**Contents of file 'lect11/lect11.tex':**

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% File: nlopt/lect11/lect11.tex [pure TeX code]2% Last change: March 17, 20033%4% Lecture No 11 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\sinc{\mathop{\rm sinc}\nolimits}% the sinc(x)=sin(x)/x function25 \def\sech{\mathop{\rm sech}\nolimits}% the sech(x)=... function26 \def\sgn{\mathop{\rm sgn}\nolimits}% sgn(x)=0, if x<0, sgn(x)=1, otherwise27 \def\lecture #1 {\hsize=150mm\hoffset=4.6mm\vsize=230mm\voffset=7mm 28 \topskip=0pt\baselineskip=12pt\parskip=0pt\leftskip=0pt\parindent=15pt 29 \headline={\ifnum\pageno>1\ifodd\pageno\rightheadline\else\leftheadline\fi 30 \else\hfill\fi} 31 \def\rightheadline{\tenrm{\it Lecture notes #1} 32 \hfil{\it Nonlinear Optics 5A5513 (2003)}} 33 \def\leftheadline{\tenrm{\it Nonlinear Optics 5A5513 (2003)} 34 \hfil{\it Lecture notes #1}} 35 \noindent\epsfxsize 100pt\epsfbox{../info/kthtext.eps} 36 \vskip-26pt\hfill\vbox{\hbox{{\it Nonlinear Optics 5A5513 (2003)}} 37 \hbox{{\it Lecture notes}}}\vskip 36pt\centerline{\twelvesc Lecture #1} 38 \vskip 24pt\noindent} 39 \def\section #1 {\medskip\goodbreak\noindent{\bf #1} 40 \par\nobreak\smallskip\noindent} 41 \def\subsection #1 {\smallskip\goodbreak\noindent{\it #1} 42 \par\nobreak\smallskip\noindent} 43 44 \lecture{11} 45 In this lecture, we will focus on configurations where the angular frequency 46 of the light is close to some transition frequency of the medium. 47 In particular, we will start with a brief outline of how the non-resonant 48 susceptibilities may be modified in such a way that weakly resonant 49 interactions can be taken into account. 50 Having formulated the susceptibilities at weakly resonant interaction, 51 we will proceed with formulating a non-perturbative approach of calculation 52 of the polarization density of the medium. For the two-level system, this 53 results in the Bloch equations governing resonant interaction between light 54 and matter. 55 \medskip 56 57 \noindent The outline for this lecture is: 58 \item{$\bullet$}{Singularities of the non-resonant susceptibilities} 59 \item{$\bullet$}{Alternatives to perturbation analysis of the 60 polarization density} 61 \item{$\bullet$}{Relaxation of the medium} 62 \item{$\bullet$}{The two-level system and the Bloch equation} 63 \item{$\bullet$}{The resulting polarization density of the medium at resonance} 64 \medskip 65 66 \section{Singularities of non-resonant susceptibilities} 67 In the theory described so far in this course, all interactions have for 68 simplicity been considered as non-resonant. 69 The explicit forms of the susceptibilities, in terms of the electric dipole 70 moments and transition frequencies of the molecules, have been obtained in 71 lecture six, of the forms 72 $$ 73 \eqalignno{ 74 \chi^{(1)}_{\mu\alpha}(-\omega;\omega) 75 &\sim{{r^{\mu}_{ab}r^{\alpha}_{ba}}\over{\Omega_{ba}-\omega}} 76 +\{{\rm similar\ terms}\}, 77 &[{\rm B.\,\&\,C.\,(4.58)}]\cr 78 \chi^{(2)}_{\mu\alpha\beta}(-\omega_{\sigma};\omega_1,\omega_2) 79 &\sim{{r^{\mu}_{ab} r^{\alpha}_{bc} r^{\beta}_{ca}} 80 \over{(\Omega_{ba}-\omega_1-\omega_2) 81 (\Omega_{ca}-\omega_2)}} 82 +\{{\rm similar\ terms}\}, 83 &[{\rm B.\,\&\,C.\,(4.63)}]\cr 84 \chi^{(3)}_{\mu\alpha\beta\gamma} 85 (-\omega_{\sigma};\omega_1,\omega_2,\omega_3) 86 &\sim{{r^{\mu}_{ab} r^{\alpha}_{bc} r^{\beta}_{cd} r^{\gamma}_{da}} 87 \over{(\Omega_{ba}-\omega_1-\omega_2-\omega_3) 88 (\Omega_{ca}-\omega_2-\omega_3) 89 (\Omega_{da}-\omega_3)}} 90 +\{{\rm similar\ terms}\}, 91 \cr&\qquad\qquad\qquad 92 &[{\rm B.\,\&\,C.\,(4.64)}]\cr 93 &\vdots\cr 94 } 95 $$ 96 To recapitulate, these forms have all been derived under the assumption 97 that the Hamiltonian (which is the general operator which describes the 98 state of the system) consist only of a thermal equilibrium part and an 99 interaction part (in the electric dipolar approximation), of the form 100 $$ 101 {\hat H}={\hat H}_0+{\hat H}_{\rm I}(t). 