Contents of file 'lect12/lect12.tex':
1 % File: nlopt/lect12/lect12.tex [pure TeX code]
2 % Last change: March 21, 2003
3 %
4 % Lecture No 12 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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12 % the Euler fraktur font.
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15 \font\ninerm=cmr9
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17 %
18 % Use AMS Euler fraktur style for short-hand notation of Fourier transform
19 %
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
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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{12}
45 In this final lecture, we will study the behaviour of the Bloch equations
46 in different regimes of resonance and relaxation. The Bloch equations
47 are formulated as a vector model, and numerical solutions to the equations
48 are discussed.
49
50 For steady-state interaction, the polarization density of the medium, as
51 obtained from the Bloch equations, is expressed in a closed form.
52 The closed solution is then expanded in a power series, which when
53 compared with the series obtained from the susceptibility formalism
54 finally tie together the Bloch theory with the susceptibilities.
55 \medskip
56
57 \noindent The outline for this lecture is:
58 \item{$\bullet$}{Recapitulation of the Bloch equations}
59 \item{$\bullet$}{The vector model of the Bloch equations}
60 \item{$\bullet$}{Special cases and examples}
61 \item{$\bullet$}{Steady-state regime}
62 \item{$\bullet$}{The intensity dependent refractive index at steady-state}
63 \item{$\bullet$}{Comparison with the susceptibility model}
64 \medskip
65
66 \section{Recapitulation of the Bloch equations for two-level systems}
67 Assuming two states $|a\rangle$ and $|b\rangle$ to be sufficiently
68 similar in order for their respective lifetimes~$T_a\approx T_b\approx T_1$
69 to hold, where $T_1$ is the {\sl longitudinal relaxation time}, the Bloch
70 equations for the two-level are given as
71 $$
72 \eqalignno{
73 {{du}\over{dt}}&=-\Delta v -u/T_2,&(1{\rm a})\cr
74 {{dv}\over{dt}}&=\Delta u+\beta(t)w-v/T_2,&(1{\rm b})\cr
75 {{dw}\over{dt}}&=-\beta(t)v-(w-w_0)/T_1,&(1{\rm c})\cr
76 }
77 $$
78 where $\beta\equiv er^{\alpha}_{ab}E^{\alpha}_{\omega}(t)/\hbar$ is the Rabi
79 frequency, being a quantity linear in the applied electric field of the
80 light, $\Delta\equiv\Omega_{ba}-\omega$ is the detuning of the
81 angular frequency of the light from the transition frequency
82 $\Omega_{ba}\equiv({\Bbb E}_b-{\Bbb E}_a)/\hbar$,
83 and where the variables $(u,v,w)$ are related to the matrix
84 elements $\rho_{mn}$ of the density operator as
85 $$
86 \eqalign{
87 u&=\rho^{\Omega}_{ba}+\rho^{\Omega}_{ab},\cr
88 v&=i(\rho^{\Omega}_{ba}-\rho^{\Omega}_{ab}),\cr
89 w&=\rho_{bb}-\rho_{aa}.\cr
90 }
91 $$
92 In these equations, $\rho^{\Omega}_{ab}$ is the {\sl temporal envelope
93 of the off-diagonal elements}, given by
94 $$
95 \rho_{ab}\equiv\rho^{\Omega}_{ab}\exp[i(\Omega_{ba}-\Delta)t].
96 $$
97 In the Bloch equations~(1), the variable $w$ describes the population
98 inversion of the two-level system, while $u$ and $v$ are related to the
99 dispersive and absorptive components of the polarization density of the
100 medium.
101 In the Bloch equations, $w_0\equiv\rho_0(b)-\rho_0(a)$
102 is the thermal equilibrium inversion of the system with no optical
103 field applied.
104
105 \section{The resulting electric polarization density of the medium}
106 The so far developed theory of the density matrix under resonant
107 interaction can now be applied to the calculation of the electric
108 polarization density of the medium, consisting of $N$ identical
109 molecules per unit volume, as
110 $$
111 \eqalign{
112 P_{\mu}({\bf r},t)&=N\langle e{\hat r}_{\mu}\rangle\cr
113 &=N\Tr[{\hat\rho} e{\hat r}_{\mu}]\cr
114 &=N\sum_{k=a,b}\langle k|{\hat\rho} e{\hat r}_{\mu}|k\rangle\cr
115 &=N\sum_{k=a,b}\sum_{j=a,b}
116 \langle k|{\hat\rho}|j\rangle
117 \langle j|e{\hat r}_{\mu}|k\rangle\cr
118 &=N\sum_{k=a,b}\left\{
119 \langle k|{\hat\rho}|a\rangle
120 \langle a|e{\hat r}_{\mu}|k\rangle
121 +\langle k|{\hat\rho}|b\rangle
122 \langle b|e{\hat r}_{\mu}|k\rangle
123 \right\}\cr
124 &=N\left\{
125 \langle a|{\hat\rho}|a\rangle
126 \langle a|e{\hat r}_{\mu}|a\rangle
127 +\langle b|{\hat\rho}|a\rangle
128 \langle a|e{\hat r}_{\mu}|b\rangle
129 +\langle a|{\hat\rho}|b\rangle
130 \langle b|e{\hat r}_{\mu}|a\rangle
131 +\langle b|{\hat\rho}|b\rangle
132 \langle b|e{\hat r}_{\mu}|b\rangle
133 \right\}\cr
134 &=N(\rho_{ba}er^{\mu}_{ab}+\rho_{ab}er^{\mu}_{ba})\cr
135 &=\{{\rm Make\ use\ of\ }\rho_{ab}=(u+iv)\exp(i\omega t)=\rho^*_{ba}\}\cr
136 &=N[(u-iv)\exp(-i\omega t)er^{\mu}_{ab}
137 +(u+iv)\exp(i\omega t)er^{\mu}_{ba}].\cr
138 }
139 $$
140 The temporal envelope $P^{\mu}_{\omega}$ of the polarization density is
141 throughout this course as well as in Butcher and Cotter's book taken as
142 $$
143 P^{\mu}({\bf r},t)=\Re[P^{\mu}_{\omega}\exp(-i\omega t)],
144 $$
145 and by identifying this expression with the right-hand side of the result
146 above, we hence finally have obtained the polarization density
147 in terms of the Bloch parameters $(u,v,w)$ as
148 $$
149 P^{\mu}_{\omega}({\bf r},t)=Ner^{\mu}_{ab}(u-iv).\eqno{(2)}
150 $$
151 This expression for the temporal envelope of the polarization density is
152 exactly in the same mode of description as the one as previously used in
153 the susceptibility theory, as in the wave equations developed in lecture
154 eight. The only difference is that now we instead consider the polarization
155 density as given by a non-perturbative analysis. Taken together with the
156 Maxwell's equations (or the propern wave equation for the envelopes of the
157 fields), the Bloch equations are known as the {\sl Maxwell--Bloch equations}.
