linarly acclratd []. Motivatd by ths complications w latr xamind mor carfully th cts of quantum uctuations for an lctron moving in a circular orbit [2

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1 OSLO-TP 3-98 April-998 Acclratd Elctrons and th Unruh Ect Jon Magn Linaas Dpartmnt of Physics P.O. Box 048 Blindrn N-036 Oslo Norway Abstract Quantum cts for lctrons in a storag ring ar studid in a co-moving, acclratd fram. Th polarization ct du to spin ip synchrotron radiation is xamind by trating th lctron as a simpl quantum mchanical two-lvl systm coupld to th orbital motion and to th radiation ld. Th xcitations of th spin systm ar rlatd to th Unruh ct, i.. th ct that an acclratd radiation dtctor is thrmally xcitd by vacuum uctuations. Th importanc of orbital uctuations is pointd out and th vrtical uctuations ar xamind. Introduction Th mting at Montry has bn ddicatd to th mmory of John Bll, and I would lik to ddicat also th prsnt papr to his mmory. Th papr is basd on a work which Idid with him svral yars ago [, 2] and on a talk I gav at CERN in Th collaboration with John Bll startd whn, as a fllow at CERN, I discussd with him th rathr xotic thortical ct oftn rfrrd to as th Unruh ct. As shown by Davis [3] and Unruh [4], an idalizd radiation dtctor which is acclratd through ordinary Minkowski vacuum gts hatd du to intractions with th vacuum uctuations of th radiation ld. For uniform linar acclration th xcitation spctrum has a univrsal, thrmal form, indpndnt of dtails of th dtctor. This ct, that vacuum sms hot, as masurd in an acclratd systm, has by Unruh and othrs bn rlatd to th phnomnon of Hawking radiation from black hols [5]. Th ida cam up during our discussion that a dpolarization ct which was known to xist for lctrons in a storag ring could hav somthing to do with th Unruh hating. W invstigatd this and found that th ct indd was rlatd, although thr wr important complications du to th fact that th lctrons wr following a circular orbit rathr than bing To appar in th Procdings of th Advancd ICFA Bam Dynamics Workshop on Quantum Aspcts of Bam Physics, Montry January 4-9, 998, d. Pisin Chn. 2 I am indbtd to Pisin Chn for his invitation to includ this papr in th procdings from th Montry workshop. A writtn vrsion of th talk at CERN has bn plannd to b publishd, but has bn long dlayd.

2 linarly acclratd []. Motivatd by ths complications w latr xamind mor carfully th cts of quantum uctuations for an lctron moving in a circular orbit [2]. In this papr I rviw th dscription of th polarization cts for circulating lctrons, in th way discussd in our two paprs. Th spin xcitations ar studid in a co-moving, acclratd fram, which follows th classical path of th circulating lctron. I rst discuss th dscription of spin motion in th acclratd fram and thn xamin spin xcitations producd ithr dirctly through uctuations in th magntic ld or indirctly through uctuations in th path. Th clos connction to th Unruh ct for a linarly acclratd systm is dmonstratd, and th polarization ct is compard to a simpl two-lvl modl with thrmal xcitation of th uppr lvl. Th vrtical uctuations in th path ar shown to giv an indpndnt dmonstration of th circular Unruh ct. 2 Quantum cts for acclratd lctrons Th motion of particls in acclrators can mostly b undrstood and dscribd in classical trms. But thr ar som quantum cts which ar non-ngligibl and which vn may b important. Ths mainly hav to do with radiation phnomna and with th radiation raction on th acclratd particls. Thrfor thy ar much mor important for th light lctron than for th much havir proton. Th acclratd lctrons mit radiation, synchrotron radiation, as it is known for particls in a magntic ld. Evn for high nrgy lctrons this procss is wll dscribd by th classical radiation formula. This was xplicitly dmonstratd by Schwingr [6] who calculatd th lowst ordr quantum corrction to th radiatd powr. Only for xtrmly high nrgtic lctrons th quantum corrctions bcom important. Th condition for this bing small can b writtn as c () with = r mc q ( vc )2 ) ; c = h (2) whr m is th lctron mass, v its vlocity and is th radius of curvatur of th particl orbit. Th important ratio thn is =( c ) 2 a a m ; () (3) whr a is th acclration of th particl in an inrtial rst fram, and a m is an acclration paramtr dtrmind by th particl mass, a = 2 v 2 ; a m = mc3 h (4) Thus, th important physical quantity is th acclration a rathr than th nrgy of th lctron. A typical valu for th paramtr in cyclic acclrators is 0 6, which shows that quantum 2

