PHYSICAL REVIEW E, VOLUME 65, 016507
Quantum theory of optical stochastic cooling
S. Heifets Stanford Linear Accelerator Center, Stanford University, Stanford, California 94309
M. Zolotorev Lawrence Berkeley National Laboratory, Berkeley, California 94729 ͑Received 18 December 2000; revised manuscript received 21 August 2001; published 20 December 2001͒ Quantum theory of optical stochastic cooling is presented. Results include a full quantum analysis of the interaction of the beam with radiation in the undulators and in the quantum amplifier. A density matrix of the whole system is constructed and the cooling rate is evaluated. It is shown that quantum fluctuations change classical results of stochastic cooling at low bunch population and set a limit on the cooling rate.
DOI: 10.1103/PhysRevE.65.016507 PACS number͑s͒: 29.27.Ϫa, 52.59.Px
I. INTRODUCTION Cooling for short bunches requires the amplifier to have a large bandwidth. For example, the bunches in the PEP-II B ͑ ͒ ϭ Recently, optical stochastic cooling OSC was proposed factory have b 33 ps. To affect different slices in the ͓ ͔ Ͻ ⌬ 1,2 as a method for the fast cooling of short bunches in bunch, Ns has to be small, Ns Nb , which requires f storage rings. In OSC, a wave of radiation is generated by a ϭ⌬/2Ͼ10 GHz. The actual PEP-II bandwidth is only particle in an undulator. After amplification in the optical 250 MHz. amplifier, the wave is sent to another undulator where its The advantage of optical stochastic cooling is large ⌬ f , interaction with the parent particle provides the desired cool- allowing for fast cooling. In this method, the bandwidth ⌬ f Ӎ␥2 ϭ ␥ ing. The first and second undulators work as a pickup-kicker 0(c/Lu) where Lu Nu u is the undulator length and 0 pair in conventional rf stochastic cooling. The phase shift of is relativistic factor of the beam in the laboratory frame. the wave, with respect to the particle, is controlled in a dis- Parameters of the undulator have to be chosen to match the persion section between the two undulators. undulator mode to the central frequency and bandwidth ⌬ f Particle radiation affects other particles in the bunch lead- of the amplifier. For the typical solid state Ti:sapphire ampli- ing to diffusion that limits the damping rate. In this respect, fier (ϭ0.8 m, ⌬ f / f Ӎ1/5). optical stochastic cooling is not different from rf stochastic Given bandwidth, fast cooling ͑for example, in the muon cooling. The number of interacting particles ͑the number of collider͒ can be achieved by reducing the number of particles ‘‘particles per slice’’͒ in the bunch with rms length b per slice Ns . For one particle per slice, cooling would be ϭc and bunch population N is b B achieved just in one turn. However, with a small Ns , classi- cal and quantum fluctuations could become dangerous. In- Nb deed, the average number of photons radiated in the main N ϭ , ͑1͒ s ⌬ͱ undulator mode is ␣ ϭ1/137. Therefore, one can expect that 2 b 0 a photon is radiated once per 137 turns, and that fluctuations and is defined by the bandwidth of the amplifier ⌬. This may be large. Concern with quantum fluctuations is the pri- interaction of particles changes the momentum of the jth mary motivation for the study presented here. particle ͓3͔, In our consideration we determine the evolution of the quantum-mechanical density matrix of the system ͑bunch ͒ ¯ ϭ Ϫ⌳ Ϫ⌳ ͑ ͒ and radiated mode through the undulators and the quantum p j p j p j ͚ pi , 2 iÞ j amplifier. The paper starts by defining the initial density ma- trix of the beam and then by describing the beam/radiation where ⌳ is the parameter of interaction between particles interaction in the undulator. Beam dynamics in the undula- proportional to the electronic gain of the amplifier. The rms tors is described in the rest frame of a bunch ͑cf. to Renieri 2ϭ ͚ ͓͗ 2͘Ϫ͗ ͘2͔ ͓ ͔ ͒ energy spread p (1/Nb) j (p j) p j for initially 4,5 and co-workers where other references can be found . uncorrelated particles is changed by The formalism we use ͓7͔ reproduces Becker and McIver ͓6͔ results but differs from their formalism. In both cases, the ⌬͓2͔ϭϪ ⌳2ϩ⌳2 2 ͑ ͒ p 2 p Ns p . 3 number of particles per bunch Ns can be arbitrary but the effects of bunching are neglected. In this sense the interac- The cooling rate is tion of particles with radiation is weak. This assumption sub- stantially simplifies consideration being quite adequate for 1 ⌬2 ϭ pϭϪ ⌳ϩ⌳2 ͑ ͒ describing optical stochastic cooling. 2 2 Ns . 4 NTurn p Next, the evolution of the density matrix in the quantum amplifier is described, taking into account the nondiagonal ϭ The maximum cooling rate of 1/Nturn 1/Ns is determined components of the density matrix. Interaction between am- by the number of particles per slice, and is achieved at ⌳ plified radiation with the bunch in the kicker is then consid- ϭ 1/Ns . ered in the same way as it was for the pickup undulator.
1063-651X/2001/65͑1͒/016507͑12͒/$20.0065 016507-1 ©2001 The American Physical Society S. HEIFETS AND M. ZOLOTOREV PHYSICAL REVIEW E 65 016507
