Nanofriction in Cold Ion Traps

A. Benassi1,2, A. Vanossi2,1 and E. Tosatti2,3,11 CNR-IOM Democritos National Simulation Center,Via Bonomea 265, I-34136 Trieste, Italy2 International School for Advanced Studies (SISSA), Via Bonomea 265, I-34136 Trieste, Italy3 International Centre for Theoretical Physics (ICTP), P.O.Box 586, I-34014 Trieste, Italy

ABSTRACT

Sliding friction between crystal lattices and the physicsof cold ion traps are so far non-overlapping fields. Twosliding lattices may either stick and show static frictionor slip with dynamic friction; cold ions are known to formstatic chains, helices, or clusters, depending on trappingconditions. Here we show, based on simulations, thatmuch could be learnt about friction by sliding, via e.g.an electric field, the trapped ion chains over a periodiccorrugated potential. Unlike infinite chains where, ac-cording to theory, the classic Aubry transition to freesliding may take place, trapped chains are always pinned.Nonetheless we find that a properly defined static fric-tion still vanishes Aubry-like at a symmetric-asymmetricstructural transition, ubiquitous for decreasing corruga-tion in both straight and zig-zag trapped chains. Dy-namic friction can also be addressed by ringdown oscilla-tions of the ion trap. Long theorized static and dynamicone dimensional friction phenomena could thus becomeexquisitely accessible in future cold ion tribology.

INTRODUCTION

The field of sliding friction has been recently revivedthanks to experimental and theoretical advances espe-cially connected with nanosystems, with brand new op-portunities to grasp the underlying phenomena of thisimportant and technologically relevant area. Unlikemacroscopic friction, where contact irregularities domi-nate, the nanoscale offers perfectly defined crystal latticefacets that can mutually slide. One fundamental piece ofphysics arising in this limit is the possibility of frictionlesssliding when two perfect lattices are mutually incommen-surate – their periodicities unrelated through any ratio-nal number– an idealized situation sometimes referred toas superlubric [1]. Graphite flakes sliding on graphitewere for example shown to be pinned by static friction– the finite threshold force Fs necessary to provoke mo-tion – when aligned, but to turn essentially free slidingonce a rotation makes them incommensurate [2]. Anadditional prediction, rigorously proven for one dimen-sional (1D) infinite harmonic chains sliding in a periodicpotential (the “corrugation” potential, in frictional lan-guage), is that the incommensurate vanishing of staticfriction will only occur so long as the sliding chain issufficiently hard compared to the corrugation strength.

FIG. 1: (a) and (b) sketch of proposed ion trap configurationin a periodic potential; (c) sketch of the straight and zig-zagchain equilibrium geometries determined by different aspectratios of the trapping quadrupolar potential. The lateral zig-zag amplitude has been artificially magnified.

When instead that strength exceeds a critical value, thechain becomes locked, or “pinned”, to the corrugation,thus developing static friction despite incommensurabil-ity. Unlike most conventional phase transitions, this cele-brated Aubry transition [3, 4] involves no structural orderparameter and no breaking of symmetry – just a changeof phase space between the two states, the unpinned andthe pinned one. The concept of pinned and unpinnedstates is by now qualitatively established in the sliding ofreal incommensurate 3D crystal surfaces [5]. Yet, the on-set of static friction in the prototypical 1D chain, wherethe Aubry transition is mathematically established, hasnever been experimentally validated. Even less is knownabout dynamic friction, about which we have no otherinsight than generic linear response [6], and some dataon 3D rare gas overlayers [7] both suggesting a quadraticincrease with corrugation.