102 $$ 103 This is a form which clearly does not contain any term related to relaxation 104 effects of the medium, that is to say, it does not contain any term describing 105 any energy flow into thermal heat. As long as we consider the interaction part 106 of the Hamiltonian to be sufficiently strong compared to any relaxation effect 107 of the medium, this is a valid approximation. 108 109 However, the problem with the non-resonant forms of the susceptibilities 110 clearly comes into light when we consider an angular frequency of the 111 light that is close to a transition frequency of the system, since for 112 the first order susceptibility, 113 $$ 114 \chi^{(1)}_{\mu\alpha}(-\omega;\omega)\to\infty, 115 \quad{\rm when\ }\omega\to\Omega_{ba}, 116 $$ 117 or for the second order susceptibility, 118 $$ 119 \chi^{(2)}_{\mu\alpha\beta}(-\omega;\omega_1,\omega_2)\to\infty, 120 \quad{\rm when\ }\omega_1+\omega_2\to\Omega_{ba} 121 {\rm\ or\ }\omega_2\to\Omega_{ca}. 122 $$ 123 This clearly non-physical behaviour is a consequence of that the denominators 124 of the rational expressions for the susceptibilities have singularities 125 at the resonances, and the aim with this lecture is to show how these 126 singularities can be removed. 127 128 \section{Modification of the Hamiltonian for resonant interaction} 129 Whenever we have to consider relaxation effects of the medium, as in the 130 case of resonant interactions, the Hamiltonian should be modified to 131 $$ 132 {\hat H}={\hat H}_0+{\hat H}_{\rm I}(t)+{\hat H}_{\rm R},\eqno{(1)} 133 $$ 134 where, as previously, ${\hat H}_0$ is the Hamiltonian in the absence of 135 external forces, ${\hat H}_{\rm I}(t)=-{\hat Q}_{\alpha}E_{\alpha}({\bf r},t)$ 136 is the interaction Hamiltonian 137 (here taken in the Schr\"odinger picture, as described in lecture four), 138 being linear in the applied electric field of the light, 139 and where the new term ${\hat H}_{\rm R}$ describes the various relaxation 140 processes that brings the system into the thermal equilibrium whenever 141 external forces are absent. 142 The state of the system (atom, molecule, or general ensemble) is then 143 conveniently described by the density operator formalism, from which 144 we can obtain macroscopically observable parameters of the medium, 145 such as the electric polarization density (as frequently encountered 146 in this course), the magnetization of the medium, current densities, etc. 147 148 The form (1) of the Hamiltonian is now to be analysed by means of the 149 equation of motion of the density operator $\hat{\rho}$, 150 $$ 151 i\hbar{{d{\hat{\rho}}}\over{dt}} 152 ={\hat H}{\hat\rho}-{\hat\rho}{\hat H} 153 =[{\hat H},{\hat\rho}],\eqno{(2)} 154 $$ 155 and depending on the setup, this equation may be solved by means of 156 perturbation analysis (for non-resonant and weakly resonant interactions), 157 or by means of non-perturbative approaches, such as the Bloch equations 158 (for strongly resonant interactions). 159 160 \section{Phenomenological representation of relaxation processes} 161 In many cases, the relaxation process of the medium towards thermal 162 equilibrium can be described by 163 $$ 164 [{\hat H}_{\rm R},{\hat\rho}] 165 =-i\hbar{\hat\Gamma}({\hat\rho}-{\hat\rho}_0), 166 $$ 167 where ${\hat\rho}_0$ is the thermal equilibrium density operator 168 of the system. The here phenomenologically introduced operator 169 ${\hat\Gamma}$ describes the relaxation of the medium, and can can be 170 considered as being independent of the interaction Hamiltonian. 171 Here the operator ${\hat\Gamma}$ has the physical dimension of an angular 172 frequency, and its matrix elements can be considered as giving the time 173 constants of decay for various states of the system. 