Contents of file 'lect6/lect6.tex':
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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