View source for Saturation of atomic transitions

= Saturation =
== The saturated transition rate ==
Consider an ensemble of <math>n</math> two-lvel atoms with levels <math>|a{\rangle}</math> and
<math>|b{\rangle}</math>:
::[[Image:Saturation_of_atomic_transitions-two-level-saturation.png|thumb|400px|none|]]
<math>\Gamma</math> is the spontaneous emission rate, and <math>R^U</math> is the transition
rate given by the coupling between incident photons and the atoms.  We
call <math>R^u</math> the unsaturated transition rate, because the actual
net transfer rate from <math>|a{\rangle}</math> to <math>|b{\rangle}</math>, is close to <math>R^u</math> only when
most atoms start in state <math>|a{\rangle}</math>.  When a sizable fraction of the <math>n</math>
atoms are already excited, the net transfer rate slows down.
Eventually, upon sufficient excitation, the number of atoms in the two
states, <math>n_a</math> and <math>n_b</math> no longer change.  Between these two extremes,
it is useful to define a saturated transition rate <math>R^s</math>, which
gives the net transfer rate from <math>|a{\rangle}</math> to <math>|b{\rangle}</math> caused by incident
photons, per atom in the ensemble.  What is a mathematical expression
for <math>R^s</math>?
The rate equations governing the atomic populations are:
:<math>\begin{array}{rcl}  
	\dot{n}_b &=& - n_b (R^u + \Gamma) + n_a R^u
\\	\dot{n}_a &=& n_b (R^u + \Gamma) - n_a R^u
\,.
\end{array}</math>
In the absence of spontaneous emission, these eqautions show that the
net transition rate between the levels is given by <math>R^u(n_a-n_b)</math>, so
that the saturated transition rate is
:<math>\begin{array}{rcl}  
	R^s &=& R^u \frac{n_a-n_b}{n_a+n_b}
\\	    &=& R^u \frac{1-\frac{n_b}{n_a}}{1+\frac{n_b}{n_a}}
\,.
\end{array}</math>
The steady state populations, obtained by setting <math>\dot{n}_a</math> and
<math>\dot{n}_b</math> to zero, satisfy
:<math> 
	\frac{n_b}{n_a} = \frac{R^u}{\Gamma + R^u}
\,,
</math>
which can be conveneintly re-expressed as
:<math> 
	\frac{n_b}{n_a} = \frac{s}{2 + s}
\,,
</math>
by defining the saturation parameter
:<math> 
	s = \frac{2 R^u}{\Gamma}
\,.
</math>
We will return later to the physical meaning of the saturation parameter.
Subsituting this into the definition of <math>R^s</math> gives
:<math>\begin{array}{rcl}  
	R^s &=& \frac{\Gamma s}{2} \frac{1-s/(2+s)}{1+s/(2+s)}
\\	 &=& \frac{\Gamma s}{2} \frac{2+s-s}{2+s+s}
\\	 &=& \frac{\Gamma s}{2} \frac{2}{2+2s}
\\	 &=& \frac{\Gamma}{2} \frac{s}{1+s}
\,.
\end{array}</math>
This is a very useful expression. In the limit of <math>R^u\ll \Gamma</math>
(small <math>s</math>), far from saturation, <math>R^s \rightarrow R^u</math>, and in the
limit of <math>R^u\gg \Gamma</math> (<math>s\rightarrow \infty</math>), complete saturation,
<math>R^s\rightarrow \Gamma/2</math>.
== Monochromatic Radiation ==
An important case is when the incident radiation is monochromatic, and
has a Lorentzian frequency distribution, <math>f(\delta)</math>, where <math>\delta</math>
is the detuning from resonance.  Recall that the unsaturated
transition rate is then given by
:<math> 
	R^u(\delta) = \frac{\pi}{2} \omega_R^2 f(\delta)
\,,
</math>
where <math>\omega_R</math> is the Rabi frequency.  Substituting for <math>f(\delta)</math>
a normalized Lorentzian distribution,
:<math> 
	f(\delta) = \frac{\pi}{2} \, \frac{1/\Gamma}{1+(2\delta/\Gamma)^2}
\,,
</math>
we obtain
:<math> 
	R^u(\delta) =\frac{\omega_R^2}{\Gamma} \,
			\frac{1}{1+(2\delta/\Gamma)^2}
\,,
</math>
and this expression for the saturation parameter
:<math> 
	s(\delta) = \frac{2\omega_R^2}{\Gamma^2} \frac{1}{1+(2\delta/\Gamma)^2}
\,.
</math>
The saturated transition rate per atom, as a function of detuning, is thus
:<math>\begin{array}{rcl}  
	R^s(\delta) = \frac{\omega_R^2}{\Gamma}
	      \frac{1}{1+(2\delta/\Gamma)^2+2(\omega_R/\Gamma)^2}
\,.