174 175 \section{Perturbation analysis of weakly resonant interactions} 176 Before entering the formalism of the Bloch equations for strongly resonant 177 interactions, we will outline the weakly resonant interactions in a 178 perturbative analysis for the susceptibilities, as previously developed 179 in lectures three, four, and five. 180 181 By taking the perturbation series for the density operator as 182 $$ 183 \hat{\rho}(t)=\underbrace{\hat{\rho}_0}_{\sim [E(t)]^0} 184 +\underbrace{\hat{\rho}_1(t)}_{\sim [E(t)]^1} 185 +\underbrace{\hat{\rho}_2(t)}_{\sim [E(t)]^2} 186 +\ldots 187 +\underbrace{\hat{\rho}_n(t)}_{\sim [E(t)]^n} 188 +\ldots, 189 $$ 190 as we previously did for the strictly non-resonant case, one obtains 191 the system of equations 192 $$ 193 \eqalign{ 194 i\hbar{{d\hat{\rho}_0}\over{dt}}&=[\hat{H}_0,\hat{\rho}_0],\cr 195 i\hbar{{d\hat{\rho}_1(t)}\over{dt}}&=[\hat{H}_0,\hat{\rho}_1(t)] 196 +[\hat{H}_{\rm I}(t),\hat{\rho}_0] 197 -i\hbar{\hat\Gamma}{\hat\rho}_1(t),\cr 198 i\hbar{{d\hat{\rho}_2(t)}\over{dt}}&=[\hat{H}_0,\hat{\rho}_2(t)] 199 +[\hat{H}_{\rm I}(t),\hat{\rho}_1(t)] 200 -i\hbar{\hat\Gamma}{\hat\rho}_2(t),\cr 201 &\vdots\cr 202 i\hbar{{d\hat{\rho}_n(t)}\over{dt}}&=[\hat{H}_0,\hat{\rho}_n(t)] 203 +[\hat{H}_{\rm I}(t),\hat{\rho}_{n-1}(t)] 204 -i\hbar{\hat\Gamma}{\hat\rho}_n(t),\cr 205 &\vdots\cr 206 } 207 $$ 208 As in the non-resonant case, one may here start with solving for the 209 zeroth order term $\hat{\rho}_0$, with all other terms obtained by 210 consecutively solving the equations of order $j=1,2,\ldots,n$, in that order. 211 212 Proceeding in exactly the same path as for the non-resonant case, 213 solving for the density operator in the interaction picture and 214 expressing the various terms of the electric polarization density 215 in terms of the corresponding traces 216 $$ 217 P_{\mu}({\bf r},t) 218 =\sum^{\infty}_{n=0} P^{(n)}_{\mu}({\bf r},t) 219 ={{1}\over{V}}\sum^{\infty}_{n=0} 220 {\rm Tr}[{\hat\rho}_n(t){\hat Q}_{\mu}], 221 $$ 222 one obtains the linear, first order susceptibility of the form 223 $$ 224 \eqalign{ 225 \chi^{(1)}_{\mu\alpha}(-\omega;\omega) 226 &={{N e^2}\over{\varepsilon_0\hbar}} 227 \sum_a\varrho_0(a)\sum_b 228 \Big({{r^{\mu}_{ab}r^{\alpha}_{ba}} 229 \over{\Omega_{ba}-\omega-i\Gamma_{ba}}} 230 +{{r^{\alpha}_{ab}r^{\mu}_{ba}} 231 \over{\Omega_{ba}+\omega-i\Gamma_{ba}}}\Big).\cr 232 } 233 $$ 234 Similarly, the second order susceptibility for weakly resonant interaction 235 is obtained as 236 $$ 237 \eqalign{ 238 \chi^{(2)}_{\mu\alpha\beta}&(-\omega_{\sigma};\omega_1,\omega_2)\cr 239 &={{N e^3}\over{\varepsilon_0 \hbar^2}} 240 {{1}\over{2!}}{\bf S} 241 \sum_a\varrho_0(a)\sum_b\sum_c 242 \Big\{ 243 {{r^{\mu}_{ab} r^{\alpha}_{bc} r^{\beta}_{ca}} 244 \over{(\Omega_{ac}+\omega_2-i\Gamma_{ac}) 245 (\Omega_{ab}+\omega_{\sigma}-i\Gamma_{ab})}} 246 \cr&\qquad 247 -{{r^{\alpha}_{ab} r^{\mu}_{bc} r^{\beta}_{ca}} 248 \over{(\Omega_{ac}+\omega_2-i\Gamma_{ac}) 249 (\Omega_{bc}+\omega_{\sigma}-i\Gamma_{bc})}} 250 -{{r^{\beta}_{ab} r^{\mu}_{bc} r^{\alpha}_{ca}} 251 \over{(\Omega_{ba}+\omega_2-i\Gamma_{ba}) 252 (\Omega_{bc}+\omega_{\sigma}-i\Gamma_{bc})}} 253 \cr&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad 254 +{{r^{\beta}_{ab} r^{\alpha}_{bc} r^{\mu}_{ca}} 255 \over{(\Omega_{ba}+\omega_2-i\Gamma_{ba}) 256 (\Omega_{ca}+\omega_{\sigma}-i\Gamma_{ca})}} 257 \Big\}.\cr 258 } 259 $$ 260 In these expressions for the susceptibilities, the singularities at resonance 261 are removed, and the spectral properties of the absolute values of the 262 susceptibilities are described by regular Lorenzian line shapes. 263 264 The values of the matrix elements $\Gamma_{mn}$ are in many cases difficult 265 to derive from a theoretical basis; however, they are often straightforward 266 to obtain by regular curve-fitting and regression analysis of experimental 267 data. 268 269 As seen from the expressions for the susceptibilities above, we still have 270 a boosting of them close to resonance (resonant enhancement). However, the 271 values of the susceptibilities reach a plateau at exact resonance, with 272 maximum values determined by the magnitudes of the involved matrix elements 273 $\Gamma_{mn}$ of the relaxation operator. 274 275 \section{Validity of perturbation analysis of the polarization density} 276 Strictly speaking, the perturbative approach is only to be considered 277 as for an infinite series expansion. 