PHYSICAL REVIEW A VOLUME 59, NUMBER 5 MAY 1999
Laser cooling of two trapped ions: Sideband cooling beyond the Lamb-Dicke limit
G. Morigi,1 J. Eschner,2 J. I. Cirac,1 and P. Zoller1 1Institut fu¨r Theoretische Physik, Universita¨t Innsbruck, A-6020 Innsbruck, Austria 2Institut fu¨r Experimentalphysik, Universita¨t Innsbruck, A-6020 Innsbruck, Austria ͑Received 8 December 1998͒ We study laser cooling of two ions that are trapped in a harmonic potential and interact by Coulomb repulsion. Sideband cooling in the Lamb-Dicke regime is shown to work analogously to sideband cooling of a single ion. Outside the Lamb-Dicke regime, the incommensurable frequencies of the two vibrational modes result in a quasicontinuous energy spectrum that significantly alters the cooling dynamics. The cooling time decreases nonlinearly with the linewidth of the cooling transition, and the effect of dark states which may slow down the cooling is considerably reduced. We show that cooling to the ground state is also possible outside the Lamb-Dicke regime. We develop the model and use quantum Monte Carlo calculations for specific examples. We show that a rate equation treatment is a good approximation in all cases. ͓S1050-2947͑99͒11605-6͔
PACS number͑s͒: 32.80.Pj, 42.50.Vk, 03.67.Lx
I. INTRODUCTION sideband cooling of two ions to the ground state has been achieved in a Paul trap that operates in the Lamb-Dicke limit The emergence of schemes that utilize trapped ions or ͓11͔. However in this experiment the Lamb-Dicke regime atoms for quantum information, and the interest in quantum required such a high trap frequency that the distance between statistics of ultracold atoms, have provided renewed interest the ions does not allow their individual addressing with a and applications for laser cooling techniques ͓1͔. The present laser. For this purpose, and also for an extension beyond two goal is to laser cool several atoms to a pure quantum state ͑to ions, linear ion traps ͓12,13͔ are the most suitable systems, the motional ground state͒, and experimental ͓2͔ and theoret- and for their physical parameters laser cooling to the ground ical efforts ͓3͔ are made in this direction. The cooling of a state is a goal yet to be achieved. Laser cooling of two ions large number of particles using lasers is a prerequisite for into the ground state is the problem that we address in this coherent control of atomic systems ͓4,5͔. In quantum infor- paper. mation, for example, laser cooling to the motional ground Theoretical studies on cooling of single ions outside the state is a fundamental step in the preparation of trapped ions Lamb-Dicke regime exist ͓14,15͔, while laser cooling of for quantum logic ͓5͔. Coherent control and manipulation of more than one ion has been analyzed only in the Lamb- information requires that each ion be individually address- Dicke regime ͓16,17͔. In this paper, we investigate laser able with a laser ͓6͔, and thus restricts the choice of the trap cooling, as developed for single ions, when it is applied to frequency, and consequently the regime in which cooling two or more ions. We focus our attention on sideband cool- must work, to relatively shallow traps ͓7͔. ing, showing and discussing new physical effects which arise Laser cooling of single ions in traps has been extensively because of the presence of two interacting particles. Doppler studied ͓8͔, and in particular sideband cooling has been dem- cooling will be discussed in a future work. We show that onstrated to be a successful technique for cooling single ions cooling of two ions outside the Lamb-Dicke regime presents to the ground state of a harmonic trap ͓9͔. Sufficient condi- novel features with respect to single-ion cooling, and we tions for sideband cooling a two level system are ͑i͒ the show how the preparation of the two ions in a pure quantum radiative linewidth ␥ is smaller than the trap frequency , state is possible. The results will allow us to obtain some such that motional sidebands, i.e., optical transitions that in- insight into the more general problem of cooling a string of volve the creation or annihilation of a specific number of N ions. This is not only relevant for quantum logic with motional quanta, can be selectively excited; and ͑ii͒ the N-ion strings but also for laser cooling of ion clusters in Paul Lamb-Dicke limit is fulfilled, i.e., the ion’s motional excur- and Penning traps ͓13,18͔. sion is much smaller than the laser wavelength. The first The paper is organized as follows. In Sec. II we introduce condition can be achieved through an adequate choice of and discuss the model which we will use throughout the atomic transition or a manipulation of the internal atomic paper, and discuss some concepts developed in sideband structure ͓10͔, while the Lamb-Dicke regime requires the cooling of one ion in relation to the presence of more than trap frequency to be much larger than the recoil frequency of one ion. In Sec. III we study and discuss sideband cooling of the optical transition. two ions inside and outside the Lamb-Dicke regime, and For more than one ion, as required in quantum logic compare the two different behaviors. Finally, in the conclu- schemes, individual addressing imposes small trap frequen- sions we summarize the main results, and discuss the prob- cies, whereas sideband cooling imposes high trap frequen- lem of cooling NϾ2 ions. cies. Furthermore, the Coulomb interaction between the par- II. MODEL ticles makes the problem much more complex, and it is not obvious whether the techniques developed for single ions We consider two ions of mass m and charge e placed in a can be transferred directly to this situation. Experimentally, one-dimensional harmonic potential of frequency .Weas-
1050-2947/99/59͑5͒/3797͑12͒/$15.00PRA 59 3797 ©1999 The American Physical Society 3798 G. MORIGI, J. ESCHNER, J. I. CIRAC, AND P. ZOLLER PRA 59
