and its interaction with the matter

Transcription

1 INTERACTION OF LIGHT WITH MATTER One of the ost portant topcs n te-dependent quantu echancs for chests s the descrpton of spectroscopy, whch refers to the study of atter through ts nteracton wth lght felds (electroagnetc radaton). Classcally, lght-atter nteractons are a result of an oscllatng electroagnetc feld resonantly nteractng wth charged partcles. Quantu echancally, lght felds wll act to couple quantu states of the atter, as we have dscussed earler. Le every other proble, our startng pont s to derve a Haltonan for the lght-atter nteracton, whch n the ost general sense would be of the for H = HM + HL + HLM. (4.1) The Haltonan for the atter H M s generally (although not necessarly) te ndependent, whereas the electroagnetc feld H L and ts nteracton wth the atter H LM are te-dependent. A quantu echancal treatent of the lght would descrbe the lght n ters of photons for dfferent odes of electroagnetc radaton, whch we wll descrbe later. We wll start wth a coon seclasscal treatent of the proble. For ths approach we treat the atter quantu echancally, and treat the feld classcally. For the feld we assue that the lght only presents a te-dependent nteracton potental that acts on the atter, but the atter doesn t nfluence the lght. (Quantu echancal energy conservaton says that we expect that the change n the atter to rase the quantu state of the syste and annhlate a photon fro the feld. We won t deal wth ths rght now). We are just nterested n the effect that the lght has on the atter. In that case, we can really gnore H, and we have a Haltonan that can be solved n the nteracton pcture representaton: L () H HM + HLM t = H + V t Here, we ll derve the Haltonan for the lght-atter nteracton, the Electrc Dpole Haltonan. It s obtaned by startng wth the force experenced by a charged partcle n an electroagnetc feld, developng a classcal Haltonan for ths syste, and then substtutng quantu operators for the atter: (4.) Andre Toaoff, MIT Departent of Chestry, /7/8

2 4- p ˆ x xˆ (4.) In order to get the classcal Haltonan, we need to wor through two steps: (1) We need to descrbe electroagnetc felds, specfcally n ters of a vector potental, and () we need to descrbe how the electroagnetc feld nteracts wth charged partcles. Bref suary of electrodynacs Let s suarze the descrpton of electroagnetc felds that we wll use. A dervaton of the plane wave solutons to the electrc and agnetc felds and vector potental s descrbed n the appendx. Also, t s helpful to revew ths ateral n Jacson 1 or Cohen-Tannoudj, et al. > Maxwell s Equatons descrbe electrc and agnetc felds( E, B ). > To construct a Haltonan, we ust descrbe the te-dependent nteracton potental (rather than a feld). > To construct the potental representaton of E and B, you need a vector potental A( r, t ) and a scalar potental ϕ ( r, t) electrostatc potental through E. For electrostatcs we norally thn of the feld beng related to the = ϕ, but for a feld that vares n te and n space, the electrodynac potental ust be expressed n ters of both A andϕ. > In general an electroagnetc wave wrtten n ters of the electrc and agnetc felds requres 6 varables (the x,y, and z coponents of E and B). Ths s an overdeterned proble; Maxwell s equatons constran these. The potental representaton has four varables ( Ax, Ay, Azandϕ ), but these are stll not unquely deterned. We choose a constrant a representaton or guage that allows us to unquely descrbe the wave. Choosng a gauge such that ϕ = (Coulob gauge) leads to a plane-wave descrpton of E and B : (, ) A r t, c 1 A( r t) + = (4.4) A = (4.5) 1 Jacson, J. D. Classcal Electrodynacs (John Wley and Sons, New Yor, 1975). Cohen-Tannoudj, C., Du, B. & Lalöe, F. Quantu Mechancs (Wley-Interscence, Pars, 1977), Appendx III.

