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    1   % File: nlopt/lect11/lect11.tex [pure TeX code]
    2   % Last change: March 17, 2003
    3   %
    4   % Lecture No 11 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{11}
   45   In this lecture, we will focus on configurations where the angular frequency
   46   of the light is close to some transition frequency of the medium.
   47   In particular, we will start with a brief outline of how the non-resonant
   48   susceptibilities may be modified in such a way that weakly resonant
   49   interactions can be taken into account.
   50   Having formulated the susceptibilities at weakly resonant interaction,
   51   we will proceed with formulating a non-perturbative approach of calculation
   52   of the polarization density of the medium. For the two-level system, this
   53   results in the Bloch equations governing resonant interaction between light
   54   and matter.
   55   \medskip
   56   
   57   \noindent The outline for this lecture is:
   58   \item{$\bullet$}{Singularities of the non-resonant susceptibilities}
   59   \item{$\bullet$}{Alternatives to perturbation analysis of the
   60     polarization density}
   61   \item{$\bullet$}{Relaxation of the medium}
   62   \item{$\bullet$}{The two-level system and the Bloch equation}
   63   \item{$\bullet$}{The resulting polarization density of the medium at resonance}
   64   \medskip
   65   
   66   \section{Singularities of non-resonant susceptibilities}
   67   In the theory described so far in this course, all interactions have for
   68   simplicity been considered as non-resonant.
   69   The explicit forms of the susceptibilities, in terms of the electric dipole
   70   moments and transition frequencies of the molecules, have been obtained in
   71   lecture six, of the forms
   72   $$
   73     \eqalignno{
   74       \chi^{(1)}_{\mu\alpha}(-\omega;\omega)
   75         &\sim{{r^{\mu}_{ab}r^{\alpha}_{ba}}\over{\Omega_{ba}-\omega}}
   76            +\{{\rm similar\ terms}\},
   77         &[{\rm B.\,\&\,C.\,(4.58)}]\cr
   78       \chi^{(2)}_{\mu\alpha\beta}(-\omega_{\sigma};\omega_1,\omega_2)
   79         &\sim{{r^{\mu}_{ab} r^{\alpha}_{bc} r^{\beta}_{ca}}
   80                \over{(\Omega_{ba}-\omega_1-\omega_2)
   81                      (\Omega_{ca}-\omega_2)}}
   82            +\{{\rm similar\ terms}\},
   83         &[{\rm B.\,\&\,C.\,(4.63)}]\cr
   84       \chi^{(3)}_{\mu\alpha\beta\gamma}
   85         (-\omega_{\sigma};\omega_1,\omega_2,\omega_3)
   86         &\sim{{r^{\mu}_{ab} r^{\alpha}_{bc} r^{\beta}_{cd} r^{\gamma}_{da}}
   87                \over{(\Omega_{ba}-\omega_1-\omega_2-\omega_3)
   88                      (\Omega_{ca}-\omega_2-\omega_3)
   89                      (\Omega_{da}-\omega_3)}}
   90            +\{{\rm similar\ terms}\},
   91       \cr&\qquad\qquad\qquad
   92         &[{\rm B.\,\&\,C.\,(4.64)}]\cr
   93         &\vdots\cr
   94     }
   95   $$
   96   To recapitulate, these forms have all been derived under the assumption
   97   that the Hamiltonian (which is the general operator which describes the
   98   state of the system) consist only of a thermal equilibrium part and an
   99   interaction part (in the electric dipolar approximation), of the form
  100   $$
  101     {\hat H}={\hat H}_0+{\hat H}_{\rm I}(t).
  102   $$
  103   This is a form which clearly does not contain any term related to relaxation
  104   effects of the medium, that is to say, it does not contain any term describing
  105   any energy flow into thermal heat. As long as we consider the interaction part
  106   of the Hamiltonian to be sufficiently strong compared to any relaxation effect
  107   of the medium, this is a valid approximation.
  108   
  109   However, the problem with the non-resonant forms of the susceptibilities
  110   clearly comes into light when we consider an angular frequency of the
  111   light that is close to a transition frequency of the system, since for
  112   the first order susceptibility,
  113   $$
  114     \chi^{(1)}_{\mu\alpha}(-\omega;\omega)\to\infty,
  115     \quad{\rm when\ }\omega\to\Omega_{ba},
  116   $$
  117   or for the second order susceptibility,
  118   $$
  119     \chi^{(2)}_{\mu\alpha\beta}(-\omega;\omega_1,\omega_2)\to\infty,
  120     \quad{\rm when\ }\omega_1+\omega_2\to\Omega_{ba}
  121     {\rm\ or\ }\omega_2\to\Omega_{ca}.
  122   $$
  123   This clearly non-physical behaviour is a consequence of that the denominators
  124   of the rational expressions for the susceptibilities have singularities
  125   at the resonances, and the aim with this lecture is to show how these
  126   singularities can be removed.