\end{array}</math>
This is an important and useful expression.  It tells us what the
transition rate is, as measured in an experiment, for example, by
absorption.  Note that this is a Lorentzian distribution, with a width
which depends on <math>\omega_R</math>.
Defining the on-resonance saturation parameter
:<math> 
	s_0 = \frac{2\omega_R^2}{\Gamma^2}
\,,
</math>
such that the saturation parameter is <math>s =
s_0/(1+(2\delta/\Gamma)^2)</math>, we find that the full width at half
maximum of <math>R^s(\delta)</math> is given by
:<math> 
	\Gamma' = \Gamma \sqrt{1+s_{0}}
\,.
</math>
indicating that the transition broadens in frequency as the
incident intensity of light (which determines <math>\omega_R</math>) increases.
Physically, this happens because of the effective shortening of the
lifetime of atoms in <math>|b{\rangle}</math> due to the stimulated emission.
The saturated transition rate per atom, expressed in terms of <math>s_0</math>, is
:<math>\begin{array}{rcl}  
	R^s(\delta) = \frac{s_0\Gamma/2}{1+s_0 + (2\delta/\Gamma)^2}
\,.
\end{array}</math>
and can be re-expressed using the power broadened linewidth <math>\Gamma'</math> as
:<math>\begin{array}{rcl}  
	R^s(\delta) = \frac{s_0}{1+s_0} \frac{\Gamma/2}{1+(2\delta/\Gamma')}
\,.
\end{array}</math>
Note that in the limit that <math>s_0\rightarrow \infty</math>, <math>R^s\rightarrow
\Gamma/2</math>.  A plot of this is shown below (figure from page 26 of {\em
Laser Cooling and Trapping}, by Metcalf and van der Straten; <math>\gamma_p
= R^s</math>):
::[[Image:Saturation_of_atomic_transitions-saturation-fig2-2-from-metcalf.png|thumb|400px|none|]]
Note that the transition rate from <math>|a{\rangle}</math> to <math>|b{\rangle}</math>, given in terms of
the absorption crosssection <math>\sigma_{\rm abs}^u</math>, and the incident
intensity <math>I_0</math>, is
:<math> 
	W_{a\rightarrow b} = \sigma_{\rm abs}^u
		\frac{1}{1+s} \frac{I_0}{\hbar\omega}
\,,
</math>
where we may identify as the saturated absorption crossection
:<math> 
	\sigma_{\rm abs}^s = \sigma_{\rm abs}^u \frac{1}{1+s}
\,.
</math>
As <math>s\rightarrow\infty</math>, <math>\sigma_{\rm abs}^s\rightarrow 0</math>.  This
process is known as bleaching.
== Saturation Intensity ==
The on-resonance saturation parameter has a useful physical meaning.
At <math>s_0=1</math>, the unsaturated transition rate becomes
:<math> 
	R^u = \Gamma/2 = \sigma_{\rm abs} \frac{I_{\rm sat}}{\hbar\omega}
\,,
</math>
were we have identified as the sauration intensity
:<math> 
	I_{\rm sat} = \frac{\hbar \omega^3}{12\pi c^2} \Gamma
\,.
</math>
This is a very useful quantity in practice.  For the sodium D line,
<math>I_{\rm sat} = 6</math> mW/cm<math>^2</math>, and that value is typical of most widely
used atomic transitions.  In terms of <math>I_{\rm sat}</math>, we may give
:<math>\begin{array}{rcl}  
	s_0 &=& \frac{I}{I_{\rm sat}}
\\	\omega_R^2 &=& \frac{\Gamma^2}{2} \frac{I}{I_{\rm sat}}
\,.
\end{array}</math>
This extrodinary <math>\sim 10</math> mW/cm<math>^2</math> value of <math>I_{\rm sat}</math> can be
apprreciated by comparing the monochromatic excitation case, which we
have just analyzed, with the case of broadband excitation.  For
broadband excitation,
:<math>\begin{array}{rcl}  
	W_{a\rightarrow b} &=& \sigma_0 \frac{\pi}{2} f(\delta) I
				\frac{\Gamma}{\hbar\omega}
\\
	&\approx& \frac{\Gamma}{2}
\,,
\end{array}</math>
where the approximation is valid in the limit that the average
intensity over the spectral density <math>\bar{I} = f(\delta) I \approx
\bar{I}_{\rm sat}</math> (a useful approximation which gives a result
independent of the details of the transition).  Here,
:<math> 
	\bar{I}_{\rm sat} = \frac{\hbar\omega^3}{6\pi^2 c^2}
\,,
</math>
which, for visible frequencies, is approximately <math>12</math> W/cm<math>^2</math> per
inverse centimeter of bandwidth.