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    1   % File: nlopt/lect4/lect4.tex [pure TeX code]
    2   % Last change: January 19, 2003
    3   %
    4   % Lecture No 4 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
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   17   %
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   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\lecture #1 {\hsize=150mm\hoffset=4.6mm\vsize=230mm\voffset=7mm
   25     \topskip=0pt\baselineskip=12pt\parskip=0pt\leftskip=0pt\parindent=15pt
   26     \headline={\ifnum\pageno>1\ifodd\pageno\rightheadline\else\leftheadline\fi
   27       \else\hfill\fi}
   28     \def\rightheadline{\tenrm{\it Lecture notes #1}
   29       \hfil{\it Nonlinear Optics 5A5513 (2003)}}
   30     \def\leftheadline{\tenrm{\it Nonlinear Optics 5A5513 (2003)}
   31       \hfil{\it Lecture notes #1}}
   32     \noindent\epsfxsize 100pt\epsfbox{../info/kthtext.eps}
   33     \vskip-26pt\hfill\vbox{\hbox{{\it Nonlinear Optics 5A5513 (2003)}}
   34     \hbox{{\it Lecture notes}}}\vskip 36pt\centerline{\twelvesc Lecture #1}
   35     \vskip 24pt\noindent}
   36   \def\section #1 {\medskip\goodbreak\noindent{\bf #1}
   37     \par\nobreak\smallskip\noindent}
   38   \def\subsection #1 {\smallskip\goodbreak\noindent{\it #1}
   39     \par\nobreak\smallskip\noindent}
   40   
   41   \lecture{4}
   42   \section{The Truth of polarization densitites}
   43   So far, we have performed the analysis in a theoretical framework that
   44   has been exclusively formulated in terms of phenomenological models, such
   45   as the anharmonic oscillator and the phenomenologically introduced
   46   polarization response function of the medium.
   47   In the real world application of nonlinear optics, however, we should not
   48   restrict the theory just to phenomenological models, but rather take
   49   advantage over the full quantum-mechanical framework of analysis of
   50   interaction between light and matter.
   51   \medskip
   52   \centerline{\epsfxsize=105mm\epsfbox{../images/dipoleop/dipoleop.1}}
   53   \medskip
   54   \centerline{Figure 1. Schematic figure of the ensemble in the
   55     ``small volume''.}
   56   \medskip
   57   \noindent
   58   In a small volume $V$ (smaller than the wavelength of the light, to ensure
   59   that the natural spatial variation of the light is not taken into account,
   60   but large enough in order to contain a sufficcient number of molecules in
   61   order to ignore the quantum-mechanical fluctuations of the dipole moment
   62   density), we consider the applied electric field to be homogeneous, and the
   63   electric polarization density of the medium is then given as the expectation
   64   value of the {\sl electric dipole operator of the ensemble of molecules}
   65   divided by the volume, as
   66   $$
   67     P_{\mu}({\bf r},t)={{\langle\hat{Q}_{\mu}\rangle}/{V}},
   68   $$
   69   where the electric dipole operator of the ensemble contained in $V$ can
   70   be written as a sum over all electrons and nuclei as
   71   $$
   72     \hat{\bf Q}=\underbrace{-e\sum_j\hat{\bf r}_j}_{\rm electrons}
   73                 +\underbrace{e\sum_k Z_k \hat{\bf r}_k}_{\rm nuclei}.
   74   $$
   75   The expectation value $\langle\hat{Q}_{\mu}\rangle$ can in principle be
   76   calculated directly from the compound, time-dependent wave function of
   77   the ensemble of molecules in the small volume, considering any kind of
   78   interaction between the molecules, which may be of an arbitrary composition.