  127   
  128   \section{Modification of the Hamiltonian for resonant interaction}
  129   Whenever we have to consider relaxation effects of the medium, as in the
  130   case of resonant interactions, the Hamiltonian should be modified to
  131   $$
  132     {\hat H}={\hat H}_0+{\hat H}_{\rm I}(t)+{\hat H}_{\rm R},\eqno{(1)}
  133   $$
  134   where, as previously, ${\hat H}_0$ is the Hamiltonian in the absence of
  135   external forces, ${\hat H}_{\rm I}(t)=-{\hat Q}_{\alpha}E_{\alpha}({\bf r},t)$
  136   is the interaction Hamiltonian
  137   (here taken in the Schr\"odinger picture, as described in lecture four),
  138   being linear in the applied electric field of the light,
  139   and where the new term ${\hat H}_{\rm R}$ describes the various relaxation
  140   processes that brings the system into the thermal equilibrium whenever
  141   external forces are absent.
  142   The state of the system (atom, molecule, or general ensemble) is then
  143   conveniently described by the density operator formalism, from which
  144   we can obtain macroscopically observable parameters of the medium,
  145   such as the electric polarization density (as frequently encountered
  146   in this course), the magnetization of the medium, current densities, etc.
  147   
  148   The form (1) of the Hamiltonian is now to be analysed by means of the
  149   equation of motion of the density operator $\hat{\rho}$,
  150   $$
  151     i\hbar{{d{\hat{\rho}}}\over{dt}}
  152       ={\hat H}{\hat\rho}-{\hat\rho}{\hat H}
  153       =[{\hat H},{\hat\rho}],\eqno{(2)}
  154   $$
  155   and depending on the setup, this equation may be solved by means of
  156   perturbation analysis (for non-resonant and weakly resonant interactions),
  157   or by means of non-perturbative approaches, such as the Bloch equations
  158   (for strongly resonant interactions).
  159   
  160   \section{Phenomenological representation of relaxation processes}
  161   In many cases, the relaxation process of the medium towards thermal
  162   equilibrium can be described by
  163   $$
  164     [{\hat H}_{\rm R},{\hat\rho}]
  165       =-i\hbar{\hat\Gamma}({\hat\rho}-{\hat\rho}_0),
  166   $$
  167   where ${\hat\rho}_0$ is the thermal equilibrium density operator
  168   of the system. The here phenomenologically introduced operator
  169   ${\hat\Gamma}$ describes the relaxation of the medium, and can can be
  170   considered as being independent of the interaction Hamiltonian.
  171   Here the operator ${\hat\Gamma}$ has the physical dimension of an angular
  172   frequency, and its matrix elements can be considered as giving the time
  173   constants of decay for various states of the system.
  174   
  175   \section{Perturbation analysis of weakly resonant interactions}
  176   Before entering the formalism of the Bloch equations for strongly resonant
  177   interactions, we will outline the weakly resonant interactions in a
  178   perturbative analysis for the susceptibilities, as previously developed
  179   in lectures three, four, and five.
  180   
  181   By taking the perturbation series for the density operator as
  182   $$
  183     \hat{\rho}(t)=\underbrace{\hat{\rho}_0}_{\sim [E(t)]^0}
  184       +\underbrace{\hat{\rho}_1(t)}_{\sim [E(t)]^1}
  185       +\underbrace{\hat{\rho}_2(t)}_{\sim [E(t)]^2}
  186       +\ldots
  187       +\underbrace{\hat{\rho}_n(t)}_{\sim [E(t)]^n}
  188       +\ldots,
  189   $$
  190   as we previously did for the strictly non-resonant case, one obtains
  191   the system of equations
  192   $$
  193     \eqalign{
  194       i\hbar{{d\hat{\rho}_0}\over{dt}}&=[\hat{H}_0,\hat{\rho}_0],\cr
  195       i\hbar{{d\hat{\rho}_1(t)}\over{dt}}&=[\hat{H}_0,\hat{\rho}_1(t)]
  196         +[\hat{H}_{\rm I}(t),\hat{\rho}_0]
  197         -i\hbar{\hat\Gamma}{\hat\rho}_1(t),\cr
  198       i\hbar{{d\hat{\rho}_2(t)}\over{dt}}&=[\hat{H}_0,\hat{\rho}_2(t)]
  199         +[\hat{H}_{\rm I}(t),\hat{\rho}_1(t)]
  200         -i\hbar{\hat\Gamma}{\hat\rho}_2(t),\cr
  201       &\vdots\cr
  202       i\hbar{{d\hat{\rho}_n(t)}\over{dt}}&=[\hat{H}_0,\hat{\rho}_n(t)]
  203         +[\hat{H}_{\rm I}(t),\hat{\rho}_{n-1}(t)]
  204         -i\hbar{\hat\Gamma}{\hat\rho}_n(t),\cr
  205      &\vdots\cr
  206     }
  207   $$
  208   As in the non-resonant case, one may here start with solving for the
  209   zeroth order term $\hat{\rho}_0$, with all other terms obtained by
  210   consecutively solving the equations of order $j=1,2,\ldots,n$, in that order.
