# 14.3: Application to Electromagnetic Perturbations

## First-Order Fermi-Wentzel "Golden Rule"

Using the earlier expressions for $$H^1_{int}$$ and for A(r,t)

$H^1_{int} = \sum\limits_j \left[ \left( \dfrac{ie\hbar}{m_ec} \right) \textbf{A}(r_j,t) \cdot{\nabla_j} \right] + \sum\limits_a \left[ \left( \dfrac{iZ_ae\hbar}{m_ac} \right) \textbf{A}(R_a,t) \cdot{\nabla_a} \right]$

and

$2\textbf{A}_o cos(\omega t - \textbf{k}\cdot{\textbf{r}}) = \textbf{A}_0 \left[ e^{i(\omega t - \textbf{k}\cdot{\textbf{r}})} + e^{-i(\omega t - \textbf{k}\cdot{\textbf{r}})} \right],$

it is relatively straightforward to carry out the above time integration to achieve a final expression for $$D_f^1(t)$$, which can then be substituted into $$C_f^1(t) = D_f^1(t) e^{(-\frac{-iE_f^0t}{\hbar})}$$ to obtain the final expression for the first-order estimate of the probability amplitude for the molecule appearing in the state $$\Phi_f e^{\frac{-iE_f^0t}{\hbar}}$$ after being subjected to electromagnetic radiation from t = 0 until t = T. This final expression reads:

$C_f^1(T) = \dfrac{1}{i\hbar} e^{\dfrac{-iE_f^0T}{\hbar}} \left[ \langle\Phi_f|\sum\limits_j \left[ \left( \dfrac{ie\hbar}{m_ec} \right) e^{-i\textbf{k}\cdot{\textbf{r}_j}}\textbf{A}_0\cdot{\nabla_j} + \sum\limits_a \left( \dfrac{iZ_ae\hbar}{m_ac} \right) e^{-i\textbf{k}\cdot{R}_a}\textbf{A}_0\cdot{\nabla_a}|\Phi_i \rangle \right] \dfrac{e^{i(\omega + \omega_{f,i})T}-1}{i(\omega + \omega_{f,i})} \right]$

$+ \dfrac{1}{i\hbar} e^{\dfrac{-iE_f^0T}{\hbar}}\left[\langle\Phi_f|\sum\limits_j \left[ \left( \dfrac{ie\hbar}{m_ec} \right) e^{i\textbf{k}\cdot{\textbf{r}_j}}\textbf{A}_0\cdot{\nabla_j} + \sum\limits_a \left( \dfrac{iZ_ae\hbar}{m_ac} \right) e^{i\textbf{k}\cdot{R}_a}\textbf{A}_0\cdot{\nabla_a}|\Phi_i \rangle \right] \dfrac{e^{i(-\omega + \omega_{f,i})T}-1}{i(-\omega + \omega_{f,i})} \right]$

where

$\omega_{f,i} = \dfrac{[E_f^0 - E_i^0]}{\hbar}$

is the resonance frequency for the transition between "initial" state $$\Phi_i \text{ and "final" state } \Phi_f$$

Defining the time-independent parts of the above expression as

$\alpha_{f,i} = \langle \Phi_f |\sum\limits_j \left[ \left( \dfrac{e}{m_ec} \right) e^{-i\textbf{k}\cdot{\textbf{r}_j}}\textbf{A}_0\cdot{\nabla_j} + \sum\limits_a \left( \dfrac{Z_ae}{m_ac} \right) e^{-i\textbf{k}\cdot{\textbf{R}_a}}\textbf{A}_0\cdot{\nabla_a}|\Phi_i \rangle, \right]$

this result can be written as

$C_f^1(T) = e^{\dfrac{-iE_f^0T}{\hbar}}\left[ \alpha_{f,i}\dfrac{e^{i(\omega+\omega_{f,i})T}-1}{i(\omega+\omega_{f,i})} + \alpha^{\text{*}}_{f,i}\dfrac{e^{-i(\omega - \omega_{f,i})T}-1}{-i(\omega-\omega_{f,i})} \right].$

