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    1   % File: nlopt/lect6/lect6.tex [pure TeX code]
    2   % Last change: February 9, 2003
    3   %
    4   % Lecture No 6 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\lecture #1 {\hsize=150mm\hoffset=4.6mm\vsize=230mm\voffset=7mm
   25     \topskip=0pt\baselineskip=12pt\parskip=0pt\leftskip=0pt\parindent=15pt
   26     \headline={\ifnum\pageno>1\ifodd\pageno\rightheadline\else\leftheadline\fi
   27       \else\hfill\fi}
   28     \def\rightheadline{\tenrm{\it Lecture notes #1}
   29       \hfil{\it Nonlinear Optics 5A5513 (2003)}}
   30     \def\leftheadline{\tenrm{\it Nonlinear Optics 5A5513 (2003)}
   31       \hfil{\it Lecture notes #1}}
   32     \noindent\epsfxsize 100pt\epsfbox{../info/kthtext.eps}
   33     \vskip-26pt\hfill\vbox{\hbox{{\it Nonlinear Optics 5A5513 (2003)}}
   34     \hbox{{\it Lecture notes}}}\vskip 36pt\centerline{\twelvesc Lecture #1}
   35     \vskip 24pt\noindent}
   36   \def\section #1 {\medskip\goodbreak\noindent{\bf #1}
   37     \par\nobreak\smallskip\noindent}
   38   \def\subsection #1 {\smallskip\goodbreak\noindent{\it #1}
   39     \par\nobreak\smallskip\noindent}
   40   
   41   \lecture{6}
   42   \section{Assembly of independent molecules}
   43   So far the description of the interaction between light and matter has been
   44   in a very general form, where it was assumed merely that the interaction
   45   is local, and in the electric dipolar approximation, where magnetic dipolar
   46   and electric quadrupolar interactions (as well as higher order terms) were
   47   neglected. The ensemble of molecules has so far no constraints in terms of
   48   composition or mutual interaction, and the electric dipolar operator is
   49   so far taken for the {\sl whole ensemble of molecules}, rather than as
   50   the dipole operator for the individual molecules.
   51   \medskip
   52   \centerline{\epsfxsize=105mm\epsfbox{../images/indepass/indepass.1}}
   53   \medskip
   54   \centerline{Figure 1. The ensemble of identical, similarly oriented,
   55     and mutually independent molecules.}
   56   \medskip
   57   \noindent
   58   For many practical applications, however, the obtained general form is
   59   somewhat inconvenient when it comes to the numerical evaluation of the
   60   susceptibilities, since tables of wave functions, transition frequencies
   61   and their corresponding electric dipole moments etc. often are tabulated
   62   exclusively for the individual molecules themselves. Thus, we will now
   63   apply the obtained general theory to an ensemble of identical molecules,
   64   making a transition from the wave function and electric dipole operator
   65   of the whole ensemble to the wave function and electric dipole operator
   66   of the individual molecule, and find an explicit form of the electric
   67   susceptibilities in terms of the {\sl quantum mechanical matrix elements
   68   of the molecular dipole operator}.
   69   
   70   We will now apply three assumptions of the ensemble of molecules,
   71   which introduce a significant simplification to the task of expressing
   72   the susceptibilities and macroscopic polarization density in terms of
   73   the individual molecular properties:
   74   \smallskip % \narrower
   75   \item{1.}{The molecules of the ensemble are all identical.}
   76   \item{2.}{The molecules of the ensemble are mutually non-interacting.}
   77   \item{3.}{The molecules of the ensemble are identically oriented.}
   78   \smallskip
   79   From a wave functional approach, it is straightforward to make the
   80   transition from the description of the ensemble down to the molecular
   81   level in a strict quantum mechanical perspective (as shown in Butcher
   82   and Cotters book). However, from a practical engineering point of view,
   83   the results are quite intuitively derived if we make the following
   84   observations, which immediately follow from the assumptions listed above:
   85   \smallskip
   86   \item{$\bullet$}{The positions of the individual molecules does not
   87     affect the electric dipole moment of the whole ensemble in $V$, if we
   88     neglect the mutual interaction between the molecules. (This holds only
   89     if the molecules are neutrally charged, since they otherwise could
   90     build up a total electric dipole moment of the ensemble.)}
   91   \item{$\bullet$}{Since the mutual interaction between the molecules is
   92     neglected, we may just as well consider the individual molecules as
   93     constituting a set of ``sub-ensembles'' of the general ensemble picture.
   94     In this picture, all quantum mechanical expectation values, involved
   95     commutators, matrix elements, etc., should be considered for each
   96     subensemble instead, and the macroscopic polarization density will
   97     in this case become
   98     $$
   99       {\bf P}({\bf r},t)=
  100         {{\left\{\matrix{{\rm the\ number\ of\ molecules}\cr
  101                   {\rm within\ the\ volume\ }V}\right\}
  102         \times\left\{\matrix{{\rm the\ expectation\ value\ of\ the}\cr
  103                              {\rm molecular\ dipole\ operator}}\right\}}
  104         \over{\left\{{\rm the\ volume\ }V\right\}}}.