   79   However, we will here describe the interactions that take place in terms
   80   of the {\sl quantum mechanical density operator} of the ensemble, in which
   81   case the expectation value is calculated from the {\sl quantum mechanical
   82   trace} as
   83   $$
   84     P_{\mu}({\bf r},t)=\Tr[\hat{\rho}(t)\hat{Q}_{\mu}]/V.
   85   $$
   86   
   87   \section{Outline}
   88   Previously, in lecture one, we applied the mathematical tool of perturbation
   89   analysis to a classical mechanical model of the dipole moment. This analysis
   90   will now essentially be repeated, but now we will instead consider a
   91   perturbation series for the quantum mechanical density operator, with
   92   the series being of the form
   93   $$
   94     \hat{\rho}(t)=\underbrace{\hat{\rho}_0}_{\sim [E(t)]^0}
   95       +\underbrace{\hat{\rho}_1(t)}_{\sim [E(t)]^1}
   96       +\underbrace{\hat{\rho}_2(t)}_{\sim [E(t)]^2}
   97       +\ldots
   98       +\underbrace{\hat{\rho}_n(t)}_{\sim [E(t)]^n}
   99       +\ldots
  100   $$
  101   As this perturbation series is inserted into the expression for the
  102   electric polarization density, we will obtain a resulting series for
  103   the polarization density as
  104   $$
  105     P_{\mu}({\bf r},t)=\sum^{\infty}_{m=0}\underbrace{\Tr[\hat{\rho}_m(t)
  106       \hat{Q}_{\mu}]/V}_{=P^{(m)}_{\mu}({\bf r},t)}
  107       \approx\sum^{n}_{m=0} P^{(m)}_{\mu}({\bf r},t).
  108   $$
  109   
  110   \section{Quantum mechanics}
  111   We consider an ensemble of molecules, where each molecule may be different
  112   from the other molecules of the ensemble, as well as being affected by some
  113   mutual interaction between the other members of the ensemble.
  114   The Hamiltonian for this ensemble is generally taken as
  115   $$
  116     \hat{H}=\hat{H}_0+\hat{H}_{\rm I}(t),
  117   $$
  118   where $\hat{H}_0$ is the Hamiltonian at thermal equilibrium, with no
  119   external forces present, and $\hat{H}_{\rm I}(t)$ is the interaction
  120   Hamiltonial (in the Schr\"{o}dinger picture), which for electric dipolar
  121   interactions take the form:
  122   $$
  123     \hat{H}_{\rm I}(t)
  124       =-\hat{\bf Q}\cdot{\bf E}({\bf r},t)
  125       =-\hat{Q}_{\alpha}E_{\alpha}({\bf r},t),
  126   $$
  127   where $\hat{\bf Q}$ is the electric dipole operator of the {\sl ensemble}
  128   of molecules contained in the small volume~$V$ (see Fig.~1). This expression
  129   may be compared with the all-classical electrostatic energy of an electric
  130   dipole moment in a electric field, $V=-{\bf p}\cdot{\bf E}({\bf r},t)$.
  131   
  132   In order to provide a proper description of the interaction between
  133   light and matter at molecular level, we must be means of some quantum
  134   mechanical description evaluate all properties of the molecule, such
  135   as electric dipole moment, magnetic dipole moment, etc., by means
  136   of {\sl quantum mechanical expectation values}.
  137   
  138   The description that we here will apply is by means of the {\sl density
  139   operator formalism}, with the density operator defined in terms of
  140   orthonormal set of wave functions $|a\rangle$ of the system as
  141   $$\hat{\rho}=\sum_a p_a|a\rangle\langle a|=\hat{\rho}(t),$$
  142   where $p_a$ are the normalized probabilities of the system to be
  143   in state $|a\rangle$, with $$\sum_a p_a=1.$$
  144   From the density operator, the expectation value of any arbitrary quantum
  145   mechanical operator $\hat{O}$ of the ensemble is obtained from the
  146   {\sl quantum mechanical trace} as
  147   $$
  148     \langle\hat{O}\rangle=\Tr(\hat{\rho}\,\hat{O})
  149       =\sum_k\langle k|\hat{\rho}\,\hat{O}|k\rangle.