3 cts indd ar vry small. 3 Evn if synchrotron radiation is ssntially a classical phnomnon for c, this dos not man that all quantum cts associatd with this phnomnon ar unimportant. Th radiation raction will xcit orbital oscillations vn for much smallr nrgis []. Th radiation ld thn acts in two ways on th particl. Quantum uctuations xcit th oscillations, whil radiation damping tnds to rduc th oscillation amplituds. Balanc btwn ths two tndncis dns a minimal, quantum limit to th bam siz. A prhaps vn clarr dmonstration of quantum cts for lctrons in a storag ring is associatd with th phnomnon of spin-ip radiation [2, 3]. Th asymmtry btwn up and down ips in th magntic ld lads to a gradual build up of transvrs polarization of th lctrons. Undr idal conditions th polarization approachs quilibrium as with a maximum polarization and a build up tim P (t) =P 0 ( t=t 0 ); (5) P 0 = 8 5 p =0:924 (6) 3 t 0 = 8 5 p m 2 c 2 3 (7) 3 2 h 5 For xisting acclrators this build up tim is of th ordr of minuts to hours. Th phnomnon of spontanous polarization of lctrons circulating in a magntic ld has bn analyzd in many publications, both for idal conditions and for th mor ralistic situation with particls moving in a variabl magntic ld. Thr also xist rviw articls on this intrsting subjct [4, 5, 6]. In particular th papr by J.D. Jackson focuss attntion on th aspcts of this phnomnon which can b dscribd in lmntary trms. My approach hr is along th sam lin. But whras Jackson rjcts th ida of a simpl dscription of th ct as a transition btwn spin nrgy lvls causd by radiation cts, which thn would lad vntually to all particls in th lowst nrgy lvl, this is in fact th pictur I will us. Th lctron spin is tratd as two-lvl quantum systm intracting with th radiation ld. But th ct of th radiation ld along th acclratd orbit of th lctrons is dirnt from th ct on an lctron sitting at rst. Transitions also to th uppr nrgy lvl ar inducd by th ld along this orbit, and that lads to a small, but non-vanishing dpolarization of th lctron bam. I will considr th idal cas of lctrons moving in a rotationally invariant magntic ld. A stabl, classical circular orbit is producd by a radial gradint in th magntic ld. This corrsponds to th situation of a wak focussing machin. 3 Thr hav bn som studis also of th quantum rgim > in connction with linar acclrators at xtrmly high nrgis [7, 8, 9, 0]. 3

