11.3. DIAGRAMMATIC PERTURBATION THEORY

Transcription

1 Andrei Tokmkoff, MIT Deprtment of Chemistry, /30/009 p DIAGRAMMATIC PERTURBATION THEORY In prctice, the nonliner response functions s written ove provide little insight into wht the moleculr origin of prticulr nonliner signls is. These multiply nested terms re difficult to understnd when fced the numerous light-mtter interctions, which cn tke on huge rnge of permuttions when performing experiments on system with multiple quntum sttes. The different terms in the response function cn led to n rry of different nonliner signls tht vry not only microscopiclly y the time-evolution of the moleculr system, ut lso differ mcroscopiclly in terms of the frequency nd wvevector of the emitted rdition. Digrmmtic perturtion theory (DPT) is simplified wy of keeping trck of the contriutions to prticulr nonliner signl given prticulr set of sttes in H 0 tht re proed in n experiment. It uses series of simple digrms to represent the evolution of the density mtrix for H 0, showing repeted interction of ρ with the fields followed y time-propgtion under H 0. From prcticl sense, DPT llows us to interpret the microscopic origin of signl with prticulr frequency nd wvevector of detection, given the specifics of the quntum system we re studying nd the detils of the incident rdition. It provides shorthnd form of the correltion functions contriuting to prticulr nonliner signl, which cn e used to understnd the microscopic informtion content of prticulr experiments. It is lso ookkeeping method tht llows us to keep trck of the contriutions of the incident fields to the frequency nd wvevector of the nonliner polriztion. There re two types of digrms we will discuss, Feynmn nd ldder digrms, ech of which hs certin dvntges nd disdvntges. For oth types of digrms, the first step in drwing digrm is to identify the sttes of H 0 tht will e interrogted y the light-fields. The digrms show n explicit series of sorption or stimulted emission events induced y the incident fields which pper s ction of the dipole opertor on the r or ket side of the density mtrix. They lso symolize the coherence or popultion stte in which the density mtrix evolves during given time intervl. The trce tken t the end following the ction of the finl dipole opertor, i.e. the signl emission, is represented y finl wvy line connecting dipole coupled sttes.

2 Andrei Tokmkoff, MIT Deprtment of Chemistry, /30/009 p. -6 Feynmn Digrms Feynmn digrms re the esiest wy of trcking the stte of coherences in different time periods, nd for noting sorption nd emission events.. Doule line represents ket nd r side of ρ.. Time-evolution is upwrd. 3. Lines intersecting digrm represent field interction. Asorption is designted through n inwrd pointing rrow. Emission is n outwrd pointing rrow. Action on the left line is ction on the ket, wheres the right line is r.. System evolves freely under H 0 etween interctions, nd density mtrix element for tht period is often explicitly written. Ldder Digrms Ldder digrms re helpful for descriing experiments on multistte systems nd/or with multiple frequencies; however, it is difficult to immeditely see the stte of the system during given time intervl. They nturlly lend themselves to description of interctions in terms of the eigensttes of H 0.. Multiple sttes rrnged verticlly y energy.. Time propgtes to right. 3. Arrows connecting levels indicte resonnt interctions. Asorption is n upwrd rrow nd emission is downwrd. A solid line is used to indicte ction on the ket, wheres dotted line is ction on the r.. Free propgtion under H 0 etween interctions, ut the stte of the density mtrix is not lwys ovious.

3 Andrei Tokmkoff, MIT Deprtment of Chemistry, /30/009 p. -7 For ech light-mtter interctions represented in digrm, there is n understnding of how this ction contriutes to the response function nd the finl nonliner polriztion stte. Ech light-mtter interction cts on one side of ρ, either through sorption or stimulted emission. Ech interction dds dipole mtrix element ij tht descries the interction mplitude nd ny orienttionl effects. Ech interction dds input electric field fctors to the polriztion, which re used to descrie the frequency nd wvevector of the rdited signl. The ction of the finl dipole opertor must return you to digonl element to contriute to the signl. Rememer tht ction on the r is the complex conjugte of ket nd sorption is complex conjugte of stimulted emission. A tle summrizing these interctions contriuting to digrm is elow. Interction Digrmmtic Representtion contri. to ( n) R contriution to k & sig ωsig KET SIDE Asorption ( En )exp ikn r iωnt E n ε ˆ n + k n +ω n Stimulted Emission ( ) E exp ik r + iω t n n n E n ε ˆ n k n ω n BRA SIDE Asorption ( En ) exp ikn r + iωnt E n ε ˆ n k n ω n Stimulted Emission ( En ) exp ikn r iωnt E n ε ˆ n + k n +ω n SIGNAL EMISSION: (Finl trce, convention: ket side) ˆ n ε