Cold atomic ions can form linear chains. Despite theirmutual Coulomb repulsion, ions can be corralled insideeffective potential traps generated by quadrupolar radio-frequency electrodes [8]. The low temperature ion equi-librium geometry is determined by the aspect ratio asym-metry of the confining potential, according to the effec-

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where ri = (xi, yi, zi) is the position of the i-th ion, m itsmass, e its charge, and the three terms represent kineticenergy, Coulomb repulsion, and the effective parabolictrapping potential respectively, with transverse andlongitudinal trapping frequencies ω⊥ and ω‖. As isknown experimentally [11] and theoretically [12, 13], bydecreasing the asymmetry aspect ratio R = ω‖/ω⊥ thetrap potential deforms from spherical to cigar-shaped,and the equilibrium ion geometry changes from 3Dclusters to helices to chains, initially zig-zag and finallylinear and straight along z, as in Fig. 1(c). As manyas a hundred ions may be stabilized in a linear con-figuration, with a few µm ion-ion spacing a0, typicalrepulsion energy e2/a0 ∼ 0.5 K, and temperatures below∼ 1µK. So far, cold ion chains raised interest in viewof promising applications for spectroscopy [14–16], fre-quency standards [17, 18], study and control of chemicalreactions [19], and quantum information [20–23]. We areconcerned here with the possibility that they could beof use in the field of friction. An interesting observationwas made by Garcia-Mata et al. [24], that ion chainsin an additional incommensurate periodic potential(produced, e.g., by suitable laser beams) would, if onlythe spatial inhomogeneity of the trapping potentialcould be neglected, resemble precisely the 1D systemwhere an Aubry transition is expected as a functionof corrugation strength. Unfortunately, the trappingpotential itself introduces an additional and brutal breakof translational invariance, at first sight spoiling thisneat idea. The ion-ion spacing is only uniform near thechain center, increasing strongly towards the two chainends, where it diverges owing to the trap confinement(see e.g., Fig. 1). Incommensurability of the chain withthe periodic potential is therefore lost, and seeminglywith that all possibilities to study phenomena such asthe vanishing of static friction at the Aubry transition,a transition known to depend critically on a preciselydefined value of incommensurability [25].

However, this conclusion is overly pessimistic. In con-texts unrelated to friction, a finite chain in a periodicpotential [26–28] is known to turn the would-be Aubrytransition to a structural phase transition. The transitionsurvives, as we will show, despite the addition of the con-fining trap potential. The distorted state, prevailing atstrong corrugation, exhibits under suitable conditions achange of symmetry and an accompanying displacive or-der parameter, measuring the breaking of inversion sym-metry about the chain center. Simulations carried out bypulling the trapped chain with a longitudinal force (suchas in principle provided by a static external electric field)

FIG. 2: Zero temperature ion chain equilibrium center-of-mass coordinate δzCM versus periodic corrugation amplitudeU0 for the linear chain (red) and zig-zag chain (blue). Posi-tion, time and energies in dimensionless units (see methods).The insets zoom on the linear chain and the zig-zag centralportions, highlighting the symmetry breaking taking place atU0 = Uc and above.

reveal that the distortion transition is associated with acontinuous onset of static friction from zero to finite, thusmimicking at finite size, and despite the trapping poten-tial, the static friction onset theoretically expected at theideal Aubry transition.

In the rest of this work explicitly simulated gedankenexperiments will demonstrate that trapped cold ions mayin fact represent an idesal system for future nanotribo-logical studies addressing both static and dynamic fric-tion. In static friction, the celebrated Aubry transitionwill become experimentally accessible for the first time.In dynamics, the friction rise with increasing corrugationshould become measurable giving new impulse to futuretheory work.

RESULTS

Model and symmetry breaking transition.

We simulate by means of standard (weakly damped)classical molecular dynamics N = 101 ions trapped bythe parabolic potential of Eq.(1), and seek first the static(T = 0) equilibrium geometry. Below a critical aspect ra-tio R0 = 1.18 · 10−3 the equilibrium geometry is a linearchain, while at R = R0 there is a transition to a pla-nar partly zig-zag structure (see Fig. 1 and Supplemen-tary Movie 1), further evolving to a helical structure, andeventually to a 3D cluster when R → 1 [12]. ChoosingR = 0.5 ·10−3 for the frictional simulation of 1D trappedlinear chain, we introduce an additional periodic “cor-rugation” potential U = U0 cos

(2πλ z). To address the

best studied case of Aubry transition, we choose the cor-rugation wavelength λ to be “golden ratio” incommen-