4 Tabl : Rotation frquncis of thr dirnt co-moving frams and th spin prcssion frquncis in ths frams for an lctron orbiting in a constant magntic ld B frams rotational frq: spin prcssion C 0 g 2mc B O mc B (g 2) 2mc B L ( ) mc B ((g 2) + 2 ) 2mc B 3 Hamiltonian for th acclratd fram Whn w introduc a co-moving fram for th dscription of th spin motion of th lctron, it is of intrst to not that this fram is uniquly dnd only up to a (tim-dpndnt) rotation. Thr ar in fact thr dirnt co-moving frams which ar charactrizd by simpl proprtis, in on way or th othr. Th rst on, which I dnot th L-fram is th on which is obtaind from th xd lab fram by a pur boost. This fram is non-rotational as sn from th lab. Th othr on is th fram which rotats with th frquncy of th orbiting particl. In this fram, which I dnot th O-fram, th acclrating ld is stationary. Finally thr is a fram, dnotd th C-fram, which is non-rotational whn viwd along th particl orbit. Th fact that this is dirnt from th L-fram is a rlativistic ct and givs ris to Thomas prcssion of th spin vctor. Th rlativ rotational frquncis of ths thr frams ar listd in Tabl for an lctron following a circular path in a magntic ld. Both th magntic ld B and th frquncis rfr to co-moving frams. Also th spin prcssion frquncis in th thr dirnt frams ar shown. This has th simplst in th C-fram. Sinc this fram is non-rotational along th orbit, all spin prcssion in this fram is du to coupling btwn th magntic momnt of th particl and th xtrnal magntic ld. Thus, th prcssion frquncy is proportional to th gyromagntic factor g, as shown in th tabl. In th O-fram on th othr hand th prcssion frquncy is proportional to g 2. This dmonstrats th wll-known fact that for g = 2 th spin prcsss xactly with th frquncy of th orbital motion. Finally, in th L-fram thr is a furthr corrction du to th rlativ rotation of th L- and th O-fram. This corrction is idntical to th rotation frquncy of th orbital motion, and this is smallr than th frquncy associatd with th Thomas prcssion by a factor of =. Evn if th spin motion is simplst in th C-fram, I shall in th following apply th O-fram for th quantum dscription. Th rason for this is that th xtrnal lds ar stationary in this fram and thrfor giv ris to a tim-indpndnt Hamiltonian. This fram will b xtndd to a local acclratd coordinat systm to allow for uctuations in th particl about th circular path, which thn is assumd to b th classical, stabl orbit of th lctrons. This coordinat systm will ncssary contain coordinat singularitis at som distanc from th orbit. But I will assum uctuations away from th stabl orbit to b small, so that linarizd quations 4

5 ar sucint. I will also assum vlocitis in this fram to b small, so that a non-rlativistic approximation can b applid. Th Hamiltonian, which govrns th tim volution in th acclratd fram is not idntical to that of th inrtial rst fram, but it can b xprssd in a simpl way in trms of obsrvabls from this fram, H 0 = H a v J z + a c K x (8) Hr H is th Dirac Hamiltonian, J z is th gnrator of rotations in th plan of th lctron orbit and K x is a boost oprator. Th coordinats in th O-fram thn ar chosn with th particl acclration in th ngativ x-dirction and with th orbit vlocity in th positiv y-dirction. Th additional trms in th xprssion for H 0 ar fairly asy to undrstand. Th prsnc of th gnrator of rotations is rlatd to th fact that th coordinat axs of th O-fram rotat along th orbit and th prsnc of th boost oprator accounts for th continuous jumping btwn inrtial frams whn th particl is acclratd. For th thr oprators includd in H 0 w hav th following xprssions, H = c~ ~ + mc h 2mc (i~ ~ E ~ ~ B) (9) J z =(~r~p) z + 2 h z (0) K x = (xh + Hx) () 2c whr a trm for th anomalous magntic momnt, = (g 2), has bn introducd in th 2 xprssion for H. All th potntials and lds in ths xprssions rfr to th inrtial rst fram of th classical orbit. Th notation ~ = ~p A ~ has bn usd for th mchanical momnt of th c lctron. Whn th uctuations around th classically stabl orbit ar assumd to b small, thn a non-rlativistic approximation maks sns. AFoldy-Wouthuysn transformation, whr w kp only th lading trms, givs a Hamiltonian which thn can b split into a spin-indpndnt and a spin dpndnt trm in th following way, H = H orb + H spin (2) ax ~2 H orb =( )( c2 2m + ) iah axm + 2mc a 2 x v (~r ~p) z + ::: (3) H spin = h ax ~ [( 4mc c )g B ~ + g ah ~E ~] 2 mc 2 v z + ah 4mc (~ ~) 2 x + ::: (4) Th Hamiltonian in th acclratd fram includs som complications compard to that of th inrtial fram. Howvr, if w considr th orbital motion only to linar ordr in th dviation from th stabl orbit, th main dirnc in th xprssions for H orb is th prsnc of th cntrifugal and Coriolis trms. For th orbital motion th spin cts only giv ris to a small prturbation. But also for th spin motion th ct of th uctuations in th orbit is small, sinc 5