† sume the ions to be strongly trapped in the other spatial where we have defined Xϭͱ1/2M0(a0ϩa0), P dimensions, so that their motion in those directions is frozen † † ϭiͱM0/2(a0Ϫa0), xϭx0ϩͱ1/2r(ar ϩar), and p out. Their internal structure is described by a two-level sys- † † ϭiͱr/2(ar Ϫar), with a0 and a0 the annihilation and cre- tem with ground state ͉g͘, excited state ͉e͘, and resonance ation operators for the COM mode, respectively, and ar and frequency 0. The ions interact with laser light at frequency † ar the corresponding ones for the relative motion ͑stretch͒ L and wave vector k. For classical laser light and in the mode. We stress that in this representation the mechanical rotating-wave approximation, the Hamiltonian of the system problem of two ions interacting through Coulomb forces is is reduced to the one of two harmonic oscillators, while the HϭH ϩH ϩV. ͑1͒ interaction of each ion with the radiation is now transformed i mec into a nonlinear coupling between the harmonic oscillators. In general, N ions in a trap can be described by a set of N Here Hi is the internal energy in the rotating frame, harmonic oscillators, coupled by laser light ͓19͔. The master equation for the density matrix of the two- HiϭϪ␦ ͉e͘ jj͗e͉, ͑2͒ jϭ͚1,2 ion system is where ␦ϭLϪ0 is the detuning, j labels the ion ( jϭ1 and 2͒, and we have taken បϭ1. Hmec is the mechanical Hamil- d i tonian, ϭϪ ͓H,͔ϩL. ͑7͒ dt ប p2 p2 1 1 e2 1 2 2 2 2 2 Hmecϭ ϩ ϩ m x1ϩ m x2ϩ , Here L is the Liouvillian describing the incoherent evolution 2m 2m 2 2 4⑀0͉x1Ϫx2͉ ͑3͒ of the system: with x j and p j the position and momentum of the jth ion ( jϭ1 and 2͒, and V describes the interaction between laser ␥ ˜ † † † and atoms, Lϭ ͓2 j j j Ϫ j jϪj j͔, ͑8͒ 2 jϭ͚1,2
⍀͑x j͒ † Ϫikxj cos Vϭ ͓ j e ϩH.c.͔. ͑4͒ jϭ͚1,2 2 where ␥ is the decay rate out of the internal excited state ͉e͘, ˜ and j describes the density matrix after a spontaneous emis- † Here ⍀(x j) is the Rabi frequency at the position x j ; j and sion for the jth ion: j are the raising and lowering dipole operators, respec- tively, defined on the jth ion ( jϭ1 and 2͒; and is the angle between the laser wave vector and the trap axis. 1 ˜ ikuxj Ϫikuxj Using the center-of-mass ͑COM͒ and relative coordinates, jϭ ͵ du N͑u͒e e , ͑9͒ the mechanical Hamiltonian in Eq. ͑3͒ is composed of two Ϫ1 separate terms: one for the COM motion which describes a particle of mass Mϭ2m interacting with a harmonic poten- tial of frequency ; the other represents the relative motion with N(u) being the dipole pattern for the decay. which describes a particle of mass ϭm/2 interacting with a In this treatment we have neglected both dipole-dipole potential, which is the sum of a harmonic potential of fre- interaction between the ions and quantum statistical proper- quency and a Coulomb-type central potential. This poten- ties. This approximation is justified in the regime that we tial may be approximated by a harmonic-oscillator potential investigate, which is characteristic of experiments using lin- of frequency rϭͱ3, obtained through the truncation at the ear ion traps for quantum information ͓12͔. In those traps the second order of its Taylor expansion around the equilibrium equilibrium distance between the ions is of the order of 2 1/3 distance x0ϭ(2e /4⑀0M) between the ions ͓16,17͔.In 10m, while the laser wavelength is typically in the visible Appendix A we discuss this approximation and we show that region and the individual ionic wave packets have spatial it is valid in the regime that we are going to study. With this widths of the order of 10–100 nm. From these considerations approximation term ͑3͒ becomes ͑apart from a constant͒ we can consider the two ions in a linear trap as two distin- guishable particles ͓20͔ in a harmonic potential which inter- P2 1 p2 1 act solely with Coulomb forces. On the basis of these con- H ϭ ϩ M2X2ϩ ϩ 2x2, ͑5͒ mec 2M 2 2 2 r siderations, we will use Eq. ͑7͒ for the numerical simulations presented below. where Xϭ(x1ϩx2)/2, Pϭp1ϩp2 are the position and mo- For the following discussion it is instructive to look at the mentum of the COM, respectively, and xϭx1Ϫx2Ϫx0 , p set of equations which one obtains from Eq. ͑7͒ in the limit ϭ(p1Ϫp2)/2 are the position and momentum of the relative of low saturation ⍀Ӷ␥, when the excited state ͉e͘ can be motion. Thus term ͑5͒, apart from a constant, can be rewrit- eliminated in second-order perturbation theory ͑we provide a ten as detailed derivation of the equations in Appendix B͒.Inthe basis of states ͉g,n͘, where nϭ(n0 ,nr) is a vector with COM vibrational number n0 and stretch mode vibrational H ϭ a†a ϩ a†a , 6 mec 0 0 r r r ͑ ͒ number nr , we have the following set of equations: PRA 59 LASER COOLING OF TWO TRAPPED IONS: . . . 3799
d ͗g,n͉͉g,m͘ϭϪi͑nϪm͒ ͗g,n͉͉g,m͘ dt •
⍀2 ͗n͉eikx1 cos ͉k͗͘k͉eϪikx1 cos ͉l͘ ͗l͉eikx1 cos ͉k͗͘k͉eϪikx1 cos ͉m͘ ϩi ͚ ͗g,l͉͉g,m͘Ϫ͗g,n͉͉g,l͘ 4 k,l ͫ ͑kϪl͒•Ϫ␦Ϫi␥/2 ͑kϪl͒•Ϫ␦ϩi␥/2 ͬ ⍀2 1 ϩ du N͑u͒͗n͉eikux1͉k͗͘k͉eϪikx1 cos ͉r͗͘s͉eikx1 cos ͉j͗͘j͉eϪikux1͉m͗͘g,r͉͉g,s͘ 4 k͚,j,r,s ͵Ϫ1 1 1 ϫ ϩ , ͑10͒ ͫ„͑jϪsϪkϩr͒•ϩi␥…„͑jϪs͒•Ϫ␦Ϫi␥/2… „͑kϪrϪjϩs͒•Ϫi␥…„͑kϪr͒•Ϫ␦ϩi␥/2…ͬ
where ϭ(,r), and where for simplicity we have assumed problem of two ions, to discuss how those techniques may be that only ion 1 is illuminated, i.e., ⍀(x1)ϭ⍀, ⍀(x2)ϭ0. applied, and whether the same concepts are still valid. For N This implies that we can address the ions individually with a harmonic-oscillator modes we can define a Lamb-Dicke pa- well-focused laser beam. It also corresponds, for example, to rameter for each mode in an analogous way to Eq. ͑11͒. For a situation where the two ions are two different isotopes, of our case, Nϭ2, the Lamb-Dicke parameters 0 for the COM which only one is resonant with light ͓21͔. mode and r for the stretch mode are defined as When treating laser cooling in a harmonic trap, an impor- tant dimensionless quantity is the Lamb-Dicke parameter , which for a single ion of mass m in a trap of frequency , ប 0ϭk ϭ , interacting with laser light of wave vector k,is ͱ2M ͱ2 ͑12͒ k ប ϭ ϭ , 1 rec r ͱ ϭk ϭ , ͑11͒ 2 2r ͱ2ͱ3 ͱ2m ͱ