3 4- Ths wave equaton allows the vector potental to be wrtten as a set of plane waves: r ( t) * r ˆ ˆ ( t) A r t = Aε e + Aε e. (4.6) (, ) Ths descrbes the wave oscllatng n te at an angular frequency and propagatng n space n the drecton along the wavevector, wth a spatal perod λ = π. The wave has an apltude A whch s drected along the polarzaton unt vector ˆε. Snce A =, we see that ˆ ε = or ˆ ε. Fro the vector potental we can obtan E and B A E = t ˆ r ( t) r ( t) = A ε e e (4.7) B = A r ( t) r ( t) = ( ˆ ε ) A e e (4.8) If we defne a unt vector along the agnetc feld polarzaton as ˆ ( ˆ ε ) b= = ˆ ˆ ε, we see that the wavevector, the electrc feld polarzaton and agnetc feld polarzaton are utually orthogonal ˆ ˆ ε b ˆ. Also, by coparng eq. (4.6) and (4.7) we see that the vector potental oscllates as cos(t), whereas the feld oscllates as sn(t). If we defne 1 E A (4.9) 1 B A (4.1) then, Note, E B = = c. (, ) ˆ ε sn ( ) E r t = E r t (4.11) (, ) ˆ sn( ) B r t = B b r t. (4.1)

4 4-4 Classcal Haltonan for radaton feld nteractng wth charged partcle Now, let s fnd a classcal Haltonan that descrbes charged partcles n a feld n ters of the vector potental. Start wth Lorentz force on a partcle wth charge q: F = q( E+ v B). (4.1) Here v s the velocty of the partcle. Wrtng ths for one drecton (x) n ters of the Cartesan coponents of E, v and B, we have: F = q E + v B v B. (4.14) x x y z z y In Lagrangan echancs, ths force can be expressed n ters of the total potental energy U as U d U Fx = + (4.15) x dt vx Usng the relatonshps that descrbe E and B n ters of A andϕ, nsertng nto eq. (4.14), and worng t nto the for of eq. (4.15), we can show that: U = qϕ qv A (4.16) Ths s derved n CTDL, 4 and you can confr by replacng t nto eq. (4.15). Now we can wrte a Lagrangan n ters of the netc and potental energy of the partcle L= T U (4.17) 1 L= v + qv A qϕ (4.18) The classcal Haltonan s related to the Lagrangan as H = p v L 1 p v v qv A qϕ = (4.19) L Recognzng p = = v + qa (4.) v we wrte v 1 ( p qa) Now substtutng (4.1) nto (4.19), we have: =. (4.1) See Schatz and Ratner, p Cohen-Tannoudj, et al. app. III, p. 149.

5 1 1 q 4-5 H = p p qa p qa p qa A+ qϕ (4.) 1 H = p qa( r, t) + qϕ ( r, t) (4.) Ths s the classcal Haltonan for a partcle n an electroagnetc feld. In the Coulob gauge ( ϕ = ), the last ter s dropped. We can wrte a Haltonan for a collecton of partcles n the absence of a external feld and n the presence of the EM feld: p H = + V r Expandng: ( ) ( ). (4.4) 1 H = ( p qa( r) ) + V ( r). (4.5) q q H = H p A+ A p + A (4.6) Generally the last ter s consdered sall copared to the cross ter. Ths ter should be consdered for extreely hgh feld strength, whch s nonperturbatve and sgnfcantly dstorts the potental bndng olecules together. One can estate that ths would start to play a role at ntensty levels >1 15 W/c, whch ay be observed for very hgh energy and tghtly focused pulsed fetosecond lasers. So, for wea felds we have an expresson that aps drectly onto solutons we can forulate n the nteracton pcture: H = H + V t (4.7) q V () t = ( p A + A p). (4.8) Quantu echancal Electrc Dpole Haltonan Now we are n a poston to substtute the quantu echancal oentu for the classcal. Here the vector potental reans classcal, and only odulates the nteracton strength.

