Spontaneous nucleation of structural defects in inhomogeneous ion chains
Gabriele De Chiara, Adolfo del Campo, Giovanna Morigi, Martin B. Plenio, Alex Retzker
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] D ec Spontaneous nucleation of structural defects ininhomogeneous ion chains
Gabriele De Chiara , , Adolfo del Campo , , GiovannaMorigi , , Martin B. Plenio , , Alex Retzker , Grup d’ `Optica, Departament de F´ısica, Universitat Aut`onoma de Barcelona,E-08193 Bellaterra, Spain F´ısica Te`orica: Informaci´o i Processos Qu`antics, Universitat Aut`onoma deBarcelona, E-08193 Bellaterra, Spain Institut f¨ur Theoretische Physik, Albert-Einstein Allee 11, Universit¨at Ulm,D-89069 Ulm, Germany QOLS, The Blackett Laboratory, Imperial College London, Prince ConsortRoad, SW7 2BW London, UK Theoretische Physik, Universit¨at des Saarlandes, D-66041 Saarbr¨ucken,Germany
Abstract.
Structural defects in ion crystals can be formed during a linearquench of the transverse trapping frequency across the mechanical instability froma linear chain to the zigzag structure. The density of defects after the sweep canbe conveniently described by the Kibble-Zurek mechanism. In particular, thenumber of kinks in the zigzag ordering can be derived from a time-dependentGinzburg-Landau equation for the order parameter, here the zigzag transversesize, under the assumption that the ions are continuously laser cooled. In alinear Paul trap the transition becomes inhomogeneous, being the charge densitylarger in the center and more rarefied at the edges. During the linear quench themechanical instability is first crossed in the center of the chain, and a front, atwhich the mechanical instability is crossed during the quench, is identified whichpropagates along the chain from the center to the edges. If the velocity of thisfront is smaller than the sound velocity, the dynamics becomes adiabatic even inthe thermodynamic limit and no defect is produced. Otherwise, the nucleation ofkinks is reduced with respect to the case in which the charges are homogeneouslydistributed, leading to a new scaling of the density of kinks with the quenchingrate. The analytical predictions are verified numerically by integrating theLangevin equations of motion of the ions, in presence of a time-dependenttransverse confinement. We argue that the non-equilibrium dynamics of an ionchain in a Paul trap constitutes an ideal scenario to test the inhomogeneousextension of the Kibble-Zurek mechanism, which lacks experimental evidence todate.PACS numbers: 03.67.-a, 37.10.Ty pontaneous nucleation of structural defects in inhomogeneous ion chains Figure 1.
Schematic representation of the “linear to zig-zag” phase transitionin a homogeneous ion chain in two dimensions illustrating the structural phasesinvolved. The high-symmetry phase corresponds to a linear chain, whilethe broken-symmetry phase is characterised by a doubly degenerate zig-zagconfiguration.
1. Introduction
Ion crystals in Paul or Penning traps represent a prominent example of a self organizedsystem that is amenable to accurate experimental characterization and manipulation[1]. Ions confined by means of static and radio-frequency electromagnetic potentialsreach crystallization when laser cooled. Crystals made up from tens to millions of ionshave been observed both in Paul traps [2, 3, 4, 5] and in Penning traps [6]. Differentstructures can be realized by varying the particle density or the trap anisotropy. InRef. [2] quasi one-dimensional structures have been experimentally characterized forfirst time. Here, the first three structures encountered when decreasing the transverseconfinement are a linear chain, a planar zigzag structure, and a helicoidal arrangementof the ions about the trap axis. Phase transitions separating two structures are usuallydiscontinuous. In this respect, the transition from the linear chain to the zigzagstructure constitutes an exception. In fact, it has been demonstrated numerically [7, 8]and later analytically using Landau theory [9] that the transition from a linear chainto the zigzag is of second order. The transition can be induced either by increasing theion density or by decreasing the transverse confinement ν t so that their value exceedsor is below, respectively, a certain critical value determining the mechanical instability.Recently, the transition linear-zigzag chain has been suggested as a test-bed for many-body quantum effects and the creation of double-well potentials [11], and is well suitedfor the distinction between the nucleation of defects and quasiparticles in a symmetrybreaking scenario [12].The aim of this paper is to study the out-of-equilibrium dynamics of a linearchain of trapped ions when ν t is lowered in time from above to below the mechanicalinstability separating the linear from the two-dimensional zigzag configuration,illustrated in Fig. 1. Due to the finiteness of the sound velocity, space-like separatedregions may develop one of the two possible zigzag orderings: odd ions up and even pontaneous nucleation of structural defects in inhomogeneous ion chains Figure 2.
