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% Lecture No 12 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{12}
In this final lecture, we will study the behaviour of the Bloch equations
in different regimes of resonance and relaxation. The Bloch equations
are formulated as a vector model, and numerical solutions to the equations
are discussed.
For steady-state interaction, the polarization density of the medium, as
obtained from the Bloch equations, is expressed in a closed form.
The closed solution is then expanded in a power series, which when
compared with the series obtained from the susceptibility formalism
finally tie together the Bloch theory with the susceptibilities.
\medskip
\noindent The outline for this lecture is:
\item{$\bullet$}{Recapitulation of the Bloch equations}
\item{$\bullet$}{The vector model of the Bloch equations}
\item{$\bullet$}{Special cases and examples}
\item{$\bullet$}{Steady-state regime}
\item{$\bullet$}{The intensity dependent refractive index at steady-state}
\item{$\bullet$}{Comparison with the susceptibility model}
\medskip
\section{Recapitulation of the Bloch equations for two-level systems}
Assuming two states $|a\rangle$ and $|b\rangle$ to be sufficiently
similar in order for their respective lifetimes~$T_a\approx T_b\approx T_1$
to hold, where $T_1$ is the {\sl longitudinal relaxation time}, the Bloch
equations for the two-level are given as
$$
\eqalignno{
{{du}\over{dt}}&=-\Delta v -u/T_2,&(1{\rm a})\cr
{{dv}\over{dt}}&=\Delta u+\beta(t)w-v/T_2,&(1{\rm b})\cr
{{dw}\over{dt}}&=-\beta(t)v-(w-w_0)/T_1,&(1{\rm c})\cr
}
$$
where $\beta\equiv er^{\alpha}_{ab}E^{\alpha}_{\omega}(t)/\hbar$ is the Rabi
frequency, being a quantity linear in the applied electric field of the
light, $\Delta\equiv\Omega_{ba}-\omega$ is the detuning of the
angular frequency of the light from the transition frequency
$\Omega_{ba}\equiv({\Bbb E}_b-{\Bbb E}_a)/\hbar$,
and where the variables $(u,v,w)$ are related to the matrix
elements $\rho_{mn}$ of the density operator as
$$
\eqalign{
u&=\rho^{\Omega}_{ba}+\rho^{\Omega}_{ab},\cr
v&=i(\rho^{\Omega}_{ba}-\rho^{\Omega}_{ab}),\cr
w&=\rho_{bb}-\rho_{aa}.\cr
}
$$
In these equations, $\rho^{\Omega}_{ab}$ is the {\sl temporal envelope
of the off-diagonal elements}, given by
$$
\rho_{ab}\equiv\rho^{\Omega}_{ab}\exp[i(\Omega_{ba}-\Delta)t].
$$
In the Bloch equations~(1), the 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, $w_0\equiv\rho_0(b)-\rho_0(a)$
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 is
throughout this course as well as in Butcher and Cotter's book 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
$$
P^{\mu}_{\omega}({\bf r},t)=Ner^{\mu}_{ab}(u-iv).\eqno{(2)}
$$
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 propern wave equation for the envelopes of the
fields), the Bloch equations are known as the {\sl Maxwell--Bloch equations}.
From Eq.~(2), it should now be clear that the Bloch variable $u$ essentially
gives the in-phase part of the polarization density (at least in this
case, where we may consider the transition dipole moments to be real-valued),
corresponding to the dispersive components of the interaction between
light and matter, while the Bloch variable $v$ on the other hand gives
terms which are shifted ninety degrees out of phase with the optical field,
hence corresponding to absorptive terms.
\vfill\eject
\section{The vector model of the Bloch equations}
In the form of Eqs.~(1), the Bloch equations can be expressed in the
form of an Euler equation as
$$
{{d{\bf R}}\over{dt}}={\bf\Omega}\times{\bf R}
-\underbrace{(u/T_2,v/T_2,(w-w_0)/T_1)}_{{\rm relaxation\ term}},
\eqno{[{\rm B.\,\&\,C.~(6.54)}]}
$$
where ${\bf R}=(u,v,w)$ is the so-called {\sl Bloch vector}, that in the
abstract $({\bf e}_u,{\bf e}_v,{\bf e}_w)$-space describes the state of
the medium, and
$$
{\bf\Omega}=(-\beta(t),0,\Delta)
$$
is the vector that gives the precession of the Bloch vector (see Fig.~1).
This form, originally proposed in 1946 by Felix
Bloch\footnote{${}^1$}{F. Bloch,
{\sl Nuclear induction}, {Phys.~Rev.} {\bf 70}, 460 (1946).
Felix Bloch was in 1952 awarded the Nobel prize in physics,
together with Edward Mills Purcell, ``for their development of new methods
for nuclear magnetic precision measurements and discoveries in connection
therewith''.} for the
motion of a nuclear spin in a magnetic field under influence of
radio-frequency electromagnetic fields, and later
on adopted by Feynman, Vernon, and Hellwarth\footnote{${}^2$}{R.~P. Feynman,
F.~L. Vernon, and R.~W. Hellwarth, {\sl Geometrical representation of the
Schr\"od\-ing\-er equation for solving maser problems}, J.~Appl.~Phys.
