2.5: Rate Equations

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.