278 For a limited number of terms, the perturbative approach is only an 279 approximative method, which though for many cases is sufficient. 280 281 The perturbation series, in the form that we have encountered it in this 282 course, defines a power series in the applied electric field of the light, 283 and as long as the lower order terms are dominant in the expansion, we 284 may safely neglect the higher order ones. 285 Whenever we encounter strong fields, however, we may run into trouble with 286 the series expansion, in particular if we are in a resonant optical regime, 287 with a boosting effect of the polarization density of the medium. 288 (This boosting effect can be seen as the equivalent to the close-to-resonance 289 behaviour of the mechanical spring model under influence of externally 290 driving forces.) 291 292 As an illustration to this source of failure of the model in the presence of 293 strong electrical fields, we may consider another, more simple example 294 of series expansions, namely the Taylor expansion of the function $\sin(x)$ 295 around $x\approx 0$, as shown in Fig.~1. 296 \bigskip 297 \centerline{\epsfxsize=110mm\epsfbox{sinapprx.eps}} 298 \centerline{Figure 1. Approximations to $f(x)=\sin(x)$ by means of power 299 series expansions of various degrees.} 300 \medskip 301 302 In analogy to the susceptibility formalism, we may consider $x$ to 303 have the role of the electric field (the variable which we make 304 the power expansion in terms of), and $\sin(x)$ to have the role 305 of the polarization density or the density operator (simply the function 306 we wish to analyze). 307 For low numerical values of $x$, up to about $x\approx 1$, 308 the $\sin(x)$ function is well described by keeping only the first two 309 terms of the expansion, corresponding to a power expansion up to and 310 including order three, 311 $$ 312 \sin(x)\approx p_3(x)=x-{{x^3}\over{3!}}. 313 $$ 314 For higher values of $x$, say up to about $x\approx 2$, the expansion 315 is still following the exact function to a good approximation if we 316 include also the third term, corresponding to a power expansion up to and 317 including order five, 318 $$ 319 \sin(x)\approx p_5(x)=x-{{x^3}\over{3!}}+{{x^5}\over{5!}}. 320 $$ 321 This necessity of including higher and higher order terms goes on 322 as we increase the value of $x$, and we can from the graph also see 323 that the breakdown at a certain level of approximation causes severe 324 difference between the approximate and exact curves. 325 In particular, if one wish to calculate the value of the function $\sin(x)$ 326 for small $x$, it might be a good idea to apply the series expansion. 327 For greater values of $x$, say $x\approx 10$, the series expansion 328 approach is, however, a bad idea, and an efficient evaluation of $\sin(x)$ 329 requires another approach. 330 331 As a matter of fact, the same arguments hold for the more complex case 332 of the series expansion of the density operator\footnote{${}^1$}{We may 333 recall that the series expansion of the density operator is {\sl the} 334 very origin of the expansion of the polarization density of the medium 335 in terms of the electric field, and hence also the very foundation for the 336 whole susceptibility formalism as described in this course.}, for which 337 we for high intensities (high electrical field strengths) must include 338 higher order terms as well. 339 340 However, we have seen that even in the non-resonant case, we may encounter 341 great algebraic complexity even in low order nonlinear terms, and since 342 the problem of formulating a proper polarization density is expanding 343 more or less exponentially with the order of the nonlinearity, the 344 usefulness of the susceptibility formalism eventually breaks down. 345 The solution to this problem is to identify the relevant transitions 346 of the ensemble, and to solve the equation of motion (2) exactly instead 347 (or at least within other levels of approximation which do not rely on 348 the perturbative foundation of the susceptibility formalism). 