The rigorous criteria for the phase transitions in Coulomb crystals
TThe rigorous criteria for the phase transitions in Coulomb crystals
The rigorous criteria for the phase transitions in Coulomb crystals
A.V. Romanova, a) S.S. Rudyi, and Y.V. Rozhdestvensky ITMO University, 49, Kronverkskiy Prospekt, Russia, 197101 (Dated: 18 February 2021)
The present work suggests rigorous criteria to determine phase transitions in Coulomb crystals in a linear ion trap. Theproposed method is based on the analysis of a cross size ρ i and relative polar angle between neighboring particles ∆ φ i asfunctions of the system parameters such as number of ions, mass and charge of ions, and trap geometry. The analyticalinterpretation of the phase transitions relies on the analysis of the cross size ρ and a metric Φ dependent on the normof the vector || ∆ φ || . We further demonstrate an analysis procedure for the numerical determination of extremes ofinterpolated functions ρ and Φ . 1D-2D and 2D-3D phase transitions points are defined as points of the greatest growthof functions ρ and Φ , respectively. I. INTRODUCTION
Coulomb crystals are unique ordered structures that form inion traps when ions are cooled below the 10 mK temperature .These structures are promising for the simulations of inacces-sible systems such as the surface of neutron stars . More-over, linear ion crystals, which are also called ion chains, areprospective for the implementation of the quantum computerbased on the ion traps .To describe the dynamics of the linear ion structures, thelinear chain oscillation formalism is used , as it considers theinteractions only between the nearest neighboring particles byanalogy with Toda chains . In this case, the criterion forthe determination of phase transitions is the change of the ra-tio of the transverse and axial frequencies . Nonetheless,the concept of “axial frequency” is associated with a cylindri-cal effective potential model, which causes a linear restoringforce in quadrupole traps . In the case of a real system, theshape of the effective potential can differ significantly from aparabolic well, for example, in the presence of end-cap elec-trodes. Toda chains allow to qualitatively describe the linear-zigzag transition in the ideal quadrupole fields . However,this method cannot describe the “zigzag – 3D structure” tran-sition and transitions in a highly deformed effective potential.Thus, the need for an alternative description of Coulomb crys-tals arises.From a physical standpoint, phase transitions result fromchanges in the total system energy. The total system energyis dependent on parameters of the system such as number ofions, trap geometry, and the elemental composition of the ioncrystal. In the general case, finding an analytical solution forthe equations of motion is complicated.On the other hand, the topological dimension of the crystalsharply changes with phase transitions occuring. Each stableconfiguration can be characterized by geometric parameters –a cross size and a relative polar angle between neighboringparticles. The cross size is the maximum deviation from thesymmetry axis z , and the relative polar angle is a differencebetween polar angles of particles with coordinates z i and z i + .In these terms, for the ion chain, the topological dimensionequals 1, and the cross size is limited by thermal vibrations. a) Electronic mail: [email protected]
In the case of the zigzag structure, the cross size is greater thanthermal vibrations of ions, and relative angles possess valuesof − π and π . It means that all units of the crystal belong tothe same plane, and the topological dimension equals 2. For3D structures the cross size is also greater than thermal vibra-tions, and relative angles belong to set [
0; 2 π ] . Thus, changesin geometric parameters of Coulomb crystals correspond tophase transitions.In the present work, we propose a rigorous criteria forthe phase transitions determination in Coulomb crystals. Weconsider a model problem where distances between parti-cles are fixed along z − axis. We show that the determina-tion of the phase transitions points reduces to ρ and Φ anal-ysis, where ρ is a cross size function and Φ is a metric,which is dependent on the norm of a relative angle vector as1 / π (cid:0) || ∆ φ || −√ N − (cid:1) , where N is a number of ions. Further,we consider a model problem where Coulomb interaction co-efficient increases. For this system, we define linear, zigzag,and three-dimensional structures in terms of ρ and Φ . Theapproach for the numerical determination of the extrema ofinterpolated functions ρ and Φ is proposed. II. THE CHARGED PARTICLES TRAPPING