6 4-6 We can show that A= A. Notce A ( A) A operatng on a wavefuncton A ψ ( A) ψ A ( ψ ) p = (4.9) V () t = q( A+ A ) (4.) = + (chan rule). For nstance, f we are = +. The frst ter s zero snce we are worng n the Coulob gauge( A = ). Now we have: V t q = A () q A p = For a sngle charge partcle our nteracton Haltonan s () V t q = A p q ˆ r ( t A ) = c.c. ε p e + (4.1) (4.) Under ost crcustances, we can neglect the wavevector dependence of the nteracton potental. If the wavelength of the feld s uch larger than the olecular denson( ) r λ, then e 1. Ths s nown as the electrc dpole approxaton. We do retan the spatal dependence for certan types of lght-atter nteractons. In that case we defne r as the center of ass of a olecule and expand r r r e = e e e r ( r ) = 1 + r r + (4.) For nteractons, wth UV, vsble, and nfrared (but not X-ray) radaton, r r << 1, and settng r r = eans that 1. We retan the second ter for quadrupole transtons: charge e dstrbuton nteractng wth gradent of electrc feld and agnetc dpole. Now, usng A = E, we wrte (4.) as qe t + t V() t = ˆ ε pe ˆ ε pe (4.4)

7 4-7 qe V () t = ( ˆ ε p) snt q (4.5) = ( Et () p ) or for a collecton of charged partcles (olecules): q E V () t ( ˆ p ) sn t ε = (4.6) Ths s nown as the electrc dpole Haltonan (EDH). Transton dpole atrx eleents We are seeng to use ths Haltonan to evaluate the transton rates nduced by V(t) fro our frst-order perturbaton theory expresson. For a perturbaton sn V t = V t the rate of transtons nduced by feld s π w = V δ ( E E ) + δ ( E E + ) Now we evaluate the atrx eleents of the EDH n the egenstates for H : (4.7) qe V ˆ = V = ε p (4.8) We can evaluate the atrx eleent p usng an expresson that holds for any one-partcle Haltonan: Ths expresson gves [ r, H ] p = rh Hr = r E E r = r. p =. (4.9) ( ) (4.4) So we have V ˆ qe = ε r (4.41)

8 4-8 or for a collecton of partcles V ˆ = E ε qr = E ˆ ε μ (4.4) = E μ l μ s the dpole operator, and μl s the transton dpole atrx eleent. We can see that t s the quantu analog of the classcal dpole oent, whch descrbes the dstrbuton of charge densty ρ n the olecule: ( r ) μ = dr r ρ. (4.4) These expressons allow us to wrte n splfed for the well nown nteracton potental for a dpole n a feld: μ E() t V t = (4.44) Then the rate of transtons between quantu states nduced by the electrc feld s π w = E μ l δ ( E E ) ( E E ) + + π = E μl δ ( ) + δ ( + ) (4.45) Equaton (4.45) s an expresson for the absorpton spectru snce the rate of transtons can be related to the power absorbed fro the feld. More generally we would express the absorpton spectru n ters of a su over all ntal and fnal states, the egenstates of H : w π f = E μ f δ f δ f + + (4.46), f The strength of nteracton between lght and atter s gven by the atrx eleent μ ˆ f f με. The scalar part f μ says that you need a change of charge dstrbuton between f and to get effectve absorpton. Ths atrx eleent s the bass of selecton rules based on the syetry of the states. The vector part says that the lght feld ust project onto the dpole oent. Ths allows nforaton to be obtaned on the orentaton of olecules, and fors the bass of rotatonal transtons.

9 4-9 Relaxaton Leads to Lne-broadenng Let s cobne the results fro the last two lectures, and descrbe absorpton to a state that s coupled to a contnuu. What happens to the probablty of absorpton f the excted state decays exponentally? relaxes exponentally... for nstance by couplng to contnuu P exp[ w t] n We can start wth the frst-order expresson: t b = dτ V t t t or equvalently b = e V () t We can add rreversble relaxaton to the descrpton ofb, followng our early approach: (4.47) (4.48) Or usng V () t = Eμ snt t w n b = e V () t b (4.49) t w n b = e sntv b t () E ( + ) ( ) t wn = e e μ b t () (4.5) The soluton to the dfferental equaton t y + ay = be α (4.51) α t at be s y() t = Ae +. (4.5) a + α

10 4-1 () b t A e ( + ) ( ) E μ wn / + ( + ) wn /+ ( ) t t wnt/ = + e Let s loo at absorpton only, n the long te lt: b () t E μ For whch the probablty of transton to s P ( ) t e = wn / E μ 1 4 ( ) + wn /4 = b = The frequency dependence of the transton probablty has a Lorentzan for: e (4.5) (4.54) (4.55) The lnewdth s related to the relaxaton rate fro nto the contnuu n. Also the lnewdth s related to the syste rather than the anner n whch we ntroduced the perturbaton.

