Contents of file 'lect4/lect4.tex':
1 % File: nlopt/lect4/lect4.tex [pure TeX code]
2 % Last change: January 19, 2003
3 %
4 % Lecture No 4 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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15 \font\ninerm=cmr9
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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
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{4}
42 \section{The Truth of polarization densitites}
43 So far, we have performed the analysis in a theoretical framework that
44 has been exclusively formulated in terms of phenomenological models, such
45 as the anharmonic oscillator and the phenomenologically introduced
46 polarization response function of the medium.
47 In the real world application of nonlinear optics, however, we should not
48 restrict the theory just to phenomenological models, but rather take
49 advantage over the full quantum-mechanical framework of analysis of
50 interaction between light and matter.
51 \medskip
52 \centerline{\epsfxsize=105mm\epsfbox{../images/dipoleop/dipoleop.1}}
53 \medskip
54 \centerline{Figure 1. Schematic figure of the ensemble in the
55 ``small volume''.}
56 \medskip
57 \noindent
58 In a small volume $V$ (smaller than the wavelength of the light, to ensure
59 that the natural spatial variation of the light is not taken into account,
60 but large enough in order to contain a sufficcient number of molecules in
61 order to ignore the quantum-mechanical fluctuations of the dipole moment
62 density), we consider the applied electric field to be homogeneous, and the
63 electric polarization density of the medium is then given as the expectation
64 value of the {\sl electric dipole operator of the ensemble of molecules}
65 divided by the volume, as
66 $$
67 P_{\mu}({\bf r},t)={{\langle\hat{Q}_{\mu}\rangle}/{V}},
68 $$
69 where the electric dipole operator of the ensemble contained in $V$ can
70 be written as a sum over all electrons and nuclei as
71 $$
72 \hat{\bf Q}=\underbrace{-e\sum_j\hat{\bf r}_j}_{\rm electrons}
73 +\underbrace{e\sum_k Z_k \hat{\bf r}_k}_{\rm nuclei}.
74 $$
75 The expectation value $\langle\hat{Q}_{\mu}\rangle$ can in principle be
76 calculated directly from the compound, time-dependent wave function of
77 the ensemble of molecules in the small volume, considering any kind of
78 interaction between the molecules, which may be of an arbitrary composition.
79 However, we will here describe the interactions that take place in terms
80 of the {\sl quantum mechanical density operator} of the ensemble, in which
81 case the expectation value is calculated from the {\sl quantum mechanical
82 trace} as
83 $$
84 P_{\mu}({\bf r},t)=\Tr[\hat{\rho}(t)\hat{Q}_{\mu}]/V.
85 $$
86
87 \section{Outline}
88 Previously, in lecture one, we applied the mathematical tool of perturbation
89 analysis to a classical mechanical model of the dipole moment. This analysis
90 will now essentially be repeated, but now we will instead consider a
91 perturbation series for the quantum mechanical density operator, with
92 the series being of the form
93 $$
94 \hat{\rho}(t)=\underbrace{\hat{\rho}_0}_{\sim [E(t)]^0}
95 +\underbrace{\hat{\rho}_1(t)}_{\sim [E(t)]^1}
96 +\underbrace{\hat{\rho}_2(t)}_{\sim [E(t)]^2}
97 +\ldots
98 +\underbrace{\hat{\rho}_n(t)}_{\sim [E(t)]^n}
99 +\ldots
100 $$
101 As this perturbation series is inserted into the expression for the
102 electric polarization density, we will obtain a resulting series for
103 the polarization density as
104 $$
105 P_{\mu}({\bf r},t)=\sum^{\infty}_{m=0}\underbrace{\Tr[\hat{\rho}_m(t)
106 \hat{Q}_{\mu}]/V}_{=P^{(m)}_{\mu}({\bf r},t)}
107 \approx\sum^{n}_{m=0} P^{(m)}_{\mu}({\bf r},t).
108 $$
109
110 \section{Quantum mechanics}
111 We consider an ensemble of molecules, where each molecule may be different
112 from the other molecules of the ensemble, as well as being affected by some
113 mutual interaction between the other members of the ensemble.