For a single particle, the field of the hyperbolic (power)electrodes provides a radial confinement. For this field wegenerally use the form U ( x , y ) = e [ U + V cos ( Ω t )] r ( x − y ) (1)where e is the charge of the particle, U is the amplitude of theDC voltage applied to the hyperbolic electrodes, V , Ω are theamplitude and the frequency of the AC voltage, respectively, r is the trap radius.If a trap comprises only power electrodes, the particle mo-tion is limited along axes x and y , but not along axis z . In reallinear traps, the axial confinement arises from the interactionof the particle with the field of end-cap electrodes U end ( x , y , z ) .Figure 1 shows a linear trap with two flat end-cap electrodes.The total potential energy of N trapped particles is the su-perposition of the interaction of each particle with fields ofpower and end-cap electrodes and Coulomb interactions be-tween all trapped particles. The trapped ions hold the local a r X i v : . [ phy s i c s . c h e m - ph ] F e b he rigorous criteria for the phase transitions in Coulomb crystals 2 FIG. 1. Schematics of the quadrupole linear trap with four power andtwo end-cap electrodes, 2 L is the length of the ion chain at equilib-rium equilibrium position when the temperature is below 10 mK .In the presence of damping force, ions form unique orderedstructures called Coulomb crystals .One can distinguish one-, two-, and three-dimensionalstructures. One-dimensional structures are linear Coulombcrystals, which are also called ion string or ion chain . Inthe case of the one-dimensional structure, equilibrium posi-tions of ions lie along one axis, and the topological dimen-sion equals 1. Two-dimensional Coulomb crystals are calledzigzag chains or simply zigzag . In this case, equilibrium po-sitions of ions belong to one plane, and the topological di-mension equals 2. In the case of three-dimensional structures,equilibrium positions of ions lie in three-dimensional space,and topological dimension equals 3. When the potential en-ergy changes, one can observe phase transitions, which areattendant with a sharp change in the topological dimension ofthe Coulomb crystal . III. MODEL PROBLEM WITH FIXED AXIAL IONCOORDINATES
Let us consider the simplest case of linear Coulomb crystal.In this model problem, equilibrium positions of ions lie alongthe z axis. In the limit of a large number of ions, the interparti-cle spacing α = z i + − z i is a smooth function of the position,and it is inversely proportional to the density of ions per unitlength n L n L ( z ) = α ( z ) (2)The density of charges for unit length can be evaluated byapplying the Gauss theorem to a continuous distribution ofcharges, which are assumed to be uniformly distributed in anelongated ellipsoid. The resulting one-dimensional densityis n L = NL (cid:18) − z L (cid:19) (3)which is defined for | z | (cid:28) L , where 2 L is the axial length of thestring at equilibrium, N is the number of ions. The density (3) is the leading term in the expansion in powers of 1 / ln N , andit gives a good estimate of the charge distribution in the centerof the chain for N sufficiently large. The length is evaluated byminimizing the energy of the crystal and at the leading orderin ln N fulfills the relation L ( N ) = ke m ν N ln N (4)Sequentially substituting Eq. (4) into Eq. (3) and Eq. (3) intoEq. (2), one finds α ( z ) = ( ke N ln N ) / N (cid:112) m i ν (cid:32) √ ke N ln N − z i (cid:112) m i ν (cid:33) (5)Thus, the positions of i + i ions are linked by therecurrence relation. If we assume that the position of the firstparticle is z =
0, then the position of the second particle isgiven by z = z + α ( z ) = + α ( ) = (cid:114) m ν e kN ln N (6)Axial positions for the next particles will be z = z + α ( z ) = α ( ) + α ( α ( )) , . . . , z i + = z i + α ( z i ) . Thus, the position ofthe particles is defined by the recurrence relation that dependsonly on α ( ) .The field of end-cap electrodes has no significant effect onthe radial motion of particles when voltages applied to theend-cap electrodes are far less than voltages applied to thepower electrodes. Consequently, we can consider a simpli-fied model – the system of material points with charges e i and masses m i . The positions of material points are fixedalong z axis, and we take into account all interactions betweenthese points. Radial confinement in this case is provided bythe