  105     $$}
  106   \bigskip
  107   \centerline{\epsfxsize=105mm\epsfbox{../images/molecass/molecass.1}}
  108   \medskip
  109   \centerline{Figure 2. The general ensemble seen as an ensemble of
  110     identical mono-molecular sub-ensembles.}
  111   \medskip
  112   Assuming there are $M$ mutually non-interacting and similarly oriented
  113   molecules in the volume $V$, the macroscopic polarization density of the
  114   medium can be written as
  115   $$
  116     \eqalign{
  117       {\bf P}({\bf r},t)
  118         &={{1}\over{V}}\langle\hat{\bf Q}\rangle
  119          ={{1}\over{V}}\langle\underbrace{-e\sum_j\hat{\bf r}_j}_{\rm electrons}
  120            +\underbrace{e\sum_k Z_k \hat{\bf r}_k}_{\rm nuclei}\rangle\cr
  121         &=\{{\rm arrange\ terms\ as\ sum\ over\ the\ molecules
  122              \ of\ the\ ensemble}\}\cr
  123         &={{1}\over{V}}
  124           \underbrace{
  125             \langle\sum^M_{m=1}
  126             \underbrace{
  127               \Big(\underbrace{-e\sum_j\hat{\bf r}^{(m)}_j}_{\rm electrons}
  128                 +\underbrace{e\sum_k Z^{(m)}_k \hat{\bf r}^{(m)}_k}_{\rm nuclei}
  129                 \Big)
  130             }_{{\rm molecular\ elec.\ dipole\ operator,\ }e\hat{\bf r}^{(m)}}
  131           \rangle
  132           }_{\rm electric\ dipole\ moment\ of\ ensemble}\cr
  133         &={{1}\over{V}}\langle\sum^M_{m=1}e\hat{\bf r}^{(m)}\rangle\cr
  134         &=\{{\rm the\ molecules\ are\ mutually\ noninteracting}\}\cr
  135         &={{1}\over{V}}\sum^M_{m=1}\langle e\hat{\bf r}^{(m)}\rangle\cr
  136         &=\{{\rm the\ molecules\ are\ identical\ and\ similarly\ oriented}\}\cr
  137         &={{1}\over{V}}\sum^M_{m=1}\langle e\hat{\bf r}\rangle\cr
  138         &=N\langle e\hat{\bf r}\rangle\cr
  139     }
  140   $$
  141   where $N=M/V$ is the number of molecules per unit volume, and
  142   $$
  143     \langle e\hat{\bf r}\rangle=\Tr[\hat{\varrho}(t) e\hat{\bf r}]
  144   $$
  145   is the expectation value of the {\sl mono-molecular} electric dipole
  146   operator, with $\hat{\varrho}(t)$ (that is to say, $\hat{\rho}(t)$ with a
  147   ``kink''\footnote{${}^1$}{{\bf kink} {\it n.} {\bf 1.} a sharp twist or
  148   bend in a wire, rope, hair, etc. [{\sl Collins Concise Dictionary},
  149   Harper-Collins (1995)].} to indicate the difference to the density
  150   operator of the general ensemble) is the molecular density operator.
  151   Notice that the form ${\bf P}({\bf r},t)=N\langle e\hat{\bf r}\rangle$
  152   is identical to the previous form for the general ensemble, though with
  153   the factor $1/V$ replaced by $N=M/V$, and with the electric dipole operator
  154   $\hat{Q}_{\alpha}$ of the ensemble replaced by the molecular dipole moment
  155   operator $e\hat{r}_{\alpha}$.
  156   
  157   When making this transition, the condition that the molecules are mutually
  158   independent is simply a statement that we {\sl locally} assume the
  159   superposition principle of the properties of the molecules (wave functions,
  160   electric dipole moments, etc.) to hold.
  161   
  162   Going in the limit of non-interacting molecules, we may picture the
  163   situation as in Fig.~2, with each molecule defining a sub-ensemble,
  164   which we are free to choose as our ``small volume'' of charged particles.
  165   As long as we do not make any claim to determine the exact individual
  166   positions of the charged particles together with their respective momentum
  167   (or any other pair of canonical variables which would violate the Heisenberg
  168   uncertainty relation), this is a perfectly valid picture, which
  169   provides the statistical expectation values of any observable property
  170   of the medium as $M/V$ times the average statistical molecular observations.
  171   
  172   The previously described operators of a general ensemble of charged
  173   particles should in this case be replaced by their corresponding
  174   mono-molecular equivalents, as listed in the following table.