  150   $$
  151   The equation of motion for the density operator is given in terms of
  152   the Hamiltonian as
  153   $$
  154     \eqalign{
  155     i\hbar{{d\hat{\rho}}\over{dt}}
  156       &=[\hat{H},\hat{\rho}]
  157        =\hat{H}\hat{\rho}-\hat{\rho}\hat{H}\cr
  158       &=[\hat{H}_0,\hat{\rho}]+[\hat{H}_{\rm I}(t),\hat{\rho}]\cr
  159     }
  160     \eqno{(1)}
  161   $$
  162   In this context, the terminology of ``equation of motion'' can be
  163   pictured as
  164   $$
  165     \bigg\{\matrix{{\rm A\ change\ of\ the\ density}\cr
  166       {\rm operator}\ \hat{\rho}(t)\ {\rm in\ time}\cr}\bigg\}
  167       \quad\Leftrightarrow\quad
  168     \bigg\{\matrix{{\rm A\ change\ of\ density}\cr
  169       {\rm of\ states\ in\ time}\cr}\bigg\}
  170       \quad\Leftrightarrow\quad
  171     \bigg\{\matrix{{\rm Change\ of\ a\ general}\cr
  172       {\rm property}\ \langle\hat{O}\rangle\ {\rm in\ time}\cr}\bigg\}
  173   $$
  174   Whenever external forces are absent, that is to say, whenever the applied
  175   electromagnetic field is zero, the equation of motion for the density
  176   operator takes the form
  177   $$i\hbar{{d\hat{\rho}}\over{dt}}=[\hat{H}_0,\hat{\rho}],$$
  178   with the solution\footnote{${}^1$}{For any macroscopic system,
  179   the probability that the system is in a particular energy eigenstate
  180   $\psi_n$, with associated energy ${\Bbb E}_n$, is given by the familiar
  181   Boltzmann distribution $$p_n=\eta\exp(-{\Bbb E}_n/k_{\rm B}T),$$
  182   where $\eta$ is a normalization constant chosen so that $\sum_n p_n=1$,
  183   $k_{\rm B}$ is the Boltzmann constant, and $T$ the absolute temperature.
  184   This probability distribution is in this course to be considered
  185   as being an axiomatic fact, and the origin of this probability distribution
  186   can readily be obtained from textbooks on thermodynamics or statistical
  187   mechanics.}
  188   $$\eqalign{\hat{\rho}(t)=\hat{\rho}_0&=\eta\exp(-\hat{H}_0/k_{\rm B}T)\cr
  189   \bigg\{&=\eta\sum^{\infty}_{j=1}{{1}\over{j!}}(-\hat{H}_0/k_{\rm B}T)^j
  190   \bigg\}\cr}$$
  191   being the time-independent density operator at thermal equilibrium,
  192   with the normalization constant $\eta$ chosen so that $\Tr(\hat{\rho})=1$,
  193   i.~e., $$\eta=1/\Tr[\exp(-\hat{H}_0/k_{\rm B}T)].$$
  194   
  195   \section{Perturbation analysis of the density operator}
  196   The task is now o obtain a solution of the equation of motion~(1) by
  197   means of a perturbation series, in similar to the analysis performed
  198   for the anharmonic oscillator in the first lecture of this course.
  199   The perturbation series is, in analogy to the mechanical spring oscillator
  200   under influence of an electromagnetic field, taken as
  201   $$
  202     \hat{\rho}(t)=\underbrace{\hat{\rho}_0}_{\sim [E(t)]^0}
  203       +\underbrace{\hat{\rho}_1(t)}_{\sim [E(t)]^1}
  204       +\underbrace{\hat{\rho}_2(t)}_{\sim [E(t)]^2}
  205       +\ldots
  206       +\underbrace{\hat{\rho}_n(t)}_{\sim [E(t)]^n}
  207       +\ldots
  208   $$
  209   The boundary condition of the perturbation series is taken
  210   as the initial condition that sometime in the past, the external
  211   forces has been absent, i.~e.