349 350 \section{The two-level system} 351 In many cases, the interaction between light and matter can be reduced 352 to that of a two-level system, consisting of only two energy eigenstates 353 $|a\rangle$ and $|b\rangle$. 354 The equation of motion of the density operator is generally given by 355 Eq.~(2) as 356 $$ 357 i\hbar{{d{\hat{\rho}}}\over{dt}}=[{\hat H},{\hat\rho}], 358 $$ 359 with 360 $$ 361 {\hat H}={\hat H}_0+{\hat H}_{\rm I}(t)+{\hat H}_{\rm R}. 362 $$ 363 For the two-level system, the equation of motion can be expressed in 364 terms of the matrix elements of the density operator as 365 $$ 366 \eqalignno{ 367 i\hbar{{d\rho_{aa}}\over{dt}} 368 &=[{\hat H}_0,{\hat\rho}]_{aa} 369 +[{\hat H}_{\rm I}(t),{\hat\rho}]_{aa} 370 +[{\hat H}_{\rm R},{\hat\rho}]_{aa},&(3{\rm a})\cr 371 i\hbar{{d\rho_{ab}}\over{dt}} 372 &=[{\hat H}_0,{\hat\rho}]_{ab} 373 +[{\hat H}_{\rm I}(t),{\hat\rho}]_{ab} 374 +[{\hat H}_{\rm R},{\hat\rho}]_{ab},&(3{\rm b})\cr 375 i\hbar{{d\rho_{bb}}\over{dt}} 376 &=[{\hat H}_0,{\hat\rho}]_{bb} 377 +[{\hat H}_{\rm I}(t),{\hat\rho}]_{bb} 378 +[{\hat H}_{\rm R},{\hat\rho}]_{bb},&(3{\rm c})\cr 379 } 380 $$ 381 where the fourth equation for $\rho_{ba}$ was omitted, since the solution 382 for this element immediately follows from 383 $$ 384 \rho_{ba}=\rho^*_{ab}. 385 $$ 386 387 \subsection{Terms involving the thermal equilibrium Hamiltonian} 388 The system of Eqs.~(3) is the starting point for derivation of the so-called 389 Bloch equations. Starting with the thermal-equilibrium part of the 390 commutators in the right-hand sides of Eqs.~(3), we have for the diagonal 391 elements 392 $$ 393 \eqalign{ 394 [{\hat H}_0,{\hat\rho}]_{aa} 395 &=\langle a|{\hat H}_0{\hat\rho}|a\rangle 396 -\langle a|{\hat\rho}{\hat H}_0|a\rangle\cr 397 &=\sum_k \underbrace{\langle a|{\hat H}_0|k\rangle}_{ 398 ={\Bbb E}_a\delta_{ak}} 399 \langle k|{\hat\rho}|a\rangle 400 -\sum_j \langle a|{\hat\rho}|j\rangle 401 \underbrace{\langle j|{\hat H}_0|a\rangle}_{ 402 ={\Bbb E}_j\delta_{ja}}\cr 403 &={\Bbb E}_a\rho_{aa}-\rho_{aa}{\Bbb E}_a\cr 404 &=0\cr 405 &=[{\hat H}_0,{\hat\rho}]_{bb},\cr 406 } 407 $$ 408 and for the off-diagonal elements 409 $$ 410 \eqalign{ 411 [{\hat H}_0,{\hat\rho}]_{ab} 412 &=\langle a|{\hat H}_0{\hat\rho}|b\rangle 413 -\langle a|{\hat\rho}{\hat H}_0|b\rangle\cr 414 &=\sum_k \underbrace{\langle a|{\hat H}_0|k\rangle}_{ 415 ={\Bbb E}_a\delta_{ak}} 416 \langle k|{\hat\rho}|b\rangle 417 -\sum_j \langle a|{\hat\rho}|j\rangle 418 \underbrace{\langle j|{\hat H}_0|b\rangle}_{ 419 ={\Bbb E}_j\delta_{jb}}\cr 420 &={\Bbb E}_a\rho_{ab}-\rho_{ab}{\Bbb E}_b\cr 421 &=-({\Bbb E}_b-{\Bbb E}_a)\rho_{ab}\cr 422 &=-\hbar\Omega_{ba}\rho_{ab}\cr 423 } 424 $$ 425 426 \subsection{Terms involving the interaction Hamiltonian} 427 For the commutators in the right-hand sides of Eqs.~(3) involving the 428 interaction Hamiltonian, we similarly have for the diagonal elements 429 $$ 430 \eqalign{ 431 [{\hat H}_{\rm I}(t),{\hat\rho}]_{aa} 432 &=\langle a|(-e{\hat r}_{\alpha}E_{\alpha}({\bf r},t)){\hat\rho}|a\rangle 433 -\langle a|{\hat\rho}(-e{\hat r}_{\alpha}E_{\alpha}({\bf r},t))|a\rangle\cr 434 &=-eE_{\alpha}({\bf r},t) 435 \bigg\{ 436 \sum_k\langle a|{\hat r}_{\alpha}|k\rangle\langle k|{\hat\rho}|a\rangle 437 -\sum_j\langle a|{\hat\rho}|j\rangle\langle j|{\hat r}_{\alpha}|a\rangle 438 \bigg\}\cr 439 &=-eE_{\alpha}({\bf r},t) 440 \bigg\{ 441 r^{\alpha}_{aa}\rho_{aa} 442 +r^{\alpha}_{ab}\rho_{ba} 443 -\rho_{aa}r^{\alpha}_{aa} 444 -\rho_{ab}r^{\alpha}_{ba} 445 \bigg\}\cr 446 &=-e(r^{\alpha}_{ab}\rho_{ba}-r^{\alpha}_{ba}\rho_{ab}) 447 E_{\alpha}({\bf r},t)\cr 448 &=-[{\hat H}_{\rm I}(t),{\hat\rho}]_{bb},\cr 449 } 450 $$ 451 and for the off-diagonal elements 452 $$ 453 \eqalign{ 454 [{\hat H}_{\rm I}(t),{\hat\rho}]_{ab} 455 &=\langle a|(-e{\hat r}_{\alpha}E_{\alpha}({\bf r},t)){\hat\rho}|b\rangle 456 -\langle a|{\hat\rho}(-e{\hat r}_{\alpha}E_{\alpha}({\bf r},t))|b\rangle\cr 457 &=-eE_{\alpha}({\bf r},t) 458 \bigg\{ 459 \sum_k\langle a|{\hat r}_{\alpha}|k\rangle\langle