  175   $$
  176   \vcenter{\halign{
  177     \quad\hfil#\hfil\quad& % Justification of first column (general ensemble)
  178     \hfil#\hfil& % Justification of second column ($\mapsto$)
  179     \hfil#\hfil& % Justification of third column (molecular representation)
  180     \qquad#\hfil\cr % Justification of last column
  181     \noalign{{\hrule width 128mm}\vskip 1pt}
  182     \noalign{{\hrule width 128mm}\smallskip}
  183     General & &Molecular &  \cr
  184     ensemble& &representation& \cr
  185     \noalign{\smallskip{\hrule width 128mm}\smallskip}
  186     $\hat{Q}_{\mu}$ & $\to$ & $e\hat{r}_{\mu}$&
  187       (Electric dipole operator)\cr
  188     $\hat{\rho}_0$ & $\to$ & $\hat{\varrho}_0$&
  189       (Density operator of thermal equilibrium)\cr
  190     $\hat{\rho}_n(t)$ & $\to$ & $\hat{\varrho}_n(t)$&
  191       ($n$th order term of density operator)\cr
  192     $\hat{H}_0$ & $\to$ & $\hat{\cal H}_0$&
  193       (Hamiltonian of thermal equilibrium)\cr
  194     $1/V$ & $\to$ & $N=M/V$ & (Number density of molecules)\cr
  195     $\rho_0(a)$ & $\to$ & $\varrho_0(a)$ & (Molecular population
  196       density at state $|a\rangle$)\cr
  197     \noalign{\medskip}
  198     \noalign{{\hrule width 128mm}\vskip 1pt}
  199     \noalign{{\hrule width 128mm}\smallskip}
  200   }}
  201   $$
  202   Whenever an operator is expressed in the interaction picture, we should
  203   keep in mind that the corresponding time development operators $\hat{U}_0(t)$,
  204   which originally were expressed in terms of the thermal equilibrium
  205   Hamiltonian $\hat{H}_0$ of the ensemble, now should be expressed in terms
  206   of the mono-molecular thermal equilibrium Hamiltonian $\hat{\cal H}_0$.
  207   
  208   The matrix elements of the mono-molecular electric dipole operator, taken
  209   in the interaction picture, become
  210   $$
  211     \eqalign{
  212       \langle a|e\hat{r}_{\alpha}(t)|b\rangle
  213         &=\langle a|\hat{U}_0(-t)e\hat{r}_{\alpha}\hat{U}_0(t)|b\rangle\cr
  214         &=\{{\rm the\ closure\ principle,\ B.\&C.\ Eq.~(3.11),\ }
  215             [\hat{O}\hat{O}']_{ab}=\sum_k O_{ak}O_{kb}\}\cr
  216         &=\sum_k\sum_l
  217           \langle a|\hat{U}_0(-t)|k\rangle
  218           \langle k|e\hat{r}_{\alpha}|l\rangle
  219           \langle l|\hat{U}_0(t)|b\rangle\cr
  220         &=\{{\rm energy\ representation,\ B.\&C.\ Eq.~(4.54),\ }
  221             [\hat{U}_0(t)]_{ab}=\exp(-i{\Bbb E}_a t/\hbar)\delta_{ab}\}\cr
  222         &=\sum_k\sum_l
  223           \exp(-i{\Bbb E}_a (-t)/\hbar)\delta_{ak}
  224           \langle k|e\hat{r}_{\alpha}|l\rangle
  225           \exp(-i{\Bbb E}_l t/\hbar)\delta_{lb}\cr
  226         &=\langle a|e\hat{r}_{\alpha}|b\rangle\exp[
  227            i\underbrace{({\Bbb E}_a-{\Bbb E}_b)t/\hbar}_{\equiv\Omega_{ab}t}]\cr
  228         &=\langle a|e\hat{r}_{\alpha}|b\rangle\exp(i\Omega_{ab}t),\cr
  229     }\eqno{(3)}
  230   $$
  231   where $\Omega_{ab}=({\Bbb E}_a-{\Bbb E}_b)/\hbar$ is the molecular transition
  232   frequency between the molecular states $|a\rangle$ and $|b\rangle$.
  233   We may thus, at least in some sense, interpret the interaction picture
  234   for the matrix elements of the molecular operators as an out-separation
  235   of the naturally occurring transition frequencies associated with a change
  236   from state $|a\rangle$ to $|b\rangle$ of the molecule.