114 The Hamiltonian for this ensemble is generally taken as
115 $$
116 \hat{H}=\hat{H}_0+\hat{H}_{\rm I}(t),
117 $$
118 where $\hat{H}_0$ is the Hamiltonian at thermal equilibrium, with no
119 external forces present, and $\hat{H}_{\rm I}(t)$ is the interaction
120 Hamiltonial (in the Schr\"{o}dinger picture), which for electric dipolar
121 interactions take the form:
122 $$
123 \hat{H}_{\rm I}(t)
124 =-\hat{\bf Q}\cdot{\bf E}({\bf r},t)
125 =-\hat{Q}_{\alpha}E_{\alpha}({\bf r},t),
126 $$
127 where $\hat{\bf Q}$ is the electric dipole operator of the {\sl ensemble}
128 of molecules contained in the small volume~$V$ (see Fig.~1). This expression
129 may be compared with the all-classical electrostatic energy of an electric
130 dipole moment in a electric field, $V=-{\bf p}\cdot{\bf E}({\bf r},t)$.
131
132 In order to provide a proper description of the interaction between
133 light and matter at molecular level, we must be means of some quantum
134 mechanical description evaluate all properties of the molecule, such
135 as electric dipole moment, magnetic dipole moment, etc., by means
136 of {\sl quantum mechanical expectation values}.
137
138 The description that we here will apply is by means of the {\sl density
139 operator formalism}, with the density operator defined in terms of
140 orthonormal set of wave functions $|a\rangle$ of the system as
141 $$\hat{\rho}=\sum_a p_a|a\rangle\langle a|=\hat{\rho}(t),$$
142 where $p_a$ are the normalized probabilities of the system to be
143 in state $|a\rangle$, with $$\sum_a p_a=1.$$
144 From the density operator, the expectation value of any arbitrary quantum
145 mechanical operator $\hat{O}$ of the ensemble is obtained from the
146 {\sl quantum mechanical trace} as
147 $$
148 \langle\hat{O}\rangle=\Tr(\hat{\rho}\,\hat{O})
149 =\sum_k\langle k|\hat{\rho}\,\hat{O}|k\rangle.
150 $$
151 The equation of motion for the density operator is given in terms of
152 the Hamiltonian as
153 $$
154 \eqalign{
155 i\hbar{{d\hat{\rho}}\over{dt}}
156 &=[\hat{H},\hat{\rho}]
157 =\hat{H}\hat{\rho}-\hat{\rho}\hat{H}\cr
158 &=[\hat{H}_0,\hat{\rho}]+[\hat{H}_{\rm I}(t),\hat{\rho}]\cr
159 }
160 \eqno{(1)}
161 $$
162 In this context, the terminology of ``equation of motion'' can be
163 pictured as
164 $$
165 \bigg\{\matrix{{\rm A\ change\ of\ the\ density}\cr
166 {\rm operator}\ \hat{\rho}(t)\ {\rm in\ time}\cr}\bigg\}
167 \quad\Leftrightarrow\quad
168 \bigg\{\matrix{{\rm A\ change\ of\ density}\cr
169 {\rm of\ states\ in\ time}\cr}\bigg\}
170 \quad\Leftrightarrow\quad
171 \bigg\{\matrix{{\rm Change\ of\ a\ general}\cr
172 {\rm property}\ \langle\hat{O}\rangle\ {\rm in\ time}\cr}\bigg\}
173 $$
174 Whenever external forces are absent, that is to say, whenever the applied
175 electromagnetic field is zero, the equation of motion for the density
176 operator takes the form
177 $$i\hbar{{d\hat{\rho}}\over{dt}}=[\hat{H}_0,\hat{\rho}],$$
178 with the solution\footnote{${}^1$}{For any macroscopic system,
179 the probability that the system is in a particular energy eigenstate
180 $\psi_n$, with associated energy ${\Bbb E}_n$, is given by the familiar
181 Boltzmann distribution $$p_n=\eta\exp(-{\Bbb E}_n/k_{\rm B}T),$$
182 where $\eta$ is a normalization constant chosen so that $\sum_n p_n=1$,
183 $k_{\rm B}$ is the Boltzmann constant, and $T$ the absolute temperature.