11 SUPPLEMENT: REVIEW OF FREE ELECTROMAGNETIC FIELD Maxwell s Equatons (SI): (1) B = () E = ρ/ () B E = t (4) B = μj + E μ E : electrc feld; B : agnetc feld; J : current densty; ρ : charge densty; : electrcal perttvty; μ : agnetc perttvty We are nterested n descrbng E and B n ters of a scalar and vector potental. Ths s requred for our nteracton Haltonan. Generally: A vector feld F assgns a vector to each pont n space. The dvergence of the feld (5) F F Fz x y z x y = + + F s a scalar. For a scalar feld φ, the gradent φ φ φ (6) φ = xˆ + yˆ + zˆ x y z s a vector for the rate of change at on pont n space. Here xˆ + yˆ + zˆ = rˆ are unt vectors. Also, the curl Andre Toaoff, MIT Departent of Chestry, 5/19/5

12 4-1 (7) F = xˆ yˆ zˆ x y z F F F x y z s a vector whose x, y, and z coponents are the crculaton of the feld about that coponent. Soe useful denttes fro vector calculus are: (8) ( F ) = (9) ( φ ) = (1) ( F) = ( F) F We now ntroduce a vector potental A( r, t ) and a scalar potental ϕ ( r, t) relate to E and B, whch we wll Snce B (11) B = A = and ( A) Usng (), we have: A E = t or A (1) E + = = : Fro (9), we see that a scalar product exsts wth: A (1) E + = ϕ ( r, t) or conventon

13 4-1 (14) A E = ϕ So we see that the potentals A and ϕ deterne the felds B and E : (15) B ( r, t) = A( r, t) t (16) E( r, t) = ϕ( r, t) A( r, t) We are nterested n deternng the wave equaton for A and ϕ. Usng (15) and dfferentatng (16) and substtutng nto (4): A ϕ + μ + μ = J Usng (1): (17) ( A) (18) A ϕ A A J + μ + + μ = μ Fro (14), we have: A E = ϕ and usng (): (19) V A ϕ ρ/ = Notce fro (15) and (16) that we only need to specfy four feld coponents ( Ax, Ay, Az, ϕ ) to deterne all sx E and B coponents. But E and B do not unquely deterne A and ϕ. So, we can construct A and ϕ n any nuber of ways wthout changng E and B. Notce that f we change A by addng of r and t, ths won t change B ( ( B) ) χ where χ s any functon, but we =. It wll change E by ( χ ) can change ϕ to ϕ = ϕ χ. Then E and B wll both be unchanged. Ths property of t

14 4-14 changng representaton (gauge) wthout changng E and B s gauge nvarance. We can transfor between gauges wth: () A ( r, t) = A( r, t) + χ ( r, t) = t (1) ϕ ( r, t) ϕ( r, t) χ( r, t) gauge transforaton Up to ths pont, A and Q are undeterned. Let s choose a χ such that: () A + μ ϕ = Lorentz condton then fro (17): () A A+ μ = μ J The RHS can be set to zero for no currents. Fro (19), we have: (4) ϕ μ ϕ = ρ Eqns. () and (4) are wave equatons for A and ϕ. Wthn the Lorentz gauge, we can stll arbtrarly add another χ (t ust only satsfy ). If we substtute () and (1) nto (4), we see: (5) χ μ χ = So we can ae further choces/constrants on A and ϕ as long as t obeys (5). For a feld far fro charges and currents, J = and ρ =. (6) μ A A + =

