Optical trapping of ion Coulomb crystals
Julian Schmidt, Alexander Lambrecht, Pascal Weckesser, Markus Debatin, Leon Karpa, Tobias Schaetz
OOptical trapping of ion Coulomb crystals
Julian Schmidt, Alexander Lambrecht, Pascal Weckesser, Markus Debatin, Leon Karpa, and Tobias Schaetz Albert-Ludwigs-Universit¨at Freiburg, Physikalisches Institut,Hermann-Herder-Straße 3, 79104 Freiburg, Germany (Dated: December 25, 2017)The electronic and motional degrees of freedom of trapped ions can be controlled and coherentlycoupled on the level of individual quanta. Assembling complex quantum systems ion by ion whilekeeping this unique level of control remains a challenging task. For many applications, linear chainsof ions in conventional traps are ideally suited to address this problem. However, driven motion dueto the magnetic or radio-frequency electric trapping fields sometimes limits the performance in onedimension and severely affects the extension to higher dimensional systems. Here, we report on thetrapping of multiple Barium ions in a single-beam optical dipole trap without radio-frequency oradditional magnetic fields. We study the persistence of order in ensembles of up to six ions within theoptical trap, measure their temperature and conclude that the ions form a linear chain, commonlycalled a one-dimensional Coulomb crystal. As a proof-of-concept demonstration, we access thecollective motion and perform spectrometry of the normal modes in the optical trap. Our systemprovides a platform which is free of driven motion and combines advantages of optical trapping,such as state-dependent confinement and nano-scale potentials, with the desirable properties ofcrystals of trapped ions, such as long-range interactions featuring collective motion. Starting withsmall numbers of ions, it has been proposed that these properties would allow the experimentalstudy of many-body physics and the onset of structural quantum phase transitions between one-and two-dimensional crystals.
PACS numbers: 37.10.Ty,03.67.Lx
I. INTRODUCTION
Coulomb crystals are an intriguing form of matter. Onthe one hand, it is believed that they make up the coreof white dwarves and the surface of neutron stars [1]. Onthe other hand, they provide a versatile solid-state-likeplatform for applications that require a magnified latticestructure. While a solid-state system with nanometer-spaced particles can hardly be observed and controlledwith single-site resolution, distances of ions in Coulombcrystals are on the order of a ≈ µ m. The correspond-ing 10 times lower densities allow to manipulate eachconstituent individually.The ions are typically trapped in Paul [2] and Pen-ning traps [3], which combine electrostatic with radio-frequency (rf) fields or magnetic fields, respectively.These traps provide stiff confinement to counteract therepulsive interaction of the positively charged ions. En-sembles with temperatures of thousands of Kelvin can betrapped and form one-component plasmas (OCP). Cool-ing the OCP leads to a phase transition to a crystal whenthe ratio of the Coulomb energy for mean ion distancesand the average kinetic energy, Γ Plasma = E Coul /E kin ,exceeds a critical value. Depending on the dimension-ality of the system, Molecular Dynamics (MD) simula-tions predict that this phase transition occurs at differ-ent Γ Plasma , e.g. Γ = q k B · T · a > > > q is the charge of theparticles, T their temperature and k B the Boltzmannconstant. For typical experimental parameters in 1D,temperatures below a critical value T c on the order of 50 mK are required to reach the phase transition.A handful of ion species available for direct laser cool-ing can be prepared at Doppler temperatures T D << T c ,and sufficiently well isolated from the environment by lev-itating them in ultra-high vacuum. Other elements andmolecular ions can also be embedded into the crystallinestructure thanks to sympathetic cooling via Coulomb in-teraction [1].Extensive work has allowed to extend the coherent con-trol and coupling of external (motional) and internal(electronic) degrees of freedom from single ions to (short)linear chains of ions [6–11].However, ions perform rf-driven motion in Paul traps anddriven cyclotron motion in Penning traps, which often isundesirable. Various methods can minimize driven mo-tion close to zero [12], but, even when assuming perfectcompensation of stray electric fields, driven motion re-mains inevitable if ions are intrinsically displaced fromthe trap center, e.g. in a 2D or 3D crystal [1, 13], or dur-ing the interaction with neutral atoms [14, 15]. In thesecases, the kinetic energy of the driven motion easily ex-ceeds the residual thermal energy by orders of magnitude.This makes it challenging to extend the unique level ofcontrol and isolation available for single ions and linearchains of ions to Coulomb crystals of larger size