184 This probability distribution is in this course to be considered
185 as being an axiomatic fact, and the origin of this probability distribution
186 can readily be obtained from textbooks on thermodynamics or statistical
187 mechanics.}
188 $$\eqalign{\hat{\rho}(t)=\hat{\rho}_0&=\eta\exp(-\hat{H}_0/k_{\rm B}T)\cr
189 \bigg\{&=\eta\sum^{\infty}_{j=1}{{1}\over{j!}}(-\hat{H}_0/k_{\rm B}T)^j
190 \bigg\}\cr}$$
191 being the time-independent density operator at thermal equilibrium,
192 with the normalization constant $\eta$ chosen so that $\Tr(\hat{\rho})=1$,
193 i.~e., $$\eta=1/\Tr[\exp(-\hat{H}_0/k_{\rm B}T)].$$
194
195 \section{Perturbation analysis of the density operator}
196 The task is now o obtain a solution of the equation of motion~(1) by
197 means of a perturbation series, in similar to the analysis performed
198 for the anharmonic oscillator in the first lecture of this course.
199 The perturbation series is, in analogy to the mechanical spring oscillator
200 under influence of an electromagnetic field, taken as
201 $$
202 \hat{\rho}(t)=\underbrace{\hat{\rho}_0}_{\sim [E(t)]^0}
203 +\underbrace{\hat{\rho}_1(t)}_{\sim [E(t)]^1}
204 +\underbrace{\hat{\rho}_2(t)}_{\sim [E(t)]^2}
205 +\ldots
206 +\underbrace{\hat{\rho}_n(t)}_{\sim [E(t)]^n}
207 +\ldots
208 $$
209 The boundary condition of the perturbation series is taken
210 as the initial condition that sometime in the past, the external
211 forces has been absent, i.~e.
212 $$
213 \hat{\rho}(-\infty)=\hat{\rho}_0,
214 $$
215 which, since the perturbation series is to be valid for {\sl all possible
216 evolutions in time of the externally applied electric field}, leads to the
217 boundary conditions for each individual term of the perturbation series as
218 $$
219 \hat{\rho}_j(-\infty)=0,\qquad j=1,2,\ldots
220 $$
221 By inserting the perturbation series for the density operator into the
222 equation of motion~(1), one hence obtains
223 $$
224 \eqalign{
225 i\hbar{{d}\over{dt}}(\hat{\rho}_0+\hat{\rho}_1(t)+\hat{\rho}_2(t)
226 +\ldots+\hat{\rho}_n(t)+\ldots)
227 &=[\hat{H}_0,\hat{\rho}_0+\hat{\rho}_1(t)+\hat{\rho}_2(t)
228 +\ldots+\hat{\rho}_n(t)+\ldots]\cr
229 &\qquad+[\hat{H}_{\rm I}(t),\hat{\rho}_0+\hat{\rho}_1(t)+\hat{\rho}_2(t)
230 +\ldots+\hat{\rho}_n(t)+\ldots],\cr
231 }
232 $$
233 and by equating terms with equal power dependence of the applied electric
234 field in the right and left hand sides, one obtains the system of equations
235 $$
236 \eqalign{
237 i\hbar{{d\hat{\rho}_0}\over{dt}}&=[\hat{H}_0,\hat{\rho}_0],\cr
238 i\hbar{{d\hat{\rho}_1(t)}\over{dt}}&=[\hat{H}_0,\hat{\rho}_1(t)]
239 +[\hat{H}_{\rm I}(t),\hat{\rho}_0],\cr
240 i\hbar{{d\hat{\rho}_2(t)}\over{dt}}&=[\hat{H}_0,\hat{\rho}_2(t)]
241 +[\hat{H}_{\rm I}(t),\hat{\rho}_1(t)],\cr
242 &\vdots\cr
243 i\hbar{{d\hat{\rho}_n(t)}\over{dt}}&=[\hat{H}_0,\hat{\rho}_n(t)]
244 +[\hat{H}_{\rm I}(t),\hat{\rho}_{n-1}(t)],\cr
245 &\vdots\cr
246 }\eqno{(2)}
247 $$
248 for the variuos order terms of the perturbation series. In Eq.~(2), we may
249 immediately notice that the first equation simply is the identity stating
250 the thermal equilibrium condition for the zeroth order term $\hat{\rho}_0$,
251 while all other terms may be obtained by consecutively solve the equations
252 of order $j=1,2,\ldots,n$, in that order.
253
254 \section{The interaction picture}
255 We will now turn our attention to the problem of actually solving the
256 obtained system of equations for the terms of the perturbation series
257 for the density operator.
