7.8 Gen­eral In­ter­ac­tion with Ra­di­a­tion

Un­der typ­i­cal con­di­tions, a col­lec­tion of atoms is not just sub­jected to a sin­gle elec­tro­mag­netic wave, as de­scribed in the pre­vi­ous sec­tion, but to broad­band in­co­her­ent ra­di­a­tion of all fre­quen­cies mov­ing in all di­rec­tions. Also, the in­ter­ac­tions of the atoms with their sur­round­ings tend to be rare com­pared to the fre­quency of the ra­di­a­tion but fre­quent com­pared to the typ­i­cal life time of the var­i­ous ex­cited atomic states. In other words, the evo­lu­tion of the atomic states is col­li­sion-dom­i­nated. The ques­tion in this sub­sec­tion is what can be said about the emis­sion and ab­sorp­tion of ra­di­a­tion by the atoms un­der such con­di­tions.

Since both the elec­tro­mag­netic field and the col­li­sions are ran­dom, a sta­tis­ti­cal rather than a de­ter­mi­nate treat­ment is needed. In it, the prob­a­bil­ity that a ran­domly cho­sen atom can be found in a typ­i­cal atomic state $\psi_{\rm {L}}$ of low en­ergy will be called $P_{\rm {L}}$. Sim­i­larly, the prob­a­bil­ity that an atom can be found in an atomic state $\psi_{\rm {H}}$ of higher en­ergy will be called $P_{\rm {H}}$. More sim­plis­tic, $P_{\rm {L}}$ can be called the frac­tion of atoms in the low en­ergy state and $P_{\rm {H}}$ the frac­tion in the high en­ergy state.

The en­ergy of the elec­tro­mag­netic ra­di­a­tion, per unit vol­ume and per unit fre­quency range, will be in­di­cated by $\rho(\omega)$. The par­tic­u­lar fre­quency $\omega_0$ that is rel­e­vant to tran­si­tions be­tween two atomic states $\psi_{\rm {L}}$ and $\psi_{\rm {H}}$ is re­lated to the en­ergy dif­fer­ence be­tween the states. In par­tic­u­lar,

$$\omega_0 = (E_{\rm {H}} - E_{\rm {L}})/\hbar$$

is the nom­i­nal fre­quency of the pho­ton re­leased or ab­sorbed in a tran­si­tion be­tween the two states.

In those terms, the frac­tions $P_{\rm {L}}$ and $P_{\rm {H}}$ of atoms in the two states evolve in time ac­cord­ing to the evo­lu­tion equa­tions, {D.41},

$$\fbox{$\displaystyle \frac{{\rm d}P_{\rm{L}}}{{\rm d}t} = \mbox{} - ... ...{L}}} \rho(\omega_0)\; P_{\rm{H}} + A_{\rm{H\to{L}}}\; P_{\rm{H}} + \ldots\; $}$$ (7.45)

$$\fbox{$\displaystyle \frac{{\rm d}P_{\rm{H}}}{{\rm d}t} = \mbox{} + ... ...}}} \rho(\omega_0)\; P_{\rm{H}} - A_{\rm{H\to{L}}}\; P_{\rm{H}} + \ldots\; $}%$$ (7.46)

In the first equa­tion, the first term in the right hand side re­flects atoms that are ex­cited from the low en­ergy state to the high en­ergy state. That de­creases the num­ber of low en­ergy atoms, ex­plain­ing the mi­nus sign. The ef­fect is of course pro­por­tional to the frac­tion $P_{\rm {L}}$ of low en­ergy atoms that is avail­able to be ex­cited. It is also pro­por­tional to the en­ergy $\rho(\omega_0)$ of the elec­tro­mag­netic waves that do the ac­tual ex­cit­ing.