k|{\hat\rho}|b\rangle 460 -\sum_j\langle a|{\hat\rho}|j\rangle\langle j|{\hat r}_{\alpha}|b\rangle 461 \bigg\}\cr 462 &=-eE_{\alpha}({\bf r},t) 463 \bigg\{ 464 r^{\alpha}_{aa}\rho_{ab} 465 +r^{\alpha}_{ab}\rho_{bb} 466 -\rho_{aa}r^{\alpha}_{ab} 467 -\rho_{ab}r^{\alpha}_{bb} 468 \bigg\}\cr 469 &=-er^{\alpha}_{ab}E_{\alpha}({\bf r},t)(\rho_{bb}-\rho_{aa}) 470 -e(r^{\alpha}_{aa}-r^{\alpha}_{bb})E_{\alpha}({\bf r},t)\rho_{ab}\cr 471 &=\{{\rm Optical\ Stark\ shift:\ } 472 \delta{\Bbb E}_k\equiv -er^{\alpha}_{kk}E_{\alpha}({\bf r},t), 473 \quad k=a,b\}\cr 474 &=-er^{\alpha}_{ab}E_{\alpha}({\bf r},t)(\rho_{bb}-\rho_{aa}) 475 +(\delta{\Bbb E}_a-\delta{\Bbb E}_b)\rho_{ab}.\cr 476 } 477 $$ 478 479 \subsection{Terms involving relaxation processes} 480 For the commutators describing relaxation processes, the diagonal elements 481 are given as 482 $$ 483 \eqalign{ 484 [{\hat H}_{\rm R},{\hat\rho}]_{aa} 485 &=-i\hbar(\rho_{aa}-\rho_0(a))/T_a,\cr 486 [{\hat H}_{\rm R},{\hat\rho}]_{bb} 487 &=-i\hbar(\rho_{bb}-\rho_0(b))/T_b,\cr 488 } 489 $$ 490 where $T_a$ and $T_b$ are the decay rates towards the thermal equilibrium 491 at respective level, and where $\rho_0(a)$ and $\rho_0(b)$ are the thermal 492 equilibrium values of $\rho_{aa}$ and $\rho_{bb}$, respectively (i.~e.~the 493 thermal equilibrium population densities of the respective level). 494 The off-diagonal elements are similarly given as 495 $$ 496 \eqalign{ 497 [{\hat H}_{\rm R},{\hat\rho}]_{ab}&=-i\hbar\rho_{ab}/T_2,\cr 498 [{\hat H}_{\rm R},{\hat\rho}]_{ba}&=-i\hbar\rho_{ba}/T_2.\cr 499 } 500 $$ 501 A common approximation is to consider the two states $|a\rangle$ 502 and $|b\rangle$ to be sufficiently similar in order to approximate 503 their lifetimes as equal, i.~e.~$T_a\approx T_b\approx T_1$, 504 where $T_1$ for historical reasons is denoted as the {\sl longitudinal 505 relaxation time}. 506 For the same historical reason, the relaxation time $T_2$ is denoted 507 as the {\sl transverse relaxation time}.\footnote{${}^2$}{For a deeper 508 discusssion and explanation of the various mechanisms involved in relaxation, 509 see for example Charles~P. Slichter, {\sl Principles of Magnetic Resonance} 510 (Springer-Verlag, Berlin, 1978), available at KTHB. This reference is 511 not mentioned in Butcher and Cotters book, but it is a very good text 512 on relaxation phenomena and how to incorporate them into a density-functional 513 description of interaction between light and matter.} 514 515 As the above matrix elements of the commutators involving the various 516 terms of the Hamiltonian are inserted into the right-hand sides of Eqs.~(3), 517 one obtains the following system of equations for the matrix elements 518 of the density operator, 519 $$ 520 \eqalignno{ 521 i\hbar{{d\rho_{aa}}\over{dt}} 522 &=-e(r^{\alpha}_{ab}\rho_{ba}-r^{\alpha}_{ba}\rho_{ab}) 523 E_{\alpha}({\bf r},t) 524 -i\hbar(\rho_{aa}-\rho_0(a))/T_a,&(4{\rm a})\cr 525 i\hbar{{d\rho_{ab}}\over{dt}} 526 &=-\hbar\Omega_{ba}\rho_{ab} 527 -er^{\alpha}_{ab}E_{\alpha}({\bf r},t)(\rho_{bb}-\rho_{aa}) 528 +(\delta{\Bbb E}_a-\delta{\Bbb E}_b)\rho_{ab} 529 -i\hbar\rho_{ab}/T_2,&(4{\rm b})\cr 530 i\hbar{{d\rho_{bb}}\over{dt}} 531 &=e(r^{\alpha}_{ab}\rho_{ba}-r^{\alpha}_{ba}\rho_{ab}) 532 E_{\alpha}({\bf r},t) 533 -i\hbar(\rho_{bb}-\rho_0(b))/T_b.&(4{\rm c})\cr 534 } 535 $$ 536 (The system of equations~(4) corresponds to Butcher and Cotter's Eqs.~(6.35).) 537 So far, the applied electric field of the light is allowed to be of arbitrary 538 form. However, in order to simplify the following analysis, we will 539 assume the light to be linearly polarized and quasimonochromatic, of the 540 form 541 $$ 542 E_{\alpha}({\bf r},t)=\Re[E^{\alpha}_{\omega}(t)\exp(-i\omega t)]. 543 $$ 544 We will in addition assume the slowly varying temporal envelope 545 $E^{\alpha}_{\omega}(t)$ to be real-valued, and we will also neglect 546 the optical Stark shifts $\delta{\Bbb E}_a$ and $\delta{\Bbb E}_b$. 547 In the absence of strong static magnetic fields, we may also assume 548 the matrix elements $er^{\alpha}_{ab}$ to be real-valued. 