TIME EVOLUTION OF SU() COHERENT STATES J ZALEŚNY Istitut of Physics Tchicl Uivrsity of Szczci Al Pistów 48 7-3 Szczci Pold Mthmticl spcts of th SU() group prmtr ξ dymics govrd by Hmiltois xhibitig som spcil typs of tim dpdc hs b prstd o lmtry lvl from th poit of viw of Möbius trsformtio of complx pl Th trjctoris of ξ i cotiuous d mppigs i discrt dymics r cosidrd Som simpl xmpls hv b xmid Alyticl cosidrtios d umricl rsults hv b giv PACS umbrs: 365Fd Sv
INTRODUCTION A grl mthod for costructig cohrt stts for rbitrry Li group hs b giv by Prlomov [3] I this ppr w r itrstig i tim volutio of th SU() cohrt stts Vrious spcts of th dymics wr xmid by my uthors Som rlvt xmpls cocrig cotiuos dymics th rdr my fid i rticls [4789] d discrt dymics i [356] For rviw ppr o costructio d clssifictio of cohrt stts s [5] Th dymics driv by SU() Hmiltoi hs b much ivstigtd mily i th cotxt of th two-photo procsss th grtio of squzd stts of light by olir opticl procsss (g dgrt prmtric mplifirs or dow-covrtrs) th propgtio of lctromgtic wv i ohomogous mdium with qudrtic dpdc of th rfrctiv idx o th trsvrs coordits d tim-dpdt hrmoic oscilltors Th rdr my fid my rfrcs i th rviw rticl [4] Th volutio of qutum stts ruld by th tim-dpdt cohrcprsrvig SU() Hmiltoi c b lyzd usig th Wi-Norm mthod which llows th possibility of rprstig th tim-volutio oprtor s fiit product of xpotil oprtors whr ch xpot cotis product of group grtor d tim-dpdt complx fuctio [4] Th fuctios oby systm of Riccti typ olir diffrtil qutios Solutios of thm cot b obtid i grl I this ppr w do ot us th Wi-Norm mthod Our tttio is focusd o th SU() group prmtr ξ Nvrthlss th qutio of motio obyd by this prmtr is lso Riccti typ qutio Bcus its grl solutios r ukow i ordr to gi th id bout th SU() dymics w xmi som itrstig xmpls of spcil timdpdc Both cotiuous d discrt dymics is studid d coctios btw th two css r show As fr s w kow th discrt dymics ws cosidrd i th cs of ifiitly rrow δ-lik fuctio pulss [36] For istc Grry t l [6] us modultd
rl δ-lik pulsig fuctio d Bchlr t l[3] cosidr so clld kickd dymics with o-modultd complx couplig prmtr I this ppr modl with modultd pulss of fiit width d mgitud is usd which is myb mor rlistic ssumptio i compriso to th kickd-lik dymics modls Though our trtmt is clssicl rthr th qutum but i th cosidrd cs of th cohrc-prsrvig Hmiltoi () it is quivlt to th gui qutum mchicl problm Th ppr is orgizd s follows I sctio w itroduc brifly th id of SU() cohrt stts d th qutio of motio of th SU() group prmtr ξ I sctio 3 w ivstigt som of th most chrctristic fturs of th cotiuous dymics bsd upo th qutio of motio for ξ W try to clssify th typs of trjctoris of th SU() group prmtr ξ o th phs spc obtid for vrious frqucis of couplig prmtr W giv hr som umricl d lyticl rsults I sctio 4 w costruct itrtio qutio for th discrt vlus of th prmtr ξ Th volutio is dscribd by Poicré-typ volutio mps W giv som umricl xmpls of th mps for vrious pulsig fuctios W ot hr tht ll stroboscopic qutios i cs of SU() group hv th form of Möbius utomorphism of th uit circl At th d w show how th chi of Möbius trsformtios my b rplcd by th chi of lir trsformtios of th complx pl W us this pproch to xprss i o-tim-dpdt pulsig cs th stp of itrtio vi th iitil coditio Th lst sctio 5 cotis summry of th ppr d vry brif discussio of Lpuov xpot d th qustio of chos i SU() systms THE MODEL W cosidr modl dscribd by th Hmiltoi giv s lir combitio of th SU() group grtors with tim-dpdt cofficits 3