4 Andrei Tokmkoff, MIT Deprtment of Chemistry, /30/009 p. -8 Once you hve written down the relevnt digrms, eing creful to identify ll permuttions of interctions of your system sttes with the fields relevnt to your signl, the correltion functions contriuting to the mteril response nd the frequency nd wvevector of the signl field cn e redily otined. It is convenient to write the correltion function s product of severl fctors for ech event during the series of interctions: ) Strt with fctor p n signifying the proility of occupying the initil stte, typiclly Boltzmnn fctor. ) Red off products of trnsition dipole moments for interctions with the incident fields, nd for the finl signl emission. 3) Multiply y terms tht descrie the propgtion under H 0 etween interctions. As strting point for understnding n experiment, it is vlule to include the effects of relxtion of the system eigensttes in the time-evolution using simple phenomenologicl pproch. Coherences nd popultions re propgted y ssigning the dmping constnt Note Γ to propgtion of the ρ element: Gˆ τ ρ = exp iω τ Γ τ ρ. () ( ) [ ] Γ =Γ nd G = G. We cn then recognize Γ ii = T s the popultion relxtion rte for stte i nd Γ ij = T the dephsing rte for the coherence ρ ij. ) Multiply y fctor of ( ) n where n is the numer of r side interctions. This fctor ccounts for the fct tht in evluting the nested commuttor, some correltion functions re sutrcted from others. 5) The rdited signl will hve frequency ωsig = ωi nd wve vector ksig = ki i i Exmple: Liner Response for Two-Level System Let s consider the digrmmtic pproch to the liner sorption prolem, using two-level system with lower level nd upper level. There is only one independent correltion function in the liner response function,

5 Andrei Tokmkoff, MIT Deprtment of Chemistry, /30/009 p. -9 ( ) ( ) ( 0) C t Tr t ρ = eq () = Tr Gˆ t ρ eq This does not need to e known efore strting, ut is useful to consider, since it should e recovered in the end. The system will e tken to strt in the ground stte ρ. Liner response only llows for one input field interction, which must e sorption, nd which we tke to e ket side interction. We cn now drw two digrms: () () Act on ket with nd tke trce. (3) Propgte under H 0 : i Γ ( τ ) G e ω τ τ =. () Act on ket with (0) to crete ρ. () Strt in ρ (dd fctor of p when reding). With this digrm, we cn egin y descriing the signl chrcteristics in terms of the induced polriztion. The product of incident fields indictes: E () i t ik i r sigt iksig r e ω + P t e ω + (3) so tht ω = ω k = k. () sig As expected the signl will rdite with the sme frequency nd in the sme direction s the incoming em. Next we cn write down the correltion function for this term. Working from ottom up: () () (3) () sig iωt Γt () = [ ] [ ] C t p e = p e iωt Γt More sophisticted wys of treting the time-evolution under H 0 in step (3) could tke the form of some of our erlier tretments of the sorption lineshpe: (5)

6 Andrei Tokmkoff, MIT Deprtment of Chemistry, /30/009 p. -0 ( τ ) ρ ρ [ ωτ] ( τ) = ρ exp iω τ g() t Gˆ ~ exp i F Note tht one could drw four possile permuttions of the liner digrm when considering r nd ket side interctions, nd initil popultion in sttes nd : (6) However, there is no new dynmicl content in these extr digrms, nd they re generlly tken to e understood through one digrm. Digrm ii is just the complex conjugte of eq. (5) so dding this signl contriution gives: () () Γ t C t C t i p ωt e = sin( ). (7) Accounting for the thermlly excited popultion initilly in leds to the expected two-level system response function tht depends on the popultion difference h () = ( ) Γt R t p p sin( ωte ). (8) Exmple: Second-Order Response for Three-Level System The second-order response is the simplest nonliner cse, ut in moleculr spectroscopy is less commonly used thn third-order mesurements. The signl genertion requires lck of inversion symmetry, which mkes it useful for studies of interfces nd chirl systems. However, let s show how one would digrmmticlly evlute the second order response for very specific system pictured t right. If we only hve popultion in the ground stte t equilirium nd if there re only resonnt interctions llowed, the permuttions of unique digrms re s follows:

