**Contents of file 'lect4/lect4.tex':**

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% File: nlopt/lect4/lect4.tex [pure TeX code]2% Last change: January 19, 20033%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 Jonsson8%9 \input epsf 10%11% Read amssym to get the AMS {\Bbb E} font (strikethrough E) and12% 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 transform19%20 \def\fourier{\mathop{\frak F}\nolimits} 21 \def\Re{\mathop{\rm Re}\nolimits}% real part22 \def\Im{\mathop{\rm Im}\nolimits}% imaginary part23 \def\Tr{\mathop{\rm Tr}\nolimits}% quantum mechanical trace24 \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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