and di-mensionality. A generic approach to entirely mitigatedriven motion is the use of optical dipole traps withoutrf and without magnetic fields. Optical dipole traps havebeen established in experiments for neutral particles fordecades [16].Recently, trapping and isolation of a single ion in a dipole a r X i v : . [ qu a n t - ph ] D ec trap was demonstrated for seconds [17, 18], comparableto the lifetime of neutral atoms for similar trapping con-ditions [19] and in agreement with theoretical predictions[20]. Several groups have also superimposed optical lat-tices with Coulomb Crystals trapped in rf traps [21–23],which, e.g., allowed to study fundamental questions inthe context of friction [24]. These experiments were re-alized providing rf confinement for the radial degrees offreedom, while axial confinement was implemented usingoptical and electrostatic fields.In this Letter, we show trapping of up to six ions in asingle-beam optical dipole trap without confinement byrf fields. We demonstrate that the ensemble remains aone-dimensional Coulomb crystal in the optical trap andreveal access to the axial motional modes. II. APPARATUS FOR OPTICAL TRAPPING OFION COULOMB CRYSTALS
Our experimental setup combines focused dipole laserbeams with a segmented linear rf trap, see Fig. 1(a). Wefollow a three step protocol for optical trapping of multi-ple ions in the absence of rf fields, during which we main-tain control over the axial confinement by electrostaticfields. In step one, we load 1 ≤ N ini ≤ rf / (2 π ) = 1 . ≈ s − , Doppler limit T D ≈ . T with T D < T (cid:28) T c ,the chosen trapping frequencies ω rfrad = 2 π ×
140 kHz and ω dcax = 2 π ×
25 kHz, and N ini <
10, the Coulomb crystalsextend as one-dimensional chains along the z -axis of therf trap with an inter-ion distance of ∼ µ m. This stepincludes the compensation of stray electric fields to the Laser VIS NIR λ
532 nm 1064 nm w (2 . ± . µ m (5 . ± . µ m z R (40 ± µ m (74 ± µ m P opt ≤ . ≤
20 W U opt /k B ≤ (110 ±
18) mK ≤ (16 ±
1) mK ω rad,opt / (2 π ) ≤ (315 ±
25) kHz ≤ (62 ±
2) kHz∆
Stark / (2 π ) ≤ (2 . ± .
4) GHz ≤ (330 ±
30) MHzTABLE I. Laser and trap parameters for visible (VIS) andnear-infrared (NIR) optical dipole traps. The symbols de-note wavelength ( λ ), 1 /e beam waist ( w ), Rayleigh length( z R ), laser power ( P ), optical trap depth ( U opt ), radial trapfrequency ( ω rad,opt ), and Stark shift in the electronic groundstate S / (∆ Stark / (2 π )). ω rad,opt denotes the radial opticaltrap frequency (neglecting electrostatic fields) at the mimi-mum beam waist and for a single ion. level of | (cid:126)E stray | < ∼ − V/m for a single ion [17, 26] anddetection by fluorescence imaging at 493 nm on the S / - P / transition (natural linewidth Γ = 2 π × . Ba + , admixed , Ba + ions remain dark due to their isotopic shifts being largecompared to Γ. However, as a consequence of sympa-thetic cooling and Coulomb repulsion, dark ions can beembedded at random sites of the lattice formed togetherwith the bright Ba + ions. From the dark gaps on thefluorescence images [Fig. 2(a)], we deduce the numberand configuration of bright ( N b ) and dark ( N d ) ions witha fidelity close to one.In step two, the ions are transferred into the optical trapby turning on either the visible (VIS) or near-infrared(NIR) dipole trap while ramping the rf field to zero[17, 26, 27]. The laser and optical dipole trap parametersare shown in Table I. The axial confinement is controlledby dc voltages applied to the endcap electrodes (yellowelectrodes in Fig. 1(a)) which remain unchanged for theremaining protocol. Both dipole traps are generated byfocusing circular Gaussian beams (see Fig. 1) with theirwave vector aligned with the linear ion chain centered atthe minimal beam waist. After a duration ∆ t opt , we turnon the rf trap while turning off the dipole trap.In step three, we detect the number N opt and analyze theconfiguration of the remaining ions. The optical trappingprobability p opt is defined as the number of successfultrapping attempts divided by total trapping attempts.We call an attempt successful if the number of ions beforeand after optical trapping is equal, N opt = N ini , and un-successful if one or more ions have been lost, N opt < N ini .In such cases, we find that typically only one or two ionsare missing after ∆ t opt . The statistical uncertainty of p opt is determined by the Wilson score interval [28]. III. KEEPING COULOMB ORDER
For our trapping parameters, theory predicts the ex-istence of one-dimensional ion crystals for temperatures
T < T c ≈
50 mK, which is on the same order as theavailable optical trap depth for a single ion, U VISopt ≤ k B ×