{\bf 28}, 49 (1957).} for solving problems in maser
theory\footnote{${}^3$}{Microwave Amplification by Stimulated Emission
of Radiation, a device for amplification of microwaves, essentially working
on the same principle as the laser.}, corresponds to the motion of a
damped gyroscope in the presence of a gravitational field.
In this analogy, the vector ${\bf \Omega}$ can be considered as the
torque vector of the spinning top of the gyroscope.
\bigskip
\centerline{\epsfxsize=90mm\epsfbox{../images/blochmod/blochmod.1}}
\medskip
{\noindent Figure 1. Evolution of the Bloch vector
${\bf R}(t)=(u(t),v(t),w(t))$ around the ``torque vector''
${\bf\Omega}=(-\beta(t),0,\Delta)$.
In the absence of optical fields, the Bloch vector relax towards
the thermal equilibrium state ${\bf R}_{\infty}=(0,0,w_0)$,
where $w_0=\rho(b)-\rho(a)$ is the molecular population inversion
at thermal equilibrium. At moderate temperatures, the thermal equilibrium
population inversion is very close to $w_0=-1$.}
\medskip
From the vector form of the Bloch equations, it is found that the
Bloch vector rotates around the torque vector ${\bf\Omega}$ as the
state of matter approaches steady state. For an adiabatically changing
applied optical field (i.~e.~a slowly varying envelope of the field),
this precession follows the torque vector.
The relaxation term in the vector Bloch equations also tells us that the
relaxation along the $w$-direction is given by the time constant~$T_1$,
while the relaxation in the $(u,v)$-plane instead is given by the time
constant $T_2$. By considering the $w$-axis as the ``longitudinal''
direction and the $(u,v)$-plane as the ``transverse'' plane, the terminology
for $T_1$ as being the ``longitudinal relaxation time'' and $T_2$
as being the ``transverse relaxation time'' should hence be clear.
\vfill\eject
\section{Transient build-up at exact resonance as the optical field
is switched on}
\subsection{The case $T_1\gg T_2$ -- Longitudinal relaxation slower than
transverse relaxation}
\bigskip
\centerline{\epsfxsize=65mm\epsfbox{fig8a.eps}\qquad
\epsfxsize=65mm\epsfbox{fig8b.eps}}
\centerline{\epsfxsize=65mm\epsfbox{fig8d.eps}\qquad
\epsfxsize=65mm\epsfbox{fig8e.eps}}
{\noindent Figure 2a. Evolution of the Bloch vector $(u(t),v(t),w(t))$
as the optical field is switched on, for the exactly resonant case
($\delta=0$), and with the longitudinal relaxation
time being much greater than the transverse relaxation time ($T_1\gg T_2$).
The parameters used in the simulation are
$\eta\equiv T_1/T_2=100$, $\delta\equiv\Delta T_2=0$, $w_0=-1$,
and $\gamma(t)\equiv\beta(t)T_2=3$, $t>0$.
The medium was initially at thermal equilibrium,
$(u(0),v(0),w(0))=(0,0,w_0)=-(0,0,1)$.}
\medskip
\bigskip
\centerline{\epsfxsize=70mm\epsfbox{fig8c.eps}}
{\noindent Figure 2b. Evolution of the magnitude of the polarization density
$|P_{\omega}(t)|\sim|u(t)-iv(t)|$ as the optical field is switched on,
corresponding to the simulation shown in Fig.~2a.}
\medskip
\vfill\eject
\subsection{The case $T_1\approx T_2$ -- Longitudinal relaxation approximately
equal to transverse relaxation}
\bigskip
\centerline{\epsfxsize=65mm\epsfbox{fig9a.eps}\qquad
\epsfxsize=65mm\epsfbox{fig9b.eps}}
\centerline{\epsfxsize=65mm\epsfbox{fig9d.eps}\qquad
\epsfxsize=65mm\epsfbox{fig9e.eps}}
{\noindent Figure 3a. Evolution of the Bloch vector $(u(t),v(t),w(t))$
as the optical field is switched on, for the exactly resonant case
($\delta=0$), and with the longitudinal relaxation
time being approximately equal to the transverse relaxation time
($T_1\approx T_2$).
The parameters used in the simulation are
$\eta\equiv T_1/T_2=2$, $\delta\equiv\Delta T_2=0$, $w_0=-1$,
and $\gamma(t)\equiv\beta(t)T_2=3$, $t>0$.