H hω K + ( K ( K hχ hχ + + () Th grtors oby th followig ruls of commuttio [ K K ± ] ± K ± [ K K + ] K () Th br ovr symbols ms complx cojugtio A Schwigr-Wigr-typ rliztio of SU() c b giv i trms of (mor fmilir to physicists) hrmoic-oscilltor crtio d ihiltio oprtors + + + ( + ) ( ) + K 4 K K (3) Bcus th Hmiltoi is lir i th grtors th cohrt chrctr of th grlizd cohrt stts ssocitd with th ocompct Li group SU() udr tim volutio is prsrvd which ms tht th qutum d clssicl volutios r sstilly idticl [3 4] Th grlizd SU() cohrt stts ξ r dfid d costructd i th wy first proposd by Prlomov [ 3] Thy r chrctrizd by th complx prmtr ξ for which ξ < W shll us th SU() group prmtr ξ s rprsttio of th phs spc Th clssicl qutio of motio for ξ is [4] ξ & { ξ H} (4) whr { } is th Poisso brckt dfid s ( ξ ) { A B} ik A B A B ξ ξ ξ ξ (5) d H ξ ( K K ξ H ± (6) Th costt k i (5) is th Brgm idx d o might tk it s k / 4 hr Th rsultig qutio from (4) is ξ& iωξ iχξ iχ (7) As w hv quivlc of clssicl d qutum dscriptio of th dymics i th ss 4
tht prmtr ξ xctly follows th qutum stt ξ w c rstrict our tttio to th motio of th poit ξ i th uit circl o th complx pl It rmis tru lso i th discrt cs Aothr drivtio of qutio (7) th rdr my fid i [3 ] I this ppr w xmi dymics i cotiuous d discrt cs followd from q (7) ssumig tht χ is tim-dpdt 3 CONTINUOUS APPROACH 3 Formultio of th problm Th phs-spc for solutios of th qutio (7) is uit circl i th complx pl Bcus of tim-dpdt cofficit χ( th qutio blogs to th clss of outoomic i f t diffrtil qutios I grl χ( is complx d my b writt i form χ ( c( whr c( f( r rl fuctios Howvr furthr w rstrict our cosidrtios to lss grl form wh f( k ω t (k is rl umbr) W c com to rottig frm with frqucy ω i ordr to limit this frqucy from th motio It is quivlt to us th itrctio pictur So w put k +k d our choic for χ( is () W sk solutio of (7) i th form χ( c( i ( + k ) ωt (8) ξ ( ( i ( + k ) ωt (9) whr ukow complx fuctio ( dscribs motio i th rottig frm It is lso limitd to th itrior of th uit circl i ( < Substitutig (9) ito (7) w obti [ ( ) + ] & ( iωk( ic( t () This qutio is som cs of th Riccti qutio A grl solutio of it is ukow To ivstigt it w cosidr som simpl css 5
3 Som xct solutios For k (k ) th solutio c b foud xplicitly W hv i this cs () t ( ) ( ) () ( i / i t i ) ( + / + i ) S t + ( i whr S( S c( τ ) dτ () d is th iitil vlu of ( I prticulr for w gt ξ ( i th iω t ( S( ) () Th bhvior of ξ( dpds o fuctio S( Wh S( fulfills th coditio t () t lim S (3) th ξ( forms mor or lss rgulr spirl i th trjctory coms up closr d closr to th uit circl without howvr rchig it Furthr w will us th m spirl to dscrib such ocompct trjctory isstil