⌬ Finally, the full density matrix is constructed and is used to corresponding to the energy spread plab in the laboratory calculate the one-pass variation of the rms energy spread of frame. the bunch. Optimum cooling rate of OSC is determined and At the entrance to the pickup, each particle is described ͑ ͒ 0 Ј ϭ compared with the cooling rate of classical theory. It is by the density matrix Eq. 9 , (pi ,pi), i 1,2,...,Nb . shown that quantum fluctuations set a limit for the damping The density matrix of the whole bunch ˆ rate of this method. ϭ͟ NB ͉ Ј͘0 Ј ͗ ͉ iϭ1 p (pi ,pi) p . In the moving frame, interaction of particles with the II. INITIAL DENSITY MATRIX OF A BUNCH mode kϭ/c is described by the Hamiltonian We consider only longitudinal motion in a bunch, assum- NB pˆ 2 ing that the Nb particle bunch is described by the Gaussian i ϩ 2ikzˆϩit ͑ ͒ Hϭ͚ ϩh͑a aϩ1/2͒Ϫihg͓ae Ϫc.c.͔, normalized distribution function DF f (p,z), iϭ1 2m ͑11͒ ͵ f ͑p,z͒dpdzϭ1. ͑5͒ ϭ ͱ ϩ 2 where m me 1 K0. If the vector potential of the radiation is normalized to one ˆ The classical DF f is related to Wigner’s density matrix .In photon per volume V ͓6,7͔ the momentum representation, the relation is 2 2 hc ˆ Ldq Aជ ϭͱ yជ͓aˆ eikzϩc.c.͔, ͉yជ͉ϭ1, ͑12͒ f ͑p,z͒ϭ ͵ ͑pϩq/2,pϪq/2͒eiqz/h, ͑6͒ V ͑2͒2 ϭ where the density matrix is normalized, then the parameter of the interaction g gk , where 2 Ldp K0 e 2 ˆ ϭ͉pЈ͗͑pЈ,p͒͘p͉, ͵ ͑p,p͒ϭ1. ͑7͒ g ϭc ͱ . ͑13͒ 2 k ͱ ϩ 2 hc kV 1 K0
For a Gaussian DF localized around the point z0 , p0 in the In the one-dimensional model, the beam interacts with phase space, one radiated mode. In this case, operators a and aϩ change the number of coherent photons in the mode, and vector 1 ͑pϪp ͒2 ͑zϪz ͒2 ͑ ͒ϭ ͫϪ 0 Ϫ 0 ͬ ͑ ͒ potential must be normalized within the phase volume ⍀ of f p,z ⌬exp ⌬2 2 . 8 2 2 2 the mode. ͓ ͔ ⍀ϭ 3 3 2 In the laboratory frame 11 , (V/(2 ) )( k /Nu). The corresponding density matrix 0 is the wave packet ͉ Ϫ͉р This can be obtained from the constrain 2 Nu on ϭ͉ Ϫ ͉ hͱ2 i 1 2 the phase slippage t kzz along the undulator and 0͑ Ј ͒ϭ ͫϪ ͑ ЈϪ ͒ Ϫ ͩ ͪ ͑ ЈϪ ͒2 from the requirement that the frequency spread ͉( p ,p exp p p z0 p p L⌬ h 2 h Ϫ ͉Ͻ r)/ r 1/(2Nu), where r is resonance frequency of 1 1 2 pϩpЈ 2 radiation at zero angle. The result for the phase volume in the Ϫ ͩ ͪ ͩ Ϫ ͪ ͬ ͑ ͒ ⌬ p0 , 9 moving frame follows from the relativistic invariance of 2 2 d3k/. ⌬ Normalized vector potential is obtained by multiplying where L is the normalization length. The rms values and ͑ ͒ ͱ⍀ of the wave packet may be small compared to the rms energy Eq. 12 by . The parameter of interaction with the mode ϭ ͱ⍀ spread ⌬ and rms length of a bunch. then becomes g gk and, using the time of the interac- B B ϭ tion in the moving frame t 2 Nu /(ck), we get gt ϭ ͱ ϩ 2 ͱ 2 2 III. PICKUP (K0 / 1 K0) e /hc. Hence, (gt) is of the order of ␣ ϭ 2 0 e /hc and is always small. We assume that there are N relativistic particles per Initial state ͉p ,n͘ϭ͉p ,p ,...,p ,n͘ of the system B i 1 2 NB bunch, there is no initial z,p correlation, and correlations that with n photons and particles with momentums pi , i may be generated in the undulators are wiped out in one turn. ϭ 1,...,NB is transformed in the undulator by the interac- The pickup and the kicker undulators are helical with the tion with the mode kϭ/cϭ␥k to the vector ͉⌿(t)͘.Asit ϭ u undulator parameter K0 and period u 2 /ku . The bunch is shown in Appendix A, Eq. ͑A21͒, ͑for more details, see ͓ ͔ is described 5 in the frame moving with the relativistic ͓7͔͒, ␥ϭ␥ ͱ ϩ 2 factor 0 / 1 K0 where the resonance frequency of the ϭ␥ mode is k ku . The bunch centroid has zero initial velocity n! d ⌬ ⌿͑ ͒Ͼϭ Ϫ ϩ ͘ͱ ͵ Ϫil⌺ and an energy spread of p, ͯ t ͚ ͯpi 2hkli ,n l⌺ ͑ ϩ ͒ e li ,pi n l⌺ ! 2 ⌬ N 1 plab B ⌬pϭ ͩ ͪ , ͑10͒ ϫ ͓Ϫ ͑ ϩ ͔͒ ͑ ͒ ͑ ͒ ͱ ϩ 2 ␥ exp i t n l⌺ ͟ Fn t,pi ,li . 14 1 K0 0 iϭ1
016507-2 QUANTUM THEORY OF OPTICAL STOCHASTIC COOLING PHYSICAL REVIEW E 65 016507
ϭ͚ Here l⌺ ili is the total number of radiated photons, ϭ Ϫ 2 Fi ͑qЈ ,q ͒ϭ͚ f Јf *0͑qЈϩ2hklЈ ,q ϩ2hkl ͒ E(pi ,li) (pi 2hkli) /(2m0), loc i i i i i i i i l,lЈ ϱ n li/2 Ϫ ai ͓͑ Ј͒2Ϫ 2͔ ͑ ͒ϭ ͵ ˆ ͩ ͪ ͑ ͉ ͉ͱ͒ qi qi t F t,p,l d e O J 2g a n li i ϫexpͭ Ϫi ͮ , ͑21͒ 0 n! ai* 2meh ϫ ͓Ϫ͑ ͒ ͑ ͔͒ ili ͉ ͑ ͒ exp it/h E pi ,li e ϭ1 . 15 ϭ Јϭ Ј Ј Ј f i f (qi ,li , ), f i f (qi ,li , ), where ͔ ץץ 2ץ ϭ ͓Ϫ ˆ The operator O exp (1/2)( / ) , Jl is the a l/2 Bessel function, and f ͑q,l,͒ϭͩ ͪ J ͓2g͉a͑t͉͒ͱ͔eil. ͑22͒ a* l ͑⑀ ͒ sin it/2 Ϫ ⑀ Ϫ ⑀ 2kpi ͑ ͒ϭ i it/2 ˙ ͑ ͒ϭ i it ⑀ ϭ It is convenient to consider Fourier transform ai t ⑀ ͒ e , ai t e , i . ͑ i/2 me ͑16͒ Ldq i ͑ ͒ϭ ͵ iqz/h i ͑ ϩ Ϫ ͒ ͑ ͒ Floc p,z e Floc p q/2,p q/2 . 23 The integration over in Eq. ͑14͒ is introduced to separate 2 h the global parameters n and l⌺ of radiation and parameters pi ⑀ Ӷ For a short undulator, parameter it 1. In this case, and l of individual particles. Ϫ ⑀ i a(t)Ӎte i it/2. Parameter (gt)2 is the average number of Expression Eq. ͑15͒ is derived neglecting terms of the 2 photons radiated in the main mode of the undulator per par- order of hk /me , ticle and is always small. This justifies the expansion of f i in 3 hk2t series over gt. Neglecting terms of the order of (gt) ,we Ӎ Ӷ ͑ ͒ Nuk Compt 1. 17 write for the ith particle 2m0 Fi ͑p,z͒ϭF0͑p,z͓͒1ϩgtF(1)ϩ͑gt͒2F(2)͔, ͑24͒ Doing this we ignore bunching due to radiation, which is loc i i i irrelevant for our purposes. where Ӷ ⑀ ⑀ Ӎ For short undulators, kpt/me 1, ͓sin( it/2)͔/( i/2) t. Ϫ ͒2 Ϫ 0Ϫ ͒2 The function Fn depends on parameter gt, where t is time of h ͑p p0 ͑z z pt/me ϭ ␥ F0͑p,z͒ϭ expͫϪ Ϫ ͬ, flight in the undulator, which is t Nu u /(c ) in the moving i ⌬ 2⌬2 22 frame. ͑25͒ Using Eq. ͑14͒, the initial density matrix and ˆ ͑0͒ϭ͉pЈ,l⌺Ј ͑͘pЈ,p,l⌺ ,l⌺Ј ͒͗p,l⌺͉ ͑18͒ hk͑pϪp0͒ (1)ϭ ͓Ϫ͑ ͒͑ ⌬͒2͔ͭ Ϫ ͫ is then transformed into Fi exp 1/2 hk/ exp ⌬2