7 Andrei Tokmkoff, MIT Deprtment of Chemistry, /30/009 p. - From the frequency conservtion conditions, it should e cler tht process i is sum-frequency signl for the incident fields, wheres digrms ii-iv refer to difference frequency schemes. To etter interpret wht these digrms refer to let s look t iii. Reding in time-ordered mnner, we cn write the correltion function corresponding to this digrm s C Tr τ ( ) ρeq( 0) ( ) Gˆ ( ) Gˆ ( ) = = τ τ ρ c c c iωτ Γτ iωcτ Γcτ cc = p e e. (9) Note tht literl interprettion of the finl trce in digrm iv would imply n sorption event n upwrd trnsition from to c. Wht does this hve to do with rditing signl? On the one hnd it is importnt to rememer tht digrm is just mthemticl shorthnd, nd tht one cn t distinguish sorption nd emission in the finl ction of the dipole opertor prior to tking trce. The other thing to rememer is tht such digrm lwys hs complex conjugte ssocited with it in the response function. The complex conjugte of iv, Q ket/r term, shown t right hs downwrd trnsition emission s the finl interction. The comintion Q Q ultimtely descries the oservle.

8 Andrei Tokmkoff, MIT Deprtment of Chemistry, /30/009 p. - Now, consider the wvevector mtching conditions for the second order signl iii. Rememering tht the mgnitude of the wvevector is k = ω c= π λ, the length of the vectors will e scled y the resonnce frequencies. When the two incident fields re crossed s slight ngle, the signl would e phse-mtched such tht the signl is rdited closest to em. Note tht the most efficient wvevector mtching here would e when fields nd re colliner.

9 Andrei Tokmkoff, MIT Deprtment of Chemistry, /30/009 p. -3 Third-Order Nonliner Spectroscopy Now let s look t exmples of digrmmtic perturtion theory pplied to third-order nonliner spectroscopy. Third-order nonlinerities descrie the mjority of coherent nonliner experiments tht re used including pump-proe experiments, trnsient grtings, photon echoes, coherent nti-stokes Rmn spectroscopy (CARS), nd degenerte four wve mixing (WM). These experiments re descried y some or ll of the eight correltion functions contriuting to 3 3 i = α α h α = ( 3) R : ( ) R R R (0) The digrms nd corresponding response first requires tht we specify the system eigensttes. The simplest cse, which llows us discuss numer of exmples of third-order spectroscopy is two-level system. Let s write out the digrms nd correltion functions for two-level system strting in ρ, where the dipole opertor couples nd. R R R3 R ket/ket/ket r/ket/r r/r/ket ket/r/r τ 3 E 3 E τ τ E + ω ω + ω3 ω+ ω + ω3 ω+ ω + ω3 + ω ω + ω3 ksig =+ k k + k3 k+ k + k3 k+ k + k3 + k k + k3

10 Andrei Tokmkoff, MIT Deprtment of Chemistry, /30/009 p. - As n exmple, let s write out the correltion function for R otined from the digrm ove. This term is importnt for understnding photon echo experiments nd contriutes to pump-proe nd degenerte four-wve mixing experiments. iωτ Γτ i Γτ iω ( ) ( ) τ3 Γτ3 ( ) ( )( ) e ( ) R = p e e ω τ = p exp iω ( τ3 τ) Γ ( τ+ τ3) Γ ( τ) The digrms show how the input field contriutions dictte the signl field frequency nd wvevector. Recognizing the dependence of product of the incident field contriutions ( 3) ( 3) ( ) 3 () E ~ P ~ R E E E, these re otined from the sig + iωt ik r iωt+ ik r + iω3t ik3 r3 ( e )( e )( e ) EEE = E E E 3 3 EE Ee 3 ωsigt+ iksig r sig 3 () ωsig = ω+ ω + ω3. (3) k = k + k + k Now, let s compre this to the response otined from R. These we otin ( ) ( ) ( ) = exp ω τ3 + τ Γ τ+ τ3 Γ τ R p i () ω = + ω ω + ω sig 3 k =+ k k + k sig 3 Note tht R nd R terms re identicl, except for the phse cquired during the initil period: [ iφ ] [ iω τ ] exp exp = ±. The (5) R term evolves in conjugte coherences during the τ nd τ 3 periods, wheres the R term evolves in the sme coherence stte during oth periods: Coherences in τ nd τ 3 Phse cquired in τ nd τ 3 R i ( + 3) e ω τ τ R i ( 3) e ω τ τ The R term hs the property of time-reversl: the phse cquired during τ is reversed in τ 3. For tht reson the term is clled rephsing. Rephsing signls re selected in photon echo experiments nd re used to distinguish line rodening mechnisms nd study spectrl diffusion. For R, the phse cquired continuously in τ nd τ 3, nd this term is clled non-