The modulus squared $$|C_f^1(T)|^2$$ gives the probability of finding the molecule in the final state $$\Phi_f$$ at time T, given that it was in $$\Phi_i$$ at time t = 0. If the light's frequency $$\omega$$ is tuned close to the transition frequency $$\omega_{f,i}$$ of a particular transition, the term whose denominator contains $$(\omega - \omega_{f,i})$$ will dominate the term with $$(\omega + \omega_{f,i})$$ in its denominator. Within this "near-resonance" condition, the above probability reduces to:

$|C_f^1|^2 = 2|\alpha_{f,i}|^2 \dfrac{1-cos(\omega - \omega_{f,i})T}{(\omega - \omega_{f,i})^2}$

$= 4|\alpha_{f,i}|^2\dfrac{sin^2(1/2(\omega - \omega_{f,i})T)}{(\omega - \omega_{f,i})^2}.$

This is the final result of the first-order time-dependent perturbation theory treatment of light-induced transitions between states $$\Phi_i \text{ and } \Phi_f$$.

The so-called sinc-function

$\dfrac{sin^2(1/2(\omega - \omega_{f,i})T)}{(\omega - \omega_{f,i})^2}$

as shown in the figure below is strongly peaked near $$\omega = \omega_{f,i}$$, and displays secondary maxima (of decreasing amplitudes) near $$\omega = \omega_{f,i} + 2\frac{n\pi}{T} , n = 1, 2$$, ... . In the $$T \rightarrow \infty$$ limit, this function becomes narrower and narrower, and the area under it

$\int\limits_{-\infty}^{\infty} \dfrac{sin^2(1/2(\omega - \omega_{f,i})T)}{(\omega - \omega_{f,i})^2}d\omega = \dfrac{T}{2}\int\limits_{-\infty}^{\infty} \dfrac{sin^2(1/2(\omega - \omega_{f,i})T)}{1/4T^2(\omega - \omega_{f,i})^2}d\left(\omega \dfrac{T}{2}\right) = \dfrac{T}{2}\int\limits_{-\infty}^{\infty} \dfrac{sin^2(x)}{x^2} = \pi\dfrac{T}{2}$

grows with T. Physically, this means that when the molecules are exposed to the light source for long times (large T), the sinc function emphasizes $$\omega$$ values near $$\omega_{f,i}$$ (i.e., the on-resonance $$\omega$$ values). These properties of the sinc function will play important roles in what follows.

In most experiments, light sources have a "spread" of frequencies associated with them; that is, they provide photons of various frequencies. To characterize such sources, it is common to introduce the spectral source function g($$\omega$$) d$$\omega$$ which gives the probability that the photons from this source have frequency somewhere between $$\omega \text{ and } \omega+d\omega$$. For narrow-band lasers, g($$\omega)$$ is a sharply peaked function about some "nominal" frequency $$\omega_o$$; broader band light sources have much broader g($$\omega$$) functions.

When such non-monochromatic light sources are used, it is necessary to average the above formula for $$|C_f^1(T)|^2$$ over the g($$\omega$$) d$$\omega$$ probability function in computing the probability of finding the molecule in state $$\Phi_f$$ after time T, given that it was in $$\Phi_i$$ up until t = 0, when the light source was turned on. In particular, the proper expression becomes:

$|C_f^1(T)|^2_{ave} = 4|\alpha_{fi}|^2 \int\limits g(\omega) \dfrac{sin^2(1/2(\omega - \omega_{f,i})T)}{(\omega - \omega_{f,i})^2}d\omega$

$= 2|\alpha_{f,i}|^2 T \int\limits_{-\infty}^{\infty} g(\omega) \dfrac{2in^2(1/2(\omega - \omega_{f,i})T)}{1/4T^2(\omega - \omega_{f,i})^2}d\left( \omega\dfrac{T}{2}\right)$