how much th curv rsmbls ordiry rgulr spirl Arisig of spirl i this cs is vry chrctristic ftur for frqucy ω idpdt of th iitil coditios s Fig s xmpl Fig: ω frqucy cs Prmtrs: c( si t + 5; ω ; k ; ξ 3+i Th spirl is formd bcus of puttig th movmt ( towrd th circumfrc o th rottio with frqucy ω O of th most rgulr spirl c b obtid for costt c d Th poit ( rus from th poit () log imgiry xis of th complx pl to th circumfrc of th uit circl Th tim dpdc of ( is giv by th fuctio 6
th(c which for smll t is simply ct Thus for smll t i ξ pictur it is so clld Archimds spirl Quit diffrt bhvior w c obsrv if S( dos ot fulfill th coditio (3) For istc if d vlus of S( r limitd to itrvl i ξ(-pictur w obsrv mor or lss (it dpds o complxity of fuctio c( ) complictd figurs drw by poit ξ( o complx pl g s Fig Fig: ω frqucy cs Prmtrs: c( si π t; ω ; k ; ξ Not tht i this cs th trjctory rmis i r of rdius lss th uit This is tru lso for othr iitil vlus Furthr w will us th m compct figurs to dscrib bhvior lik this Aothr xct solutio of q() c b foud for rbitrry prmtr k d costt c It is k ( is ( ( k / k ) ( k )/( k ) is ( t ) ( k )/( k ) (4) whr k + k 4 α α α c α St () ct k ω For iitil x +iy th trjctory of th poit ( x(+iy( is giv s ( x B A) + y whr: x + y A x α B A + A α (5) Spirl solutios ppr oly wh circl (5) hs crossovr poits with uit circl Th o 7
of thm is ttrctig fixd poit d th othr o is rpulsig Th poit ( rus log th circl (5) but it cot chiv th ttrctig poit i fiit tim O th mps for ξ( w obsrv th spirl bcus of th rottig trm i q(9) Th coordits of th fix poits r giv by α x y ± 4α (6) Th obvious coditio x < givs us importt iqulity k < c /ω (7) For giv costt mplitud c it dtrmis th rottiol frqucy k ω for which spirl solutios ppr Eg for k i i frqucy ω cs th iqulity is vlid for y ozro c For k i zro frqucy it follows tht ω < c For vrious iitil vlus q(5) givs us whol mp of trjctoris i rottig frm which r ococtric circls with ctrs lyig o th rl xis For α < / ll th circls r tirly isid th uit circl Thr r lso two lliptic fixd poits o th rl xis but oly o of thm tht lyig isid th uit circl is itrstig for us (Fig3) Fig3: Th fmily of trjctoris i th rottig frm Prmtrs: c ; ω ; k ; i iitil vlus o rl xis: Th lgr is α th closr is th lliptic fixd poit to th boudry of th uit circl For α / th fixd poit chivs th uit circl d chgs its chrctr bcomig prbolic 8
fixd poit For lgr α th prbolic fixd poit splits ito two hyprbolic fixd poits ttrctor d rpulsr (Fig4) Fig4: Th fmily of trjctoris i th rottig frm Prmtrs: c 6; ω ; k ; sv iitil vlus o rl xis: At th d of this poit w xmi som lir pproximtio of q() This pproximtio my b usd r th ctr of th uit circl whr << Istd of q () o my us & ( iωk( ic( (8) A grl solutio of this qutio c b sily foud I prticulr for th most turl i this pproximtio iitil vlu th solutio (icludig lso th rottig trm) is ξ ( t iω t ikωτ ir( k whr R( k c( τ ) dτ (9) Not tht R(k S( For d costt c th trjctory obtid i th lir pproximtio is th sm s tht corrspodig to th xct solutio (th circl (5) ) Th diffrc btw xct d pproximtd solutios pprs oly i tim dpdc of th movmt log th trjctory 9