The medium was initially at thermal equilibrium,
$(u(0),v(0),w(0))=(0,0,w_0)=-(0,0,1)$.}
\medskip
\bigskip
\centerline{\epsfxsize=70mm\epsfbox{fig9c.eps}}
{\noindent Figure 3b. Evolution of the magnitude of the polarization density
$|P_{\omega}(t)|\sim|u(t)-iv(t)|$ as the optical field is switched on,
corresponding to the simulation shown in Fig.~3a.}
\medskip
\vfill\eject
\section{Transient build-up at off-resonance as the optical field
is switched on}
\subsection{The case $T_1\approx T_2$ -- Longitudinal relaxation approximately
equal to transverse relaxation}
\bigskip
\centerline{\epsfxsize=65mm\epsfbox{fig10a.eps}\qquad
\epsfxsize=65mm\epsfbox{fig10b.eps}}
\centerline{\epsfxsize=65mm\epsfbox{fig10d.eps}\qquad
\epsfxsize=65mm\epsfbox{fig10e.eps}}
{\noindent Figure 4a. Evolution of the Bloch vector $(u(t),v(t),w(t))$
as the optical field is switched on, for the off-resonant case
($\delta\ne 0$), and with the longitudinal relaxation
time being approximately equal to the transverse relaxation time
($T_1\approx T_2$).
The parameters used in the simulation are
$\eta\equiv T_1/T_2=2$, $\delta\equiv\Delta T_2=4$, $w_0=-1$,
and $\gamma(t)\equiv\beta(t)T_2=3$, $t>0$.
The medium was initially at thermal equilibrium,
$(u(0),v(0),w(0))=(0,0,w_0)=-(0,0,1)$.}
\medskip
\bigskip
\centerline{\epsfxsize=70mm\epsfbox{fig10c.eps}}
{\noindent Figure 4b. Evolution of the magnitude of the polarization density
$|P_{\omega}(t)|\sim|u(t)-iv(t)|$ as the optical field is switched on,
corresponding to the simulation shown in Fig.~4a.}
\medskip
\vfill\eject
\section{Transient decay for a process tuned to exact resonance}
\subsection{The case $T_1\gg T_2$ -- Longitudinal relaxation slower than
transverse relaxation}
\bigskip
\centerline{\epsfxsize=70mm\epsfbox{fig1a.eps}
\epsfxsize=70mm\epsfbox{fig1b.eps}}
{\noindent Figure 5. Evolution of the Bloch vector $(u(t),v(t),w(t))$
after the optical field is switched off, for the case of tuning to
exact resonance ($\delta=0$), and with the longitudinal relaxation
time being much greater than the transverse relaxation time ($T_1\gg T_2$).
The parameters used in the simulation are
$\eta\equiv T_1/T_2=100$, $\delta\equiv\Delta T_2=0$, $w_0=-1$,
and $\gamma(t)\equiv\beta(t)T_2=0$.}
\medskip
\subsection{The case $T_1\approx T_2$ -- Longitudinal relaxation approximately
equal to transverse relaxation}
\bigskip
\centerline{\epsfxsize=70mm\epsfbox{fig2a.eps}
\epsfxsize=70mm\epsfbox{fig2b.eps}}
{\noindent Figure 6. Evolution of the Bloch vector $(u(t),v(t),w(t))$
after the optical field is switched off, for the case of tuning to
exact resonance ($\delta=0$), and with the longitudinal relaxation
time being approximately equal to the transverse relaxation time
($T_1\approx T_2$).
The parameters used in the simulation are
$\eta\equiv T_1/T_2=2$, $\delta\equiv\Delta T_2=0$, $w_0=-1$,
and $\gamma(t)\equiv\beta(t)T_2=0$.}
\medskip
\vfill\eject
\section{Transient decay for a slightly off-resonant process}
\subsection{The case $T_1\gg T_2$ -- Longitudinal relaxation slower than
transverse relaxation}
\bigskip
\centerline{\epsfxsize=65mm\epsfbox{fig3a.eps}\qquad
\epsfxsize=65mm\epsfbox{fig3b.eps}}
\centerline{\epsfxsize=65mm\epsfbox{fig3d.eps}\qquad
\epsfxsize=65mm\epsfbox{fig3e.eps}}
{\noindent Figure 7a. Evolution of the Bloch vector $(u(t),v(t),w(t))$
after the optical field is switched off, for the off-resonant case
($\delta\ne 0$), and with the longitudinal relaxation
time being much greater than the transverse relaxation time ($T_1\gg T_2$).
The parameters used in the simulation are
$\eta\equiv T_1/T_2=100$, $\delta\equiv\Delta T_2=2$, $w_0=-1$,
and $\gamma(t)\equiv\beta(t)T_2=0$.