  237   
  238   \subsection{First order electric susceptibility}
  239   From the general form of the first order (linear) electric susceptibility,
  240   as derived in the previous lecture, one obtains
  241   $$
  242     \eqalign{
  243       \chi^{(1)}_{\mu\alpha}(-\omega;\omega)
  244         &=-{{N}\over{\varepsilon_0 i\hbar}}\int^0_{-\infty}
  245           \Tr\{\hat{\varrho}_0[e\hat{r}_{\mu},e\hat{r}_{\alpha}(\tau)]\}
  246           \exp(-i\omega\tau)\,d\tau\cr
  247         &=-{{N e^2}\over{\varepsilon_0 i\hbar}}\int^0_{-\infty}
  248           \sum_a\langle a|\hat{\varrho}_0(\hat{r}_{\mu}\hat{r}_{\alpha}(\tau)
  249                               -\hat{r}_{\alpha}(\tau)\hat{r}_{\mu})|a\rangle
  250           \exp(-i\omega\tau)\,d\tau\cr
  251         &=\{{\rm the\ closure\ principle}\}\cr
  252         &=-{{N e^2}\over{\varepsilon_0 i\hbar}}\int^0_{-\infty}
  253           \sum_a\sum_k
  254             \langle a|\hat{\varrho}_0|k\rangle
  255             \langle k|\hat{r}_{\mu}\hat{r}_{\alpha}(\tau)
  256                       -\hat{r}_{\alpha}(\tau)\hat{r}_{\mu}|a\rangle
  257           \exp(-i\omega\tau)\,d\tau\cr
  258         &=\{{\rm use\ that\ }
  259             \langle a|\hat{\varrho}_0|k\rangle=\varrho_0(a)\delta_{ak}\}\cr
  260         &=-{{N e^2}\over{\varepsilon_0 i\hbar}}\int^0_{-\infty}
  261           \sum_a\varrho_0(a)
  262             \big[\langle a|\hat{r}_{\mu}\hat{r}_{\alpha}(\tau)|a\rangle
  263              -\langle a|\hat{r}_{\alpha}(\tau)\hat{r}_{\mu}|a\rangle\big]
  264           \exp(-i\omega\tau)\,d\tau\cr
  265         &=\{{\rm the\ closure\ principle\ again}\}\cr
  266         &=-{{N e^2}\over{\varepsilon_0 i\hbar}}\int^0_{-\infty}
  267           \sum_a\varrho_0(a)\sum_b
  268             \big[\langle a|\hat{r}_{\mu}|b\rangle
  269                  \langle b|\hat{r}_{\alpha}(\tau)|a\rangle
  270              -\langle a|\hat{r}_{\alpha}(\tau)|b\rangle
  271               \langle b|\hat{r}_{\mu}|a\rangle\big]
  272           \exp(-i\omega\tau)\,d\tau\cr
  273         &=\{{\rm Use\ results\ of\ Eq.~(3)\ and\ shorthand\ notation\ }
  274             \langle a|\hat{r}_{\alpha}|b\rangle=r^{\alpha}_{ab}\}\cr
  275         &=-{{N e^2}\over{\varepsilon_0 i\hbar}}
  276           \sum_a\varrho_0(a)\sum_b\int^0_{-\infty}
  277             \big[r^{\mu}_{ab}r^{\alpha}_{ba}\exp(i\Omega_{ba}\tau)
  278               -r^{\alpha}_{ab}r^{\mu}_{ba}\exp(-i\Omega_{ba}\tau)\big]
  279           \exp(-i\omega\tau)\,d\tau\cr
  280         &=\{{\rm evaluate\ integral\ }
  281             \int^0_{-\infty}\exp(iC\tau)\,d\tau\to 1/(iC)\}\cr
  282         &=-{{N e^2}\over{\varepsilon_0 i\hbar}}
  283           \sum_a\varrho_0(a)\sum_b
  284           \Big[{{r^{\mu}_{ab}r^{\alpha}_{ba}}\over{i(\Omega_{ba}-\omega)}}
  285             -{{r^{\alpha}_{ab}r^{\mu}_{ba}}\over{-i(\Omega_{ba}+\omega)}}\Big]\cr
  286         &={{N e^2}\over{\varepsilon_0\hbar}}
  287           \sum_a\varrho_0(a)\sum_b
  288           \Big({{r^{\mu}_{ab}r^{\alpha}_{ba}}\over{\Omega_{ba}-\omega}}
  289             +{{r^{\alpha}_{ab}r^{\mu}_{ba}}\over{\Omega_{ba}+\omega}}\Big).\cr
  290     }
  291   $$
  292   This result, which was derived entirely under the assumption that all
  293   resonances of the medium are located far away from the angular frequency
  294   of the light, possesses a new type of symmetry, which can be seen
  295   if we make the interchange
  296   $$
  297     (-\omega,\mu)\leftrightharpoons(\omega,\alpha),
  298   $$
  299   which by using the above result for the linear susceptibility gives
  300   $$
  301     \eqalign{
  302       \chi^{(1)}_{\alpha\mu}(\omega;-\omega)
  303         &={{N e^2}\over{\varepsilon_0\hbar}}
  304           \sum_a\varrho_0(a)\sum_b
  305           \Big({{r^{\alpha}_{ab}r^{\mu}_{ba}}\over{\Omega_{ba}+\omega}}
  306             +{{r^{\mu}_{ab}r^{\alpha}_{ba}}\over{\Omega_{ba}-\omega}}\Big)\cr
  307         &=\{{\rm change\ order\ of\ appearance\ of\ the\ terms}\}\cr
  308         &={{N e^2}\over{\varepsilon_0\hbar}}
  309           \sum_a\varrho_0(a)\sum_b
  310           \Big({{r^{\mu}_{ab}r^{\alpha}_{ba}}\over{\Omega_{ba}-\omega}}
  311             +{{r^{\alpha}_{ab}r^{\mu}_{ba}}\over{\Omega_{ba}+\omega}}\Big)\cr
  312         &=\chi^{(1)}_{\mu\alpha}(-\omega;\omega),\cr
  313     }
  314   $$
  315   which is a signature of the {\sl overall permutation symmetry} that applies
  316   to nonresonant interactions. With this result, the reason for the peculiar
  317   notation with ``$(-\omega;\omega)$'' should be all clear, namely that it
  318   serves as an explicit way of notation of the overall permutation symmetry,
  319   whenever it applies.