33 Arisig of spirls Udoubtdly w should xpct risig spirl if ξ( ( for t Tkig q() d qutio cojugtd to it o c obti qutio of motio for ( s d dt ( ( ) c( Im ( ( ( ) This qutio is k idpdt so it hv to b vlid for ll k Th forml solutio of it is () T ( ( ) t ( whr T ( c( τ )Im ( τ ) dτ () Th spirl-lik solutio occurs if T( ) Th most obvious wy i which th coditio c b fulfilld is c ( Im ( < for vry t () Th simpl xmpl is th cs of c costt i vlu but chgig its sig vry tim wh th poit ( (rottig with frqucy ω c c kω 4α ) crosss th x-x It rsults i spirl bhvior v for α < / Th coditio () is lso fulfilld i th ω frqucy cs It is itrstig howvr tht som ffctiv ω frqucis c b itroducd Eg lt th cofficit χ( is giv s χ ik ωt ( c si( κω whr c is costt (3) Th si(κω twic chgs sig i its priod τ i thr is π phs gi i tim itrvl τ Durig this tim th vctor -ik ωt turs by gl k ωτ Th totl gl α k ωτ + π so ffctiv frqucy ω ff α /τ k ω + π /τ i ω ff (k + κ) ω Not tht w c thik bout π phs ot s gi but s loss It lds to ffctiv frqucy ω ff (k κ) ω If w tk k d κ i th followig wy k + κ or k κ i κ k or κ k (4) th w hv ω ffctiv frqucy cs Morovr th coditio (3) is fulfilld Idd
i umricl xprimts w obsrv spirls similr to tht o show i Fig 34 Costt χ For compltss w giv th solutio of q (7) for costt χ It will b usd ltr to costruct th itrtio qutio i discrt cs For iitil vlu ξ() ξ th solutio is P( p ω) + ( p + ω) ξ ( ipt χ ( P ) ipt (5) whr: p ω χ χξ P χξ + ω + + ω p p For rl p th trjctoris of q (5) r o-coctric circls Thr is o lliptic fixd poit isid th uit circl For imgiry p two hyprbolic fixd poits o stbl d o ustbl ppr Both lyig o th uit circl I othr wy th rsult (5) my b obtid from q(4) if k d χc Th shp of trjctoris for rl p d imgiry p is xctly th sm s thos show i Fig3 d Fig4 Crtily t prst thy rprst trjctoris of poit ξ( For y complx χ c iβ w d oly to rott th bov pictur giv for c bout gl β At lst it is worth to mtio tht oly for ω > χ th dymics is wll dfid sic oly th th Hmiltoi () is boudd from blow [] 4 DISCRETE APPROACH 4 Formultio of th problm Our discrt modl is s follows W divid tim o sgmts of lgth T I ch sgmt volutio from ( )T to T t is fr (i χ ) d dscribd by ξ ( τ ) ξ (6) iω ( τ ) whr T > t d ξ is costt i sgmt For tim from T t to T w impos costt χ diffrt from zro so volutio is govrd
by th q (5) Diffrt vlus of χ c b i diffrt sgmts W xmi discrt vlus of ξ just ftr th pulss As rsult th pulsig dymics is dscribd by th itrtio qutio A ξ + B ξ Bξ + A (7) whr A B [ p iχ cos( p si( p iω ( T iω si( p] p iω ( T ω χ This form of A B is vlid for rl p (ω >χ) If o uss imgiry p g i kickd dymics th trigoomtric fuctios chg ito hyprbolic fuctios Formul (7) is spcil cs of so clld Möbius trsformtios wll kow i th complx pl thory Ths trsformtios form group Sic our phs spc is limitd to th itrior of th uit circl it is ough to xmi subgroup of ll Möbius trsformtios tht mp uit circl ito itslf i utomorphisms of th uit circl z + ( ) iθ z ( ) ( ) z (8) Th trsformtio dpds o o rl prmtr θ () d o complx prmtr () for which () < Th uppr idx idicts tht th prmtrs my b stp-dpdt Eq(7) is spcil cs of q(8) d th group prmtrs r dtrmid by ( i ) θ A A ( ) B A (9) Th form (8) is vry grl It mbrcs th modls of pulsd SU() dymics discussd prviously i litrtur g [3 6] Ev cotiuous dymics my b trtd s spcil cs of it if o puts T t d T Th group proprty of th utomorphism bls i pricipl to writ dow th form of th solutio ftr stps vi th iitil coditio z
