Contents of file 'lect12/lect12.tex':

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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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14   \input amssym
15   \font\ninerm=cmr9
16   \font\twelvesc=cmcsc10
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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
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{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
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
240     \epsfxsize=65mm\epsfbox{fig8b.eps}}
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
267     \epsfxsize=65mm\epsfbox{fig9b.eps}}
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
297     \epsfxsize=65mm\epsfbox{fig10b.eps}}
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
358     \epsfxsize=65mm\epsfbox{fig3b.eps}}
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
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
383     \epsfxsize=65mm\epsfbox{fig4b.eps}}
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
411     \epsfxsize=65mm\epsfbox{fig5b.eps}}
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
438     \epsfxsize=65mm\epsfbox{fig6b.eps}}
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
466     \epsfxsize=65mm\epsfbox{fig7b.eps}}
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
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