ˆ ͑t͒ϭ͉⌿Ј͑t͒͑͘pЈ,p,l⌺ ,lЈ ͒͗⌿͑t͉͒. ͑19͒ pt ⌺ ϩiϪ2ikͩ zϪ ͪͬ 2me IV. DENSITY MATRIX OF THE UNDULATOR hk͑pϪp0͒ pt ϪЈ ͫ Ϫ Јϩ ͩ Ϫ ͪͬ exp ⌬2 i 2ik z Let us obtain the explicit form of the density matrix Eq. 2me ͑19͒. hk͑pϪp0͒ pt We assume that at the entrance to the pickup there is no ϩ ͫϪ Ϫ ϩ ͩ Ϫ ͪͬ exp ⌬2 i 2ik z radiation, nϭ0. In this case, initial density matrix ˆ 2me N 0 ϭ͟ B ͉ Ј͘ Ј ͗ ͉ 0 iϭ1 p (pi ,pi) p is transformed according to Eqs. hk͑pϪp ͒ pt ϩЈ expͫϪ ϩiЈϪ2ikͩ zϪ ͪͬͮ . ͑14͒ and ͑19͒ to ˆ (t)ϭ͉qЈ,l⌺Ј ͘(qЈ,q,l⌺ ,l⌺Ј )͗q,l⌺ , where ⌬2 2me ͑ ͒ 1 ddЈ 26 ͑ Ј Ј ͒ϭ ͵ ͓Ϫ ͑ Ј ЈϪ ͔͒ q ,q,l⌺ ,l⌺ exp i l⌺ l⌺ 2 (2) ͱl⌺!l⌺Ј ! ͑2͒ F has a similar structure. With the same accuracy, ϫ ͓ ͑ Ϫ Ј ͔͒ ͵ Ј exp i t l⌺ l⌺ d d NB ͑ ͒ϭͭ ͟ 0͑ ͒ͮ ͓ ⌺ (1)ϩ͑ ͒2⌺ corr͔ Floc p,z Fi pi ,zi exp gt iFi gt iFi , ϪϪЈ iϭ1 ϫe Oˆ Oˆ F ͑qЈ,q͒. ͑20͒ Ј Ј loc ͑27͒ Here ͉q͘ stands for the set ͉q ,...,q ͘, 1 NB corrϭ (2)Ϫ ͓ (1)͔2 ͑ ͒ where Fi Fi (1/2) Fi . Equation 27 takes into 2 NB account all terms of the order of Nbgt and Nb(gt) neglect- 3 ͑ Ј ͒ϭ i ing terms Nb(gt) . Floc q ,q ͟ Floc , iϭ1 The sum
016507-3 S. HEIFETS AND M. ZOLOTOREV PHYSICAL REVIEW E 65 016507
ϵ (1) ͑ ͒ Nu L f 0 gt͚ Fi 28 N ϭN ͩ ͪ . ͑34͒ i s b ͱ 0 3 2 B in the exponent of Eq. ͑27͒ is defined by parameters In terms of averaged Ϯ introduced in Eq. ͑30͒, Eq. ͑28͒ takes the form of NB p t hk͑p Ϫp0͒ ͑ ͒ϭ ͫ Ϫ ͩ Ϫ i ͪ Ϯ i i ͬ Ϯ p,z gt͚ exp 2ik z i ⌬2 ͒ϭϪ iϪ ϪiЈϩ Ϫiϩ iЈ iϭ1 2me f 0͑ , Ј ϩe Ј ϩ*e Ϫ*e Ј Ϫe . ͑35͒ 1 hk 2 ϫ ͫϪ ͩ ͪ ͬ ͑ ͒ exp ⌬ . 29 2⌺ corr 2 The second terms (gt) iFi in the exponent of Eq. ͑27͒ can be expanded over h. Expansion starts with the term This expression must be averaged over the frequency spread proportional to h2. It can be split in two parts: the first of ¯ϭ␥ which, in the mode around k ku,
2 N hk B ¯͑ Ϫ 0͒ (1) 2 i iЈ ϪiЈ Ϫi pit hk pi pi ϭϪ ͑ ͒ ͩ ͪ ͑ ϩЈ ͒͑Ј ϩ ͒ ͑ ͒ϭ ͚ ͫ Ϫ ¯ͩ Ϫ ͪ Ϯ ͬ f cor Ns gt ⌬ e e e e , Ϯ p,z gt exp 2ik zi ϭ 2m ⌬2 i 1 e ͑36͒ 2 1 hk¯ ϫ ͫ Ϫ ͩ ͪ ͬ ͑ ͒ is proportional to the number of particles Ns , and the second exp ⌬ si , 30 (2) 2 one f cor , is proportional to the sum over oscillating factors. ϭ⌺ Ϯ Ϫ Introducing rϮ i exp͓ 4ik(zi pit/2me)͔, we can write where ͑gt͒2 hk 2 (2) ϭϪ ͩ ͪ ͓͑ iϩЈ iЈ͒2 dk f cor e e rϪ ϭ ͵ ͓Ϫ ͑ Ϫ¯ ͒ 2 ⌬ si exp 2i k k ϪiЈ Ϫi 2 ϩ͑Јe ϩe ͒ rϩ͔. ͑37͒ 2 Ϫ¯ ͒ ¯ ͒ sin ͑ Nu͑k k /k ϫ͑z Ϫp t/2m ͔͒ . ͑31͒ i i e ¯ ͒ Ϫ¯ ͒2 In these notations, ͑ Nu /k ͑k k
NB Factors s restrict summation in Eq. ͑30͒ over particles 0 f (,Ј)ϩ f (1) ϩ f (2) i ͑ ͒ϭͭ ͟ ͑ ͒ͮ 0 cor cor ͑ ͒ Floc p,z Fi pi ,zi e . 38 ¯ ͑ ϭ within the length proportional to 2 Nu /(2k) the length of a i 1 ͒ ϭ ‘‘slice’’ , or, in the laboratory system, within ls Nu lab . ϭӶ ӷ 2 The first factor is the product of unperturbed single particle Parameter Ns Ϫ Ϫ* /(gt) is the fundamental param- eter of the theory defining the number of interacting particles distribution functions while the exponential factor describes (2) within the bandwidth of the mode ͑number of particles per particle interaction. The last term, f cor , is small. Equation (2) (2) slice͒. Here double averaging means averaging with the den- ͑ ͒ f corϭ ϩ 38 can be simplified by writing e (1 f cor) and re- ͑ ͒ 0 0 sity matrix of the wave packet Eq. 25 and over z ,p within placing ϪgtЈeϪiЈ, gteϪi, Ϫgtei, and gtЈeiЈ by the Gaussian bunch (z ,p )ϭ(1/2 ⌬ )exp͓Ϫp2/2⌬2 B 0 0 B B 0 B the derivatives over * , * , ϩ , and Ϫ , respectively. Ϫ 2 2 ͔ ͑ ͒ ϩ Ϫ z0/2 B . If the width of packet in Eq. 25 is of the order f (1) The result is the differential operator Pˆ (Ϯ). The factor e cor ӷ ¯ӷ of the length of a slice, and Nu 1, then k 1, and can be written as
2 2 dx sin x dy sin y f (1) i iЈ ϭ ͵ e corϭOˆ exp͓Ϫ͑e ϩЈe ͒ Ns ͚ 2 2 , i x y ϪiЈ Ϫi Ϫ͑Јe ϩe ͔͉͒ϭϭ , ͑39͒ 2ik p t 0 ϫͳͳ ͫϪ ͑ Ϫ ͒ͩ Ϫ i ͪͬʹʹ ͑ ͒ exp x y zi . 32 2ϭ 2 ⌬ 2ץץ 2ץ N 2m ˆ ϭ Ϫ2 u e where O, exp͓ ( / )͔, and Ns(gt) (hk/ ) . Then, Neglecting terms of the order of h,weget NB ͱ ͑ ͒ϭͭ ͟ 0͑ ͒ͮ ͑ ϩ ˆ ͒ ˆ N 2 Floc p,z Fi pi ,zi 1 P O, ϭ u ͑ ͒ ϭ Ns Nb , 33 i 1 3k B i ϪiЈ ϫexp͓Ϫ͑ϩϩ͒e ϪЈ͑ϩ*ϩ͒e where B is the rms bunch length in the moving frame and 4 Ϫi iЈ we use ͐(dx/)(sin x/x) ϭ0.6666. In terms of the wave- ϩ͑Ϫ*Ϫ͒e ϩЈ͑ϪϪ͒e ͔. ͑40͒ length of the mode and the bunch length in the laboratory frame, Now it is easy to calculate
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ˆ ˆ Ϫ ϩ͒ iϩ Ϫ͒ Ϫi V. SOME RESULTS FOR THE UNDULATOR O,OЈ,Ј exp͓ ͑ ϩ e ͑ Ϫ* e First, let us show that the total density matrix Eq. ͑47͒ of ϪЈ͑ ϩ͒ ϪiЈϩЈ͑ Ϫ͒ iЈ͔͉ ϩ* e Ϫ e ϭЈϭ1 the system ͑particles and radiation͒ allows us to reproduce results obtained by different methods. ϭexp͓͑1/2͒͑ϩϩ͒͑*Ϫ͒ Ϫ A trace of the density matrix Eq. ͑47͒ defines energy loss i ϩ͑1/2͒͑ϩ*ϩ͒͑ϪϪ͔͒exp͓Ϫ͑ϩϩ͒e ͗p͘ϭTr͑pˆ ˆ ͒ϭϪ2hk͑gt͒2. ͑49͒ Ϫi ϪiЈ ϩ͑Ϫ*Ϫ͒e Ϫ͑ϩ*ϩ͒e The density matrix of radiation, rad , can be obtained as the iЈ ϩЈ͑ϪϪ͒e ͔. ͑41͒ trace of the full density matrix Eq. ͑47͒ over particle indexes,
