Raman cooling and heating of two trapped Ba ions

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1 PHYSICAL REVIEW A, VOLUME 65, Raman cooing and heating of two trapped Ba ions D. Reiß, K. Abich, W. Neuhauser, Ch. Wunderich, and P. E. Toschek Institut für Laser-Physik, Universität Hamburg, Jungiusstraße 9, Hamburg, Germany Received 15 March 2001; pubished 12 Apri 2002 We study cooing of the coective vibrationa motion of two 138 Ba ions confined in an eectrodynamic trap and irradiated by aser ight cose to the resonances S 1/2 -P 1/2 493 nm and P 1/2 -D 3/2 650 nm. The motiona state of the ions is monitored by a spatiay resoving photomutipier. Depending on detuning and intensity of the cooing asers, macroscopicay different motiona states corresponding to different ion temperatures are observed. We aso derive the ions temperature from detaied anaytica cacuations of aser cooing taking into account the Zeeman structure of the energy eves invoved. The observed motiona states perfecty match the cacuated temperatures. Significant heating is observed in the vicinity of the dark resonances of the Zeemanspit S 1/2 -D 3/2 Raman transitions. Here two-photon processes dominate the interaction between asers and ions. Parameter regimes of aser ight are identified that impy most efficient aser cooing. DOI: /PhysRevA PACS numbers: Pj, Vk, Lx I. INTRODUCTION In order to impement conditiona quantum dynamics of qubits represented by the interna structure of individua ions in a Pau trap, a vibrationa mode of the entire ion string is used. Cirac and Zoer 1 have suggested a method for conditiona dynamics that requires the ions to be cooed to their vibrationa ground state. Here, the vibrationa state represents a bus qubit that gets entanged with the ions interna states by stimuated transitions, where the absorption or emission of a photon is couped with the emission or absorption of a vibrationa quantum. The recenty proposed So rensen-mo mer scheme 2 incorporates two-photon transitions in order to directy entange the interna states of the ions. Here, the vibrationa mode is ony virtuay excited. In this way the contribution of higher vibrationa excitations can be made to destructivey interfere. In this way, the scheme does not require cooing to the vibrationa ground state, but admits popuation in higher vibrationa eves. In genera, the manipuation of the motion of trapped partices by ight reies on the couping of eectronic and motiona degrees of freedom via the scattering of photons. Here, the important figure of merit is the Lamb-Dicke parameter LDP, its square given by the ratio of the recoi energy transfered when scattering a photon, and the energy of the vibrationa quantum 3. For sma, transition processes invoving the creation or annihiation of n vibrationa quanta i.e. nth order sideband transitions are suppressed by the factor 2n. Therefore, a sma LDP aows ony inear couping invoving ony singe-phonon transitions between eectronic and vibrationa degrees of freedom. This Jaynes- Cummings type of couping 4 is prerequisite for many schemes suggested in quantum information processing incuding the mentioned impementations 1,2. Penio and Knight 5 have proposed to drive the interaction between eectronic and vibrationa degrees of freedom by tuning strong aser ight to the resonance frequency, instead of the sideband. In this case, the ight intensity imposes a specific ight shift on the ions that seects the desired transition. By doing so, the time needed for conditiona dynamics is reduced, as compared with the Cirac-Zoer scheme, thus reducing the effect of decoherence. Here, too, the LDP pays a crucia roe as the fideity of such a fast quantum gate is f Whie the So rensen-mo mer scheme does not require ground-state cooing, it sti demands ow vibrationa quantum numbers i.e., 2 (n1)1. Thus, aser cooing is essentia for quantum information processing in ion traps, and it is desirabe to have at hand robust cooing techniques that are suited for reaching ow vibrationa excitation at moderate technoogica expenditure on the side of the aser sources. In particuar, cooing of the coective motion of severa partices is prerequisite for impementing conditiona quantum dynamics with trapped ions. It has been known for some time that the motion of trapped partices strongy depends on aser cooing. With a singe trapped partice, distinct macroscopic motiona states