(Compare with Fig.~5 for the exactly resonant case.)}
\medskip
\bigskip
\centerline{\epsfxsize=65mm\epsfbox{fig3f.eps}\qquad
\epsfxsize=65mm\epsfbox{fig3g.eps}}
{\noindent Figure 7b. Same as Fig.~7a, but with $\delta=-2$ as negative.}
\medskip
\vfill\eject
\subsection{The case $T_1\approx T_2$ -- Longitudinal relaxation approximately
equal to transverse relaxation}
\bigskip
\centerline{\epsfxsize=65mm\epsfbox{fig4a.eps}\qquad
\epsfxsize=65mm\epsfbox{fig4b.eps}}
\centerline{\epsfxsize=65mm\epsfbox{fig4d.eps}\qquad
\epsfxsize=65mm\epsfbox{fig4e.eps}}
{\noindent Figure 8a. Evolution of the Bloch vector $(u(t),v(t),w(t))$
after the optical field is switched off, for the off-resonant case
($\delta\ne 0$), and with the longitudinal relaxation
time being approximately equal to the transverse relaxation time
($T_1\approx T_2$).
The parameters used in the simulation are
$\eta\equiv T_1/T_2=2$, $\delta\equiv\Delta T_2=2$, $w_0=-1$,
and $\gamma(t)\equiv\beta(t)T_2=0$.
(Compare with Fig.~6 for the exactly resonant case.)}
\medskip
\bigskip
\centerline{\epsfxsize=70mm\epsfbox{fig4c.eps}}
{\noindent Figure 8b. Evolution of the magnitude of the polarization density
$|P_{\omega}(t)|\sim|u(t)-iv(t)|$ as the optical field is switched on,
corresponding to the simulation shown in Fig.~8a.}
\medskip
\vfill\eject
\section{Transient decay for a far off-resonant process}
\subsection{The case $T_1\gg T_2$ -- Longitudinal relaxation slower than
transverse relaxation}
\bigskip
\centerline{\epsfxsize=65mm\epsfbox{fig5a.eps}\qquad
\epsfxsize=65mm\epsfbox{fig5b.eps}}
\centerline{\epsfxsize=65mm\epsfbox{fig5d.eps}\qquad
\epsfxsize=65mm\epsfbox{fig5e.eps}}
{\noindent Figure 9a. Evolution of the Bloch vector $(u(t),v(t),w(t))$
after the optical field is switched off, for the far off-resonant case
($\delta\ne 0$), and with the longitudinal relaxation
time being much greater than the transverse relaxation time ($T_1\gg T_2$).
The parameters used in the simulation are
$\eta\equiv T_1/T_2=100$, $\delta\equiv\Delta T_2=20$, $w_0=-1$,
and $\gamma(t)\equiv\beta(t)T_2=0$.
(Compare with Fig.~5 for the exactly resonant case,
and with Fig.~7a for the slightly off-resonant case.)}
\medskip
\bigskip
\centerline{\epsfxsize=70mm\epsfbox{fig5c.eps}}
{\noindent Figure 9b. Evolution of the magnitude of the polarization density
$|P_{\omega}(t)|\sim|u(t)-iv(t)|$ as the optical field is switched on,
corresponding to the simulation shown in Fig.~9a.}
\medskip
\vfill\eject
\subsection{The case $T_1\approx T_2$ -- Longitudinal relaxation approximately
equal to transverse relaxation}
\bigskip
\centerline{\epsfxsize=65mm\epsfbox{fig6a.eps}\qquad
\epsfxsize=65mm\epsfbox{fig6b.eps}}
\centerline{\epsfxsize=65mm\epsfbox{fig6d.eps}\qquad
\epsfxsize=65mm\epsfbox{fig6e.eps}}
{\noindent Figure 10a. Evolution of the Bloch vector $(u(t),v(t),w(t))$
after the optical field is switched off, for the far off-resonant case
($\delta\ne 0$), and with the longitudinal relaxation
time being approximately equal to the transverse relaxation time
($T_1\approx T_2$).
The parameters used in the simulation are
$\eta\equiv T_1/T_2=2$, $\delta\equiv\Delta T_2=20$, $w_0=-1$,
and $\gamma(t)\equiv\beta(t)T_2=0$.
(Compare with Fig.~6 for the exactly resonant case,
and with Fig.~8a for the slightly off-resonant case.)}
\medskip
\bigskip
\centerline{\epsfxsize=70mm\epsfbox{fig6c.eps}}
{\noindent Figure 10b. Evolution of the magnitude of the polarization density
$|P_{\omega}(t)|\sim|u(t)-iv(t)|$ as the optical field is switched on,
corresponding to the simulation shown in Fig.~10a.}
\medskip
\vfill\eject
\subsection{The case $T_1\ll T_2$ -- Longitudinal relaxation faster than
transverse relaxation}
\bigskip
\centerline{\epsfxsize=65mm\epsfbox{fig7a.eps}\qquad
\epsfxsize=65mm\epsfbox{fig7b.eps}}
\centerline{\epsfxsize=65mm\epsfbox{fig7d.eps}\qquad
\epsfxsize=65mm\epsfbox{fig7e.eps}}
{\noindent Figure 11a. Same parameter values as in Fig.~6, but with
the longitudinal relaxation
time being much smaller than the transverse relaxation time
($T_1\ll T_2$), $\eta\equiv T_1/T_2=0.1$.