Integration over and Ј can be carried out using ͑ Ј ͒ϭ ͑ Ј Ј ͒ ͑ ͒ rad l⌺ ,l⌺ ͚ pi ,pi ,l⌺ ,l⌺ . 50 p d l/2 i ͵ eil exp͓eϪiϪei͔ϭͩ ͪ J ͑2ͱ͒. ͑42͒ 2 l Equations ͑49͒ and ͑50͒ reproduce results ͓10͔ obtained by the operator formalism method. ͓ ͔ After that, integrals over and Ј are 9 The density matrix of radiation for a single electron, Nb ϭ1, with no initial photons, ϱ Ϫ Ϫ ͵ de l/2J ͑2ͱa͒ϭal/2e a. ͑43͒ 2l l ͑gt͒ 2 0 w ͑l͒ϭ ͑l,l͒ϭ eϪ(gt) , ͑51͒ rad rad l! The distribution function at the end of the pickup is a density matrix of a coherent state. The average number Ldq of radiated photons in a single mode is small, ͗l͘ϭ(gt)2 ͑ Ј ͒ϭ ͵ iqz/h͑ ϩ Ϫ Ј ͒ 2 2 2 p,z,l⌺ ,l⌺ e p q/2,p q/2,l⌺ ,l⌺ ϭ␣ ϭ ͗ ⌬ ͘ϭ͗ ͘Ϫ͗ ͘ 2h 0 1/137. Fluctuations are large ( l) l l ϭ͗ ͘ ͑44͒ l , and ͗͑⌬l͒2͘ 1 takes the form of ϭ ϭ137. ͑52͒ ͗l͘2 ͑gt͒2 1 NB ͑ Ј ͒ϭ ͓ ͑ Ϫ Ј ͔͒ͭ ͟ 0͑ ͒ͮ ͓ ͔ p,z,l⌺ ,l⌺ exp i t l⌺ l⌺ Fi pi ,zi Radiation of a bunch corresponds to thermal statistics 6 ͱl⌺!l⌺Ј ! iϭ1 ͑␣͒l⌺ ϫ͑ ϩ ˆ ͒ ͑ ͒ ͑ ͒ Ј ͒ϭ␦ ␣ϭ ͒2 ͑ ͒ 1 P R p,z , 45 rad͑l⌺ ,l⌺ l ,lЈ , Nb͑gt . 53 ⌺ ⌺ ͑1ϩ␣͒l⌺ϩ1 where For the following, it is convenient to transform the density Ј matrix (p,z,lЈ ,l⌺) back to the momentum representation, ͒ϭ ˆ Ϫ͒l⌺ Ϫ͒l⌺ Ϫ ͒ Ϫ͒ ⌺ R͑p,z O,͑ Ϫ* ͑ Ϫ exp͓ ͑1/2 ͑ Ϫ* ϫ͑ ϩ͒Ϫ͑ ͒͑ Ϫ͒͑*ϩ͔͒ ͑ ͒ ϩ 1/2 Ϫ ϩ . 46 ͑pϩq/2,pϪq/2͒ϭͭ͟ ͵ ͑dz /L͒exp͓Ϫi͑qЈϪq͒z ͔ͮ i i i For small , the density matrix at the end of the pickup is qЈϩq ϫͩ i i ͪ ,zi . 1 NB 2 ͑ Ј ͒ϭ ͭ ͟ 0͑ ͒ͮ p,z,l⌺ ,l⌺ Fi pi ,zi ͱ Ј ϭ l⌺!l⌺! i 1 The result then becomes
ϫ͑1ϩPˆ ͒R͑p,z,N,͒exp͓it͑l⌺Ϫl⌺Ј ͔͒, ˆ ϭ͉qЈ,l⌺Ј ͑͘qЈq͒͗q,l⌺͉, ͑54͒ ͑47͒ where
where Rϭexp͓Ϫ(1/2)(Ϫ*ϩϩc.c.)͔˜R(p,z,N,), and 1 dzi ͑qЈ,q͒ϭ ͭ͟ ͵ Fi͑qЈ,q,z͒ͮ ͑1ϩPˆ ͒ Ϫ* l⌺ϩl⌺Ј l⌺Ϫl⌺Ј ͱ Ј L ˜ ͑ ͒ϭͩ ͪ ͉ ͉2N ϭ ϭ l⌺!l⌺! i R p,z,N, Ϫ , N , . Ϫ 2 2 qЈϩq ͑48͒ ϫRͩ ,z,N,ͪ e2it, ͑55͒ 2 2͉ ͉2 2 ⌬ 2 Correction Ϫ is of the order of ͓Ns(gt) (hk/ )͔ and is always negligible. and
016507-5 S. HEIFETS AND M. ZOLOTOREV PHYSICAL REVIEW E 65 016507
2 h 1 qЈϩq ͑NϪ͒! ␣* i͑ Ј ͒ϭ ͫϪ ͑ ЈϪ ͒ Ϫ ͩ ͪ 2 Ϫ͉␣͉2 F q ,q,z ⌬ exp i q q z/h ⌬2 ͑n,nЈ,t͒ϭ ͩ ͪ ͉␣͉ e 2 2 ͱn!nЈ! ␣ 2 1 qЈϩq 2 Ϫ ͩ Ϫ ͪ ͬ ͑ ͒ ϩ Ϫ 2 z t . 56 ϫ 1͑1Ϫ͒N L2 ͩ Ϫ ͯ␣ͯ ͪ . 2 2me NϪ 1Ϫ ͓ Ј ͑ ͒ Note that Ϯ are functions of the coordinates (zi , qi 63 ϩ qi͔/2) of all particles. Then VI. OPTICAL AMPLIFIER AND DISPERSION SECTION ␣ The density matrix Eqs. ͑47͒ and ͑48͒ at the exit of the ͗a͑t͒͘ϭ , ͑64͒ ͱ pickup undulator is the superposition of coherent states. Transformation of such a state in the optical amplifier can be obtained in the following way ͓8͔: which shows that parameter defines the gain G of the am- Let us consider the two level model of the amplifier with plifier, Gϭ1/. Ͼ The lowest moments are inverse population Nu Nd . The equation describing the time evolution of the density matrix is well known ͓12͔, ͗a͑t͒͘ϭͱG␣, ͗aϩa͘ϭG͉␣͉2ϩ͑GϪ1͒, ͑65͒ ˙ ϭϪ ϩϩ ϩϪ ϩ gNu͓aa aa 2a a͔ and the signal-to-noise ratio is independent of G for Gӷ1 Ϫ ϩ ϩ ϩ Ϫ ϩ ͑ ͒ gNd͓ a a a a 2a a ͔. 57 ͗͑aϩa͒2͘Ϫ͗aϩa͘2 1ϩ2͉␣͉2 Let us use this representation with a fixed number of pho- ϭ Ӎ1. ͑66͒ ͗aϩa͘2 ͑1ϩ͉␣͉2͒2 tons, (t)ϭ͉nЈ͘(nЈ,n,t)͗n͉, and define F(N,,t), Let us use these results to transform the density matrix Eq. F͑N,,t͒ ͑nЈ,n,t͒ϭ , Nϭ͑nϩnЈ͒/2, ϭ͑nϪnЈ͒/2. ͑48͒ in the amplifier. ͱ Ј n!n ! The Mellin transform ˜R (N,)of˜R(͓qЈϩq͔/2,z,N,), ͑58͒ M iϱ dN qЈϩq The function F is a solution of Eq. ͑57͒. Parameter is the ˜ ͑ ͒ϭ ͵ ϪN˜ ͩ ͪ ͑ ͒ RM , R ,z,N, , 67 integral of motion. Dependence on N can be obtained using a Ϫiϱ2i 2 Mellin transform 2 is proportional to ␦(Ϫ ), ϭ͉Ϫ͉ , ϱ 0 0 ͑ ͒ϭ ͵ Ј ͑ Ј ͒ ͑ Ј ͒ ͑ ͒ F N, ,t dz Gm N,z , f 0 z ,m , 59 0 Ϫ* ˜ ͑ ͒ϭ ͫ ͬ ␦͑Ϫ ͒ ͑ ͒ RM , 0 0 . 68 ϭ Ϫ where gNut, t is the amplification time and f 0 is given by the initial condition, After the amplifier, ˜R(͓qЈϩq͔/2,z,N,) should be re- ϱ ͓ ͔ i dN placed 8 by Fampl , ͑ ͒ϭ ͵ ϪN ͑ ͒ ͑ ͒ f 0 z, z F N, ,0 . 60 Ϫ ϱ i 2 i ͉͉ N 1 Ϫ* Ϫ*Ϫ GϪ1 ͑ ͒ϭ͑ Ϫ͉͉͒ ͫ ͬ ͫ ͬ ͩ ͪ ͓ ͔ Fampl N, N ! Ϫ The kernel Gm can be obtained 8 for an arbitrary ratio of G Ϫ G 1 G N /N . In the case of a fully inverted population, N ϭ0, u d d ͉ ͉2 ͉ ͉ Ϫ ϫL2 ͩ Ϫ ͪ . ͑69͒ NϪ͉͉ GϪ1 ͑ Ј ͒ϭ͑ Ϫ͒ ͑ Ϫ͒N 2 ͑Ϫ ͒ ͑ ͒ Gm N,z , N ! 1 b LNϪ b , 61 zЈ m Here G is the power gain of the amplifier, LN are Laguerre Ϫ2N gt ϭ ϩ Ј ϭ Ϫ Ј where bϭzЈ/(1Ϫ), ϭe u , and t is the the time of polynomials, and N (l⌺ l⌺)/2, (l⌺ l⌺)/2. amplification. Equation ͑69͒ describes the amplification of the main term Parameter is related to a gain of the amplifier. Consider, in Eq. ͑47͒. Calculation of the derivatives in the correction for example, radiation from the undulator of a single electron term, Pˆ R(p,z,N,) where Pˆ is a differential operator of the described by initial coherent state second order in Ϯ , gives a polynomial of the second order in N multiplied by R(p,z,N,). The result can be written as nЈ n ˆ N ͔ ץ ץ͓ ˆ 2 ͒*␣͑ ␣ ϭ Ϫ͉␣͉ ͑ ͒ P(y / y )x y , where P is now a differential operator of ͑nЈ,n,0͒ e . 62 2 ͱn!nЈ! the second order in y, independent of N, and yϭ͉Ϫ͉ , x ϭϪ*/Ϫ . This expression can be transformed in the ampli- The result of amplification is described by the density matrix fier in the same way as the main term above.