have been observed with different parameters of the cooing asers 6. With a coud of ions, changes in the temperature induced by aser cooing cause phase transitions between a crystaine and an evaporated state 7,8. These macroscopicay distinct states indicate the specific temperature achieved by aser cooing. In this paper we investigate the macroscopic motiona states of two trapped 138 Ba under the action of aser cooing. In Sec. II, the theory of aser cooing of the coective motion of two charged partices trapped in an anisotropic harmonic potentia is outined. If the partices obey the Lamb-Dicke imit, the cooing rate and the achievabe temperature are cacuabe without any free parameter. We fuy take into account the mutieve structure of the invoved ions, thus aowing for any type of cooing process, such as Dopper 9, sideband 10, and Raman cooing 11. In Sec. III we describe the experimenta setup and present the resuts of the measurements. Phase transitions between macroscopicay distinct states are observed. The parameter vaues at which these transitions take pace perfecty agree with the vaues cacuated from the theoretica mode. In Sec. IV we use the mode to find optimized aser parameters for cooing the coective motion of the ions /2002/655/ /$ The American Physica Society

2 REIß, ABICH, NEUHAUSER, WUNDERLICH, AND TOSCHEK PHYSICAL REVIEW A TABLE I. Frequencies and Lamb-Dicke parameters for the different vibrationa modes. Note that the geometrica factor (k e q )is incuded in the LDPs Eq. 15. In the present experiment the ange of incidence is 45 to a directions, yieding a factor of 1/3. Mode X Y Z x ỹ z /2 MHz II. THEORY A. Coective motion We consider the coective motion of two ions of mass m in a three-dimensiona 3D harmonic trapping potentia V trap r j m 2 x 2 x j 2 y 2 y j 2 z 2 z j 2, which is the same for both ions ( j1,2). Since each of the ions carries one positive eementary charge e, they repe each other by the Couomb interaction V C r 1,r 2 e r 1 r 2. The potentia is simpified by using center-of-mass and reative coordinates R r 1r and rr 2 r 1. 3 With the definition of tota and reduced masses M2m and m/2, one gets VR M 2 x 2 X 2 y 2 Y 2 z 2 Z 2 for the center-of-mass motion, corresponding to the motion of a singe trapped partice with the origina osciation frequencies, but twice the ions mass. The potentia for the reative motion is Vr 2 x 2 x 2 y 2 y 2 z 2 z 2 e r. If the trap frequencies sighty differ as they do in our experiment, see Tabe I this potentia does not show spherica symmetry. With x y z hoding, the potentia has absoute minima on the x axis and sadde points on the y and z axis at r 0,q 3 e q 2 e q, qx,y,z. 6 Figure 1 shows the potentia in the (x,y) pane. It is cear FIG. 1. Effective doube-we potentia V(x,y) for the reative motion of two trapped 138 Ba ions trap frequencies as given in Tabe I. The contour ines correspond to V2,4,6,8 ev. that for high enough temperature, i.e., at a mean motiona energy E ex comparabe with the potentia barrier at the sadde point on the y axis V y 3 3 e /3 y 2/3 x, 7 the ions can move freey in a ring-shaped region in the (x,y) pane. However, at ow temperature, the reative motion is restricted to the wes, i.e., the ions wi appear at approximatey fixed positions. The same argument can be appied to the z direction: For even higher energies E ex V z the motiona state changes from a ring to a sphere. A three motiona states have been observed in our experiment. However, in the foowing we sha focus on the transition between the ocaized and the ring state, as we are mainy interested in aser cooing in the ow-temperature regime. In the ocaized state the potentia can be approximated around the minimum at r 0,x. Expanding up to second order in the dispacement r rr 0,x one gets VrVr 0,x 2 x2 x 2 yỹ 2 zz 2 with oca frequencies cf. Tabe I. x3 x, y,z 2 2 y,z x B. Laser cooing of the coective motion The mechanica effect of irradiating ions with aser ight eads to thermaization of the motiona degrees of freedom. The corresponding cooing rate and steady-state temperature depend on the eectronic eve scheme and aser parameters;they are cacuated in this section. The key to the theory presented here is the we-known Lamb-Dicke imit. As ong as the ions are ocaized near the minima of the potentia,