  211   
  212   Proceeding in exactly the same path as for the non-resonant case,
  213   solving for the density operator in the interaction picture and
  214   expressing the various terms of the electric polarization density
  215   in terms of the corresponding traces
  216   $$
  217     P_{\mu}({\bf r},t)
  218       =\sum^{\infty}_{n=0} P^{(n)}_{\mu}({\bf r},t)
  219       ={{1}\over{V}}\sum^{\infty}_{n=0}
  220        {\rm Tr}[{\hat\rho}_n(t){\hat Q}_{\mu}],
  221   $$
  222   one obtains the linear, first order susceptibility of the form
  223   $$
  224     \eqalign{
  225       \chi^{(1)}_{\mu\alpha}(-\omega;\omega)
  226         &={{N e^2}\over{\varepsilon_0\hbar}}
  227           \sum_a\varrho_0(a)\sum_b
  228           \Big({{r^{\mu}_{ab}r^{\alpha}_{ba}}
  229               \over{\Omega_{ba}-\omega-i\Gamma_{ba}}}
  230             +{{r^{\alpha}_{ab}r^{\mu}_{ba}}
  231               \over{\Omega_{ba}+\omega-i\Gamma_{ba}}}\Big).\cr
  232     }
  233   $$
  234   Similarly, the second order susceptibility for weakly resonant interaction
  235   is obtained as
  236   $$
  237     \eqalign{
  238       \chi^{(2)}_{\mu\alpha\beta}&(-\omega_{\sigma};\omega_1,\omega_2)\cr
  239         &={{N e^3}\over{\varepsilon_0 \hbar^2}}
  240           {{1}\over{2!}}{\bf S}
  241           \sum_a\varrho_0(a)\sum_b\sum_c
  242           \Big\{
  243               {{r^{\mu}_{ab} r^{\alpha}_{bc} r^{\beta}_{ca}}
  244                \over{(\Omega_{ac}+\omega_2-i\Gamma_{ac})
  245                      (\Omega_{ab}+\omega_{\sigma}-i\Gamma_{ab})}}
  246           \cr&\qquad
  247               -{{r^{\alpha}_{ab} r^{\mu}_{bc} r^{\beta}_{ca}}
  248                 \over{(\Omega_{ac}+\omega_2-i\Gamma_{ac})
  249                       (\Omega_{bc}+\omega_{\sigma}-i\Gamma_{bc})}}
  250               -{{r^{\beta}_{ab} r^{\mu}_{bc} r^{\alpha}_{ca}}
  251                 \over{(\Omega_{ba}+\omega_2-i\Gamma_{ba})
  252                       (\Omega_{bc}+\omega_{\sigma}-i\Gamma_{bc})}}
  253           \cr&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad
  254               +{{r^{\beta}_{ab} r^{\alpha}_{bc} r^{\mu}_{ca}}
  255                 \over{(\Omega_{ba}+\omega_2-i\Gamma_{ba})
  256                       (\Omega_{ca}+\omega_{\sigma}-i\Gamma_{ca})}}
  257           \Big\}.\cr
  258     }
  259   $$
  260   In these expressions for the susceptibilities, the singularities at resonance
  261   are removed, and the spectral properties of the absolute values of the
  262   susceptibilities are described by regular Lorenzian line shapes.
  263   
  264   The values of the matrix elements $\Gamma_{mn}$ are in many cases difficult
  265   to derive from a theoretical basis; however, they are often straightforward
  266   to obtain by regular curve-fitting and regression analysis of experimental
  267   data.
  268   
  269   As seen from the expressions for the susceptibilities above, we still have
  270   a boosting of them close to resonance (resonant enhancement). However, the
  271   values of the susceptibilities reach a plateau at exact resonance, with
  272   maximum values determined by the magnitudes of the involved matrix elements
  273   $\Gamma_{mn}$ of the relaxation operator.
  274   
  275   \section{Validity of perturbation analysis of the polarization density}
  276   Strictly speaking, the perturbative approach is only to be considered
  277   as for an infinite series expansion.
  278   For a limited number of terms, the perturbative approach is only an
  279   approximative method, which though for many cases is sufficient.
  280   
  281   The perturbation series, in the form that we have encountered it in this
  282   course, defines a power series in the applied electric field of the light,
  283   and as long as the lower order terms are dominant in the expansion, we
  284   may safely neglect the higher order ones.
  285   Whenever we encounter strong fields, however, we may run into trouble with
  286   the series expansion, in particular if we are in a resonant optical regime,
  287   with a boosting effect of the polarization density of the medium.
  288   (This boosting effect can be seen as the equivalent to the close-to-resonance
  289   behaviour of the mechanical spring model under influence of externally
  290   driving forces.)
  291   
  292   As an illustration to this source of failure of the model in the presence of
  293   strong electrical fields, we may consider another, more simple example
  294   of series expansions, namely the Taylor expansion of the function $\sin(x)$
  295   around $x\approx 0$, as shown in Fig.~1.
  296   \bigskip
  297   \centerline{\epsfxsize=110mm\epsfbox{sinapprx.eps}}
  298   \centerline{Figure 1. Approximations to $f(x)=\sin(x)$ by means of power
  299     series expansions of various degrees.}
  300   \medskip
  301   
  302   In analogy to the susceptibility formalism, we may consider $x$ to
  303   have the role of the electric field (the variable which we make
  304   the power expansion in terms of), and $\sin(x)$ to have the role
  305   of the polarization density or the density operator (simply the function
  306   we wish to analyze).
  307   For low numerical values of $x$, up to about $x\approx 1$,
  308   the $\sin(x)$ function is well described by keeping only the first two
  309   terms of the expansion, corresponding to a power expansion up to and
  310   including order three,
  311   $$
  312     \sin(x)\approx p_3(x)=x-{{x^3}\over{3!}}.