016507-6 QUANTUM THEORY OF OPTICAL STOCHASTIC COOLING PHYSICAL REVIEW E 65 016507
VII. DISPERSION SECTION Here l⌺ϭNϩ, l⌺Ј ϭNϪ, m⌺ϭMϪ, m⌺Ј ϭMϩ. Be- Ј The dispersion section with momentum compaction ␣ cause l⌺ and l⌺ are positive, the range of summation is 0 MC Ͻ Ͻϱ Ϫ Ͻ Ͻϱ ͉͉Ͻ and length L introduces (z,p) correlation by changing the N , N M , and N. Functions Ϯ in Fampl ds Јϩ ⌬ ϭ␣ Ϫ 0 depend on coordinates of individual particles (q qi)/2,zi . path length in the lab frame by z MCLds(p p )/q0.In i the moving frame, such a correlation is described by the The operator Fout is classical distribution function 1 d Ϫim⌺ ͑ Ϫ ͒2 ͑ Ϫ Ϫ ͒2 F ͑q,l⌺ ,m⌺͒ϭ ͵ e 1 p p0 z z0 p out ͱ f ͑p,z͒ϭ expͫϪ Ϫ ͬ, ͑l⌺ϩm⌺͒! 2 2⌬ 2⌬2 22 ͑ ͒ l⌺ 70 Ϫ ϫexp͓it͑l⌺ϩm⌺͔͒ ͵ d e Oˆ , ϭ␥ ␣ l⌺! where parameter 0 MCLds /mec. The corresponding density matrix is different from Eq. ͑9͒ by the factor ͑75͒ exp͓Ϫ(i/h)(qЈ2Ϫq2)/2͔. Hence, the dispersion section modifies Fi(qЈ,q,z) in Eq. and ͑55͒. Equation ͑56͒ has to be replaced with dzi i 2 2 i ⌽ ͑ ͒ϭ (i)͑ ͒ F ͑qЈ,q,z͒exp͓Ϫ͑i/h͒͑qЈ Ϫq ͒/2͔e . ͑71͒ loc q,qЈ, , Ј ͟ F qЈ,q,z i L Here, a phase slip of a bunch centroid is added and should i be controlled in the experiment. ϫexpͭ Ϫ ͓͑qЈ͒2Ϫ͑q ͒2͔ͮ 2h i i VIII. KICKER ϫS* ͑q,,͒S ͑qЈ,Ј,Ј͒, ͑76͒ mi i The density matrix at the entrance to the kicker is ob- tained by combining Eqs. ͑47͒, ͑48͒, ͑69͒, and ͑71͒, where
mi/2 Fin a Ј ͑ ͒ϭͩ ͪ ͓ ͉ ͑ ͉͒ͱ͔ imi ˆ ͑t͒ϭ͉qЈ,l ͘ ͗q,l⌺͉, ͑72͒ Sm q, , Jm 2g ai t e in ⌺ i a* i ͱl⌺!l⌺Ј ! Ϫi͓͑qϪ2hkm͒2͔t where ϫexpͭ ͮ . ͑77͒ 2meh ϭ ͒ ϩ ˆ ͒ ͒ 2it Fin Fds͑qЈ,q ͑1 P Fampl͑N, e To describe stochastic cooling, it is sufficient to calculate the 1 ϫ ͩ Ϫ ͓ ϩ ͔ͪ momentum of a particle at the end of the kicker. The average exp * ϩ c.c. 2 Ϫ moments for the jth particle after a bunch passes through the ͗ k͘ϭ ͓ ˆ kˆ ͔ ϭ ˆ system are p j Tr p j out(t) , k 0,1,..., where p is and momentum operator and brackets ͗•••͘ signify averaging over the wave packet. In the momentum representation, only dz ϭ ͵ i͑ Ј ͒ ͭ Ϫ ͓͑ Ј͒2Ϫ 2͔ͮ the diagonal components, qЈϪ2hkmЈϭq Ϫ2hkm , i Fds ͟ F q ,q,z exp i q q . i i i i i L 2h k ϭ1,2,...,N , and l⌺Ј ϩm⌺Ј ϭl⌺ϩm⌺ , contribute in ͗p ͘. ͑73͒ b j We can utilize the fact that Ϯ are functions only of the sum Јϩ Јϭ ϩ ЈϪ ϭ The transform of the density matrix at the end of the kicker q q and introduce P, qi Pi hk(mi mi), qi Pi Ϫ ЈϪ is given by Eq. ͑14͒ where n has to be replaced by the num- hk(mi mi). This allows us to write ber of photons l⌺ . We use notation mi for the number of ͗ n͘ϭ͓ Ϫ ͑ ϩ Ј͔͒n⌽ ͑ ϩ ˆ ͒ photons radiated by the ith electron in the kicker and m⌺ p P j hk m j m j loc 1 P ϭ͚ m for the total number of photons. We also assume that i i ϫ * ͑ ͒ ͑ Ј Ј Ј ͒ ͑ ͒ the parameters of both undulators are the same. Fout q,l⌺ ,m⌺ Fout q ,l⌺ ,m⌺ Fampl N, , Given the above, the density matrix at the exit of the ͑78͒ kicker where ˆ ͒ϭ͉ Ϫ Ј ϩ Ј ⌽ ͒ out͑t qЈ 2hkmЈ,l⌺ m⌺͘ loc͑q,qЈ, , Ј dzidPi ϫF* ͑q,l⌺ ,m⌺͒F* ͑qЈ,lЈ ,mЈ ͒͑1ϩPˆ ͒ ⌽ ϭ͟ ͑P ,z ͒ ͚ S ͑,͒S Ј͑Ј,Ј͒ out out ⌺Ј ⌺ loc ⌬ 0 i i mi m i 2 Ј i mi ,mi 1 ϫ ͑ ͒ 2it ͩ Ϫ ͓ ϩ ͔ͪ F N, e exp * ϩ c.c. ϫ ͓Ϫ ͑ ϩ ͒͑ ЈϪ ͔͒ ͑ ͒ ampl 2 Ϫ exp 2ik zi Pi mi mi , 79
ϫ͗qϪ2hkm,l⌺ϩm⌺͉. ͑74͒ and
016507-7 S. HEIFETS AND M. ZOLOTOREV PHYSICAL REVIEW E 65 016507
͑ ͒ϭ ͓Ϫ͑ Ϫ 0͒2 ⌬2Ϫ͑ Ϫ 0Ϫ ͒2͔ can neglect terms that are independent of G. In this approxi- 0 Pi ,zi exp Pi pi /2 zi zi Pit/m0 . ˜ ͑80͒ mation, momentum p j of the jth particle at the end of the kicker is Note that Fampl depends on Ϯ that are given now by Eq. ͑ ͒ ˜ ϭ Ϫ ͑ ͒2 ͱ ͓ ͕Ϫ ͑ ϩ ͒ϩ ͖ 32 where pi are replaced by Pi . p j p j 2 gt hk G 0* exp 2ik z j effp j i Similarly to what was done for the pickup, we expand ϩ ͔ ͑ ͒ S(,) in series over gt, neglecting terms O(gt)3. Here we c.c. , 85 skip over the details of the calculations and give the final where ϭϩt/2m , and is the phase slip of the bunch result eff e ˜ 2 centroid. Calculation of p j at the end of the kicker gives ͗ n͘ϭ ˆ ͑ ϩ ˆ ͒ ͑ ͒ ͑ Ј Ј Ј ͒ ˜ 2ϭ 2Ϫ ͑ ͒2͑ ͒ ͑ ͓Ϫ ͑ ϩ ͒ϩ ͔ p j ͚ Kn 1 P Fout q,l⌺ ,m⌺ Fout q ,l⌺ ,m⌺ p j p j 4G gt hk p j 0* exp 2ik z j effp j i iÞ j ϩ ͒ϩ ͑ ͒2͑ ͒2͓ ϩ͑ ͒2 ϩ ͔ c.c. 8G gt hk 1 gt 0 0* c.c. ϫQ͑b ,b ͒F ͑P,z͉͒ Ϫ͘ . ͑81͒ 1 2 ampl b2 b1 ϩ ͱ ͑ ͒4͑ ͒2͑ iϩ ͒ ͑ ͒ 4 G gt hk b1 0*e c.c. . 