so that the kick operator for the jth ion ( jϭ1 and 2͒ is 2 where recϭk /2m is the recoil frequency. The parameter written as appears in the kick operator exp(ikx) in the term describing ikx i (a†ϩa ) (Ϫ1)jϪ1i (a†ϩa ) the exchange of momentum between radiation and atoms, e jϭe 0 0 0 e r r r . ͑13͒ which, using the relation xϭͱ1/2m(a†ϩa) and definition † In general, for N ions ϭ/ͱN ͓19͔. In the following, ͑11͒, is rewritten as exp(ikx)ϭexp„i(a ϩa)…. The Lamb- 0 Dicke regime corresponds to the condition ͱnӶ1, with n when we refer to the Lamb-Dicke regime, we will consider vibrational number; in other words, to the situation in which, the situation where the conditions ͱn00Ӷ1 and ͱnrrӶ1 during a spontaneous emission, a change in the vibrational are fulfilled. From Eq. ͑13͒ we see that in Eq. ͑4͒ the Lamb- number of the atomic state is unlikely due to energy conser- Dicke parameters appear multiplied by the factor cos . vation. In this regime the kick operator may be expanded in Therefore, the Lamb-Dicke parameters for the coherent ex- powers of , and with good approximation the expansion citation, i cos , are always less than or equal to the Lamb- may be truncated at the first order ͓8͔. Another important Dicke parameters defined in Eq. ͑12͒, which characterize the parameter, as known from cooling of single ions, is the ratio spontaneous emission. between the radiative linewidth ␥ and the trap frequency : To discuss the importance of the ratio ␥/ in the case of in the so-called strong confinement regime ␥/Ӷ1 the laser two ions, we first consider the bare spectrum of energies of can selectively excite sidebands of the optical transition our system with frequencies and ͱ3. In Fig. 1 we plot the which involve a well-defined change of the vibrational num- ber n. In this regime, together with the Lamb-Dicke regime, sideband cooling works efficiently: when the laser is red de- tuned with ␦ϭϪ, the system is cooled by approximately one phonon of energy in each fluorescence cycle, finally reaching the vibrational ground state nϭ0 ͓9͔. In contrast, in the weak confinement regime ␥/у1, transitions which in- volve different changes of the vibrational number n are ex- cited simultaneously. This is the Doppler cooling regime, where the achievable minimum energy for a single ion is approximately ␥/2 for a detuning ␦ϭϪ␥/2 ͓22͔. FIG. 1. Number of states D(E) in the energy interval ͓E,E Having introduced these basic concepts and methods of ϩ␦E͔ plotted as a function of energy E in units of . The grid is laser cooling of single ions in traps, we now return to the ␦Eϭ/3. 3800 G. MORIGI, J. ESCHNER, J. I. CIRAC, AND P. ZOLLER PRA 59
FIG. 2. ͑a͒ Absorption spectrum I(␦) vs detuning ␦ ͑in units of ) for 0ϭ0.1 for a thermal distribution with average energy per mode ¯ nϭ7.5, plotted on a grid of width /10. ͑b͒ Plot of the average COM mode vibrational number ͗n0͘ ͑solid line͒ and average stretch mode 2 vibrational number ͗nr͘ ͑dashed line͒ vs time in units tFϭ2␥/⍀ , for 0ϭ0.1, ␥ϭ0.2, ⍀ϭ0.034, ␦ϭϪ, and atoms initially in a flat distribution on the states with energy Eр15. ͑c͒ Population of the COM mode P ͑onset͒ and of stretch mode P ͑inset͒ vs the respective n0 nr vibrational state number at tϭ600tF . function D(E) vs the energy E, defined as the number of I͑␦͒ϭ n exp͑ikx͒ l 2P͑n͒, ͑14͒ states in the interval of energy E,Eϩ␦E . From this figure ͚ ͉͗ ͉ ͉ ͉͘ ͓ ͔ (nϪl)•ϭ␦ we see that due to the incommensurate character of the fre- quencies the spectrum does not exhibit a well-distinguished where P(n) is a normalized distribution of the states ͉n͘.In series of energy levels, but rather tends toward a quasicon- the Lamb-Dicke regime, we find that I(␦) exhibits two main tinuum. Therefore, the strong confinement requirement for pairs of sidebands around the optical frequency 0: one at sideband cooling needs to be reconsidered. The main ques- frequencies Ϯ corresponding to the transition n n 0 0 0→ 0 tion which we will address in the following is whether it is Ϯ1, the other at frequencies 0Ϯr corresponding to nr still possible to cool one mode to the ground state by means nrϮ1 ͓see Fig. 2͑a͔͒. The strength of these sidebands rela- of sideband cooling. As we will show, the Lamb-Dicke pa- → 2 2 tive to the carrier n n are proportional to 0 and r , re- rameter distinguishes two regimes which exhibit dramatic spectively. All the other→ sidebands have strengths of higher differences. 2 2 orders in 0 and r . This implies that by selecting one of these four sidebands by laser excitation we will induce the III. SIDEBAND COOLING OF TWO IONS corresponding phononic transition; for example by choosing the sideband corresponding to (n0 ,nr) (n0Ϫ1,nr) , we can In the following we study sideband cooling of two ions, cool the COM mode to its vibrational→ ground state, as for a first in the Lamb-Dicke regime and then outside of this re- single ion. This has been experimentally demonstrated by the gime. We will show that in this latter case two-ion effects NIST group at Boulder ͓11͔. In Fig. 2 we plot the results of appear due to the dense spectrum of energy levels. In our a quantum Monte Carlo ͑QMC͒ wave-function simulation calculations we first consider sideband cooling when laser ͓23͔ of Eq. ͑7͒ for two ions in a trap with Lamb-Dicke pa- light excites only one of the two ions directly. Thus in Eq. rameter 0ϭ0.1, radiative linewidth ␥ϭ0.2, detuning ␦ϭ ͑4͒ we take ⍀(x1)ϭ⍀, and ⍀(x2)ϭ0. Afterwards we com- Ϫ and an initially flat distribution for the states with energy pare this case to the one in which both are driven by light, Eр15. In Fig. 2͑b͒ the average vibrational numbers of the i.e., ⍀(x1)ϭ⍀(x2)ϭ⍀, showing that the only difference be- COM mode ͑solid line͒ and of the stretch mode ͑dashed line͒ tween the two cases is the cooling time, which in the latter are plotted as a function of time in unit of fluorescence 1 2 case scales by a factor 2 . In the following we assume that the cycles tFϭ2␥/⍀ . The system behaves as if the two modes laser wave vector is parallel to the trap axis, i.e., cos ϭ1. This assumption facilitates the analysis and it is justified by the simple scaling just described. Furthermore, it corre- sponds to the case in which the two ions are two different ionic isotopes, of which one is driven by light ͓21͔.Atthe end of this section we will briefly discuss cooling of two identical ions when the wave vector is not parallel to the trap axis.