158
159 From Eq.~(2), it should now be clear that the Bloch variable $u$ essentially
160 gives the in-phase part of the polarization density (at least in this
161 case, where we may consider the transition dipole moments to be real-valued),
162 corresponding to the dispersive components of the interaction between
163 light and matter, while the Bloch variable $v$ on the other hand gives
164 terms which are shifted ninety degrees out of phase with the optical field,
165 hence corresponding to absorptive terms.
166
167 \vfill\eject
168
169 \section{The vector model of the Bloch equations}
170 In the form of Eqs.~(1), the Bloch equations can be expressed in the
171 form of an Euler equation as
172 $$
173 {{d{\bf R}}\over{dt}}={\bf\Omega}\times{\bf R}
174 -\underbrace{(u/T_2,v/T_2,(w-w_0)/T_1)}_{{\rm relaxation\ term}},
175 \eqno{[{\rm B.\,\&\,C.~(6.54)}]}
176 $$
177 where ${\bf R}=(u,v,w)$ is the so-called {\sl Bloch vector}, that in the
178 abstract $({\bf e}_u,{\bf e}_v,{\bf e}_w)$-space describes the state of
179 the medium, and
180 $$
181 {\bf\Omega}=(-\beta(t),0,\Delta)
182 $$
183 is the vector that gives the precession of the Bloch vector (see Fig.~1).
184
185 This form, originally proposed in 1946 by Felix
186 Bloch\footnote{${}^1$}{F. Bloch,
187 {\sl Nuclear induction}, {Phys.~Rev.} {\bf 70}, 460 (1946).
188 Felix Bloch was in 1952 awarded the Nobel prize in physics,
189 together with Edward Mills Purcell, ``for their development of new methods
190 for nuclear magnetic precision measurements and discoveries in connection
191 therewith''.} for the
192 motion of a nuclear spin in a magnetic field under influence of
193 radio-frequency electromagnetic fields, and later
194 on adopted by Feynman, Vernon, and Hellwarth\footnote{${}^2$}{R.~P. Feynman,
195 F.~L. Vernon, and R.~W. Hellwarth, {\sl Geometrical representation of the
196 Schr\"od\-ing\-er equation for solving maser problems}, J.~Appl.~Phys.
197 {\bf 28}, 49 (1957).} for solving problems in maser
198 theory\footnote{${}^3$}{Microwave Amplification by Stimulated Emission
199 of Radiation, a device for amplification of microwaves, essentially working
200 on the same principle as the laser.}, corresponds to the motion of a
201 damped gyroscope in the presence of a gravitational field.
202 In this analogy, the vector ${\bf \Omega}$ can be considered as the
203 torque vector of the spinning top of the gyroscope.
204
205 \bigskip
206 \centerline{\epsfxsize=90mm\epsfbox{../images/blochmod/blochmod.1}}
207 \medskip
208 {\noindent Figure 1. Evolution of the Bloch vector
209 ${\bf R}(t)=(u(t),v(t),w(t))$ around the ``torque vector''
210 ${\bf\Omega}=(-\beta(t),0,\Delta)$.
211 In the absence of optical fields, the Bloch vector relax towards
212 the thermal equilibrium state ${\bf R}_{\infty}=(0,0,w_0)$,
213 where $w_0=\rho(b)-\rho(a)$ is the molecular population inversion
214 at thermal equilibrium. At moderate temperatures, the thermal equilibrium
215 population inversion is very close to $w_0=-1$.}
216 \medskip
217
218 From the vector form of the Bloch equations, it is found that the
219 Bloch vector rotates around the torque vector ${\bf\Omega}$ as the
220 state of matter approaches steady state. For an adiabatically changing
221 applied optical field (i.~e.~a slowly varying envelope of the field),
222 this precession follows the torque vector.
223
224 The relaxation term in the vector Bloch equations also tells us that the
225 relaxation along the $w$-direction is given by the time constant~$T_1$,
226 while the relaxation in the $(u,v)$-plane instead is given by the time
227 constant $T_2$. By considering the $w$-axis as the ``longitudinal''
228 direction and the $(u,v)$-plane as the ``transverse'' plane, the terminology
229 for $T_1$ as being the ``longitudinal relaxation time'' and $T_2$
230 as being the ``transverse relaxation time'' should hence be clear.
