p. Supp. 9- Suppleent to Rate of Absorpton and Stulated Esson Here are a ouple of ore detaled dervatons: Let s look a lttle ore arefully at the rate of absorpton w k ndued by an sotrop, broadband lght soure w k w k ωρ E ωdω where, for a onohroat lght soure π w k ω ω E k ˆ µ δω k ω For a broadband sotrop lght soure ρω dω represents a nuber densty of eletroagnet odes n a freueny range dω ths s the nuber of standng eletroagnet waves n a unt volue. For one freueny we wrote: but ore generally: ˆ A A e k r ωt +.. ωt A A k ˆ e kr +.. k, where the su s over the k odes and s the polarzaton oponent. By sung over wave vetors for a box of fxed volue, the nuber densty of odes n a freueny range dω radated nto a sold angle dω s ω dn π dωd Ω and we get ρ E by ntegratng over all Ω ρ ω ω ω dω π dω dω π dω E 4π nuber densty at ω
p. Supp. 9- We an now wrte the total transton rate between two dsrete levels sued over all freuenes, dreton, polarzatons w k dω π E ω δωk ω π ω dω k ˆ µ 8π µ k E ω k ω µ 6π k We an wrte an energy densty whh s the nuber densty n a range dω # of polarzaton oponents energy densty per ode. U ω k ω E π 8π rate of energy flow/ w k B k U ω k 4π B k µ k s the Ensten B oeffent for the rate of absorpton U s the energy densty and an also be wrtten n a uantu for, by wrtng t n ters of the nuber of photons N Nω 8π E ω U ω k N π The golden rule rate for absorpton also gves the sae rate for stulated esson. We fnd for two levels and n : w n w n B n Uω n B n Uω n sne Uω n Uω n B n B n The absorpton probablty per unt te euals the stulated esson probablty per unt te.
p. Supp. 9- Verson : Let s alulate the rate of transtons ndued by an sotrop broadband soure we ll do t a bt dfferently ths te. The unts are gs. The power transported through a surfae s gven by the Poyntng vetor and depends on k. S E B ω A kˆ ω E 4π 8π π and the energy densty for ths sngle ode wave s the te average of S/. The vetor potental for a sngle ode s ˆ A A e k r ωt +.. wth ω k. More generally any wave an be expressed as a su over Fourer oponents of the wave vetor: A A k ˆ k, kr ωt e V +.. The fator of V noralzes for the energy densty of the wave whh depends on k. The nteraton Haltonan for a sngle partle s: Vt A ρ or for a olleton of partles Vt A p Now, the oentu depends on the poston of partles, and we an express p n ters of an ntegral over the dstrbuton of partles: p d r p r p r p δ r r So f we assue that all partles have the sae ass and harge say eletrons:
p. Supp. 9-4 Vt d r A r,t p r The rate of transtons ndued by a sngle ode s: w δ ω ω A k ˆ p r k k, π V And the total transton rate for an sotrop broadband soure s: w k k, k, We an replae the su over odes for a fxed volue wth an ntegral over k : V k w k k k, d k π dk k dω π dω ω dω π C So for the rate we have: dω snθ dθ dø π ω w d ω δ ω ω d Ω k ˆ p r A π k k k, an be wrtten as k The atrx eleent an be evaluated n a anner slar to before: k ˆ r p k ˆ p δ r r ˆ k r, H δ r r [ ] ω ˆ k k r ω k µ ˆ where µ r k For the feld k A k k E k ω E 4ω
p. Supp. 9-5 π W k dω ω 4 π δω k ω ω k E dω ω k ˆ µ 8π/ µ k for sotrop ω 6π E µ k For a broadband soure, the energy densty of the lght U I ω E 8π W k B k U ω k 4π B k µ k We an also wrte the ndent energy densty n ters of the uantu energy per photon. For N photons n a sngle ode: ω Nω B k N π where B k has oleular uanttes and no dependene or feld. Note B k B k rato of S.E. absorpton. The rato of absorpton an be related to the absorpton ross-seton, δ A P total energy absorbed/unt te σ A I total ntensty energy/unt te/area P ω W k ω B k U ω k I U ω k ω σ a B k or ore generally, when you have a freueny-dependent absorpton oeffent desrbed by a lneshape funton g ω ω σ ω ω a B k g unts of