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% Lecture No 11 in the course ``Nonlinear optics'', held January-March,
% 2003, at the Royal Institute of Technology, Stockholm, Sweden.
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% Copyright (C) 2002-2003, Fredrik Jonsson
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\def\sinc{\mathop{\rm sinc}\nolimits} % the sinc(x)=sin(x)/x function
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\hfil{\it Nonlinear Optics 5A5513 (2003)}}
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\hfil{\it Lecture notes #1}}
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\lecture{11}
In this lecture, we will focus on configurations where the angular frequency
of the light is close to some transition frequency of the medium.
In particular, we will start with a brief outline of how the non-resonant
susceptibilities may be modified in such a way that weakly resonant
interactions can be taken into account.
Having formulated the susceptibilities at weakly resonant interaction,
we will proceed with formulating a non-perturbative approach of calculation
of the polarization density of the medium. For the two-level system, this
results in the Bloch equations governing resonant interaction between light
and matter.
\medskip
\noindent The outline for this lecture is:
\item{$\bullet$}{Singularities of the non-resonant susceptibilities}
\item{$\bullet$}{Alternatives to perturbation analysis of the
polarization density}
\item{$\bullet$}{Relaxation of the medium}
\item{$\bullet$}{The two-level system and the Bloch equation}
\item{$\bullet$}{The resulting polarization density of the medium at resonance}
\medskip
\section{Singularities of non-resonant susceptibilities}
In the theory described so far in this course, all interactions have for
simplicity been considered as non-resonant.
The explicit forms of the susceptibilities, in terms of the electric dipole
moments and transition frequencies of the molecules, have been obtained in
lecture six, of the forms
$$
\eqalignno{
\chi^{(1)}_{\mu\alpha}(-\omega;\omega)
&\sim{{r^{\mu}_{ab}r^{\alpha}_{ba}}\over{\Omega_{ba}-\omega}}
+\{{\rm similar\ terms}\},
&[{\rm B.\,\&\,C.\,(4.58)}]\cr
\chi^{(2)}_{\mu\alpha\beta}(-\omega_{\sigma};\omega_1,\omega_2)
&\sim{{r^{\mu}_{ab} r^{\alpha}_{bc} r^{\beta}_{ca}}
\over{(\Omega_{ba}-\omega_1-\omega_2)
(\Omega_{ca}-\omega_2)}}
+\{{\rm similar\ terms}\},
&[{\rm B.\,\&\,C.\,(4.63)}]\cr
\chi^{(3)}_{\mu\alpha\beta\gamma}
(-\omega_{\sigma};\omega_1,\omega_2,\omega_3)
&\sim{{r^{\mu}_{ab} r^{\alpha}_{bc} r^{\beta}_{cd} r^{\gamma}_{da}}
\over{(\Omega_{ba}-\omega_1-\omega_2-\omega_3)
(\Omega_{ca}-\omega_2-\omega_3)
(\Omega_{da}-\omega_3)}}
+\{{\rm similar\ terms}\},
\cr&\qquad\qquad\qquad
&[{\rm B.\,\&\,C.\,(4.64)}]\cr
&\vdots\cr
}
$$
To recapitulate, these forms have all been derived under the assumption
that the Hamiltonian (which is the general operator which describes the
state of the system) consist only of a thermal equilibrium part and an
interaction part (in the electric dipolar approximation), of the form
$$
{\hat H}={\hat H}_0+{\hat H}_{\rm I}(t).
$$
This is a form which clearly does not contain any term related to relaxation
effects of the medium, that is to say, it does not contain any term describing
any energy flow into thermal heat. As long as we consider the interaction part
of the Hamiltonian to be sufficiently strong compared to any relaxation effect
of the medium, this is a valid approximation.
However, the problem with the non-resonant forms of the susceptibilities
clearly comes into light when we consider an angular frequency of the
light that is close to a transition frequency of the system, since for
the first order susceptibility,
$$
\chi^{(1)}_{\mu\alpha}(-\omega;\omega)\to\infty,
\quad{\rm when\ }\omega\to\Omega_{ba},
$$
or for the second order susceptibility,
$$
\chi^{(2)}_{\mu\alpha\beta}(-\omega;\omega_1,\omega_2)\to\infty,
\quad{\rm when\ }\omega_1+\omega_2\to\Omega_{ba}
{\rm\ or\ }\omega_2\to\Omega_{ca}.
$$
This clearly non-physical behaviour is a consequence of that the denominators
of the rational expressions for the susceptibilities have singularities
at the resonances, and the aim with this lecture is to show how these
singularities can be removed.