  320   
  321   It should be noticed that while intrinsic permutation symmetry is a general
  322   property that solely applies to nonlinear optical interactions, the overall
  323   permutation symmetry applies to linear interactions as well.
  324   
  325   \subsection{Second order electric susceptibility}
  326   In similar to the linear electric susceptibility, starting from the general
  327   form of the second order (quadratic) electric susceptibility, one obtains
  328   $$
  329     \eqalign{
  330       \chi^{(2)}_{\mu\alpha\beta}&(-\omega_{\sigma};\omega_1,\omega_2)\cr
  331         &={{N e^3}\over{\varepsilon_0 (i\hbar)^2}}
  332           {{1}\over{2!}}{\bf S}
  333           \int^0_{-\infty}\int^{\tau_1}_{-\infty}
  334           \Tr\{\hat{\varrho}_0[[\hat{r}_{\mu},\hat{r}_{\alpha}(\tau_1)],
  335             \hat{r}_{\beta}(\tau_2)]\}
  336           \exp[-i(\omega_1\tau_1+\omega_2\tau_2)]
  337           \,d\tau_2\,d\tau_1\cr
  338         &={{N e^3}\over{\varepsilon_0 (i\hbar)^2}}
  339           {{1}\over{2!}}{\bf S}
  340           \int^0_{-\infty}\int^{\tau_1}_{-\infty}
  341           \sum_a\langle a|\hat{\varrho}_0[
  342             \underbrace{[\hat{r}_{\mu},\hat{r}_{\alpha}(\tau_1)]}_{
  343               \hat{r}_{\mu}\hat{r}_{\alpha}(\tau_1)
  344                 -\hat{r}_{\alpha}(\tau_1)\hat{r}_{\mu}},
  345             \hat{r}_{\beta}(\tau_2)]|a\rangle
  346           \exp[-i(\omega_1\tau_1+\omega_2\tau_2)]
  347           \,d\tau_2\,d\tau_1\cr
  348         &=\{{\rm expand\ the\ commutator}\}\cr
  349         &={{N e^3}\over{\varepsilon_0 (i\hbar)^2}}
  350           {{1}\over{2!}}{\bf S}
  351           \int^0_{-\infty}\int^{\tau_1}_{-\infty}
  352           \sum_a\Big(
  353               \langle a|\hat{\varrho}_0\hat{r}_{\mu}
  354                  \hat{r}_{\alpha}(\tau_1)\hat{r}_{\beta}(\tau_2)|a\rangle
  355               -\langle a|\hat{\varrho}_0\hat{r}_{\alpha}(\tau_1)
  356                  \hat{r}_{\mu}\hat{r}_{\beta}(\tau_2)|a\rangle
  357           \cr&\qquad\qquad
  358               -\langle a|\hat{\varrho}_0\hat{r}_{\beta}(\tau_2)
  359                  \hat{r}_{\mu}\hat{r}_{\alpha}(\tau_1)|a\rangle
  360               +\langle a|\hat{\varrho}_0\hat{r}_{\beta}(\tau_2)
  361                  \hat{r}_{\alpha}(\tau_1)\hat{r}_{\mu}|a\rangle
  362           \Big)\exp[-i(\omega_1\tau_1+\omega_2\tau_2)]
  363           \,d\tau_2\,d\tau_1\cr
  364         &=\{{\rm apply\ the\ closure\ principle\ and\ Eq.~(3)}\}\cr
  365         &={{N e^3}\over{\varepsilon_0 (i\hbar)^2}}
  366           {{1}\over{2!}}{\bf S}
  367           \int^0_{-\infty}\int^{\tau_1}_{-\infty}
  368           \sum_a\sum_k\sum_b\sum_c\Big(
  369               \langle a|\hat{\varrho}_0|k\rangle
  370               \langle k|\hat{r}_{\mu}|b\rangle
  371               \langle b|\hat{r}_{\alpha}(\tau_1)|b\rangle
  372               \langle c|\hat{r}_{\beta}(\tau_2)|a\rangle
  373           \cr&\qquad\qquad\qquad\qquad
  374               -\langle a|\hat{\varrho}_0|k\rangle
  375                \langle k|\hat{r}_{\alpha}(\tau_1)|b\rangle
  376                \langle b|\hat{r}_{\mu}|c\rangle
  377                \langle c|\hat{r}_{\beta}(\tau_2)|a\rangle
  378           \cr&\qquad\qquad\qquad\qquad
  379               -\langle a|\hat{\varrho}_0|k\rangle
