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    1   % File: nlopt/lect12/lect12.tex [pure TeX code]
    2   % Last change: March 21, 2003
    3   %
    4   % Lecture No 12 in the course ``Nonlinear optics'', held January-March,
    5   % 2003, at the Royal Institute of Technology, Stockholm, Sweden.
    6   %
    7   % Copyright (C) 2002-2003, Fredrik Jonsson
    8   %
    9   \input epsf
   10   %
   11   % Read amssym to get the AMS {\Bbb E} font (strikethrough E) and
   12   % the Euler fraktur font.
   13   %
   14   \input amssym
   15   \font\ninerm=cmr9
   16   \font\twelvesc=cmcsc10
   17   %
   18   % Use AMS Euler fraktur style for short-hand notation of Fourier transform
   19   %
   20   \def\fourier{\mathop{\frak F}\nolimits}
   21   \def\Re{\mathop{\rm Re}\nolimits}     % real part
   22   \def\Im{\mathop{\rm Im}\nolimits}     % imaginary part
   23   \def\Tr{\mathop{\rm Tr}\nolimits}     % quantum mechanical trace
   24   \def\sinc{\mathop{\rm sinc}\nolimits} % the sinc(x)=sin(x)/x function
   25   \def\sech{\mathop{\rm sech}\nolimits} % the sech(x)=... function
   26   \def\sgn{\mathop{\rm sgn}\nolimits}   % sgn(x)=0, if x<0, sgn(x)=1, otherwise
   27   \def\lecture #1 {\hsize=150mm\hoffset=4.6mm\vsize=230mm\voffset=7mm
   28     \topskip=0pt\baselineskip=12pt\parskip=0pt\leftskip=0pt\parindent=15pt
   29     \headline={\ifnum\pageno>1\ifodd\pageno\rightheadline\else\leftheadline\fi
   30       \else\hfill\fi}
   31     \def\rightheadline{\tenrm{\it Lecture notes #1}
   32       \hfil{\it Nonlinear Optics 5A5513 (2003)}}
   33     \def\leftheadline{\tenrm{\it Nonlinear Optics 5A5513 (2003)}
   34       \hfil{\it Lecture notes #1}}
   35     \noindent\epsfxsize 100pt\epsfbox{../info/kthtext.eps}
   36     \vskip-26pt\hfill\vbox{\hbox{{\it Nonlinear Optics 5A5513 (2003)}}
   37     \hbox{{\it Lecture notes}}}\vskip 36pt\centerline{\twelvesc Lecture #1}
   38     \vskip 24pt\noindent}
   39   \def\section #1 {\medskip\goodbreak\noindent{\bf #1}
   40     \par\nobreak\smallskip\noindent}
   41   \def\subsection #1 {\smallskip\goodbreak\noindent{\it #1}
   42     \par\nobreak\smallskip\noindent}
   43   
   44   \lecture{12}
   45   In this final lecture, we will study the behaviour of the Bloch equations
   46   in different regimes of resonance and relaxation. The Bloch equations
   47   are formulated as a vector model, and numerical solutions to the equations
   48   are discussed.
   49   
   50   For steady-state interaction, the polarization density of the medium, as
   51   obtained from the Bloch equations, is expressed in a closed form.
   52   The closed solution is then expanded in a power series, which when
   53   compared with the series obtained from the susceptibility formalism
   54   finally tie together the Bloch theory with the susceptibilities.
   55   \medskip
   56   
   57   \noindent The outline for this lecture is:
   58   \item{$\bullet$}{Recapitulation of the Bloch equations}
   59   \item{$\bullet$}{The vector model of the Bloch equations}
   60   \item{$\bullet$}{Special cases and examples}
   61   \item{$\bullet$}{Steady-state regime}
   62   \item{$\bullet$}{The intensity dependent refractive index at steady-state}
   63   \item{$\bullet$}{Comparison with the susceptibility model}
   64   \medskip
   65   
   66   \section{Recapitulation of the Bloch equations for two-level systems}
   67   Assuming two states $|a\rangle$ and $|b\rangle$ to be sufficiently
   68   similar in order for their respective lifetimes~$T_a\approx T_b\approx T_1$
   69   to hold, where $T_1$ is the {\sl longitudinal relaxation time}, the Bloch
   70   equations for the two-level are given as
   71   $$
   72     \eqalignno{
   73       {{du}\over{dt}}&=-\Delta v -u/T_2,&(1{\rm a})\cr
   74       {{dv}\over{dt}}&=\Delta u+\beta(t)w-v/T_2,&(1{\rm b})\cr
   75       {{dw}\over{dt}}&=-\beta(t)v-(w-w_0)/T_1,&(1{\rm c})\cr
   76     }
   77   $$
   78   where $\beta\equiv er^{\alpha}_{ab}E^{\alpha}_{\omega}(t)/\hbar$ is the Rabi
   79   frequency, being a quantity linear in the applied electric field of the
   80   light, $\Delta\equiv\Omega_{ba}-\omega$ is the detuning of the
   81   angular frequency of the light from the transition frequency
   82   $\Omega_{ba}\equiv({\Bbb E}_b-{\Bbb E}_a)/\hbar$,
   83   and where the variables $(u,v,w)$ are related to the matrix
   84   elements $\rho_{mn}$ of the density operator as
   85   $$
   86     \eqalign{
   87       u&=\rho^{\Omega}_{ba}+\rho^{\Omega}_{ab},\cr
   88       v&=i(\rho^{\Omega}_{ba}-\rho^{\Omega}_{ab}),\cr
   89       w&=\rho_{bb}-\rho_{aa}.\cr
   90     }
   91   $$
   92   In these equations, $\rho^{\Omega}_{ab}$ is the {\sl temporal envelope
   93   of the off-diagonal elements}, given by
   94   $$
   95     \rho_{ab}\equiv\rho^{\Omega}_{ab}\exp[i(\Omega_{ba}-\Delta)t].
   96   $$
   97   In the Bloch equations~(1), the variable $w$ describes the population
   98   inversion of the two-level system, while $u$ and $v$ are related to the
   99   dispersive and absorptive components of the polarization density of the
  100   medium.
  101   In the Bloch equations, $w_0\equiv\rho_0(b)-\rho_0(a)$
  102   is the thermal equilibrium inversion of the system with no optical
  103   field applied.