6 th spin prcssion mainly is dtrmind by th strong magntic ld along th classical orbit. In principl on could thn dtrmin th particl motion in th following way: On rst solvs for th orbital motion, nglcting th spin, and thn on dtrmins th spin motion, trating th orbital uctuations and th quantum lds as prturbations. Howvr, whn calculating th polarization of th particl bam, on can simplify this approach somwhat, sinc it is only th uctuations in th particl orbit which ar drivn by th coupling to th radiation ld which ar important. 4 Spin transitions Th spin Hamiltonian can b writtn in th form with ~! 0 giving ris to th classical part of th prcssion, H spin = h~! ~ (5) 2 ~! = ~! 0 + ~! (6) ~! 0 = and ~! as th uctuation part, ~! = 2mc g ~ B 0 2mc [g ~ B a v ~ k = 2mc (g 2)B 0 ~ k (7) ax c g B ~ 2 0 (g 2) a ~ i ~] (8) c In th last two xprssions ~ k is th unit vctor orthogonal to th plan of motion and ~ i a unit vctor in th x-dirction. ~ B0 = B 0 ~ k is th xtrnal magntic ld on th classical orbit, and ~ B accounts for th uctuations in th magntic ld. This can b sparatd into two parts, ~ B = ~ B q + ~ B c (9) whr B ~ q dnots th quantum ld along th classical orbit and B ~ c is th variation in th xtrnal ld du to uctuations in th orbit. Th spin motion now can b dtrmind by tim dpndnt prturbation thory. ~! 0 thn dns th unprturbd part of th spin Hamiltonian and ~! th prturbation. To rst ordr, th transition probabilitis pr unit tim btwn th lvls of th unprturbd Hamiltonian ar givn by = lim = 4 T! Z + Z T=2 4T j i! 0! ()j0 > j 2 T=2 d i! 0 < 0j! (=2)! ( =2)j0 > (20) 6

7 j0 > in this quation dnots th stat of th combind systm of radiation ld and orbit variabls, unprturbd by th spin.! is a linar combination of th x- and y-componnt of~!,! =! x i! y (2) Th sam notation will b usd for othr variabls in th following. A usful substitution rul which can b usd in th xprssion for! is th following on, d d F!i! 0F (22) Th dirnc btwn ths two xprssions only givs ris to nd cts in th intgral for th transition amplitud, and for larg T this is supprssd in du to th prfactor =T. This substitution rul now can b usd to liminat th orbital variabls in th xprssion for!, which can b writtn in th form! = 2mc [gb m q + gb c 2iv! 0 _z] (23) c With th stabl orbit in th symmtry plan of th magntic ld, B c gts contribution only from th gradint in th z-dirction. This implis that (to lowst ordr)! only dpnds on th vrtical uctuations in th particl orbit. Ths uctuations in turn ar dtrmind by coupling to th radiation ld in th following way, z 2 2 z 3mc 3 (::: a 2 c 2 _z)+2 z= m E qz (24) whr a radiation raction trm has bn introducd and whr non-linar trms hav bn nglctd. Th rstoring lctric forc in th z dirction can b rlatd to th gradint in th magntic ld, 2 = a n (25) n = B = B is th radius of th (classical) lctron orbit, and n th fall-o paramtr of th magntic ld. By us of th substitution rul q.(22) now can b solvd for z, (26) z =[ 2! 2 0 i! 0] m E qz + (27) = 22 3mc 3 (a2 c 2 +!2 0) (28) hr dnots a trm which is supprssd for larg T. Whn th xprssion for z is insrtd in q.(23), this givs (for v c),! = 2mc [gb q +(2+f (g))e qz ] (29) 7