z z i θ z (3) whr θ my b clld s ffctiv prmtrs Not tht thy hv lowr idics i cotrry to th currt prmtrs θ () () It is sy to fid itrtio qutios for ffctiv prmtrs iθ + ( + ) iθ iθ + + ( + ) ( + ) iθ + ( + ) iθ + i ( + ) θ + (3) From qs (3) d (3) w ot tht if z for th lso d ivrsly Th cs z is crtily log of th spirl bhvior i th cotiuous dymics I th cotrry if th itrtig poit rmis i th clos r of rdius lss th uit th it is log of th compct bhvior Numricl rsults giv us both typs of th bhvior I Fig5 d Fig6 χ rotts with d simultously its bsolut vlu chgs priodiclly with itrtio Fig5: Discrt dymics for 55si (T) xp(it) Prmtrs: T ; t T; ω ; ξ 3
Fig6: Discrt dymics for 56si (T) xp(it) Prmtrs: T ; t T; ω ; ξ Ths picturs illustrt tht thr xist som criticl vlu of cofficit χ (for giv tim sgmts) wh bhvior drsticlly chgs from compct to spirl bhvior 4 Fixd poits Fixd poits for modls which r spcil css of Möbius utomorphisms of th uit circl hs b lrdy discussd i litrtur g [3] Hr w brifly xmi th grl itrtio q(8) Its fixd poits giv by th coditio z z + fulfill th qutio iθ iθ ( ) z whr θ std hr for () θ () Th solutios r z + (3) i z i θ θ si ± θ si (33) For si (θ /) > thr r two lliptic fixd poits ivrs to ch othr with rspct to th uit circl So oly o of thm lis i itrstig us r of th uit circl I prbolic cs wh si (θ /) both poits mt ch othr i th sm plc of th uit circl Ad for si (θ /) < thr r two hyprbolic fixd poits o th uit circl O of thm is ttrctiv d th othr is rpulsiv I grl χ is itrtio-dpdt d so r prmtrs () θ () Th th bov formul givs poits which r fixd oly i trsitio 4
from stp to + I fct w c trt q (33) s qutio of motio for fixd poits Filly w xplicitly writ dow th criticl qutio: si (θ /) For th spcil modl dscribd by q(7) it tks th form { p cos( p si[ ( T ] + ω si( p cos[ ω( T ] } χ si ( p ω (34) W plot th right sid of th qutio s fuctio of χ i th physicl rg ω > χ (Fig7) Fig7: Plot of fuctio f f ( χ ) (s q34) i rg ω > χ Prmtrs T ; t 5 T; ω O of th solutios of q(34) is χ ω idpdtly of prmtrs T t Not tht i cotrry to th cotius dymics thr r rgs whr hyprbolic poits ppr d rgs whr oly lliptic poit xists v for th itrtio-idpdt χ (for giv T 43 Lir trsformtios W show hr tht i som ss w my shift th problm from Möbius mppigs of th uit circl to th lir trsformtios of it Th volutio of th itrtd poit is ruld by th chi M of Möbius mppigs (8) M W W W W W (35) whr ch W k k ms uit circl Th bov chi c b xprssd by w o L 5