  212   $$
  213     \hat{\rho}(-\infty)=\hat{\rho}_0,
  214   $$
  215   which, since the perturbation series is to be valid for {\sl all possible
  216   evolutions in time of the externally applied electric field}, leads to the
  217   boundary conditions for each individual term of the perturbation series as
  218   $$
  219     \hat{\rho}_j(-\infty)=0,\qquad j=1,2,\ldots
  220   $$
  221   By inserting the perturbation series for the density operator into the
  222   equation of motion~(1), one hence obtains
  223   $$
  224     \eqalign{
  225       i\hbar{{d}\over{dt}}(\hat{\rho}_0+\hat{\rho}_1(t)+\hat{\rho}_2(t)
  226         +\ldots+\hat{\rho}_n(t)+\ldots)
  227         &=[\hat{H}_0,\hat{\rho}_0+\hat{\rho}_1(t)+\hat{\rho}_2(t)
  228           +\ldots+\hat{\rho}_n(t)+\ldots]\cr
  229         &\qquad+[\hat{H}_{\rm I}(t),\hat{\rho}_0+\hat{\rho}_1(t)+\hat{\rho}_2(t)
  230           +\ldots+\hat{\rho}_n(t)+\ldots],\cr
  231     }
  232   $$
  233   and by equating terms with equal power dependence of the applied electric
  234   field in the right and left hand sides, one obtains the system of equations
  235   $$
  236     \eqalign{
  237       i\hbar{{d\hat{\rho}_0}\over{dt}}&=[\hat{H}_0,\hat{\rho}_0],\cr
  238       i\hbar{{d\hat{\rho}_1(t)}\over{dt}}&=[\hat{H}_0,\hat{\rho}_1(t)]
  239         +[\hat{H}_{\rm I}(t),\hat{\rho}_0],\cr
  240       i\hbar{{d\hat{\rho}_2(t)}\over{dt}}&=[\hat{H}_0,\hat{\rho}_2(t)]
  241         +[\hat{H}_{\rm I}(t),\hat{\rho}_1(t)],\cr
  242       &\vdots\cr
  243       i\hbar{{d\hat{\rho}_n(t)}\over{dt}}&=[\hat{H}_0,\hat{\rho}_n(t)]
  244         +[\hat{H}_{\rm I}(t),\hat{\rho}_{n-1}(t)],\cr
  245       &\vdots\cr
  246     }\eqno{(2)}
  247   $$
  248   for the variuos order terms of the perturbation series. In Eq.~(2), we may
  249   immediately notice that the first equation simply is the identity stating
  250   the thermal equilibrium condition for the zeroth order term $\hat{\rho}_0$,
  251   while all other terms may be obtained by consecutively solve the equations
  252   of order $j=1,2,\ldots,n$, in that order.
  253   
  254   \section{The interaction picture}
  255   We will now turn our attention to the problem of actually solving the
  256   obtained system of equations for the terms of the perturbation series
  257   for the density operator.
  258   In a classical picture, the obtained equations are all of the form
  259   similar to
  260   $$
  261     {{d\rho}\over{dt}}=f(t)\rho+g(t),\eqno{(3)}
  262   $$
  263   for known functions $f(t)$ and $g(t)$. To solve these equations,
  264   we generally look for an integrating factor $I(t)$ satisfying
  265   $$
  266     I(t){{d\rho}\over{dt}}-I(t)f(t)\rho={{d}\over{dt}}[I(t)\rho].\eqno{(4)}
  267   $$
  268   By carrying out the differentiation in the right hand side of
  269   the equation, we find that the integrating factor should satisfy
  270   $$
  271     {{d I(t)}\over{dt}}=-I(t)f(t),
  272   $$
  273   which is solved by\footnote{${}^2$}{Butcher and Cotter have in their
  274   classical description of integrating factors chosen to put $I(0)=1$.}
  275   $$
  276     I(t)=I(0)\exp\big[-\int^t_0 f(\tau)\,d\tau\big].
  277   $$
  278   The original ordinary differential equation~(3) is hence solved by
  279   multiplying with the intagrating factor $I(t)$ and using the
  280   property~(4) of the integrating factor, giving the equation
  281   $$
  282     {{d}\over{dt}}[I(t)\rho]=I(t)g(t),
  283   $$
  284   from which we hence obtain the solution for $\rho(t)$ as
  285   $$
  286     \rho(t)={{1}\over{I(t)}}\int^t_0 I(\tau)g(\tau)\,d\tau.
  287   $$
  288   From this preliminary discussion we may anticipate that equations of motion
  289   for the various order perturbation terms of the density operator can be
  290   solved in a similar manner, using integrating factors. However, it should
  291   be kept in mind that we here are dealing with {\sl operators} and not
  292   classical quantities, and since we do not know if the integrating factor
  293   is to be multiplied from left or right.