3 RAMAN COOLING AND HEATING OF TWO TRAPPED Ba IONS PHYSICAL REVIEW A their spatia extension is much smaer than the waveength of exciting aser ight. In this case, the photon recoi operator e ik r, i.e., the couping of the interna dynamics to the ions motion, can be expanded in the LDP. Since the LDP is sma, dynamics due to this couping is sow compared with the interna dynamics that may be adiabaticay eiminated. This reduction eads to a master equation for the motiona degrees of freedom, where the invoved transition rates depend on steady-state expectation vaues of interna operators. This procedure and its appication to a singe Ba ion has been described in Ref. 12. We do not account for dipoe-dipoe interaction between the two ions, since the separation of the two ions is arge as compared to the optica waveengths used (r 0 /8). Even though processes caused by this interaction possiby coud be observed 13, they are rarey compared with the singe-ion scattering, and their infuence on the coective motion is negigibe. Thus, we do not consider correations between the two ions interna dynamics that woud ead to cooperative effects such as superradiance. We rather cacuate the cooing effect that each of the two ions individuay has on the common motion. This effect is the same for both ions, since the foci of the asers are wide compared with the distance of the ions the waist of the Gaussian beams being w10 r 0 ), and therefore, the aser intensities, and consequenty aser cooing do not depend on the ions positions. We sha now give a short recap of the theory presented in Ref. 12 and appy its resuts to the coective motion of two ions in the current experiment. We start from the density operator ϱ for the motiona and interna degrees of freedom. The dynamics is described by the master equation where Lϱ j1,2 d dt ϱt 1 i H,ϱtLϱt, 2 1, j dw e ik r j ϱe ik r j, j 1, j, j ϱϱ, j, j is the Liouviean aowing for spontaneous emission. It has the usua Lindbad form 14, where dissipation acts on the interna via ) and on the motiona via e ik r ) degrees of freedom. The index j abes the two ions, and the index runs over a atomic transitions invoved. 2 is the spontaneous emission rate, and are the corresponding transition operators. W () is the anguar distribution of spontaneous emission for this transition. The Hamitonian is given by H P2 p2 VR 2M 2 Vr j1,2 H in, j V Dip, j. 12 H in, j is the Hamitonian for the interna eves of the jth ion. The ion s dipoe interaction with the aser ight can be written as V Dip, j e ik r j, j H.c., 13 where is the Rabi frequency of the ight driving the transition. Summing over both ions, and rewriting the dipoe interaction in terms of center-of-mass and reative coordinates Eq. 3, gives V Dip e ik R e i/2k r,1 e (i/2)k r,2 H.c. 14 Note that the center-of-mass and reative motions are couped via the interaction with the aser ight. However, in the Lamb-Dicke imit this couping can be negected, as k R, and k r are sma and the potentia can be expanded. This expansion is characterized by the Lamb-Dicke parameters,q k e q /2m q 15 for the center-of-mass modes and,q k e q /2m q for the norma modes of the reative motion. In the experiment, a LDPs are much smaer than unity, see Tabe I. Now, we rewrite the coordinates in terms of the owering and raising operators of the corresponding vibrationa eigenmodes R q 2M q a q a q, r q ã q ã q. 2 q 16 Since the expansion is around the potentia minimum, there is a constant phase factor exp(i/2)k r 0, that is incorporated in the atomic transition operators, j. Now, keeping ony terms up to first order in any Lamb-Dicke parameter yieds V dip j1,2 1j 2 x, j 1 2 q q,q a y q a q, j,q ã q ã q y, j, 17 where x and y ( )/i have been used. The first term of Eq. 17 describes the driving of the interna dynamics by the interaction with the aser ight. Since it is of zeroth order in any LDP it does not contain the mechanica effect of ight, i.e., no aser cooing or heating. The second and third term of Eq. 17 first order in a LDP coupe the interna dynamics to the center-of-mass and reative motions, respectivey. The ion-dependent sign in the third ine of Eq. 17 arises from the opposite impact of the two ions recoi on the reative motion. Since the vibrationa modes decoupe in the Lamb-Dicke imit, we can restrict ourseves to a 1D description. Therefore, the index q characterizing the specific mode is omitted in what foows