linear restoring force with the proportionality coefficient K (cylindrical pseudopotential model) , and K x = K y = K = · ( e V ) / ( m Ω r ) . Positions along z axis can be found byusing Eq. (5).Within this approximation, we consider the system of N particles with equal masses m = m = . . . = m i = m and equalcharges e = e = . . . = e i = e . The particles are trapped bythe cylindrical pseudopotential in the ion trap in ( x , y ) plane.The equations of motion of the i -th particle have the form, andthe gravity force is neglected here d x i d τ = − Ax i + B N ∑ i = ∑ i (cid:54) = j (cid:2) ( x i − x j ) κ ( i , j ) − (cid:3) − dx i d τ (7) d y i d τ = − Ay i + B N ∑ i = ∑ i (cid:54) = j (cid:2) ( y i − y j ) κ ( i , j ) − (cid:3) − dy i d τ (8)where κ ( i , j ) = (cid:2) ( x i − x j ) + ( y i − y j ) + ( z i − z j ) (cid:3) / , A =( m ν ) / β , B = mke / β , ν = K / m , τ = t β / m , k = / ( πε ) , ε is a dielectric constant, e is the charge of i and j particles, β is the effective friction coefficient . Finally,he rigorous criteria for the phase transitions in Coulomb crystals 3substituting the variables A and B into (6), we obtain the fol-lowing α ( ) = (cid:114) A BN ln N (9)Thus, the dynamical system depends only on three parameters– A, B, and the number of ions, N. IV. CONDITIONS OF PHASE TRANSITIONS INCOULOMB CRYSTALS
The dynamic system (7)–(8) is dissipative. Thus, particlesaspire to occupy isolated local minima of potential energy.When τ → ∞ , one can observe a formation of stable con-figurations . The formation of a certain stable configurationdepends on the system parameters as well as on the initialconditions (the initial coordinates q n i [ ] and velocities ˙ q n i [ ] ).Within the above model problem, we can describe the stableconfigurations by two geometric parameters. These geomet-ric parameters are the cross size of the Coulomb crystal andrelative polar angle. The cross size of Coulomb crystal is amaximum deviation of an ion from the symmetry axis ρ = N max i = (cid:18)(cid:113) x i + y i (cid:19) = N max i = ρ i (10)The relative polar angle is the difference between polar angles ∆ φ of two particles with z i and z i + coordinates. N − N particles, can bedefined as ∆ φ = ( ∆ φ , ∆ φ , ..., ∆ φ i , ..., ∆ φ N − ) == ( arctan y x − arctan y x , ..., arctan y N − x N − − arctan y N x N ) (11)The scalar describing a vector ∆ φ is the norm of this vector || ∆ φ || = (cid:115) N − ∑ i = ( ∆ φ i ) (12)On the one hand, phase transitions are attended by a sharpchange in the relative angles and cross size of the Coulombcrystal. On the other hand, phase transitions occur as a resultof a change in the potential system energy. In terms of equa-tions of motion (7)–(8), the energy of interaction with the trap-ping potential is proportional to A, and the energy of Coulombinteraction is proportional to B. Thus, ρ and || ∆ φ || are func-tions of the parameters of the system ρ ( A , B , q n i [ ] , ˙ q n i [ ]) , || ∆ φ || ( A , B , q n i [ ] , ˙ q n i [ ]) , where q n i [ ] , ˙ q n i [ ] are initial con-ditions of generalized coordinates for i ∈ [ , N ] . Each of thestable configurations can be characterized with certain valuesof ρ and || ∆ φ || . Points of the phase transitions are the pointswhere the functions ρ and || ∆ φ || have the fastest growth rate.To prove this statement, we consider the phase transitionswhen the parameter A is fixed. Figure 2a shows the linearion crystal when B (cid:28)
1. In this case, the cross radius ρ → z . The vector ∆ φ is indetermi-nate for a linear chain since ( x i , y i ) = ( , ) . By this one can see that the linear structure is fully described by the cross sizefunction ρ ( B ) . When the parameter B increases, ρ also in-creases. Moreover, the function ρ ( B ) has a cusp at the point B crit . Thus, B crit is a point of the phase transition “string-zigzag”.We can describe the zigzag configuration with two param-eters. Ideally, the relative angles ∆ φ i can take discrete values − π and π (Figure 2b). The norm of the relative angle vector || ∆ φ || takes the form || ∆ φ || = || − π , π , − π ... || = (cid:115) N − ∑ i = [( − ) i π ] = π √ N − B in the equations ofmotion (7)–(8) results in the growth of || ∆ φ || . The function || ∆ φ || has a cusp at the point B crit . Hence, the point of phasetransition “zigzag–three-dimensional structure” is the mini-mal value of B , at which || ∆ φ || − π √ N − (
0; 2 π ] . (a)(b)FIG. 2. Simulation of stable configurations in the linear trap a) linearion crystal in the case of N = N =
10 particles.