  380                \langle k|\hat{r}_{\beta}(\tau_2)|b\rangle
  381                \langle b|\hat{r}_{\mu}|c\rangle
  382                \langle c|\hat{r}_{\alpha}(\tau_1)|a\rangle
  383           \cr&\qquad\qquad\qquad\qquad
  384               +\langle a|\hat{\varrho}_0|k\rangle
  385                \langle k|\hat{r}_{\beta}(\tau_2)|b\rangle
  386                \langle b|\hat{r}_{\alpha}(\tau_1)|c\rangle
  387                \langle c|\hat{r}_{\mu}|a\rangle
  388           \Big)\exp[-i(\omega_1\tau_1+\omega_2\tau_2)]
  389           \,d\tau_2\,d\tau_1\cr
  390         &=\{{\rm use\ that\ }
  391             \langle a|\hat{\varrho}_0|k\rangle=\varrho_0(a)\delta_{ak}
  392             {\rm\ and\ apply\ Eq.~(3)}\}\cr
  393         &={{N e^3}\over{\varepsilon_0 (i\hbar)^2}}
  394           {{1}\over{2!}}{\bf S}
  395           \sum_a\varrho_0(a)\sum_b\sum_c
  396           \int^0_{-\infty}\int^{\tau_1}_{-\infty}\Big\{
  397               r^{\mu}_{ab} r^{\alpha}_{bc} r^{\beta}_{ca}
  398               \exp[i(\Omega_{bc}\tau_1+\Omega_{ca}\tau_2)]
  399           \cr&\qquad\qquad\qquad\qquad
  400               -r^{\alpha}_{ab} r^{\mu}_{bc} r^{\beta}_{ca}
  401               \exp[i(\Omega_{ab}\tau_1+\Omega_{ca}\tau_2)]
  402   %        \cr&\qquad\qquad\qquad\qquad
  403               -r^{\beta}_{ab} r^{\mu}_{bc} r^{\alpha}_{ca}
  404               \exp[i(\Omega_{ca}\tau_1+\Omega_{ab}\tau_2)]
  405           \cr&\qquad\qquad\qquad\qquad
  406               +r^{\beta}_{ab} r^{\alpha}_{bc} r^{\mu}_{ca}
  407               \exp[i(\Omega_{bc}\tau_1+\Omega_{ab}\tau_2)]
  408           \Big\}\exp[-i(\omega_1\tau_1+\omega_2\tau_2)]
  409           \,d\tau_2\,d\tau_1\cr
  410         &={{N e^3}\over{\varepsilon_0 (i\hbar)^2}}
  411           {{1}\over{2!}}{\bf S}
  412           \sum_a\varrho_0(a)\sum_b\sum_c
  413           \int^0_{-\infty}\Big\{
  414               {{r^{\mu}_{ab} r^{\alpha}_{bc} r^{\beta}_{ca}}
  415                \over{i(\Omega_{ca}-\omega_2)}}
  416               \exp[i\underbrace{(\Omega_{bc}+\Omega_{ca})}_{=\Omega_{ba}}\tau_1]
  417           \cr&\qquad\qquad\qquad\qquad
  418               -{{r^{\alpha}_{ab} r^{\mu}_{bc} r^{\beta}_{ca}}
  419                 \over{i(\Omega_{ca}-\omega_2)}}
  420               \exp[i\underbrace{(\Omega_{ab}+\Omega_{ca})}_{=\Omega_{cb}}\tau_1]
  421   %        \cr&\qquad\qquad\qquad\qquad
  422               -{{r^{\beta}_{ab} r^{\mu}_{bc} r^{\alpha}_{ca}}
  423                 \over{i(\Omega_{ab}-\omega_2)}}
  424               \exp[i\underbrace{(\Omega_{ca}+\Omega_{ab})}_{=\Omega_{cb}}\tau_1)]
  425           \cr&\qquad\qquad\qquad\qquad
  426               +{{r^{\beta}_{ab} r^{\alpha}_{bc} r^{\mu}_{ca}}
  427                 \over{i(\Omega_{ab}-\omega_2)}}
  428               \exp[i\underbrace{(\Omega_{bc}+\Omega_{ab})}_{=-\Omega_{ca}}\tau_1]
  429           \Big\}\exp[-i\underbrace{(\omega_1+\omega_2)}_{=\omega_{\sigma}}\tau_1]
  430           \,d\tau_1\cr
  431         &={{N e^3}\over{\varepsilon_0 (i\hbar)^2}}
  432           {{1}\over{2!}}{\bf S}
  433           \sum_a\varrho_0(a)\sum_b\sum_c
  434           \Big\{
  435               -{{r^{\mu}_{ab} r^{\alpha}_{bc} r^{\beta}_{ca}}
  436                \over{(\Omega_{ca}-\omega_2)(\Omega_{ba}-\omega_{\sigma})}}
  437               +{{r^{\alpha}_{ab} r^{\mu}_{bc} r^{\beta}_{ca}}