4 REIß, ABICH, NEUHAUSER, WUNDERLICH, AND TOSCHEK PHYSICAL REVIEW A Now, the fast interna dynamics of the ions is adiabaticay eiminated and one obtains a master equation for the externa degrees of freedom. This procedure finay eads to the we-known time evoution of the mean phonon number 15 with the cooing rate d dt na A na, WA A and if the cooing rate is positive the mean motiona energy A E Ex. A A The transition rates A are given by A 2ReSD The first constituent is the fuctuation spectrum of the eectronic dipoe couping S, dte it y y t 0 0, 22 where is the vibrationa frequency of the considered mode. The expectation vaue is to be taken in the steady state of the interna dynamics that in turn is cacuated from the master equation d dt ϱ int 1 i H in x,ϱ in tl 0 ϱ in t, 23 derived from Eq. 10, by using adiabatic eimination, and keeping ony the zeroth order in any LDP. Here, L 0 ϱ in 2 ϱ in ϱ in ϱ in 24 describes the spontaneous emission in zeroth order. Via the driving by aser ight, the steady state depends on the parameters intensity, detuning, poarization of the ight fied. The second contribution to the transition rates originates from the spontaneousy emitted photons as described by the Liouviean L. So far, spontaneous emission has ony been considered up to the zeroth order in the LDP. The secondorder term is readiy cacuated and eads to the additiona transition rate D k e q both from n to n1 and vice versa. Here, the geometrica factor (k e q ) 2 in the LDP Eq. 15 is repaced by 1 1 dw () 2 that accounts for the anguar distribution of the ight emission. III. EXPERIMENTAL OBSERVATION The two Ba -ions are confined in a 1-mm-diameter Pau trap and irradiated by aser ight at 493 nm for excitation of resonance scattering on the 2 S 1/2-2 P 1/2 transition. This aser is detuned sighty beow resonance for cooing the ions. The fuorescence signa is recorded by photon counting. A second aser at 650 nm prevents optica pumping into the 2 D 3/2 eve. A static magnetic fied defines the quantization direction and ifts the degeneracy of the magnetic subeves. Its direction is at 45 with respect to the pane defined by the ring eectrode. The directions of propagation and the poarizations of both ight beams are set perpendicuar to the magnetic fied. The power eves of the ight fieds are stabiized by eectro-optic moduators. Precise detuning of the 650-nm aser is accompished using a doube-pass acousto-optic moduator. So far, the setup has been described esewhere 16. In addition the fuorescent ight is monitored by a spatiay resoving photo mutipier reveaing the position of the ions in the trap. The ions are dynamicay trapped by superimposed eectric dc and rf fieds 17 and represent partices inside a neary spheroida harmonic quasipotentia. In order to determine the geometry of this quasipotentia osciations of a singe trapped ion are excited by appying an additiona rf fied. Three sharp resonances are found at the frequencies given in Tabe I, i.e., macroscopic motion of the ion in the x, y, and/or z direction of the trap is observed. The x direction, within the radia pane, is found at approximatey 45 degrees to the poarization of the asers. Whie the nondegeneracy of z is obvious, the asymmetry of x and y originates from a sight asymmetry of the ring eectrode. The distance of the two ions in the potentia minima, as cacuated from Eq. 6 using the actua vaues of and q is r m and agrees very we with the experimenta vaue 3.65 m obtained from the spatiay resoved measurement. Keeping a other parameters constant, the frequency of the 650-nm aser is scanned across the 2 D 3/2-2 P 1/2 ine. The intensity of the scattered ight vs detuning of the 650-nm ight, P D, is shown in the upper part of Fig. 2, where 650 is the aser frequency and P D is the atomic transition frequency. The structure of dark resonances is we known from the fuorescence of a singe ion 18 and corresponds to four dark states, i.e., coherent superpositions of magnetic substates of the 2 S 1/2 and 2 D 3/2 eves, respectivey. From a fit of this excitation spectrum with eight-eve optica Boch equations, accounting for the Zeeman substructure, we obtain the Rabi frequencies of both asers, the detuning S P of the 493-nm aser as we as the strength of the magnetic fied. This is a standard procedure in our Ba experiments. The parameters obtained from the spectrum shown in Fig. 2 are MHz, MHz, MHz, and