  104   
  105   \section{The resulting electric polarization density of the medium}
  106   The so far developed theory of the density matrix under resonant
  107   interaction can now be applied to the calculation of the electric
  108   polarization density of the medium, consisting of $N$ identical
  109   molecules per unit volume, as
  110   $$
  111     \eqalign{
  112       P_{\mu}({\bf r},t)&=N\langle e{\hat r}_{\mu}\rangle\cr
  113         &=N\Tr[{\hat\rho} e{\hat r}_{\mu}]\cr
  114         &=N\sum_{k=a,b}\langle k|{\hat\rho} e{\hat r}_{\mu}|k\rangle\cr
  115         &=N\sum_{k=a,b}\sum_{j=a,b}
  116           \langle k|{\hat\rho}|j\rangle
  117           \langle j|e{\hat r}_{\mu}|k\rangle\cr
  118         &=N\sum_{k=a,b}\left\{
  119           \langle k|{\hat\rho}|a\rangle
  120           \langle a|e{\hat r}_{\mu}|k\rangle
  121          +\langle k|{\hat\rho}|b\rangle
  122           \langle b|e{\hat r}_{\mu}|k\rangle
  123         \right\}\cr
  124         &=N\left\{
  125           \langle a|{\hat\rho}|a\rangle
  126           \langle a|e{\hat r}_{\mu}|a\rangle
  127          +\langle b|{\hat\rho}|a\rangle
  128           \langle a|e{\hat r}_{\mu}|b\rangle
  129          +\langle a|{\hat\rho}|b\rangle
  130           \langle b|e{\hat r}_{\mu}|a\rangle
  131          +\langle b|{\hat\rho}|b\rangle
  132           \langle b|e{\hat r}_{\mu}|b\rangle
  133         \right\}\cr
  134         &=N(\rho_{ba}er^{\mu}_{ab}+\rho_{ab}er^{\mu}_{ba})\cr
  135         &=\{{\rm Make\ use\ of\ }\rho_{ab}=(u+iv)\exp(i\omega t)=\rho^*_{ba}\}\cr
  136         &=N[(u-iv)\exp(-i\omega t)er^{\mu}_{ab}
  137             +(u+iv)\exp(i\omega t)er^{\mu}_{ba}].\cr
  138     }
  139   $$
  140   The temporal envelope $P^{\mu}_{\omega}$ of the polarization density is
  141   throughout this course as well as in Butcher and Cotter's book taken as
  142   $$
  143     P^{\mu}({\bf r},t)=\Re[P^{\mu}_{\omega}\exp(-i\omega t)],
  144   $$
  145   and by identifying this expression with the right-hand side of the result
  146   above, we hence finally have obtained the polarization density
  147   in terms of the Bloch parameters $(u,v,w)$ as
  148   $$
  149     P^{\mu}_{\omega}({\bf r},t)=Ner^{\mu}_{ab}(u-iv).\eqno{(2)}
  150   $$
  151   This expression for the temporal envelope of the polarization density is
  152   exactly in the same mode of description as the one as previously used in
  153   the susceptibility theory, as in the wave equations developed in lecture
  154   eight. The only difference is that now we instead consider the polarization
  155   density as given by a non-perturbative analysis. Taken together with the
  156   Maxwell's equations (or the propern wave equation for the envelopes of the
  157   fields), the Bloch equations are known as the {\sl Maxwell--Bloch equations}.
  158   
  159   From Eq.~(2), it should now be clear that the Bloch variable $u$ essentially
  160   gives the in-phase part of the polarization density (at least in this
  161   case, where we may consider the transition dipole moments to be real-valued),
  162   corresponding to the dispersive components of the interaction between
  163   light and matter, while the Bloch variable $v$ on the other hand gives
  164   terms which are shifted ninety degrees out of phase with the optical field,
  165   hence corresponding to absorptive terms.
  166   
  167   \vfill\eject
  168   
  169   \section{The vector model of the Bloch equations}
  170   In the form of Eqs.~(1), the Bloch equations can be expressed in the
  171   form of an Euler equation as
  172   $$
  173     {{d{\bf R}}\over{dt}}={\bf\Omega}\times{\bf R}
  174       -\underbrace{(u/T_2,v/T_2,(w-w_0)/T_1)}_{{\rm relaxation\ term}},
  175       \eqno{[{\rm B.\,\&\,C.~(6.54)}]}
  176   $$
  177   where ${\bf R}=(u,v,w)$ is the so-called {\sl Bloch vector}, that in the
  178   abstract $({\bf e}_u,{\bf e}_v,{\bf e}_w)$-space describes the state of
  179   the medium, and
  180   $$
  181     {\bf\Omega}=(-\beta(t),0,\Delta)
  182   $$
  183   is the vector that gives the precession of the Bloch vector (see Fig.~1).
  184   
  185   This form, originally proposed in 1946 by Felix
  186   Bloch\footnote{${}^1$}{F. Bloch,
  187   {\sl Nuclear induction}, {Phys.~Rev.} {\bf 70}, 460 (1946).
  188   Felix Bloch was in 1952 awarded the Nobel prize in physics,
  189   together with Edward Mills Purcell, ``for their development of new methods
  190   for nuclear magnetic precision measurements and discoveries in connection
  191   therewith''.} for the
  192   motion of a nuclear spin in a magnetic field under influence of
  193   radio-frequency electromagnetic fields, and later
  194   on adopted by Feynman, Vernon, and Hellwarth\footnote{${}^2$}{R.~P. Feynman,
  195   F.~L. Vernon, and R.~W. Hellwarth, {\sl Geometrical representation of the
  196   Schr\"od\-ing\-er equation for solving maser problems}, J.~Appl.~Phys.
  197   {\bf 28}, 49 (1957).} for solving problems in maser
  198   theory\footnote{${}^3$}{Microwave Amplification by Stimulated Emission
  199   of Radiation, a device for amplification of microwaves, essentially working
  200   on the same principle as the laser.}, corresponds to the motion of a
  201   damped gyroscope in the presence of a gravitational field.
  202   In this analogy, the vector ${\bf \Omega}$ can be considered as the
  203   torque vector of the spinning top of the gyroscope.
  204   
  205   \bigskip
  206   \centerline{\epsfxsize=90mm\epsfbox{../images/blochmod/blochmod.1}}
  207   \medskip
  208   {\noindent Figure 1. Evolution of the Bloch vector
  209   ${\bf R}(t)=(u(t),v(t),w(t))$ around the ``torque vector''
  210   ${\bf\Omega}=(-\beta(t),0,\Delta)$.