Sim­i­larly, the sec­ond term in the right hand side of the first equa­tion re­flects the frac­tion of low en­ergy atoms that is cre­ated through de-ex­ci­ta­tion of ex­cited atoms by the elec­tro­mag­netic ra­di­a­tion. The fi­nal term re­flects the low en­ergy atoms cre­ated by spon­ta­neous de­cay of ex­cited atoms. The con­stant $A_{\rm {H\to{L}}}$ is the spon­ta­neous emis­sion rate. (It is re­ally the de­cay rate $\lambda$ as de­fined ear­lier in sec­tion 7.5.3, but in the present con­text the term spon­ta­neous emis­sion rate and sym­bol $A$ tend to be used.)

The sec­ond equa­tion can be un­der­stood sim­i­larly as the first. If there are tran­si­tions with states other than $\psi_{\rm {L}}$ and $\psi_{\rm {H}}$, all their ef­fects should be summed to­gether; that is in­di­cated by the dots in (7.45) and (7.46).

The con­stants in the equa­tions are col­lec­tively re­ferred to as the “Ein­stein $A$ and $B$ co­ef­fi­cients.” Imag­ine that some big shot in en­gi­neer­ing was too lazy to se­lect ap­pro­pri­ate sym­bols for the quan­ti­ties used in a pa­per and just called them $A$ and $B$. Ref­er­ees and stan­dards com­mit­tees would be on his/her back, big shot or not. How­ever, in physics they still stick with the stu­pid sym­bols al­most a cen­tury later. At least in this con­text.

Any­way, the $B$ co­ef­fi­cients are, {D.41},

$$\fbox{$\displaystyle B_{\rm{L\to{H}}} = B_{\rm{H\to{L}}} =... ...vert e{\skew0\vec r}\vert\psi_{\rm{H}}\rangle\vert^2}{3} $} %$$ (7.47)

Here $\epsilon_0$ $\vphantom0\raisebox{1.5pt}{$=$}$ 8.854 19 10$\POW9,{-12}$ C$\POW9,{2}$/J m is the per­mit­tiv­ity of space. Note from the ap­pear­ance of the Planck con­stant that the emis­sion and ab­sorp­tion of ra­di­a­tion is truly a quan­tum ef­fect. The sec­ond ra­tio is the av­er­age atomic ma­trix el­e­ment dis­cussed in the pre­vi­ous sec­tion. The fact that $B_{\rm {L\to{H}}}$ equals $B_{\rm {H\to{L}}}$ re­flects that the elec­tric field is equally ef­fec­tive for ab­sorp­tion as for stim­u­lated emis­sion. It is a con­se­quence of the sym­me­try prop­erty of two-state sys­tems men­tioned in the pre­vi­ous sec­tion.

The spon­ta­neous emis­sion rate was found by Ein­stein us­ing a dirty trick, {D.42}. It is

$$\fbox{$\displaystyle A_{\rm{H\to{L}}} = B_{\rm{H\to{L}}} \... ...ho_{\rm{equiv}}(\omega) = \frac{\hbar\omega^3}{\pi^2c^3} $} %$$ (7.48)

One way of think­ing of the mech­a­nism of spon­ta­neous emis­sion is that it is an ef­fect of the ground state elec­tro­mag­netic field. Just like nor­mal par­ti­cle sys­tems still have nonzero en­ergy left in their ground state, so does the elec­tro­mag­netic field. You could there­fore think of this ground state elec­tro­mag­netic field as the source of the atomic per­tur­ba­tions that cause the atomic de­cay. If that pic­ture is right, then the term $\rho_{\rm {equiv}}$ in the ex­pres­sion above should be the en­ergy of the field in the ground state. In terms of the analy­sis of chap­ter 6.8, that would mean that in the ground state, there is ex­actly one pho­ton left in each ra­di­a­tion mode. Just drop the fac­tor (6.10) from (6.11).

