**Contents of file 'lect12/lect12.tex':**

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% File: nlopt/lect12/lect12.tex [pure TeX code]2% Last change: March 21, 20033%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 Jonsson8%9 \input epsf 10%11% Read amssym to get the AMS {\Bbb E} font (strikethrough E) and12% the Euler fraktur font.13%14 \input amssym 15 \font\ninerm=cmr9 16 \font\twelvesc=cmcsc10 17%18% Use AMS Euler fraktur style for short-hand notation of Fourier transform19%20 \def\fourier{\mathop{\frak F}\nolimits} 21 \def\Re{\mathop{\rm Re}\nolimits}% real part22 \def\Im{\mathop{\rm Im}\nolimits}% imaginary part23 \def\Tr{\mathop{\rm Tr}\nolimits}% quantum mechanical trace24 \def\sinc{\mathop{\rm sinc}\nolimits}% the sinc(x)=sin(x)/x function25 \def\sech{\mathop{\rm sech}\nolimits}% the sech(x)=... function26 \def\sgn{\mathop{\rm sgn}\nolimits}% sgn(x)=0, if x<0, sgn(x)=1, otherwise27 \def\lecture #1 {\hsize=150mm\hoffset=4.6mm\vsize=230mm\voffset=7mm 28 \topskip=0pt\baselineskip=12pt\parskip=0pt\leftskip=0pt\parindent=15pt 29 \headline={\ifnum\pageno>1\ifodd\pageno\rightheadline\else\leftheadline\fi 30 \else\hfill\fi} 31 \def\rightheadline{\tenrm{\it Lecture notes #1} 32 \hfil{\it Nonlinear Optics 5A5513 (2003)}} 33 \def\leftheadline{\tenrm{\it Nonlinear Optics 5A5513 (2003)} 34 \hfil{\it Lecture notes #1}} 35 \noindent\epsfxsize 100pt\epsfbox{../info/kthtext.eps} 36 \vskip-26pt\hfill\vbox{\hbox{{\it Nonlinear Optics 5A5513 (2003)}} 37 \hbox{{\it Lecture notes}}}\vskip 36pt\centerline{\twelvesc Lecture #1} 38 \vskip 24pt\noindent} 39 \def\section #1 {\medskip\goodbreak\noindent{\bf #1} 40 \par\nobreak\smallskip\noindent} 41 \def\subsection #1 {\smallskip\goodbreak\noindent{\it #1} 42 \par\nobreak\smallskip\noindent} 43 44 \lecture{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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