At this point, we can define all stable structures in the con-sidered model as follows1) If for each ion with coordinates ( x i , y i ) the conditions ρ ( B ) ≤ ε , lim t → ∞ ε = linear structure .2) If for each ion with coordinates ( x i , y i ) the conditions ρ ( B ) > ε , Φ ( B ) = (cid:12)(cid:12)(cid:12)(cid:12) π || ( ∆ φ ( B )) || − √ N − (cid:12)(cid:12)(cid:12)(cid:12) ≤ ξ , lim t → ∞ ξ = zigzag
3) If for each ion with coordinates ( x i , y i ) the conditions ρ ( B ) > ε , Φ ( B ) > ξ (16)are satisfied, then this structure is called three-dimensionalstructure . V. SPECIAL ASPECTS OF THE NUMERICAL ANALYSISOF PHASE TRANSITIONS
We can apply the definitions (14)–(16) to the models sim-ilar to (7)–(8) when T → ∞ . Within these models, we ne-glect the radiofrequency heating, stochastic effects, which re-sult from the ion re-emission during the laser cooling, and thedetection aspects. However, we cannot neglect these effects inthe case of real systems. Therefore, ρ ( B ) and Φ ( B ) from thedefinitions (14)–(16) always have nonzero values. It meansthat ε and ξ are limited from below by nonzero ε and ξ ,respectively. The points of the “linear–zigzag” and “zigzag–three dimensional structure” phase transitions are determinedat points ρ ( B crit ) = ε and Φ ( B crit ) = ξ .Similar reasoning is also correct for studying the dynamicsof ideal models after the limited amount of time τ max . In thepresence of the linear friction, the oscillation amplitude de-cays exponentially and takes a nonzero value at any τ (cid:54) = ∞ .We further consider the values of ε and ξ in the equations ofmotion (7)–(8) for τ max = T .Figure 3 shows functions ρ i ( B ) and ∆ φ i ( B ) for the systemof N = A = ρ i ( B ) and ∆ φ i ( B ) for ions that have “neighboring” z coordinates.The step δ B equals 10 − . The linear structure forms when B is less than B ≈ . ρ i are low, and ∆ φ i areundetermined. The cross size of the crystal increases substan-tially when the value of B tends to B (Fig. 3a), and ∆ φ i take onvalues ± π (Fig. 3b). Note that one can see only four branchesof ρ i in Fig. 3a since zigzag has the central symmetry. Whenthe value of B crosses a B ≈ .
012 point, ρ i continue to in-crease, and ∆ φ i belong to [
0; 2 π ] . Thus, one can observe phasetransitions in terms (14)–(16) near the points B ≈ .