  313   $$
  314   For higher values of $x$, say up to about $x\approx 2$, the expansion
  315   is still following the exact function to a good approximation if we
  316   include also the third term, corresponding to a power expansion up to and
  317   including order five,
  318   $$
  319     \sin(x)\approx p_5(x)=x-{{x^3}\over{3!}}+{{x^5}\over{5!}}.
  320   $$
  321   This necessity of including higher and higher order terms goes on
  322   as we increase the value of $x$, and we can from the graph also see
  323   that the breakdown at a certain level of approximation causes severe
  324   difference between the approximate and exact curves.
  325   In particular, if one wish to calculate the value of the function $\sin(x)$
  326   for small $x$, it might be a good idea to apply the series expansion.
  327   For greater values of $x$, say $x\approx 10$, the series expansion
  328   approach is, however, a bad idea, and an efficient evaluation of $\sin(x)$
  329   requires another approach.
  330   
  331   As a matter of fact, the same arguments hold for the more complex case
  332   of the series expansion of the density operator\footnote{${}^1$}{We may
  333   recall that the series expansion of the density operator is {\sl the}
  334   very origin of the expansion of the polarization density of the medium
  335   in terms of the electric field, and hence also the very foundation for the
  336   whole susceptibility formalism as described in this course.}, for which
  337   we for high intensities (high electrical field strengths) must include
  338   higher order terms as well.
  339   
  340   However, we have seen that even in the non-resonant case, we may encounter
  341   great algebraic complexity even in low order nonlinear terms, and since
  342   the problem of formulating a proper polarization density is expanding
  343   more or less exponentially with the order of the nonlinearity, the
  344   usefulness of the susceptibility formalism eventually breaks down.
  345   The solution to this problem is to identify the relevant transitions
  346   of the ensemble, and to solve the equation of motion (2) exactly instead
  347   (or at least within other levels of approximation which do not rely on
  348   the perturbative foundation of the susceptibility formalism).
  349   
  350   \section{The two-level system}
  351   In many cases, the interaction between light and matter can be reduced
  352   to that of a two-level system, consisting of only two energy eigenstates
  353   $|a\rangle$ and $|b\rangle$.
  354   The equation of motion of the density operator is generally given by
  355   Eq.~(2) as
  356   $$
  357     i\hbar{{d{\hat{\rho}}}\over{dt}}=[{\hat H},{\hat\rho}],
  358   $$
  359   with
  360   $$
  361     {\hat H}={\hat H}_0+{\hat H}_{\rm I}(t)+{\hat H}_{\rm R}.
  362   $$
  363   For the two-level system, the equation of motion can be expressed in
  364   terms of the matrix elements of the density operator as
  365   $$
  366     \eqalignno{
  367       i\hbar{{d\rho_{aa}}\over{dt}}
  368         &=[{\hat H}_0,{\hat\rho}]_{aa}
  369            +[{\hat H}_{\rm I}(t),{\hat\rho}]_{aa}
  370            +[{\hat H}_{\rm R},{\hat\rho}]_{aa},&(3{\rm a})\cr
  371       i\hbar{{d\rho_{ab}}\over{dt}}
  372         &=[{\hat H}_0,{\hat\rho}]_{ab}
  373            +[{\hat H}_{\rm I}(t),{\hat\rho}]_{ab}
  374            +[{\hat H}_{\rm R},{\hat\rho}]_{ab},&(3{\rm b})\cr
  375       i\hbar{{d\rho_{bb}}\over{dt}}
  376         &=[{\hat H}_0,{\hat\rho}]_{bb}
  377            +[{\hat H}_{\rm I}(t),{\hat\rho}]_{bb}
  378            +[{\hat H}_{\rm R},{\hat\rho}]_{bb},&(3{\rm c})\cr
  379     }
  380   $$
  381   where the fourth equation for $\rho_{ba}$ was omitted, since the solution
  382   for this element immediately follows from
  383   $$
  384     \rho_{ba}=\rho^*_{ab}.