Ion chain in a zigzag configuration exhibiting structural defects. Theions are confined in a ring trap, and we report here the distribution of charges intwo dimensions along the ring mapped to a line for clarity. The solid line joiningthe ions serves as a guide to the eye. In the example reported there are 4 defects.According to the Kibble-Zurek mechanism, the average size of the domains isgiven by the correlation length at the freeze-out time, ˆ ξ (see text for details). ions down or odd ions down and even ions up as depicted in Fig. 2. These regions arethe analogs of magnetic domains in a ferromagnetic material and the interface betweenthese domains is a structural defect. The classical and quantum properties of thesedefects and their possible use for quantum information processing were studied in [13].Defects formation can be understood from simple statistical mechanics considerations.If the rate of change of ν t is larger than the relaxation rate of the crystal, the latterdoes not have enough time to relax to the minimum energy configuration, which ischaracterized by a perfectly ordered zigzag structure, thus exhibiting no defects andcorresponding to the state that would have been obtained after a perfect adiabatictransition. In principle, for an infinite system whose phase transition is described byLandau theory [14, 15], the dynamical relaxation time scale diverges at the criticalpoint and therefore no matter how slowly ν t changes, there will always be proliferationof defects. This is the celebrated Kibble-Zurek mechanism (KZM) which can generallyaccount for the nucleation of topological defects in scenarios of spontaneous symmetrybreaking [16, 17]. Moreover, Zurek predicted the scaling of the number of defects as afunction of the rate of passing through the phase transition [17]. The KZM scenarioand Zurek’s prediction for the scaling of the number of defects have been verified in avariety of systems numerically [18] and experimentally [19]. Recently, the KZ scalingprediction has also been extended to quantum phase transitions [20].In this work we extend the results in our proposal [23] and study the productionof kinks in the linear to zigzag transition in ion crystals by deriving a time dependentGinzburg-Landau theory for the transition. With this result, we determine the scalingof the number of defects with the rate of change of the transverse frequency whichfollows from the KZM and compare it with numerical simulations. In a ring trap whenthe interparticle distance is uniform, one recovers the standard KZM. By contrast, ina linear Paul trap the density of ions is inhomogeneous, higher at the center and morerarefied at the edges. When decreasing in time the transverse trap frequency, thevalue of the mechanical instability is first crossed in the center of the chain, and onecan identify a front crossing the transition from the linear to the zigzag phase whichmoves from the center towards the edges, and whose velocity depends on the rate ofthe quench. If the velocity of this front is smaller than the sound velocity, at whicha perturbation propagates along the chain, the dynamics becomes adiabatic. Thisresult holds even in the thermodynamic limit. Otherwise, the nucleation of defects ispartially suppressed with respect to the homogeneous case and a novel scaling of thedensity of kinks with the rate of the transition emerges in the inhomogeneous extensionof KZM [21, 22]. The whole body of experimental results in [19] has been aimed pontaneous nucleation of structural defects in inhomogeneous ion chains
2. Ginzburg-Landau equation in presence of laser cooling for the orderparameter
In this section we start from the theory presented in [24, 9], and derive a Ginzburg-Landau equation for the order parameter of the linear-zigzag structural transition.The order parameter corresponds to the position offset of the ions from the trap axis.For a large number of ions, using a local density approximation we can approximate thecrystal as a continuum, so that the order parameter is a field. The theory is extendedto the case in which the crystal motion is laser cooled, and the coupling to an externalreservoir is described using the theoretical model developed in [25]. Defect formationby quenching the control parameter, here the transverse trap frequency, across thestructural instability is studied by using the corresponding Langevin equation for theorder parameter. This theoretical model allows us to estimate the density of defectsat the end of the quench.
The system we consider consists of N ions of mass m , charge Q and coordinates r n = ( x n , y n , z n ) which are confined in a quasi one dimensional trap with tightharmonic confinement at frequency ν t in the plane yz . In this section we assumea ring trap with large radius R , so that we may impose periodic boundary conditionsalong the x -axis. The Lagrangian describing the dynamics of the ions is L = T − V , (1)where the kinetic and potential energies take the form, respectively, T = 12 m X n ˙ r n , (2) V = 12 mν t X n ( y n + z n ) + Q X n = n ′ | r n − r n ′ | . (3)At sufficiently low temperatures and sufficiently large transverse confinement the ionscrystallize around the stable equilibrium points of the potential V , which are alignedalong the x -axis, r n = ( na, , a the interparticle distance in the x -directionsuch that a = 2 πR/N .The stability of the linear chain along the x axis requires a transverse trapfrequency exceeding a threshold value ν ( c ) t , which scales with the characteristicfrequency ω = p Q /ma . At ν ( c ) t the configuration has a structural instability, suchthat for ν t < ν ( c ) t the ions are organized in a planar structure with equilibrium positions r n = ( na, ( − n ( b/
2) cos θ, ( − n ( b/