3

surate relative to the chain center ion-ion spacing a0,λ = 2a0/(1+

√5) [3, 4]. Preserving the chain’s inversion

symmetry, the periodic potential phase is chosen so thatthe central ion sits at a potential maximum. At van-ishing external force, the chain equilibrium configurationcan then be followed while adiabatically increasing thecorrugation amplitude U0. As shown in Fig. 2, the chainground state remains fully left-right symmetric right upto a critical corrugation amplitude Uc = 0.01627, (in di-mensionless units, see Methods), where it sharply turnsasymmetric, the center of mass (CM) moving right orleft by an amount δzCM (U0) (see Supplementary Movie2). This transition, similar but not identical to thatknown for free finite chains [26, 28] is the remnant ofthe Aubry transition of the ideal infinite chain. In boththe finite and the infinite chain, the probability to find anion (here the central ion) sitting exactly at a periodic po-tential maximum vanishes above the critical corrugationstrength, where the ion moves out to a minimum distanceψ away from the potential maximum [4, 29]. Unlike theAubry transition that has no structural order parameter,the (T = 0) finite chain transition is structural, char-acterized by a displacive order parameter δzCM , Ising-like since the central ion can identically fall left or rightof the centre. This order parameter grows continuouslyas |U0 − Uc|β at the transition, (see Fig. 2). We findβ = 0.496±0.005 for the trapped linear ion chain – prac-tically coincident with the exponent κ = 0.5 known forfinite chains with short range forces and without trap-ping potential [28]. Moreover, upon reducing the confin-ing shape asymmetry by increasing the aspect ratio toR = 1.2 · 10−3, the chain equilibrium geometry evolvesfrom linear to a partly planar zig-zag (Fig. 2), still sym-metric about z = 0 at U0 = 0. For increasing corru-gation a symmetry-breaking transition similar to that ofthe linear chain takes place here too at Uc = 0.01357 –a smaller value since in the zig-zag the ions are furtherapart – and for such case β = 0.504± 0.005 (see Supple-mentary Movie 3). For both the linear and the zig-zagchain the critical amplitude Uc also depends upon theincommensurability ratio λ/a0 (not shown). While theseresults are valid at T=0, it should be stressed that as inall small size systems, temperature will cancel all sym-metry breaking and order parameters, owing to thermaljumps taking place above the energy barrier ∆ which sep-arates the two opposite Ising like order parameter valleys.However, the mean time lapse between jumps divergesexponentially when T << Tb ∼ ∆/kB , the trap blockingtemperature. Except too close to Uc, ion traps are rou-tinely cooled down to temperatures of order 10−7 (in ourunits e2/d ∼ 6.3 K) way below the blocking temperature(for example Tb ∼ 180µK even for U0 = 0.014, barelyabove the critical Uc = 0.01357). Even at finite temper-atures therefore, the ion chain will still break inversionsymmetry and exhibit a distortive order parameter formacroscopically long times similar to the T = 0 results

detailed above.

Determination of the static friction force.