(Compare with Figs.~9a and~10a for the cases $T_1\gg T_2$
and $T_1\approx T_2$, respectively.)}
\medskip
\bigskip
\centerline{\epsfxsize=70mm\epsfbox{fig7c.eps}}
{\noindent Figure 11b. Evolution of the magnitude of the polarization density
$|P_{\omega}(t)|\sim|u(t)-iv(t)|$ as the optical field is switched on,
corresponding to the simulation shown in Fig.~11a.}
\medskip
\vfill\eject
\section{The connection between the Bloch equations and the susceptibility}
As an example of the connection between the polarization density obtained
from the Bloch equations and the one obtained from the susceptibility
formalism, we will now -- once again -- consider the intensity-dependent
refractive of the medium.
\subsection{The intensity-dependent refractive index in the susceptibility
formalism}
Previously in this course, the intensity-dependent refractive index has
been obtained from the optical Kerr-effect in isotropic media, in the form
$$
n=n_0+n_2|{\bf E}_{\omega}|^2,
$$
where $n_0=[1+\chi^{(1)}_{xx}(-\omega;\omega)]^{1/2}$ is the linear
refractive index, and
$$
n_2={{3}\over{8n_0}}\chi^{(3)}_{xxxx}(-\omega;\omega,\omega,-\omega)
$$
is the parameter of the intensity dependent contribution.
However, since we by now are fully aware that the polarization density
in the description of the susceptibility formalism originally is given
as an infinity series expansion, we may expect that the general form
of the intensity dependent refractive index rather would
be as a power series in the intensity,
$$
n=n_0+n_2|{\bf E}_{\omega}|^2
+n_4|{\bf E}_{\omega}|^4
+n_6|{\bf E}_{\omega}|^6+\ldots
$$
For linearly polarized light, say along the $x$-axis of a Cartesian
coordinate system, we know that such a series is readily possible to
derive in terms of the susceptibility formalism, with the different
order terms of the refractive index expansion given by the elements
$$
\eqalign{
n_2&\sim\chi^{(3)}_{xxxx}
(-\omega;\omega,\omega,-\omega),\cr
n_4&\sim\chi^{(5)}_{xxxxxx}
(-\omega;\omega,\omega,-\omega,\omega,-\omega),\cr
n_6&\sim\chi^{(7)}_{xxxxxxxx}
(-\omega;\omega,\omega,-\omega,\omega,-\omega,\omega,-\omega),\cr
&\qquad\vdots\cr
}
$$
Such an analysis would, however, be extremely cumbersome when it comes
to the analysis of higher-order effects, and the obtained sum of various
order terms would also be almost impossible to obtain a closed expression
for.
For future reference, to be used in the interpretation of the polarization
density given by the Bloch equations, the intensity dependent polarization
density is though shown in its explicit form below, including up to the
seventh order interaction term in the Butcher and Cotter convention,
$$
\eqalignno{
P^x_{\omega}
=\varepsilon_0&\chi^{(1)}_{xx}
(-\omega;\omega)E^x_{\omega}
&({\rm order}\ n=1)\cr
&+\varepsilon_0(3/4)\chi^{(3)}_{xxxx}
(-\omega;\omega,\omega,-\omega)|E^x_{\omega}|^2 E^x_{\omega}
&({\rm order}\ n=3)\cr
&+\varepsilon_0(5/8)\chi^{(5)}_{xxxxxx}
(-\omega;\omega,\omega,-\omega,\omega,-\omega)
|E^x_{\omega}|^4 E^x_{\omega}
&({\rm order}\ n=5)\cr
&+\varepsilon_0(35/64)\chi^{(7)}_{xxxxxxxx}
(-\omega;\omega,\omega,-\omega,\omega,-\omega,\omega,-\omega)
|E^x_{\omega}|^6 E^x_{\omega}
&({\rm order}\ n=7)\cr
&+\ldots&\cr
}
$$
The other approach to calculation of the polarization density, as we
next will outline, is to use the steady-state solutions to the Bloch
equations.
\vfill\eject
\subsection{The intensity-dependent refractive index in the Bloch-vector
formalism}
For steady-state interaction between light and matter, the solutions
to the Bloch equations yield
$$
\eqalignno{
&u-iv={{-\beta w}\over{\Delta-i/T_2}},&[{\rm B.\,\&\,C.~(6.53a)}],\cr
&w={{w_0[1+(\Delta T_2)^2]}
\over{1+(\Delta T_2)^2+\beta^2 T_1 T_2}},&[{\rm B.\,\&\,C.~(6.53b)}],\cr
}
$$
where, as previously, $\beta=er^{\alpha}_{ab}E^{\alpha}_{\omega}(t)/\hbar$
is the Rabi frequency, though now considered to be a slowly varying
(adiabatically following) quantity, due to the assumption of steady-state
behaviour.