2) sin θ ) with θ ∈ [0; 2 π ] the angle between thecrystal plane and the plane xy . The structure has the form of a zigzag line, which joinsthe charges along the plane, with b the transverse size of the zigzag. An appropriatethermodynamic limit can be defined, letting the number of ions N → ∞ whilekeeping the value ν ( c ) t and the characteristic frequency ω fixed. This corresponds pontaneous nucleation of structural defects in inhomogeneous ion chains a or equivalently the charge density[24]. One finds that the structural instability is a second-order phase transition,with control field ν t (or alternatively, a ) and order parameter b [9]. In particular,the critical value of the transverse frequency in the thermodynamic limit is givenby ν ( c ) t = ω p ζ (3) / . . . . ) ω , ζ being the Riemann-zeta function. Fromthe Landau theory of the structural phase transition in Ref. [9], one can developa continuum model for the field describing the order parameter applying standardassumptions, thereby obtaining the Ginzburg-Landau equation. We sketch below theassumptions made, in order to clarify the range of validity of the model which formsthe basis of this work.In the linear chain, in the harmonic limit, transverse and axial modes aredecoupled. For periodic boundary conditions the modes have a well defined quasi-momentum k which takes values within the Brillouin zone. For N ions, the transverse,normal modes at wave vector k are the real and imaginary part of the modeΨ σk = 1 √ N X n e i kna σ n (4)where σ n = y n , z n is the transverse displacement of the ion at the equilibrium position r n and k ∈ [0 , π/a ]. Using decomposition (4), as shown in Ref. [9], where all thedetails of the derivation are provided, the transverse potential, obtained by expandingEq. (3) around the equilibrium positions, takes the form V ≃ V (0) + V (2) + V (3) + V (4) (5)where the label indicates the order of the expansion. In particular, V (2) = 12 X k ∈ (0 ,π/a ] X σ = y,z mβ ( k ) (cid:0) Re { Ψ σk } + Im { Ψ σk } (cid:1) (6)with β ( k ) = ν t − ω X j> j sin (cid:18) jka (cid:19) (7)and where we omitted to write the potential term for the axial modes. The terms V (3) and V (4) correspond to third- and fourth-order expansion in the fluctuations aroundthe equilibrium position, and contain the coupling between the axial and radial modes.We now make the assumption that the ions are pinned in the axial direction and canonly oscillate in the transverse direction and discard the coupling to the axial modes.In this regime, it was found in Ref. [9] that V (3) = 0 and the fourth order term reads: V (4) = X k + k + k + k =0 X σ,τ = y,z A ( k , k , k , k ) ψ σk ψ σk ψ τk ψ τk (8)with A ( k , k , k , k ) = 32 N Q a X m> m Y p =1 sin mk p a β ( k ) is minimum at k = π/a , correspondingto the so-called zigzag mode. The critical value of the transverse trap frequency ν ( c ) t is found from the relation β ( k ) = 0. When ν t < ν ( c ) t the linear chain is unstable, andthe ions order in a zigzag structure across the trap axis. pontaneous nucleation of structural defects in inhomogeneous ion chains ν ( c ) t , the mode of the linear chainwhich first becomes unstable and determines the equilibrium positions of the zigzagstructure (soft mode) is the transverse mode of the linear chain with the shortestwave length (zigzag mode), at wave vector k = π/a . When ν t is sufficiently close tothe critical value ν ( c ) t , an effective potential can be derived for the transverse normalmodes Ψ σ ˜ k with wave vector ˜ k = k − δk , such that aδk ≪
1. The effective potential iscomposed by a quadratic component, given in Eq. (6), with coefficient β (˜ k ), Eq. (7)such that [10] β (˜ k ) (cid:12)(cid:12)(cid:12) ˜ k = k − δk ≈ δ + h δk , (10)where δ = ν t − ν ( c )2 t and which can take negative values depending on the value of ν t .Within the same assumptions, we can approximate: A ( k , k , k , k ) ≈ N Q a X m> m − = m N A , (11)where A = (93 ζ (5) / ω /a and we obtain for the fourth order term: V (4) = m N A ′ X ˜ k +˜ k +˜ k +˜ k =0 X σ,τ = y,z ψ σ ˜ k ψ σ ˜ k ψ τ ˜ k ψ τ ˜ k . (12)Note that the prime index in the sum over the quasimomenta refers to the fact thatthe sum is restricted to the quasi momenta ˜ k j = k ± δk , with δka ≪
1. The potentialform V in Eq. (5) is valid at second order in the small parameter δka ≪
1, when theamplitude of the transverse oscillations is much smaller than the interparticle distance a . The limit aδk ≪ x n → x , where x is a continuous variable, and( − n σ n → ψ σ ( x ), where ψ σ ( x ) is now the position-dependent order parameter, whichis related to ψ σ ˜ k through the Fourier transform ψ σ ˜ k = 1 √ N Z d xa e − i δkx ψ σ ( x ) , σ = y, z (13)while the factor 1 /a in the right-hand side gives the density of states. For a largenumber of ions the discrete sum over ˜ k vectors in the potential energy becomes anintegral according to the rule P ˜ k → R d( δk ) N a/ (2 π ). The second order potentialterm becomes: V (2) = m X σ Z d ( δk ) dx dx πa ( δ + h δk ) e − iδk ( x − x ) ψ σ ( x ) ψ σ ( x ) , (14)and using the integral representation of the Dirac function: Z d ( δk ) e iδkx = 2 πδ ( x ) (15)we obtain: V (2) = m X σ Z dxa h δψ σ ( x ) + h ( ∂ x ψ σ ( x )) i . (16) pontaneous nucleation of structural defects in inhomogeneous ion chains
7A similar calculations leads to: V (4) = m A X σ,τ Z dxa ψ σ ( x ) ψ τ ( x ) . (17)Finally, we obtain the Lagrangian L = R d x L ( x ) where L ( x ) is the Lagrangiandensity and reads L ( x ) = 12 ma X σ h ( ∂ t ψ σ ( x )) − h ( ∂ x ψ σ ( x )) − δψ σ ( x ) − A ψ σ ( x ) X τ ψ τ ( x ) i . (18)We now discuss individually the parameters entering Eq. (18). The parameter δ = ν t − ν ( c )2 t , (19)determines whether the ground state of the system is a linear chain or a zigzag,depending on its sign. The parameter h = ω a √ log 2 is a velocity, and determinesthe speed with which a transverse perturbation propagates along the chain. Finally,the parameter A is positive and determines the value of the order parameter when δ < ψ σ [ δ + 2 A ( ψ y + ψ z )] = 0 . (20)Equation (20) always admits the solution ψ σ = 0, corresponding to all the ions layingon the x axis. This solution is clearly stable only for δ >
