Absorption and Stimulated Emission of Radiation

Let us employ time-dependent perturbation theory to investigate the interaction of an electron in a hydrogen-like atom with classical (i.e., non-quantized) electromagnetic radiation. The Hamiltonian of such an electron is

$$H = \frac{\left({\bf p} + e\, {\bf A}\right)^{\,2} }{2\,m_e} - e \,\phi+ {\mit\Phi}(r),$$ (8.122)

where ${\mit\Phi}(r)$ is the atomic potential energy, and ${\bf A}({\bf x})$ and $\phi({\bf x})$ are the vector and scalar potentials associated with the incident radiation. (See Section 3.6.) The previous equation can be written

$$H = \frac{ \left(p^{\,2} +e \,{\bf A}\cdot {\bf p} +e \,{\bf p}\cdot{\bf A} + e^{\,2} A^{\,2}\right)}{2\,m_e}- e \,\phi + {\mit\Phi}(r).$$ (8.123)

Now,

$${\bf p}\cdot {\bf A} = {\bf A}\cdot {\bf p},$$ (8.124)

provided that we adopt the Coulomb gauge, $\nabla\cdot{\bf A} = 0$ . (See Exercise 6.) Hence,

$$H = \frac{p^{\,2}}{2\,m_e} +\frac{e\,{\bf A}\cdot{\bf p}}{m_e} +\frac{ e^{\,2} A^{\,2}}{2\,m_e}- e\, \phi + {\mit\Phi}(r).$$ (8.125)

Suppose that the perturbation corresponds to a monochromatic plane-wave of angular frequency $\omega$ , for which [49]

$$\phi =0,$$ (8.126)

$${\bf A} = 2\, A_0 \, \mbox{\boldmath$\epsilon$} \,\cos\left (\frac{\omega}{c} \,{\bf n}\cdot{\bf x} - \omega\, t\right),$$ (8.127)

where $\epsilon$ and ${\bf n}$ are unit vectors that specify the wave's electric polarization direction (i.e., the direction of its electric component) and its direction of propagation, respectively. [The wavevector is ${\bf k}=(\omega/c)\,{\bf n}$ .] Here, $c$ is the velocity of light in vacuum. Note that, according to standard electromagnetic theory, $\epsilon$ $\cdot{\bf n} = 0$ [49]. The Hamiltonian becomes

$$H = H_0 + H_1(t),$$ (8.128)

with

$$H_0 = \frac{p^{\,2}}{2\,m_e} + {\mit\Phi}(r),$$ (8.129)

and

$$H_1 \simeq \frac{e\,{\bf A}\cdot{\bf p}}{m_e},$$ (8.130)

where the term involving $A^{\,2}$ , which is second order in $A_0$ , has been neglected.

The perturbing Hamiltonian can be written

$$H_1 = \frac{e \,A_0}{m_e} \left(\exp\left[\,{\rm i}\left(\frac{\o... ...frac{\omega}{c}\right) {\bf n}\cdot{\bf x} + {\rm i}\, \omega\, t\right]\right) \mbox{\boldmath$\epsilon$} \cdot{\bf p}.$$ (8.131)

This has the same form as Equation (8.113), provided that

$$V = \frac{e \,A_0}{m_e}\, \exp\left[-{\rm i}\left(\frac{\omega}{c}\right) {\bf n}\cdot{\bf x}\right] \mbox{\boldmath$\epsilon$} \cdot{\bf p}.$$ (8.132)

It is clear, by analogy with the analysis of Section 8.8, that the first term on the right-hand side of Equation (8.134) describes a process by which the atom absorbs energy $\hbar\,\omega$ from the electromagnetic field. On the other hand, the second term describes a process by which the atom emits energy $\hbar\,\omega$ to the electromagnetic field. We can interpret the former and latter processes as the absorption and stimulated emission, respectively, of a photon of energy $\hbar\,\omega$ by the atom.

It is convenient to define

$${\bf d}_{if} = \frac{-{\rm i}}{m_e\,\omega_{if}}\left\langle i\le... ...\omega}{c}\right){\bf n}\cdot{\bf x}\right] {\bf p} \right\vert f\right\rangle,$$ (8.133)

which has the dimensions of length. Note that

$$\mbox{\boldmath$\epsilon$} \cdot {\bf d}_{if} = \frac{-{\rm i}}{e\,A_0\,\omega_{if}}\,\langle i\vert\,V^{\,\dag }\,\vert f\rangle.$$ (8.134)

It follows, from Equations (8.122) and (8.123), that the rates of absorption and stimulated emission are

$$w_{i\rightarrow f}^{\,\rm abs} = 2\pi\, \frac{e^{\,2}\,\omega_{fi... ...ldmath$\epsilon$}\cdot{\bf d}_{fi}\right\vert^{\,2} \delta(\omega-\omega_{fi}),$$ (8.135)

and

$$w_{i\rightarrow f}^{\,\rm stm} = 2\pi\, \frac{e^{\,2}\,\omega_{if... ...ldmath$\epsilon$}\cdot{\bf d}_{if}\right\vert^{\,2} \delta(\omega-\omega_{if}),$$ (8.136)

respectively. (See Exercise 19.) It can be seen that absorption involves the absorption by the atom of a photon of angular frequency

$$\omega_{fi} = \frac{E_f-E_i}{\hbar},$$ (8.137)

and energy $\hbar\,\omega_{fi} =E_f-E_i$ , causing a transition from an atomic state with an initial energy $E_i$ to a state with a final energy $E_f>E_i$ . On the other hand, stimulated emission involves the emission by the atom of a photon of angular frequency

$$\omega_{if} = \frac{E_i-E_f}{\hbar},$$ (8.138)

and energy $\hbar\,\omega_{if}= E_i-E_f$ , causing a transition from an atomic state with an initial energy $E_i$ to a state with a final energy $E_f<E_i$ . In both cases, ${\bf n}$ and $\epsilon$ specify the direction of propagation and electric polarization direction, respectively, of the photon. For the case of stimulated emission, ${\bf n}$ and $\epsilon$ also specify the direction of propagation and polarization direction, respectively, of the radiation that stimulates the atomic transition.