If the light-source function is "tuned" to peak near $$\omega = \omega_{f,i}$$ and if $$g(\omega)$$ is much broader (in $$\omega$$-space) than the $$\dfrac{sin^2(1/2(\omega - \omega_{f,i})T)}{(\omega - \omega_{f,i})^2}$$ function, g($$\omega$$) can be replaced by its value at the peak of the $$\dfrac{sin^2(1/2(\omega - \omega_{f,i})T)}{(\omega - \omega_{f,i})^2}$$ function, yielding:

$|C_f^1(T)_{ave} = 2g(\omega_{f,i})|\alpha_{f,i}|^2T \int\limits^{\infty}_{-\infty}\dfrac{sin^2(1/2(\omega - \omega_{f,i})T}{1/4T^2(\omega - \omega_{f,i})^2}d\left( \omega\dfrac{T}{2} \right) = 2g(\omega_{f,i})|\alpha_{f,i}|^2 T\int\limits_{-\infty}^{\infty} \dfrac{sin^2(x)}{x^2}dx = 2\pi g(\omega_{f,i})|\alpha_{f,i}|^2T.$

The fact that the probability of excitation from $$\Phi_i \text{ to } \Phi_f$$ grows linearly with the time T over which the light source is turned on implies that the rate of transitions between these two states is constant and given by:

$\textbf{R}_{i,f} = 2\pi g(\omega_{f,i})|\alpha_{f,i}|^2;$

this is the so-called first-order Fermi-Wentzel "golden rule" expression for such transition rates. It gives the rate as the square of a transition matrix element between the two states involved, of the first order perturbation multiplied by the light source function $$g(\omega)$$ evaluated at the transition frequency $$\omega_{f,i}$$.

## Higher Order Results

Solution of the second-order time-dependent perturbation equations,

$i\hbar\dfrac{\partial \Psi^2}{\partial t} = (H^0\Psi^2 + H^2_{int}\Psi^0 + H^1_{int}\Psi^1)$

which will not be treated in detail here, gives rise to two distinct types of contributions to the transition probabilities between $$\Phi_i \text{ and } \Phi_f$$:

There will be matrix elements of the form

$\langle \Phi_f | \sum\limits_j \left[ \left( \dfrac{e^2}{2m_ec^2} \right)| \textbf{A}(\textbf{r}_j,t)|^2 \right] + \sum\limits_a\left[ \left( \dfrac{Z_a^2e^2}{2m_ac^2} \right)|\textbf{A}(R_a,t)|^2 \right]|\Phi_i \rangle$

arising when $$H^2_{int} \text{ couples } \Phi_i \text{ to } \Phi_f$$.

There will be matrix elements of the form

$\sum\limits_k <\Phi_f |\sum\limits_j \left[ \left( \dfrac{ie\hbar}{m_ec} \right)\textbf{A}(r_j,t)\cdot{\nabla_j} \right] + \sum\limits_a \left[ \left( \dfrac{iZ_ae\hbar}{m_ac} \right)\textbf{A}(R_a,t)\cdot{\nabla_a} \right]| \Phi_k \rangle$

$\langle\Phi_k |\sum\limits_j \left[ \left( \dfrac{ie\hbar}{m_ec} \right)\textbf{A}(r_j,t)\cdot{\nabla_j} \right] + \sum\limits_a \left[ \left( \dfrac{iZ_ae\hbar}{m_ac} \right)\textbf{A}(R_a,t)\cdot{\nabla_a} \right]| \Phi_i \rangle$

arising from expanding $$H^1_{int}\Psi^1 = \sum\limits_kC_k^1H^1_{int}|\Phi_k \rangle$$ and using the earlier result for the first-order amplitudes $$C_k^1$$. Because both types of second-order terms vary quadratically with the A(r,t) potential, and because A has time dependence of the form $$cos(\omega t - \textbf{k}\cdot{\textbf{r}})$$, these terms contain portions that vary with time as $$cos(2\omega t).$$ As a result, transitions between initial and final states $$\Phi_i \text{ and } \Phi_f$$ whose transition frequency is $$\omega_{f,i}$$ can be induced when $$2\omega = \omega_{f,i}$$; in this case, one speaks of coherent two-photon induced transitions in which the electromagnetic field produces a perturbation that has twice the frequency of the "nominal" light source frequency $$\omega$$.