86 The sum stands for integrals ͟ ͓(dz dP ) i i i ϭ ͉ /(2⌬)͔ (P ,z ) over all particles in a bunch, and Here 0 Ϯ h→0. Double averaging over the wave packet 0 i i 0(p j ,z j) and over a Gaussian distribution of particles in the ⌬2ϭ͗͗ 2͘͘Ϫ͗͗ ͘͘2 ͑ ͒ϭ ͑Ј iЈϪЈ ϪiЈϩ Ϫi bunch gives rms p p at the end of the kicker Q b1 ,b2 exp b1e b2*e b1*e 2 2 Ϫb ei͒, ͑82͒ ⌬˜ Ϫ⌬ hk 2 ϭϪ16ͱG͑gt͒2 ⌳sin 2 ⌬ ⌬ B Ϫ where b ϭgt͚ e i i, and phase ϭ2k͓z ϩp ͔. Opera- 1 i j j j 2 ˆ ϭ ˆ ϭ ˆ ϭ hk tors Kn for different n 0,1,2 are K0 1, K1 q j ϩ ͑ ͒2ͩ ͪ ͓ ϩ ͑ ͒2͔ 8G gt ⌬ 1 Ns gt Ϫ Ϫ ˆ ϭ ˆ 2ϩ 2 ϩ B hkgt(a a), K2 K1 (hk) gt(a a), where 2 hk 2 4 Ϫ2(k⌬ )2 ϩ ͱ ͑ ͒ ͩ ͪ B eff ץ ץ ץ ץ Ϫ Ϫ 8 G gt Ns cos e . ϭ i j ϩ i j ϭ i j ϩ i j ⌬ B . ץ e , a e e ץ a e b* b1ץ *bץ b2 2 1 ͑87͒ ͑83͒ ⌳ϭ ⌬ ͓Ϫ ⌬ 2͔ ͑ ͒ ͑ ͒ Here k B eff exp 2(k B eff) . Equation 81 after some calculations see Appendix B can To get damping, we have to choose sin ϭ1. The damping be written as is maximized if the power gain G of the amplifier is equal to Ϫ* ⌳ ͗ n͘ϭ ˆ ͑ ϩ ˆ ͒ͩ ͪ ͱ ϭ ͑ ͒ p j ͚ Kn͚ 1 P G ⌬ ͒ ϩ ͒2 . 88 iÞ j Ϫ ͑hk/ B ͓1 Ns͑gt ͔ b Ϫb Parameter ⌳ as a function of xϭk⌬ has a maximum ϫ ͓ ͱ ͉ Ϫ ͉2͉ Ϫ ͉2͔ 1 2 B eff I2 2 G b2 b1 Ϫ ϩ ͩ Ϫ ͪ value of ⌳ Ӎ0.3 at xӍ2. This defines the optimum pa- b1* b2* max rameter of the dispersion section. ϫ ͕͑ Ϫ ͉͒ Ϫ ͉2ϩ͑ ͓͒ ͑ Ϫ ͒ϩ ͔͖ exp G 1 b2 b1 1/2 b2 b2* b1* c.c. The optimized reduction of the rms energy spread in one pass through the system is ϫexp͕͑1/2͓͒Ϫ͑Ϫ*Ϫϩ͒ϩc.c͔͖͉ ϭ . ͑84͒ b2 b1 ⌬˜ 2Ϫ⌬2 ⌳2 8 max ϭϪ . ͑89͒ The operators Kˆ are not more than the second order dif- ⌬2 ͒Ϫ2ϩ n ͑gt Ns ferential operators in b2 , b1, and the function depends on Ϫ b2,1 only through powers of b2 b1. Therefore, it is sufficient to take into account only terms ϭ0, ϭϮ1/2, and ϭ IX. CONCLUSION Ϯ1 in the sum over . Additionally, we can expand the One-pass reduction of the energy spread rms is derived answer in series over gt and neglect terms O(h3). following the evolution of the density matrix through all ͗ n͘ To check the result, we can calculate the average p j for components of the system. The consideration is fully quan- nϭ0. This quantity is just the norm of the distribution func- tum mechanical both for the beam and for the radiation. The tion and, therefore, has to be equal to 1. Indeed, the answer bunching effect is neglected and length of a slice of the order is different from one by the term of the order of of Nu lab is assumed to be small compared to the bunch 2 4 ⌬ 4 0 Ns (gt) (hk/ ) . length in the laboratory frame B . The result for the moments nϭ1 and nϭ2 were obtained It is shown that the constructed density matrix may be with MATHEMATICA. As it will be shown below, the power used to obtain results on the photon statistics and distribution ⌬ ӷ gain G must be of the order of b /(hk). Because G 1, we of particles due to the beam-radiation interaction in the un-
016507-8 QUANTUM THEORY OF OPTICAL STOCHASTIC COOLING PHYSICAL REVIEW E 65 016507
ϫ Ϫ ЈϮ ͔ dulator. The time evolution of the density matrix of radiation (lj lj 1) , and write explicitly only those quantum num- Ϯ in the amplifier is described, including the nondiagonal com- bers that are changed by the interaction, Fn(t,p j ,l j 1) ponents of the matrix. ϭF ͓t,(p ,l ),...,(p ,l Ϯ1),...,(p ,l )͔. n 1 1 j j NB NB The final result Eq. ͑89͒ is equivalent to the classical 2 ͑ 2 Neglecting terms hk /2m0 in the exponent i.e., in the equation of stochastic cooling with quantum noise 1/(gt) . laboratory frame, terms of the order of hk k /m Ӷ1), we ͑ ͒ ӷ 2 u L 0 Equation 89 for large Ns 1/(gt) reproduces the main re- can solve Eq. ͑A2͒ by the Fourier transform sult of the classical theory of stochastic cooling. The damp- ing rate is given by the number of particles N per slice. N s B 2d i il However, for small Ns the damping rate goes to a constant ͑ ͒ϭͭ ͟ ͵ i iͮ Fn t,p j ,l j e 2 2ϰ͓ 2 ϩ 2 ͔␣ iϭ1 0 2 proportional to 1/(gt) , where (gt) K0/(1 K0) 0. The quantum fluctuations set the limit on the damping rate, the ϫF͑t,p ,...,p , ,..., ͒. ͑A3͒ ␣ 1 NB 1 NB minimum number of turns for cooling is of the order of 1/ 0. The term 1/(gt)2 is equivalent to the noise induced by 1/␣ 0 Function F(t,)ϭF(t,p ,...,p , ,..., )is particles and is related to the quantum limit of the input noise 1 NB 1 NB of the amplifier equal to one photon in a mode. The other given by quantum-mechanical corrections are small, of the order of Fץ ͒ ⌬ ͑ ⌬ (hk)/ B i.e., hkL / pL in the laboratory frame and are ˙ ͑ ͒ϭϪ * ͑ ͒ϩ ͑ ϩ ͒ ϩ , ץ F t, gv0 F t, g n 1 v0F igv0͚ noticeable only in very cold beams where the energy spread i i is comparable to photon energy. ͑A4͒