15 4-15 (7) ϕ μ ϕ+ = We now choose ϕ = (Coulob gauge), and fro () we see: (8) A = So, the wave equaton for our vector potental s: (9) μ A A + = The solutons to ths equaton are plane waves. () A A sn ( t r α ) = + α : phase (1) = A cos ( t r + α ) s the wave vector whch ponts along the drecton of propagaton and has a agntude: () = μ = / c Snce (8) A = ( α ) A cos t r + = () A = A A s the drecton of the potental polarzaton. Fro (15) and (16), we see that for ϕ = : A E = = A cos( t r + α) ( ) cos( α ) B = A= A t r + E B

16 Andre Toaoff, MIT Departent of Chestry, 5/19/ EINSTEIN B COEFFICIENT AND ABSORPTION CROSS-SECTION The rate of absorpton nduced by the feld s π w E ( ) = ( ) ε μ δ ( ) ˆ (4.56) The rate s clearly dependent on the strength of the feld. The varable that you can ost easly easure s the ntensty I, the energy flux through a unt area, whch s the te-averaged value of the Poyntng vector, S c S = ( E B) (4.57) 4π c c I = S = E E 4π = 8π. (4.58) (Note, I ve rather abruptly swtched unts to cgs). Usng ths we can wrte 4π w = I c ( ) ˆ ε μ δ ( ), (4.59) where I have also ade use of the unfor dstrbuton of polarzatons applcable to an sotropc 1 feld: E xˆ = E ˆ y = E zˆ = E. An equvalent representaton of the apltude of a onochroatc feld s the energy densty U I 1 = = E. (4.6) c 8π whch allows the rates of transton to be wrtten as w = B U (4.61) The frst factor contans the ters n the atter that dctate the absorpton rate. B s ndependent of the propertes of the feld and s called the Ensten B coeffcent 4π B = μ. (4.6) You ay see ths wrtten elsewhere as = ( ) B π μ of a wave s expressed n Hz nstead of angular frequency., whch holds when the energy densty Andre Toaoff, MIT Departent of Chestry, /1/8

17 4-17 If we assocate the energy densty wth a nuber of photons N, then U can also be wrtten n a quantu for E N = U = N. (4.6) 8π π c Now let s relate the rates of absorpton to a quantty that s drectly easured, an absorpton cross-secton α: total energy absorbed / unt te α = total ncdent nt ensty energy / unt te / area w I = = B U cu 4π = μ = B c c ( ) ( ) (4.64) More generally, you ay have a frequency-dependent absorpton coeffcent α ( ) B ( ) B g( ) = where g() s a unt noralzed lneshape functon. The golden rule rate for absorpton also gves the sae rate for stulated esson. Gven two levels and n : w n = w n ( ) = ( ) snce ( ) = ( ) B U B U U U n n n n n n (4.65) B n = B n The absorpton probablty per unt te equals the stulated esson probablty per unt te. Also, the cross-secton for absorpton s equal to an equvalent cross-secton for stulated α = α. esson, A n SE n

18 4-18 We can now use a phenoenologcal approach to calculate the change n the ntensty of ncdent lght, due to absorpton and stulated esson passng through a saple of length L where the levels are therally populated. Gven that we have a theral dstrbuton of dentcal non-nteractng partcles, wth quantu states such that the level s hgher n energy than n : di I di I = N α dx+ N α dx (4.66) n A SE = N N α dx (4.67) n Here N n and N are populaton of the upper and lower states, but expressed as a populaton denstes. If N s the olecule densty, N n e β E n = N Z (4.68) Integratng (4.67) over a pathlength L we have T I ΔNαL = = e (4.69) I e N α L N : c α : c L: c We see that the transsson of lght through the saple decays exponentally as a functon of path length. ΔN = N n N s the theral populaton dfference between states. The second expresson coes fro the hgh frequency approxaton applcable to optcal spectroscopy, but certanly not for agnetc resonance: ΔN N. Wrtten as the falar Beer-Labert Law: A= log I = εcl. (4.7) I C : ol / lter ε : lter / ol c ε = N α

19 SPONTANEOUS EMISSION What doesn t coe naturally out of se-classcal treatents s spontaneous esson transtons when the feld sn t present. To treat t properly requres a quantu echancal treatent of the feld, where energy s conserved, such that annhlaton of a quantu leads to creaton of a photon wth the sae energy. We need to treat the partcles and photons both as quantzed objects. You can deduce the rates for spontaneous esson fro statstcal arguents (Ensten). For a saple wth a large nuber of olecules, we wll consder transtons between two states and n wth E > En. The Boltzann dstrbuton gves us the nuber of olecules n each state. N N = e (4.71) n / / T n For the syste to be at equlbru, the te-averaged transtons up W n. In the presence of a feld, we would want to wrte for an enseble ( ) ( ) Wn ust equal those down? N B U = N B U (4.7) n n n n n but clearly ths can t hold for fnte teperature, where N < Nn, so there ust be another type of esson ndependent of the feld. So we wrte W n = W ( + ( )) = ( ) n n n n n n n N A B U N B U (4.7) Andre Toaoff, MIT Departent of Chestry, 5/19/5

20 4- If we substtute the Boltzann equaton nto ths and use Bn = Bn, we can solve for A n : n n n n / ( )( T 1) A = B U e (4.74) For the energy densty we wll use Planc s blacbody radaton dstrbuton: U ( ) 1 = n / π c e T 1 U N (4.75) U s the energy densty per photon of frequency. N s the ean nuber of photons at a frequency. An = B n Ensten A coeffcent (4.76) π c The total rate of esson fro the excted state s = ( ) + usng U( ) w B U A n n n n = + π c B 1 n ( N ) n = N π C (4.77) (4.78) Notce, even when the feld vanshes ( N ), we stll have esson. Reeber, for the seclasscal treatent, the total rate of stulated esson was wn = B n N π c (4.79) If we use the statstcal analyss to calculate rates of absorpton we have wn = B nn π c (4.8) The A coeffcent gves the rate of esson n the absence of a feld, and thus s the nverse of the radatve lfete: 1 τ rad = (4.81) A

21 QUANTIZED RADIATION FIELD Bacground Our treatent of the vector potental has drawn on the onochroatc plane-wave soluton to the wave-equaton for A. The quantu treatent of lght as a partcle descrbes the energy of the lght source as proportonal to the frequency, and the photon of ths frequency s assocated wth a cavty ode wth wavevector =/cthat descrbes the nuber of oscllatons that the wave can ae n a cube wth length L. For a very large cavty you have a contnuous range of allowed. The cavty s portant for consderng the energy densty of a lght feld, snce the electroagnetc feld energy per unt volue wll clearly depend on the wavelength λ = π/ of the lght. Boltzann used a descrpton of the lght radated fro a blacbody source of fnte volue at constant teperature n ters of a superposton of cavty odes to coe up wth the statstcs for photons. The classcal treatent of ths proble says that the energy densty (odes per unt volue) ncreases rapdly wth ncreasng wavelength. For an equlbru body, the energy absorbed has to equal the energy radated, but clearly as frequency ncreases, the energy of the radated lght should dverge. Boltzann used the detaled balance condton to show that the partcles that ade up lght ust obey Bose-Ensten statstcs. That s the equlbru probablty of fndng a photon n a partcular cavty ode s gven by f 1 e 1 ( ) = /T Fro our perspectve (n retrospect), ths should be expected, because the quantu treatent of any partcle has to follow ether Bose-Ensten statstcs or Fer-Drac statstcs, and clearly lght energy s soethng that we want to be able to ncrease arbtrarly. That s, we want to be able to add ode and ore photons nto a gven cavty ode. By sung over the nuber of cavty odes n a cubcal box (usng perodc boundary condtons) we can deterne that the densty of cavty odes (a photon densty of states), g ( ) = π c Usng the energy of a photon, the energy densty per ode s g ( ) = π c and so the probablty dstrbuton that descrbes the quantu frequency dependent energy densty s = = π /T u g f 1 c e 1 Andre Toaoff, MIT Departent of Chestry, 4/