\section{Modification of the Hamiltonian for resonant interaction}
Whenever we have to consider relaxation effects of the medium, as in the
case of resonant interactions, the Hamiltonian should be modified to
$$
{\hat H}={\hat H}_0+{\hat H}_{\rm I}(t)+{\hat H}_{\rm R},\eqno{(1)}
$$
where, as previously, ${\hat H}_0$ is the Hamiltonian in the absence of
external forces, ${\hat H}_{\rm I}(t)=-{\hat Q}_{\alpha}E_{\alpha}({\bf r},t)$
is the interaction Hamiltonian
(here taken in the Schr\"odinger picture, as described in lecture four),
being linear in the applied electric field of the light,
and where the new term ${\hat H}_{\rm R}$ describes the various relaxation
processes that brings the system into the thermal equilibrium whenever
external forces are absent.
The state of the system (atom, molecule, or general ensemble) is then
conveniently described by the density operator formalism, from which
we can obtain macroscopically observable parameters of the medium,
such as the electric polarization density (as frequently encountered
in this course), the magnetization of the medium, current densities, etc.
The form (1) of the Hamiltonian is now to be analysed by means of the
equation of motion of the density operator $\hat{\rho}$,
$$
i\hbar{{d{\hat{\rho}}}\over{dt}}
={\hat H}{\hat\rho}-{\hat\rho}{\hat H}
=[{\hat H},{\hat\rho}],\eqno{(2)}
$$
and depending on the setup, this equation may be solved by means of
perturbation analysis (for non-resonant and weakly resonant interactions),
or by means of non-perturbative approaches, such as the Bloch equations
(for strongly resonant interactions).
\section{Phenomenological representation of relaxation processes}
In many cases, the relaxation process of the medium towards thermal
equilibrium can be described by
$$
[{\hat H}_{\rm R},{\hat\rho}]
=-i\hbar{\hat\Gamma}({\hat\rho}-{\hat\rho}_0),
$$
where ${\hat\rho}_0$ is the thermal equilibrium density operator
of the system. The here phenomenologically introduced operator
${\hat\Gamma}$ describes the relaxation of the medium, and can can be
considered as being independent of the interaction Hamiltonian.
Here the operator ${\hat\Gamma}$ has the physical dimension of an angular
frequency, and its matrix elements can be considered as giving the time
constants of decay for various states of the system.
\section{Perturbation analysis of weakly resonant interactions}
Before entering the formalism of the Bloch equations for strongly resonant
interactions, we will outline the weakly resonant interactions in a
perturbative analysis for the susceptibilities, as previously developed
in lectures three, four, and five.
By taking the perturbation series for the density operator as
$$
\hat{\rho}(t)=\underbrace{\hat{\rho}_0}_{\sim [E(t)]^0}
+\underbrace{\hat{\rho}_1(t)}_{\sim [E(t)]^1}
+\underbrace{\hat{\rho}_2(t)}_{\sim [E(t)]^2}
+\ldots
+\underbrace{\hat{\rho}_n(t)}_{\sim [E(t)]^n}
+\ldots,
$$
as we previously did for the strictly non-resonant case, one obtains
the system of equations
$$
\eqalign{
i\hbar{{d\hat{\rho}_0}\over{dt}}&=[\hat{H}_0,\hat{\rho}_0],\cr
i\hbar{{d\hat{\rho}_1(t)}\over{dt}}&=[\hat{H}_0,\hat{\rho}_1(t)]
+[\hat{H}_{\rm I}(t),\hat{\rho}_0]
-i\hbar{\hat\Gamma}{\hat\rho}_1(t),\cr
i\hbar{{d\hat{\rho}_2(t)}\over{dt}}&=[\hat{H}_0,\hat{\rho}_2(t)]
+[\hat{H}_{\rm I}(t),\hat{\rho}_1(t)]
-i\hbar{\hat\Gamma}{\hat\rho}_2(t),\cr
&\vdots\cr
i\hbar{{d\hat{\rho}_n(t)}\over{dt}}&=[\hat{H}_0,\hat{\rho}_n(t)]
+[\hat{H}_{\rm I}(t),\hat{\rho}_{n-1}(t)]
-i\hbar{\hat\Gamma}{\hat\rho}_n(t),\cr
&\vdots\cr
}
$$
As in the non-resonant case, one may here start with solving for the
zeroth order term $\hat{\rho}_0$, with all other terms obtained by
consecutively solving the equations of order $j=1,2,\ldots,n$, in that order.
