Okamato, H.

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Lawrence Berkeley National LaboratoryLawrence Berkeley National LaboratoryTitleThree-Dimensional Laser CoolingPermalinkhttps://escholarship.org/uc/item/820331sfAuthorOkamato, H.Publication Date2008-09-19eScholarship.org Powered by the California Digital LibraryUniversity of CaliforniaLBL-35899UC-414Lawrence Berkeley LaboratoryUNIVERSITY OF CALIFORNIAAccelerator & FusionResearch DivisionPresented at the European Particle Accelerator Conference,London, England, June 27-July 1, 1994, and to be publishedin the ProceedingsThree-Dimensional Laser CoolingH. Okamoto, A.M. Sessler, and D. MohlJune 1994Prepared for the U.S. Department of Energy under Contract Number DE-AC03-76SF00098-n (jo ~. r") -:> f~n :p+::'Sz:: III nro .-+- ("IIII ([) 'DT U1 _<InrnV I:)-) '~'[LJ l::~) (,,~.t.:e.:::-Three-Dimensional Laser CoolingtHirorni Okamoto*, Andrew M. SesslerLawrence Berkeley Laboratory1 Cyclotron Road, Berkeley, CA 94720, U.S.A.Dieter MoWCERN-PS, CH 1211Geneva 23, Switzerland2.2 Equations ofnwtion(3)2. A COUPLING CAVITI2.1 Rf cavitiesAs a source for developing syncbro-betatron coupling, letus first consider a coupling cavity excited with TM210 mode.Assuming a simple rectangular pill-box geometry, the vectorpotential is given byAt =(0' 0, ~: sin(7)cos(;)sin(ffiet)} (1)where 2a and 2b are, respectively, the width and height of thecavity, V c is the voltage amplitude of the rf field, and ffi cdenotes the rf angular frequency (harmonic number h c).Since the longitudinal electric field derived from Eq.(l) nearthe center has a linear horizontal (x) dependence, we obtain acoupling between the synchrotron motion and horizontalbetatron motion.In order to achieve enhanced coupling, it is necessary toforce stored ions to execute synchrotron oscillations. Thus weneed to introduce an rf bunching cavity which is an ordinarycavity excited either with a coaxial or TMOIO mode. Wewrite the vector potential asAb = (0, 0, ~: Sin(ffibt»). (2)where ffib and Vb are, respectively, the angular frequency(harmonic number hb) and voltage amplitude of the bunchingcavity.Taking a linear approximation, and assuming zero-dispersion and zero-derivative of it at the cavity, theHamiltonian of the system including the coupling andbunching cavites can be given as [8]1H1 =-(p/+v/x2)2~OW2 21tVL2. 5: (8 8)---+--sm('V+'Vb)Up - b2 ~o21thb l e . (he )1: (8 8)---"-.-=;...x-sm -'V+'Ve up - c .he h bHere q is the charge state of stored ions, ~ O=YII'-2 -Y -2 is thephase slip (or off-momentum) factor which always takes anegative value in our applications, Vx is the horizontal tune,vL is related to the longitudinal tune vL as cos(21tV0=I-21t2VL2, 'Vb and 'If c are the synchronous phases at thebunching and coupling cavity respectively. The canonicalAbstractTo realize effective transverse laser coolingsimultaneously with longitudinal laser cooling, twopossibilities are theoretically studied. The schemes are bothbased on forced synchro-betatron coupling, which transfer theextremely effective longitudinal laser cooling action totransverse directions. The coupling is prOVided by an rf-cavity with a transverse variation of the longitudinal electricfield (e.g. 1M21O mode) or by a normal rf-cavity located at aposition of [mite dispersion. It is shown that, when a workingpoint is close to the difference resonance, cooling in all threedegrees of freedom is simultaneously obtained.1. IN1RODUCTIONElectron cooling and stochastic cooling are well-established techniques, which have been successfully appliedto decreasing the temperature of stored beams of anti-protonsas well as other particles. A new method, called lasercooling[I], has recently been employed, experimentally, in theTSR Storage Ring of Heidelberg[2] and in the ASTRID Ringof Aarhus[3]. Longitudinal temperature of close to 1 mK, thelowest circulating beam temperature ever reported, have beenachieved for 7li+. The laser cooling mechanism is, however,applicable only to longitudinal motion, and no effectivemethod to realize simultaneous cooling in both longitUdinaland transverse motion has, so far, been developed. Thus in theexperiments mentioned the longitudinal temperature was 1mK, but the transverse temperature was 1000 KIn the present paper, we explore two possible schemes toachieve three-dimensional laser cooling based on forcedsynchro-betatron coupling.