  211   In the absence of optical fields, the Bloch vector relax towards
  212   the thermal equilibrium state ${\bf R}_{\infty}=(0,0,w_0)$,
  213   where $w_0=\rho(b)-\rho(a)$ is the molecular population inversion
  214   at thermal equilibrium. At moderate temperatures, the thermal equilibrium
  215   population inversion is very close to $w_0=-1$.}
  216   \medskip
  217   
  218   From the vector form of the Bloch equations, it is found that the
  219   Bloch vector rotates around the torque vector ${\bf\Omega}$ as the
  220   state of matter approaches steady state. For an adiabatically changing
  221   applied optical field (i.~e.~a slowly varying envelope of the field),
  222   this precession follows the torque vector.
  223   
  224   The relaxation term in the vector Bloch equations also tells us that the
  225   relaxation along the $w$-direction is given by the time constant~$T_1$,
  226   while the relaxation in the $(u,v)$-plane instead is given by the time
  227   constant $T_2$. By considering the $w$-axis as the ``longitudinal''
  228   direction and the $(u,v)$-plane as the ``transverse'' plane, the terminology
  229   for $T_1$ as being the ``longitudinal relaxation time'' and $T_2$
  230   as being the ``transverse relaxation time'' should hence be clear.
  231   
  232   \vfill\eject
  233   
  234   \section{Transient build-up at exact resonance as the optical field
  235     is switched on}
  236   \subsection{The case $T_1\gg T_2$ -- Longitudinal relaxation slower than
  237     transverse relaxation}
  238   \bigskip
  239   \centerline{\epsfxsize=65mm\epsfbox{fig8a.eps}\qquad
  240     \epsfxsize=65mm\epsfbox{fig8b.eps}}
  241   \centerline{\epsfxsize=65mm\epsfbox{fig8d.eps}\qquad
  242     \epsfxsize=65mm\epsfbox{fig8e.eps}}
  243   {\noindent Figure 2a. Evolution of the Bloch vector $(u(t),v(t),w(t))$
  244     as the optical field is switched on, for the exactly resonant case
  245     ($\delta=0$), and with the longitudinal relaxation
  246     time being much greater than the transverse relaxation time ($T_1\gg T_2$).
  247     The parameters used in the simulation are
  248     $\eta\equiv T_1/T_2=100$, $\delta\equiv\Delta T_2=0$, $w_0=-1$,
  249     and $\gamma(t)\equiv\beta(t)T_2=3$, $t>0$.
  250     The medium was initially at thermal equilibrium,
  251     $(u(0),v(0),w(0))=(0,0,w_0)=-(0,0,1)$.}
  252   \medskip
  253   
  254   \bigskip
  255   \centerline{\epsfxsize=70mm\epsfbox{fig8c.eps}}
  256   {\noindent Figure 2b. Evolution of the magnitude of the polarization density
  257     $|P_{\omega}(t)|\sim|u(t)-iv(t)|$ as the optical field is switched on,
  258     corresponding to the simulation shown in Fig.~2a.}
  259   \medskip
  260   
  261   \vfill\eject
  262   
  263   \subsection{The case $T_1\approx T_2$ -- Longitudinal relaxation approximately
  264     equal to transverse relaxation}
  265   \bigskip
  266   \centerline{\epsfxsize=65mm\epsfbox{fig9a.eps}\qquad
  267     \epsfxsize=65mm\epsfbox{fig9b.eps}}
  268   \centerline{\epsfxsize=65mm\epsfbox{fig9d.eps}\qquad
  269     \epsfxsize=65mm\epsfbox{fig9e.eps}}
  270   {\noindent Figure 3a. Evolution of the Bloch vector $(u(t),v(t),w(t))$
  271     as the optical field is switched on, for the exactly resonant case
  272     ($\delta=0$), and with the longitudinal relaxation
  273     time being approximately equal to the transverse relaxation time
  274     ($T_1\approx T_2$).
  275     The parameters used in the simulation are
  276     $\eta\equiv T_1/T_2=2$, $\delta\equiv\Delta T_2=0$, $w_0=-1$,
  277     and $\gamma(t)\equiv\beta(t)T_2=3$, $t>0$.
  278     The medium was initially at thermal equilibrium,
  279     $(u(0),v(0),w(0))=(0,0,w_0)=-(0,0,1)$.}
  280   \medskip
  281   
  282   \bigskip
  283   \centerline{\epsfxsize=70mm\epsfbox{fig9c.eps}}
  284   {\noindent Figure 3b. Evolution of the magnitude of the polarization density
  285     $|P_{\omega}(t)|\sim|u(t)-iv(t)|$ as the optical field is switched on,
  286     corresponding to the simulation shown in Fig.~3a.}
  287   \medskip
  288   
  289   \vfill\eject
  290   
  291   \section{Transient build-up at off-resonance as the optical field
  292     is switched on}
  293   \subsection{The case $T_1\approx T_2$ -- Longitudinal relaxation approximately
  294     equal to transverse relaxation}
  295   \bigskip
  296   \centerline{\epsfxsize=65mm\epsfbox{fig10a.eps}\qquad
  297     \epsfxsize=65mm\epsfbox{fig10b.eps}}
  298   \centerline{\epsfxsize=65mm\epsfbox{fig10d.eps}\qquad
  299     \epsfxsize=65mm\epsfbox{fig10e.eps}}
  300   {\noindent Figure 4a. Evolution of the Bloch vector $(u(t),v(t),w(t))$
  301     as the optical field is switched on, for the off-resonant case
  302     ($\delta\ne 0$), and with the longitudinal relaxation
  303     time being approximately equal to the transverse relaxation time
  304     ($T_1\approx T_2$).
  305     The parameters used in the simulation are
  306     $\eta\equiv T_1/T_2=2$, $\delta\equiv\Delta T_2=4$, $w_0=-1$,
  307     and $\gamma(t)\equiv\beta(t)T_2=3$, $t>0$.