Proceeding in exactly the same path as for the non-resonant case,
solving for the density operator in the interaction picture and
expressing the various terms of the electric polarization density
in terms of the corresponding traces
$$
P_{\mu}({\bf r},t)
=\sum^{\infty}_{n=0} P^{(n)}_{\mu}({\bf r},t)
={{1}\over{V}}\sum^{\infty}_{n=0}
{\rm Tr}[{\hat\rho}_n(t){\hat Q}_{\mu}],
$$
one obtains the linear, first order susceptibility of the form
$$
\eqalign{
\chi^{(1)}_{\mu\alpha}(-\omega;\omega)
&={{N e^2}\over{\varepsilon_0\hbar}}
\sum_a\varrho_0(a)\sum_b
\Big({{r^{\mu}_{ab}r^{\alpha}_{ba}}
\over{\Omega_{ba}-\omega-i\Gamma_{ba}}}
+{{r^{\alpha}_{ab}r^{\mu}_{ba}}
\over{\Omega_{ba}+\omega-i\Gamma_{ba}}}\Big).\cr
}
$$
Similarly, the second order susceptibility for weakly resonant interaction
is obtained as
$$
\eqalign{
\chi^{(2)}_{\mu\alpha\beta}&(-\omega_{\sigma};\omega_1,\omega_2)\cr
&={{N e^3}\over{\varepsilon_0 \hbar^2}}
{{1}\over{2!}}{\bf S}
\sum_a\varrho_0(a)\sum_b\sum_c
\Big\{
{{r^{\mu}_{ab} r^{\alpha}_{bc} r^{\beta}_{ca}}
\over{(\Omega_{ac}+\omega_2-i\Gamma_{ac})
(\Omega_{ab}+\omega_{\sigma}-i\Gamma_{ab})}}
\cr&\qquad
-{{r^{\alpha}_{ab} r^{\mu}_{bc} r^{\beta}_{ca}}
\over{(\Omega_{ac}+\omega_2-i\Gamma_{ac})
(\Omega_{bc}+\omega_{\sigma}-i\Gamma_{bc})}}
-{{r^{\beta}_{ab} r^{\mu}_{bc} r^{\alpha}_{ca}}
\over{(\Omega_{ba}+\omega_2-i\Gamma_{ba})
(\Omega_{bc}+\omega_{\sigma}-i\Gamma_{bc})}}
\cr&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad
+{{r^{\beta}_{ab} r^{\alpha}_{bc} r^{\mu}_{ca}}
\over{(\Omega_{ba}+\omega_2-i\Gamma_{ba})
(\Omega_{ca}+\omega_{\sigma}-i\Gamma_{ca})}}
\Big\}.\cr
}
$$
In these expressions for the susceptibilities, the singularities at resonance
are removed, and the spectral properties of the absolute values of the
susceptibilities are described by regular Lorenzian line shapes.
The values of the matrix elements $\Gamma_{mn}$ are in many cases difficult
to derive from a theoretical basis; however, they are often straightforward
to obtain by regular curve-fitting and regression analysis of experimental
data.
As seen from the expressions for the susceptibilities above, we still have
a boosting of them close to resonance (resonant enhancement). However, the
values of the susceptibilities reach a plateau at exact resonance, with
maximum values determined by the magnitudes of the involved matrix elements
$\Gamma_{mn}$ of the relaxation operator.
\section{Validity of perturbation analysis of the polarization density}
Strictly speaking, the perturbative approach is only to be considered
as for an infinite series expansion.
For a limited number of terms, the perturbative approach is only an
approximative method, which though for many cases is sufficient.
The perturbation series, in the form that we have encountered it in this
course, defines a power series in the applied electric field of the light,
and as long as the lower order terms are dominant in the expansion, we
may safely neglect the higher order ones.
Whenever we encounter strong fields, however, we may run into trouble with
the series expansion, in particular if we are in a resonant optical regime,
with a boosting effect of the polarization density of the medium.
(This boosting effect can be seen as the equivalent to the close-to-resonance
behaviour of the mechanical spring model under influence of externally
driving forces.)
As an illustration to this source of failure of the model in the presence of
strong electrical fields, we may consider another, more simple example
of series expansions, namely the Taylor expansion of the function $\sin(x)$
around $x\approx 0$, as shown in Fig.~1.
\bigskip
\centerline{\epsfxsize=110mm\epsfbox{sinapprx.eps}}
\centerline{Figure 1. Approximations to $f(x)=\sin(x)$ by means of power
series expansions of various degrees.}
\medskip
In analogy to the susceptibility formalism, we may consider $x$ to
have the role of the electric field (the variable which we make
the power expansion in terms of), and $\sin(x)$ to have the role
of the polarization density or the density operator (simply the function
we wish to analyze).