258 In a classical picture, the obtained equations are all of the form
259 similar to
260 $$
261 {{d\rho}\over{dt}}=f(t)\rho+g(t),\eqno{(3)}
262 $$
263 for known functions $f(t)$ and $g(t)$. To solve these equations,
264 we generally look for an integrating factor $I(t)$ satisfying
265 $$
266 I(t){{d\rho}\over{dt}}-I(t)f(t)\rho={{d}\over{dt}}[I(t)\rho].\eqno{(4)}
267 $$
268 By carrying out the differentiation in the right hand side of
269 the equation, we find that the integrating factor should satisfy
270 $$
271 {{d I(t)}\over{dt}}=-I(t)f(t),
272 $$
273 which is solved by\footnote{${}^2$}{Butcher and Cotter have in their
274 classical description of integrating factors chosen to put $I(0)=1$.}
275 $$
276 I(t)=I(0)\exp\big[-\int^t_0 f(\tau)\,d\tau\big].
277 $$
278 The original ordinary differential equation~(3) is hence solved by
279 multiplying with the intagrating factor $I(t)$ and using the
280 property~(4) of the integrating factor, giving the equation
281 $$
282 {{d}\over{dt}}[I(t)\rho]=I(t)g(t),
283 $$
284 from which we hence obtain the solution for $\rho(t)$ as
285 $$
286 \rho(t)={{1}\over{I(t)}}\int^t_0 I(\tau)g(\tau)\,d\tau.
287 $$
288 From this preliminary discussion we may anticipate that equations of motion
289 for the various order perturbation terms of the density operator can be
290 solved in a similar manner, using integrating factors. However, it should
291 be kept in mind that we here are dealing with {\sl operators} and not
292 classical quantities, and since we do not know if the integrating factor
293 is to be multiplied from left or right.
294
295 In order not to loose any generality, we may look for a set of two
296 integrating factors $\hat{V}_0(t)$ and $\hat{U}_0(t)$, in operator sense,
297 that we left and right multiply the unknown terms of the $n$th order
298 equation by, and we require these operators to have the effective impact
299 $$
300 \hat{V}_0(t)\bigg\{i\hbar{{d\hat{\rho}_n(t)}\over{dt}}
301 -[\hat{H}_0,\hat{\rho}_n(t)]\bigg\}\hat{U}_0(t)
302 =i\hbar{{d}\over{dt}}[\hat{V}_0(t)\hat{\rho}_n(t)\hat{U}_0(t)].
303 \eqno{(5)}
304 $$
305 By carrying out the differentiation in the right-hand side, expanding
306 the commutator in the left hand side, and rearranging terms, one then
307 obtains the equation
308 $$
309 \bigg\{i\hbar{{d}\over{dt}}\hat{V}_0(t)+\hat{V}_0(t)\hat{H}_0\bigg\}
310 \hat{\rho}_n(t)\hat{U}_0(t)+\hat{V}_0(t)\hat{\rho}_n(t)
311 \bigg\{i\hbar{{d}\over{dt}}\hat{U}_0(t)-\hat{H}_0\hat{U}_0(t)\bigg\}=0
312 $$
313 for the operators $\hat{V}_0(t)$ and $\hat{U}_0(t)$. This equation clearly
314 is satisfied if both of the braced expressions simultaneously are zero for
315 all times, in other words, if the so-called {\sl time-development operators}
316 $\hat{V}_0(t)$ and $\hat{U}_0(t)$ are chosen to satisfy
317 $$
318 \eqalign{
319 &i\hbar{{d\hat{V}_0(t)}\over{dt}}+\hat{V}_0(t)\hat{H}_0=0,\cr
320 &i\hbar{{d\hat{U}_0(t)}\over{dt}}-\hat{H}_0\hat{U}_0(t)=0,\cr
321 }
322 $$
323 with solutions
324 $$
325 \eqalign{
326 \hat{U}_0(t)&=\exp(-i\hat{H}_0 t/\hbar),\cr
327 \hat{V}_0(t)&=\exp(i\hat{H}_0 t/\hbar)=\hat{U}_0(-t).\cr
328 }
329 $$
330 In these expressions, the exponentials are to be regarded as being defined
331 by their series expansion.
332 In particular, each term of the series expansion contains an operator part
333 being a power of the thermal equilibrium Hamiltonian $\hat{H}_0$, which
334 commute with any of the other powers.