231
232 \vfill\eject
233
234 \section{Transient build-up at exact resonance as the optical field
235 is switched on}
236 \subsection{The case $T_1\gg T_2$ -- Longitudinal relaxation slower than
237 transverse relaxation}
238 \bigskip
239 \centerline{\epsfxsize=65mm\epsfbox{fig8a.eps}\qquad
240 \epsfxsize=65mm\epsfbox{fig8b.eps}}
241 \centerline{\epsfxsize=65mm\epsfbox{fig8d.eps}\qquad
242 \epsfxsize=65mm\epsfbox{fig8e.eps}}
243 {\noindent Figure 2a. Evolution of the Bloch vector $(u(t),v(t),w(t))$
244 as the optical field is switched on, for the exactly resonant case
245 ($\delta=0$), and with the longitudinal relaxation
246 time being much greater than the transverse relaxation time ($T_1\gg T_2$).
247 The parameters used in the simulation are
248 $\eta\equiv T_1/T_2=100$, $\delta\equiv\Delta T_2=0$, $w_0=-1$,
249 and $\gamma(t)\equiv\beta(t)T_2=3$, $t>0$.
250 The medium was initially at thermal equilibrium,
251 $(u(0),v(0),w(0))=(0,0,w_0)=-(0,0,1)$.}
252 \medskip
253
254 \bigskip
255 \centerline{\epsfxsize=70mm\epsfbox{fig8c.eps}}
256 {\noindent Figure 2b. Evolution of the magnitude of the polarization density
257 $|P_{\omega}(t)|\sim|u(t)-iv(t)|$ as the optical field is switched on,
258 corresponding to the simulation shown in Fig.~2a.}
259 \medskip
260
261 \vfill\eject
262
263 \subsection{The case $T_1\approx T_2$ -- Longitudinal relaxation approximately
264 equal to transverse relaxation}
265 \bigskip
266 \centerline{\epsfxsize=65mm\epsfbox{fig9a.eps}\qquad
267 \epsfxsize=65mm\epsfbox{fig9b.eps}}
268 \centerline{\epsfxsize=65mm\epsfbox{fig9d.eps}\qquad
269 \epsfxsize=65mm\epsfbox{fig9e.eps}}
270 {\noindent Figure 3a. Evolution of the Bloch vector $(u(t),v(t),w(t))$
271 as the optical field is switched on, for the exactly resonant case
272 ($\delta=0$), and with the longitudinal relaxation
273 time being approximately equal to the transverse relaxation time
274 ($T_1\approx T_2$).
275 The parameters used in the simulation are
276 $\eta\equiv T_1/T_2=2$, $\delta\equiv\Delta T_2=0$, $w_0=-1$,
277 and $\gamma(t)\equiv\beta(t)T_2=3$, $t>0$.
278 The medium was initially at thermal equilibrium,
279 $(u(0),v(0),w(0))=(0,0,w_0)=-(0,0,1)$.}
280 \medskip
281
282 \bigskip
283 \centerline{\epsfxsize=70mm\epsfbox{fig9c.eps}}
284 {\noindent Figure 3b. Evolution of the magnitude of the polarization density
285 $|P_{\omega}(t)|\sim|u(t)-iv(t)|$ as the optical field is switched on,
286 corresponding to the simulation shown in Fig.~3a.}
287 \medskip
288
289 \vfill\eject
290
291 \section{Transient build-up at off-resonance as the optical field
292 is switched on}
293 \subsection{The case $T_1\approx T_2$ -- Longitudinal relaxation approximately
294 equal to transverse relaxation}
295 \bigskip
296 \centerline{\epsfxsize=65mm\epsfbox{fig10a.eps}\qquad
297 \epsfxsize=65mm\epsfbox{fig10b.eps}}
298 \centerline{\epsfxsize=65mm\epsfbox{fig10d.eps}\qquad
299 \epsfxsize=65mm\epsfbox{fig10e.eps}}
300 {\noindent Figure 4a. Evolution of the Bloch vector $(u(t),v(t),w(t))$
301 as the optical field is switched on, for the off-resonant case
302 ($\delta\ne 0$), and with the longitudinal relaxation
303 time being approximately equal to the transverse relaxation time
304 ($T_1\approx T_2$).
305 The parameters used in the simulation are
306 $\eta\equiv T_1/T_2=2$, $\delta\equiv\Delta T_2=4$, $w_0=-1$,
307 and $\gamma(t)\equiv\beta(t)T_2=3$, $t>0$.
308 The medium was initially at thermal equilibrium,
309 $(u(0),v(0),w(0))=(0,0,w_0)=-(0,0,1)$.}
310 \medskip
311
312 \bigskip
313 \centerline{\epsfxsize=70mm\epsfbox{fig10c.eps}}
314 {\noindent Figure 4b. Evolution of the magnitude of the polarization density
315 $|P_{\omega}(t)|\sim|u(t)-iv(t)|$ as the optical field is switched on,
316 corresponding to the simulation shown in Fig.~4a.}
317 \medskip
318
319 \vfill\eject
320
321 \section{Transient decay for a process tuned to exact resonance}
322 \subsection{The case $T_1\gg T_2$ -- Longitudinal relaxation slower than
323 transverse relaxation}
324 \bigskip
325 \centerline{\epsfxsize=70mm\epsfbox{fig1a.eps}
326 \epsfxsize=70mm\epsfbox{fig1b.eps}}
327 {\noindent Figure 5. Evolution of the Bloch vector $(u(t),v(t),w(t))$
328 after the optical field is switched off, for the case of tuning to
329 exact resonance ($\delta=0$), and with the longitudinal relaxation
330 time being much greater than the transverse relaxation time ($T_1\gg T_2$).
331 The parameters used in the simulation are
332 $\eta\equiv T_1/T_2=100$, $\delta\equiv\Delta T_2=0$, $w_0=-1$,
333 and $\gamma(t)\equiv\beta(t)T_2=0$.}
334 \medskip
335
336 \subsection{The case $T_1\approx T_2$ -- Longitudinal relaxation approximately
337 equal to transverse relaxation}
338 \bigskip
339 \centerline{\epsfxsize=70mm\epsfbox{fig2a.eps}
340 \epsfxsize=70mm\epsfbox{fig2b.eps}}
341 {\noindent Figure 6. Evolution of the Bloch vector $(u(t),v(t),w(t))$
342 after the optical field is switched off, for the case of tuning to
343 exact resonance ($\delta=0$), and with the longitudinal relaxation
344 time being approximately equal to the transverse relaxation time
345 ($T_1\approx T_2$).