  308     The medium was initially at thermal equilibrium,
  309     $(u(0),v(0),w(0))=(0,0,w_0)=-(0,0,1)$.}
  310   \medskip
  311   
  312   \bigskip
  313   \centerline{\epsfxsize=70mm\epsfbox{fig10c.eps}}
  314   {\noindent Figure 4b. Evolution of the magnitude of the polarization density
  315     $|P_{\omega}(t)|\sim|u(t)-iv(t)|$ as the optical field is switched on,
  316     corresponding to the simulation shown in Fig.~4a.}
  317   \medskip
  318   
  319   \vfill\eject
  320   
  321   \section{Transient decay for a process tuned to exact resonance}
  322   \subsection{The case $T_1\gg T_2$ -- Longitudinal relaxation slower than
  323     transverse relaxation}
  324   \bigskip
  325   \centerline{\epsfxsize=70mm\epsfbox{fig1a.eps}
  326     \epsfxsize=70mm\epsfbox{fig1b.eps}}
  327   {\noindent Figure 5. Evolution of the Bloch vector $(u(t),v(t),w(t))$
  328     after the optical field is switched off, for the case of tuning to
  329     exact resonance ($\delta=0$), and with the longitudinal relaxation
  330     time being much greater than the transverse relaxation time ($T_1\gg T_2$).
  331     The parameters used in the simulation are
  332     $\eta\equiv T_1/T_2=100$, $\delta\equiv\Delta T_2=0$, $w_0=-1$,
  333     and $\gamma(t)\equiv\beta(t)T_2=0$.}
  334   \medskip
  335   
  336   \subsection{The case $T_1\approx T_2$ -- Longitudinal relaxation approximately
  337     equal to transverse relaxation}
  338   \bigskip
  339   \centerline{\epsfxsize=70mm\epsfbox{fig2a.eps}
  340     \epsfxsize=70mm\epsfbox{fig2b.eps}}
  341   {\noindent Figure 6. Evolution of the Bloch vector $(u(t),v(t),w(t))$
  342     after the optical field is switched off, for the case of tuning to
  343     exact resonance ($\delta=0$), and with the longitudinal relaxation
  344     time being approximately equal to the transverse relaxation time
  345     ($T_1\approx T_2$).
  346     The parameters used in the simulation are
  347     $\eta\equiv T_1/T_2=2$, $\delta\equiv\Delta T_2=0$, $w_0=-1$,
  348     and $\gamma(t)\equiv\beta(t)T_2=0$.}
  349   \medskip
  350   
  351   \vfill\eject
  352   
  353   \section{Transient decay for a slightly off-resonant process}
  354   \subsection{The case $T_1\gg T_2$ -- Longitudinal relaxation slower than
  355     transverse relaxation}
  356   \bigskip
  357   \centerline{\epsfxsize=65mm\epsfbox{fig3a.eps}\qquad
  358     \epsfxsize=65mm\epsfbox{fig3b.eps}}
  359   \centerline{\epsfxsize=65mm\epsfbox{fig3d.eps}\qquad
  360     \epsfxsize=65mm\epsfbox{fig3e.eps}}
  361   {\noindent Figure 7a. Evolution of the Bloch vector $(u(t),v(t),w(t))$
  362     after the optical field is switched off, for the off-resonant case
  363     ($\delta\ne 0$), and with the longitudinal relaxation
  364     time being much greater than the transverse relaxation time ($T_1\gg T_2$).
  365     The parameters used in the simulation are
  366     $\eta\equiv T_1/T_2=100$, $\delta\equiv\Delta T_2=2$, $w_0=-1$,
  367     and $\gamma(t)\equiv\beta(t)T_2=0$.
  368     (Compare with Fig.~5 for the exactly resonant case.)}
  369   \medskip
  370   
  371   \bigskip
  372   \centerline{\epsfxsize=65mm\epsfbox{fig3f.eps}\qquad
  373     \epsfxsize=65mm\epsfbox{fig3g.eps}}
  374   {\noindent Figure 7b. Same as Fig.~7a, but with $\delta=-2$ as negative.}
  375   \medskip
  376   
  377   \vfill\eject
  378   
  379   \subsection{The case $T_1\approx T_2$ -- Longitudinal relaxation approximately
  380     equal to transverse relaxation}
  381   \bigskip
  382   \centerline{\epsfxsize=65mm\epsfbox{fig4a.eps}\qquad
  383     \epsfxsize=65mm\epsfbox{fig4b.eps}}
  384   \centerline{\epsfxsize=65mm\epsfbox{fig4d.eps}\qquad
  385     \epsfxsize=65mm\epsfbox{fig4e.eps}}
  386   {\noindent Figure 8a. Evolution of the Bloch vector $(u(t),v(t),w(t))$
  387     after the optical field is switched off, for the off-resonant case
  388     ($\delta\ne 0$), and with the longitudinal relaxation
  389     time being approximately equal to the transverse relaxation time
  390     ($T_1\approx T_2$).
  391     The parameters used in the simulation are
  392     $\eta\equiv T_1/T_2=2$, $\delta\equiv\Delta T_2=2$, $w_0=-1$,
  393     and $\gamma(t)\equiv\beta(t)T_2=0$.
  394     (Compare with Fig.~6 for the exactly resonant case.)}
  395   \medskip
  396   
  397   \bigskip
  398   \centerline{\epsfxsize=70mm\epsfbox{fig4c.eps}}
  399   {\noindent Figure 8b. Evolution of the magnitude of the polarization density
  400     $|P_{\omega}(t)|\sim|u(t)-iv(t)|$ as the optical field is switched on,
  401     corresponding to the simulation shown in Fig.~8a.}
  402   \medskip
  403   
  404   \vfill\eject
  405   
  406   \section{Transient decay for a far off-resonant process}
  407   \subsection{The case $T_1\gg T_2$ -- Longitudinal relaxation slower than
  408     transverse relaxation}
  409   \bigskip
  410   \centerline{\epsfxsize=65mm\epsfbox{fig5a.eps}\qquad
  411     \epsfxsize=65mm\epsfbox{fig5b.eps}}
  412   \centerline{\epsfxsize=65mm\epsfbox{fig5d.eps}\qquad
  413     \epsfxsize=65mm\epsfbox{fig5e.eps}}
  414   {\noindent Figure 9a. Evolution of the Bloch vector $(u(t),v(t),w(t))$
  415     after the optical field is switched off, for the far off-resonant case
  416     ($\delta\ne 0$), and with the longitudinal relaxation
  417     time being much greater than the transverse relaxation time ($T_1\gg T_2$).