0, i.e. in the linear chainphase. For δ < ̺ = ± p − δ/ A ,with ̺ = p ψ y + ψ z [9]. These solutions correspond to zigzag structures with thesame amplitude but laid on different planes.From the Ginzburg-Landau equation one can derive the critical exponent of thecorrelation length close to the critical point and at T = 0. In particular, under theassumption of a small spatial and static deformation of the order parameter at acertain position, the field autocorrelation function at distance x decays exponentiallyas ∼ exp( −| x | /ξ ) [14], where ξ = h/δ / is the correlation length, such that ξ ∼ a ω ν ( c ) t (cid:16) ν t − ν ( c ) t (cid:17) / (21)and it diverges as ξ ∼ ( ν t − ν ( c ) t ) − / with the corresponding critical exponent is 1 / We now consider the physical situation, in which the crystal motion is laser cooled.For the moment, we consider the discrete distribution of ion charges forming a linearchain (so that the effective potential for the transverse modes is essentially quadratic).For Doppler cooling, an effective equation for the crystal modes can be derived, which pontaneous nucleation of structural defects in inhomogeneous ion chains ∂ t ψ σk + η σ,k ∂ t ψ σk + β ( k ) ψ k = ε σk ( t ) , (22)which is here reported for the transverse modes. The damping rate η σ,k dependson the corresponding mode frequency and it can take different values dependingon the propagation direction of the cooling lasers. The scalar ε σk ( t ) represents thecorresponding Langevin force, such that its moments fulfill the relations h ε σk ( t ) i = 0 , (23) h ε σk ( t ) ε σk ′ ( t ′ ) i = 2 η σ,k κ B T δ k,k ′ δ ( t − t ′ ) /m, (24)where κ B is the Boltzmann constant and T is the temperature which determines thethermal state due to cooling. Equation (22) relies on the assumption that the dynamicsof the electronic degrees of freedom is much faster than that of the motional degrees offreedom so they can be adiabatically eliminated from the equations for the degrees offreedom of the crystal normal modes, which is generally true when all ions are drivenby the cooling laser [26].In the following we focus on the situation in which the trap frequency along the z -axis is much larger than that along the y -axis. In this limit we drop the σ -label, andwrite an equation for the transverse modes along y and close to the instability point,at the transition from a linear chain to a zigzag structure in the xy plane. Here, theparameters η k and ε k are slowly varying and can be assumed to be constant. Underthese assumptions, the Euler-Lagrange equation for ψ = ψ y ( x ) obtained from Eq. (18)reads ∂ t ψ − h ∂ x ψ + η∂ t ψ + δψ + 2 A ψ = ε ( t ) . (25)Note that h can be obtained from the group velocity s = h k √ δ + h k at the critical point δ = 0. Damping introduces a characteristic relaxation time scale τ , with which thesystem reaches equilibrium. Sufficiently close to the transition point, η ≫ δ and τ ≈ η/δ, (26)see [14] and the appendix. Therefore the relaxation time τ diverges when thetransverse frequency ν t approaches the critical value ν ( c ) t as τ ∼ ( ν t − ν ( c ) t ) − . Thesame scaling is valid also in the zigzag phase δ <
0. The quantities ξ and τ and theirscaling with δ are crucial for determining the scaling of the defect production duringa quench of the transverse frequency as explained in the next section. Within the Ginzburg-Landau description we now assume that the transverse trapfrequency ν t undergoes a change in time in the interval [ − τ Q , τ Q ], such that ν t = q ν ( c )2 t + δ ( t ) , (27)and δ ( t ) = − δ tτ Q , (28)with δ > δ ≪ ν ( c ) t . The transverse trap frequency value is swept throughthe mechanical instability of the linear-zigzag chain, such that δ ( − τ Q ) = δ > δ ( τ Q ) = − δ <
0. Correspondingly, the equation for the field now reads ∂ t ψ − h ∂ x ψ + η∂ t ψ + δ ( t ) ψ + 2 A ψ = ε ( t ) , (29) pontaneous nucleation of structural defects in inhomogeneous ion chains t the ground state and thermodynamic properties at the phasetransition are well defined. The time-dependence of δ ( t ) gives now a non-equilibriumproblem. In principle, this should be solved considering the differential equation withtime-dependent parameters. Nevertheless one might reach an understanding of theessential features by separating it into two domains. In the first domain, one can usethe equilibrium solution provided that one may assume the adiabatic approximation,namely, | δ ( t ) / ˙ δ ( t ) | ≫ τ ( t ), where τ ( t ) is the relaxation time for the equilibriumsituation at the given value δ = δ ( t ). The second domain is known as the impulseregion , where as a result of the critical slowing down the order paramer is assumedto be frozen, ceasing to react to the external quench [17]. This stage is separatedfrom the adiabatic regime by the freeze-out time scale ˆ t , i.e., a time scale in which theadiabatic condition ceases to be valid. This time scale can be estimated by setting | δ (ˆ t ) / ˙ δ (ˆ t ) | = τ (ˆ t ), which clearly sets a lower limit to ˆ t . On this time-scale the orderparameter is correlated over domains of characteristic length ˆ ξ = ξ ( δ (ˆ t )) given by thecorrelation length at the freeze-out time. The density of defects d (number of defectsover the total number of ions) can then be estimated by the relation d ∼ / ˆ ξ .We now determine the density of defects for the chosen time variation of theparameter δ , Eq. (28). In this case, an estimate for the freeze-out time ˆ t is given bythe instant of time ˆ t at which the rate of change of δ equals the relaxation time: (cid:12)(cid:12)(cid:12)(cid:12) δ (ˆ t )˙ δ (ˆ t ) (cid:12)(cid:12)(cid:12)(cid:12) = ˆ t = τ (ˆ t ) (30)Equation (30) allows for simple solutions in two specific limits, which will be analyzedin this paper. In the limit, in which q δ (ˆ t ) ≪ η or equivalently η ≫ δ /τ Q , thedamping overcomes the oscillations associated with the frequency at the freeze-outtime scale, q δ (ˆ t ). Note that this condition is imposed at the freeze-out-time beforecrossing the transition. It is then when the properties of the broken symmetry phaseare determined. We denote this regime by “overdamped limit”, following the definitionintroduced in Ref. [28]. In this case, using Eqs. (28) and (26) one finds ˆ t = √ τ τ Q with τ = η/δ , which sets δ (ˆ t ) = p ηδ /τ Q . The density of defects in the overdampedlimit takes the form d o ∼ ξ o = 1 a ω (cid:18) δ ητ Q (cid:19) / . (31)where we used Eq. (21), assuming that the temperature is sufficiently low to neglectfinite temperature effects. Another limit, which we will consider, is the one in which q δ (ˆ t ) ≫ η , namely, the Ginzburg-Landau equation is the one of an underdampedoscillator at the freeze-out time. We denote this regime by “underdamped” limitaccording to Ref. [28]. In this case, ˆ t = ( τ τ Q ) / with τ = 1 / √ δ , δ (ˆ t ) = ( δ /τ Q ) / ,and the density of defects is given by d u ∼ ξ u = 1 a ω (cid:18) δ τ Q (cid:19) / . (32)The algebraic scaling in Eq. (32) coincides with the one found in Ref. [28]. Thedifferent power scaling of δ , with respect to Ref. [28], is due to the fact that inthe present expressions we have written explicitly the proportionality factors (whichin [28] are summarized in the parameter ξ ). We finally note that, the fact that the pontaneous nucleation of structural defects in inhomogeneous ion chains -14 -12 -10 -8 -6 log[1/ ντ Q ] -4.5-4.0-3.5-3.0 l og [ d o ] HKZM
Figure 3.