Thus far the description of the ion trap in a periodiccorrugation, while still free of external forces. To studystatic friction we apply a uniform force F , such as thatof an electric field parallel to the chain axis. The usualprocedure used to determine static friction – the small-est force capable of causing a pinned-to-sliding transition,following which the chain is depinned with unlimited slid-ing – does not work here, since the field-induced cen-ter of mass displacement δzCM (F ) is finite by necessity,owing to the confining trap potential. We circumventthat problem by monitoring another quantity, namelythe restoring force FR that must be applied to the chainin order to shift its CM coordinate back exactly to zero– keeping the chain’s central ion exactly on top of thecorrugation maximum at z = 0. In the infinite free chainthis alternative definition of static friction coincides withthe standard pinned-to-sliding Peierls-Nabarro thresholdforce FPN [4, 30], identifiable with the static friction Fsof the system. In the pinned Aubry state the critical forceFPN is exactly such as to drive to zero the minimum equi-librium deviation ψ of one “kink” away from the near-est corrugation maximum, eventually triggering the freemotion which initiates the pinned-to-sliding transition ofthe whole system [4, 31]. Analogously, for trapped ionsa growing external force F gradually causes the centralatom deviation from the potential maximum to decrease,and eventually to vanish at F = FR. We conclude thatFR measures the static friction even in a trap, whereproper free sliding is impeded. As shown in Fig.3 (inset),for U0 < Uc the central ion of the trap spontaneouslyremains precisely at zero, whence FR = 0. However atU0 > Uc where the force-free displacive order parame-ter δzCM turns spontaneously nonzero, a finite restor-ing force FR must be applied in order to cancel it. Thegrowth of the symmetry restoring force FR with U0 (Fig. 3) is, despite the larger threshold, generally close tothe Aubry static friction expected theoretically for the in-finite ideal chain [1, 3] , when calculated for same chainparameters a0, U0 and golden ratio λ, as those at thetrapped chain center. We have thus established a con-nection between the trapped chain restoring force FR andthe Peierls-Nabarro static friction of the infinite chain,of tribological significance [1]. Numerical differences be-tween the two are of course hardly surprising. The finitesize Na0 cuts off the Aubry critical region where the cor-relation length exceeds this size. For the same reason thechain depinning exponent χ, FR ∼ |U0 − Uc|χ, which wefind to be about 1.85 ± 0.2 differs from the higher ex-ponent ∼ 3 expected for the infinite free chain for shortrange forces [4, 29, 30]. The effective static friction oflinear and zig-zag chains also differ slightly (see Fig. 3),

4

FIG. 3: Effective static friction measured by the restoringforce FR necessary to symmetrize the system about the cen-ter, versus corrugation amplitude U0 for the trapped linearion chain (red) and zig-zag chain (blue). The insets illustratethe procedure used. Black dashed line: calculated theoreti-cal static friction at the Aubry transition of the ideal infinitegolden ratio incommensurate chain. Arrows mark the vanish-ing point of effective static friction in each case.

the softer zig-zag chain easier to pin, owing to its largercompliance than the linear chain. Overall, both casesprovide a measurable realization of the sharp onset ofstatic friction at the Aubry transition, valid beyond thelimitations of a trapped chain.

Dynamic friction.

Besides static friction and its Aubry-like vanishingtransition, ion traps could in addition provide nontriv-ial insight on dynamic friction. The onset of dynamicalfriction in depinned infinite chains is a very subtle eventas discussed in literature [32]; the pinned-to-sliding onsethas been simulated and elegantly described by Braun andcoworkers [4, 31]. In tribology we are primarily interestedin the growth, for incommensurate sliders at finite slidingspeed, of dynamical friction with increasing corrugation.

To simulate the chain dynamical behaviour we initiallydisplace the ions away from their equilibrium configura-tion with a static field, then remove the field and followthe spontaneous “ringdown” damping of the bodily os-cillation of the whole chain in the trap potential. For aninitial large CM displacement of order 10 a0 the chain os-cillation damping is readily observable, and can for weakcorrugation U0 � Uc be measured even in the absence ofan external heat bath. The initial potential energy (hereabout 0.1) is converted to Joule heating of the chain, at aringdown rate which is a direct measure of dynamic fric-tion. Actually, a trapped chain has an intrinsic dynamicfriction even in the absence of corrugation, because of thetrap-potential-induced transfer of kinetic energy from theCM motion to internal chain phonon modes. However,

FIG. 4: Oscillatory kinetic ringdowns for the linear (upperpanel) and the zig-zag (lower panel) chains. The two panelsshow the zCM coordinate as a function of time for differentvalues of U0 < Uc: red U0 = 0, green U0 = 0.0001, blueU0 = 0.0002, magenta U0 = 0.0003 and cyan U0 = 0.0004.

this effect is small; the conversion and the correspondingdynamic friction is strongly enhanced by the corruga-tion potential, as shown in Fig. 4 and in SupplementaryMovies 4 and 5 (for the linear and zig-zag configurationsof the chain, respectively). Simulation of the chain oscil-lations yields a ringdown amplitude decay for both linearand zig-zag chains, of the form A0exp − [t/τ ]. We findthat the lifetime τ(U0) strongly decreases for increasingU0. The corresponding increase of dynamic friction FD,directly proportional to τ−1, is to a good approximationquadratic, FD ∼ a + bU2