8 P γ /γ r Figur : Th quilibrium polarization P as a function of clos to th rsonanc with th vrtical oscillations. Th dashd lin corrsponds to th limiting valu P =0:92, away from th rsonanc. Th scal for is rlativ to th rsonanc valu r. with f (g) as a rsonanc trm, (g 2) 2 f (g) = 2! 2 0 i! 0 This trm blows up whn th frquncy of th fr oscillations in th z-dirction is clos to th classical spin prcssion frquncy, but it tnds rapidly to zro away from th rsonanc. Th nw xprssion for! (29) now only dpnds on th fr quantum lds, and th transition probabilitis can b xprssd in trms of corrlation functions of ths lds along th particl orbit, = 4 Z + (30) d i! 0 < 0j! (=2)! ( =2)j0 > (3) Now j0 > rfrs to th vacuum stat of th radiation ld. To calculat ths probabilitis now is straightforward. Th lds in th co-moving fram most convnintly ar xprssd in trms of lab fram lds and th corrlation functions of ths ar found by xprssing th ld oprators in trms of cration and annihilation oprators. To lading ordr in = th rlvant fourir intgrals can b calculatd analytically. Th polarization is dtrmind by th population of th two spin lvls, and this in turn is found by th standard argumnt of quilibrium btwn transitions up and down. W hav, P = + (32) + + In Fig. th polarization is shown as a function of. Excpt for valus clos to th rsonanc with th vrtical motion, th standard rsult for th polarization is found, P = 0:924. Th ct of th rsonanc is mainly to dpolariz th bam, but an intrsting dtail is th cohrnt ct which givs a maximum valu of P =0:992 clos to th rsonanc. Thus, at last in principl it is possibl to xcd th limiting valu of

9 5 Quantum uctuations and th Unruh ct Th xprssion for th transition probabilitis now maks it possibl to s th clos rlation btwn th polarization ct and th Unruh ct [4]. Lt m rwrit it in th form with Z + = i! 0 C () (33) C + () =<D y (0)D() > (34) C () =<D()D y (0) > (35) Ihav hr introducd th nw notation D =(=2)! + ;D y =(=2)!. Th oprator D thn is a linar combination of lctric and magntic lds in th co-moving fram, D() =~ ~ E(x()) + ~ ~ B(x()); (36) whr x() is th spac-tim orbit of th particl. Th xprssion (33) is similar to that which dns transitions in a point dtctor in th cas of th Unruh ct. Th main dirnc is that th world lin of th dtctor thn corrsponds to linar acclration rathr than to circular motion as in th prsnt cas. Howvr, th xcitations of th acclratd systms in th two cass can b undrstood qualitativly in th sam way. Th corrlation functions C givs a masur of vacuum uctuations of th lctromagntic ld along th orbit x(), and ths uctuations giv ris to xcitations in th dtctor whn C + includs a spctral componnt which coincids with th xcitation nrgy. To s this mor clarly lt m rst discuss th simplst cas, namly with a two lvl dtctor at rst. Th transitions btwn th lvls ar givn by th sam st of quations, (33-36), but now simply with x() =(;0) (37) Sinc D() is a linar combination of lctromagntic lds it can b dcomposd in th form D() = Z d 3 k X r [c r ( ~ k) ikx a r ( ~ k)+d r ( ~ k) +ikx a y r( ~ k)] (38) whr a r ( ~ k) and a y r( ~ k) ar photon annihilation and cration oprators and c r ( ~ k) and d r ( ~ k) fourir cocints. According to q.(33) only th positiv frquncy parts of this oprator ar rlvant for th transitions. Positiv frquncy thn is masurd rlativ to th propr tim along th orbit x(). But with th dtctor at rst this coincids with positiv frquncis masurd in th lab fram. And, as is wll known, th positiv frquncy part of lab fram lds only contains annihilation oprators. This is simply bcaus all xcitations in th lab fram hav positiv nrgy. So th rlvant componnt ofd() is Z + Z X d i! 0 D() =2 d k! 2 0 c r ( ~ k) i~ k~x ar ( ~ k) (39) 9 r