  294   
  295   In order not to loose any generality, we may look for a set of two
  296   integrating factors $\hat{V}_0(t)$ and $\hat{U}_0(t)$, in operator sense,
  297   that we left and right multiply the unknown terms of the $n$th order
  298   equation by, and we require these operators to have the effective impact
  299   $$
  300     \hat{V}_0(t)\bigg\{i\hbar{{d\hat{\rho}_n(t)}\over{dt}}
  301       -[\hat{H}_0,\hat{\rho}_n(t)]\bigg\}\hat{U}_0(t)
  302       =i\hbar{{d}\over{dt}}[\hat{V}_0(t)\hat{\rho}_n(t)\hat{U}_0(t)].
  303     \eqno{(5)}
  304   $$
  305   By carrying out the differentiation in the right-hand side, expanding
  306   the commutator in the left hand side, and rearranging terms, one then
  307   obtains the equation
  308   $$
  309     \bigg\{i\hbar{{d}\over{dt}}\hat{V}_0(t)+\hat{V}_0(t)\hat{H}_0\bigg\}
  310       \hat{\rho}_n(t)\hat{U}_0(t)+\hat{V}_0(t)\hat{\rho}_n(t)
  311       \bigg\{i\hbar{{d}\over{dt}}\hat{U}_0(t)-\hat{H}_0\hat{U}_0(t)\bigg\}=0
  312   $$
  313   for the operators $\hat{V}_0(t)$ and $\hat{U}_0(t)$. This equation clearly
  314   is satisfied if both of the braced expressions simultaneously are zero for
  315   all times, in other words, if the so-called {\sl time-development operators}
  316   $\hat{V}_0(t)$ and $\hat{U}_0(t)$ are chosen to satisfy
  317   $$
  318     \eqalign{
  319       &i\hbar{{d\hat{V}_0(t)}\over{dt}}+\hat{V}_0(t)\hat{H}_0=0,\cr
  320       &i\hbar{{d\hat{U}_0(t)}\over{dt}}-\hat{H}_0\hat{U}_0(t)=0,\cr
  321     }
  322   $$
  323   with solutions
  324   $$
  325     \eqalign{
  326       \hat{U}_0(t)&=\exp(-i\hat{H}_0 t/\hbar),\cr
  327       \hat{V}_0(t)&=\exp(i\hat{H}_0 t/\hbar)=\hat{U}_0(-t).\cr
  328     }
  329   $$
  330   In these expressions, the exponentials are to be regarded as being defined
  331   by their series expansion.
  332   In particular, each term of the series expansion contains an operator part
  333   being a power of the thermal equilibrium Hamiltonian $\hat{H}_0$, which
  334   commute with any of the other powers.
  335   We may easily verify that the obtained solutions, in a strict operator sense,
  336   satisfy the relations
  337   $$
  338     \hat{U}_0(t)\hat{U}_0(t')=\hat{U}_0(t+t'),
  339   $$
  340   with, in particular, the corollary
  341   $$
  342     \hat{U}_0(t)\hat{U}_0(-t)=\hat{U}_0(0)=1.
  343   $$
  344   Let us now again turn our attention to the original equation of motion
  345   that was the starting point for this discussion.
  346   By multiplying the $n$th order subequation of Eq.~(2) with $\hat{U}_0(-t)$
  347   from the left, and multiplying with $\hat{U}_0(t)$ from the right, we
  348   by using the relation~(5) obtain
  349   $$
  350     i\hbar{{d}\over{dt}}\bigg\{\hat{U}_0(-t)\hat{\rho}_n(t)\hat{U}_0(t)\bigg\}
  351     =\hat{U}_0(-t)[\hat{H}_{\rm I}(t),\hat{\rho}_{n-1}(t)]\hat{U}_0(t),
  352   $$
  353   which is integrated to yield the solution
  354   $$
  355     \hat{U}_0(-t)\hat{\rho}_n(t)\hat{U}_0(t)
  356     ={{1}\over{i\hbar}}\int^t_{-\infty}\hat{U}_0(-\tau)
  357       [\hat{H}_{\rm I}(\tau),\hat{\rho}_{n-1}(\tau)]\hat{U}_0(\tau)\,d\tau,
  358   $$
  359   where the lower limit of integration was fixed in accordance with
  360   the initial condition $\hat{\rho}_n(-\infty)=0$, $n=1,2,\ldots\,$.