For low numerical values of $x$, up to about $x\approx 1$,
the $\sin(x)$ function is well described by keeping only the first two
terms of the expansion, corresponding to a power expansion up to and
including order three,
$$
\sin(x)\approx p_3(x)=x-{{x^3}\over{3!}}.
$$
For higher values of $x$, say up to about $x\approx 2$, the expansion
is still following the exact function to a good approximation if we
include also the third term, corresponding to a power expansion up to and
including order five,
$$
\sin(x)\approx p_5(x)=x-{{x^3}\over{3!}}+{{x^5}\over{5!}}.
$$
This necessity of including higher and higher order terms goes on
as we increase the value of $x$, and we can from the graph also see
that the breakdown at a certain level of approximation causes severe
difference between the approximate and exact curves.
In particular, if one wish to calculate the value of the function $\sin(x)$
for small $x$, it might be a good idea to apply the series expansion.
For greater values of $x$, say $x\approx 10$, the series expansion
approach is, however, a bad idea, and an efficient evaluation of $\sin(x)$
requires another approach.
As a matter of fact, the same arguments hold for the more complex case
of the series expansion of the density operator\footnote{${}^1$}{We may
recall that the series expansion of the density operator is {\sl the}
very origin of the expansion of the polarization density of the medium
in terms of the electric field, and hence also the very foundation for the
whole susceptibility formalism as described in this course.}, for which
we for high intensities (high electrical field strengths) must include
higher order terms as well.
However, we have seen that even in the non-resonant case, we may encounter
great algebraic complexity even in low order nonlinear terms, and since
the problem of formulating a proper polarization density is expanding
more or less exponentially with the order of the nonlinearity, the
usefulness of the susceptibility formalism eventually breaks down.
The solution to this problem is to identify the relevant transitions
of the ensemble, and to solve the equation of motion (2) exactly instead
(or at least within other levels of approximation which do not rely on
the perturbative foundation of the susceptibility formalism).
\section{The two-level system}
In many cases, the interaction between light and matter can be reduced
to that of a two-level system, consisting of only two energy eigenstates
$|a\rangle$ and $|b\rangle$.
The equation of motion of the density operator is generally given by
Eq.~(2) as
$$
i\hbar{{d{\hat{\rho}}}\over{dt}}=[{\hat H},{\hat\rho}],
$$
with
$$
{\hat H}={\hat H}_0+{\hat H}_{\rm I}(t)+{\hat H}_{\rm R}.
$$
For the two-level system, the equation of motion can be expressed in
terms of the matrix elements of the density operator as
$$
\eqalignno{
i\hbar{{d\rho_{aa}}\over{dt}}
&=[{\hat H}_0,{\hat\rho}]_{aa}
+[{\hat H}_{\rm I}(t),{\hat\rho}]_{aa}
+[{\hat H}_{\rm R},{\hat\rho}]_{aa},&(3{\rm a})\cr
i\hbar{{d\rho_{ab}}\over{dt}}
&=[{\hat H}_0,{\hat\rho}]_{ab}
+[{\hat H}_{\rm I}(t),{\hat\rho}]_{ab}
+[{\hat H}_{\rm R},{\hat\rho}]_{ab},&(3{\rm b})\cr
i\hbar{{d\rho_{bb}}\over{dt}}
&=[{\hat H}_0,{\hat\rho}]_{bb}
+[{\hat H}_{\rm I}(t),{\hat\rho}]_{bb}
+[{\hat H}_{\rm R},{\hat\rho}]_{bb},&(3{\rm c})\cr
}
$$
where the fourth equation for $\rho_{ba}$ was omitted, since the solution
for this element immediately follows from
$$
\rho_{ba}=\rho^*_{ab}.