Scaling of the density of defects for in an ion chain in a ring trap asa function of the rate of quenching the frequency of the transverse confinement.The fit is log d = − .
889 + 0 .
239 log r (regression coefficient 0 . N = 50,and it involves average over 200 realizations. The parameters are η = 185 ω , ν ( c ) t = 2 . ω , δ = 0 . ω , ǫ = 2 . × − aω / . l = Na is the length of the trapand ǫ is the parameter which controls the strength of the noise which is chosen byadding the following term to the equation of motion ǫN (0 , √ ∆ t, where ∆ t is thetime step. This noise corresponds to a temperature of k B T = 6 . × − ma ω η ,which in our case corresponds to a temperature of the order of 20 phonons. Notethat the ratio of the number of defects over the total number of ions N is relatedto the spatial density of defects by a factor l/N . density of defects exhibits a power-law behaviour as a function of τ Q is obviouslydue to the specific choice of the quench as a function of time and of the consideredregime (determined by the ratio η/δ (ˆ t )). The result can be understood as the domainsize, determining the density of defects is given by the speed at which a perturbationpropagates along the chain, aω , multiplied by the relaxation time scale.The scenario just discussed describes the dynamics of the structural phasetransition for ions placed on a ring with equilibrium positions r n , where theinterparticle spacing is homogeneous. We now verify numerically the HKZM predictionby integrating numerically the Euler-Lagrange equations obtained by minimizing theLagrangian in Eq. (1) for a finite number of ions, and whose motion is damped inpresence of a Langevin force. Assuming a ring confinement and pinning of one ion,the relevant degrees of freedom in the critical region are the transverse coordinates y n . At t = 0 the ions are in a linear chain configuration with h y n i = 0. The systemis then driven by a linear quench of the form given in Eq. (28), and the density ofdefects d is computed at some asymptotic time, after which d remains practicallyconstant. A typical evolution instance with 4 defects is shown in Fig. 2. These defectsresemble the non-massive kinks of the Frenkel-Kontorova model with a transversaldegree of freedom that can be described by an effective φ theory for the translationaldisplacement [29]. The classical and quantum behavior of these kinks was studied in[13]. In the numerics, for each set of parameters, the density of defects is averaged overmany different realizations such as the one in Fig. 2. We then study the dependence of pontaneous nucleation of structural defects in inhomogeneous ion chains d on τ Q . A least-squares fit to the list of data is used to compute the scaling exponent.Figure 3 shows the scaling of the density of defects d as a function of the frequencyquench time τ Q in the overdamped regime, in agreement with the homogeneous KZM.Deviations from KZM occur for fast quenches, for which the density of defects saturatesas a result of the interactions between different kinks. In addition, finite size effectslimits also the validity of the KZM scaling for very slow quenches that can lead to anadiabatic dynamics. Nucleation of defects is expected to be suppressed whenever thecorrelation length at the freeze-out time scale exceeds the size of the system, ˆ ξ > N a ,both in the underdamped and overdamped regimes.
3. Ion chain inside a linear Paul trap: inhomogeneous effects
The standard scaling of topological defects for homogeneous phase transitions shouldbe revised whenever the quench is local or there is a spatial dependence of the relativefrequency δ [22, 21]. In this section we discuss the dynamics of the structural phasetransition for ion chains with open boundaries, for which both transverse and axialtrapping potentials are harmonic. The ion chain in a linear Paul trap is characterizedby a linear charge distribution which is inhomogeneous, with ion density increasingtowards the center of the trap [30]. This leads to a stronger Coulomb repulsion forthe ions near the center of the chain. Correspondingly, the radial short-wavelengthmodes have amplitude which is larger at the center, and vanishes at the edges [24].The structural instability is hence first visible at the center of the trap, such thatat the critical value of the transverse frequency the central ions are displaced fromthe trap axis forming a zigzag chain, as illustrated in Fig. 4. At lower values of thetransverse frequency the number of ions which are displaced from the center increasestowards the edges of the chain, until all the chain is in a planar configuration [7]. Fora sufficiently long chain, one could associate with this behaviour a spatially-dependentcritical frequency, determining the transition to the zigzag, such that it is largest atthe center and smaller at the edges. Correspondingly, if a quench of the transversefrequency is applied, the transition point is crossed at different instant of times alongthe chain (from the center to the edges). The velocity, at which this front propagates,is the so-called front velocity [22, 23]. In this case, the ratio between the front velocityand the sound velocity determines nucleation of defects. In the following, we computethe number of kinks formed in an ion crystal in a linear Paul trap in such scenario.The mechanism resembles the formation of solitons in a cigar-shaped Bose-Einsteincondensate recently discussed by Zurek [21].More precisely, we now consider the case, in which the ions are also trapped inthe x -direction by a harmonic potential of frequency ν , such that the aspect ratio ν/ν t is sufficiently small to allow for low-dimensional crystalline structures. We will nowconsider the case in which the number N of particles is finite, but it is sufficientlylarge to allow for a local density approximation. In this limit and away from the chainedges the linear density n ( x ) is well approximated by the function [30] n ( x ) = 34 NL (cid:18) − x L (cid:19) , (33)with L the half-length of the chain and x the distance from the center. Within thistreatment, the interparticle spacing a ( x ) is a slowly-varying function of the position,such that a ( x ) = 1 /n ( x ). In the thermodynamic limit, in which a (0) is fixed as thenumber of particles goes to infinity, N → ∞ , one recovers the statistical mechanics pontaneous nucleation of structural defects in inhomogeneous ion chains Figure 4.