110 mK. However, this criterion is only valid undercertain assumptions, such as homogeneous radial con-finement and similar temperatures and heating rates inall spatial degrees of freedom. These approximations arenot fully justified in our experiment. Therefore, more de-tailed studies are required to confirm the survival of thecrystal during ∆ t opt .We first investigate the feasibility of confining more thanone ion in the VIS dipole trap, which provides the deeperpotential U VISopt . However, the VIS trap only creates anattractive potential for Ba + in its electronic ground state S / . In the metastable D states, which can be populatedby off-resonant scattering, the potential is repulsive and (a) xy z (b) zxyy x z (c) x z dipoletrapradial opticalpotentialdc electrodesrf electrodesDoppler coolingBa + radial dc defocusing Ba + (d) a x i a l d c c o n fi n e m e n t dipole trap axial dc con fi nement116 µm FIG. 1. Schematic of the apparatus and contributions to the total trapping potential. (a) We load chains of up to six Doppler-cooled Barium ions, with lengths of up to 135 µ m (5 ions: 116 µ m), into a linear rf trap. dc electrodes are used to adjustthe external electrostatic potential. The ions are then transferred from the rf trap into either a visible (VIS) or near-infrared(NIR) optical dipole trap by ramping up the optical potential and simultaneously turning off the rf fields. Axial dc confinementin the external electrostatic potential [grey surfaces in (b) and (d)] as well as the ions’ mutual Coulomb interaction (c) leadto additional defocusing forces in the radial directions. In our experiment, defocusing is chosen to predominantly lie in the x direction [red arrow in (b)] while defocusing in the y direction is negligible (thin gray arrow). With Rayleigh lengths z VIS R = 40 µ m and z NIR R = 74 µ m for the respective dipole laser beams, the radial optical potential depicted by the blue surfacesin (d) also depends on the ion position along the z -axis. the ion is lost. At P VISopt = 9 . /e lifetime of a single ion in the center of the VIStrap to about 1 . t opt ≈ µ s and turn on additional repumpinglasers [17]. As shown in Fig. 2(a), we demonstrate reli-able optical trapping for N ini ≤ p opt ≈
1. For N ini = 6, we find p opt ≤ .
2. Note thatfor P VISopt = 0 W, no trapping is observed. Currently,the setup does not allow for direct imaging during ∆ t opt ,since the Stark shift inside the VIS laser ∆ VISStark exceedsΓ by orders of magnitude. The images of the crystalsare therefore taken during steps 1 and 3 of the protocol[see Fig. 2(a)]. Note that there is no direct evidence thatthe crystal survived the transfers between the traps. Theensemble might melt and turn into a gas-phase OCP dur-ing optical trapping. Then, at the beginning of step 3, itcould re-crystalize under the effect of the detection laserand the associated Doppler cooling.To gain deeper insight into the dynamics during trans-fers and ∆ t opt , we study the ion ensemble indirectly byembedding , Ba + as markers to witness changes ofthe crystalline configuration, see Fig. 2(a). For N ini = N opt ≤ N d ≤
2, we observe that close to 100 % ofthe image pairs show identical configurations of brightand dark ions. After random reorganization, a givenconfiguration only occurs with a probability of p rand = N d ! N b ! /N ini !. We typically observe that the configura-tions of 4-ion crystals ( N ini = 4, N d = 1) remain un-changed over the course of 15 consecutive experiments, yielding ( p rand ) = 9 × − . We attribute events withchanged configuration ( < N opt < N ini ).The persistence of Coulomb order is evidence that thethermal excitation of the ensemble remains below T c . Ad-ditionally, the method demonstrates that even isotopeswhich are not Doppler-cooled (e.g. , Ba + ), can beoptically trapped when embedded into the ensemble, de-spite the intrinsically reduced cooling rate. IV. TEMPERATURE OF MULTIPLE IONSIN AN OPTICAL TRAP
To access the mean kinetic energy within the ensem-ble during ∆ t opt and to study the dominant loss mecha-nisms, we further investigate the dependence of p opt onour experimental parameters and measure p opt for differ-ent P VISopt .The dependence of p opt on the optical trap depth U VISopt has previously been exploited to determine the tempera-ture of a single ion [17, 26, 27]. U VISopt defines the cut-offenergy for a 3D Boltzmann distribution of indeterminatetemperature T . By integrating the distribution up to U VISopt , one obtains the approximate analytic expression
Doppler cooling Δ t opt Radio frequencyPrepare & detect N ini
Detect N opt
OpticaltrappingTime(a)(b) Dipole laser(1) (2) (3)
FIG. 2. Demonstrating the persistence of Coulomb order foran increasing number of optically trapped ions. (a) Fluores-cence images of Coulomb crystals with N ini = 1 ... + ionsare recorded before and after optical trapping of N opt . For N ini ≥