From the steady-state solutions, the $\mu$-component ($\mu=x,y,z$) of the
electric polarization
density ${\bf P}({\bf r},t)=\Re[{\bf P}_{\omega}\exp(-i\omega t)]$ of the
medium hence is given as
$$
\eqalign{
P^{\mu}_{\omega}
&=Ner^{\mu}_{ab}(u-iv)\cr
&=-Ner^{\mu}_{ab}{{\beta w}\over{\Delta-i/T_2}}\cr
&=-Ner^{\mu}_{ab}{{\beta}\over{(\Delta-i/T_2)}}
{{w_0[1+(\Delta T_2)^2]}\over{[1+(\Delta T_2)^2+\beta^2 T_1 T_2]}}\cr
&=-New_0{{r^{\mu}_{ab}}\over{(\Delta-i/T_2)}}
{{\beta}\over
{\left[1+{{T_1 T_2}\over{(1+(\Delta T_2)^2)}}\beta^2\right]}}.\cr
}\eqno{(3)}
$$
In this expression for the polarization density, it might at a first glance
seem as it is negative for a positive Rabi frequency $\beta$, henc giving a
polarization density that is directed anti-parallel to the electric field.
However, the quantity $w_0=\rho_0(b)-\rho_0(a)$, the population inversion
at thermal equilibrium, is always negative (since we for sure do not have
any population inversion at thermal equilibrium, for which we rather expect
the molecules to occupy the lower state), hence ensuring that the off-resonant,
real-valued polarization density always is directed along the direction of the
electric field of the light.
Next observation is that the polarization density no longer is expressed
as a power series in terms of the electric field, but rather as a rational
function,
$$
P^{\mu}_{\omega}\sim X/(1+X^2),\eqno{(4)}
$$
where
$$
\eqalign{
X&=\sqrt{{T_1 T_2}/{(1+(\Delta T_2)^2)}}\beta\cr
&=\sqrt{{T_1 T_2}/{(1+(\Delta T_2)^2)}}
er^{\alpha}_{ab}E^{\alpha}_{\omega}(t)/\hbar\cr
}
$$
is a parameter linear in the electric field. The principal shape of the
rational function in Eq.~(4) is shown in Fig.~12.
From Eq.~(4), the polarization density is found to increase with increasing
$X$ up to $X=1$, as we expect for an increasing power of an optical beam.
However, for $X>1$, we find the somewhat surprising fact that the
polarization density instead {\sl decrease} with an increasing intensity;
this peculiar suggested behaviour should hence be explained before continuing.
The first observation we may do is that the linear polarizability
(i.~e.~what we usually associate with linear optics) follows the
first order approximation $p(X)=X$.
In the region where the peculiar decrease of the polarization density
appear, the difference between the suggested nonlinear polarization
density and the one given by the linear approximation is {\sl huge},
and since we {\sl a priori} expect nonlinear contributions to be small
compared to the alsways present linear ones, this is already an indication
of that we in all practical situations do not have to consider the
descrease of polarization density as shown in Fig.~12.
For optical fields of the strength that would give rise to nonlinearities
exceeding the linear terms, the underlying physics will rather belong
to the field of plasma and high-energy physics, rather than a bound-charge
description of gases and solids. This implies that the validity of the
models here applied (bound charges, Hamiltonians being linear in the
optical field, etc.) are limited to a range well within $X\le 1$.
\vfill\eject
\centerline{\epsfxsize=120mm\epsfbox{polplot.eps}}
{\noindent Figure 12. The principal shape of the electric polarization
density of the medium, as function of the applied electric field of
the light.
In this figure, $X=\sqrt{{T_1 T_2}/{(1+(\Delta T_2)^2)}}\beta$
is a normalized parameter describing the field strength of the electric
field of the light.}
\bigskip
Another interesting point we may observe is a more mathematically
related one.
In Fig.~12, we see that even for very high order terms (such as the
approximating power series of degree 31, as shown in the figure),
all power series expansions fail before reaching $X=1$.
The reason for this is that the power series that approximate the
rational function $X/(1+X^2)$,
$$
X/(1+X^2)=X-X^3+X^5-X^7+\ldots,
$$
is convergent only for $|X|<1$; for all other values, the series is
divergent.
This means that no matter how many terms we include in the power series
in $X$, it will nevertheless fail when it comes to the evaluation
for $|X|>1$.
Since this power series expansion is equivalent to the expansion
of the nonlinear polarization density in terms of the electrical field
of the light (keeping in mind that $X$ here actually is linear in the
electric field and hence strictly can be considered as the field variable),
this also is an indication that at this point the whole susceptibility
formalism fail to give a proper description at this working point.
This is an excellent illustration of the downturn of the susceptibility
description of interaction between light and matter; no matter how
many terms we may include in the power series of the electrcal field,
{\sl it will at some point nevertheless fail to give the total picture of
the interaction}, and we must then instead seek other tools.