[4,5]. In principle, with only minormodifications in the hardware of the ring, it should bepossible to cool transverse beam temperature to the sameorder as the longitudinal temperature.If the present schemes work as theoretically expected, itshould be possible to observe beam crystallization[6,7]. Inaddition, laser-cooled beams with ultra-low temperaturescould be employed to cool another beam, just analogous to theelectron cooling scheme.This work was supported in part by the Director, Office of EnergyResearch, Office of High Energy and Nuclear Physics, of the U.S.Department of Energy under ocntract no. DE-AC03-76 SF00098.*On leave from Accelerator Laboratory, Institute for ChemicalResearch, Kyoto Unaiversity, Gokanoshou, Uji, Kyoto 611, Japan.(6)-- Horizontal••••• Longitudinal········...~.....,~' ..o 50 100150200Number of turns2648-3VJ 10x10j.~VJE~(,)CI)Since the motions governed by Eqs.(4) and (5) with thedamping term are linear, we can construct the transfer matrixof each element. Figs.I show the damping rates evaluatedfrom the eigenvalues of the one-tum matrix. It is seen that,only when the operating point is on a difference resonance,Le. v x-vL=integer, the damping of both modes saturate at thelevel which is exactly half of the maximum longitudinal rate.The condition of resonance must, therefore, be closelysatisfied to enhance the horizontal damping.The dispersion relation from the one-tum transfer matrixcan be approximately solved to give the damping rates of thelongitudinal and horizontal motion{1t21I;olr/ COth( AD }Im(v) == vx 2 2 2AD 1t Il;olrc COth(AD),41t V x 2where, we have assumed the resonance condition, takenAu=A x (the extent of the cooling section), and employed apertwbation analysis assuming r c 2«1.The tracking result corresponding to the case (c) in Fig.lis shown in Fig.2, using the coupling constant r c=O.OO6. Thedamping of the both motions is beautifully demonstrated.3.1 Hamiltonian3. DISPERSION-INDUCED COUPLING2.3 Damping ratesFig.2 Tracking result corresponding to the parametersemployed in Fig.l(c) with rc=O.OO6.o 2 .c 6 a 10X10')Coupling coostanl r co 2 4 6 a 10xl0"CoupIi.Dg constant r c2(c)(a)i /~••/~ RESCRVlCEo 2 4 II. 8 10>:10"CoopliDg coostmt r c.. IVL~13I'3variables (x,Px,) correspond to the horizontal displacement andmomentum, and ('V,W) to the phase and energy deviationfrom the synchronous values. The coupling strength has beenintroduced as rc=qVcRl2~oCIJoa, where R represents theaverage ring radius, and PO is the reference particlemomentum. In Eq.(3), the independent variable is taken asS=sIR with s being the distance along the reference particleorbit, and S b and Sc indicating the location of the bunchingand coupling cavity respectively.To couple the vertical motion to the other two motions, it isonly necessary to install a skew quadrupole or a verticalcoupling cavity in the ring, but, in this section, we consider,for simplicity, only longitudinal-horizontal coupling.The Hamiltonian (3) yields the equations of motiond2xdS2 + v/x = 21trc,!,Op(S - Sc), (4)2d Y+21tVL2'V0p(S-Sb)=-2~rcxOp(0-Oc)' (5)dOwhere we have assume that 'Vb=1tI2 and 'Vc=0. In our model,the simple frictionalteim, EA(d\p'kl8), is added to the l.h.s. ofEq.(5) to incoq:x>rale the laser cooling effect Here, £=1 in thelaser cooling section, while £=0 in other regions. Thismodeling of laser cooling is only approximate (it does notinclude the variation of cooling force with longitudinalvelocity and it does not differentiate between the cooling oflongitudinal velocity and the cooling of energy), but theapproximation made is adequate for the analysis of this paper.5xI0" -r------_005Xl0··:;- 3:§Fig.l The damping rates, Im(v), of the longitudinal andtransverse modes plotted as a function of the couplingstrength rc for three different synchrotron tunes. Thehorizontal tune and the damping parameter are fixed,respectively, at vx=2.l7 and A [)'21t=O.OI. The couplingcavity and bunching cavity are next to each other, and are setat the position opposite to the cooling section. The dOlledcurves in Fig. I (c) correspond to Eqs.