346 The parameters used in the simulation are
347 $\eta\equiv T_1/T_2=2$, $\delta\equiv\Delta T_2=0$, $w_0=-1$,
348 and $\gamma(t)\equiv\beta(t)T_2=0$.}
349 \medskip
350
351 \vfill\eject
352
353 \section{Transient decay for a slightly off-resonant process}
354 \subsection{The case $T_1\gg T_2$ -- Longitudinal relaxation slower than
355 transverse relaxation}
356 \bigskip
357 \centerline{\epsfxsize=65mm\epsfbox{fig3a.eps}\qquad
358 \epsfxsize=65mm\epsfbox{fig3b.eps}}
359 \centerline{\epsfxsize=65mm\epsfbox{fig3d.eps}\qquad
360 \epsfxsize=65mm\epsfbox{fig3e.eps}}
361 {\noindent Figure 7a. Evolution of the Bloch vector $(u(t),v(t),w(t))$
362 after the optical field is switched off, for the off-resonant case
363 ($\delta\ne 0$), and with the longitudinal relaxation
364 time being much greater than the transverse relaxation time ($T_1\gg T_2$).
365 The parameters used in the simulation are
366 $\eta\equiv T_1/T_2=100$, $\delta\equiv\Delta T_2=2$, $w_0=-1$,
367 and $\gamma(t)\equiv\beta(t)T_2=0$.
368 (Compare with Fig.~5 for the exactly resonant case.)}
369 \medskip
370
371 \bigskip
372 \centerline{\epsfxsize=65mm\epsfbox{fig3f.eps}\qquad
373 \epsfxsize=65mm\epsfbox{fig3g.eps}}
374 {\noindent Figure 7b. Same as Fig.~7a, but with $\delta=-2$ as negative.}
375 \medskip
376
377 \vfill\eject
378
379 \subsection{The case $T_1\approx T_2$ -- Longitudinal relaxation approximately
380 equal to transverse relaxation}
381 \bigskip
382 \centerline{\epsfxsize=65mm\epsfbox{fig4a.eps}\qquad
383 \epsfxsize=65mm\epsfbox{fig4b.eps}}
384 \centerline{\epsfxsize=65mm\epsfbox{fig4d.eps}\qquad
385 \epsfxsize=65mm\epsfbox{fig4e.eps}}
386 {\noindent Figure 8a. Evolution of the Bloch vector $(u(t),v(t),w(t))$
387 after the optical field is switched off, for the off-resonant case
388 ($\delta\ne 0$), and with the longitudinal relaxation
389 time being approximately equal to the transverse relaxation time
390 ($T_1\approx T_2$).
391 The parameters used in the simulation are
392 $\eta\equiv T_1/T_2=2$, $\delta\equiv\Delta T_2=2$, $w_0=-1$,
393 and $\gamma(t)\equiv\beta(t)T_2=0$.
394 (Compare with Fig.~6 for the exactly resonant case.)}
395 \medskip
396
397 \bigskip
398 \centerline{\epsfxsize=70mm\epsfbox{fig4c.eps}}
399 {\noindent Figure 8b. Evolution of the magnitude of the polarization density
400 $|P_{\omega}(t)|\sim|u(t)-iv(t)|$ as the optical field is switched on,
401 corresponding to the simulation shown in Fig.~8a.}
402 \medskip
403
404 \vfill\eject
405
406 \section{Transient decay for a far off-resonant process}
407 \subsection{The case $T_1\gg T_2$ -- Longitudinal relaxation slower than
408 transverse relaxation}
409 \bigskip
410 \centerline{\epsfxsize=65mm\epsfbox{fig5a.eps}\qquad
411 \epsfxsize=65mm\epsfbox{fig5b.eps}}
412 \centerline{\epsfxsize=65mm\epsfbox{fig5d.eps}\qquad
413 \epsfxsize=65mm\epsfbox{fig5e.eps}}
414 {\noindent Figure 9a. Evolution of the Bloch vector $(u(t),v(t),w(t))$
415 after the optical field is switched off, for the far off-resonant case
416 ($\delta\ne 0$), and with the longitudinal relaxation
417 time being much greater than the transverse relaxation time ($T_1\gg T_2$).
418 The parameters used in the simulation are
419 $\eta\equiv T_1/T_2=100$, $\delta\equiv\Delta T_2=20$, $w_0=-1$,
420 and $\gamma(t)\equiv\beta(t)T_2=0$.
421 (Compare with Fig.~5 for the exactly resonant case,
422 and with Fig.~7a for the slightly off-resonant case.)}
423 \medskip
424
425 \bigskip
426 \centerline{\epsfxsize=70mm\epsfbox{fig5c.eps}}
427 {\noindent Figure 9b. Evolution of the magnitude of the polarization density
428 $|P_{\omega}(t)|\sim|u(t)-iv(t)|$ as the optical field is switched on,
429 corresponding to the simulation shown in Fig.~9a.}
430 \medskip
431
432 \vfill\eject
433
434 \subsection{The case $T_1\approx T_2$ -- Longitudinal relaxation approximately
435 equal to transverse relaxation}
436 \bigskip
437 \centerline{\epsfxsize=65mm\epsfbox{fig6a.eps}\qquad
438 \epsfxsize=65mm\epsfbox{fig6b.eps}}
439 \centerline{\epsfxsize=65mm\epsfbox{fig6d.eps}\qquad
440 \epsfxsize=65mm\epsfbox{fig6e.eps}}
441 {\noindent Figure 10a. Evolution of the Bloch vector $(u(t),v(t),w(t))$
442 after the optical field is switched off, for the far off-resonant case
443 ($\delta\ne 0$), and with the longitudinal relaxation
444 time being approximately equal to the transverse relaxation time
445 ($T_1\approx T_2$).