$$
\subsection{Terms involving the thermal equilibrium Hamiltonian}
The system of Eqs.~(3) is the starting point for derivation of the so-called
Bloch equations. Starting with the thermal-equilibrium part of the
commutators in the right-hand sides of Eqs.~(3), we have for the diagonal
elements
$$
\eqalign{
[{\hat H}_0,{\hat\rho}]_{aa}
&=\langle a|{\hat H}_0{\hat\rho}|a\rangle
-\langle a|{\hat\rho}{\hat H}_0|a\rangle\cr
&=\sum_k \underbrace{\langle a|{\hat H}_0|k\rangle}_{
={\Bbb E}_a\delta_{ak}}
\langle k|{\hat\rho}|a\rangle
-\sum_j \langle a|{\hat\rho}|j\rangle
\underbrace{\langle j|{\hat H}_0|a\rangle}_{
={\Bbb E}_j\delta_{ja}}\cr
&={\Bbb E}_a\rho_{aa}-\rho_{aa}{\Bbb E}_a\cr
&=0\cr
&=[{\hat H}_0,{\hat\rho}]_{bb},\cr
}
$$
and for the off-diagonal elements
$$
\eqalign{
[{\hat H}_0,{\hat\rho}]_{ab}
&=\langle a|{\hat H}_0{\hat\rho}|b\rangle
-\langle a|{\hat\rho}{\hat H}_0|b\rangle\cr
&=\sum_k \underbrace{\langle a|{\hat H}_0|k\rangle}_{
={\Bbb E}_a\delta_{ak}}
\langle k|{\hat\rho}|b\rangle
-\sum_j \langle a|{\hat\rho}|j\rangle
\underbrace{\langle j|{\hat H}_0|b\rangle}_{
={\Bbb E}_j\delta_{jb}}\cr
&={\Bbb E}_a\rho_{ab}-\rho_{ab}{\Bbb E}_b\cr
&=-({\Bbb E}_b-{\Bbb E}_a)\rho_{ab}\cr
&=-\hbar\Omega_{ba}\rho_{ab}\cr
}
$$
\subsection{Terms involving the interaction Hamiltonian}
For the commutators in the right-hand sides of Eqs.~(3) involving the
interaction Hamiltonian, we similarly have for the diagonal elements
$$
\eqalign{
[{\hat H}_{\rm I}(t),{\hat\rho}]_{aa}
&=\langle a|(-e{\hat r}_{\alpha}E_{\alpha}({\bf r},t)){\hat\rho}|a\rangle
-\langle a|{\hat\rho}(-e{\hat r}_{\alpha}E_{\alpha}({\bf r},t))|a\rangle\cr
&=-eE_{\alpha}({\bf r},t)
\bigg\{
\sum_k\langle a|{\hat r}_{\alpha}|k\rangle\langle k|{\hat\rho}|a\rangle
-\sum_j\langle a|{\hat\rho}|j\rangle\langle j|{\hat r}_{\alpha}|a\rangle
\bigg\}\cr
&=-eE_{\alpha}({\bf r},t)
\bigg\{
r^{\alpha}_{aa}\rho_{aa}
+r^{\alpha}_{ab}\rho_{ba}
-\rho_{aa}r^{\alpha}_{aa}
-\rho_{ab}r^{\alpha}_{ba}
\bigg\}\cr
&=-e(r^{\alpha}_{ab}\rho_{ba}-r^{\alpha}_{ba}\rho_{ab})
E_{\alpha}({\bf r},t)\cr
&=-[{\hat H}_{\rm I}(t),{\hat\rho}]_{bb},\cr
}
$$
and for the off-diagonal elements
$$
\eqalign{
[{\hat H}_{\rm I}(t),{\hat\rho}]_{ab}
&=\langle a|(-e{\hat r}_{\alpha}E_{\alpha}({\bf r},t)){\hat\rho}|b\rangle
-\langle a|{\hat\rho}(-e{\hat r}_{\alpha}E_{\alpha}({\bf r},t))|b\rangle\cr
&=-eE_{\alpha}({\bf r},t)
\bigg\{
\sum_k\langle a|{\hat r}_{\alpha}|k\rangle\langle k|{\hat\rho}|b\rangle
-\sum_j\langle a|{\hat\rho}|j\rangle\langle j|{\hat r}_{\alpha}|b\rangle
\bigg\}\cr
&=-eE_{\alpha}({\bf r},t)
\bigg\{
r^{\alpha}_{aa}\rho_{ab}
+r^{\alpha}_{ab}\rho_{bb}
-\rho_{aa}r^{\alpha}_{ab}
-\rho_{ab}r^{\alpha}_{bb}
\bigg\}\cr
&=-er^{\alpha}_{ab}E_{\alpha}({\bf r},t)(\rho_{bb}-\rho_{aa})
-e(r^{\alpha}_{aa}-r^{\alpha}_{bb})E_{\alpha}({\bf r},t)\rho_{ab}\cr
&=\{{\rm Optical\ Stark\ shift:\ }
\delta{\Bbb E}_k\equiv -er^{\alpha}_{kk}E_{\alpha}({\bf r},t),
\quad k=a,b\}\cr
&=-er^{\alpha}_{ab}E_{\alpha}({\bf r},t)(\rho_{bb}-\rho_{aa})
+(\delta{\Bbb E}_a-\delta{\Bbb E}_b)\rho_{ab}.\cr
}
$$
\subsection{Terms involving relaxation processes}
For the commutators describing relaxation processes, the diagonal elements
are given as
$$
\eqalign{
[{\hat H}_{\rm R},{\hat\rho}]_{aa}
&=-i\hbar(\rho_{aa}-\rho_0(a))/T_a,\cr
[{\hat H}_{\rm R},{\hat\rho}]_{bb}
&=-i\hbar(\rho_{bb}-\rho_0(b))/T_b,\cr
}
$$
where $T_a$ and $T_b$ are the decay rates towards the thermal equilibrium
at respective level, and where $\rho_0(a)$ and $\rho_0(b)$ are the thermal
equilibrium values of $\rho_{aa}$ and $\rho_{bb}$, respectively (i.~e.~the
thermal equilibrium population densities of the respective level).