M W L W whr (36) L Z W W Z Z W W W Z W Z Z k Wk Wk Z k (37) d trsformtios W k Z k d Z k W k r ch othr rciprocl W lso ssum tht Z k r uit circls d mppigs W k Z k r som Möbius utomorphisms W wt to mk th trsformtio Z Z Z W W Z k k k k k k (38) lir i it should mp ifiity o ifiity Th L bcoms th chi of lir mppigs d lir mppigs of uit circl r simply rottios of th circl L Z Z Z Z Z (39) Th lir mppig L my b sily foud Th usig (36) o my fid xprssio for w (w ) Ufortutly i grl it is impossibl to ccomplish Th xcptio is if prmtrs θ do ot dpd of itrtio th (i vry stp) it is possibl to mk tht th first mppig i (right sid) (38) mps ifiity i fixd poit of th scod mppig i (38) d th th third mppig i (38) is rciprocl to th first Usig formul (36) w obti w vi iitil coditio w It tks o th form of q (3) with th followig ffctiv prmtrs ( ) K iθ / θ i iθ (4) ( + ) ( ) whr K d + + ( + ) ( ) ( + ) cos( θ / ) + ( ) cos( θ / ) si ( θ / ) (4) si ( θ / ) Th bov rsult c b obtid i my diffrt wys I th spcil cs of kickd dymics it ws giv i [3] but without prov 6
5 FINAL REMARKS I th first prt of th ppr som fturs of th cotius dymics of SU() hv b dscribd W fid two distictiv typs of bhvior Th first o whr trjctoris td to th uit circl s limit (spirl bhvior) d th scod o whr trjctoris form mor or lss complictd figurs isid uit circl d do ot td to th limit (compct bhvior) I th xt prt of th ppr w hv show tht it is covit to xmi ll pulsd SU() group modls from th grl poit of viw of Möbius utomorphism of th uit circl Th two distictiv typs of bhvior occurrig i cotiuous cs c lso b idtifid i pulsig css As xmpl of pulsd systm srvs us th modl of fiit width d mgitud of pulsig pk Som umriclly obtid picturs g Fig5 look lik chotic Nvrthlss it c b sily provd tht th motio is i fct rgulr bcus th Lpuov xpot is ithr zro or gtiv I th lttr cs it my b fiit or ifiit Th zro Lpuov xpot corrspods to th compct bhvior Th gtiv Lpuov xpots corrspod to spirls Thr is o positiv Lpuov xpot d it ms thr is o clssicl chos i th systm Thr wr som ttmpts to look for figrprits of qutum chos i SU() systms bcus it sms tht th qutum utocorrltio fuctio xhibits som dcy [6] (s lso []) Howvr tht dcy pprs wh th typ of motio chgs from th dymics dscribd by zro Lpuov xpot to th dymics dscribd by gtiv Lpuov xpot (compr Fig5 d 6) Th clssicl d qutum mppigs bsd o th Hmiltoi () r quivlt W thik it is rthr strg to look for figrprits of qutum chos wh th clssicl coutrprt bcoms v mor rgulr th bfor trsitio 7
ACKNOWLEDGMENTS I would lik to thk Prof A Bchlr for my usful discussios durig prprtio of th ppr Without his ssistc this ppr would vr hv com ito xistc Rfrcs [] AFR d Toldo Piz PhysRv A5 6 (995) [] T Lisowski J Phys A: Mth G 5 L95 (99) [3] A Bchlr T Lisowski Phys Ltt A6 6 (99) [4] CC Grry J Kifr J Phys A: Mth G 4 353 (99) [5] CC Grry RGrob ER Vrscy Phys Rv A43 36 (99) [6] CC Grry ER Vrscy Phys Rv A39 577 (989) [7] CC Grry PhK M ER Vrscy Phys Rv A39 668 (989) [8] A Orłowski K Wódkiwicz Jour Mod Optics 37 95 (99) [9] PK Arvid JOptSocAm B5 545 (988) [] PW Miloi JR Ackrhlt ME Goggi Phys Rv A35 74 (987) [] HP Yu Phys Rv A3 6 (976) [] AM Prlomov Commu Mth Phys 6 (97) [3] AM Prlomov Usp Fiz Nuk 3 3 (977) (i Russi) [4] G Dttoli J C Gllrdo d A Torr Rvist dl Nuovo Cimto (988) [5] W M Zhg D H Fg d R Gilmor Rv Mod Phys 6 868 (99) 8