446 The parameters used in the simulation are
447 $\eta\equiv T_1/T_2=2$, $\delta\equiv\Delta T_2=20$, $w_0=-1$,
448 and $\gamma(t)\equiv\beta(t)T_2=0$.
449 (Compare with Fig.~6 for the exactly resonant case,
450 and with Fig.~8a for the slightly off-resonant case.)}
451 \medskip
452
453 \bigskip
454 \centerline{\epsfxsize=70mm\epsfbox{fig6c.eps}}
455 {\noindent Figure 10b. Evolution of the magnitude of the polarization density
456 $|P_{\omega}(t)|\sim|u(t)-iv(t)|$ as the optical field is switched on,
457 corresponding to the simulation shown in Fig.~10a.}
458 \medskip
459
460 \vfill\eject
461
462 \subsection{The case $T_1\ll T_2$ -- Longitudinal relaxation faster than
463 transverse relaxation}
464 \bigskip
465 \centerline{\epsfxsize=65mm\epsfbox{fig7a.eps}\qquad
466 \epsfxsize=65mm\epsfbox{fig7b.eps}}
467 \centerline{\epsfxsize=65mm\epsfbox{fig7d.eps}\qquad
468 \epsfxsize=65mm\epsfbox{fig7e.eps}}
469 {\noindent Figure 11a. Same parameter values as in Fig.~6, but with
470 the longitudinal relaxation
471 time being much smaller than the transverse relaxation time
472 ($T_1\ll T_2$), $\eta\equiv T_1/T_2=0.1$.
473 (Compare with Figs.~9a and~10a for the cases $T_1\gg T_2$
474 and $T_1\approx T_2$, respectively.)}
475 \medskip
476
477 \bigskip
478 \centerline{\epsfxsize=70mm\epsfbox{fig7c.eps}}
479 {\noindent Figure 11b. Evolution of the magnitude of the polarization density
480 $|P_{\omega}(t)|\sim|u(t)-iv(t)|$ as the optical field is switched on,
481 corresponding to the simulation shown in Fig.~11a.}
482 \medskip
483
484 \vfill\eject
485
486 \section{The connection between the Bloch equations and the susceptibility}
487 As an example of the connection between the polarization density obtained
488 from the Bloch equations and the one obtained from the susceptibility
489 formalism, we will now -- once again -- consider the intensity-dependent
490 refractive of the medium.
491
492 \subsection{The intensity-dependent refractive index in the susceptibility
493 formalism}
494 Previously in this course, the intensity-dependent refractive index has
495 been obtained from the optical Kerr-effect in isotropic media, in the form
496 $$
497 n=n_0+n_2|{\bf E}_{\omega}|^2,
498 $$
499 where $n_0=[1+\chi^{(1)}_{xx}(-\omega;\omega)]^{1/2}$ is the linear
500 refractive index, and
501 $$
502 n_2={{3}\over{8n_0}}\chi^{(3)}_{xxxx}(-\omega;\omega,\omega,-\omega)
503 $$
504 is the parameter of the intensity dependent contribution.
505 However, since we by now are fully aware that the polarization density
506 in the description of the susceptibility formalism originally is given
507 as an infinity series expansion, we may expect that the general form
508 of the intensity dependent refractive index rather would
509 be as a power series in the intensity,
510 $$
511 n=n_0+n_2|{\bf E}_{\omega}|^2
512 +n_4|{\bf E}_{\omega}|^4
513 +n_6|{\bf E}_{\omega}|^6+\ldots
514 $$
515 For linearly polarized light, say along the $x$-axis of a Cartesian
516 coordinate system, we know that such a series is readily possible to
517 derive in terms of the susceptibility formalism, with the different
518 order terms of the refractive index expansion given by the elements
519 $$
520 \eqalign{
521 n_2&\sim\chi^{(3)}_{xxxx}
522 (-\omega;\omega,\omega,-\omega),\cr
523 n_4&\sim\chi^{(5)}_{xxxxxx}
524 (-\omega;\omega,\omega,-\omega,\omega,-\omega),\cr
525 n_6&\sim\chi^{(7)}_{xxxxxxxx}
526 (-\omega;\omega,\omega,-\omega,\omega,-\omega,\omega,-\omega),\cr
527 &\qquad\vdots\cr
528 }
529 $$
530 Such an analysis would, however, be extremely cumbersome when it comes
531 to the analysis of higher-order effects, and the obtained sum of various
532 order terms would also be almost impossible to obtain a closed expression
533 for.
534 For future reference, to be used in the interpretation of the polarization
535 density given by the Bloch equations, the intensity dependent polarization
536 density is though shown in its explicit form below, including up to the
537 seventh order interaction term in the Butcher and Cotter convention,
538 $$
539 \eqalignno{
540 P^x_{\omega}
541 =\varepsilon_0&\chi^{(1)}_{xx}
542 (-\omega;\omega)E^x_{\omega}
543 &({\rm order}\ n=1)\cr
544 &+\varepsilon_0(3/4)\chi^{(3)}_{xxxx}
545 (-\omega;\omega,\omega,-\omega)|E^x_{\omega}|^2 E^x_{\omega}
546 &({\rm order}\ n=3)\cr
547 &+\varepsilon_0(5/8)\chi^{(5)}_{xxxxxx}
548 (-\omega;\omega,\omega,-\omega,\omega,-\omega)
549 |E^x_{\omega}|^4 E^x_{\omega}
550 &({\rm order}\ n=5)\cr
551 &+\varepsilon_0(35/64)\chi^{(7)}_{xxxxxxxx}
552 (-\omega;\omega,\omega,-\omega,\omega,-\omega,\omega,-\omega)
553 |E^x_{\omega}|^6 E^x_{\omega}
554 &({\rm order}\ n=7)\cr
555 &+\ldots&\cr
556 }
557 $$
558 The other approach to calculation of the polarization density, as we
559 next will outline, is to use the steady-state solutions to the Bloch
560 equations.