Sequence of classical ground states of a harmonically trapped 1Dion crystal for decreasing values of the transverse trap frequency ν t (from topto bottom), across the mechanical instability from a linear to a zigzag chain.Due to the harmonic axial potential, the density of ions in the center is larger.Correspondingly, this is the first region where the zigzag structure is formed whendecreasing ν t . and dynamical properties of the ion chain in a infinite-radius ring trap [24, 9]. For N finite, the transition from a linear to a zigzag chain can be estimated with the value ν ( c ) t ≈ N ν/ (4 √ log N ), which was found by taking only nearest-neighbours couplingand where the corrections scale with powers of 1 / log N [24]. Apart from a factor oforder unity, it corresponds to the relation ν ( c )2 t = 72 ζ (3) Q ma (0) (34)with a (0) = 1 /n (0).In order to determine a Ginzburg-Landau equation for this case, we first assumethat the spatial variation is very slow, such that sufficiently far away from the edges,the length scale over which the interparticle distance changes is much larger than thecoarse graining length δx ≫ a ( x ) with which we study the dynamics: δx = a ( x ) / (cid:12)(cid:12)(cid:12)(cid:12) dadx (cid:12)(cid:12)(cid:12)(cid:12) ≫ a ( x ) (35)In this limit, we can make a slowly-varying ansatz for the short-wavelength eigenmodesof the linear chain, such that for a given eigenmode we extend the treatment for thehomogeneous case starting from Eq. (4) and write σ n = α n e i kna , with α n slowly-varying amplitude [24]. Within a local-density approximation, the ions contained in aregion of size δx and centered at position x , become unstable at the position-dependentcritical transverse frequency given by ν ( c ) t ( x ) = 72 ζ (3) Q ma ( x ) , (36) pontaneous nucleation of structural defects in inhomogeneous ion chains a ( x ). Moreover, extending to the non-homogeneous case theLagrangian in Eq. (18), we find the Lagrangian L ′ = R d x L ′ ( x ), with the Lagrangiandensity L ′ ( x ) = 12 ρ ( x ) h ( ∂ t ψ ( x )) − h ( x ) ( ∂ x ψ ( x )) − δ ( x ) ψ ( x ) − A ( x ) ψ ( x ) i . (37)In Eq. (37), ρ ( x ) = mn ( x ) is the linear mass density, and the spatial dependence of thecoefficients δ ( x ), h ( x ), and A ( x ) is found by using the position-dependent interparticledistance a ( x ) in the corresponding formulas for the homogeneous case. When derivingthe Lagrangian density of Eq. (37) we have neglected the coupling between axial andtransverse degrees of freedom, assuming very small fluctuations about the criticalpoint.In order to derive the Euler-Lagrange equation for the field, we assume thatthe linear density, and hence the interparticle distance a ( x ), does not depend on thevalue of the transverse trap frequency and thus remains constant when quenching ν t through the critical point. Terms proportional to the spatial gradient d a/ d x are also neglected, assuming that sufficiently far away from the edges the inequality | ψ ′′ ( x ) | ≫ | ψ ′ ( x ) a ′ ( x ) /a ( x ) | holds (which is consistent with the approximation madefor deriving the Lagrange density). Within this limit, the equation of motion for thefield ψ = ψ ( x, t ) reads ∂ t ψ − h ( x ) ∂ x ψ + η∂ t ψ + δ ( x, t ) ψ + 2 A ( x ) ψ = ε ( t ) , (38)where now δ ( x, t ) = ν t ( t ) − ν ct ( x ) = ν ( c ) t (0) − ν ( c ) t ( x ) − δ tτ Q , (39)and we considered the time-dependence given in Eq. (28).We now consider defect formation when the value of the transverse frequency isquenched through the critical point. This will happen in the central region of the chainfirst, giving rise to a propagating front along the axis, whose coordinates ( x F , t F )satisfy δ ( x F , t F ) = 0. The front velocity v F , at which the instability propagates,can be found by taking the ratio between the characteristic length of the controlparameter, ( ∂ x δ ( x, t ) /δ ( x, t )) − , over the characteristic time scale at which it changes,( ∂ t δ ( x, t ) /δ ( x, t )) − . It takes the form giving v F ∼ ∂ t δ ( x, t ) ∂ x δ ( x, t ) . (40)An explicit dependence on the physical parameter can be found using the spatially-dependent critical frequency in the expression ν c ( x ) = ν c (0)[1 − X ] , (41)with X = x/L . For the front velocity one obtains v F ∼ δ τ Q (cid:12)(cid:12)(cid:12)(cid:12) dν c ( x ) dx (cid:12)(cid:12)(cid:12)(cid:12) − x F = Lδ ν ( c ) t (0) τ Q | X | (1 − X ) − . (42)Whenever the transition is homogeneous, v F becomes infinite and the standard KZMapplies, allowing for the nucleation of defects in the whole system. Otherwise, as pontaneous nucleation of structural defects in inhomogeneous ion chains X ∗ L , where the front velocity v F is larger than thecharacteristic velocity ˆ v x with which a perturbation propagates along the chain at thefreeze-out time. We will find that the scaling of kinks density with the quenching ratewill be in this case different from the scaling found in the homogeneous case.In order to determine the density of defects, we now evaluate the velocity ˆ v x withwhich a perturbation propagates along the chain at the freeze-out time. An upper-bound for ˆ v x can be found by considering the ratio of the correlation length ˆ ξ x andthe relaxation time ˆ τ x at ˆ t ,ˆ v x ∼ ˆ ξ x ˆ τ x . (43)where the x -subindex underlines the spatial dependence due to the inhomogeneousnature of the system. This also corresponds to the speed of sound at the freeze-out point at an energy which is one over the relaxation time. In order to computeit, first note that the relative frequency can be written with reference to t F as δ ( x, t ) = − δ ( t − t F ) /τ Q , where the spatial dependence is encoded in t F = τ [ ν ( c ) t (0) − ν ( c ) t ( x ) ] /δ . One can find the instant ˆt, relative to t F , at which the dynamics stopsbeing adiabatic by equating the time scale δ/ ˙ δ to the relaxation time τ x = η/δ ( x, t ).In the overdamped regime, ˆt = ( ητ Q /δ ) / , which sets the freezed-out correlationlength ˆ ξ x = aω / q | δ ( x, ˆt) | = aω ( ηδ /τ Q ) − / . Hence, the characteristic velocity ofa perturbation becomes ˆ v x = ˆ ξ x / ˆ τ x = aω ( δ /η τ Q ) / .The key insight in the IKZM is that nucleation of defects is only expectedwhenever the front velocity is larger than ˆ v x , namely, v F > ˆ v x , (44)so that spatially separated regions of the chain are causally disconnected. The violationof this inequality opens the possibility of driving a truly adiabatic transition. Thereason is that the choice of the ground state in the broken symmetry phase is notindependent in different regions of the system whenever v F is small enough withrespect to ˆ v x . As a result the inhomogeneous nature of the transition allows for anadiabatic crossing, where a zigzag chain results after the quench with no defects. Thisholds even in thermodynamic limit and dramatically differs from the HKZM wheredefects would always be expected for infinite systems.In the overdamped regime, nucleation of kinks is still possible whenever v F ˆ v x = A o | X | (1 − X ) − > , (45)with A o = L ν ( c ) t (0) aω (cid:18) ηδ τ Q (cid:19) . (46)This will generally be fulfilled in a limited region of the system, 2 ˆ X ∗ L , where thehomogeneous KZM applies. One can estimate the effective size of this region wheredefect nucleation is possible by setting v F / ˆ v x = 1, and assuming ˆ X ∗ ≪
1. Then, itfollows that ˆ X ∗ o ≃ A o which leads to the scaling law of the density of kinks d o ∼ X ∗ o ˆ ξ o = L ν ( c ) t (0) a ω ηδ τ Q . (47) pontaneous nucleation of structural defects in inhomogeneous ion chains -4 -3 -2 -1 0 log[1/ ντ Q ] -4.5-4.0-3.5-3.0-2.5-2.0-1.5 l og [ d o ] IKZM (a) -3 -2 -1 0 log[1/ ντ Q ] -5.0-4.5-4.0-3.5-3.0-2.5 l og [ d u ] IKZM (b)
Figure 5.
Density of defects for a harmonically trapped ion chain as a functionof the inverse of the sweeping rate (a) in the overdamped regime ( η = 100 ν ),where the slope in the fit is 1 .
006 with regression coefficient 0 . η = 10 ν ), where the slope in the fit is 1 .
384 with regressioncoefficient 0 . N C = 30 ions, inorder to minimize defect losses ( N = 50, 2000 realizations). The parameters are ν ( c ) t (0) ≃ ν , δ = 36 ν , ǫ = 0 . l ν / , with l = Q /mν and ν being theaxial frequency. Though the absolute density of defects is reduced with respect to the homogeneouscase (since ˆ X ∗ < τ x = 1 / p | δ ( x, t ) | , leading tothe freeze-out time ˆt = ( τ Q /δ ) / . In this time scale, the correlation length freezes,with respect to the quench time scale τ Q , at a value ˆ ξ x = aω ( τ Q /δ ) / leading to auniform sound velocity ˆ v x = ˆ ξ x / ˆ τ x = aω . Again, the transition remains adiabatic aslong as v F ˆ v x = A u | X | (1 − X ) − > , (48)where we introduced the parameter A u = L ν ( c ) t (0) aω δ τ Q . (49)Using the same argument employed in the overdamped regime, kinks are found in aregion of size ˆ X ∗ u ≃ A u , so that d u ∼ X ∗ u ˆ ξ u = L ν ( c ) t (0) a ω (cid:18) δ τ Q (cid:19) / . (50)To test the IKZM, we study numerically in Figure 5 the scaling of the density ofkinks as a function of the quenching rate both in the overdamped and underdampedregimes. The results show a good agreement with the predictions in Eqs. (47) and(50). Nonetheless, there is a saturation of the density of kinks at high quenching ratesdue to the interactions between the kinks. pontaneous nucleation of structural defects in inhomogeneous ion chains N that nucleateaccording to the IKZM and HKZM obeys the relation N IKZM N HKZM = 2 ˆ X ∗ , (51)with a well define scaling with respect to the quenching rate, and which applies bothin the underdamped and overdamped regimes. In addition, the HKZM is known tooverestimate the number of defects by a numerical factor f of order unity, f ∼ . In this section we discuss different mechanisms responsible for deviations from theIKZM scaling. Two new effects come into play with respect to the ring configuration:a) coupling between axial and transverse modes b) enhanced defect transport. Weshall dwell on the implications of these two effects in KZM scaling.
Axial-transverse mode coupling - The (inhomogeneous) trapping potential allowsthe ions to shift in the axial direction as the structural phase transition takes place.As a result, during the course of the transition the axial density of ions increases nearthe center of the trap, an effect which hinders the study of the IKZM, as expectedfrom the derivation of the GLE in the thermodynamic limit and further suggested bynumerical simulations.