22 4- The Quantu Vector Potental So, for a quantzed feld, the feld wll be descrbed by a photon nuber N, whch represents the j nuber of photons n a partcular ode (, j) wth frequency = c n a cavty of volue v. For lght of a partcular frequency, the energy of the lght wll be N j. So, the state of the electroagnetc feld can be wrtten: ϕ = N, N, N, EM,j 1 1,j,j If y atter absorbs a photon fro ode, then the state of y syste would be ϕ = N, N 1, N, EM,j 1,j,j What I want to do s to wrte a quantu echancal Haltonan that ncludes both the atter and the feld, and then use frst order perturbaton theory to tell e about the rates of absorpton and stulated esson. So, I a gong to partton y Haltonan as a su of a contrbuton fro the atter and the feld: H = HEM + HM If the atter s descrbed by expressed as product states: ϕ M, then the total state of the E.M. feld and atter can be ϕ = ϕ ϕ EM M And we have egenenerges E= E + E EM M

23 4- Now, f I a watchng transtons fro an ntal state to a fnal state, then I can express the ntal and fnal states as: ϕ I = ;N 1,N,N,N, atter feld ϕ = ; N, N, N,, N ± 1, F 1 N + : esson : absorpton N 1 Where I have abbrevated N N, j, the energes of these two states are: E E N = + = c I j j j j j E = E + N ± F j j j So loong at absorpton states as:, we can wrte the Golden Rule Rate for transtons between π w E E V t = δ ϕf ϕi Now, let s copare ths to the absorpton rate n ters of the classcal vector potental: π q w A p ( ) ˆ = δ,j j v j If these are to be the sae, then clearly V ( t) ust have part that loos le ( ˆ p) that acts on the atter, but t wll also need another part that acts to lower and rase the photons n the feld. Based on analogy wth our electrc dpole Haltonan, we wrte: q 1 * () = ( ˆ ˆ ˆ j + ˆ,j j,j) V t p A p A v,j

24 4-4 where Aˆ and A ˆ are lowerng/rasng operators for photons n ode. These are operators n,j,j the feld states, whereas p reans only an operator n the atter states. So, we can wrte out the atrx eleents of V as q 1 ϕf V() t ϕ ˆ ˆ I = p,n 1, A,N, v 1 = ˆ μ v A ( ) Coparng wth our Golden Rule expresson for absorpton, We see that the atrx eleent w π δ μ = ( ) E ( ) E E A = but = N 4v 8π = So we can wrte π v N Â,j = π a v, j  = π a v,j j, where a,a are lowerng, rasng operators. So ˆ π r ( t) r ( t) A= ˆ j ( a e + a e j j ) v,j

25 4-5 So what we have here s a syste where the lght feld loos le an nfnte nuber of haronc oscllators, one per ode, and the feld rases and lowers the nuber of quanta n the feld whle the oentu operator lowers and rases the atter: H () 1 ( a a ) () H= H + H + V t = H + V t EM M EM j j,j p HM = + V r,t V t = + q = A p,j q π r ( t ) r ( t) = ( ˆ j p) a e a e +,j,j v ( ) ( + ) = V + V Let s loo at the atrx eleents for absorpton ( > ) ( ) q π,n 1 V,N =, N 1 ( ˆ p) a, N v q π = N ˆ p v π v N = μ ˆ and for stulated esson ( < )

26 4-6 ( + ) q π,n + 1 V, N =, N + 1 ( ˆ p) a, N v q π = N ˆ ˆ + 1 p v π N 1 ˆ v = + μ We have spontaneous esson! Even f there are no photons n the ode ( N = ) have transtons downward n the atter whch creates a photon. Let s play ths bac nto the suaton-over-odes expresson for the rates of absorpton/esson by sotropc feld. π ( + ) w = d δ dω,n + 1V,N ( π ) π c j π 8π = ( π )( N + 1) μ ( πc) nuberdenstyper ode 4 N + 1 c = μ = B N + 1 energydensty per ode averageover polarzaton, you can stll So we have the result we deduced before.