4, the gaps marked by orange circles reveal the pres-ence of dark ions which appear at initial random lattice sitesafter Doppler and sympathetic cooling. (b) The experimentalprotocol (not to scale) consists of three steps: (1) we detectthe initial configuration and ion number N ini while Dopplercooling the ions; (2) the ions are transferred into the dipoletrap by turning off the rf field and cooling lasers for the opticaltrapping duration ∆ t opt , keeping the electrostatic potential;(3) we again detect the number and final configuration of allremaining ions in the rf trap. An intermittent gaseous phasefollowed by recrystallization or enhanced diffusion should beobservable with high fidelity via changes of the positions ofthe dark ions within the crystal. p opt ( ξ ) = 1 − e − ξ − ξe − ξ , where ξ = U VISopt /k B T , whichcan be used to derive T .Since we have to consider additional effects contribut-ing to the total trapping potential shown in Fig. 1, it isevident that this approach is not suitable for N ini > N ini and theaxial equilibrium position z i of ion i . These effects are offundamental importance for optical trapping of chargedparticles and have to be considered when deriving the lo-cal trap depth ∆ U tot for ion i . First, the ensemble of ionswill extend at least up to the length of the Coulomb crys-tal in the rf trap for the chosen trapping parameters and T < T c . Because the axial extensions are comparable tothe Rayleigh length, e.g. | z − z | ≈ µ m > × z VISR for N ini = 6 [see fig. 1(d) and table I], the trap depthdepends on the axial position(s) of the ion(s). Evenwhen only considering the optical potential, the increas-ing laser beam size w VIS ( z i ) results in a reduced trapdepth U VISopt ( z i ), see Fig. 1(d).Additionally, the electrostatic potential set by dc elec- P VISopt (W) p op t (a) U tot ( z ) / k B (mK) p op t (b) N ini T ( m K ) FIG. 3. Measuring the temperature of multiple ions in thesingle-beam VIS dipole trap. (a) We measure the trappingprobability p opt for N ini = 1 ... P VISopt (red, blue, green, yellow and black squares). The solid linesindicate fits with the radial-cutoff model [29], which relatesthe trapping probability and finite trap depth for atoms ora single ion to their temperature. We observe that trapping N ini > U tot ( z ,N ini ). Solid lines depict fits assumingthe altered radial-cutoff model for ∆ U tot ( z i ) (see text). Weobtain temperatures of T N ini = (0 . ± .
1) mK for N ini ≤
3, aswell as T = (1 . ± .
5) mK and T = (1 . ± .
5) mK (inset). trodes and the mutual Coulomb interaction lead to a po-tential energy U el ( (cid:126)r i = ( x i , y i , z i )) = U dc ( (cid:126)r i ) + U coul ( (cid:126)r i ).This results in effective radial defocusing, which can beseen by expanding U el ( (cid:126)r i ) to second order, yielding po-tential curvatures m ˜ ω x, el ( z i ) and m ˜ ω y, el ( z i ) for ion i with mass m near (cid:126)r i = (0 , , z i ). The contribution bythe electrostatic potential energy U dc ( (cid:126)r i ) expanded tosecond order defines characteristic potential curvatures m ˜ ω x,y,z ) , dc , related via the Laplace equation, ˜ ω x, dc +˜ ω y, dc + ˜ ω z, dc = 0. Positive terms can be interpreted astrapping frequencies and imply confinement, e.g. in theaxial direction ˜ ω z, dc >
0, whereas negative terms cor-respond to defocusing, inevitable in at least one radialdirection. In our setup, we find that the defocusing al-most exclusively occurs along the x direction (see Fig. 1)such that ˜ ω x, dc ≈ − ˜ ω z, dc . We therefore neglect the resid-ual defocusing along the y direction, making the x direc-tion the preferred escape path for the ions. We thenapproximate the Coulomb interaction U coul ( (cid:126)r i ) | z i = z i ofion i with all other ions, assuming that the ions remainat their equilibrium positions. This results in an addi-tional defocusing in the x and y directions ˜ ω x, coul ( z i ) =˜ ω y, coul ( z i ) <
0. Finally, we approximate the curvature of U el ( (cid:126)r i ) via ˜ ω x, el ( z i ) = ˜ ω x, dc + ˜ ω x, coul ( z i ) for ion i in theweakest confined direction x . The total radial potentialenergy for ion i at the axial position z i is then written as U tot ( x i , y i , z i ) = U VISopt ( x i , y i , z i ) + U el ( x i , y i , z i ). In thefollowing, the difference between the local maximum andminimum of U tot ( x i , , z i ) along the x direction will bereferred to as the local radial trap depth ∆ U tot ( z i ) (seeSupplemental Material for details).The measured optical trapping probability for N ini ionsand laser power P VISopt is modeled as the product of theindividual trapping probabilities of the ions, p opt ( N ini ) = (cid:81) i ≤ N ini p opt,ind ( ξ i ), where ξ i = ∆ U tot ( z i ) /k B T . This al-lows fitting p opt ( N ini ) for each N ini ≤
5. In Fig. 3(b), weshow p opt ( N ini ) in dependence on the smallest trap depth,∆ U tot ( z ) = ∆ U tot ( z N ini ) at the edges of the ensemble.We derive temperatures near T N ini = (0 . ± .