ACKNOWLEDGMENTS where We are thankful to A. Zholentz for useful discussions and Ϫ ⑀ Ϫ 2kpi to O. Heifets for editing of the text. This work is supported v ͑t,͒ϭ͚ e i it i i, ⑀ ϭ . ͑A5͒ 0 i m by U.S. Department of Energy Contract No. DE-AC03- i 0 76SF00515. If the system is initially in the n-photon state ͉⌿(0)͘ ϭ͉p ,n͘, then F (0,p ,l )ϭ͟ ␦ and F(0,)ϭ1. APPENDIX A: BEAM DYNAMICS IN THE UNDULATOR i n i i i li,0 Characteristics (t) of the Eq. ͑A4͒ are defined by Interaction of particles with the mode described by ˙ (t)ϭϪigv (t,), with the solutions (t)ϭ0ϩV(t), ͑ ͒ i 0 i i Hamiltonian Eq. 11 is just backscattering of equivalent where V(0)ϭ0, 0ϭ (0) are constants. The function V(t) ͉ ͘ϭ͉ ͘ i i photons. The initial state pi ,n p1 ,p2 ,...,pN ,n of the B to be determined is the same for all i(t). system with n photons and particles with momentums pi , i ˙ ϭϪ ϭ V(t) satisfies V(t) igv0. The substitution of (t)in 1,...,NB is transformed by the interaction with the mode ͑ ͒ ϭ ϭ␥ ͉⌿ Eq. A5 gives k /c ku to the vector (t)͘, v ͑t,͒ϭu ͑t,0͒eϪiV(t), n! 0 0 ͉⌿͑ ͒͘ϭ ͉ Ϫ ϩ ͘ͱ t ͚ pi 2hkli ,n l⌺ ͑ ϩ ͒ li ,pi n l⌺ ! where d Ϫil ϫ ͵ e ⌺ ͑ 0͒ϵ ͓Ϫ ⑀ Ϫ 0͔ ͑ ͒ 2 u0 t, ͚ exp i it i i . A6 i NB iV(t)ϭ iVϭ 0 ץ ץ ͒ ͑ ͒ ͒ ϫ Ϫ ϩ exp͓ i t͑n l⌺ ͔͟ Fn͑t,pi ,li . A1 Hence, ( / t)e gv0e gu0(t, ), with the solution iϭ1
Ϫ 0 ϭ͚ iV(t)ϭ ϩ ͒ i j ͑ ͒ Here l⌺ ili is the total number of radiated pho- e 1 g͚ a j͑t e , A7 ϭ Ϫ 2 tons, E(pi ,li) (pi 2hkli) /(2m0), and the amplitudes F(t,͓p ,l ͔) are defined by the equation i i where
˙ ͑ ͒ϭ ͓͑ ϩ ͒ ͑ Ϫ ͒ ͑⑀ ͒ Fn t,p j ,l j g͚ n l⌺ Fn t,p j ,l j 1 sin it/2 Ϫ ⑀ Ϫ ⑀ ͑ ͒ϭ i it/2 ˙ ͑ ͒ϭ i it ͑ ͒ j ai t ⑀ ͒ e , ai t e . A8 ͑ i/2 ϫ Ϫ ͒ Ϫ ͒ exp͕ 2i͑k/m0 t͑p 2hklj ͖ ͑ ͒ Ϫ ϩ ͒ Equation A7 defines characteristics i(t) Fn͑t,p j,l j 1
ϫexp͕2i͑k/m ͒t͑p Ϫ2hkl ͖͔͒, ͑A2͒ 0ϩ 0 Ϫ 0 0 j j i iϭ i i iVϭ i i ͫ ϩ ͑ ͒ i j ͬ e e e 1 g͚ a j t e , following from the Schro¨dinger equation. Here, we use ͑A9͒ Ϯ ˆ Ϫ Ϫ ͗ Ϫ ͉ 2ikz͉ ЈϪ Ј͘ϭ ␦͓ Ј Ϫ ϩ i iϭ͓ i i͔ 1 p j 2hklj e p j 2hklj (2 h / L) p j p j 2hk e e .
016507-9 S. HEIFETS AND M. ZOLOTOREV PHYSICAL REVIEW E 65 016507
͑ ͒ i0 Equation A9 can be reversed to get constants of motion F(t,)ϭ⌽(t,e i ), where ⌽ is arbitrary function of the ar- 0 0 i 0 in terms of . Defining ⌳( ,..., )ase i i i i 1 NB guments (t,)ϵe i given by Eq. ͑A11͒. Equation ͑A4͒ in i ϭ⌳ei i(t) and substituting this in the right-hand side of Eq. terms of t, takes form ͑A9͒ to get ⌽͑t,͒ץ ϭϪ ⌽͑ ͒ϩ ͑ ϩ ͒ ⌽ ͑ ͒ Ϫ gv0* t, g n 1 v0 , A12 tץ ͒ ͑ ϭ Ϫ ͒ i j⌳ 1 g͚ a j͑t e . A10 j where Hence,
0 Ϫ Ϫ 0 0 Ϫ v*͑t͒ϭu*ͫ 1ϩg͚ a ͑t͒/ ͬ. ͑A13͒ i i ϭ i iͫ Ϫ ͑ ͒ i jͬ i i ϭ͓ i i ͔ 1 0 0 i i e e 1 g͚ a j t e , e e . i ͑A11͒ ͑ ͒ Integrating Eq. A12 over t and substituting i from Eq. The general solution of the Eq. ͑A4͒ can be found as ͑A11͒ gives
t Ϫ expͭ Ϫg͚ a*ei ϩ͑g2/2͚͒ͯ a*ei ͯ2Ϫ͑g2/2͚͒ ei( i j) ͵ d͓a a˙ *Ϫa*a˙ ͔ͮ i i j i i j i i i, j 0 ͑ ͒ϭ⌽ ͑͒ ͑ ͒ F t, 0 nϩ1 . A14 Ϫ ͫ 1Ϫg͚ a e i iͬ i i
Note, a(0)ϭ0. Thus, the initial condition F(0,)ϭ1 allows d ⌽ ϭ ͵ exp͓ilϪa*eiϩaeϪi͔ us to choose 0( ) 1. 2 Parameter ⑀t is of the order of ⑀t Ӎ22N (␦p/p) /ͱ1ϩK2 where N is the number of peri- a l/2 u L 0 u ϭͩ ͪ ͑ ͱ ͒ ͑ ͒ ␦ Jl 2 aa* . A18 ods in the undulator and ( p/p)L is the rms energy spread in a* the lab system. We assume that ⑀tӶ1 that means that the rms energy spread of the bunch is small compared to the ⌬ Ӎ The replacement of k→Ϫk on the left-hand side of Eq. width / 1/Nu of the mode. In this case, the integral term in the numerator is ͑A18͒ changes the right-hand side of the equation by → Jl(•••) Il(•••) leaving the rest intact. t it3 t4 Thus, ͵ ͓ ˙ Ϫ ˙ ͔Ӎ ͑⑀ ϩ⑀ ͒Ϫ ͑⑀2Ϫ⑀2͒ d a jai* ai*a j i j i j , 0 6 12 N ͑A15͒ ϱ n B a li/2 ͑ ͒ϭ ͵ Ϫ ˆ ͩ i ͪ ͑ ͉ ͉ͱ͒ ͭ ͟ Fn t,p,l d e O Jl 2g ai 0 n! iϭ1 a* i i.e., ⑀ t times smaller than the other terms in the exponent i 2 ͓which are of the order of gt and (gt) ͔ and can be ne- t glected. The remaining factors can be written introducing ϫexpͫ Ϫ͑g2/2͒ ͵ d͑a a˙ *Ϫc.c.͒ͬͮͯ , i i Ϫ ϭ ͚ i i 0 ϭ additional integration over in terms of b g iai(t)e 1 ͑A19͒ ϱ n F͑t,͒ϭ ͵ d eϪ exp͓Ϫb*ϩbϩ͑1/2͒bb*͔. 0 n! where Jl is the Bessel function. ͑A16͒ ϭ For a single particle, NB 1, identity Let us factorize this expression using identity l/2 ˆ ͩ ͪ ͑ ͉ ͉ͱ͉͒ O J 2g a ϭ Ϫ ϩ ϩ ͒ ϭ ˆ bϪb*͉ l 1 exp͓ b* b ͑1/2 bb*͔ Oe ϭ1 , ͑A17͒ ϭl/2 ͒ 2͉ ͉2 ͉ ͉ͱ ͑ ͒ exp͓͑1/2 g a ͔Jl͓2g a ͔ A20 ͔ ץץ 2ץ ϭ ͓Ϫ 1 ˆ where operator O exp ( 2)( / ) . ͑ ͒ Now integration in Eq. A3 over i can be easily carried can be verified using the series for the Bessel function. This out for each particle using identity allows us to write