The off-diagonal elements are similarly given as
$$
\eqalign{
[{\hat H}_{\rm R},{\hat\rho}]_{ab}&=-i\hbar\rho_{ab}/T_2,\cr
[{\hat H}_{\rm R},{\hat\rho}]_{ba}&=-i\hbar\rho_{ba}/T_2.\cr
}
$$
A common approximation is to consider the two states $|a\rangle$
and $|b\rangle$ to be sufficiently similar in order to approximate
their lifetimes as equal, i.~e.~$T_a\approx T_b\approx T_1$,
where $T_1$ for historical reasons is denoted as the {\sl longitudinal
relaxation time}.
For the same historical reason, the relaxation time $T_2$ is denoted
as the {\sl transverse relaxation time}.\footnote{${}^2$}{For a deeper
discusssion and explanation of the various mechanisms involved in relaxation,
see for example Charles~P. Slichter, {\sl Principles of Magnetic Resonance}
(Springer-Verlag, Berlin, 1978), available at KTHB. This reference is
not mentioned in Butcher and Cotters book, but it is a very good text
on relaxation phenomena and how to incorporate them into a density-functional
description of interaction between light and matter.}
As the above matrix elements of the commutators involving the various
terms of the Hamiltonian are inserted into the right-hand sides of Eqs.~(3),
one obtains the following system of equations for the matrix elements
of the density operator,
$$
\eqalignno{
i\hbar{{d\rho_{aa}}\over{dt}}
&=-e(r^{\alpha}_{ab}\rho_{ba}-r^{\alpha}_{ba}\rho_{ab})
E_{\alpha}({\bf r},t)
-i\hbar(\rho_{aa}-\rho_0(a))/T_a,&(4{\rm a})\cr
i\hbar{{d\rho_{ab}}\over{dt}}
&=-\hbar\Omega_{ba}\rho_{ab}
-er^{\alpha}_{ab}E_{\alpha}({\bf r},t)(\rho_{bb}-\rho_{aa})
+(\delta{\Bbb E}_a-\delta{\Bbb E}_b)\rho_{ab}
-i\hbar\rho_{ab}/T_2,&(4{\rm b})\cr
i\hbar{{d\rho_{bb}}\over{dt}}
&=e(r^{\alpha}_{ab}\rho_{ba}-r^{\alpha}_{ba}\rho_{ab})
E_{\alpha}({\bf r},t)
-i\hbar(\rho_{bb}-\rho_0(b))/T_b.&(4{\rm c})\cr
}
$$
(The system of equations~(4) corresponds to Butcher and Cotter's Eqs.~(6.35).)
So far, the applied electric field of the light is allowed to be of arbitrary
form. However, in order to simplify the following analysis, we will
assume the light to be linearly polarized and quasimonochromatic, of the
form
$$
E_{\alpha}({\bf r},t)=\Re[E^{\alpha}_{\omega}(t)\exp(-i\omega t)].
$$
We will in addition assume the slowly varying temporal envelope
$E^{\alpha}_{\omega}(t)$ to be real-valued, and we will also neglect
the optical Stark shifts $\delta{\Bbb E}_a$ and $\delta{\Bbb E}_b$.
In the absence of strong static magnetic fields, we may also assume
the matrix elements $er^{\alpha}_{ab}$ to be real-valued.
When these assumptions and approximations are applied to the
equations of motion~(4), one obtains
$$
\eqalignno{
{{d\rho_{aa}}\over{dt}}
&=i(\rho_{ba}-\rho_{ab})\beta(t)\cos(\omega t)
-(\rho_{aa}-\rho_0(a))/T_a,&(5{\rm a})\cr
{{d\rho_{ab}}\over{dt}}
&=i\Omega_{ba}\rho_{ab}
+i\beta(t)\cos(\omega t)(\rho_{bb}-\rho_{aa})
-\rho_{ab}/T_2,&(5{\rm b})\cr
{{d\rho_{bb}}\over{dt}}
&=-i(\rho_{ba}-\rho_{ab})\beta(t)\cos(\omega t)
-(\rho_{bb}-\rho_0(b))/T_b,&(5{\rm c})\cr
}
$$
where the {\sl Rabi frequency} $\beta(t)$, defined in terms of the spatial
envelope of the electrical field and the transition dipole moment as
$$
\beta(t)=er^{\alpha}_{ab}E^{\alpha}_{\omega}(t)/\hbar
=e{\bf r}_{ab}\cdot{\bf E}_{\omega}(t)/\hbar,
$$
was introduced.