335 We may easily verify that the obtained solutions, in a strict operator sense,
336 satisfy the relations
337 $$
338 \hat{U}_0(t)\hat{U}_0(t')=\hat{U}_0(t+t'),
339 $$
340 with, in particular, the corollary
341 $$
342 \hat{U}_0(t)\hat{U}_0(-t)=\hat{U}_0(0)=1.
343 $$
344 Let us now again turn our attention to the original equation of motion
345 that was the starting point for this discussion.
346 By multiplying the $n$th order subequation of Eq.~(2) with $\hat{U}_0(-t)$
347 from the left, and multiplying with $\hat{U}_0(t)$ from the right, we
348 by using the relation~(5) obtain
349 $$
350 i\hbar{{d}\over{dt}}\bigg\{\hat{U}_0(-t)\hat{\rho}_n(t)\hat{U}_0(t)\bigg\}
351 =\hat{U}_0(-t)[\hat{H}_{\rm I}(t),\hat{\rho}_{n-1}(t)]\hat{U}_0(t),
352 $$
353 which is integrated to yield the solution
354 $$
355 \hat{U}_0(-t)\hat{\rho}_n(t)\hat{U}_0(t)
356 ={{1}\over{i\hbar}}\int^t_{-\infty}\hat{U}_0(-\tau)
357 [\hat{H}_{\rm I}(\tau),\hat{\rho}_{n-1}(\tau)]\hat{U}_0(\tau)\,d\tau,
358 $$
359 where the lower limit of integration was fixed in accordance with
360 the initial condition $\hat{\rho}_n(-\infty)=0$, $n=1,2,\ldots\,$.
361 In some sense, we may consider the obtained solution as being the
362 end point of this discussion; however, we may simplify the expression
363 somewhat by making a few notes on the properties of the time development
364 operators.
365 By expanding the right hand side of the solution, and inserting
366 $\hat{U}_0(\tau)\hat{U}_0(-\tau)=1$ between $\hat{H}_{\rm I}(\tau)$
367 and $\hat{\rho}_{n-1}(\tau)$ in the two terms, we obtain
368 $$
369 \eqalign{
370 \hat{U}_0(-t)\hat{\rho}_n(t)\hat{U}_0(t)
371 &={{1}\over{i\hbar}}\int^t_{-\infty}\hat{U}_0(-\tau)
372 [\hat{H}_{\rm I}(\tau)\hat{\rho}_{n-1}(\tau)
373 -\hat{\rho}_{n-1}(\tau)\hat{H}_{\rm I}(\tau)]\hat{U}_0(\tau)\,d\tau\cr
374 &={{1}\over{i\hbar}}\int^t_{-\infty}
375 \hat{U}_0(-\tau)\hat{H}_{\rm I}(\tau)\underbrace{\hat{U}_0(\tau)
376 \hat{U}_0(-\tau)}_{=1}\hat{\rho}_{n-1}(\tau)\hat{U}_0(\tau)\,d\tau\cr
377 &\qquad\qquad-{{1}\over{i\hbar}}\int^t_{-\infty}
378 \hat{U}_0(-\tau)\hat{\rho}_{n-1}(\tau)\underbrace{\hat{U}_0(\tau)
379 \hat{U}_0(-\tau)}_{=1}\hat{H}_{\rm I}(\tau)\hat{U}_0(\tau)\,d\tau\cr
380 &={{1}\over{i\hbar}}\int^t_{-\infty}
381 [\underbrace{\hat{U}_0(-\tau)\hat{H}_{\rm I}(\tau)\hat{U}_0(\tau)}_{
382 \equiv\hat{H}'_{\rm I}(t)},
383 \underbrace{\hat{U}_0(-\tau)\hat{\rho}_{n-1}(\tau)\hat{U}_0(\tau)}_{
384 \equiv\hat{\rho}'_{n-1}(t)}]\,d\tau,\cr
385 }
386 $$
387 and hence, by introducing the primed notation in the {\sl interaction picture}
388 for the quantum mechanical operators,
389 $$
390 \eqalign{
391 \hat{\rho}'_n(t)&=\hat{U}_0(-t)\hat{\rho}_n(t)\hat{U}_0(t),\cr
392 \hat{H}'_{\rm I}(t)&=\hat{U}_0(-t)\hat{H}_{\rm I}(t)\hat{U}_0(t),\cr