0 for weak corrugation (Fig. 5).A quadratic increase is just what the linear response the-ory predicts based on Fermi’s golden rule [6, 33]. It alsoagrees with experimental Quartz Crystal Microbalancemonolayer friction data of Coffey and Krim [7]. Thereis in the literature no dynamic friction theory for theinfinite linear chain to be used for comparison with thetrapped chain results of Fig. 5; the present results cantherefore be considered a first exploratory result in thisdirection. The exceedingly small values of a speak forthe strong harmonicity of the chain, whose CM motioncan only with great difficulty excite the higher frequencyintra-chain phonons in the absence of corrugation. Thequadratic increase with U0 reflects the effective open-ing of anharmonic excitation channels through Umklappscattering. Larger oscillation amplitudes could not bereadily simulated because the Joule heat liberated wouldin that case be excessive and would destroy the chainin the lack of some external dissipation mechanism. Afully nonlinear behavior of dynamic friction would be ex-pected to emerge in that case, with outcomes stronglydependent upon the heat disposal mechanism.

5

FIG. 5: (a) Inverse decay time τ−1 of CM ringdown oscilla-tions, measuring the dynamic friction of the ion chain, linear(red) and zig-zag (blue). The inset illustrates the procedureused to obtain it. Continuos lines represent fits of the forma + bU2

0 (a = 0.014 and b = 867.82 for the linear chain anda = 0.0 and b = 250.31 for the zig-zag).

DISCUSSION AND CONCLUSIONS

The remarkable static and dynamic friction propertiesemerging for trapped ion chains seem well amenable toexperimental study. The exceedingly small values of thedistortion δzCM necessary to establish an accurate valuefor the effective static friction FR will require a fine tun-ing of corrugation and applied force. Upon increasingthe force however, the instant F = FR is reached andthe symmetric position δzCM = 0 is attained, the chainwill spontaneously jump over the barrier to the oppo-site Ising order parameter valley. By e.g., alternating intime rightward and leftward forces, the effective staticfriction FR could become readily detectable in experi-ment as the threshold force magnitude for the chain’sCM to sway visibly leftward and rightward. The presentstatic friction study is conducted at zero temperature,ignoring quantum effects, without external sources of ei-ther heating, cooling or damping, and assuming that asufficiently strong effective periodic corrugation poten-tial can be laser generated. While each of these pointsrequires careful consideration, none, as will be discussedbelow, is in our view fatal to the results presented forstatic friction. The proposed dynamic friction experi-ment would heat up the trapped ion chains, and causetheir destruction once the Joule heating is excessive. Weverified however that even at a temperature of ∼ 0.55K, orders of magnitude higher than the effective chainDebye temperature of ∼ 145µK (considering Mg+ ionsseparated by a0 = 5µm), the chain is still stable in thestrong trapping potential.

In conclusion, the field of cold atoms has had a tremen-dous development, recently covering important interdis-ciplinary areas such as Anderson localization [34]. Now

the potential tribological impact of cold ion results couldalso be significant. The onset of static friction is a richproblem of current interest [35, 36], and so is the dynamicfriction growth with corrugation[7]. Most importantly,ion chains represent a new clean design, free of most com-plications of real crystal lattices and with many more con-trollable parameters for a study of basic nanofrictionalproperties.