A. Lamb-Dicke regime FIG. 3. ͑a͒ Comparison between the rate equation ͑solid line͒ In the Lamb-Dicke regime the Franck-Condon coeffi- and QMC calculation ͑dashed line͒. Same parameters as in Fig. cients ͗n͉exp(ikx)͉l͘ in the numerators of the right-hand side 2͑b͒. ͑b͒ Comparison between the time dependence of the COM terms of Eq. ͑10͒ may be expanded in terms of the Lamb- average vibrational number as in Fig. 2͑b͒͑solid line͒ and the av- Dicke parameters 0 and r . The response of the system to erage vibrational number for the case in which a single ion is cooled laser light is governed by its absorption spectrum I(␦), ͑dashed line͒. For the single ion the mass has been rescaled so that (1) (1) which is evaluated by summing all contributions to laser- ϭ0 , ϭ, ␥ϭ0.2, and ⍀ϭ0.034, with an initially flat excited transitions at frequency L , distribution for the first 15 states. PRA 59 LASER COOLING OF TWO TRAPPED IONS: . . . 3801
FIG. 4. ͑a͒ Absorption spectrum I(␦) vs detuning ␦ ͑in units of ) for 0ϭ0.6 of ions in a thermal distribution with average energy per ¯ 2 mode nϭ7.5, plotted on a grid of width /10. ͑b͒ Plot of ͗n0͘ ͑solid line͒ and ͗nr͘ ͑dashed line͒ vs time in unit tFϭ2␥/⍀ , for 0 ϭ0.6, ␥ϭ0.2, ⍀ϭ0.034, ␦ϭϪ2, and atoms initially in a flat distribution for states with energy Eр15. ͑c͒ Population of the COM mode P ͑onset͒ and the stretch mode P ͑inset͒ vs the respective vibrational number state at tϭ600t . n0 nr F were decoupled, since only one mode is cooled while the reducing the density-matrix equation in the low saturation other remains almost frozen. Nevertheless the stretch mode limit to rate equations. In fact the coherences in Eq. ͑10͒ is cooled on a much longer time scale, as an effect of off- have either an oscillation frequency which is much larger resonant excitation. In Fig. 2͑c͒ the populations of the vibra- than the fluorescence rate 1/tF , or a coupling to the popula- tional states of the two modes are plotted at time tϭ600tF , tion which is of higher order in the Lamb-Dicke parameter showing the COM mode in the ground state and the nearly expansion, or both. Therefore, they can be neglected in the uncooled stretch mode. In this limit we can neglect the cou- equations of the populations, and we obtain the set of rate pling of the population to the coherences in Eq. ͑10͒, thus equations
d ⍀2 ͉͗n͉eikx1͉k͉͘2 ͗n͉͉n͘ϭϪ␥ ͗n͉͉n͘ dt 4 ͚ 2 2 k ͓͑kϪn͒•Ϫ␦͔ ϩ␥ /4 ⍀2 1 ͉͗n͉eikux1͉k͉͘2͉͗k͉eϪikx1͉r͉͘2 ϩ␥ du N͑u͒ ͗r͉͉r͘, ͑15͒ 4 ͚ ͵ 2 2 k,r Ϫ1 ͓͑kϪr͒•Ϫ␦͔ ϩ␥ /4
where we have omitted the label g of the states. The validity are excited increases with ␥. In Figs. 4͑b͒ and 4͑c͒ we con- of this approximation is shown for the above case in Fig. sider sideband cooling for Lamb-Dicke parameter 0ϭ0.6 3͑a͒, where the results of Fig. 2͑b͒ are compared with those and detuning ␦ϭϪ2 ͓14͔, where the other parameters are of a rate equation simulation according to Eq. ͑15͒.Asa the same as in Figs. 2͑b͒ and 2͑c͒. As one can see, the two further proof that the two modes can be considered decou- modes are coupled and cooled together. Thus as a first big pled during the time in which the COM motion is cooled, in difference with respect to single-ion cooling, we see that here Fig. 3͑b͒ we compare the time dependence of the average the energy is not taken away from one mode only; rather, it vibrational number of the COM mode with the one of a is subtracted from the system as a whole. Another striking single trapped ion which is cooled under the same Lamb- difference appears in the cooling time, which is significantly Dicke parameter, radiative linewidth, trap frequency, Rabi longer in comparison with the time necessary to cool one frequency, and initial distribution as the COM mode. We see single ion outside the Lamb-Dicke regime ͓14͔. This slowing that the two curves overlap appreciably, justifying the picture down is partly due to the increase of the dimension of the of sideband cooling of two ions in the Lamb-Dicke regime as phase space where the cooling takes place: the presence of if the modes were decoupled from one another. two modes makes the problem analogous to cooling in a two-dimensional trap, whose axes are coupled by the laser. B. Outside the Lamb-Dicke regime The ions thus make a random walk in a larger phase space, To illustrate the physical features of the system outside and the cooling becomes slower. However, the cooling time the Lamb-Dicke regime, in Fig. 4͑a͒ we plot the absorption is even considerably longer than one would expect taking the spectrum I(␦) as defined in Eq. ͑14͒ for two ions in a har- dimensionality into account. This can be explained by look- monic trap with 0ϭ0.6. We see that the spectrum exhibits ing again at the spectrum in Fig. 4͑a͒: despite the high den- many sidebands whose density increases as the detuning in- sity of resonances, the coupling between the states is still creases. The main consequence is that we cannot select a governed by the Franck-Condon coupling, i.e., by the terms given sideband by choosing the laser frequency, but rather in the numerator of Eq. ͑10͒ which outside the Lamb-Dicke excite a group of resonances that correspond to transitions to limit oscillate with the vibrational numbers of the states. In a set of quasidegenerate states. The range of transitions that the limit of linewidth ␥Ӷ, where a single sideband can be 3802 G. MORIGI, J. ESCHNER, J. I. CIRAC, AND P. ZOLLER PRA 59
FIG. 5. ͑a͒ Population Pnϭ͗n͉͉n͘ as a function of n0 and nr at a time tϭ1000tF and for ␥ϭ0.02, ⍀ϭ0.17␥ϭ0.0034, and tF ϭ2␥/⍀2ϭ3460/. All the other physical parameters are the same as in Figs. 4͑b͒ and 4͑c͒. ͑b͒ and ͑c͒ Modulus square of the Franck- FIG. 6. Population P as a function of n and n at a time t Condon coefficients for the relative motion F n 0 r l,nr 2 † ϭ600tF and for ␥ϭ, ⍀ϭ0.17␥, and tFϭ2␥/⍀ ϭ69/. All other ir(a ϩar) 2 ϭ͉͗l͉e r ͉nr͉͘ with lϭ6 and 7. physical parameters are the same as in Figs. 4͑b͒ and 4͑c͒. selected, we may encounter dark states like in cooling of on the linewidth is a two-ion effect. In contrast, in sideband single ions 14 , i.e., states whose coupling to the resonantly ͓ ͔ cooling of single ions, the fluorescence time tF determines excited state is very small since their motional wave function the cooling time scale for ␥/р1, and the curves for differ- after the absorption of a laser photon happens to have a very ent values of ␥ vs the time in units of the respective tF do not small overlap with the motional wave function of the excited show striking differences. states. This effect limits the cooling efficiency, since the at- The presence of dark states and the coupling of each state oms may remain trapped in these states and not be cooled to more than one state at almost the same transition fre- further, or else cooled much more slowly, toward the ground quency might also lead to the formation of dark coherences state. For two ions the probability of finding zeroes of the between quasidegenerate states, i.e., to superpositions of Franck-Condon coupling is larger than for one ion, as the states which decouple from laser excitation because of quan- coupling to the excited state is constituted by two integrals, tum interference. However, for the considered system those one for the COM and the other for the relative motion wave dark coherences do not play any significant role. We prove functions. Thus the probability of having dark states is this numerically in Fig. 8, where we plot the comparison higher. To illustrate this phenomenon, in Fig. 5͑a͒ we plot between a QMC and a rate equation simulation. We see that the occupation of the states Pn as a