Contents of file 'lect11/lect11.tex':
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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18 % Use AMS Euler fraktur style for short-hand notation of Fourier transform
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20 \def\fourier{\mathop{\frak F}\nolimits}
21 \def\Re{\mathop{\rm Re}\nolimits} % real part
22 \def\Im{\mathop{\rm Im}\nolimits} % imaginary part
23 \def\Tr{\mathop{\rm Tr}\nolimits} % quantum mechanical trace
24 \def\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
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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