11 Andrei Tokmkoff, MIT Deprtment of Chemistry, /30/009 p. -5 rephsing. Anlysis of R nd R 3 revels tht these terms re non-rephsing nd rephsing, respectively. phse cquired φ 0 e ω τ e ω τ i 3 i i 3 e + ω τ τ τ 3 non-rephsing rephsing P() t t t 3 t t For the present cse of third-order spectroscopy pplied to two-level system, we oserve tht the two rephsing functions R nd R 3 hve the sme emission frequency nd wvevector, nd would therefore oth contriute eqully to given detection geometry. The two terms differ in which popultion stte they propgte during the τ vrile. Similrly, the nonrephsing functions R nd R ech hve the sme emission frequency nd wvevector, ut differ y the τ popultion. For trnsitions etween more thn two system sttes, these terms could e seprted y frequency or wvevector (see ppendix). Since the rephsing pir R nd R 3 oth contriute eqully to signl scttered in the k+ k + k3 direction, they re lso referred to s S I. The nonrephsing pir R nd R oth sctter in the + k k + k3 direction nd re leled s S II. Our findings for the four independent correltion functions re summrized elow. S I S II rephsing non-rephsing R ω ω ω3 R 3 ω ω ω3 R ω ω ω3 R ω ω ω3 ω sig k sig τ popultion + + k+ k + k3 excited stte + + k+ k + k3 ground stte k k + k3 ground stte k k + k3 excited stte

12 Andrei Tokmkoff, MIT Deprtment of Chemistry, /30/009 p. -6 Frequency Domin Representtion 3 A Fourier-Lplce trnsform of P ( 3 ) ( ) t with respect to the time intervls llows us to otin n ( ) ( ) 3 expression for the third order nonliner susceptiility, χ ω, ω, ω : P ( 3 ) ( 3 ( ω ) ) sig χ ( ω ;,, sig ω ω ω ) 3 = E E E (6) 3 3 where ( n ) i n n i ( n d e τ Ω Ω τ d e R ) (,, ) χ τ L τ τ τ K τ. (7) = 0 n 0 n Here the Fourier trnsform conjugte vriles Ω m to the time-intervl τ m re the sum over ll frequencies for the incident field interctions up to the period for which you re evolving: m m Ω = ω (8) i= For instnce, the conjugte vrile for the third time-intervl of + k k + k3 experiment is the sum over the three preceding incident frequencies Ω 3 = ω ω + ω3. In generl, χ (3) is sum over mny correltion functions nd includes sum over sttes: 3 3 i,, 3 = p α α 6 h cd α = ( χ ) ( ω ω ω ) χ χ (9) Here is the initil stte nd the sum is over ll possile intermedite sttes. Also, to descrie frequency domin experiments, we hve to permute over ll possile time orderings. Most generlly, the eight terms in i ( 3) ( 3) R led to 8 terms for χ, s result of the 3!=6 permuttions of the time-ordering of the input fields. Given set of digrms, we cn write the nonliner susceptiility directly s follows: ) Red off products of light-mtter interction fctors. ) Multiply y resonnce denomintor terms tht descrie the propgtion under H 0. In the frequency domin, if we pply eq. (7) to response functions tht use phenomenologicl time-propgtors of the form eq. (), we otin Gˆ Ω m is defined in eq. (8). ( τ ) m ρ Ω i. (0) Γ ( ω ) m 3) As for the time domin, multiply y fctor of ( ) n for n r side interctions. ) The rdited signl will hve frequency ωsig = ωi nd wvevector ksig = ki. i i

13 Andrei Tokmkoff, MIT Deprtment of Chemistry, /30/009 p. -7 As n exmple, consider the term for R pplied to two-level system tht we wrote in the time domin in eq. () ( ) ( ) iγ ( ω ω ) χ = ω ω ω iγ ( ) ( ) ( ) ( ) 3 = ω ω iγ ω ω iγ ω + ω ω ω iγ 3 ω ω + ω ω iγ The terms re written from digrm with ech interction nd propgtion dding resonnt denomintor term (here reding left to right). The full frequency domin response is sum over multiple terms like these. ()