393 }
394 $$
395 the solutions of the system of equations for the terms of the perturbation
396 series for the density operator {\sl in the interaction picture} take the
397 simplified form
398 $$
399 \hat{\rho}'_n(t)={{1}\over{i\hbar}}\int^t_{-\infty}
400 [\hat{H}'_{\rm I}(\tau),\hat{\rho}'_{n-1}(\tau)]\,d\tau,
401 \qquad n=1,2,\ldots,
402 $$
403 with the variuos order solutions expressed in the original Schr\"{o}dinger
404 picture by means of the inverse transformation
405 $$\hat{\rho}_n(t)=\hat{U}_0(t)\hat{\rho}'_n(t)\hat{U}_0(-t).$$
406
407 \section{The first order polarization density}
408 With the quantum mechanical perturbative description of the interaction
409 between light and matter in fresh mind, we are now in the position of
410 formulating the polarization density of the medium from a quantum mechanical
411 description. A minor note should though be made regarding the Hamiltonian,
412 which now is expressed in the interaction picture, and hence the electric
413 dipolar operator (since the electric field here is considered to be a
414 macroscopic, classical quantity) is given in the interaction picture as
415 well,
416 $$
417 \eqalign{
418 \hat{H}'_{\rm I}(\tau)&=\hat{U}_0(-\tau)\underbrace{
419 [-\hat{Q}_{\alpha}E_{\alpha}(\tau)]}_{=\hat{H}_{\rm I}(\tau)}
420 \hat{U}_0(\tau)\cr
421 &=-\hat{U}_0(-\tau)\hat{Q}_{\alpha}\hat{U}_0(\tau)E_{\alpha}(\tau)\cr
422 &=-\hat{Q}_{\alpha}(\tau)E_{\alpha}(\tau)\cr
423 }
424 $$
425 where $\hat{Q}_{\alpha}(\tau)$ denotes the electric dipolar operator of the
426 ensemble, taken {\sl in the interaction picture}.
427 \vfill\eject
428
429 By inserting the expression for the first order term of the
430 perturbation series for the density operator into the quantum mechanical
431 trace of the first order electric polarization density of the medium,
432 one obtains
433 $$
434 \eqalign{
435 P^{(1)}_{\mu}({\bf r},t)&={{1}\over{V}}\Tr[\hat{\rho}_1(t)\hat{Q}_{\mu}]\cr
436 &={{1}\over{V}}\Tr\Big[
437 \underbrace{\Big(\hat{U}_0(t)
438 \underbrace{{{1}\over{i\hbar}}\int^t_{-\infty}
439 [\hat{H}'_{\rm I}(\tau),\hat{\rho}_0]\,d\tau
440 }_{=\hat{\rho}'_1(t)}\hat{U}_0(-t)\Big)
441 }_{=\hat{\rho}_1(t)}
442 \hat{Q}_{\mu}\Big]\cr
443 &=\Bigg\{
444 \matrix{
445 E_{\mu}(\tau){\rm\ is\ a\ classical\ field
446 \ (omit\ space\ dependence\ {\bf r})},\cr
447 [\hat{H}'_{\rm I}(\tau),\hat{\rho}_0]
448 =[-\hat{Q}_{\alpha}(\tau)E_{\alpha}(\tau),\hat{\rho}_{0}]
449 =-E_{\alpha}(\tau)[\hat{Q}_{\alpha}(\tau),\hat{\rho}_{0}]
450 }
451 \Bigg\}\cr
452 &=-{{1}\over{V i\hbar}}\Tr\Big\{
453 \hat{U}_0(t)
454 \int^t_{-\infty} E_{\alpha}(\tau)
455 [\hat{Q}_{\alpha}(\tau),\hat{\rho}_0]\,d\tau\,
456 \hat{U}_0(-t)\,
457 \hat{Q}_{\mu}\Big\}\cr
458 &=\{{\rm Pull\ out\ }E_{\alpha}(\tau)