10 which only annihilats photons with th sam nrgy as th nrgy splitting of th two-lvl systm. As a consqunc of this th probability for transitions up in nrgy is zro, sinc th D oprator acts on th vacuum stat. Howvr, transitions down may b dirnt from zro, sinc in this cas it is instad D y which acts on th vacuum stat. Thus, th vacuum uctuations only induc transitions to lowr nrgis. This clarly is rlatd to nrgy consrvation in th combind systm of dtctor and radiation ld. If th two-lvl systm movs with constant vlocity th pictur is th sam, sinc th sign of th zro componnt of th photon momntum k is th sam in all inrtial frams. Th only way to hav a non-zro probability for xcitations to highr nrgis is to includ othr stats than th vacuum stat in th on which th oprator D acts on. In particular th probability is non-zro for stats with tmpratur T 6= 0. Howvr, for acclratd motion this is no longr th cas. x() thn is no longr a linar function of and both functions ikx( ) and +ikx( ) will in gnral hav positiv frquncy parts in trms of th variabl. In addition, for th lctromagntic ld, thr will b a -dpndnt Lorntz transformation conncting th lds in th co-moving fram with th lab fram lds. Th nt ct is to introduc a mixing btwn th positiv and ngativ frquncy parts, so that both th anihilation and th cration parts of th oprator D will hav positiv frquncy componnts in trms of th tim variabl. As a consqunc of this thr will b in gnral non-vanishing probabilitis for xcitations both up and down in nrgy for th acclratd systm, vn with th quantum ld in th vacuum stat. For uniform linar acclration along th z-axis, th acclratd path x() is dscribd by t = a c sinh( a c ); z = a c cosh( a ); x = y =0; (40) c Th trajctory x() in this cas dpnds only on on fr paramtr, which is th rst fram acclration a. An intrsting symmtry which is prsnt for this motion corrsponds to a shift in th -paramtr in th imaginary dirction, x() =x(+i 2c a ) (4) This symmtry, togthr with gnral symmtris from ld thory, rlatd to PCT-invarianc [7, 8, 9], givs a simpl rlation btwn th corrlation functions corrsponding to transitions up and down in nrgy, C + () =C ( i 2c a ) (42) This rlation is similar to on which is prsnt for corrlation functions at non-zro tmpratur, and it lads to a similar rsult for th ratio btwn probabilitis for transitions up and down, + = = Z + d i! 0 C ( Z + = xp( i 2c a ) d i! 0( +i 2c a ) C () h! 0 ah=2c ) (43) 0