  385   $$
  386   
  387   \subsection{Terms involving the thermal equilibrium Hamiltonian}
  388   The system of Eqs.~(3) is the starting point for derivation of the so-called
  389   Bloch equations. Starting with the thermal-equilibrium part of the
  390   commutators in the right-hand sides of Eqs.~(3), we have for the diagonal
  391   elements
  392   $$
  393     \eqalign{
  394       [{\hat H}_0,{\hat\rho}]_{aa}
  395         &=\langle a|{\hat H}_0{\hat\rho}|a\rangle
  396           -\langle a|{\hat\rho}{\hat H}_0|a\rangle\cr
  397         &=\sum_k \underbrace{\langle a|{\hat H}_0|k\rangle}_{
  398                    ={\Bbb E}_a\delta_{ak}}
  399                  \langle k|{\hat\rho}|a\rangle
  400           -\sum_j \langle a|{\hat\rho}|j\rangle
  401                   \underbrace{\langle j|{\hat H}_0|a\rangle}_{
  402                    ={\Bbb E}_j\delta_{ja}}\cr
  403         &={\Bbb E}_a\rho_{aa}-\rho_{aa}{\Bbb E}_a\cr
  404         &=0\cr
  405         &=[{\hat H}_0,{\hat\rho}]_{bb},\cr
  406     }
  407   $$
  408   and for the off-diagonal elements
  409   $$
  410     \eqalign{
  411       [{\hat H}_0,{\hat\rho}]_{ab}
  412         &=\langle a|{\hat H}_0{\hat\rho}|b\rangle
  413           -\langle a|{\hat\rho}{\hat H}_0|b\rangle\cr
  414         &=\sum_k \underbrace{\langle a|{\hat H}_0|k\rangle}_{
  415                    ={\Bbb E}_a\delta_{ak}}
  416                  \langle k|{\hat\rho}|b\rangle
  417           -\sum_j \langle a|{\hat\rho}|j\rangle
  418                   \underbrace{\langle j|{\hat H}_0|b\rangle}_{
  419                    ={\Bbb E}_j\delta_{jb}}\cr
  420         &={\Bbb E}_a\rho_{ab}-\rho_{ab}{\Bbb E}_b\cr
  421         &=-({\Bbb E}_b-{\Bbb E}_a)\rho_{ab}\cr
  422         &=-\hbar\Omega_{ba}\rho_{ab}\cr
  423     }
  424   $$
  425   
  426   \subsection{Terms involving the interaction Hamiltonian}
  427   For the commutators in the right-hand sides of Eqs.~(3) involving the
  428   interaction Hamiltonian, we similarly have for the diagonal elements
  429   $$
  430     \eqalign{
  431       [{\hat H}_{\rm I}(t),{\hat\rho}]_{aa}
  432       &=\langle a|(-e{\hat r}_{\alpha}E_{\alpha}({\bf r},t)){\hat\rho}|a\rangle
  433       -\langle a|{\hat\rho}(-e{\hat r}_{\alpha}E_{\alpha}({\bf r},t))|a\rangle\cr
  434       &=-eE_{\alpha}({\bf r},t)
  435         \bigg\{
  436           \sum_k\langle a|{\hat r}_{\alpha}|k\rangle\langle k|{\hat\rho}|a\rangle
  437          -\sum_j\langle a|{\hat\rho}|j\rangle\langle j|{\hat r}_{\alpha}|a\rangle
  438         \bigg\}\cr
  439       &=-eE_{\alpha}({\bf r},t)
  440         \bigg\{
  441           r^{\alpha}_{aa}\rho_{aa}
  442           +r^{\alpha}_{ab}\rho_{ba}
  443           -\rho_{aa}r^{\alpha}_{aa}
  444           -\rho_{ab}r^{\alpha}_{ba}
  445         \bigg\}\cr
  446       &=-e(r^{\alpha}_{ab}\rho_{ba}-r^{\alpha}_{ba}\rho_{ab})
  447          E_{\alpha}({\bf r},t)\cr
  448       &=-[{\hat H}_{\rm I}(t),{\hat\rho}]_{bb},\cr
  449     }
  450   $$
  451   and for the off-diagonal elements
  452   $$
  453     \eqalign{
  454       [{\hat H}_{\rm I}(t),{\hat\rho}]_{ab}
  455       &=\langle a|(-e{\hat r}_{\alpha}E_{\alpha}({\bf r},t)){\hat\rho}|b\rangle
  456       -\langle a|{\hat\rho}(-e{\hat r}_{\alpha}E_{\alpha}({\bf r},t))|b\rangle\cr
  457       &=-eE_{\alpha}({\bf r},t)
  458         \bigg\{
  459           \sum_k\langle a|{\hat r}_{\alpha}|k\rangle\langle k|{\hat\rho}|b\rangle
  460          -\sum_j\langle a|{\hat\rho}|j\rangle\langle j|{\hat r}_{\alpha}|b\rangle
  461         \bigg\}\cr
  462       &=-eE_{\alpha}({\bf r},t)
  463         \bigg\{
  464           r^{\alpha}_{aa}\rho_{ab}
  465           +r^{\alpha}_{ab}\rho_{bb}
  466           -\rho_{aa}r^{\alpha}_{ab}
  467           -\rho_{ab}r^{\alpha}_{bb}
  468         \bigg\}\cr
  469       &=-er^{\alpha}_{ab}E_{\alpha}({\bf r},t)(\rho_{bb}-\rho_{aa})
  470         -e(r^{\alpha}_{aa}-r^{\alpha}_{bb})E_{\alpha}({\bf r},t)\rho_{ab}\cr
  471       &=\{{\rm Optical\ Stark\ shift:\ }
  472           \delta{\Bbb E}_k\equiv -er^{\alpha}_{kk}E_{\alpha}({\bf r},t),
  473           \quad k=a,b\}\cr
  474       &=-er^{\alpha}_{ab}E_{\alpha}({\bf r},t)(\rho_{bb}-\rho_{aa})
  475         +(\delta{\Bbb E}_a-\delta{\Bbb E}_b)\rho_{ab}.\cr
  476     }
  477   $$
  478   
  479   \subsection{Terms involving relaxation processes}
  480   For the commutators describing relaxation processes, the diagonal elements
  481   are given as
  482   $$
  483     \eqalign{
  484       [{\hat H}_{\rm R},{\hat\rho}]_{aa}
  485         &=-i\hbar(\rho_{aa}-\rho_0(a))/T_a,\cr
  486       [{\hat H}_{\rm R},{\hat\rho}]_{bb}
  487         &=-i\hbar(\rho_{bb}-\rho_0(b))/T_b,\cr
  488     }
  489   $$
  490   where $T_a$ and $T_b$ are the decay rates towards the thermal equilibrium
  491   at respective level, and where $\rho_0(a)$ and $\rho_0(b)$ are the thermal
  492   equilibrium values of $\rho_{aa}$ and $\rho_{bb}$, respectively (i.~e.~the
  493   thermal equilibrium population densities of the respective level).