561 \vfill\eject
562
563 \subsection{The intensity-dependent refractive index in the Bloch-vector
564 formalism}
565 For steady-state interaction between light and matter, the solutions
566 to the Bloch equations yield
567 $$
568 \eqalignno{
569 &u-iv={{-\beta w}\over{\Delta-i/T_2}},&[{\rm B.\,\&\,C.~(6.53a)}],\cr
570 &w={{w_0[1+(\Delta T_2)^2]}
571 \over{1+(\Delta T_2)^2+\beta^2 T_1 T_2}},&[{\rm B.\,\&\,C.~(6.53b)}],\cr
572 }
573 $$
574 where, as previously, $\beta=er^{\alpha}_{ab}E^{\alpha}_{\omega}(t)/\hbar$
575 is the Rabi frequency, though now considered to be a slowly varying
576 (adiabatically following) quantity, due to the assumption of steady-state
577 behaviour.
578 From the steady-state solutions, the $\mu$-component ($\mu=x,y,z$) of the
579 electric polarization
580 density ${\bf P}({\bf r},t)=\Re[{\bf P}_{\omega}\exp(-i\omega t)]$ of the
581 medium hence is given as
582 $$
583 \eqalign{
584 P^{\mu}_{\omega}
585 &=Ner^{\mu}_{ab}(u-iv)\cr
586 &=-Ner^{\mu}_{ab}{{\beta w}\over{\Delta-i/T_2}}\cr
587 &=-Ner^{\mu}_{ab}{{\beta}\over{(\Delta-i/T_2)}}
588 {{w_0[1+(\Delta T_2)^2]}\over{[1+(\Delta T_2)^2+\beta^2 T_1 T_2]}}\cr
589 &=-New_0{{r^{\mu}_{ab}}\over{(\Delta-i/T_2)}}
590 {{\beta}\over
591 {\left[1+{{T_1 T_2}\over{(1+(\Delta T_2)^2)}}\beta^2\right]}}.\cr
592 }\eqno{(3)}
593 $$
594 In this expression for the polarization density, it might at a first glance
595 seem as it is negative for a positive Rabi frequency $\beta$, henc giving a
596 polarization density that is directed anti-parallel to the electric field.
597 However, the quantity $w_0=\rho_0(b)-\rho_0(a)$, the population inversion
598 at thermal equilibrium, is always negative (since we for sure do not have
599 any population inversion at thermal equilibrium, for which we rather expect
600 the molecules to occupy the lower state), hence ensuring that the off-resonant,
601 real-valued polarization density always is directed along the direction of the
602 electric field of the light.
603
604 Next observation is that the polarization density no longer is expressed
605 as a power series in terms of the electric field, but rather as a rational
606 function,
607 $$
608 P^{\mu}_{\omega}\sim X/(1+X^2),\eqno{(4)}
609 $$
610 where
611 $$
612 \eqalign{
613 X&=\sqrt{{T_1 T_2}/{(1+(\Delta T_2)^2)}}\beta\cr
614 &=\sqrt{{T_1 T_2}/{(1+(\Delta T_2)^2)}}
615 er^{\alpha}_{ab}E^{\alpha}_{\omega}(t)/\hbar\cr
616 }
617 $$
618 is a parameter linear in the electric field. The principal shape of the
619 rational function in Eq.~(4) is shown in Fig.~12.
620
621 From Eq.~(4), the polarization density is found to increase with increasing
622 $X$ up to $X=1$, as we expect for an increasing power of an optical beam.
623 However, for $X>1$, we find the somewhat surprising fact that the
624 polarization density instead {\sl decrease} with an increasing intensity;
625 this peculiar suggested behaviour should hence be explained before continuing.
626
627 The first observation we may do is that the linear polarizability
628 (i.~e.~what we usually associate with linear optics) follows the
629 first order approximation $p(X)=X$.
630 In the region where the peculiar decrease of the polarization density
631 appear, the difference between the suggested nonlinear polarization
632 density and the one given by the linear approximation is {\sl huge},
633 and since we {\sl a priori} expect nonlinear contributions to be small
634 compared to the alsways present linear ones, this is already an indication
635 of that we in all practical situations do not have to consider the
636 descrease of polarization density as shown in Fig.~12.
637
638 For optical fields of the strength that would give rise to nonlinearities
639 exceeding the linear terms, the underlying physics will rather belong
640 to the field of plasma and high-energy physics, rather than a bound-charge
641 description of gases and solids. This implies that the validity of the
642 models here applied (bound charges, Hamiltonians being linear in the
643 optical field, etc.) are limited to a range well within $X\le 1$.
644 \vfill\eject
645
646 \centerline{\epsfxsize=120mm\epsfbox{polplot.eps}}
647 {\noindent Figure 12. The principal shape of the electric polarization
648 density of the medium, as function of the applied electric field of
649 the light.
650 In this figure, $X=\sqrt{{T_1 T_2}/{(1+(\Delta T_2)^2)}}\beta$
651 is a normalized parameter describing the field strength of the electric
652 field of the light.}
653 \bigskip
654
655 Another interesting point we may observe is a more mathematically
656 related one.
657 In Fig.~12, we see that even for very high order terms (such as the
658 approximating power series of degree 31, as shown in the figure),
659 all power series expansions fail before reaching $X=1$.