5 RAMAN COOLING AND HEATING OF TWO TRAPPED Ba IONS PHYSICAL REVIEW A FIG. 2. Two trapped Ba ions show different motiona states depending on aser parameters. Top: fuorescence of two trapped ions as a function of aser detuning, coected in 0.1 s. Bottom: spatia distribution of the two ions at the detunings indicated above. B/ B 7.72 MHz, where B is the Bohr magneton. Additiona information is derived from monitoring the motion of the two ions. Depending on the detuning of the 650-nm aser, different motiona states are observed. Whereas the ions appear at fixed positions if the red aser is tuned beow the owest or we above the highest dark resonance, they form a ring-shaped distribution in the (x, y) pane with certain intermediate detunings. The ower part of Fig. 2 shows the motiona state at four different detunings, corresponding to the fat dots in the spectrum. Note the cear distinction of the ocaized state at MHz from the nonocaized state ony 2.62 MHz away: Phase transitions between the two observed motiona states appear at very definite aser frequencies. Obviousy, the motiona state of the ions depends on the ion temperature set by the aser cooing. In Fig. 3, the motiona states observed at different detunings are shown together with the motiona energies and cooing rates cacuated from theory. The different motiona states appear in distinct regions of detuning: The ions are observed in the fixed position for the detuning 650 beow 612 MHz. At this imiting vaue a sharp phase transition occurs, and the ions form a ring when 612 MHz MHz. At the upper boundary, they are cooed again to the fixed-position state. Such a hot region appears with each dark resonance. At some vaues of the detuning, e.g., at MHz, the motion fuctuated between the fixed and the ring-shaped state, thus remaining indefinite. The motiona energy has been cacuated from the mode presented in the preceding section, using the experimenta vaues for the ỹ mode. This energy shoud be compared with the height of the potentia barrier Eq. 7, which for the given data is V y 7.15 ev. Wherever the predicted temperature varies steepy upon detuning, a sharp transition is observed. This is the case, when the corresponding cooing FIG. 3. Top: observed motiona states for different detunings of the 650-nm ight. The dots correspond to individua observations. Midde: mean motiona energy in the ỹ mode cacuated from theory. Bottom: cooing rate for the ỹ mode cacuated from theory. rate changes from positive to negative, i.e., when the asers actuay heat the ion. Laser heating appears at negative detunings of both asers when Raman transitions dominate the interaction. In the vicinity of the dark resonances, the interna state consists of a coherent superposition of the S 1/2 and the D 3/2 states, forming an effective two eve system. Now, with both detunings being negative, and zee, the effective detuning of this Raman transition is positive, and causes heating of the motion. Here, zee is the difference in Zeeman shifts of the invoved subeves that defines the resonance condition of the considered dark state. Note that the transition to the ring state, at positive detunings from the Raman resonances, appears where the expected energy is sti consideraby ower than the potentia barrier. However, this does not contradict the mode, which assumes the ions cose to the potentia minima. As soon as they are smeared out over a region of considerabe size, the harmonic approximation around the minimum no onger hods. The cooing gets ess effective for the broadened veocity distribution. This happens when the therma energy is substantiay ower than V y. Sti, the observed phase transitions agree very we with the resuts of the mode. At ow speed of scanning, for exampe, d 650 /dt 12 MHz/s, we observed hysteresis effects: The motiona state upon up-tuning the aser frequency differed from the state observed upon down-tuning. However, this effect vanished when the scanning rate was decreased. Scans at a very ow speed (d 650 /dt0.12 MHz/s showed that the motiona state soey depends on the aser parameters. Therefore, a steady state exists, which is approached in a finite motiona reaxation time. The fina temperature is, in principe, not ony set by the aser cooing, but it is subject to additiona heating mechanisms, such as fuctuations in the trapping eectric fied. However, the cacuated transition rates A s 1 equa the number of scattered phonons per unit time, and thus exceed the rate of parasitic heating ( s 1 ) ob