  361   In some sense, we may consider the obtained solution as being the
  362   end point of this discussion; however, we may simplify the expression
  363   somewhat by making a few notes on the properties of the time development
  364   operators.
  365   By expanding the right hand side of the solution, and inserting
  366   $\hat{U}_0(\tau)\hat{U}_0(-\tau)=1$ between $\hat{H}_{\rm I}(\tau)$
  367   and $\hat{\rho}_{n-1}(\tau)$ in the two terms, we obtain
  368   $$
  369     \eqalign{
  370       \hat{U}_0(-t)\hat{\rho}_n(t)\hat{U}_0(t)
  371       &={{1}\over{i\hbar}}\int^t_{-\infty}\hat{U}_0(-\tau)
  372         [\hat{H}_{\rm I}(\tau)\hat{\rho}_{n-1}(\tau)
  373         -\hat{\rho}_{n-1}(\tau)\hat{H}_{\rm I}(\tau)]\hat{U}_0(\tau)\,d\tau\cr
  374       &={{1}\over{i\hbar}}\int^t_{-\infty}
  375         \hat{U}_0(-\tau)\hat{H}_{\rm I}(\tau)\underbrace{\hat{U}_0(\tau)
  376         \hat{U}_0(-\tau)}_{=1}\hat{\rho}_{n-1}(\tau)\hat{U}_0(\tau)\,d\tau\cr
  377       &\qquad\qquad-{{1}\over{i\hbar}}\int^t_{-\infty}
  378         \hat{U}_0(-\tau)\hat{\rho}_{n-1}(\tau)\underbrace{\hat{U}_0(\tau)
  379         \hat{U}_0(-\tau)}_{=1}\hat{H}_{\rm I}(\tau)\hat{U}_0(\tau)\,d\tau\cr
  380       &={{1}\over{i\hbar}}\int^t_{-\infty}
  381         [\underbrace{\hat{U}_0(-\tau)\hat{H}_{\rm I}(\tau)\hat{U}_0(\tau)}_{
  382           \equiv\hat{H}'_{\rm I}(t)},
  383         \underbrace{\hat{U}_0(-\tau)\hat{\rho}_{n-1}(\tau)\hat{U}_0(\tau)}_{
  384           \equiv\hat{\rho}'_{n-1}(t)}]\,d\tau,\cr
  385     }
  386   $$
  387   and hence, by introducing the primed notation in the {\sl interaction picture}
  388   for the quantum mechanical operators,
  389   $$
  390     \eqalign{
  391       \hat{\rho}'_n(t)&=\hat{U}_0(-t)\hat{\rho}_n(t)\hat{U}_0(t),\cr
  392       \hat{H}'_{\rm I}(t)&=\hat{U}_0(-t)\hat{H}_{\rm I}(t)\hat{U}_0(t),\cr
  393     }
  394   $$
  395   the solutions of the system of equations for the terms of the perturbation
  396   series for the density operator {\sl in the interaction picture} take the
  397   simplified form
  398   $$
  399     \hat{\rho}'_n(t)={{1}\over{i\hbar}}\int^t_{-\infty}
  400       [\hat{H}'_{\rm I}(\tau),\hat{\rho}'_{n-1}(\tau)]\,d\tau,
  401     \qquad n=1,2,\ldots,
  402   $$
  403   with the variuos order solutions expressed in the original Schr\"{o}dinger
  404   picture by means of the inverse transformation
  405   $$\hat{\rho}_n(t)=\hat{U}_0(t)\hat{\rho}'_n(t)\hat{U}_0(-t).$$
  406   
  407   \section{The first order polarization density}
  408   With the quantum mechanical perturbative description of the interaction
  409   between light and matter in fresh mind, we are now in the position of
  410   formulating the polarization density of the medium from a quantum mechanical
  411   description. A minor note should though be made regarding the Hamiltonian,
  412   which now is expressed in the interaction picture, and hence the electric
  413   dipolar operator (since the electric field here is considered to be a
  414   macroscopic, classical quantity) is given in the interaction picture as
  415   well,
  416   $$
  417     \eqalign{
  418       \hat{H}'_{\rm I}(\tau)&=\hat{U}_0(-\tau)\underbrace{
  419         [-\hat{Q}_{\alpha}E_{\alpha}(\tau)]}_{=\hat{H}_{\rm I}(\tau)}
  420         \hat{U}_0(\tau)\cr
  421       &=-\hat{U}_0(-\tau)\hat{Q}_{\alpha}\hat{U}_0(\tau)E_{\alpha}(\tau)\cr
  422       &=-\hat{Q}_{\alpha}(\tau)E_{\alpha}(\tau)\cr
  423     }
  424   $$
  425   where $\hat{Q}_{\alpha}(\tau)$ denotes the electric dipolar operator of the
  426   ensemble, taken {\sl in the interaction picture}.