660 The reason for this is that the power series that approximate the
661 rational function $X/(1+X^2)$,
662 $$
663 X/(1+X^2)=X-X^3+X^5-X^7+\ldots,
664 $$
665 is convergent only for $|X|<1$; for all other values, the series is
666 divergent.
667 This means that no matter how many terms we include in the power series
668 in $X$, it will nevertheless fail when it comes to the evaluation
669 for $|X|>1$.
670 Since this power series expansion is equivalent to the expansion
671 of the nonlinear polarization density in terms of the electrical field
672 of the light (keeping in mind that $X$ here actually is linear in the
673 electric field and hence strictly can be considered as the field variable),
674 this also is an indication that at this point the whole susceptibility
675 formalism fail to give a proper description at this working point.
676
677 This is an excellent illustration of the downturn of the susceptibility
678 description of interaction between light and matter; no matter how
679 many terms we may include in the power series of the electrcal field,
680 {\sl it will at some point nevertheless fail to give the total picture of
681 the interaction}, and we must then instead seek other tools.
682
683 Returning to the polarization density given by Eq.~(3), we may now express
684 this in an explicit form by inserting $\Delta\equiv\Omega_{ba}-\omega$ for
685 the angular frequency detuning, the Rabi frequency
686 $\beta=er^{\alpha}_{ab}E^{\alpha}_{\omega}(t)/\hbar$,
687 and the thermal equilibrium inversion $w_0=\rho_0(b)-\rho_0(a)$.
688 This gives the polarization density of the medium as
689 $$
690 P^{\mu}_{\omega}
691 =\varepsilon_0
692 \underbrace{
693 \underbrace{
694 {{Ne^2}\over{\varepsilon_0\hbar}}(\rho_0(a)-\rho_0(b))
695 {{r^{\mu}_{ab}r^{\alpha}_{ab}}\over{(\Omega_{ba}-\omega-i/T_2)}}
696 }_{=\chi^{(1)}_{\mu\alpha}(-\omega;\omega)
697 \ {\rm for\ a\ two\ level\ medium}}
698 \underbrace{
699 {{1}\over{\left[1+{{T_1 T_2}\over{(1+(\Omega_{ba}-\omega)^2 T^2_2)}}
700 (er^{\gamma}_{ab}E^{\gamma}_{\omega}/\hbar)^2\right]}}
701 }_{{\rm nonlinear\ correction\ factor\ to}
702 \ \chi^{(1)}_{\mu\alpha}(-\omega;\omega)}
703 }_{{\rm the\ field\ corrected\ susceptibility,}
704 \ {\bar{\chi}}(\omega;{\bf E}_{\omega})
705 \ [{\rm see\ Butcher\ and\ Cotter,\ section~6.3.1}]}
706 E^{\alpha}_{\omega}.
707 $$
708 In this form, the polarization density is given as the product with
709 a term which is identical to the linear
710 susceptibility\footnote{${}^4$}{In the explicit expressions for the
711 linear susceptibility, for example Butcher and Cotter's Eqs.~(4.58)
712 and~(4.111) for the non-resonant and resonant cases, respectively,
713 there are two terms, one with $\Omega_{ba}-\omega$ in the denominator
714 and the other one with $\Omega_{ba}+\omega$.
715 The reason why the second form does not appear in the expression for the
716 field corrected susceptibility, as derived from the Bloch equations, is
717 that {\sl we have used the rotating wave approximation in the derivation
718 of the final expression.} (Recapitulate that in the rotating wave
719 approximation, terms with oscillatory dependence of
720 $\exp[i(\Omega_{ba}+\omega)t]$ were neglected.)
721 As a result, all temporally phase-mismatched terms are neglected, and in
722 particular only terms with $\Omega_{ba}-\omega$ in the denominator will
723 remain. This, however, is a most acceptable approximation, especially when
724 it comes to resonant interactions, where terms with $\Omega_{ba}-\omega$
725 in the denominator by far will dominate over non-resonant terms.}
726 (as obtained in the perturbation analysis in the frame of the susceptibility
727 formalism), and a correction factor which is a nonlinear function of the
728 electric field.
729
730 The nonlinear correction factor, of the form $1/(1+X^2)$, with
731 $X=\sqrt{{T_1 T_2}/{(1+(\Delta T_2)^2)}}\beta$ as previously, can now be
732 expanded in a power series around the small-signal limit $X=0$, using
733 $$
734 1/(1+X^2)=1-X^2+X^4-X^6+\ldots,
735 $$
736 from which we obtain the polarization density as a power series in the
737 electric field (which for the sake of simplicitly now is taken as linearly
738 polarized along the $x$-axis) as
739 $$
740 \eqalign{
741 P^x_{\omega}
742 \approx\varepsilon_0
743 &{{Ne^2}\over{\varepsilon_0\hbar}}(\rho_0(a)-\rho_0(b))
744 {{r^x_{ab}r^x_{ab}}\over{(\Omega_{ba}-\omega-i/T_2)}}
745 E^x_{\omega}\cr
746 &-\varepsilon_0
747 {{Ne^4}\over{\varepsilon_0\hbar^3}}(\rho_0(a)-\rho_0(b))
748 {{r^x_{ab}r^x_{ab}}\over{(\Omega_{ba}-\omega-i/T_2)}}
749 {{(r^x_{ab})^2}\over{[1/T^2_2+(\Omega_{ba}-\omega)^2](T_2/T_1)}}
750 |E^x_{\omega}|^2
751 E^x_{\omega}\cr
752 &+\varepsilon_0
753 {{Ne^6}\over{\varepsilon_0\hbar^5}}(\rho_0(a)-\rho_0(b))
754 {{r^x_{ab}r^x_{ab}}\over{(\Omega_{ba}-\omega-i/T_2)}}
755 {{(r^x_{ab})^4}\over{[1/T^2_2+(\Omega_{ba}-\omega)^2]^2(T_2/T_1)^2}}
756 |E^x_{\omega}|^4
757 E^x_{\omega}\cr
758 &+\ldots\cr
759 }\eqno{(5)}
760 $$
761 This form is identical to one as obtained in the susceptibility formalism;
762 however, the steps that led us to this expression for the polarization
763 density {\sl do not rely on the perturbation theory of the density
764 operator}, but rather on the explicit form of the steady-state solutions
765 to the Bloch equations.