  418     The parameters used in the simulation are
  419     $\eta\equiv T_1/T_2=100$, $\delta\equiv\Delta T_2=20$, $w_0=-1$,
  420     and $\gamma(t)\equiv\beta(t)T_2=0$.
  421     (Compare with Fig.~5 for the exactly resonant case,
  422     and with Fig.~7a for the slightly off-resonant case.)}
  423   \medskip
  424   
  425   \bigskip
  426   \centerline{\epsfxsize=70mm\epsfbox{fig5c.eps}}
  427   {\noindent Figure 9b. Evolution of the magnitude of the polarization density
  428     $|P_{\omega}(t)|\sim|u(t)-iv(t)|$ as the optical field is switched on,
  429     corresponding to the simulation shown in Fig.~9a.}
  430   \medskip
  431   
  432   \vfill\eject
  433   
  434   \subsection{The case $T_1\approx T_2$ -- Longitudinal relaxation approximately
  435     equal to transverse relaxation}
  436   \bigskip
  437   \centerline{\epsfxsize=65mm\epsfbox{fig6a.eps}\qquad
  438     \epsfxsize=65mm\epsfbox{fig6b.eps}}
  439   \centerline{\epsfxsize=65mm\epsfbox{fig6d.eps}\qquad
  440     \epsfxsize=65mm\epsfbox{fig6e.eps}}
  441   {\noindent Figure 10a. Evolution of the Bloch vector $(u(t),v(t),w(t))$
  442     after the optical field is switched off, for the far off-resonant case
  443     ($\delta\ne 0$), and with the longitudinal relaxation
  444     time being approximately equal to the transverse relaxation time
  445     ($T_1\approx T_2$).
  446     The parameters used in the simulation are
  447     $\eta\equiv T_1/T_2=2$, $\delta\equiv\Delta T_2=20$, $w_0=-1$,
  448     and $\gamma(t)\equiv\beta(t)T_2=0$.
  449     (Compare with Fig.~6 for the exactly resonant case,
  450     and with Fig.~8a for the slightly off-resonant case.)}
  451   \medskip
  452   
  453   \bigskip
  454   \centerline{\epsfxsize=70mm\epsfbox{fig6c.eps}}
  455   {\noindent Figure 10b. Evolution of the magnitude of the polarization density
  456     $|P_{\omega}(t)|\sim|u(t)-iv(t)|$ as the optical field is switched on,
  457     corresponding to the simulation shown in Fig.~10a.}
  458   \medskip
  459   
  460   \vfill\eject
  461   
  462   \subsection{The case $T_1\ll T_2$ -- Longitudinal relaxation faster than
  463     transverse relaxation}
  464   \bigskip
  465   \centerline{\epsfxsize=65mm\epsfbox{fig7a.eps}\qquad
  466     \epsfxsize=65mm\epsfbox{fig7b.eps}}
  467   \centerline{\epsfxsize=65mm\epsfbox{fig7d.eps}\qquad
  468     \epsfxsize=65mm\epsfbox{fig7e.eps}}
  469   {\noindent Figure 11a. Same parameter values as in Fig.~6, but with
  470     the longitudinal relaxation
  471     time being much smaller than the transverse relaxation time
  472     ($T_1\ll T_2$), $\eta\equiv T_1/T_2=0.1$.
  473     (Compare with Figs.~9a and~10a for the cases $T_1\gg T_2$
  474     and $T_1\approx T_2$, respectively.)}
  475   \medskip
  476   
  477   \bigskip
  478   \centerline{\epsfxsize=70mm\epsfbox{fig7c.eps}}
  479   {\noindent Figure 11b. Evolution of the magnitude of the polarization density
  480     $|P_{\omega}(t)|\sim|u(t)-iv(t)|$ as the optical field is switched on,
  481     corresponding to the simulation shown in Fig.~11a.}
  482   \medskip
  483   
  484   \vfill\eject
  485   
  486   \section{The connection between the Bloch equations and the susceptibility}
  487   As an example of the connection between the polarization density obtained
  488   from the Bloch equations and the one obtained from the susceptibility
  489   formalism, we will now -- once again -- consider the intensity-dependent
  490   refractive of the medium.
  491   
  492   \subsection{The intensity-dependent refractive index in the susceptibility
  493     formalism}
  494   Previously in this course, the intensity-dependent refractive index has
  495   been obtained from the optical Kerr-effect in isotropic media, in the form
  496   $$
  497     n=n_0+n_2|{\bf E}_{\omega}|^2,
  498   $$
  499   where $n_0=[1+\chi^{(1)}_{xx}(-\omega;\omega)]^{1/2}$ is the linear
  500   refractive index, and
  501   $$
  502     n_2={{3}\over{8n_0}}\chi^{(3)}_{xxxx}(-\omega;\omega,\omega,-\omega)
  503   $$
  504   is the parameter of the intensity dependent contribution.
  505   However, since we by now are fully aware that the polarization density
  506   in the description of the susceptibility formalism originally is given
  507   as an infinity series expansion, we may expect that the general form
  508   of the intensity dependent refractive index rather would
  509   be as a power series in the intensity,
  510   $$
  511     n=n_0+n_2|{\bf E}_{\omega}|^2
  512          +n_4|{\bf E}_{\omega}|^4
  513          +n_6|{\bf E}_{\omega}|^6+\ldots
  514   $$
  515   For linearly polarized light, say along the $x$-axis of a Cartesian
  516   coordinate system, we know that such a series is readily possible to
  517   derive in terms of the susceptibility formalism, with the different
  518   order terms of the refractive index expansion given by the elements
  519   $$
  520     \eqalign{
  521       n_2&\sim\chi^{(3)}_{xxxx}
  522         (-\omega;\omega,\omega,-\omega),\cr
  523       n_4&\sim\chi^{(5)}_{xxxxxx}
  524         (-\omega;\omega,\omega,-\omega,\omega,-\omega),\cr
  525       n_6&\sim\chi^{(7)}_{xxxxxxxx}
  526         (-\omega;\omega,\omega,-\omega,\omega,-\omega,\omega,-\omega),\cr
  527       &\qquad\vdots\cr
  528     }
  529   $$
  530   Such an analysis would, however, be extremely cumbersome when it comes
  531   to the analysis of higher-order effects, and the obtained sum of various
  532   order terms would also be almost impossible to obtain a closed expression
  533   for.
