Contents of file 'lect6/lect6.tex':




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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
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12   % the Euler fraktur font.
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14   \input amssym
15   \font\ninerm=cmr9
16   \font\twelvesc=cmcsc10
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18   % Use AMS Euler fraktur style for short-hand notation of Fourier transform
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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
27       \else\hfill\fi}
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 (nth 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