Contents of file 'lect11/lect11.tex':




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1   % File: nlopt/lect11/lect11.tex [pure TeX code]
2   % Last change: March 17, 2003
3   %
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 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\sinc{\mathop{\rm sinc}\nolimits} % the sinc(x)=sin(x)/x function
25   \def\sech{\mathop{\rm sech}\nolimits} % the sech(x)=... function
26   \def\sgn{\mathop{\rm sgn}\nolimits}   % sgn(x)=0, if x<0, sgn(x)=1, otherwise
27   \def\lecture #1 {\hsize=150mm\hoffset=4.6mm\vsize=230mm\voffset=7mm
28     \topskip=0pt\baselineskip=12pt\parskip=0pt\leftskip=0pt\parindent=15pt
30       \else\hfill\fi}
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