Now, the energy density of an electromagnetic radiation field is [49]

$$U = \frac{1}{2}\left(\frac{\epsilon_0\,E_0^{\,2}}{2}+ \frac{B_0^{\,2}}{2\,\mu_0} \right),$$ (8.139)

where $E_0$ and $B_0=E_0/c= 2\,A_0\,\omega/c$ are the peak electric and magnetic field-strengths, respectively. Hence,

$$U = 2\,\epsilon_0\,\omega^{\,2}\,\vert A_0\vert^{\,2},$$ (8.140)

and expressions (8.138) and (8.139) become

$$w_{i\rightarrow f}^{\,\rm abs} = 4\pi^{\,2}\, \alpha\,\frac{c}{\h... ...oldmath$\epsilon$}\cdot{\bf d}_{fi}\right\vert^{\,2}\delta(\omega-\omega_{fi}),$$ (8.141)

and

$$w_{i\rightarrow f}^{\,\rm stm} = 4\pi^{\,2} \,\alpha\,\frac{c}{\h... ...oldmath$\epsilon$}\cdot{\bf d}_{if}\right\vert^{\,2}\delta(\omega-\omega_{if}),$$ (8.142)

respectively, where $\alpha$ is the fine structure constant.

Let us suppose that the incident radiation has a range of different angular frequencies, so that

$$U = \int_{-\infty}^\infty d\omega\,u(\omega),$$ (8.143)

where $u(\omega)\,d\omega$ is the energy density of radiation whose angular frequency lies in the range $\omega$ to $\omega+d\omega$ . Equations (8.144) and (8.145) imply that

$$\frac{dw_{i\rightarrow f}^{\,\rm abs}}{d\omega} = 4\pi^{\,2} \,\a... ...oldmath$\epsilon$}\cdot{\bf d}_{fi}\right\vert^{\,2}\delta(\omega-\omega_{fi}),$$ (8.144)

and

$$\frac{dw_{i\rightarrow f}^{\,\rm stm}}{d\omega} = 4\pi^{\,2} \,\a... ...oldmath$\epsilon$}\cdot{\bf d}_{if}\right\vert^{\,2}\delta(\omega-\omega_{if}).$$ (8.145)

Here, $(dw_{i\rightarrow f}^{\,\rm abs}/d\omega)\,d\omega$ is the rate of absorption associated with radiation whose angular frequency lies in the range $\omega$ to $\omega+d\omega$ , et cetera. Incidentally, we are assuming that the radiation is incoherent, so that intensities can be added [64]. Of course,

$$w_{i\rightarrow f}^{\,\rm abs} =\int_{-\infty}^\infty d\omega\, \frac{dw_{i\rightarrow f}^{\,\rm... ...ga_{fi})\left\vert\mbox{\boldmath$\epsilon$}\cdot{\bf d}_{fi}\right\vert^{\,2},$$ (8.146)

$$w_{i\rightarrow f}^{\,\rm stm} =\int_{-\infty}^\infty d\omega\, \frac{dw_{i\rightarrow f}^{\,\rm... ...ga_{if})\left\vert\mbox{\boldmath$\epsilon$}\cdot{\bf d}_{if}\right\vert^{\,2}.$$ (8.147)

The rate at which the atom gains energy from the electromagnetic field as a consequence of absorption can be written

$$P^{\,\rm abs}_{i\rightarrow f} = \int_{-\infty}^\infty d\omega\,\... ...y}^\infty d\omega\,\,c\,u(\omega)\,\sigma_{i\rightarrow f}^{\,\rm abs}(\omega),$$ (8.148)

where $\sigma_{i\rightarrow f}^{\,\rm abs}$ is the so-called absorption cross-section, and $c\,u(\omega)\,d\omega$ the electromagnetic energy flux associated with radiation whose angular frequency lies in the range $\omega$ to $\omega+d\omega$ [49]. It follows that

$$\sigma_{i\rightarrow f}^{\,\rm abs}(\omega)=4\pi^{\,2}\,\alpha\,\... ...oldmath$\epsilon$}\cdot{\bf d}_{fi}\right\vert^{\,2}\delta(\omega-\omega_{fi}).$$ (8.149)

Similarly, the stimulated emission cross-section takes the form

$$\sigma_{i\rightarrow f}^{\,\rm stm}(\omega)=4\pi^{\,2}\,\alpha\,\... ...oldmath$\epsilon$}\cdot{\bf d}_{if}\right\vert^{\,2}\delta(\omega-\omega_{if}).$$ (8.150)

Finally, comparing Equations (8.149) and (8.150) with the previous two equations, we deduce that

$$w_{i\rightarrow f}^{\,\rm abs} =\int_{-\infty}^\infty d\omega\,c\,n(\omega)\,\sigma_{\i\rightarrow f}^{\,\rm abs}(\omega),$$ (8.151)

$$w_{i\rightarrow f}^{\,\rm stm} =\int_{-\infty}^\infty d\omega\,c\,n(\omega)\,\sigma_{i\rightarrow f}^{\,\rm stm}(\omega).$$ (8.152)

Here, we have written $u(\omega)=n(\omega)\,\hbar\,\omega$ , where $n(\omega)\,d\omega$ is the number density of photons whose angular frequencies lie in the range $\omega$ to $\omega+d\omega$ .