766
767 \section{Summary of the Bloch and susceptibility polarization densities}
768 To summarize this last lecture on the Bloch equations, expressing the involved
769 parameters in the same style as previously used in the description of the
770 susceptibility formalism, the polarization density obtained from the
771 steady-state solutions to the Bloch equations is
772 $$
773 P^{\mu}_{\omega}
774 =\varepsilon_0
775 {{Ne^2}\over{\varepsilon_0\hbar}}(\rho_0(a)-\rho_0(b))
776 {{r^{\mu}_{ab}r^{\alpha}_{ab}}\over{(\Omega_{ba}-\omega-i/T_2)}}
777 {{1}\over{\left[1+{{T_1 T_2}\over{(1+(\Omega_{ba}-\omega)^2 T^2_2)}}
778 (er^{\alpha}_{ab}E^{\alpha}_{\omega}/\hbar)^2\right]}}
779 E^{\alpha}_{\omega}.
780 $$
781 By expanding this in a power series in the electrical field, one obtains
782 the form (5), in which we from the same description of the polarization
783 density in the susceptibility formalism can identify
784 $$
785 \eqalign{
786 \chi^{(1)}_{xx}(-\omega;\omega)
787 &={{Ne^2}\over{\varepsilon_0\hbar}}(\rho_0(a)-\rho_0(b))
788 {{r^x_{ab}r^x_{ab}}\over{(\Omega_{ba}-\omega-i/T_2)}},\cr
789 \chi^{(3)}_{xxxx}(-\omega;\omega,\omega,-\omega)
790 &=-{{4Ne^4}\over{3\varepsilon_0\hbar^3}}(\rho_0(a)-\rho_0(b))
791 {{r^x_{ab}r^x_{ab}}\over{(\Omega_{ba}-\omega-i/T_2)}}
792 {{(r^x_{ab})^2}\over{[1/T^2_2+(\Omega_{ba}-\omega)^2](T_2/T_1)}},\cr
793 \chi^{(5)}_{xxxxxx}(-\omega;\omega,\omega,-\omega,&\omega,-\omega)\cr
794 &={{8Ne^6}\over{5\varepsilon_0\hbar^5}}(\rho_0(a)-\rho_0(b))
795 {{r^x_{ab}r^x_{ab}}\over{(\Omega_{ba}-\omega-i/T_2)}}
796 {{(r^x_{ab})^4}\over{[1/T^2_2+(\Omega_{ba}-\omega)^2]^2(T_2/T_1)^2}},\cr
797 }
798 $$
799 as being the first contributions to the two-level polarization density,
800 including up to fifth order interactions.
801 For a summary of the non-resonant forms of the susceptibilities of
802 two-level systems, se Butcher and Cotter, Eqs.~(6.71)--(6.73).
803 \vfill\eject
804
805 \section{Appendix: Notes on the numerical solution to the Bloch equations}
806 In their original form, the Bloch equations for a two-level system are
807 given by Eqs.~(1) as
808 $$
809 \eqalignno{
810 {{du}\over{dt}}&=-\Delta v -u/T_2,\cr
811 {{dv}\over{dt}}&=\Delta u+\beta(t)w-v/T_2,\cr
812 {{dw}\over{dt}}&=-\beta(t)v-(w-w_0)/T_1.\cr
813 }
814 $$
815 By taking the time in units of the transverse relaxation time $T_2$, as
816 $$
817 \tau=t/T_2,
818 $$
819 the Bloch equations in this normalized time scale become
820 $$
821 \eqalignno{
822 {{du}\over{d\tau}}&=-\Delta T_2 v -u,\cr
823 {{dv}\over{d\tau}}&=\Delta T_2 u+\beta(t)T_2 w-v,\cr
824 {{dw}\over{d\tau}}&=-\beta(t)T_2 v-(w-w_0)T_2/T_1.\cr
825 }
826 $$
827 In this system of equations, all coefficients are now normalized and
828 physically dimensionless, expressed as relevant quotes between relaxation
829 times and products of the Rabi frequency or detuning frequency with
830 the transverse relaxation time.
831 Hence, by taking the normalized parameters
832 $$
833 \eqalign{
834 \delta&=\Delta T_2,\cr
835 \gamma(t)&=\beta(t)T_2,\cr
836 \eta&=T_1/T_2,\cr
837 }
838 $$
839 where $\delta$ can be considered as the normalized detuning from molecular
840 resonance of the medium, $\gamma(t)$ as the normalized Rabi frequency,
841 and $\eta$ as a parameter which describes the relative impact of the
842 longitudinal vs transverse relaxation times, the Bloch equations take
843 the normalized final form
844 $$
845 \eqalignno{
846 {{du}\over{d\tau}}&=-\delta v -u,\cr
847 {{dv}\over{d\tau}}&=\delta u+\gamma(t) w-v,\cr
848 {{dw}\over{d\tau}}&=-\gamma(t) v-(w-w_0)/\eta.\cr
849 }
850 $$
851 This normalized form of the Bloch equations has been used throughout the
852 generation of graphs in Figs.~2--11 of this lecture, describing the
853 qualitative impact of different regimes of resonance and relaxation.
854 The normalized Bloch equations were in the simulations shown in Figs.~2--11
855 integrated by using the standard routine {\tt ODE45()} in MATLAB.
856 \bye
857
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