function of the COM and there are no striking differences between the two curves. We relative vibrational numbers n0 and nr , respectively, at a point out that outside the Lamb-Dicke regime a rate equation time tϭ1000tF after sideband cooling of the COM with ␦ treatment is not justified in principle, since secular approxi- ϭϪ2. Here ␥ϭ0.02, and we are in the limit in which the mation arguments and Lamb-Dicke limit arguments cannot single resonances are resolved. As a consequence, the most be applied. Here the rate equations are used to highlight the likely coherent transitions are n0 n0Ϫ2 and nr nr . The effect of neglecting the coherences in the dynamical evolu- → → effect of the dark states is visible in the tail of occupied states tion of the cooling, while these coherences are fully ac- of Pn , with nrϭ6 and 7. In Figs. 5͑b͒ and 5͑c͒, we plot the counted for in the QMC treatment. In order to see why co- modulus square of the Franck-Condon coefficients for the herences do not play any significant role in the cooling relative motion corresponding to the coupling of the states dynamics, we look at the definition of a dark coherence. Let nrϭ6 and 7 to the other motional states: here it is clearly us consider a state ͉␣͘ at tϭ0 defined for simplicity as linear shown that for the transitions nrϭ6 6 and nrϭ7 7 the superposition of two quasidegenerate states ͉␣͘ϭa1͉n͘ coupling is reduced nearly to zero.→ As the linewidth→␥ in- creases, the number of states to which a single state is coupled increases. Thus the number of channels through which the atom may be cooled is larger. As an effect the dark states disappear. This is shown in Fig. 6, where the popula- tion Pn is plotted for tϭ600tF and ␥ϭ, and otherwise the same parameters as before. Here we see that the system is cooled homogeneously. The effect of varying ␥ is summa- rized in Fig. 7, where we compare the average COM vibra- tional number vs time in unit tF for various values of ␥. The results of Fig. 7 show clearly that as the linewidth increases the number of fluorescence cycles needed for cooling the FIG. 7. Time dependence of the average vibrational number of system decreases dramatically. It is important to note that in the COM mode for ␥ϭ0.02 ͑dashed line͒, ␥ϭ0.2 ͑solid line͒, this diagram the time is measured in units of fluorescence ␥ϭ0.4 ͑dash-dotted line͒ and ␥ϭ ͑dotted line͒, keeping constant 2 cycles for each ␥, so that the absolute cooling time clearly the ratio ⍀/␥ϭ0.17. The time is in unit tF(␥)ϭ2␥/⍀ Ϸ70/␥. All reduces more strongly. We stress that this strong dependence other parameters are the same as in Figs. 4͑b͒ and 4͑c͒. PRA 59 LASER COOLING OF TWO TRAPPED IONS: . . . 3803
FIG. 9. Plot of the average vibrational number vs time for a FIG. 8. Comparison between the rate equation ͑solid line͒ and harmonic oscillator of frequency coupled to a second one with QMC calculation ͑dashed line͒. Same physical parameters as in Fig. frequency 2. Comparison between rate equation ͑solid line͒ and 4͑b͒. The onset refers to the COM, and the inset to the relative QMC calculation ͑dashed line͒. The Lamb-Dicke parameter for the motion vibrational number. mode is ϭ0.6. ␥ϭ0.2, ⍀ϭ0.034, and ␦ϭϪ2, and atoms i are initially flatly distributed on the states with an energy Eр15. ϩa2e ͉m͘, with a1 , a2, and real coefficients and with n and m states almost degenerate in energy, so that ͉(nϪm) modes simultaneously, as shown in Eqs. ͑4͒ and ͑13͒. When •͉ϭ⌬E with ⌬Eр␥. In principle a dark coherence can be a linear superposition of any number of states. However, as both ions are excited by laser light, the system is described we will see from the arguments below, our restriction to two by a four-level scheme, corresponding to the four internal states does not affect the generality of the result. The evolu- states ͉a1 ,b2͘ with a,bϭe,g, where we assume that when a photon is emitted, we detect from which ion the event has tion ͉␣(t)͘ in the Schro¨dinger picture, apart from a global phase factor, is written as occurred, as a consequence of the spatial resolution of the ions. We expect that the effect on the cooling will be a dou- i(t) bling in the number of quantum jumps and hence of the ͉␣͑t͒͘ϭa1͉n͘ϩa2e ͉m͘, ͑16͒ cooling rate. This is shown in Fig. 10, where we compare the with (t)ϭ0ϩ⌬Et. The superposition state is dark when time dependence of the COM vibrational number for the the following condition is fulfilled: cases in which only one ion is illuminated ͑dashed line͒ or both ions are illuminated with the same laser intensity ͑solid ͗l͉eik(Xϩx/2)͉␣͑t͒͘ϳ0 ͑17͒ line with label 1͒. In the latter case cooling is visibly faster, and the time dependence scales with a factor of 2 with re- for any state ͉l͘ belonging to the set of states ͕͉l͖͘ to which spect to the case with one ion illuminated, as we can see it is resonantly or almost resonantly coupled. If condition when we replot the solid line 1 vs t/2tF ͑solid line with index ͑17͒ holds at tϭ0, it will hold up to a time t such that 2͒. The two-ion effects found above are clearly independent ⌬Etϳ/2. For the system we are dealing with we do not of the number of scattering points, with the only difference have an exact degeneracy, thus we check whether the state that the dark coherence condition is now written as ͉␣͘ can remain dark for a time sufficiently long to affect the ͗l͉(eik(Xϩx/2)ϩeik(XϪx/2))͉␣(t)͘ϳ0. We point out again that cooling dynamics appreciably. The smallest possible value effects due to interference between the internal excitation of ⌬E in the range of energies of our calculations is ⌬E paths have been neglected, as we consider the ions to be two ϭ0.07, and we find that rotates by an angle /2 in less distinguishable particles ͓25͔. than one fluorescence cycle for the values of ␥ that we have We would like to stress that in the above calculations we considered. The dark coherences are then washed away dur- have considered the case of only one ion driven by radiation ing the evolution as an effect of the incommensurate fre- while the laser wave vector is parallel to the trap axis. How- quencies between the two modes. This result together with ever, if one wants to cool two identical ions by shining light the numerical results suggests that rate equations can be used in the study of cooling ͓24͔. In order to highlight that the absence of dark coherences is a signature of the peculiar spectrum of the system, in Fig. 9 we plot the cooling of one mode outside the Lamb-Dicke regime for the case of a discrete spectrum where we have exact degeneracy. More precisely we consider two harmonic oscillators with commensurate frequencies and 2, where all the other physical parameters are the same as before. In this case the different outcome between the QMC and the rate equation treatment is dramatic, giving evidence of the role of coherences in the evolution. FIG. 10. Plot of the average vibrational number of the COM C. Light on both ions mode vs the time in unit tF for the case in which both ions are illuminated ͑solid line, index 1͒, and only one ion is illuminated The calculations that we have shown refer to the case in ͑dashed line͒. The solid line with index 2 corresponds to line 1 which only one ion is illuminated. As we have seen, al- rescaled, where the time has been divided by a factor 2. The other though light interacts with one ion it couples with both parameters are the same as in Figs. 4͑b͒ and 4͑c͒. 