016507-10 QUANTUM THEORY OF OPTICAL STOCHASTIC COOLING PHYSICAL REVIEW E 65 016507
ϱ nϩl/2 a l/2 be obtained by complex conjugation. ͑ ͒ϭ ͵ Ϫͩ i ͪ ͑ ͉ ͉ͱ͒ Ϫ Fn t,p,l d e Jl 2g a The sum S( ) can be split into two parts, one, for n! a* 0 i ϽMϽϱ, ϽNϽϱ, and another one for ϪϱϽMϽϪ, ϫexp͓͑1/2͒g2͉a͉2͔ ϪMϽNϽϱ. In the first sum we may start summation from ϭϪ MϪ N because the maximum power of z in LNϩ (z)is t ϩ Ͻ ϫexpͫ Ϫ͑g2/2͒ ͵ d͑a a˙ *Ϫc.c.͒ͬ. ͑A21͒ N . Therefore, derivatives over z are equal to if N . i i 0 After this, the sum can be calculated first by expressing 2 ͓Ϫ ͔ LNϪ y in terms of the confluent hypergeometric func- Integration over gives ͓9͔ the result in terms of La- tion and using the integral representation of the latter, l guerre polynomials Ln Ϫ ͑ ϩ͒ iϱ N N ! Ϫ Ϫ ds s l 2 2 l 2 2 2 ͓Ϫ ͔ϭ 2 y ͵ sy ͑ ͒ϭ͑ ͒ ͓Ϫ͑ ͒ ͉ ͉ ͔ ͑ ͉ ͉ ͒ L Ϫ y y e e . Fn t,p,l ga exp 1/2 g a Ln g a N ͑ Ϫ͒ Nϩϩ1 N ! Ϫiϱ2 i ͑sϪ1͒ t ͑B5͒ ϫexpͫ Ϫ͑g2/2͒ ͵ d͑a a˙ *Ϫc.c.͒ͬ. i i 0 Mϩ͓ ͔ Second, we can write LNϪ x* ͔ ͓ ͉͔ 2 MϪ͓ ͔ ץ ץϭ Ϫ͓ ͒ ͑ A22 ( / z ) LNϩ z zϭx*, and use 9 ϱ Equation ͑A22͒ reproduces Dattoli-Renieri’s ͓4͔ result for ͑Nϩ͒! Nϩ MϪ͑ ͒ MϪ͑ ͒ a single particle. Equation ͑A19͒ defines the evolution of the ͚ LNϩ x LNϩ z NϭϪM ͑NϩM ͒! initial state of the system for a bunch with Nb particles. ͑xz͒Ϫ(MϪ)/2 ϭ exp͓Ϫ͑xϩz͒/͑1Ϫ͔͒ APPENDIX B: DETAILS OF THE DERIVATION 1Ϫ OF EQ. „84… 2ͱxz ͑ ͒ ϫ ͩ ͪ ͑ ͒ Equation 81 can be simplified, first, by integrating over I͉ Ϫ͉ , B6 M 1Ϫ and Ј and then by and Ј using formula ϭ͕͓ Ϫ Ϫ ͔͖ l m/2 where s(G 1)/(s 1)G . In this form, the answer is ͵ Ϫ ˆ ͩ ͪ ͑ ͱ ͉͒ ϭ m/2 Ϫb/2 m͑ ͒ valid also for the second part of the sum, Ϫϱ Ͻ M ϽϪ , d e O Jm 2 b ϭ1 b e Ll b . l! ϪMϽNϽϱ. ͑ ͒ B1 The sum over M, This can be obtained by using the series represntation of the 2 ϱ M ץ Bessel function. The result is given in terms of Laguerre x0 m S͑͒ϭͩ Ϫ ͪ ͚ 2 ץ polynomials L . Equation ͑B1͒ is valid both for mϾ0 and n z MϭϪϱ y Ͻ Ϫ͉m͉ m 0, where Ll (b) has to be understood as Ϫ Ϫ Ϫ iϱ ds ͑s͒ esy ͑xz͒ (M )/2 ͑lϪ͉m͉͒! ϫeϪy ͵ Ϫ͉m͉ m ͉m͉ ͉m͉ ϩ1 Ϫ L ͑b͒ϭ͑Ϫ1͒ b L Ϫ͉ ͉͑b͒. ͑B2͒ Ϫiϱ2 i ͑sϪ1͒ 1 l l! l m 2ͱxz In this way we obtain ϫ Ϫ[(xϩz)/(1Ϫ)] ͩ ͪ ͑ ͒ e I͉ Ϫ͉ , B7 M 1Ϫ
F ͑l⌺ ,M ⌺͒F ͑l⌺Ј ,M ⌺Ј ͒Q͑b ,b ͒ out out 1 2 can be calculated using 1 ϭ͑ ͒MϪ͑ ͒Mϩ ͫϪ ͑ ϩ ͒ͬ ϱ b1* b1 exp b2b1* c.c. 2 ␣k/2 ͒ϭ (/ͱ␣)ϩͱ␣ ͑ ͒ ͚ I͉k͉͑2 e . B8 ϭϪϱ ϫ MϪ͑ ͒ Mϩ͑ ͒ ͑ ͒ k LNϩ b2b1* LNϪ b2*b1 , B3 After that, each derivative over z gives the factor (/͓1 ϭ ͚ ͓Ϫ ϩ ͔ k where b1 gt exp 2ik(zj pj) . The average, ͗p ͘ is Ϫ Ϫ j ͔)(1 ͓x/x0͔). S takes the form of proportional to the sum
ϱ ϱ N iϱ ds ͑NϪ͒! GϪ1 ͑͒ϭ Ϫ2 Ϫy ͵ sy ͑͒ϭ M ͩ ͪ S y x0 e e S ͚ x0 ͚ Ϫiϱ2i MϭϪϱ Nϭmax(ϪM,N) ͑NϩM ͒! G ϫexp͕͓x*͑x/x Ϫ1͒ϩx Ϫx͔/͑1Ϫ͖͒ Ϫ ϩ y 0 0 ϫLM ͓x͔LM ͓x*͔L2 ͫϪ ͬ, ͑B4͒ Nϩ NϪ NϪ GϪ1 x 2 Gϩ1͑GϪ1͒ ϫͩ 1Ϫ ͪ . ͑B9͒ ϭ͉ ͉2 ϭ ϭ͉ ͉2 Ͻ x ͑ Ϫ ͒2ϩ1 where y Ϫ , x b2b1* , and x0 b1 . Terms 0 can 0 s G
016507-11 S. HEIFETS AND M. ZOLOTOREV PHYSICAL REVIEW E 65 016507
The integral is given here by the residues of the poles at s Ϫ* ϭ ͗ n͘ϭ ˆ ͑ ϩ ˆ ͒ͩ ͪ G, p j ͚ Kn͚ 1 p iÞ j Ϫ ϫ ͱ ͉ Ϫ ͉2͉ Ϫ ͉2 ͑ ͒ I2͓2 G b2 b1 Ϫ ϩ ͔, B11 2 x0 x Ϫ ͑͒ϭ ͑ Ϫ ͒ͩ ͪ ͩ Ϫ ͪ ͑ ͱ ͒ b1 b2 S G G 1 1 I2 2 GAy ͩ ͪ exp͕͑GϪ1͉͒b Ϫb ͉2 yA x0 Ϫ 2 1 b1* b2* ϫ ͓ ϩ ϩ͑ Ϫ ͒ ͔ ͑ ͒ exp x0 y G 1 A , B10 ϩ͑ ͓͒ ͑ Ϫ ͒ϩ ͔͖ 1/2 b2 b2* b1* c.c.
ϫexp͕͑1/2͓͒Ϫ͑Ϫ*Ϫϩ͒ϩc.c.͔͖͉ ϭ . b2 b1 ϭ͉ Ϫ ͉2 where A b2 b1 . Finally, ͑B12͒
͓1͔ M.S. Zolotorev and A. Zholentz, Phys. Rev. E 50, 3087 ͑1994͒. ͓7͔ S. Heifets, Report No. SLAC-PUB-8593, 2000 ͑unpublished͒. ͓2͔ A.A. Mikhailichenko and M.S. Zolotorev, Phys. Rev. Lett. 71, ͓8͔ S. Heifets, Report No. SLAC-PUB-8575, 2000 ͑unpublished͒. 4146 ͑1993͒. ͓9͔ I.S. Gradshteyn and I.M. Ryzhik, Tables of Integrals, Series, ͓3͔ D. Mohl, Stochastic Cooling for Beginners ͑CERN School, and Products ͑Academic Press, New York, 1980͒, p. 1038. Geneva, 1977͒. ͓10͔ W. Becker and J.K. McIver, Phys. Rev. A 28, 1838 ͑1983͒. ͓4͔ G. Dattoli and A. Renieri, Laser Handbook, edited by W.B. ͓11͔ K.J. Kim Brightness, Nucl. Instrum. Methods Phys. Res. A ͑ ͒ Colson and A. Renieri Elsevier, New York, 1990 , Vol 6. 246,71͑1986͒. ͓ ͔ 5 A. Bambini and A. Renieri, Lett. Nuovo Cimento 21, 399 ͓12͔ M.O. Scully and M. Suhail Zubairy, Quantum Optics ͑Cam- ͑ ͒ 1978 . bridge University Press, Cambridge, England, 1997͒. ͓6͔ W. Becker and J.K. McIver, Phys. Rev. A 27, 1030 ͑1983͒.
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