14 Andrei Tokmkoff, MIT Deprtment of Chemistry, /30/009 p. -8 Appendix: Third-order digrms for four-level system The third order response function cn descrie interction with up to four eigensttes of the system Hmiltonin. These re exmples of correltion functions within R (3) for four-level system representtive of vironic trnsitions ccompnying n electronic excittion, s relevnt to resonnce Rmn spectroscopy. Note tht these digrms present only one exmple of multiple permuttions tht must e considered given prticulr time-sequence of incident fields tht my hve vrile frequency. R ket / ket / ket R r/ket/r R 3 r/r/ket R ket/r/r d τ 3 d c d c E 3 E c τ τ d c E d d d d c c c c c dc d d c cd cdcd dcd c ω =+ ω ω + ω sig 3 = ω + ω + ω = ω c dc d ω + ω + ω 3 ω + ω ω = ω d c dc ω + ω + ω 3 ω ω + ω = ω c d dc + ω ω + ω 3 ω ω ω = ω d cd c The signl frequency comes from summing ll incident resonnce frequencies ccounting for the sign of the excittion. The products of trnsition mtrix elements re written in time-ordered fshion without the projection onto the incident field polriztion needed to properly ccount for orienttionl effects. The R term is more properly written ( ˆ ε )( ˆ ε )( ˆ ε )( ˆ ε ). c dc 3 d n Note tht the product of trnsition dipole mtrix elements otined from the sequence of interctions cn lwys e re-written in the cycliclly invrint form ccd d. This is one further mnifesttion of closed loops formed y the sequence of interctions.

15 Andrei Tokmkoff, MIT Deprtment of Chemistry, /30/009 p. -9 Appendix: Third-order digrms for virtion The third-order nonliner response functions for infrred virtionl spectroscopy re often pplied to wekly nhrmonic virtion. For high frequency virtions in which only the v = 0 stte is initilly populted, when the incident fields re resonnt with the fundmentl virtionl trnsition, we generlly consider digrms involving the system eigensttes v = 0, nd, nd which include v=0- nd v=- resonnces. Then, there re three distinct signl contriutions: Signl k sig Digrms nd Trnsition Dipole Scling R/NR S I k+ k + k3 rephsing S II + k k + k3 non-rephsing S III + k+ k k3 non-rephsing 0 0

16 Andrei Tokmkoff, MIT Deprtment of Chemistry, /30/009 p. -30 Note tht for the S I nd S II signls there re two types of contriutions: two digrms in which ll interctions re with the v=0- trnsition (fundmentl) nd one digrm in which there re two interctions with v=0- nd two with v=- (the overtone). These two types of contriutions hve opposite signs, which cn e seen y counting the numer of r side interctions, nd hve emission frequencies of ω 0 or ω. Therefore, for hrmonic oscilltors, which hve ω 0 = ω nd 0 =, we cn see tht the signl contriutions should destructively interfere nd vnish. This is mnifesttion of the finding tht hrmonic systems disply no nonliner response. Some devition from hrmonic ehvior is required to oserve signl, such s virtionl nhrmonicity ω 0 ω, electricl nhrmonicity 0, or level-dependent dmping Γ 0 Γ or Γ 00 Γ.. D. Lee nd A. C. Alrecht, A unified view of Rmn, resonnce Rmn, nd fluorescence spectroscopy (nd their nlogues in two-photon sorption). Adv. Infrred nd Rmn Spectr., 79 (985).. To properly ccount for ll orienttionl fctors, the trnsition dipole moment must e projected onto the incident electric field polriztion ˆε leding to the terms in the tle. This leds to nonliner polriztion tht cn hve x, y, nd z polriztion components in the l frme. The re otined y projecting the mtrix element prior to the finl trce onto the desired nlyzer xis ˆn ε. 3. Prior, Y. A complete expression for the third order susceptiility χ ( 3) -perturtive nd digrmtic pproches. IEEE J. Quntum Electron. QE-0, 37 (98). See lso, Dick, B. Response functions nd susceptiilities for multiresonnt nonliner opticl spectroscopy: Perturtive computer lger solution including feeding. Chem. Phys. 7, 59 (993).. Bloemergen, N., Lotem, H. & Lynch, R. T. Lineshpes in coherent resonnt Rmn scttering. Indin J. Pure Appl. Phys. 6, 5 (978).