11 If th ratio btwn th two transition probabilitis now is intrprtd as a Boltzmann factor, thn thr is a simpl linar rlation btwn th tmpratur associatd with this factor and th acclration a, kt a = ah (44) 2c T a thn is th Unruh tmpratur for th acclratd systm and k th Boltzmann constant. Th drivation shows that th thrmal proprty of th xcitation spctrum dpnds only on gnral proprtis of th quantum lds and on spcial proprtis of th acclratd trajctory x(). Dtails of th acclratd systm is not important. In th cas of lctron polarization on may considr th qustion whthr linarly acclratd lctrons could b usd to dtct th Unruh ct. An additional magntic ld along th lctron path could provid th ncssary splitting of th spin nrgy lvls. In principl this should giv a clanr dmonstration of th hating by acclration ct than th polarization ct for circulating lctrons. Howvr, as discussd in rf. [] th tim ndd to rach quilibrium is far too long to mak this ct rlvant for th motion of lctrons in linar acclrators. For lctrons in cyclic acclrators much largr acclrations can b obtaind and corrspondingly much smallr tim constants for th approach to quilibrium. 6 Unruh ct for circulating lctrons Lt m now considr th lctron polarization for circulating lctrons from th point of viw of th Unruh ct. On main dirnc btwn circular motion and linar acclration is that th formr dpnds on two indpndnt paramtrs, which wmay tak tobaand. Howvr, for ultrarlativistic lctrons th quantum uctuations acting th spin motion in th co-moving fram ssntially only dpnd on a. If w disrgard all complications rlatd to th circular motion and simply assum th tmpratur formula (44) to b valid, w nd th following. In th non-rotational C-fram th nrgy splitting of th spin systm is for, This givs a Boltzmann factor and th polarization is xp( E =h! 0 = g ha 2c E (45) kt a ) = xp[ g] (46) P (g) = tanh[g=2] (47) For th physical valu g = 2 this givs P =0:996 as compard with th prviously citd corrct valu P =0:924. It is of intrst to notic th similaritis and th dirncs in th g-dpndnc of th corrct function P (g) and th function obtaind from th simpl tmpratur formula. Th two functions ar displayd as curv A and B in Fig. 2. Th main dirnc btwn ths two curvs is a shift along th g-axis. Such a shift may loosly b associatd with an angular vlocity prsnt in th

12 B 0.5 C A P g Figur 2: Th lctron polarization P as a function of th gyromagntic factor g. Th curv A is basd on th dtaild calculations. Curv B is basd on a two-lvl spin modl with thrmal xcitations du to th Unruh tmpratur. Curv C is obtaind from th dtaild calculation (curv A) whn trms idntid as du to orbital uctuations ar supprssd. systm which coupls to th spin. Thus, if w assum th spin xcitations to hav a thrmal form in th rotating O-fram rathr than in th C-fram, this would shift th polarization curv with two units along th g-axis. Th corrct curv is locatd somwhr inbtwn, and thr sms not to b a simpl xplanation for this. On should also not som important dirncs in th dtails of th two curvs. In th tmpratur formula givn abov w hav mad th simpl assumption that th spin systm can b considrd as a thrmally xcitd two-lvl systm, indpndnt of th othr dgrs of frdom of th lctron. This may b a too simpl modl vn if th Unruh tmpratur formula, in som approximat maning, should b valid. Aftr all th uctuations in th path ar important for th polarization ct, as Ihavprviously pointd out. In ordr to stimat th ct of ths uctuations, w may simply lav out th two trms in (23) associatd with uctuations in th path and kp only th trm which accounts for th dirct coupling of th magntic momnt to th quantum uctuations of th magntic ld. Th rsulting function is displayd as curv C in Fig. 2. Now th shift rlativ to th thrmal curv has disappard and th form is also quit similar in th two cass. Th nw curv can in fact b approximatd wll by a formula lik (46), but with a tmpratur somwhat highr than th Unruh tmpratur, T ff :3T a. Sinc th orbital uctuations ar important for th polarization ct, a bttr comparison with th Unruh ct would b obtaind by trating th orbital motion togthr with th spin motion. Howvr, as alrady pointd out, th lctron spin is not so important for th orbital motion. As a nal point I will thrfor considr th vrtical motion without taking into account th cts of th spin. 2