1) mK for N ini ≤ T = (1 . ± .
5) mK and T = (1 . ± .
5) mK.The apparent increase in temperature for larger N ini maystem from a residual angle between the rf and opticalaxis, which becomes increasingly relevant for larger ionnumbers. In addition, the trap depth, and therefore p opt ( N ini ), is affected by stray fields (which are currentlycompensated at the position of the center ion only) anddeviations from the assumed beam profile for increasingdistance | z i | . Nonetheless, including the spatial depen-dence of the total radial confinement improves the de-scription of the system and yields T < (cid:28) T c dur-ing ∆ t opt . For these temperatures and our experimentalparameters, MD simulations show that the amplitudesof the ions’ axial motion are small ( <
10 %) comparedto the distance of neighboring ions. Thus, applying theLindemann criterion for a small number of lattice sites,we conclude that ensembles of up to 5 ions form crystalsduring ∆ t opt [1, 30].In our system, the number of ions forming a crystal in theVIS trap is limited by our beam geometry and the cho-sen trapping parameters. Further improvement could beachieved by adapting the laser beam geometry, replacingaxial electrostatic confinement with optical confinementor by using different ion species featuring either a smallerdecay rate into repulsive D states or no such states at all.In the next section, we will use the further detuned NIRoptical trap to reduce off-resonant scattering, at the ex-pense of the confinement, see table I. V. DETECTING MOTIONAL MODES OFOPTICALLY TRAPPED IONS
To gain further insight into the dynamics of the ensem-ble during ∆ t opt , we investigate the vibrational spectrumof the ions. Here, we choose to explore the axial degree of freedom. Coupling to the charge allows exciting themotion of trapped ions by applying oscillating voltagesto specific electrodes. Since the position of the ionsin the rf trap can be observed directly on the CCDcamera, we observe motional excitation as an effectivelyincreased ion image size caused by the integration of thefluorescence of the ion along its trajectory [see Fig. 4(a)]. in rf trap COM stretch(a)(b) in dipole trap
FIG. 4. Spectrometry of normal modes, demonstrating ac-cess to the axial phonons of the crystal during optical trap-ping. In (a), we show typical fluorescence images of ionsin the rf trap with dc axial confinement, resonantly modu-lated with oscillating electric fields (two-ion distance 43 µ m,other images to scale). For N ini = 1, we observe a singleresonance for the axial motion at ω COMax . For N ini = 2 , ω strax = √ ω COMax correspondingto out-of-phase motion appears. (b) Optical trapping prob-ability for N ini = 1 (blue squares) and N ini = 2 (red cir-cles) ion(s) in the NIR trap, as a function of the frequency ω mod / π of the oscillating electric field. We observe a dropin p opt at ω COMax for both N ini = 1 and N ini = 2, in agree-ment with the expected axial confinement. For the COMmode, the solid lines show fits to the data. The resonance at ω strax = 2 π × (43 . ± .
15) kHz ≈ √ ω COMax (binned data pointsweighted with their statistical significance, for details on un-certainty see text) only emerges in the case N ini = 2 and showsaccess to the motional degrees of freedom of the Coulombcrystal during ∆ t opt . In the case of the stretch mode, wenumerically simulate the out-of-phase motion driven by anoscillating electric field with amplitude E = 1 . / m (nofree parameters). We depict the amplitude | z str | = | z − z | / t opt = 10 ms by the solid red line (axison the right-hand side). The nonlinearity of the Coulombinteraction leads to an asymmetric frequency response (seeSupplemental Material). We emphasize that the dependenceof p opt on | z str | is nontrivial and not taken into account here. The blurring and its dependence on the frequency ofexcitation, ω mod , allows to calibrate the parameters ofthe trapping potential. The motion of more than oneion forming a Coulomb crystal is typically described interms of normal modes [31]. In Fig. 4(a), the collectivemotion in the rf trap is presented exemplarily for twoaxial modes of up to three ions. The mode of lowest axialfrequency, the center-of-mass (COM) mode, describesthe in-phase oscillation of all ions at ω COMax . The nexthigher frequency at ω strax = √ ω COMax corresponds to thestretch mode, where two ions oscillate opposite in phase.For this mode and odd ion numbers, the center ion hasto remain at rest.To perform motional spectrometry of ions in the opticaltrap, we repeat the protocol shown in Fig. 2(b) for N ini = 1 , t opt only. We choose ∆ t opt = 10 ms within the NIRtrap to improve the frequency resolution while furthermitigating off-resonant scattering to ≤
10 s − . At specific ω mod , we observe reduced optical trapping probabilities.We identify these frequencies as resonances within theoscillation spectrum of the ensemble during ∆ t opt . For N ini = 1, we observe a single resonance, centered at ω COMax = 2 π × (25 . ± .