  534   For future reference, to be used in the interpretation of the polarization
  535   density given by the Bloch equations, the intensity dependent polarization
  536   density is though shown in its explicit form below, including up to the
  537   seventh order interaction term in the Butcher and Cotter convention,
  538   $$
  539     \eqalignno{
  540       P^x_{\omega}
  541          =\varepsilon_0&\chi^{(1)}_{xx}
  542             (-\omega;\omega)E^x_{\omega}
  543         &({\rm order}\ n=1)\cr
  544         &+\varepsilon_0(3/4)\chi^{(3)}_{xxxx}
  545             (-\omega;\omega,\omega,-\omega)|E^x_{\omega}|^2 E^x_{\omega}
  546         &({\rm order}\ n=3)\cr
  547         &+\varepsilon_0(5/8)\chi^{(5)}_{xxxxxx}
  548             (-\omega;\omega,\omega,-\omega,\omega,-\omega)
  549             |E^x_{\omega}|^4 E^x_{\omega}
  550         &({\rm order}\ n=5)\cr
  551         &+\varepsilon_0(35/64)\chi^{(7)}_{xxxxxxxx}
  552             (-\omega;\omega,\omega,-\omega,\omega,-\omega,\omega,-\omega)
  553             |E^x_{\omega}|^6 E^x_{\omega}
  554         &({\rm order}\ n=7)\cr
  555         &+\ldots&\cr
  556     }
  557   $$
  558   The other approach to calculation of the polarization density, as we
  559   next will outline, is to use the steady-state solutions to the Bloch
  560   equations.
  561   \vfill\eject
  562   
  563   \subsection{The intensity-dependent refractive index in the Bloch-vector
  564     formalism}
  565   For steady-state interaction between light and matter, the solutions
  566   to the Bloch equations yield
  567   $$
  568     \eqalignno{
  569       &u-iv={{-\beta w}\over{\Delta-i/T_2}},&[{\rm B.\,\&\,C.~(6.53a)}],\cr
  570       &w={{w_0[1+(\Delta T_2)^2]}
  571         \over{1+(\Delta T_2)^2+\beta^2 T_1 T_2}},&[{\rm B.\,\&\,C.~(6.53b)}],\cr
  572     }
  573   $$
  574   where, as previously, $\beta=er^{\alpha}_{ab}E^{\alpha}_{\omega}(t)/\hbar$
  575   is the Rabi frequency, though now considered to be a slowly varying
  576   (adiabatically following) quantity, due to the assumption of steady-state
  577   behaviour.
  578   From the steady-state solutions, the $\mu$-component ($\mu=x,y,z$) of the
  579   electric polarization
  580   density ${\bf P}({\bf r},t)=\Re[{\bf P}_{\omega}\exp(-i\omega t)]$ of the
  581   medium hence is given as
  582   $$
  583     \eqalign{
  584       P^{\mu}_{\omega}
  585         &=Ner^{\mu}_{ab}(u-iv)\cr
  586         &=-Ner^{\mu}_{ab}{{\beta w}\over{\Delta-i/T_2}}\cr
  587         &=-Ner^{\mu}_{ab}{{\beta}\over{(\Delta-i/T_2)}}
  588           {{w_0[1+(\Delta T_2)^2]}\over{[1+(\Delta T_2)^2+\beta^2 T_1 T_2]}}\cr
  589         &=-New_0{{r^{\mu}_{ab}}\over{(\Delta-i/T_2)}}
  590           {{\beta}\over
  591            {\left[1+{{T_1 T_2}\over{(1+(\Delta T_2)^2)}}\beta^2\right]}}.\cr
  592     }\eqno{(3)}
  593   $$
  594   In this expression for the polarization density, it might at a first glance
  595   seem as it is negative for a positive Rabi frequency $\beta$, henc giving a
  596   polarization density that is directed anti-parallel to the electric field.
  597   However, the quantity $w_0=\rho_0(b)-\rho_0(a)$, the population inversion
  598   at thermal equilibrium, is always negative (since we for sure do not have
  599   any population inversion at thermal equilibrium, for which we rather expect
  600   the molecules to occupy the lower state), hence ensuring that the off-resonant,
  601   real-valued polarization density always is directed along the direction of the
  602   electric field of the light.
  603   
  604   Next observation is that the polarization density no longer is expressed
  605   as a power series in terms of the electric field, but rather as a rational
  606   function,
  607   $$
  608     P^{\mu}_{\omega}\sim X/(1+X^2),\eqno{(4)}
  609   $$
  610   where
  611   $$
  612     \eqalign{
  613       X&=\sqrt{{T_1 T_2}/{(1+(\Delta T_2)^2)}}\beta\cr
  614        &=\sqrt{{T_1 T_2}/{(1+(\Delta T_2)^2)}}
  615          er^{\alpha}_{ab}E^{\alpha}_{\omega}(t)/\hbar\cr
  616     }
  617   $$
  618   is a parameter linear in the electric field. The principal shape of the
  619   rational function in Eq.~(4) is shown in Fig.~12.
  620   
  621   From Eq.~(4), the polarization density is found to increase with increasing
  622   $X$ up to $X=1$, as we expect for an increasing power of an optical beam.
  623   However, for $X>1$, we find the somewhat surprising fact that the
  624   polarization density instead {\sl decrease} with an increasing intensity;
  625   this peculiar suggested behaviour should hence be explained before continuing.
  626   
  627   The first observation we may do is that the linear polarizability
  628   (i.~e.~what we usually associate with linear optics) follows the
  629   first order approximation $p(X)=X$.
  630   In the region where the peculiar decrease of the polarization density
  631   appear, the difference between the suggested nonlinear polarization
  632   density and the one given by the linear approximation is {\sl huge},
  633   and since we {\sl a priori} expect nonlinear contributions to be small
  634   compared to the alsways present linear ones, this is already an indication
  635   of that we in all practical situations do not have to consider the
  636   descrease of polarization density as shown in Fig.~12.
  637   
  638   For optical fields of the strength that would give rise to nonlinearities
  639   exceeding the linear terms, the underlying physics will rather belong
  640   to the field of plasma and high-energy physics, rather than a bound-charge
  641   description of gases and solids. This implies that the validity of the
  642   models here applied (bound charges, Hamiltonians being linear in the
  643   optical field, etc.) are limited to a range well within $X\le 1$.
  644   \vfill\eject
  645   
  646   \centerline{\epsfxsize=120mm\epsfbox{polplot.eps}}
  647   {\noindent Figure 12. The principal shape of the electric polarization
  648   density of the medium, as function of the applied electric field of
  649   the light.