\section{The rotating-wave approximation}
In the middle equation of the system~(5), we have a time-derivative
of $\rho_{ab}$ in the left-hand side, while we in the right-hand side
have a term $i\Omega_{ba}\rho_{ab}$. Seen as the homogeneous part of
a linear differential equation, this suggests that we may further
simplify the equations of motion by taking a new variable
$\rho^{\Omega}_{ab}$ according to the variable substitution
$$
\rho_{ab}=\rho^{\Omega}_{ab}\exp[i(\Omega_{ba}-\Delta)t],\eqno{(6)}
$$
where $\Delta\equiv\Omega_{ba}-\omega$ is the detuning of the
angular frequency of the light from the transition frequency
$\Omega_{ba}\equiv({\Bbb E}_b-{\Bbb E}_a)/\hbar$.
By inserting Eq.~(6) into Eqs.~(5), keeping in mind that
$\rho_{ba}=\rho^*_{ab}$, one obtains the system
$$
\eqalignno{
{{d\rho_{aa}}\over{dt}}
&=i(\rho^{\Omega}_{ba}\exp[-i(\Omega_{ba}-\Delta)t]
-\rho^{\Omega}_{ab}\exp[i(\Omega_{ba}-\Delta)t])
\beta(t)\cos(\omega t)
-(\rho_{aa}-\rho_0(a))/T_a,&(6{\rm a})\cr
{{d\rho^{\Omega}_{ab}}\over{dt}}
&=i\Delta\rho^{\Omega}_{ab}
+i\beta(t)\cos(\omega t)\exp[-i(\Omega_{ba}-\Delta)t]
(\rho_{bb}-\rho_{aa})
-\rho^{\Omega}_{ab}/T_2,&(6{\rm b})\cr
{{d\rho_{bb}}\over{dt}}
&=-i(\rho^{\Omega}_{ba}\exp[-i(\Omega_{ba}-\Delta)t]
-\rho^{\Omega}_{ab}\exp[i(\Omega_{ba}-\Delta)t])
\beta(t)\cos(\omega t)
-(\rho_{bb}-\rho_0(b))/T_b,&(6{\rm c})\cr
}
$$
The idea with the rotating-wave approximation is now to separate out
rapidly oscillating terms of angular frequencies $\omega+\Omega_{ba}$
and $-(\omega+\Omega_{ba})$, and neglect these terms, compared with
more slowly varying terms. The motivation for this approximation is that
whenever high-frequency components appear in the equations of motions,
the high-frequency terms will when integrated contain large denominators,
and will hence be minor in comparison with terms with a slow variation.
In some sense we can also see this as a temporal averaging procedure,
where rapidly oscillating terms average to zero rapidly compared
to slowly varying (or constant) components.
For example, in Eq.~(6b), the product of the $\cos(\omega t)$
and the exponential function is approximated as
$$
\eqalign{
\cos(\omega t)\exp[-i(\Omega_{ba}-\Delta)t]
&={{1}\over{2}}[\exp(i\omega t)+\exp(-i\omega t)]
\exp[-i\underbrace{(\Omega_{ba}-\Delta)}_{=\omega}t]\cr
&={{1}\over{2}}[1+\exp(-i2\omega t)]\to{{1}\over{2}},
}
$$
while in Eqs.~(6a) and~(6c), the same argument gives
$$
\eqalign{
\exp[i(\Omega_{ba}-\Delta)t]\cos(\omega t)
&={{1}\over{2}}[\exp(i\omega t)+\exp(-i\omega t)]
\exp[-i\underbrace{(\Omega_{ba}-\Delta)}_{=\omega}t]\cr
&={{1}\over{2}}[\exp(i2\omega t)+1]\to{{1}\over{2}}.
}
$$
By applying this {\sl rotating-wave approximation}, the equations
of motion~(6) hence take the form
$$
\eqalignno{
{{d\rho_{aa}}\over{dt}}
&={{i}\over{2}}(\rho^{\Omega}_{ba}-\rho^{\Omega}_{ab})\beta(t)
-(\rho_{aa}-\rho_0(a))/T_a,&(7{\rm a})\cr
{{d\rho^{\Omega}_{ab}}\over{dt}}
&=i\Delta\rho^{\Omega}_{ab}
+{{i}\over{2}}\beta(t)(\rho_{bb}-\rho_{aa})
-\rho^{\Omega}_{ab}/T_2,&(7{\rm b})\cr
{{d\rho_{bb}}\over{dt}}
&=-{{i}\over{2}}(\rho^{\Omega}_{ba}-\rho^{\Omega}_{ab})\beta(t)
-(\rho_{bb}-\rho_0(b))/T_b.&(7{\rm c})\cr
}
$$
In this final form, before entering the Bloch vector description of the
interaction, these equations correspond to Butcher and Cotter's Eqs.~(6.41).