  494   The off-diagonal elements are similarly given as
  495   $$
  496     \eqalign{
  497       [{\hat H}_{\rm R},{\hat\rho}]_{ab}&=-i\hbar\rho_{ab}/T_2,\cr
  498       [{\hat H}_{\rm R},{\hat\rho}]_{ba}&=-i\hbar\rho_{ba}/T_2.\cr
  499     }
  500   $$
  501   A common approximation is to consider the two states $|a\rangle$
  502   and $|b\rangle$ to be sufficiently similar in order to approximate
  503   their lifetimes as equal, i.~e.~$T_a\approx T_b\approx T_1$,
  504   where $T_1$ for historical reasons is denoted as the {\sl longitudinal
  505   relaxation time}.
  506   For the same historical reason, the relaxation time $T_2$ is denoted
  507   as the {\sl transverse relaxation time}.\footnote{${}^2$}{For a deeper
  508   discusssion and explanation of the various mechanisms involved in relaxation,
  509   see for example Charles~P. Slichter, {\sl Principles of Magnetic Resonance}
  510   (Springer-Verlag, Berlin, 1978), available at KTHB. This reference is
  511   not mentioned in Butcher and Cotters book, but it is a very good text
  512   on relaxation phenomena and how to incorporate them into a density-functional
  513   description of interaction between light and matter.}
  514   
  515   As the above matrix elements of the commutators involving the various
  516   terms of the Hamiltonian are inserted into the right-hand sides of Eqs.~(3),
  517   one obtains the following system of equations for the matrix elements
  518   of the density operator,
  519   $$
  520     \eqalignno{
  521       i\hbar{{d\rho_{aa}}\over{dt}}
  522         &=-e(r^{\alpha}_{ab}\rho_{ba}-r^{\alpha}_{ba}\rho_{ab})
  523            E_{\alpha}({\bf r},t)
  524            -i\hbar(\rho_{aa}-\rho_0(a))/T_a,&(4{\rm a})\cr
  525       i\hbar{{d\rho_{ab}}\over{dt}}
  526         &=-\hbar\Omega_{ba}\rho_{ab}
  527            -er^{\alpha}_{ab}E_{\alpha}({\bf r},t)(\rho_{bb}-\rho_{aa})
  528            +(\delta{\Bbb E}_a-\delta{\Bbb E}_b)\rho_{ab}
  529            -i\hbar\rho_{ab}/T_2,&(4{\rm b})\cr
  530       i\hbar{{d\rho_{bb}}\over{dt}}
  531         &=e(r^{\alpha}_{ab}\rho_{ba}-r^{\alpha}_{ba}\rho_{ab})
  532            E_{\alpha}({\bf r},t)
  533            -i\hbar(\rho_{bb}-\rho_0(b))/T_b.&(4{\rm c})\cr
  534     }
  535   $$
  536   (The system of equations~(4) corresponds to Butcher and Cotter's Eqs.~(6.35).)
  537   So far, the applied electric field of the light is allowed to be of arbitrary
  538   form. However, in order to simplify the following analysis, we will
  539   assume the light to be linearly polarized and quasimonochromatic, of the
  540   form
  541   $$
  542     E_{\alpha}({\bf r},t)=\Re[E^{\alpha}_{\omega}(t)\exp(-i\omega t)].
  543   $$
  544   We will in addition assume the slowly varying temporal envelope
  545   $E^{\alpha}_{\omega}(t)$ to be real-valued, and we will also neglect
  546   the optical Stark shifts $\delta{\Bbb E}_a$ and $\delta{\Bbb E}_b$.
  547   In the absence of strong static magnetic fields, we may also assume
  548   the matrix elements $er^{\alpha}_{ab}$ to be real-valued.
  549   When these assumptions and approximations are applied to the
  550   equations of motion~(4), one obtains
  551   $$
  552     \eqalignno{
  553       {{d\rho_{aa}}\over{dt}}
  554         &=i(\rho_{ba}-\rho_{ab})\beta(t)\cos(\omega t)
  555            -(\rho_{aa}-\rho_0(a))/T_a,&(5{\rm a})\cr
  556       {{d\rho_{ab}}\over{dt}}
  557         &=i\Omega_{ba}\rho_{ab}
  558            +i\beta(t)\cos(\omega t)(\rho_{bb}-\rho_{aa})
  559            -\rho_{ab}/T_2,&(5{\rm b})\cr
  560       {{d\rho_{bb}}\over{dt}}
  561         &=-i(\rho_{ba}-\rho_{ab})\beta(t)\cos(\omega t)
  562            -(\rho_{bb}-\rho_0(b))/T_b,&(5{\rm c})\cr
  563     }
  564   $$
  565   where the {\sl Rabi frequency} $\beta(t)$, defined in terms of the spatial
  566   envelope of the electrical field and the transition dipole moment as
  567   $$
  568     \beta(t)=er^{\alpha}_{ab}E^{\alpha}_{\omega}(t)/\hbar
  569             =e{\bf r}_{ab}\cdot{\bf E}_{\omega}(t)/\hbar,
  570   $$
  571   was introduced.