3804 G. MORIGI, J. ESCHNER, J. I. CIRAC, AND P. ZOLLER PRA 59 on one of them, the laser beam must necessarily be at a also play a role against the creation of dark coherences. In certain angle with respect to the ion string. Thus the Lamb- fact, although on one hand the large density of states may Dicke parameter characterizing the coherent excitation will favor the appearance of stable dark coherences between ‘‘ac- be smaller than the Lamb-Dicke parameter for the spontane- cidentally’’ degenerate states, on the other hand for values of ous decay, and, depending on the minimum angle required, ␥ large enough the coherent effects will wash out because of the coherent laser will excite with some selectivity one of the the coupling to a ‘‘continuum’’ of states. From these consid- two modes. However, for cooling purposes it is preferable to erations we expect laser cooling to the ground state still to be have the two Lamb-Dicke parameters values, corresponding possible for NϾ2 ions. Furthermore, one can cool a given to the spontaneous emission and the coherent excitation, as set of modes to the ground state through the choice of the close as possible. laser detuning outside the Lamb-Dicke regime. For example, taking a detuning ␦ϭϪk , the modes with frequency j IV. CONCLUSIONS уk may be cooled to their vibrational ground states. It should be noted that as the number of ions N increases, the In this paper we have studied the question of cooling two Lamb-Dicke parameter of each mode decreases approxi- ions in a linear trap to the ground state by means of sideband mately as 1/ͱN ͓19͔, also allowing one to reach the Lamb- cooling. We have studied sideband cooling in the Lamb- Dicke regime when this condition is not fulfilled for single Dicke regime, and have shown that in this limit the two ions. In this limit the modes may be considered decoupled, harmonic oscillators can be considered decoupled when one and sideband cooling is particularly efficient. of the two is cooled by means of sideband cooling. We have As a last consideration, we note that the behavior of a found that the cooling dynamics in the low-intensity limit number of ions Nу3 cooled by light depends on which ions may be described by rate equations, and essentially that all of the string are driven. In fact each position of the string the considerations developed for sideband cooling of single couples with the different modes with amplitudes that de- ions apply. This regime was considered in Ref. ͓16͔ for pend on the position itself ͓19͔. Only if all the ions are illu- studying the effects of dipole-dipole interaction in laser cool- minated may we consider all the modes as coupled and ing of two ions, and by Javanainen in Ref. ͓17͔ in his study cooled simultaneously. But this ‘‘coupling’’ changes as we on laser cooling of ion clusters. select and drive only certain ions of the string. In this case a We have then investigated laser cooling outside the certain amount of modes may be cooled, while the others Lamb-Dicke regime, finding striking differences with the will remain hot or be cooled on a longer time scale. Lamb-Dicke limit. Here the energy spectrum may be consid- ered a quasicontinuum, though the coupling between the states is still governed by the Franck-Condon coupling. A consequence is that the cooling efficiency depends strongly ACKNOWLEDGMENTS on the radiative linewidth. For very small ratio ␥/ the effect The authors would like to thank D. Leibfried, F. Schmidt- of dark states is appreciable, and manifests itself in a drastic Kaler, H. Baldauf, and W. Lange for many stimulating dis- increase of the cooling time, i.e., the number of fluorescence cussions. One of us ͑G.M.͒ wishes to thank S. Stenholm for cycles needed to cool the system. For ␥/Ͻ1, but large stimulating discussions. This work was supported by the enough, the effect of dark states is washed away, the modes Austrian Fond zur Fo¨rderung der wissenschaftlichen Fors- are cooled simultaneously, and the cooling time is consider- chung and the TMR under Network No. ERBFMRX-CT96- ably shorter and comparable to the time needed for cooling 0002. single ions outside of the Lamb-Dicke regime. These effects are all consequences of the density of states in the energy spectrum. A further property of the system is the absence of APPENDIX A: HARMONIC APPROXIMATION dark coherences, as a consequence of the incommensurate frequencies of the harmonic oscillators, i.e., of the absence of We consider term ͑3͒ and rewrite it in COM and relative perfect degeneracy. This implies that in the low-intensity motion canonical variables: limit rate equations still provide a good description of the cooling dynamics. Finally, we have compared cooling when P2 1 p2 1 e2 light is shone on one ion only or on both ions, finding a 2 2 2 2 simple difference of a factor 2 in the rate of cooling. Hmecϭ ϩ M X ϩ ϩ x ϩ 2M 2 2 2 4⑀0͉x͉ On the basis of the obtained results we would like to comment on cooling of a string of NϾ2 ions. N ions in a ϭH͑X,P͒ϩH͑x,p͒, ͑A1͒ harmonic trap may be described by N harmonic oscillators. The mode frequencies 0 ,1 ,...,NϪ1 are all incommen- surate, and the number of states in the interval of energy with ϭm/2 the reduced mass, and Mϭ2m the total mass. NϪ1 ͓E,Eϩ␦E͔ is D(E)ϭE ␦E/01•••NϪ1(NϪ1)!. Out- The mechanical problem is separable into center-of-mass side the Lamb-Dicke regime the absorption spectrum is then motion and relative motion, where H(X,P) describes the even denser than in the two-ion case, and the probability of harmonic motion of a particle of mass M interacting with a dark states will be larger. However, a suitable increase of the harmonic oscillator of frequency , and H(x,p) the motion linewidth will cancel their effects, since each state will see of a particle of mass interacting with a potential V(x) 2 2 2 an even higher number of states, and thus of possible transi- ϭ x /2ϩe /4⑀0͉x͉, i.e., a harmonic potential of fre- tions, than for two ions. We expect that this last property will quency and a central repulsive Coulomb-type potential. PRA 59 LASER COOLING OF TWO TRAPPED IONS: . . . 3805
We focus our attention on the potential V(x), restricting ͉͗ j͉A͑x͉͒jϮ1͉͘Ӷបr , ͑A5͒ its domain on the semiaxis xϾ0. The equilibrium point is 2 2 1/3 found to be x0ϭ(e /4⑀0 ) . Expanding V(x) around with ͉j͘ the eigenstate of the harmonic oscillator of fre- x0,wefind quency r with eigenvalue jបr . The first condition means that the energy shift due to the anharmonic term is much 2 1/3 3 e 3 smaller than the spectrum separation, whereas the second V͑x͒ϭ 2 ϩ 2͑xϪx ͒2 2ͩ4⑀ ͪ 2 0 condition means that the coupling between the states is a 0 small perturbation, and we will show that it may be ne- ϱ e2 n n glected. The coupling between the state j and the state jϩk ϩ ͑Ϫ1͒ ϩ ͑xϪx0͒ ͑A2͒ ͚ϭ n 1 is not taken here into account for simplicity, but it may be n 3 4⑀0x0 shown that it is much smaller than kបr in a similar way to 3 e2 1/3 1 the one we discuss below. Let us rewrite the relations in Eq. ϭ 2 ϩ 2͑xϪx ͒2ϩA͑x͒, ͑A5͒ as 2 ͩ 4⑀ ͪ 2 r 0 0 ͑A3͒ e2 ϱ ͗ j͉A͑x͉͒j͘ϭ ͚ j,2m , where rϭͱ3 and A(x) sum over the higher order terms, 4⑀0x0 mϭ2 which we call the anharmonic terms. We quantize the oscil- ͑A6͒ 2 ϱ lation around the equilibrium position x0, e ͗ j͉A͑x͉͒jϩ1͘ϭϪ ͚ j,2mϩ1 , 4⑀0x0 mϭ1 ប † xϭx0ϩͱ ͑ar ϩar͒, ͑A4͒ 2mr with where a and a† are annihilation and creation operators, re- 2m r r ប spectively. From perturbation theory we may consider V(x) ϭ ͗ j͉͑a†ϩa ͒2m͉j͘, ͑A7͒ j,2m ͱ 2 r r harmonic when the following conditions are fulfilled: ͩ x 2r ͪ 0
͗ j͉A͑x͉͒j͘Ӷបr , where j,2mϩ1 is analogously defined. From the relations
͑ jϩm͒! ͑2m͒! ͑ jϩm͒! ͑ jϩm͒! Ͻ͗ j͉͑a†ϩa ͒2m͉j͘Ͻ Ͻ22m , j! r r m!m! j! j! ͑A8͒ ͑ jϩm͒! ͑2mϩ1͒! ͑ jϩm͒! ͑ jϩm͒! ͱjϩmϩ1Ͻ͗ j͉͑a†ϩa ͒2mϩ1͉jϮ1͘Ͻ ͱjϩmϩ1Ͻ22mϩ1 ͱjϩmϩ1, j! r r m!͑mϩ1͒! j! j!