02) kHz. For N ini = 2, in addi-tion to the resonance at ω COMax = 2 π × (24 . ± .
02) kHz,we find another pronounced drop of p opt centeredat ω strax = 2 π × (43 . ± .
15) kHz ≈ √ ω COMax . Theexperimental uncertainty of ω strax is estimated via thespacing between adjacent data points which is consistentwith the frequency resolution of the excitation. Weinterpret the additional resonance as the spectrometricfingerprint of the stretch mode. Based on the agreementwith the theoretical prediction we confirm the survivalof the crystal in absence of any rf while demonstratingthe possibility to address and exploit its normal modes.To study the dynamics of the motional excitation andrelated loss mechanisms, we compare our experimentalresults with numerical simulations for N ini = 1 , VI. CONCLUSIONS AND OUTLOOK
In summary, we demonstrate trapping of Coulombcrystals in a single-beam optical trap. We reveal theimportance of Coulomb interaction and the electrostaticfield along the axis of the dipole trap and demonstrateaccess to the collective motion. For neutral particles,these effects can usually be neglected. On the one hand,Coulomb interaction establishes the axial and radial mo-tional modes in ion crystals. Control of these modeson the single-phonon level in optical traps would per-mit to couple electronic degrees of freedom of the ions.As in rf traps, phonons could mediate spin-spin inter-action between the ions and act as a data-bus [31]. Inaddition, they can feature as quasi-particles which spanthe bosonic degree of freedom in open quantum systems[32] and extend experimental quantum simulations, e.g.by tunneling between the lattice sites defined by the ions[33, 34]. On the other hand, ion interaction and electro-static forces currently limit the size of optically trappableCoulomb crystals. These contributions modify the trap-ping potential itself and reduce its depth. We aim toincrease the number of ions and the dimensionality ofthe crystals using the range of readily available dipoletrap geometries, e.g. Bessel beams, optical lattices oradditional (crossed) laser beams.We also embed sympathetically cooled ions, here Bariumisotopes, without substantially affecting the temperatureof the crystal. Given suitable electronic transitions andsufficient coupling strengths, it is possible to opticallytrap ions of different electronic states and exploit thestate-dependent potential [17]. Co-trapping other ionicspecies and molecular ions [1] should also be considered.We argue that systems in which rf micromotion existsdue to intrinsic displacement from the center of the trap,as in higher-dimensional Coulomb crystals or during theinteraction with (cold) neutral atoms, could substantiallybenefit from optical trapping of ions. While still in its in-fancy, the technique presented here could provide a cleanplatform to experimentally investigate systems with pre-dicted quantum phase transitions and feature quantummany-body effects, briefly described in the following.The number of ions and the ratio of radial and axialconfinement determine whether a crystal exists in a 1Dchain or 2D zigzag structure. The two symmetric con-figurations of zigzag and “zagzig” are trapped within aneffective double-well potential with well-controllable bar-rier height and are predicted to allow for experimentalstudies of a wide range of physical effects, starting with N ≥
3. Adiabatically reducing the radial confinement tocross the structural quantum phase transition from 1D to2D has been proposed to create a superposition of zigzagand zagzig [35]. The impact of quantum fluctuations atcriticality is predicted to dominate the structure adoptedby ions cooled close to the motional ground state, thatis, even at finite temperatures [36]. It has also been pro-posed to create an entangled state, incorporating bothstructural phases, linear and zigzag, simultaneously [37–39]. Preparing one ion of a linear chain in a coherentsuperposition of two electronic states and exploiting thestate-dependence of the optical trap [17] could directlyimplement this proposal.Additionally, embedding a single ion in a BEC has beenproposed as a controlled quantum many-body systemdriven by the nucleation of tens or hundreds of atomspolarized in the ion’s electric field [40, 41]. In hybridtraps, which combine an rf trap for the ion with an op-tical trap for the atoms, micromotion limits sympatheticcooling of the ion to a regime above the ultracold temper-atures required for the formation the clusters [14, 15, 42].Optical trapping of ions may be a generic solution toovercome this limitation [42] even for
N > N ≥
2) by coherent scatter-ing inside a light field without an optical cavity [45].