27 4-7 Appendx: Rates of Absorpton and Stulated Esson Here are a couple of ore detaled dervatons: Verson 1: Let s loo a lttle ore carefully at the rate of absorpton lght source w = w ρe d where, for a onochroatc lght source π w E ( ) = ( ) ˆ μ δ ( ) w nduced by an sotropc, broadband For a broadband sotropc lght source ρ ( ) d represents a nuber densty of electroagnetc odes n a frequency range d ths s the nuber of standng electroagnetc waves n a unt volue. For one frequency we wrote: but ore generally: ˆ r ( t) A= A c.c. e + r ( t) A = A ˆ + c.c.,j je where the su s over the odes and j s the polarzaton coponent. By sung over wave vectors for a box of fxed volue, the nuber densty of odes n a frequency range d radated nto a sold angle dω s and we get ρ E by ntegratng over all Ω 1 dn = ddω c ( π) 1 ρ ( ) d= d dω= d π ( ) E c π c 4π nuber densty at

28 4-8 We can now wrte the total transton rate between two dscrete levels sued over all frequences, drecton, polarzatons π 1 w ( ) ˆ = d E δ d Ω j μ ( π ) c j ( ) E = μ 6π c 8π μ We can wrte an energy densty whch s the nuber densty n a range d # of polarzaton coponents energy densty per ode. E U( ) = π c 8π w = B U rate of energy flow/c B 4π = μ s the Ensten B coeffcent for the rate of absorpton U s the energy densty and can also be wrtten n a quantu for, by wrtng t n ters of the nuber of photons N E N = U = N 8π π c ( ) 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 ( ) = ( ) snce ( ) = ( ) B U B U U U n n n n n n B n = B n The absorpton probablty per unt te equals the stulated esson probablty per unt te.

29 4-9 Verson : Let s calculate the rate of transtons nduced by an sotropc broadband source we ll do t a bt dfferently ths te. The unts are cgs. The power transported through a surface s gven by the Poyntng vector and depends on. c c A ˆ E S= E B= = 4π 8π π and the energy densty for ths sngle ode wave s the te average of S/c. The vector potental for a sngle ode s r ( t ˆ ) A = A + c.c. e wth = c. More generally any wave can be expressed as a su over Fourer coponents of the wave vector: r ( t) e A= A ˆ j + c.c. j V,j The factor of V noralzes for the energy densty of the wave whch depends on. The nteracton Haltonan for a sngle partcle s: () V t q = A ρ or for a collecton of partcles V t q = A p () Now, the oentu depends on the poston of partcles, and we can express p n ters of an ntegral over the dstrbuton of partcles: p= d r p r p r = pδ r r So f we assue that all partcles have the sae ass and charge say electrons:

30 4- q V t d r A r t p r () = (, ) The rate of transtons nduced by a sngle ode s: π q V,j,j j ( w ) = δ( ) A ˆ p( r) And the total transton rate for an sotropc broadband source s: w = ( w ),j,j We can replace the su over odes for a fxed volue wth an ntegral over : 1 V Ω d d d d d ( π) ( π) ( πc ) Ω So for the rate we have: dω = snθ dθ dø π q w d d p r A = δ ( ) Ω ˆ,j j ( πc) can be wrtten as The atrx eleent can be evaluated n a anner slar to before: q q ˆ p r = pδ r r ˆ j = q q ˆ r,h 1 δ r r [ ] = q ˆ r 1 = μ ˆ where μ= q r For the feld E j E A = = 4 j j j

31 4-1 π W ˆ = d δ E dω j μ 4 ( π c) j = 6π c E μ For a broadband source, the energy densty of the lght 8π/ μ for sotropc I E U = = c 8π c 4π = ( ) = μ W B U B We can also wrte the ncdent energy densty n ters of the quantu energy per photon. For N photons n a sngle ode: N = B N c π where B has olecular quanttes and no dependence or feld. Note B = B rato of S.E. = absorpton. The rato of absorpton can be related to the absorpton cross-secton, δ A P total energy absorbed/unt te σ A = = I total ntensty (energy/unt te/area) P= W = B U I= cu( ) σ a = B c or ore generally, when you have a frequency-dependent absorpton coeffcent descrbed by a lneshape functon g ( ) σa = c B g unts of c