  650   In this figure, $X=\sqrt{{T_1 T_2}/{(1+(\Delta T_2)^2)}}\beta$
  651   is a normalized parameter describing the field strength of the electric
  652   field of the light.}
  653   \bigskip
  654   
  655   Another interesting point we may observe is a more mathematically
  656   related one.
  657   In Fig.~12, we see that even for very high order terms (such as the
  658   approximating power series of degree 31, as shown in the figure),
  659   all power series expansions fail before reaching $X=1$.
  660   The reason for this is that the power series that approximate the
  661   rational function $X/(1+X^2)$,
  662   $$
  663     X/(1+X^2)=X-X^3+X^5-X^7+\ldots,
  664   $$
  665   is convergent only for $|X|<1$; for all other values, the series is
  666   divergent.
  667   This means that no matter how many terms we include in the power series
  668   in $X$, it will nevertheless fail when it comes to the evaluation
  669   for $|X|>1$.
  670   Since this power series expansion is equivalent to the expansion
  671   of the nonlinear polarization density in terms of the electrical field
  672   of the light (keeping in mind that $X$ here actually is linear in the
  673   electric field and hence strictly can be considered as the field variable),
  674   this also is an indication that at this point the whole susceptibility
  675   formalism fail to give a proper description at this working point.
  676   
  677   This is an excellent illustration of the downturn of the susceptibility
  678   description of interaction between light and matter; no matter how
  679   many terms we may include in the power series of the electrcal field,
  680   {\sl it will at some point nevertheless fail to give the total picture of
  681   the interaction}, and we must then instead seek other tools.
  682   
  683   Returning to the polarization density given by Eq.~(3), we may now express
  684   this in an explicit form by inserting $\Delta\equiv\Omega_{ba}-\omega$ for
  685   the angular frequency detuning, the Rabi frequency
  686   $\beta=er^{\alpha}_{ab}E^{\alpha}_{\omega}(t)/\hbar$,
  687   and the thermal equilibrium inversion $w_0=\rho_0(b)-\rho_0(a)$.
  688   This gives the polarization density of the medium as
  689   $$
  690     P^{\mu}_{\omega}
  691       =\varepsilon_0
  692       \underbrace{
  693         \underbrace{
  694           {{Ne^2}\over{\varepsilon_0\hbar}}(\rho_0(a)-\rho_0(b))
  695           {{r^{\mu}_{ab}r^{\alpha}_{ab}}\over{(\Omega_{ba}-\omega-i/T_2)}}
  696         }_{=\chi^{(1)}_{\mu\alpha}(-\omega;\omega)
  697             \ {\rm for\ a\ two\ level\ medium}}
  698         \underbrace{
  699           {{1}\over{\left[1+{{T_1 T_2}\over{(1+(\Omega_{ba}-\omega)^2 T^2_2)}}
  700             (er^{\gamma}_{ab}E^{\gamma}_{\omega}/\hbar)^2\right]}}
  701         }_{{\rm nonlinear\ correction\ factor\ to}
  702            \ \chi^{(1)}_{\mu\alpha}(-\omega;\omega)}
  703       }_{{\rm the\ field\ corrected\ susceptibility,}
  704          \ {\bar{\chi}}(\omega;{\bf E}_{\omega})
  705          \ [{\rm see\ Butcher\ and\ Cotter,\ section~6.3.1}]}
  706         E^{\alpha}_{\omega}.
  707   $$
  708   In this form, the polarization density is given as the product with
  709   a term which is identical to the linear
  710   susceptibility\footnote{${}^4$}{In the explicit expressions for the
  711   linear susceptibility, for example Butcher and Cotter's Eqs.~(4.58)
  712   and~(4.111) for the non-resonant and resonant cases, respectively,
  713   there are two terms, one with $\Omega_{ba}-\omega$ in the denominator
  714   and the other one with $\Omega_{ba}+\omega$.
  715   The reason why the second form does not appear in the expression for the
  716   field corrected susceptibility, as derived from the Bloch equations, is
  717   that {\sl we have used the rotating wave approximation in the derivation
  718   of the final expression.} (Recapitulate that in the rotating wave
  719   approximation, terms with oscillatory dependence of
  720   $\exp[i(\Omega_{ba}+\omega)t]$ were neglected.)
  721   As a result, all temporally phase-mismatched terms are neglected, and in
  722   particular only terms with $\Omega_{ba}-\omega$ in the denominator will
  723   remain. This, however, is a most acceptable approximation, especially when
  724   it comes to resonant interactions, where terms with $\Omega_{ba}-\omega$
  725   in the denominator by far will dominate over non-resonant terms.}
  726   (as obtained in the perturbation analysis in the frame of the susceptibility
  727   formalism), and a correction factor which is a nonlinear function of the
  728   electric field.
  729   
  730   The nonlinear correction factor, of the form $1/(1+X^2)$, with
  731   $X=\sqrt{{T_1 T_2}/{(1+(\Delta T_2)^2)}}\beta$ as previously, can now be
  732   expanded in a power series around the small-signal limit $X=0$, using
  733   $$
  734     1/(1+X^2)=1-X^2+X^4-X^6+\ldots,
  735   $$
  736   from which we obtain the polarization density as a power series in the
  737   electric field (which for the sake of simplicitly now is taken as linearly
  738   polarized along the $x$-axis) as
  739   $$
  740     \eqalign{
  741       P^x_{\omega}
  742         \approx\varepsilon_0
  743          &{{Ne^2}\over{\varepsilon_0\hbar}}(\rho_0(a)-\rho_0(b))
  744           {{r^x_{ab}r^x_{ab}}\over{(\Omega_{ba}-\omega-i/T_2)}}
  745         E^x_{\omega}\cr
  746         &-\varepsilon_0
  747           {{Ne^4}\over{\varepsilon_0\hbar^3}}(\rho_0(a)-\rho_0(b))
  748           {{r^x_{ab}r^x_{ab}}\over{(\Omega_{ba}-\omega-i/T_2)}}
  749           {{(r^x_{ab})^2}\over{[1/T^2_2+(\Omega_{ba}-\omega)^2](T_2/T_1)}}
  750           |E^x_{\omega}|^2
  751         E^x_{\omega}\cr
  752         &+\varepsilon_0
  753           {{Ne^6}\over{\varepsilon_0\hbar^5}}(\rho_0(a)-\rho_0(b))
  754           {{r^x_{ab}r^x_{ab}}\over{(\Omega_{ba}-\omega-i/T_2)}}
  755           {{(r^x_{ab})^4}\over{[1/T^2_2+(\Omega_{ba}-\omega)^2]^2(T_2/T_1)^2}}
  756           |E^x_{\omega}|^4
  757         E^x_{\omega}\cr
  758         &+\ldots\cr
  759     }\eqno{(5)}
  760   $$
  761   This form is identical to one as obtained in the susceptibility formalism;
  762   however, the steps that led us to this expression for the polarization
  763   density {\sl do not rely on the perturbation theory of the density
  764   operator}, but rather on the explicit form of the steady-state solutions
  765   to the Bloch equations.