ACKNOWLEDGEMENTS
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46 mK i = 2, 4 − x i ( µ m)
14 mK i = 1, 5 z FIG. 5. (a) Sketch of a five ion crystal and (b) trap depths forthe optical and electrostatic potential at the equilibrium po-sitions z i of the individual ions. The ions with index i = 1 ... z . The optical potential is calculated as-suming a Gaussian laser beam with power P VISopt = 8 W andminimal beam waist radius w VIS x = 2 . µ m. The electro-static potential U el ( (cid:126)r i ) (dashed gray line) is approximated as aquadrupole potential and consists of an external electrostaticpotential U dc ( (cid:126)r i ) with maximum defocusing in the x directionand mutual Coulomb interaction U coul ( (cid:126)r i ) which leads to thestrongest defocusing for the center ion. The total potential isthen evaluated at the position of the ions with i = 3 (red), i = 2 , i = 1 , U el ( (cid:126)r i ).[39] J. D. Baltrusch, C. Cormick, and G. Morigi, Phys. Rev.A , 032104 (2012).[40] R. Cˆot´e, V. Kharchenko, and M. D. Lukin, Phys. Rev.Lett. , 093001 (2002).[41] J. M. Schurer, A. Negretti, and P. Schmelcher, Phys.Rev. Lett. , 063001 (2017).[42] M. Tomza, K. Jachymski, R. Gerritsma, A. Negretti,T. Calarco, Z. Idziaszek, and P. S. Julienne, arXivpreprint arXiv:1708.07832 (2017).[43] U. Bissbort, D. Cocks, A. Negretti, Z. Idziaszek,T. Calarco, F. Schmidt-Kaler, W. Hofstetter, andR. Gerritsma, Phys. Rev. Lett. , 080501 (2013).[44] R. Gerritsma, A. Negretti, H. Doerk, Z. Idziaszek,T. Calarco, and F. Schmidt-Kaler, Phys. Rev. Lett. ,080402 (2012).[45] S. Ostermann, F. Piazza, and H. Ritsch, Phys. Rev. X , 021026 (2016). SUPPLEMENTAL MATERIAL
Calculation of the local radial trap depth ∆ U tot ( z i ) : We calculate the local radial trap depth at the position of the individual ions. Ions are intrin-sically sensitive to electric fields due to their charge.When applying axial electrostatic confinement U dc ( (cid:126)r i )and taking into account mutual Coulomb interaction U coul ( (cid:126)r i ), Laplace’s equation predicts defocusing in atleast one radial direction. To derive the ions’ trapdepth in the absence of rf fields, we have to consider U el ( (cid:126)r i ) = U dc ( (cid:126)r i ) + U coul ( (cid:126)r i ). Only taking into accountsecond-order terms (stray field compensation makes thefirst-order terms negligible) leads to a quadrupole po-tential U el ( (cid:126)r i ) = m/ (cid:16) ˜ ω x, el x i + ˜ ω y, el y i + ˜ ω z, el z i (cid:17) with˜ ω x, el + ˜ ω y, el + ˜ ω z, el = 0 where m denotes the mass ofthe ion. Coulomb interaction and radial optical poten-tial depend on the axial equilibrium position z i . Forthe defocusing of U el ( (cid:126)r i ) in x -direction, represented by˜ ω x, el ( z i ) <
0, a Gaussian laser beam with radius w ( z i )along x and optical trap depth U opt ( z i ) <
0, the totalpotential along x is given by U tot ( x, , z i ) = 12 m ˜ ω x, el ( z i ) x + U opt ( z i ) exp − x w ( z i ) . (1)This function is plotted in Fig. 5 for typical experimentalvalues. The locations of the local minima and maximaare determined by x i, min = 0 and (2) x i, max ( z i ) = w ( z i )2 log (cid:32) U opt ( z i ) m ˜ ω x, el ( z i ) w ( z i ) (cid:33) . (3)Calculating the trap depth as ∆ U tot ( z i ) = U tot ( x i, max , , z i ) − U tot ( x i, min , , z i ) yields the fol-lowing expression:∆ U tot ( z i ) = − U opt ( z i )++ m ˜ ω x, el ( z i ) w ( z i )4 (cid:34) (cid:32) U opt ( z i ) m ˜ ω x, el ( z i ) w ( z i ) (cid:33)(cid:35) . (4)The second term in this expression, which correspondsto the offset of U tot ( ± x i, max , , z i ) from zero visible inFig. 5(b), is only well defined when m ˜ ω x, el ( z i ) w ( z i ) < U opt ( z i ). This condition is violated when the electro-static defocusing is stronger than the optical confine-ment, prohibiting the existence of a local minimum at x = 0 such that ∆ U tot (0 , , z i ) = 0. Numerical simulations of the normal mode de-tection:
In the following, we discuss in more detail theresults of the numerical model shown in Fig. 4(b), de-picting the motional amplitude of the axial out-of-phase(stretch) mode for large excitation amplitudes. The ax-ial potential is dominated by electrostatic forces whichprovide near-harmonic confinement, and Coulomb inter-action leads to equilibrium positions z , . In the exper-iment, we drive the system with an oscillating electric FIG. 6. Behavior of the ions’ out-of-phase oscillation in theoptical trap. (a) Numerical simulation of the frequency re-sponse of the out-of-phase oscillation (solid lines) and exper-imental results for spectrometry of two ions, also shown inFig. 4. The simulation is repeated for different amplitudes ofthe driving field, E = (0 . , . , .
