2}} \hat{d} (z, t) \nonumber \]
\[\dfrac{\partial}{\partial t} d(z, t) = -\dfrac{1}{T_2} \hat{d} + \dfrac{\sqrt{2Z_{F_0}}}{2j\hbar} (\vec{M} \vec{e}) A(t) w \nonumber \]
\[\dfrac{\partial}{\partial t} w(z,t) = -\dfrac{w - w_0}{T_1} + \dfrac{\sqrt{2Z_{F_0}}}{j\hbar}((\vec{M}^* \vec{e}^*)A^*(t) \hat{d} - (\vec{M} \vec{e})A(t)\hat{d}^*) \nonumber \]
Furthermore, in the limit, where the dephasing time \(T_2\) is much faster than the variation in the envelope of the electric field, one can adiabatically eliminate the rapidly decaying dipole moment, i.e.
\[\hat{d} = T_2 \dfrac{\sqrt{2Z_{F_0}}}{2j\hbar} (\vec{M} \vec{e}) A(t) w \nonumber \]
\[\hat{w} = -\dfrac{w - w_0}{T_1} + \dfrac{|A(t)|^2}{E_s}w, \nonumber \]
where \(E_s = I_sT_1\), is called the saturation fluence, \([J/cm^2]\), of the medium. Note, now we don’t have to care anymore about the dipole moment and we are left over with a rate equation for the population difference of the
medium and the complex field amplitude of the wave.
\[\left (\dfrac{\partial}{\partial z} + \dfrac{1}{c_0} \dfrac{\partial}{\partial t} \right ) A(z, t) = \dfrac{N\hbar}{4T_2 E_s} w (z, t) A(z,t),\label{eq2.5.13} \]
\[\dot{w} = -\dfrac{w - w_0}{T_1} + \dfrac{|A(z, t)|^2}{E_s} w(z, t) \nonumber \]
Equation (\(\ref{eq2.5.13}\)) clearly shows that we obtain gain for an inverted medium and that the gain saturates with the electromagnetic power density flowing through the medium.