Contents of file 'lect4/lect4.tex':




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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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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{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