0) mV/m for the (blue, red,green) curves. The solid vertical line corresponds to √ ω COM ,the expected position of ω str for the measured resonance of thein-phase motion for two ions ω COM . The dashed vertical line isthe position of the minimum of p opt and the gray shaded areacorresponds to its estimated uncertainty. The measured fre-quency and the asymmetric shape of the data are in excellentagreement with the simulation results (no free parameters).(b) Radial trap depths for one ion (gray line) and two ions(black line) as a function of axial position z and trajectoriesof ions for varying driving amplitudes (blue, red and greenhorizontal lines). For a single ion, the trap depth is given bythe combination of an electrostatic and an optical potentialand is largest at z = 0 due to divergence of the beam. Fortwo ions at z , (equilibrium positions z , depicted by verticalblack dashed lines), the additional defocusing from Coulombinteraction lowers the trap depth as the ions approach eachother, e.g. when oscillating at large amplitudes and out-of-phase. The lowered trap depth radially expels the ions fromthe optical trap. Due to the anharmonicity of the potential,the oscillation is asymmetric. field E ( t ) = E sin ( ω mod t ) by applying voltages to theendcap electrodes.Now considering the time-dependent positions z , ( t ) ofthe two ions, we obtain a system of coupled harmonicoscillators with nonlinear coupling given by Coulomb in-teraction ∓ e / π(cid:15) | z ( t ) − z ( t ) | (negative for z and positive for z when z < < z ). Damping can beneglected as laser cooling is inactive during optical trap-ping. The driving terms for z and z have the form+ eE sin ( ω mod t ) and ± eE sin ( ω mod t ), where the identi-cal signs correspond to a homogeneous field used to ex-cite the COM mode and the opposite signs correspondto a field gradient used to excite the stretch mode. Byadding or subtracting the equations of motion and defin-ing z COM = ( z + z ) / z str = ( z − z ) /
2, we obtainthe decoupled equations:¨ z COM ( t ) + ω z z COM ( t ) − eEm sin ( ω mod t ) = 0 (5)and¨ z str ( t ) + ω z z str ( t ) + e π(cid:15) m | z str ( t ) | − eEm sin ( ω mod t ) = 0 . (6)The out-of-phase motion behaves like a harmonic oscilla-tor for small amplitudes, when the change of the distancebetween the ions, and thereby their Coulomb interaction,is small. For large driving amplitudes, we solve the dif-ferential equation (6) numerically for a varying frequency ω mod , see Fig. 6 (a).At t = 0, the ions are assumed to be at rest at theirequilibrium positions in a harmonic potential describedby the measured frequency of the two-ion COM mode ω COMax = 2 π × (24 . ± .
02) kHz. We then let the sys-tem evolve for ∆ t = 10 ms (time step δt = 1 µ s). Theoscillation amplitude is defined as half the distance be-tween the maximal and minimal separation of the ionsmax( z str ( t )) − min( z str ( t )). The asymmetric shape of theresonance appears when the ions are sufficiently close toexperience modified Coulomb interaction. Additionally,the largest oscillation amplitude is shifted to higher fre-quencies with increased driving amplitude. In the exper-iment, the amplitude of the driving electric field is chosenas E = (1 . ± .
1) mV/m. To investigate the sensitivityof this parameter, we vary E in the simulation, see Fig.6(a).The detection of the oscillation is achieved by measuringthe trapping probability. This raises the question how theions are lost from the trap. As the axial position changesduring the resonant excitation, the radial optical trapdepth changes as well. We distinguish two possible lossmechanisms. (a) If the ions move away from the focus ofthe laser beam, the beam waist increases and the radialtrap depth approaches zero. This may lead to loss whenexciting the COM or stretch mode. (b) When resonantlyexciting the stretch mode, the ions perform out-of-phasemotion. When their distance is minimal, Coulomb re-pulsion leads to stronger radial defocusing, reducing thetrap depth. More intuitively, the effect can be describedas a “collision” of the ions in the guide created by theoptical trap. When this happens, they avoid each other,moving to the side and leaving the trapping region. In0figure 6(b), the axial oscillation amplitude of the out-of-phase motion is shown together with the dependence of the radial trap depth on the ion distance dd