It is a pretty rea­son­able de­scrip­tion, but it is not quite true. In the ground state of the elec­tro­mag­netic field there is half a pho­ton in each mode, not one. It is just like a har­monic os­cil­la­tor, which has half an en­ergy quan­tum $\hbar\omega$ left in its ground state, chap­ter 4.1. Also, a ground state en­ergy should not make a dif­fer­ence for the evo­lu­tion of a sys­tem. In­stead, be­cause of a twi­light ef­fect, the pho­ton that the ex­cited atom in­ter­acts with is the one that it will emit, ad­den­dum {A.24}.

As a spe­cial ex­am­ple of the given evo­lu­tion equa­tions, con­sider a closed box whose in­side is at ab­solute zero tem­per­a­ture. Then there is no am­bi­ent black­body ra­di­a­tion, $\rho$ $\vphantom0\raisebox{1.5pt}{$=$}$ 0. Now as­sume that ini­tially there is a thin gas of atoms in the box in an ex­cited state $\psi_{\rm {H}}$. These atoms will de­cay to what­ever are the avail­able atomic states of lower en­ergy. In par­tic­u­lar, ac­cord­ing to (7.46) the frac­tion $P_{\rm {H}}$ of ex­cited atoms left will evolve as

$$\frac{{\rm d}P_{\rm {H}}}{{\rm d}t} = - \left[ A_{\rm {H... ...\to{L}}_2} + A_{\rm {H\to{L}}_3} + \ldots \right] P_{\rm {H}}$$

where the sum is over all the lower en­ergy states that ex­ist. It de­scribes the ef­fect of all pos­si­ble spon­ta­neous emis­sion processes that the ex­cited state is sub­ject to. (The above equa­tion is a rewrite of (7.28) of sec­tion 7.5.3 in the present no­ta­tions.)

The above ex­pres­sion as­sumed that the ex­cited atoms are in a box that is at ab­solute zero tem­per­a­ture. Atoms in a box that is at room tem­per­a­ture are bathed in ther­mal black­body ra­di­a­tion. In prin­ci­ple you would then have to use the full equa­tions (7.45) and (7.46) to fig­ure out what hap­pens to the num­ber of ex­cited atoms. Stim­u­lated emis­sion will add to spon­ta­neous emis­sion and new ex­cited atoms will be cre­ated by ab­sorp­tion. How­ever, at room tem­per­a­ture black­body ra­di­a­tion has neg­li­gi­ble en­ergy in the vis­i­ble light range, chap­ter 6.8 (6.10). Tran­si­tions in this range will not re­ally be af­fected.

Key Points

$\begin{picture}(15,5.5)(0,-3) \put(2,0){\makebox(0,0){\scriptsize\bf0}} \put(12... ...\thicklines \put(3,0){\line(1,0){12}}\put(11.5,-2){\line(1,0){3}} \end{picture}$

This sec­tion de­scribed the gen­eral evo­lu­tion equa­tions for a sys­tem of atoms in an in­co­her­ent am­bi­ent elec­tro­mag­netic field.

$\begin{picture}(15,5.5)(0,-3) \put(2,0){\makebox(0,0){\scriptsize\bf0}} \put(12... ...\thicklines \put(3,0){\line(1,0){12}}\put(11.5,-2){\line(1,0){3}} \end{picture}$

The con­stants in the equa­tions are called the Ein­stein $A$ and $B$ co­ef­fi­cients.

$\begin{picture}(15,5.5)(0,-3) \put(2,0){\makebox(0,0){\scriptsize\bf0}} \put(12... ...\thicklines \put(3,0){\line(1,0){12}}\put(11.5,-2){\line(1,0){3}} \end{picture}$

The $B$ co­ef­fi­cients de­scribe the rel­a­tive re­sponse of tran­si­tions to in­co­her­ent ra­di­a­tion. They are given by (7.47).

$\begin{picture}(15,5.5)(0,-3) \put(2,0){\makebox(0,0){\scriptsize\bf0}} \put(12... ...\thicklines \put(3,0){\line(1,0){12}}\put(11.5,-2){\line(1,0){3}} \end{picture}$

The $A$ co­ef­fi­cients de­scribe the spon­ta­neous emis­sion rate. They are given by (7.48).