  427   \vfill\eject
  428   
  429   By inserting the expression for the first order term of the
  430   perturbation series for the density operator into the quantum mechanical
  431   trace of the first order electric polarization density of the medium,
  432   one obtains
  433   $$
  434     \eqalign{
  435       P^{(1)}_{\mu}({\bf r},t)&={{1}\over{V}}\Tr[\hat{\rho}_1(t)\hat{Q}_{\mu}]\cr
  436         &={{1}\over{V}}\Tr\Big[
  437           \underbrace{\Big(\hat{U}_0(t)
  438             \underbrace{{{1}\over{i\hbar}}\int^t_{-\infty}
  439               [\hat{H}'_{\rm I}(\tau),\hat{\rho}_0]\,d\tau
  440             }_{=\hat{\rho}'_1(t)}\hat{U}_0(-t)\Big)
  441           }_{=\hat{\rho}_1(t)}
  442           \hat{Q}_{\mu}\Big]\cr
  443         &=\Bigg\{
  444           \matrix{
  445               E_{\mu}(\tau){\rm\ is\ a\ classical\ field
  446               \ (omit\ space\ dependence\ {\bf r})},\cr
  447               [\hat{H}'_{\rm I}(\tau),\hat{\rho}_0]
  448                =[-\hat{Q}_{\alpha}(\tau)E_{\alpha}(\tau),\hat{\rho}_{0}]
  449                =-E_{\alpha}(\tau)[\hat{Q}_{\alpha}(\tau),\hat{\rho}_{0}]
  450           }
  451           \Bigg\}\cr
  452         &=-{{1}\over{V i\hbar}}\Tr\Big\{
  453           \hat{U}_0(t)
  454             \int^t_{-\infty} E_{\alpha}(\tau)
  455               [\hat{Q}_{\alpha}(\tau),\hat{\rho}_0]\,d\tau\,
  456               \hat{U}_0(-t)\,
  457           \hat{Q}_{\mu}\Big\}\cr
  458         &=\{{\rm Pull\ out\ }E_{\alpha}(\tau)
  459             {\rm\ and\ the\ integral\ outside\ the\ trace}\}\cr
  460         &=-{{1}\over{V i\hbar}}
  461           \int^t_{-\infty} E_{\alpha}(\tau)\Tr\{
  462           \hat{U}_0(t)[\hat{Q}_{\alpha}(\tau),\hat{\rho}_0]\hat{U}_0(-t)\,
  463           \hat{Q}_{\mu}\}\,d\tau\cr
  464         &=\Bigg\{{\rm Express\ }E_{\alpha}(\tau){\rm\ in\ frequency\ domain,\ }
  465             E_{\alpha}(\tau)=\int^{\infty}_{-\infty}E_{\alpha}(\omega)
  466             \exp(-i\omega\tau)\,d\omega\Bigg\}\cr
  467         &=-{{1}\over{V i\hbar}}
  468           \int^{\infty}_{-\infty} \int^t_{-\infty}
  469           E_{\alpha}(\omega)\Tr\{
  470           \hat{U}_0(t)[\hat{Q}_{\alpha}(\tau),\hat{\rho}_0]\hat{U}_0(-t)\,
  471           \hat{Q}_{\mu}\}\exp(-i\omega\tau)\,d\tau\,d\omega\cr
  472         &=\{{\rm Use\ }\exp(-i\omega\tau)=\exp(-i\omega t)
  473             \exp[-i\omega(\tau-t)]\}\cr
  474         &=-{{1}\over{V i\hbar}}
  475           \int^{\infty}_{-\infty} \int^t_{-\infty}