  438                 \over{(\Omega_{ca}-\omega_2)(\Omega_{cb}-\omega_{\sigma})}}
  439           \cr&\qquad\qquad\qquad\qquad
  440               +{{r^{\beta}_{ab} r^{\mu}_{bc} r^{\alpha}_{ca}}
  441                 \over{(\Omega_{ab}-\omega_2)(\Omega_{cb}-\omega_{\sigma})}}
  442               -{{r^{\beta}_{ab} r^{\alpha}_{bc} r^{\mu}_{ca}}
  443                 \over{(\Omega_{ab}-\omega_2)(-\Omega_{ca}-\omega_{\sigma})}}
  444           \Big\}\cr
  445         &\hskip 90mm\ldots{\it continued\ on\ next\ page}\ldots\cr
  446     }
  447   $$
  448   \vfill\eject
  449   $$
  450     \eqalign{
  451       \ldots{\it continuing}&{\it\ from\ previous\ page}\ldots\cr
  452       \phantom{\chi^{(2)}_{\mu\alpha\beta}}
  453         &={{N e^3}\over{\varepsilon_0 \hbar^2}}
  454           {{1}\over{2!}}{\bf S}
  455           \sum_a\varrho_0(a)\sum_b\sum_c
  456           \Big\{
  457               {{r^{\mu}_{ab} r^{\alpha}_{bc} r^{\beta}_{ca}}
  458                \over{(\Omega_{ac}+\omega_2)(\Omega_{ab}+\omega_{\sigma})}}
  459               -{{r^{\alpha}_{ab} r^{\mu}_{bc} r^{\beta}_{ca}}
  460                 \over{(\Omega_{ac}+\omega_2)(\Omega_{bc}+\omega_{\sigma})}}
  461           \cr&\qquad\qquad\qquad\qquad
  462               -{{r^{\beta}_{ab} r^{\mu}_{bc} r^{\alpha}_{ca}}
  463                 \over{(\Omega_{ba}+\omega_2)(\Omega_{bc}+\omega_{\sigma})}}
  464               +{{r^{\beta}_{ab} r^{\alpha}_{bc} r^{\mu}_{ca}}
  465                 \over{(\Omega_{ba}+\omega_2)(\Omega_{ca}+\omega_{\sigma})}}
  466           \Big\}.\cr
  467     }
  468   $$
  469   Now it is easily seen that if we, for example, interchange
  470   $$
  471     (-\omega_{\sigma},\mu)\rightleftharpoons(\omega_2,\beta),
  472   $$
  473   one obtains
  474   $$
  475     \eqalign{
  476       \chi^{(2)}_{\beta\alpha\mu}&(\omega_2;\omega_1,-\omega_{\sigma})\cr
  477         &={{N e^3}\over{\varepsilon_0 \hbar^2}}
  478           {{1}\over{2!}}{\bf S}
  479           \sum_a\varrho_0(a)\sum_b\sum_c
  480           \Big\{
  481               {{r^{\beta}_{ab} r^{\alpha}_{bc} r^{\mu}_{ca}}
  482                \over{(\Omega_{ac}-\omega_{\sigma})(\Omega_{ab}-\omega_2)}}
  483               -{{r^{\alpha}_{ab} r^{\beta}_{bc} r^{\mu}_{ca}}
  484                 \over{(\Omega_{ac}-\omega_{\sigma})(\Omega_{bc}-\omega_2)}}
  485           \cr&\qquad\qquad\qquad\qquad
  486               -{{r^{\mu}_{ab} r^{\beta}_{bc} r^{\alpha}_{ca}}
  487                 \over{(\Omega_{ba}-\omega_{\sigma})(\Omega_{bc}-\omega_2)}}
  488               +{{r^{\mu}_{ab} r^{\alpha}_{bc} r^{\beta}_{ca}}
  489                 \over{(\Omega_{ba}-\omega_{\sigma})(\Omega_{ca}-\omega_2)}}
  490           \Big\}\cr
  491         &=\{{\rm use\ }\Omega_{ab}=-\Omega_{ba},{\rm\ etc.}\}\cr
  492         &={{N e^3}\over{\varepsilon_0 \hbar^2}}
  493           {{1}\over{2!}}{\bf S}
  494           \sum_a\varrho_0(a)\sum_b\sum_c
  495           \Big\{
  496               \underbrace{{{r^{\beta}_{ab} r^{\alpha}_{bc} r^{\mu}_{ca}}
  497                \over{(\Omega_{ca}+\omega_{\sigma})(\Omega_{ba}+\omega_2)}}
  498               }_{\rm identify\ 4th\ term}
  499               -{{r^{\alpha}_{ab} r^{\beta}_{bc} r^{\mu}_{ca}}
  500                 \over{(\Omega_{ca}+\omega_{\sigma})(\Omega_{cb}+\omega_2)}}