6 REIß, ABICH, NEUHAUSER, WUNDERLICH, AND TOSCHEK PHYSICAL REVIEW A FIG. 5. Mean number of vibrationa quanta for fixed aser parameters dotted and averaged over nine combinations of both detunings see text soid. Laser parameters: MHz, MHz, MHz. Other parameters as in Sec. III FIG. 4. Fuorescence, mean number of vibrationa quanta in the x mode, and cooing rate vs aser detuning. The vertica dotted ine indicates the minima n, aser parameters: MHz, MHz, MHz. Other parameters as in Sec. III served in ion trap experiments 19. Therefore, this additiona heating can be negected as ong as the ions are kept aser cooed. IV. OPTIMIZATION OF LASER COOLING The discussed mode is now empoyed to numericay investigate the infuence of various parameters on the aser cooing, and to find optimized conditions for effective and robust cooing. We concentrate on the x stretch mode, since it is ess affected by perturbations from stray fieds of the trap 20 and, therefore, is suitabe for serving as a bus mode between the ions. A promising candidate for impementing qubits in trapped Ba is the S 1/2 -D 5/ nm transition 21. The LDP corresponding to this infrared transition is smaer than those of other transitions see Tabe I, and the So rensen-mo mer criterion 2 (n1)1 yieds the condition n10 4 that is easiy fufied in practice. In fact, a mean phonon number n1 can be reached: This is shown in the midde part of Fig. 4, where n is potted vs the detuning of the 650-nm ight. The owest vibrationa excitation was found by randomy searching the 4D parameter space for the owest achievabe n. Then, three aser parameters were kept constant, whie the red-aser detuning was varied, in order to generate the graph. The minima temperature is achieved around MHz. Note that the owest vaue n0.9 indicated by the vertica dotted ine does not match a particuary arge cooing rate, as is shown in the ower part of Fig. 4. Looking at the expected fuorescence signa upper part of the figure, which corresponds to the popuation of the P 1/2 eve, we note that the optimum detuning is on the red side of the dark resonance at MHz, as is expected for Raman cooing. The dark resonance is power broadened by the high intensity at 650 nm. On the other hand, we find ow temperatures ony for ow intensity of the 493-nm ight. This fact indicates that scattering of the bue ight is the dominant heating mechanism. We do not caim to have found the absoute minimum of achievabe temperature: Other sets of aser parameters may yied simiary ow temperatures, and aso the appied magnetic fied to some degree infuences the achievabe temperature. We have restricted the parameter search to the range of negative detuning. Even ower energies may be reached with both asers detuned to the bue side of the atomic resonance and the parameters adjusted to yied so-caed eectromagneticay induced transparency cooing 22. Here, the dynamic Stark contribution to the noninear susceptibiity dominates the interaction of ions and ight 23. However, Dopper precooing is sti necessary and the asers must either be switched from negative to positive detuning, or ese additiona ight sources are required. Moreover, the intensities of the asers must be precisey controed, since for this type of cooing to work, the dynamic Stark shift shoud match the vibrationa frequency particuary we. However, cooing mechanisms are desirabe, that are more robust against variations of intensity and detuning of the aser ight, and thus are easier to impement experimentay. The robustness of the cooing is affected by the sharp boundaries between parameter regimes of very ow and high temperature, because in these regions any drift of the asers frequencies may cause drastic deterioration of the cooing. In order to find safe parameters we repeated the search, incorporating a tentative drift in the foowing way: For a given point in the ( 493, 650 ) parameter pane, n was cacuated at this point and at eight additiona points, where one or both of the detunings differ by 12 MHz, and then averaged over the n vaues at a nine points. The best robust parameters found differ from the ones used to cacuate Fig. 4 and the achievabe ion temperature is not as ow as in the idea situation, but one sti finds n3.3. The resut is shown in Fig. 5, where the mean phonon number is shown as a function of 650 soid ine. The dotted ine shows, for comparison, the vibrationa excitation that is achieved with the same parameters, when the averaging over nine points is not carried out. The regions of safe cooing are narrower, since detuning cose to a boundary between cooing and heating is penaized. It seems not advisabe to use a rather narrow region of cooing as, for exampe, cose to MHz, because spurious aser drift might