  766   
  767   \section{Summary of the Bloch and susceptibility polarization densities}
  768   To summarize this last lecture on the Bloch equations, expressing the involved
  769   parameters in the same style as previously used in the description of the
  770   susceptibility formalism, the polarization density obtained from the
  771   steady-state solutions to the Bloch equations is
  772   $$
  773     P^{\mu}_{\omega}
  774       =\varepsilon_0
  775           {{Ne^2}\over{\varepsilon_0\hbar}}(\rho_0(a)-\rho_0(b))
  776           {{r^{\mu}_{ab}r^{\alpha}_{ab}}\over{(\Omega_{ba}-\omega-i/T_2)}}
  777           {{1}\over{\left[1+{{T_1 T_2}\over{(1+(\Omega_{ba}-\omega)^2 T^2_2)}}
  778             (er^{\alpha}_{ab}E^{\alpha}_{\omega}/\hbar)^2\right]}}
  779         E^{\alpha}_{\omega}.
  780   $$
  781   By expanding this in a power series in the electrical field, one obtains
  782   the form (5), in which we from the same description of the polarization
  783   density in the susceptibility formalism can identify
  784   $$
  785     \eqalign{
  786       \chi^{(1)}_{xx}(-\omega;\omega)
  787         &={{Ne^2}\over{\varepsilon_0\hbar}}(\rho_0(a)-\rho_0(b))
  788           {{r^x_{ab}r^x_{ab}}\over{(\Omega_{ba}-\omega-i/T_2)}},\cr
  789       \chi^{(3)}_{xxxx}(-\omega;\omega,\omega,-\omega)
  790         &=-{{4Ne^4}\over{3\varepsilon_0\hbar^3}}(\rho_0(a)-\rho_0(b))
  791           {{r^x_{ab}r^x_{ab}}\over{(\Omega_{ba}-\omega-i/T_2)}}
  792           {{(r^x_{ab})^2}\over{[1/T^2_2+(\Omega_{ba}-\omega)^2](T_2/T_1)}},\cr
  793       \chi^{(5)}_{xxxxxx}(-\omega;\omega,\omega,-\omega,&\omega,-\omega)\cr
  794         &={{8Ne^6}\over{5\varepsilon_0\hbar^5}}(\rho_0(a)-\rho_0(b))
  795           {{r^x_{ab}r^x_{ab}}\over{(\Omega_{ba}-\omega-i/T_2)}}
  796          {{(r^x_{ab})^4}\over{[1/T^2_2+(\Omega_{ba}-\omega)^2]^2(T_2/T_1)^2}},\cr
  797     }
  798   $$
  799   as being the first contributions to the two-level polarization density,
  800   including up to fifth order interactions.
  801   For a summary of the non-resonant forms of the susceptibilities of
  802   two-level systems, se Butcher and Cotter, Eqs.~(6.71)--(6.73).
  803   \vfill\eject
  804   
  805   \section{Appendix: Notes on the numerical solution to the Bloch equations}
  806   In their original form, the Bloch equations for a two-level system are
  807   given by Eqs.~(1) as
  808   $$
  809     \eqalignno{
  810       {{du}\over{dt}}&=-\Delta v -u/T_2,\cr
  811       {{dv}\over{dt}}&=\Delta u+\beta(t)w-v/T_2,\cr
  812       {{dw}\over{dt}}&=-\beta(t)v-(w-w_0)/T_1.\cr
  813     }
  814   $$
  815   By taking the time in units of the transverse relaxation time $T_2$, as
  816   $$
  817     \tau=t/T_2,
  818   $$
  819   the Bloch equations in this normalized time scale become
  820   $$
  821     \eqalignno{
  822       {{du}\over{d\tau}}&=-\Delta T_2 v -u,\cr
  823       {{dv}\over{d\tau}}&=\Delta T_2 u+\beta(t)T_2 w-v,\cr
  824       {{dw}\over{d\tau}}&=-\beta(t)T_2 v-(w-w_0)T_2/T_1.\cr
  825     }
  826   $$
  827   In this system of equations, all coefficients are now normalized and
  828   physically dimensionless, expressed as relevant quotes between relaxation
  829   times and products of the Rabi frequency or detuning frequency with
  830   the transverse relaxation time.
  831   Hence, by taking the normalized parameters
  832   $$
  833     \eqalign{
  834       \delta&=\Delta T_2,\cr
  835       \gamma(t)&=\beta(t)T_2,\cr
  836       \eta&=T_1/T_2,\cr
  837     }
  838   $$
  839   where $\delta$ can be considered as the normalized detuning from molecular
  840   resonance of the medium, $\gamma(t)$ as the normalized Rabi frequency,
  841   and $\eta$ as a parameter which describes the relative impact of the
  842   longitudinal vs transverse relaxation times, the Bloch equations take
  843   the normalized final form
  844   $$
  845     \eqalignno{
  846       {{du}\over{d\tau}}&=-\delta v -u,\cr
  847       {{dv}\over{d\tau}}&=\delta u+\gamma(t) w-v,\cr
  848       {{dw}\over{d\tau}}&=-\gamma(t) v-(w-w_0)/\eta.\cr
  849     }
  850   $$
  851   This normalized form of the Bloch equations has been used throughout the
  852   generation of graphs in Figs.~2--11 of this lecture, describing the
  853   qualitative impact of different regimes of resonance and relaxation.
  854   The normalized Bloch equations were in the simulations shown in Figs.~2--11
  855   integrated by using the standard routine {\tt ODE45()} in MATLAB.
  856   \bye
  857   

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