  501           \cr&\qquad\qquad\qquad\qquad
  502               -{{r^{\mu}_{ab} r^{\beta}_{bc} r^{\alpha}_{ca}}
  503                 \over{(\Omega_{ab}+\omega_{\sigma})(\Omega_{cb}+\omega_2)}}
  504               +\underbrace{{{r^{\mu}_{ab} r^{\alpha}_{bc} r^{\beta}_{ca}}
  505                 \over{(\Omega_{ab}+\omega_{\sigma})(\Omega_{ac}+\omega_2)}}
  506                }_{\rm identify\ 1st\ term}
  507           \Big\}\cr
  508         &=\{{\rm interchange\ dummy\ indices\ }a\to c\to b\to a
  509             {\rm\ in\ 2nd\ term}\}\cr
  510         &=\{{\rm interchange\ dummy\ indices\ }a\to b\to c\to a
  511             {\rm\ in\ 3rd\ term}\}\cr
  512         &={{N e^3}\over{\varepsilon_0 \hbar^2}}
  513           {{1}\over{2!}}{\bf S}
  514           \sum_a\varrho_0(a)\sum_b\sum_c
  515           \Big\{
  516               \underbrace{{{r^{\beta}_{ab} r^{\alpha}_{bc} r^{\mu}_{ca}}
  517                \over{(\Omega_{ca}+\omega_{\sigma})(\Omega_{ba}+\omega_2)}}
  518               }_{\rm identify\ 4th\ term}
  519               -\underbrace{{{r^{\alpha}_{ca} r^{\beta}_{ab} r^{\mu}_{bc}}
  520                 \over{(\Omega_{bc}+\omega_{\sigma})(\Omega_{ba}+\omega_2)}}
  521                }_{\rm identify\ as\ 3rd\ term}
  522           \cr&\qquad\qquad\qquad\qquad
  523               -\underbrace{{{r^{\mu}_{bc} r^{\beta}_{ca} r^{\alpha}_{ab}}
  524                 \over{(\Omega_{bc}+\omega_{\sigma})(\Omega_{ac}+\omega_2)}}
  525                }_{\rm identify\ as\ 2nd\ term}
  526               +\underbrace{{{r^{\mu}_{ab} r^{\alpha}_{bc} r^{\beta}_{ca}}
  527                 \over{(\Omega_{ab}+\omega_{\sigma})(\Omega_{ac}+\omega_2)}}
  528                }_{\rm identify\ 1st\ term}
  529           \Big\}\cr
  530         &=\chi^{(2)}_{\mu\beta\alpha}(-\omega_{\sigma};\omega_1,\omega_2),\cr
  531     }
  532   $$
  533   that is to say, the second order susceptibility is left invariant
  534   under any of the $(2+1)!=6$ possible pairwise permutations of
  535   $(-\omega_{\sigma},\mu)$, $(\omega_1,\alpha)$, and $(\omega_2,\beta)$;
  536   this is the {\sl overall permutation symmetry for the second order
  537   susceptibility}, and applies whenever the interaction is moved
  538   far away from any resonance.
  539   
  540   We recapitulate that when deriving the form of the nonlinear
  541   susceptibilities that lead to intrinsic permutation symmetry,
  542   either in terms of polarization response functions in time domain
  543   or in terms of a mechanical spring model, nothing actually had to be stated
  544   regarding the nature of interaction.
  545   This is rather different from what we just obtained for the overall
  546   permutation symmetry, as being a signature of a nonresonant interaction
  547   between the light and matter, and this symmetry cannot (in contrary to
  548   the intrinsic permutation symmetry) be expressed unless the origin
  549   of interaction is considered.\footnote{${}^2$}{It should though be
  550   noticed that the derivation of the susceptibilities still may
  551   be performed within a mechanical spring model, as long as the
  552   resonance frequencies of the oscillator are removed far from the
  553   angular frequencies of the present light.}
  554   
  555   \bye
  556   

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