\section{The Bloch equations}
Assuming the two states $|a\rangle$ and $|b\rangle$ to be sufficiently
similar in order to approximate~$T_a\approx T_b\approx T_1$,
where $T_1$ is the longitudinal relaxation time, and by taking new
variables $(u,v,w)$ according to
$$
\eqalign{
u&=\rho^{\Omega}_{ba}+\rho^{\Omega}_{ab},\cr
v&=i(\rho^{\Omega}_{ba}-\rho^{\Omega}_{ab}),\cr
w&=\rho_{bb}-\rho_{aa},\cr
}
$$
the equations of motion (7) are cast in the {\sl Bloch equations}
$$
\eqalignno{
{{du}\over{dt}}&=-\Delta v -u/T_2,&(8{\rm a})\cr
{{dv}\over{dt}}&=\Delta u+\beta(t)w-v/T_2,&(8{\rm b})\cr
{{dw}\over{dt}}&=-\beta(t)v-(w-w_0)/T_1.&(8{\rm c})\cr
}
$$
In these equations, the introduced variable $w$ describes the population
inversion of the two-level system, while $u$ and $v$ are related to the
dispersive and absorptive components of the polarization density of the
medium.
In the Bloch equations above, $w_0=\rho_0(b)-\rho_0(a)$
is the thermal equilibrium inversion of the system with no optical
field applied.
\section{The resulting electric polarization density of the medium}
The so far developed theory of the density matrix under resonant
interaction can now be applied to the calculation of the electric
polarization density of the medium, consisting of $N$ identical
molecules per unit volume, as
$$
\eqalign{
P_{\mu}({\bf r},t)&=N\langle e{\hat r}_{\mu}\rangle\cr
&=N\Tr[{\hat\rho} e{\hat r}_{\mu}]\cr
&=N\sum_{k=a,b}\langle k|{\hat\rho} e{\hat r}_{\mu}|k\rangle\cr
&=N\sum_{k=a,b}\sum_{j=a,b}
\langle k|{\hat\rho}|j\rangle
\langle j|e{\hat r}_{\mu}|k\rangle\cr
&=N\sum_{k=a,b}\left\{
\langle k|{\hat\rho}|a\rangle
\langle a|e{\hat r}_{\mu}|k\rangle
+\langle k|{\hat\rho}|b\rangle
\langle b|e{\hat r}_{\mu}|k\rangle
\right\}\cr
&=N\left\{
\langle a|{\hat\rho}|a\rangle
\langle a|e{\hat r}_{\mu}|a\rangle
+\langle b|{\hat\rho}|a\rangle
\langle a|e{\hat r}_{\mu}|b\rangle
+\langle a|{\hat\rho}|b\rangle
\langle b|e{\hat r}_{\mu}|a\rangle
+\langle b|{\hat\rho}|b\rangle
\langle b|e{\hat r}_{\mu}|b\rangle
\right\}\cr
&=N(\rho_{ba}er^{\mu}_{ab}+\rho_{ab}er^{\mu}_{ba})\cr
&=\{{\rm Make\ use\ of\ }\rho_{ab}=(u+iv)\exp(i\omega t)=\rho^*_{ba}\}\cr
&=N[(u-iv)\exp(-i\omega t)er^{\mu}_{ab}
+(u+iv)\exp(i\omega t)er^{\mu}_{ba}].\cr
}
$$
The temporal envelope $P^{\mu}_{\omega}$ of the polarization density,
throughout this course as well as in Butcher and Cotter's book, is taken as
$$
P^{\mu}({\bf r},t)=\Re[P^{\mu}_{\omega}\exp(-i\omega t)],
$$
and by identifying this expression with the right-hand side of the result
above, we hence finally have obtained the polarization density
in terms of the Bloch parameters $(u,v,w)$ as
$$
P^{\mu}_{\omega}({\bf r},t)=Ner^{\mu}_{ab}(u-iv).
$$
This expression for the temporal envelope of the polarization density is
exactly in the same mode of description as the one as previously used in
the susceptibility theory, as in the wave equations developed in lecture
eight. The only difference is that now we instead consider the polarization
density as given by a non-perturbative analysis. Taken together with the
Maxwell's equations (or the proper wave equation for the envelopes of the
fields), the Bloch equations are known as the {\sl Maxwell--Bloch equations}.
\bye