549 When these assumptions and approximations are applied to the 550 equations of motion~(4), one obtains 551 $$ 552 \eqalignno{ 553 {{d\rho_{aa}}\over{dt}} 554 &=i(\rho_{ba}-\rho_{ab})\beta(t)\cos(\omega t) 555 -(\rho_{aa}-\rho_0(a))/T_a,&(5{\rm a})\cr 556 {{d\rho_{ab}}\over{dt}} 557 &=i\Omega_{ba}\rho_{ab} 558 +i\beta(t)\cos(\omega t)(\rho_{bb}-\rho_{aa}) 559 -\rho_{ab}/T_2,&(5{\rm b})\cr 560 {{d\rho_{bb}}\over{dt}} 561 &=-i(\rho_{ba}-\rho_{ab})\beta(t)\cos(\omega t) 562 -(\rho_{bb}-\rho_0(b))/T_b,&(5{\rm c})\cr 563 } 564 $$ 565 where the {\sl Rabi frequency} $\beta(t)$, defined in terms of the spatial 566 envelope of the electrical field and the transition dipole moment as 567 $$ 568 \beta(t)=er^{\alpha}_{ab}E^{\alpha}_{\omega}(t)/\hbar 569 =e{\bf r}_{ab}\cdot{\bf E}_{\omega}(t)/\hbar, 570 $$ 571 was introduced. 572 573 \section{The rotating-wave approximation} 574 In the middle equation of the system~(5), we have a time-derivative 575 of $\rho_{ab}$ in the left-hand side, while we in the right-hand side 576 have a term $i\Omega_{ba}\rho_{ab}$. Seen as the homogeneous part of 577 a linear differential equation, this suggests that we may further 578 simplify the equations of motion by taking a new variable 579 $\rho^{\Omega}_{ab}$ according to the variable substitution 580 $$ 581 \rho_{ab}=\rho^{\Omega}_{ab}\exp[i(\Omega_{ba}-\Delta)t],\eqno{(6)} 582 $$ 583 where $\Delta\equiv\Omega_{ba}-\omega$ is the detuning of the 584 angular frequency of the light from the transition frequency 585 $\Omega_{ba}\equiv({\Bbb E}_b-{\Bbb E}_a)/\hbar$. 586 587 By inserting Eq.~(6) into Eqs.~(5), keeping in mind that 588 $\rho_{ba}=\rho^*_{ab}$, one obtains the system 589 $$ 590 \eqalignno{ 591 {{d\rho_{aa}}\over{dt}} 592 &=i(\rho^{\Omega}_{ba}\exp[-i(\Omega_{ba}-\Delta)t] 593 -\rho^{\Omega}_{ab}\exp[i(\Omega_{ba}-\Delta)t]) 594 \beta(t)\cos(\omega t) 595 -(\rho_{aa}-\rho_0(a))/T_a,&(6{\rm a})\cr 596 {{d\rho^{\Omega}_{ab}}\over{dt}} 597 &=i\Delta\rho^{\Omega}_{ab} 598 +i\beta(t)\cos(\omega t)\exp[-i(\Omega_{ba}-\Delta)t] 599 (\rho_{bb}-\rho_{aa}) 600 -\rho^{\Omega}_{ab}/T_2,&(6{\rm b})\cr 601 {{d\rho_{bb}}\over{dt}} 602 &=-i(\rho^{\Omega}_{ba}\exp[-i(\Omega_{ba}-\Delta)t] 603 -\rho^{\Omega}_{ab}\exp[i(\Omega_{ba}-\Delta)t]) 604 \beta(t)\cos(\omega t) 605 -(\rho_{bb}-\rho_0(b))/T_b,&(6{\rm c})\cr 606 } 607 $$ 608 The idea with the rotating-wave approximation is now to separate out 609 rapidly oscillating terms of angular frequencies $\omega+\Omega_{ba}$ 610 and $-(\omega+\Omega_{ba})$, and neglect these terms, compared with 611 more slowly varying terms. The motivation for this approximation is that 612 whenever high-frequency components appear in the equations of motions, 613 the high-frequency terms will when integrated contain large denominators, 614 and will hence be minor in comparison with terms with a slow variation. 615 In some sense we can also see this as a temporal averaging procedure, 616 where rapidly oscillating terms average to zero rapidly compared 617 to slowly varying (or constant) components. 618 619 For example, in Eq.~(6b), the product of the $\cos(\omega t)$ 620 and the exponential function is approximated as 621 $$ 622 \eqalign{ 623 \cos(\omega t)\exp[-i(\Omega_{ba}-\Delta)t] 624 &={{1}\over{2}}[\exp(i\omega t)+\exp(-i\omega t)] 625 \exp[-i\underbrace{(\Omega_{ba}-\Delta)}_{=\omega}t]\cr 626 &={{1}\over{2}}[1+\exp(-i2\omega t)]\to{{1}\over{2}}, 627 } 628 $$ 629 while in Eqs.~(6a) and~(6c), the same argument gives 630 $$ 631 \eqalign{ 632 \exp[i(\Omega_{ba}-\Delta)t]\cos(\omega t) 633 &={{1}\over{2}}[\exp(i\omega t)+\exp(-i\omega t)] 634 \exp[-i\underbrace{(\Omega_{ba}-\Delta)}_{=\omega}t]\cr 635 &={{1}\over{2}}[\exp(i2\omega t)+1]\to{{1}\over{2}}. 636 } 637 $$ 638 By applying this {\sl rotating-wave approximation}, the equations 639 of motion~(6) hence take the form 640 $$ 641 \eqalignno{ 642 {{d\rho_{aa}}\over{dt}} 643 &={{i}\over{2}}(\rho^{\Omega}_{ba}-\rho^{\Omega}_{ab})\beta(t) 644 -(\rho_{aa}-\rho_0(a))/T_a,&(7{\rm a})\cr 645 {{d\rho^{\Omega}_{ab}}\over{dt}} 646 &=i\Delta\rho^{\Omega}_{ab} 647 +{{i}\over{2}}\beta(t)(\rho_{bb}-\rho_{aa}) 648 -\rho^{\Omega}_{ab}/T_2,&(7{\rm b})\cr 649 {{d\rho_{bb}}\over{dt}} 650 &=-{{i}\over{2}}(\rho^{\Omega}_{ba}-\rho^{\Omega}_{ab})\beta(t) 651 -(\rho_{bb}-\rho_0(b))/T_b.