13 Th (Hisnbrg) quation of motion for th vrtical oscillations can b writtn as z +2 _z+ 2 z= m E qz (48) In this quation = ( 2 a 2 )=(3mc 5 ) and only th most important part of th radiation damping trm has bn kpt. Making us of th fact that th damping is small,, on can solv th quation to nd an (approximat) xprssion for z() in trms of th quantum ld E qz. For th uctuation in th z-coordinat on nds th following xprssion <z 2 >= Z 2 ( + m )2 d jj cos <E qz (=2)E qz ( =2) > (49) This shows that th uctuations in th vrtical dirction ar dtrmind by th corrlation function of th z-componnt of th lctric ld along th classical orbit. Th vrtical uctuations in fact can b intrprtd as bing du to th circular Unruh ct in a similar way as th polarization ct. Th man nrgy associatd with th uctuations is for larg, <E> vrt = m 2 <z 2 >= 3 p ah 3 96 c It is proportional to th acclration a, but with a dirnt prfactor as compard with th linar Unruh ct. It corrsponds to a somwhat highr tmpratur (50) T ff :5T a (5) To linar ordr th xcitation spctrum in this cas in fact has a thrmal form, and thr is no complication with rotating frams. So in this rspct th vrtical orbit xcitations giv a simplr dmonstration of th Unruh hating in th circular cas than th dpolarization of th lctrons do. But th uctuations ar small and to masur thm may bmuch mor dicult task. 7 Concluding rmarks Th quantum cts for lctrons in a storag ring hav hr bn studid within a simpl idalizd modl for a cyclic acclrator. Th lctron spin has bn tratd, in th co-moving acclratd fram of th classical orbit, as a two-lvl systm coupld linarly to th quantum lds and to th orbital uctuations. In this dscription th xtrnal magntic ld along th orbit dns th (unprturbd) spin lvls of th lctrons, and th radiation ld causs transitions btwn ths two lvls. Th radiation ld acts both dirctly on th spin, through th coupling to th magntic momnt, and also indirctly, through th uctuations it introducs in th particl orbit. Th quilibrium polarization calculatd in this way agrs with th classical rsults known from th litratur. Howvr, a rsonanc btwn th spin and orbital uctuations givs an ct which mainly is dpolarizing, but clos to th rsonanc lads to a small incras in th polarization. 3

14 In a mor ralistic dscription of a cyclic acclrator thr will b svral modications of this pictur. In th cas of a strong focussing machin th magntic ld will no longr b uniform along th orbit. Th unprturbd part of th spin Hamiltonian thn will b tim dpndntinth co-moving fram, and as a consqunc of this th prturbations cannot b dscribd in trms of transitions btwn stationary spin lvls. Thr may b othr prturbations in th magntic ld that caus a coupling btwn vrtical and horizontal oscillations. Also non-linar cts may b important. This will in gnral lad to a much richr structur of spin-orbit rsonancs than in th idalizd modl whr only on rsonanc is prsnt. All ths cts crtainly hav to b takn into account whn on wants to modl th spin bhaviour in a ral acclrator [20, 2, 22]. Nvrthlss, to undrstand th main aspcts of th quantum cts for th acclratd lctrons, th simpl idalizd modl usd hr may bofintrst. As discussd in this papr, th xprssion for th spin ip probabilitis can b rducd to a form whr thy ar dtrmind only by vacuum corrlation functions of th lctromagntic lds along th classical orbit. Exprssd in this way thr is a clar similaritybtwn th polarization ct and th Unruh ct for a linarly acclratd two-lvl systm. But for circular motion th corrlation functions do not hav a truly thrmal form. And for th circulating lctrons thr ar complications du to th rotations of frams along th orbit and du to coupling btwn th spin and orbital uctuations. Howvr, whn th cts of th orbital uctuations ar suprssd w obtain a polarization curv which iswll approximatd by th curv obtind from a thrmally xcitd two-lvl systm. A mor carful comparison btwn th polarization ct and th Unruh ct would man to includ th orbital motion in th dscription. Sinc th uctuations in th orbital motion can b tratd as bing indpndnt of th spin, it is maningful to xamin th vrtical uctuations sparatly. Also ths utuations ar dtrmind by vacuum uctuations of th lctromagntic ld along th classical orbit. Th rsult for th vrtical uctuations is that thy do hav a thrmal xcitation spctrum, but th tmpratur is slightly highr than th Unruh tmpratur dtrmind from th acclration alon. 4

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