  476           E_{\alpha}(\omega)\Tr\{
  477           \hat{U}_0(t)[\hat{Q}_{\alpha}(\tau),\hat{\rho}_0]\hat{U}_0(-t)\,
  478           \hat{Q}_{\mu}\}
  479      \cr&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\times
  480           \exp[-i\omega(\tau-t)]\,d\tau\,\exp(-i\omega t)\,d\omega\cr
  481         &=\varepsilon_0\int^{\infty}_{-\infty}
  482           \chi^{(1)}_{\mu\alpha}(-\omega;\omega)E_{\alpha}(\omega)
  483           \exp(-i\omega t)\,d\omega,\cr
  484     }
  485   $$
  486   where the first order (linear) electric susceptibility is defined as
  487   $$
  488     \eqalign{
  489       \chi^{(1)}_{\mu\alpha}(-\omega;\omega)
  490         &=-{{1}\over{\varepsilon_0 V i\hbar}}\int^t_{-\infty}
  491           \Tr\{\underbrace{\hat{U}_0(t)[\hat{Q}_{\alpha}(\tau),\hat{\rho}_0]
  492             \hat{U}_0(-t)}_{=[\hat{Q}_{\alpha}(\tau-t),\hat{\rho}_0]}\,
  493           \hat{Q}_{\mu}\}\exp[-i\omega(\tau-t)]\,d\tau\cr
  494         &=\Bigg\{\matrix{{\rm Expand\ the\ commutator\ and\ insert\ }
  495                          \hat{U}_0(-t)\hat{U}_0(t)\equiv 1\cr
  496                    {\rm\ in\ middle\ of\ each\ of\ the\ terms,\ using\ }
  497                          [\hat{U}_0(t),\hat{\rho}_0]=0\cr
  498                    \Rightarrow\hat{U}_0(t)[\hat{Q}_{\alpha}(\tau),\hat{\rho}_0]
  499                          \hat{U}_0(-t)=[\hat{Q}_{\alpha}(\tau-t),\hat{\rho}_0]}
  500           \Bigg\}\cr
  501         &=-{{1}\over{\varepsilon_0 V i\hbar}}\int^t_{-\infty}
  502           \Tr\{[\hat{Q}_{\alpha}(\tau-t),\hat{\rho}_0]\hat{Q}_{\mu}\}
  503           \exp[-i\omega(\tau-t)]\,d\tau\cr
  504         &=\Bigg\{{\rm Change\ variable\ of\ integration\ }\tau'=\tau-t;
  505           \int^t_{-\infty}\cdots d\tau\to\int^0_{-\infty}\cdots d\tau'\Bigg\}\cr
  506         &=-{{1}\over{\varepsilon_0 V i\hbar}}\int^0_{-\infty}
  507           \underbrace{\Tr\{[\hat{Q}_{\alpha}(\tau'),\hat{\rho}_0]
  508             \hat{Q}_{\mu}\}}_{
  509               =\Tr\{\hat{\rho}_0[\hat{Q}_{\mu},\hat{Q}_{\alpha}(\tau')]\}}
  510           \exp(-i\omega\tau')\,d\tau'\cr
  511         &=\{{\rm Cyclic\ permutation\ of\ the\ arguments\ in\ the\ trace}\}\cr
  512         &=-{{1}\over{\varepsilon_0 V i\hbar}}\int^0_{-\infty}
  513           \Tr\{\hat{\rho}_0[\hat{Q}_{\mu},\hat{Q}_{\alpha}(\tau)]\}
  514           \exp(-i\omega\tau)\,d\tau.\cr
  515     }
  516   $$
  517   \bye
  518   

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Last modified Wednesday 16 Nov 2011