  572   
  573   \section{The rotating-wave approximation}
  574   In the middle equation of the system~(5), we have a time-derivative
  575   of $\rho_{ab}$ in the left-hand side, while we in the right-hand side
  576   have a term $i\Omega_{ba}\rho_{ab}$. Seen as the homogeneous part of
  577   a linear differential equation, this suggests that we may further
  578   simplify the equations of motion by taking a new variable
  579   $\rho^{\Omega}_{ab}$ according to the variable substitution
  580   $$
  581     \rho_{ab}=\rho^{\Omega}_{ab}\exp[i(\Omega_{ba}-\Delta)t],\eqno{(6)}
  582   $$
  583   where $\Delta\equiv\Omega_{ba}-\omega$ is the detuning of the
  584   angular frequency of the light from the transition frequency
  585   $\Omega_{ba}\equiv({\Bbb E}_b-{\Bbb E}_a)/\hbar$.
  586   
  587   By inserting Eq.~(6) into Eqs.~(5), keeping in mind that
  588   $\rho_{ba}=\rho^*_{ab}$, one obtains the system
  589   $$
  590     \eqalignno{
  591       {{d\rho_{aa}}\over{dt}}
  592         &=i(\rho^{\Omega}_{ba}\exp[-i(\Omega_{ba}-\Delta)t]
  593              -\rho^{\Omega}_{ab}\exp[i(\Omega_{ba}-\Delta)t])
  594              \beta(t)\cos(\omega t)
  595            -(\rho_{aa}-\rho_0(a))/T_a,&(6{\rm a})\cr
  596       {{d\rho^{\Omega}_{ab}}\over{dt}}
  597         &=i\Delta\rho^{\Omega}_{ab}
  598            +i\beta(t)\cos(\omega t)\exp[-i(\Omega_{ba}-\Delta)t]
  599              (\rho_{bb}-\rho_{aa})
  600            -\rho^{\Omega}_{ab}/T_2,&(6{\rm b})\cr
  601       {{d\rho_{bb}}\over{dt}}
  602         &=-i(\rho^{\Omega}_{ba}\exp[-i(\Omega_{ba}-\Delta)t]
  603              -\rho^{\Omega}_{ab}\exp[i(\Omega_{ba}-\Delta)t])
  604              \beta(t)\cos(\omega t)
  605            -(\rho_{bb}-\rho_0(b))/T_b,&(6{\rm c})\cr
  606     }
  607   $$
  608   The idea with the rotating-wave approximation is now to separate out
  609   rapidly oscillating terms of angular frequencies $\omega+\Omega_{ba}$
  610   and $-(\omega+\Omega_{ba})$, and neglect these terms, compared with
  611   more slowly varying terms. The motivation for this approximation is that
  612   whenever high-frequency components appear in the equations of motions,
  613   the high-frequency terms will when integrated contain large denominators,
  614   and will hence be minor in comparison with terms with a slow variation.
  615   In some sense we can also see this as a temporal averaging procedure,
  616   where rapidly oscillating terms average to zero rapidly compared
  617   to slowly varying (or constant) components.
  618   
  619   For example, in Eq.~(6b), the product of the $\cos(\omega t)$
  620   and the exponential function is approximated as
  621   $$
  622     \eqalign{
  623       \cos(\omega t)\exp[-i(\Omega_{ba}-\Delta)t]
  624         &={{1}\over{2}}[\exp(i\omega t)+\exp(-i\omega t)]
  625            \exp[-i\underbrace{(\Omega_{ba}-\Delta)}_{=\omega}t]\cr
  626         &={{1}\over{2}}[1+\exp(-i2\omega t)]\to{{1}\over{2}},
  627     }
  628   $$
  629   while in Eqs.~(6a) and~(6c), the same argument gives
  630   $$
  631     \eqalign{
  632       \exp[i(\Omega_{ba}-\Delta)t]\cos(\omega t)
  633         &={{1}\over{2}}[\exp(i\omega t)+\exp(-i\omega t)]
  634            \exp[-i\underbrace{(\Omega_{ba}-\Delta)}_{=\omega}t]\cr
  635         &={{1}\over{2}}[\exp(i2\omega t)+1]\to{{1}\over{2}}.
  636     }
  637   $$
  638   By applying this {\sl rotating-wave approximation}, the equations
  639   of motion~(6) hence take the form
  640   $$
  641     \eqalignno{
  642       {{d\rho_{aa}}\over{dt}}
  643         &={{i}\over{2}}(\rho^{\Omega}_{ba}-\rho^{\Omega}_{ab})\beta(t)
  644            -(\rho_{aa}-\rho_0(a))/T_a,&(7{\rm a})\cr
  645       {{d\rho^{\Omega}_{ab}}\over{dt}}
  646         &=i\Delta\rho^{\Omega}_{ab}
  647            +{{i}\over{2}}\beta(t)(\rho_{bb}-\rho_{aa})
  648            -\rho^{\Omega}_{ab}/T_2,&(7{\rm b})\cr
  649       {{d\rho_{bb}}\over{dt}}
  650         &=-{{i}\over{2}}(\rho^{\Omega}_{ba}-\rho^{\Omega}_{ab})\beta(t)
  651            -(\rho_{bb}-\rho_0(b))/T_b.&(7{\rm c})\cr
  652     }
  653   $$
  654   In this final form, before entering the Bloch vector description of the
  655   interaction, these equations correspond to Butcher and Cotter's Eqs.~(6.41).