Dynamical losses of defects - In order to minimize the axial-transverse modecoupling it is desirable to drive the transition just in the center of the chain, so thatthe ions at the edges of the trap remain in the linear configuration. For the groundstate, the amplitude of the transverse displacement of the ions increases monotonicallyfrom each of the edges of the chain towards the center of the trap. Hence, theeffective Peierls-Nabarro potential [31, 29] seen by a kink decays as one approachesthe ends of the chain. As a consequence there is transport of defects, which providesa mechanism for their losses near the edges of the trap. Note that defect transportremains even if the longitudinal degrees of freedom of the ions are frozen on a lattice:the transverse motion suffices for its dynamics. A way of minimizing these defectlosses is by making the inter-ion spacing homogeneous, which in turn makes the Peierls-Nabarro potential periodic along the chain. Different potentials have been proposed toachieve homogeneous inter-ion spacing [32]. For an ion chain in a linear Paul trap, theoptimal axial trapping potential can be found by fitting the local Coulomb potentialin an homogeneous chain, U ( { x n } ) = P n = n ′ | x n − x n ′ | = a P n = n ′ | n − n ′ | where a is thedesired value of the inter-ion spacing. Under such axial confinement, the transitionbecomes then more homogeneous. Nonetheless, there is a local correction to thetransverse critical frequency, ν r,eff ( n ) = ν r ( t ) − P n = n ′ e /m | x n − x n ′ | which varies as afunction of the position along the chain. Hence even when the ions are homogeneouslyspaced, the transition remains inhomogeneous. The underlying defect dynamics is alsodriven by the fact that pairs of defects with the same topological charge repel eachother and attract otherwise. Scattering between kinks and anti-kinks (of positiveand negative topological charge) can occur leading to their annihilation. Nonethelessdefects stop seeing each other when they are separated by few ions ( ∼ ∼ ν − which suffices for its imaging inthe laboratory. The mechanisms for defect losses are particularly relevant in the pontaneous nucleation of structural defects in inhomogeneous ion chains ξ = ξ (ˆ t ), equals thelength of the part of the chain where defects are counted, and more fundamentally,where defects can nucleate 2 ˆ X ∗ . Defects nucleate as long as 2 ˆ X ∗ L > ˆ ξ . b) Thebreakdown of the IKZM scaling at fast rates due to a saturation of the average densityof defects. If one wishes to check the scaling by varying the quenching time between τ i and τ f (ideally ranging over few orders of magnitude), it should be possible to achievethe corresponding densities of defects ( d i and d f ) related as d i = ( τ f /τ i ) α d f , where α is the IKZM scaling (1 in the overdamped regime, 4 / × ions was created with an axial trappingfrequency of 1 MHz. Ring configuration were created dynamically in [33]. In the samesetup static structures of up to 50 ions were created at the temperature of 1 mK.
4. Conclusions
In this work we analyzed the formation of defects in a one dimensional Coulombcrystal during a frequency quench from the linear chain to the zigzag structure. Westudied the cases of ions crystals in a linear Paul trap and in a ring-shaped trap andpredicted the defects production rate. Our study shows that a Coulomb crystal is aparticularly neat and controllable system where the homogeneous and inhomogeneousKZM can be tested. Despite the experimental work in [19] addressing the homogeneousKZM (HKZM), the inhomogeneous KZM (IKZM) [22, 23] lacks to date experimentalverification, and the ion chains in a Paul trap are put forward here as an ideal systemfor such goal.
Acknowledgements.
We thank T. Calarco, S. Fishman, H. Rieger, H.Landa, S.Marcovitch and B. Reznik for fruitful discussions and R. Rivers and J.Dziarmaga for useful comments. We further ackowledge support by the EuropeanCommission (AQUTE, SCALA and QAP, STREPs HIP and PICC), the EPSRC,ESF EUROQUAM CMMC, the Generalitat de Catalunya Grant No. 2005SGR-00343and the Spanish Ministerio de Educaci´on y Ciencia (FIS2007-66944; FIS2008-01236;Juan de la Cierva; Ramon-y-Cajal, Consolider Ingenio 2010 ”QOIT”). G.M. andM.P. acknowledge the support of a Heisenberg Professorship and an Alexander-von-Humboldt Professorship, respectively.
Appendix A. Relaxation time
In order to compute the relaxation time, we consider the following autocorrelationfunction: g t ( τ ) = h y n ( t + τ ) y n ( t ) i = 1 N X kk ′ e i ( k + k ′ ) na h Ψ k ( t + τ )Ψ k ′ ( t ) i (A.1) pontaneous nucleation of structural defects in inhomogeneous ion chains y andwe inverted Eq. (4) for the expansion of the particles coordinates in terms of normalmodes. In the overdamped regime: η ≫ p | β ( k ) | we can neglect the second orderderivative in Eq. (22) whose solution becomesΨ k ( t ) = Ψ k e − β ( k ) η t + 1 η Z t e − β ( k ) η ( t − s ) ε k ( s ) ds (A.2)Substituting this solution in the definition of g t ( τ ) Eq.(A.1) we get: g t ( τ ) = 1 N X k e − β ( k ) η (2 t + τ ) (cid:20) h Ψ k Ψ − k i − κ B Tmβ ( k ) (cid:16) e − β ( k ) η t − (cid:17)(cid:21) (A.3)Now for fixed time t , all the components with different momentum in theautocorrelation function g t ( τ ) decay exponentially with the time separation τ . Thelargest time scale, corresponding to the minimum frequency δ = min k β ( k ), definesthe relaxation time: τ o = ηδ (A.4)where the subscript o stresses that this result has been obtained in the overdampedregime. Note also that defects are localised excitations and a correction depending on h can be obtained from the two-point correlation function. References [1] Dubin D H E and O’Neil T M 1999
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