7 RAMAN COOLING AND HEATING OF TWO TRAPPED Ba IONS PHYSICAL REVIEW A easiy impose unexpected heating. However, some regions e.g., 650 (22 to 172 MHz and 650 (5 to 52 MHz are broad enough to aow for some aser drift with satisfactory cooing. V. CONCLUSION We have observed the effect of aser cooing and heating on the coective motion of two trapped Ba ions by monitoring the macroscopic motiona states of the ions. These states indicate the vibrationa excitation, or temperature of the ions. In this way, Raman heating is demonstrated that dominates on the bue side of the dark resonances i.e., Zee ). The observed phase transitions are a remarkabe macroscopic manifestation of dark states, i.e., of the coherent quantum superposition of atomic eves. Raman cooing on the red side of the dark resonance is impied, though it is not directy observed with this method, since no macroscopic change of the motiona state shows up. The good agreement of the cacuated excitation with the experimenta observations shows that the outined theory describes we the aser cooing of the coective motion of the ions. The predicted cooing rates greaty exceed spurious heating rates of the vibrationa motion, for exampe, from fuctuations of the trapping eectric fied. Thus, such heating is negigibe as ong as the cooing ight is on. The mode comes with no free parameters; however, it is appicabe ony if the ions are ocaized near their potentia wes. Nevertheess, this is not a strong restriction since this regime is supposed to be used in any experiment impementing conditiona quantum dynamics. This aows us to identify parameter regimes that impy effective and robust aser cooing. We find that the coective motion can become cooed cose to the ground state, if the dark resonances are not much broadened by aser power or fuctuations. But even when fuctuations in aser parameters are aowed for, ow eves of temperature emerge that are suitabe for quantum information processing using the scheme for conditiona quantum dynamics suggested by So rensen and Mo mer 2. We note that the eve scheme of Ba has the same structure as, for exampe, Ca 24, a type of ion often suggested, and the resuts obtained here are appicabe, with appropriate modifications, to the cooing of these ions, too. 1 J.I. Cirac and P. Zoer, Phys. Rev. Lett. 74, A. So rensen and K. Mo mer, Phys. Rev. Lett. 82, S. Stenhom, Rev. Mod. Phys. 58, B.W. Shore and P.L. Knight, J. Mod. Opt. 40, D. Jonathan, M.B. Penio, and P.L. Knight, Phys. Rev. A 62, Th. Sauter, H. Gihaus, W. Neuhauser, R. Batt, and P.E. Toschek, Europhys. Lett. 7, R. Büme, J.M. Chen, E. Peik, W. Quint, W. Scheich, Y.R. Shen, and H. Water, Nature London 334, Th. Sauter, H. Gihaus, I. Siemers, R. Batt, W. Neuhauser, and P.E. Toschek, Z. Phys. D: At., Mo. Custers 10, W. Neuhauser, M. Hohenstatt, P.E. Toschek, and H.G. Dehmet, Phys. Rev. Lett. 41, C. Monroe, D.M. Meekhof, B.E. King, S.R. Jefferts, W.M. Itano, D.J. Wineand, and P. Goud, Phys. Rev. Lett. 75, H. Gihaus, T. Sauter, W. Neuhauser, R. Batt, and P.E. Toschek, Opt. Commun. 69, D. Reiß, A. Lindner, and R. Batt, Phys. Rev. A 54, A. Beige and G.C. Hegerfedt, Phys. Rev. A 59, G. Lindbad, Commun. Math. Phys. 48, M. Lindberg and J. Javanainen, J. Opt. Soc. Am. B 3, M. Schubert, I. Siemers, R. Batt, W. Neuhauser, and P.E. Toschek, Phys. Rev. A 52, W. Neuhauser, M. Hohenstatt, P.E. Toschek, and H.G. Dehmet, Phys. Rev. A 22, Y. Stagies, I. Siemers, B. Appasamy, and P.E. Toschek, J. Opt. Soc. Am. B 15, H. Rohde, S.T. Gude, C.F. Roos, P.A. Barton, D. Leibfried, J. Eschner, F. Schmidt-Kaer, and R. Batt, e-print quant-ph/ unpubished. 20 B.E. King, C.S. Wood, C.J. Myatt, Q.A. Turchette, D. Leibfried, W.M. Itano, C. Monroe, and D.J. Wineand, Phys. Rev. Lett. 81, B. Appasamy, Y. Stagies, and P.E. Toschek, Phys. Rev. Lett. 80, G. Morigi, J. Eschner, C.H. Keite, Phys. Rev. Lett. 85, ; C.F. Roos, D. Leibfried, A. Mundt, F. Schmidt-Kaer, J. Eschner, and R. Batt, ibid. 85, Th. Hänsch and P.E. Toschek, Z. Phys. 236, For exampe, A. Steane, App. Phys. B: Lasers Opt. 64,