Returning to the polarization density given by Eq.~(3), we may now express
this in an explicit form by inserting $\Delta\equiv\Omega_{ba}-\omega$ for
the angular frequency detuning, the Rabi frequency
$\beta=er^{\alpha}_{ab}E^{\alpha}_{\omega}(t)/\hbar$,
and the thermal equilibrium inversion $w_0=\rho_0(b)-\rho_0(a)$.
This gives the polarization density of the medium as
$$
P^{\mu}_{\omega}
=\varepsilon_0
\underbrace{
\underbrace{
{{Ne^2}\over{\varepsilon_0\hbar}}(\rho_0(a)-\rho_0(b))
{{r^{\mu}_{ab}r^{\alpha}_{ab}}\over{(\Omega_{ba}-\omega-i/T_2)}}
}_{=\chi^{(1)}_{\mu\alpha}(-\omega;\omega)
\ {\rm for\ a\ two\ level\ medium}}
\underbrace{
{{1}\over{\left[1+{{T_1 T_2}\over{(1+(\Omega_{ba}-\omega)^2 T^2_2)}}
(er^{\gamma}_{ab}E^{\gamma}_{\omega}/\hbar)^2\right]}}
}_{{\rm nonlinear\ correction\ factor\ to}
\ \chi^{(1)}_{\mu\alpha}(-\omega;\omega)}
}_{{\rm the\ field\ corrected\ susceptibility,}
\ {\bar{\chi}}(\omega;{\bf E}_{\omega})
\ [{\rm see\ Butcher\ and\ Cotter,\ section~6.3.1}]}
E^{\alpha}_{\omega}.
$$
In this form, the polarization density is given as the product with
a term which is identical to the linear
susceptibility\footnote{${}^4$}{In the explicit expressions for the
linear susceptibility, for example Butcher and Cotter's Eqs.~(4.58)
and~(4.111) for the non-resonant and resonant cases, respectively,
there are two terms, one with $\Omega_{ba}-\omega$ in the denominator
and the other one with $\Omega_{ba}+\omega$.
The reason why the second form does not appear in the expression for the
field corrected susceptibility, as derived from the Bloch equations, is
that {\sl we have used the rotating wave approximation in the derivation
of the final expression.} (Recapitulate that in the rotating wave
approximation, terms with oscillatory dependence of
$\exp[i(\Omega_{ba}+\omega)t]$ were neglected.)
As a result, all temporally phase-mismatched terms are neglected, and in
particular only terms with $\Omega_{ba}-\omega$ in the denominator will
remain. This, however, is a most acceptable approximation, especially when
it comes to resonant interactions, where terms with $\Omega_{ba}-\omega$
in the denominator by far will dominate over non-resonant terms.}
(as obtained in the perturbation analysis in the frame of the susceptibility
formalism), and a correction factor which is a nonlinear function of the
electric field.
The nonlinear correction factor, of the form $1/(1+X^2)$, with
$X=\sqrt{{T_1 T_2}/{(1+(\Delta T_2)^2)}}\beta$ as previously, can now be
expanded in a power series around the small-signal limit $X=0$, using
$$
1/(1+X^2)=1-X^2+X^4-X^6+\ldots,
$$
from which we obtain the polarization density as a power series in the
electric field (which for the sake of simplicitly now is taken as linearly
polarized along the $x$-axis) as
$$
\eqalign{
P^x_{\omega}
\approx\varepsilon_0
&{{Ne^2}\over{\varepsilon_0\hbar}}(\rho_0(a)-\rho_0(b))
{{r^x_{ab}r^x_{ab}}\over{(\Omega_{ba}-\omega-i/T_2)}}
E^x_{\omega}\cr
&-\varepsilon_0
{{Ne^4}\over{\varepsilon_0\hbar^3}}(\rho_0(a)-\rho_0(b))
{{r^x_{ab}r^x_{ab}}\over{(\Omega_{ba}-\omega-i/T_2)}}
{{(r^x_{ab})^2}\over{[1/T^2_2+(\Omega_{ba}-\omega)^2](T_2/T_1)}}
|E^x_{\omega}|^2
E^x_{\omega}\cr
&+\varepsilon_0
{{Ne^6}\over{\varepsilon_0\hbar^5}}(\rho_0(a)-\rho_0(b))
{{r^x_{ab}r^x_{ab}}\over{(\Omega_{ba}-\omega-i/T_2)}}
{{(r^x_{ab})^4}\over{[1/T^2_2+(\Omega_{ba}-\omega)^2]^2(T_2/T_1)^2}}
|E^x_{\omega}|^4
E^x_{\omega}\cr
&+\ldots\cr
}\eqno{(5)}
$$
This form is identical to one as obtained in the susceptibility formalism;
however, the steps that led us to this expression for the polarization
density {\sl do not rely on the perturbation theory of the density
operator}, but rather on the explicit form of the steady-state solutions
to the Bloch equations.