  656   
  657   \section{The Bloch equations}
  658   Assuming the two states $|a\rangle$ and $|b\rangle$ to be sufficiently
  659   similar in order to approximate~$T_a\approx T_b\approx T_1$,
  660   where $T_1$ is the longitudinal relaxation time, and by taking new
  661   variables $(u,v,w)$ according to
  662   $$
  663     \eqalign{
  664       u&=\rho^{\Omega}_{ba}+\rho^{\Omega}_{ab},\cr
  665       v&=i(\rho^{\Omega}_{ba}-\rho^{\Omega}_{ab}),\cr
  666       w&=\rho_{bb}-\rho_{aa},\cr
  667     }
  668   $$
  669   the equations of motion (7) are cast in the {\sl Bloch equations}
  670   $$
  671     \eqalignno{
  672       {{du}\over{dt}}&=-\Delta v -u/T_2,&(8{\rm a})\cr
  673       {{dv}\over{dt}}&=\Delta u+\beta(t)w-v/T_2,&(8{\rm b})\cr
  674       {{dw}\over{dt}}&=-\beta(t)v-(w-w_0)/T_1.&(8{\rm c})\cr
  675     }
  676   $$
  677   In these equations, the introduced variable $w$ describes the population
  678   inversion of the two-level system, while $u$ and $v$ are related to the
  679   dispersive and absorptive components of the polarization density of the
  680   medium.
  681   In the Bloch equations above, $w_0=\rho_0(b)-\rho_0(a)$
  682   is the thermal equilibrium inversion of the system with no optical
  683   field applied.
  684   
  685   \section{The resulting electric polarization density of the medium}
  686   The so far developed theory of the density matrix under resonant
  687   interaction can now be applied to the calculation of the electric
  688   polarization density of the medium, consisting of $N$ identical
  689   molecules per unit volume, as
  690   $$
  691     \eqalign{
  692       P_{\mu}({\bf r},t)&=N\langle e{\hat r}_{\mu}\rangle\cr
  693         &=N\Tr[{\hat\rho} e{\hat r}_{\mu}]\cr
  694         &=N\sum_{k=a,b}\langle k|{\hat\rho} e{\hat r}_{\mu}|k\rangle\cr
  695         &=N\sum_{k=a,b}\sum_{j=a,b}
  696           \langle k|{\hat\rho}|j\rangle
  697           \langle j|e{\hat r}_{\mu}|k\rangle\cr
  698         &=N\sum_{k=a,b}\left\{
  699           \langle k|{\hat\rho}|a\rangle
  700           \langle a|e{\hat r}_{\mu}|k\rangle
  701          +\langle k|{\hat\rho}|b\rangle
  702           \langle b|e{\hat r}_{\mu}|k\rangle
  703         \right\}\cr
  704         &=N\left\{
  705           \langle a|{\hat\rho}|a\rangle
  706           \langle a|e{\hat r}_{\mu}|a\rangle
  707          +\langle b|{\hat\rho}|a\rangle
  708           \langle a|e{\hat r}_{\mu}|b\rangle
  709          +\langle a|{\hat\rho}|b\rangle
  710           \langle b|e{\hat r}_{\mu}|a\rangle
  711          +\langle b|{\hat\rho}|b\rangle
  712           \langle b|e{\hat r}_{\mu}|b\rangle
  713         \right\}\cr
  714         &=N(\rho_{ba}er^{\mu}_{ab}+\rho_{ab}er^{\mu}_{ba})\cr
  715         &=\{{\rm Make\ use\ of\ }\rho_{ab}=(u+iv)\exp(i\omega t)=\rho^*_{ba}\}\cr
  716         &=N[(u-iv)\exp(-i\omega t)er^{\mu}_{ab}
  717             +(u+iv)\exp(i\omega t)er^{\mu}_{ba}].\cr
  718     }
  719   $$
  720   The temporal envelope $P^{\mu}_{\omega}$ of the polarization density,
  721   throughout this course as well as in Butcher and Cotter's book, is taken as
  722   $$
  723     P^{\mu}({\bf r},t)=\Re[P^{\mu}_{\omega}\exp(-i\omega t)],
  724   $$
  725   and by identifying this expression with the right-hand side of the result
  726   above, we hence finally have obtained the polarization density
  727   in terms of the Bloch parameters $(u,v,w)$ as
  728   $$
  729     P^{\mu}_{\omega}({\bf r},t)=Ner^{\mu}_{ab}(u-iv).
  730   $$
  731   This expression for the temporal envelope of the polarization density is
  732   exactly in the same mode of description as the one as previously used in
  733   the susceptibility theory, as in the wave equations developed in lecture
  734   eight. The only difference is that now we instead consider the polarization
  735   density as given by a non-perturbative analysis. Taken together with the
  736   Maxwell's equations (or the proper wave equation for the envelopes of the
  737   fields), the Bloch equations are known as the {\sl Maxwell--Bloch equations}.
  738   \bye
  739   

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