we find that dimensions is relatively tight. In principle one could build up a potential that does not diverge in a specific point of the x 2m axis, whose behavior for xϾ0 tends to the Coulomb one. In ͉ j,2mϩ1͉ ប ϳ 2 ͱjϩmϩ1. ͑A9͒ this limit the series should converge for low number states j. ͱ 2 ͉ j,2m͉ ͩ x 2r ͪ However, such a detailed study is beyond the scope of this 0 paper. A discussion can be found in Ref. ͓20͔. We restrict From Eq. ͑A9͒ we see that the series does not converge, as ourselves to the case in which Ӷ1, showing that in the the term of the expansion depends on the term m. However chosen range the harmonic approximation is quite good. From this consideration it is then sufficient to compare the for a certain interval corresponding to j,mрM 0 such that 2 third-order term with the harmonic potential in order to show ϭͱប/x02rͱM 0Ӷ1, each term is bounded by the corre- sponding term of a geometrical series with factor Ӷ1. For that the approximation is sensible. The relation to be satisfied is then typical values of a linear ion trap ͓12͔, ϳͱM 0/100, so that M can assume very large values, M Ӷ104. 0 0 2 3 The divergence at orders mϾM is a signature of the e ប 0 ប ӷ j3/2, ͑A10͒ divergence of the Coulomb potential at xϭ0: such diver- r 4 ͩͱ2 ͪ 4⑀0x0 r gence requires the wave functions to be zero in xр0, whereas the harmonic-oscillator wave functions are different with jрM 0, which poses a further condition on j and M 0. from zero on all the space, although their occupation on the Manipulating the expression we find semiaxis xр0 is very small for Ӷ1. The divergence at x ϭ0 is a mathematical consequence, and does not correspond 2/3 2 បr x0 to the physical situation, in which the ions move in a three- jӶ jmaxϷ , ͑A11͒ ͩ e2/4⑀ x ͪ ͩ ͱប/2 ͪ dimensional space, although the confinement in the other two 0 0 r 3806 G. MORIGI, J. ESCHNER, J. I. CIRAC, AND P. ZOLLER PRA 59 which substantially agrees with the qualitative estimate in We introduce the Liouvillians Ref. ͓26͔. For jӶ jmax the potential can be considered har- monic. For linear traps ͓12͔, jmaxϳ120, and the harmonic i † approximation is valid for the region of energy we are con- L0ϭϪ ͓HeffϪHeff͔, ͑B4͒ sidering. As an example the case jϭ100 corresponds to a ប Ϫ3 correction of the order of 10 បr . i L ϭϪ ͓V,͔, APPENDIX B: ADIABATIC ELIMINATION 1 ប OF THE EXCITED STATE We rewrite Eq. ͑7͒ in the following way: L2ϭJ, ͑B5͒ d i i ϭϪ ͓H ϪH† ͔Ϫ ͓V,͔ϩJ, ͑B1͒ dt ប eff eff ប so that Eq. ͑B1͒ can be rewritten as where Heff is the effective Hamiltonian d ϭ͓L ϩL ϩL ͔. ͑B6͒ dt 0 1 2 ␥ † HeffϭHiϩHmecϪi i i , ͑B2͒ 2 iϭ͚1,2 In the limit ⍀Ӷ␥ we can eliminate the excited state in and J the jump operator second-order perturbation theory. Calling P the projector onto the internal ground state ͉g͘, and using a standard deri- ˜ † vation based on projectors ͓27͔, we obtain the following Jϭ␥ ii . ͑B3͒ iϭ͚1,2 equation for the ground state of the system:
d t P͑t͒ϭPL0P͑t͒ϩ d PL1͑1ϪP͒exp͑L0͒͑1ϪP͒L1P͑tϪ͒ dt ͵0
t t ϩ ͵ d1 ͵ d2PL2͑1ϪP͒exp͑L01͒͑1ϪP͒L1exp„L0͑2Ϫ1͒…P͑tϪ2͒, ͑B7͒ 0 1 where P is a projector so defined on a density operator X, PXϭ͉g͗͘g͉X͉g͗͘g͉. The Markov approximation can be applied in the limit in which we may consider the coupling to the excited state to evolve at a higher rate respect to the time scale which characterizes the ground-state evolution. This is true once we have moved to the interaction picture with respect to the trap frequency. We define
iHmect ϪiHmect vI͑t͒ϭe P͑t͒e , ͑B8͒ and in the interaction picture Eq. ͑B7͒ has the form
d t iHmect ϪiHmec(tϪ) iHmec(tϪ) ϪiHmect vI͑t͒ϭe d PL1exp͑L0͒L1e vI͑tϪ͒e e dt ͫ ͵0 ͬ
t t iHmect ϪiHmec(tϪ2) iHmec(tϪ2) ϪiHmect ϩe ͵ d1 ͵ d2PL2exp͑L01͒L1exp„L0͑2Ϫ1͒…L1e vI͑tϪ2͒e e . ͑B9͒ ͫ 0 1 ͬ
Now we can neglect the change of vI during the time on which the excited state evolves. Going back to the original reference frame, we have now the equation in the Markov approximation:
d ϱ iH0 ϪiH0 P͑t͒ϭ d PL1exp͑L0͒L1e P͑t͒e dt ͵0
ϱ ϱ iH02 ϪiH02 ϩ͵ d1 ͵ d2PL2 exp͑L01͒L1 exp„L0͑2Ϫ1͒…L1e P͑t͒e . ͑B10͒ 0 1
We substitute now into Eq. ͑B10͒ the explicit form of the operators. After some algebra and application of the commutation rules, we obtain the following equation ͑where we have neglected the interference terms between the two ions͒: PRA 59 LASER COOLING OF TWO TRAPPED IONS: . . . 3807
2 ϱ d ⍀i (i␦-␥/2) ikxi ϪiHmec Ϫikxi iHmec P͑t͒ϭPL0͑t͒Ϫ P d ͓e e e e e P1͑t͒ dt iϭ͚1,2 4 ͵0
ϩeϪ(i␦ϩ␥/2)P͑t͒eϪiHmeceikxieiHmeceϪikxi͔
2 ϱ ϱ 1 ⍀i ikuxi ϩ ͚ P ͵ d1 ͵ d2 ͵ du N͑u͒e iϭ1,2 4 0 1 Ϫ1
ϫ͓ei␦(1Ϫ2)Ϫ␥/2(1ϩ2)eϪiHmec1eϪikxieiHmec1P͑t͒eϪiHmec2eikxieiHmec2
ϩe-i␦(1Ϫ2)Ϫ␥/2(1ϩ2)eϪiHmec2eϪikxieiHmec2P͑t͒eϪiHmec1eikxieiHmec1͔eϪikuxi. ͑B11͒
Projecting Eq. ͑B11͒ on the basis of states ͕͉n͖͘, we obtain Eq. ͑10͒.