\section{Summary of the Bloch and susceptibility polarization densities}
To summarize this last lecture on the Bloch equations, expressing the involved
parameters in the same style as previously used in the description of the
susceptibility formalism, the polarization density obtained from the
steady-state solutions to the Bloch equations is
$$
P^{\mu}_{\omega}
=\varepsilon_0
{{Ne^2}\over{\varepsilon_0\hbar}}(\rho_0(a)-\rho_0(b))
{{r^{\mu}_{ab}r^{\alpha}_{ab}}\over{(\Omega_{ba}-\omega-i/T_2)}}
{{1}\over{\left[1+{{T_1 T_2}\over{(1+(\Omega_{ba}-\omega)^2 T^2_2)}}
(er^{\alpha}_{ab}E^{\alpha}_{\omega}/\hbar)^2\right]}}
E^{\alpha}_{\omega}.
$$
By expanding this in a power series in the electrical field, one obtains
the form (5), in which we from the same description of the polarization
density in the susceptibility formalism can identify
$$
\eqalign{
\chi^{(1)}_{xx}(-\omega;\omega)
&={{Ne^2}\over{\varepsilon_0\hbar}}(\rho_0(a)-\rho_0(b))
{{r^x_{ab}r^x_{ab}}\over{(\Omega_{ba}-\omega-i/T_2)}},\cr
\chi^{(3)}_{xxxx}(-\omega;\omega,\omega,-\omega)
&=-{{4Ne^4}\over{3\varepsilon_0\hbar^3}}(\rho_0(a)-\rho_0(b))
{{r^x_{ab}r^x_{ab}}\over{(\Omega_{ba}-\omega-i/T_2)}}
{{(r^x_{ab})^2}\over{[1/T^2_2+(\Omega_{ba}-\omega)^2](T_2/T_1)}},\cr
\chi^{(5)}_{xxxxxx}(-\omega;\omega,\omega,-\omega,&\omega,-\omega)\cr
&={{8Ne^6}\over{5\varepsilon_0\hbar^5}}(\rho_0(a)-\rho_0(b))
{{r^x_{ab}r^x_{ab}}\over{(\Omega_{ba}-\omega-i/T_2)}}
{{(r^x_{ab})^4}\over{[1/T^2_2+(\Omega_{ba}-\omega)^2]^2(T_2/T_1)^2}},\cr
}
$$
as being the first contributions to the two-level polarization density,
including up to fifth order interactions.
For a summary of the non-resonant forms of the susceptibilities of
two-level systems, se Butcher and Cotter, Eqs.~(6.71)--(6.73).
\vfill\eject
\section{Appendix: Notes on the numerical solution to the Bloch equations}
In their original form, the Bloch equations for a two-level system are
given by Eqs.~(1) as
$$
\eqalignno{
{{du}\over{dt}}&=-\Delta v -u/T_2,\cr
{{dv}\over{dt}}&=\Delta u+\beta(t)w-v/T_2,\cr
{{dw}\over{dt}}&=-\beta(t)v-(w-w_0)/T_1.\cr
}
$$
By taking the time in units of the transverse relaxation time $T_2$, as
$$
\tau=t/T_2,
$$
the Bloch equations in this normalized time scale become
$$
\eqalignno{
{{du}\over{d\tau}}&=-\Delta T_2 v -u,\cr
{{dv}\over{d\tau}}&=\Delta T_2 u+\beta(t)T_2 w-v,\cr
{{dw}\over{d\tau}}&=-\beta(t)T_2 v-(w-w_0)T_2/T_1.\cr
}
$$
In this system of equations, all coefficients are now normalized and
physically dimensionless, expressed as relevant quotes between relaxation
times and products of the Rabi frequency or detuning frequency with
the transverse relaxation time.
Hence, by taking the normalized parameters
$$
\eqalign{
\delta&=\Delta T_2,\cr
\gamma(t)&=\beta(t)T_2,\cr
\eta&=T_1/T_2,\cr
}
$$
where $\delta$ can be considered as the normalized detuning from molecular
resonance of the medium, $\gamma(t)$ as the normalized Rabi frequency,
and $\eta$ as a parameter which describes the relative impact of the
longitudinal vs transverse relaxation times, the Bloch equations take
the normalized final form
$$
\eqalignno{
{{du}\over{d\tau}}&=-\delta v -u,\cr
{{dv}\over{d\tau}}&=\delta u+\gamma(t) w-v,\cr
{{dw}\over{d\tau}}&=-\gamma(t) v-(w-w_0)/\eta.\cr
}
$$
This normalized form of the Bloch equations has been used throughout the
generation of graphs in Figs.~2--11 of this lecture, describing the
qualitative impact of different regimes of resonance and relaxation.
The normalized Bloch equations were in the simulations shown in Figs.~2--11
integrated by using the standard routine {\tt ODE45()} in MATLAB.
\bye