METHODS

Simulation Details. The equations of motion de-rived from (1) turn dimensionless by defining the quan-tity d3 = e2/mω2

⊥, and expressing distance in units ofd, time in units of 1/ω⊥ and the mass in units of m.With this definition, energy and force are also dimension-less in units of e2/d and e2/d2 respectively. Simulationsare carried out using classical molecular dynamics, whennecessary with a weak frictional damping, and mostlyat zero or low temperature. The equations of motionhave been integrated using a velocity-Verlet algorithmwith a time step ∆t = 5 · 10−3 time units. The thresh-old trap potential asymmetry R0 for linear/zig-zag struc-ture transition is determined with adiabatic increments∆R = 10−4 starting from R = 0.5 · 10−3. The criti-cal periodic corrugation potential amplitude Uc for thesymmetric-asymmetric transition of linear and zig-zagchains is obtained by adiabatic increments ∆U0 = 10−5

starting from an initial value U0 = 0.012 for the zig-zagconfiguration and U0 = 0.015 for the linear chain. Inall the adiabatic increment procedures a small viscousdamping −γvi, with γ = 0.01, is applied to all ions inorder to eliminate the excess energy. With our choice ofγ a total time of 2500 between successive increments isenough to damp all the excess energy away.

Physical parameters and orders of magnitudea)Temperature and quantum effects. The present

study is conducted conventionally at zero temperature,ignoring quantum effects, without external sources ofheating, cooling or damping, and assuming that a suf-ficiently strong periodic potential can be laser generatedfor the transition to take place at the given incommensu-rability provided by the laser field. Even though temper-ature is in reality finite (typically of order 100 nK), thepreviously reported comparison between left and rightchain distortion energies with kBT implies a symmetry-broken state lifetime much longer than any measurementtime, justifiying the zero temperature treatment, wheredata and their significance are clearer. In this regimemoreover, quantum zero-point effects could in principlealter some of the details close to the depinning thresh-old, but are otherwise not expected to affect the overallfriction picture, both static and dynamic.

b) Corrugation potential magnitude. To produce theeffective periodic corrugation potential, the wavelength

6

ξ of a laser light must fit an electronic excitation in thespectrum of the chosen ion species (e.g. 280 nm for Mg+,729 nm for Ca+ or 399 nm for Yb+), with a typicala0 = 5µm the golden mean incommensurate periodic po-tential wavelength is λ = 3000nm � ξ, nevertheless thedesired λ value for the effective periodic potential can beobtained by laser beam crossing at a chosen angle [24].Experimentally, it is mandatory to realize a sufficient cor-rugation amplitude U = U0 for the Aubry transition tooccur. Assuming for example Mg+ ions and golden ra-tio incommensurability, the critical amplitude of the ef-fective periodic potential Uc requires a laser intensity ofsomeKW/m2, which seems entirely within reach. On theother hand, the Aubry pinning transition becomes, as iswell known, easier with any chain-corrugation incommen-surability different from the golden ratio [25], whereby ingeneral the transition would occur with even with weakercorrugation amplitudes than U0.

c) Phase misalignment effects. We assumed so far thepossibility to position the corrugation potential maxi-mum precisely at the minimum of the harmonic trap, afeat which may be experimentally difficult. Slight shiftsof the corrugation potential phase away from that pointwould however simply smear the Aubry pinning onset inthe same way a weak symmetry breaking field transitionwould affect a phase transition.

d) Chain heating during ringdown. In the oscillationringdown simulations, the initial potential energy differ-ence 8.9, due to the chain displacement of 10 a0, is grad-ually converted to heating of the chain. With the mass ofMg+, the final temperature reached is approximately 0.55K, orders of magnitude higher than the effective chainDebye temperature of ∼ 145µK. Despite that, we ver-ified by direct simulation that at this temperature thechain is still stable in the strong trapping potential.

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AcknowledgementsThis work is part of Eurocores Projects FANAS/AFRIsponsored by the Italian Research Council (CNR), andof FANAS/ACOF. It is also sponsored in part by theItalian Ministry of University and Research, throughPRIN/COFIN contracts 20087NX9Y7 and 2008y2p573.

ET thanks M. Inguscio for telling him about cold iontraps.

Author contributionsA.B. and A.V. carried out the MD simulations. E.T.coordinated the work. All authors contributed to thenumerical data analysis, paper writing and revising.

Additional informationSupplementary information accompanies this paper onwww.nature.com/naturecommunications.

Correspondence and requests for materials should beaddressed to E.T., tosatti@sissa.it.

Competing financial interests: The authors declare nocompeting financial interests.

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