͓1͔ An overview of laser cooling can be found in S. Chu, Rev. ͑1998͒; R. J. Hughes, D. F. V. James, J. J. Gomez, M. S. Mod. Phys. 70, 685 ͑1998͒; C. Cohen-Tannoudij, ibid. 70, 707 Gulley, M. H. Holzscheiter, P. G. Kwiat, S. K. Lamoreaux, C. ͑1998͒; W. D. Phillips, ibid. 70, 721 ͑1998͒. G. Peterson, V. D. Sandberg, M. M. Schauer, C. M. Simmons, ͓2͔ H. Perrin, A. Kuhn, I. Bouchoule, and C. Salomon, Europhys. C. E. Thorbum, D. Tupa, P. Z. Wang, and A. G. White, Lett. 42, 395 ͑1998͒; S. E. Hamann, D.L. Haycock, G. Klose, Fortschr. Phys. 46, 329 ͑1998͒. P. H. Pax, I. H. Deutsch, and P. S. Jessen, Phys. Rev. Lett. 80, ͓13͔ M. G. Raizen, J. M. Gilligan, J. C. Bergquist, W. M. Itano, and 4149 ͑1998͒. D. J. Wineland, Phys. Rev. A 45, 6493 ͑1992͒; M. Drewsen, C. ͓3͔ A theoretical study about possibilities of achieving Bose- Brodersen, L. Hornekaer, J. S. Hangst, and J. P. Schiffer, Phys. Einstein condensation with laser cooling of neutral atoms can Rev. Lett. 81, 2878 ͑1998͒; W. Alt, M. Block, P. Seibert, and be found in Y. Castin, J. I. Cirac, and M. Lewenstein, Phys. G. Werth, Phys. Rev. A 58, R23 ͑1998͒; I. Waki, S. Kassner, G. Birkl, and H. Walther, Phys. Rev. Lett. 68, 2007 ͑1992͒. Rev. Lett. 80, 5305 ͑1998͒, and references therein. ͓14͔ G. Morigi, J. I. Cirac, M. Lewenstein, and P. Zoller, Europhys. ͓4͔ A discussion about the advantages of using laser cooling for Lett. 39,13͑1997͒. achieving a quantum statistical regime can be found in J. I. ͓15͔ G. Morigi, J. I. Cirac, K. Ellinger, and P. Zoller, Phys. Rev. A Cirac, M. Lewenstein, and P. Zoller, Phys. Rev. A 51, 2899 57, 2909 ͑1998͒; D. Stevens, A. Brochards, and A. M. Steane, ͑1995͒, and references therein. Phys. Rev. A 58, 2750 ͑1998͒; L. Santos and M. Lewenstein, ͓5͔ J. I. Cirac and P. Zoller, Phys. Rev. Lett. 74, 4091 ͑1995͒. For preprint quant-ph/9808014. a review on the ion trap quantum computer, see A. Steane, ͓16͔ A. W. Vogt, J. I. Cirac, and P. Zoller, Phys. Rev. A 53, 950 Appl. Phys. B: Photophys. Laser Chem. 64, 623 ͑1997͒. ͑1996͒. ͓6͔ H. C. Naegerl, D. Leibfried, H. Rohde, G. Thalhammer, J. ͓17͔ J. Javanainen, J. Opt. Soc. Am. B 5,73͑1988͒. Eschner, F. Schmidt-Kaler, and R. Blatt, Phys. Rev. A ͑to be ͓18͔ J. N. Tan, J. J. Bollinger, B. Jelenkovic, and D. J. Wineland, published͒. Phys. Rev. Lett. 75, 4198 ͑1995͒; F. Diedrich, E. Peik, J. M. ͓7͔ Recently a scheme of addressing two ions individually has Chen, W. Quint, and H. Walther, ibid. 59, 2935 ͑1987͒;W.M. been demonstrated, where the ions are not optically resolved Itano, J. J. Bollinger, J. N. Tan, B. Jelenkovic, X.-P. Huang, but instead their micromotion is used; see Q. A. Turchette, C. and D. J. Wineland, Science 279, 686 ͑1998͒; G. Birkl, S. S. Wood, B. E. King, C. J. Myatt, D. Leibfried, W. M. Itano, Kassner, and H. Walther, Nature ͑London͒ 357, 310 ͑1992͒. C. Monroe, and D. J. Wineland, Phys. Rev. Lett. 81, 3631 ͓19͔ D. F. V. James, Appl. Phys. B: Photophys. Laser Chem. 66, ͑1998͒. A generalization of this technique for more than two 181 ͑1998͒. ions has also been proposed; see D. Leibfried, Los Alamos ͓20͔ J. Yin and J. Javanainen, Phys. Rev. A 51, 3959 ͑1995͒. archives, e-print quant-ph/9812033. ͓21͔ W. Lange ͑private communication͒. ͓8͔ S. Stenholm, Rev. Mod. Phys. 58, 699 ͑1986͒. ͓22͔ The final energy obtained by means of Doppler cooling in one ͓9͔ F. Diedrich, J. C. Berquist, W. M. Itano, and D. J. Wineland, dimension of single atoms in a trap depends on the internal Phys. Rev. Lett. 62, 403 ͑1989͒; C. Monroe, D. M. Meekhof, transition quantum numbers, and for hyperfine states with se-
B. E. King, S. R. Jefferts, W. M. Itano, D. J. Wineland, and P. lection rule ⌬mFϭϮ1itis͗n͘ϭ7␥/20; see Ref. ͓8͔. Gould, ibid. 75, 4011 ͑1995͒. ͓23͔ J. Dalibard, Y. Castin, and K. Mølmer, Phys. Rev. Lett. 68, ͓10͔ I. Marzoli, J. I. Cirac, R. Blatt, and P. Zoller, Phys. Rev. A 49, 580 ͑1992͒. 2771 ͑1994͒. ͓24͔ This argument is valid in the limit of low saturation. At satu- ͓11͔ B. E. King, C. S. Wood, C. J. Myatt, Q. A. Turchette, D. ration the antibunched nature of the atoms’ spontaneous emis- Leibfried, W. M. Itano, C. Monroe, and D. J. Wineland, Phys. sion plays a role by modifying the diffusion. See Ref. ͓8͔. Rev. Lett. 81, 1525 ͑1998͒. ͓25͔ Phenomena due to interference between the internal paths of ͓12͔ H. C. Naegerl, W. Bechter, J. Eschner, F. Schmidt-Kaler, and excitation, which lead to superradiance and subradiance ef- R. Blatt, Appl. Phys. B: Photophys. Laser Chem. 66, 603 fects, occur for distances between the ions of the order of the 3808 G. MORIGI, J. ESCHNER, J. I. CIRAC, AND P. ZOLLER PRA 59
wavelength, when dipole-dipole interaction cannot be ne- ͓26͔ V. V. Dodonov, V. I. Manko, and L. Rosa, Phys. Rev. A 57, glected. See R. G. DeVoe and R. G. Brewer, Phys. Rev. Lett. 2851 ͑1998͒. 76, 2049 ͑1996͒; R. G. Brewer, ibid. 77, 5153 ͑1996͒; and Ref. ͓27͔ C. W. Gardiner, Quantum Noise ͑Springer-Verlag, Berlin, ͓16͔. 1991͒.