&(7{\rm c})\cr 652 } 653 $$ 654 In this final form, before entering the Bloch vector description of the 655 interaction, these equations correspond to Butcher and Cotter's Eqs.~(6.41). 656 657 \section{The Bloch equations} 658 Assuming the two states $|a\rangle$ and $|b\rangle$ to be sufficiently 659 similar in order to approximate~$T_a\approx T_b\approx T_1$, 660 where $T_1$ is the longitudinal relaxation time, and by taking new 661 variables $(u,v,w)$ according to 662 $$ 663 \eqalign{ 664 u&=\rho^{\Omega}_{ba}+\rho^{\Omega}_{ab},\cr 665 v&=i(\rho^{\Omega}_{ba}-\rho^{\Omega}_{ab}),\cr 666 w&=\rho_{bb}-\rho_{aa},\cr 667 } 668 $$ 669 the equations of motion (7) are cast in the {\sl Bloch equations} 670 $$ 671 \eqalignno{ 672 {{du}\over{dt}}&=-\Delta v -u/T_2,&(8{\rm a})\cr 673 {{dv}\over{dt}}&=\Delta u+\beta(t)w-v/T_2,&(8{\rm b})\cr 674 {{dw}\over{dt}}&=-\beta(t)v-(w-w_0)/T_1.&(8{\rm c})\cr 675 } 676 $$ 677 In these equations, the introduced variable $w$ describes the population 678 inversion of the two-level system, while $u$ and $v$ are related to the 679 dispersive and absorptive components of the polarization density of the 680 medium. 681 In the Bloch equations above, $w_0=\rho_0(b)-\rho_0(a)$ 682 is the thermal equilibrium inversion of the system with no optical 683 field applied. 684 685 \section{The resulting electric polarization density of the medium} 686 The so far developed theory of the density matrix under resonant 687 interaction can now be applied to the calculation of the electric 688 polarization density of the medium, consisting of $N$ identical 689 molecules per unit volume, as 690 $$ 691 \eqalign{ 692 P_{\mu}({\bf r},t)&=N\langle e{\hat r}_{\mu}\rangle\cr 693 &=N\Tr[{\hat\rho} e{\hat r}_{\mu}]\cr 694 &=N\sum_{k=a,b}\langle k|{\hat\rho} e{\hat r}_{\mu}|k\rangle\cr 695 &=N\sum_{k=a,b}\sum_{j=a,b} 696 \langle k|{\hat\rho}|j\rangle 697 \langle j|e{\hat r}_{\mu}|k\rangle\cr 698 &=N\sum_{k=a,b}\left\{ 699 \langle k|{\hat\rho}|a\rangle 700 \langle a|e{\hat r}_{\mu}|k\rangle 701 +\langle k|{\hat\rho}|b\rangle 702 \langle b|e{\hat r}_{\mu}|k\rangle 703 \right\}\cr 704 &=N\left\{ 705 \langle a|{\hat\rho}|a\rangle 706 \langle a|e{\hat r}_{\mu}|a\rangle 707 +\langle b|{\hat\rho}|a\rangle 708 \langle a|e{\hat r}_{\mu}|b\rangle 709 +\langle a|{\hat\rho}|b\rangle 710 \langle b|e{\hat r}_{\mu}|a\rangle 711 +\langle b|{\hat\rho}|b\rangle 712 \langle b|e{\hat r}_{\mu}|b\rangle 713 \right\}\cr 714 &=N(\rho_{ba}er^{\mu}_{ab}+\rho_{ab}er^{\mu}_{ba})\cr 715 &=\{{\rm Make\ use\ of\ }\rho_{ab}=(u+iv)\exp(i\omega t)=\rho^*_{ba}\}\cr 716 &=N[(u-iv)\exp(-i\omega t)er^{\mu}_{ab} 717 +(u+iv)\exp(i\omega t)er^{\mu}_{ba}].\cr 718 } 719 $$ 720 The temporal envelope $P^{\mu}_{\omega}$ of the polarization density, 721 throughout this course as well as in Butcher and Cotter's book, is taken as 722 $$ 723 P^{\mu}({\bf r},t)=\Re[P^{\mu}_{\omega}\exp(-i\omega t)], 724 $$ 725 and by identifying this expression with the right-hand side of the result 726 above, we hence finally have obtained the polarization density 727 in terms of the Bloch parameters $(u,v,w)$ as 728 $$ 729 P^{\mu}_{\omega}({\bf r},t)=Ner^{\mu}_{ab}(u-iv). 730 $$ 731 This expression for the temporal envelope of the polarization density is 732 exactly in the same mode of description as the one as previously used in 733 the susceptibility theory, as in the wave equations developed in lecture 734 eight. The only difference is that now we instead consider the polarization 735 density as given by a non-perturbative analysis. Taken together with the 736 Maxwell's equations (or the proper wave equation for the envelopes of the 737 fields), the Bloch equations are known as the {\sl Maxwell--Bloch equations}. 738 \bye 739

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