459 {\rm\ and\ the\ integral\ outside\ the\ trace}\}\cr
460 &=-{{1}\over{V i\hbar}}
461 \int^t_{-\infty} E_{\alpha}(\tau)\Tr\{
462 \hat{U}_0(t)[\hat{Q}_{\alpha}(\tau),\hat{\rho}_0]\hat{U}_0(-t)\,
463 \hat{Q}_{\mu}\}\,d\tau\cr
464 &=\Bigg\{{\rm Express\ }E_{\alpha}(\tau){\rm\ in\ frequency\ domain,\ }
465 E_{\alpha}(\tau)=\int^{\infty}_{-\infty}E_{\alpha}(\omega)
466 \exp(-i\omega\tau)\,d\omega\Bigg\}\cr
467 &=-{{1}\over{V i\hbar}}
468 \int^{\infty}_{-\infty} \int^t_{-\infty}
469 E_{\alpha}(\omega)\Tr\{
470 \hat{U}_0(t)[\hat{Q}_{\alpha}(\tau),\hat{\rho}_0]\hat{U}_0(-t)\,
471 \hat{Q}_{\mu}\}\exp(-i\omega\tau)\,d\tau\,d\omega\cr
472 &=\{{\rm Use\ }\exp(-i\omega\tau)=\exp(-i\omega t)
473 \exp[-i\omega(\tau-t)]\}\cr
474 &=-{{1}\over{V i\hbar}}
475 \int^{\infty}_{-\infty} \int^t_{-\infty}
476 E_{\alpha}(\omega)\Tr\{
477 \hat{U}_0(t)[\hat{Q}_{\alpha}(\tau),\hat{\rho}_0]\hat{U}_0(-t)\,
478 \hat{Q}_{\mu}\}
479 \cr&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\times
480 \exp[-i\omega(\tau-t)]\,d\tau\,\exp(-i\omega t)\,d\omega\cr
481 &=\varepsilon_0\int^{\infty}_{-\infty}
482 \chi^{(1)}_{\mu\alpha}(-\omega;\omega)E_{\alpha}(\omega)
483 \exp(-i\omega t)\,d\omega,\cr
484 }
485 $$
486 where the first order (linear) electric susceptibility is defined as
487 $$
488 \eqalign{
489 \chi^{(1)}_{\mu\alpha}(-\omega;\omega)
490 &=-{{1}\over{\varepsilon_0 V i\hbar}}\int^t_{-\infty}
491 \Tr\{\underbrace{\hat{U}_0(t)[\hat{Q}_{\alpha}(\tau),\hat{\rho}_0]
492 \hat{U}_0(-t)}_{=[\hat{Q}_{\alpha}(\tau-t),\hat{\rho}_0]}\,
493 \hat{Q}_{\mu}\}\exp[-i\omega(\tau-t)]\,d\tau\cr
494 &=\Bigg\{\matrix{{\rm Expand\ the\ commutator\ and\ insert\ }
495 \hat{U}_0(-t)\hat{U}_0(t)\equiv 1\cr
496 {\rm\ in\ middle\ of\ each\ of\ the\ terms,\ using\ }
497 [\hat{U}_0(t),\hat{\rho}_0]=0\cr
498 \Rightarrow\hat{U}_0(t)[\hat{Q}_{\alpha}(\tau),\hat{\rho}_0]
499 \hat{U}_0(-t)=[\hat{Q}_{\alpha}(\tau-t),\hat{\rho}_0]}
500 \Bigg\}\cr
501 &=-{{1}\over{\varepsilon_0 V i\hbar}}\int^t_{-\infty}
502 \Tr\{[\hat{Q}_{\alpha}(\tau-t),\hat{\rho}_0]\hat{Q}_{\mu}\}
503 \exp[-i\omega(\tau-t)]\,d\tau\cr
504 &=\Bigg\{{\rm Change\ variable\ of\ integration\ }\tau'=\tau-t;
505 \int^t_{-\infty}\cdots d\tau\to\int^0_{-\infty}\cdots d\tau'\Bigg\}\cr
506 &=-{{1}\over{\varepsilon_0 V i\hbar}}\int^0_{-\infty}
507 \underbrace{\Tr\{[\hat{Q}_{\alpha}(\tau'),\hat{\rho}_0]
508 \hat{Q}_{\mu}\}}_{
509 =\Tr\{\hat{\rho}_0[\hat{Q}_{\mu},\hat{Q}_{\alpha}(\tau')]\}}
510 \exp(-i\omega\tau')\,d\tau'\cr
511 &=\{{\rm Cyclic\ permutation\ of\ the\ arguments\ in\ the\ trace}\}\cr
512 &=-{{1}\over{\varepsilon_0 V i\hbar}}\int^0_{-\infty}
513 \Tr\{\hat{\rho}_0[\hat{Q}_{\mu},\hat{Q}_{\alpha}(\tau)]\}
514 \exp(-i\omega\tau)\,d\tau.\cr
515 }
516 $$
517 \bye
518
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