(6). The slip factor istaken to be -0.947 roughly corresponding to the ASTRID ringin Denmark[3].Let us now investigate transverse laser cooling inducedthrough dispersion at an rf cavity. We here take verticalmotion in to account, introducing a pure skew quadrupole.The Hamiltonian of the system is given byI I ):. W 2H =-(p 2+ V 2x2)+_(p 2+ v 2y2) __SO__2 2 x x 2 Y Y 2hbqV b+ r qXY' Op (0 - Sq) - -R- cos('V - ~bPx )op (8 - 8b ),POI-'Oc(7)where 8q is the location of the skew quadrupole magnethaving the coupling strength r q. The dispersion functionevaluated at the skew quadrupole position 8q and the2derivative of the dispersion at the cavity position 8b have beenassumed to be zero, and Sb=11(8=8b)1R.The equation of motion can be readily derived fromEq.(7). The equation for the variable W is, however,modified, incorporating the laser damping effect, asdW 2m;2.- == -j::-sm('I'- SbPx)op(8-8b) - fAW. (8)de ~O3.2 Damping ratesIahle 1Stored Ions: 100 keV 24Mg+Ring Circumference: 40 [m] (R=6.366 [m] )Transition energy Ytr : 4.58Tunes: vx==2.27, vy=2.27, vL=0.27 (The same as Fig.3b )RF Voltage Vb : 2.9 [leV]RF frequency: 580.7 [kHz] (h=26)Dispersion at the cavity 11b : 0.3 [m]Transverse coupling strength r q : 0.1vx=2.27, vy=2.27, VL=O.27-- Horizontal.•..•. Vertical.•.•.. Longitudinal50 100 150200Number of turnso48To estimate the optimum dispersion T]opt, we just equateEq.(9) with Ad12n. To obtain:11 2 = ll;olR2(A 2 + r q2')cot Jlo (10)opt 61tV v 2 D V v 2x L x ywhere we have assumed A 0«1.The tracking result based on the same parameters used inFig.3(a) is given in Fig.4; effective damping is achieved. Anexample of parameters roughly corresponding to those of theASTRID Ring are presented in Table 1.FigA Tracking result corresponding to the parametersemployed in Fig.3(a) with Tlb=1 m.o-l,----r----r--....,:.:::~~2.0(a)0.5 1.0 1.5Dispersion llb [m]vx=2.l2, vy=3.12, VL=O·1221O-;'~:=:"'-r---....---r-----.0.05x 10- 343Two separate resonance conditions are necessary now;VX-VL==integer for the longitudinal-horizontal coupling, andvrvy=integer for the horiwntal-vertical coupling.Then, using perturbation analysis with respect to smallSb2, one obtains the damping rates of the transverse modesIm(V) = VxVL2Sb2 tan J.!O/(tanh AD + r q2 ) (9)41l;ol 2 2 2Vxv y sinhAD 'where ~e have assumed r q l..Jvxvy «1, putting that vx=J..IO,vy.=J.Io+2m1t, and v L=J.Io+2n1t (m,n=integer). From Figs.3,which give numerical evaluation of all three damping rates,one concludes that there is an optimum dispersion for whichthe three rates become equal.We have presented two possible schemes which providethree-dimensional laser cooling. It is found that, when theoperating point is on resonance, the coupling is considerablyenhanced leading to large damping rates of the transversemodes. Thus, it is possible to reduce the transverse beamtemperatures to a similar level to the longitudinal one.4. SUMMARY21o ~~----.~:::::::::;:==:::::::;::====;5x1 0- 343Fig.3 The damping rates, Im(v), of the longitudinal andtransverse modes plotted as a function of the dispersion at therf-cavity ''1b=T](8=8b). The If-cavity sits at 8b==1t, the positionopposite to the cooling section, while the quadrupole with thecoupling strength r q=0.1 is set at 8 q=3 rc/2. The dotted curvescorrespond to Eq.(9), and the line parallel to the abscissaindicates the level Im(v)=Arfl2tt.0.0 0.5 1.0 1.5Dispersion llb [m]2.0 5. REFERENCES[1]. D. 1. Wineland and H. Dehmelt, Bull. Am. Phys. Soc. 20, 637(1975); T. Hanchand A, Schawlow, Opt. Comm. 13,68 (1975).[2]. S. SchrOder et aI., Phys. Rev. Lett. 64, 2901 (1990).[3].1. S. Hangst et ai., Phys. Rev. Lett. 67, 1238 (1991).[4]. H. Okamoto, A. M. Sessler, and D. Mohl, to be published inPhys. Rev. Lett.[5]. H. Okamoto, to be published.[6].1. P. Schiffer and P. Kienle, Z. Phys. A321, 181 (1985).[7]. 1. Wei, X-Po Ii, and A. M. Sessler, submitted to Phys. Rev. Lett.[8]. See, for example, C. 1. A. Corsten and H. L. Hagedoorn, Nucl.Instrum. Methods A212, 37 (1983).3

Three-Dimensional Laser Cooling

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