007 and B ≈ . B and B ” and donot call B and B the critical points, as ρ ( B ) and Φ ( B ) haveno cusps at the points of phase transitions when T is limited.According to the definitions (14–16), ε tends to ε at the pointof the “linear chain-zigzag” phase transition only in the eventof T → ∞ . In other words, the determination accuracy of thepoints of phase transitions is reliant on the limited period oftime T . The values B → B crit and B → B crit under thecondition T → ∞ .For clarity, we consider the numerical calculation of ρ ( B ) in the case of N = A = B ) for dimensionless T = , , ρ ( B ) has no explicit cusp near the point B on a finite time interval. According to the definition (14),the Coulomb crystal is linear when ρ →
0. In terms of nu-merical calculation, the minimal value of ρ is limited by an (a)(b)FIG. 3. Functions of geometrical parameters in the case of N = A = ρ i ( B ) ; b) the relative angle betweenneighboring particles ∆ φ i ( B ) accuracy rate. The red line in Figure 4 represents the ErrorRate (ER). Thus, B is limited by the maximum value B atthe point ρ ( B ) = ER , and ρ < ER is true for all B < B inconsideration of the monotonic increase of the function ρ ( B ) .Nonetheless, the value of B can be significantly different forvarious T and B → B crit for T → ∞ . The values of B for T = , , ρ ( B ) when B → B crit . Moreover, the value ofthe cross size at which the derivative ( ∂ ρ / ∂ B ) has maximum is independent from T . From this perspective, the point ofthe phase transition can be defined as the value of B at which ( ∂ ρ / ∂ B ) has the maximum and consequently ρ ( B ) has themaximum growth rate. As shown in Figure 5a, the function ρ ( B ) monotonically increases in the range of B ≈ .
007 to B ≈ B but the crystal is still linear in terms (14)–(16) andthe phase transition does not occur here. The growth of ρ ( B ) in this range represents the fact that the phase transition is notinstantaneous. Therefore, one can observe the phase transition“linear chain–zigzag” at the point B = . ρ ( B ) = ε = . ε tends to infinity when T → ∞ . One can find the point B = . FIG. 4. Numerical calculation of the cross size function ρ ( B ) in thecase of N = A = T = T = T = ρ ( B ) = ER = − for T = ξ in the similar way by the analysis of functions Φ ( B ) and ∂ Φ / ∂ B (Figure 5b). VI. CONCLUSION
In the present work we have proposed a new method to de-termine phase transitions in Coulomb crystals in a linear iontrap. This method is based on the studying of the geomet-rical parameters of the ion crystal – the cross size ρ ( B ) andthe metric Φ ( B ) . We have considered a model where ionspositions along the z axis are fixed. In terms of this model,the linear structure can be characterized with ρ ( B ) ≤ ε . Thefunction ρ ( B ) has a cusp at the point B , which is the pointof the “linear structure–zigzag” phase transition. The zigzagstructure is described by two parameters ρ and Φ ( B ) , and ρ > ε , Φ ( B ) ≤ ξ . The function Φ ( B ) has a cusp at the point B which is the point of the “zigzag–three-dimensional struc-ture” phase transition. In the case of a three-dimensionalstructure, ρ > ε , Φ ( B ) > ξ . Hence, functions ∂ ρ / ∂ B | B = B and ∂ Φ / ∂ B | B = B have asymptotes at the points of the “linearstructure–zigzag” ( B ) and “zigzag–three-dimensional struc-ture” ( B ) phase transitions, respectively. The remarkable factis that ε and ξ are limited from below by nonzero ε and ξ ,when one considers a system for a limited period of time T .Moreover, the determination accuracy of ε , ξ and, hence,the points of phase transitions, is reliant on the limited periodof time T : the more the time T , the closer the value of B tothe true value B crit .The method proposed is necessary for the correct experi-mental data analysis and determination of points of phase tran-sitions in real systems. For instance, one can describe theoret-ically unusual structures such as radial two-dimensional ion (a)(b)FIG. 5. The numerical calculation of the geometrical parametersfunctions a) the cross size ρ ( B ) (solid line) and the first derivative ∂ ρ / ∂ B (dashed line); b) the metric Φ ( B ) (solid line) and the firstderivative ∂ Φ / ∂ B (dashed line) crystals that were experimentally observed recently . More-over, these criteria allow to discover previously unknownstructures. M. Drewsen, “Ion coulomb crystals,” Physica B: Condensed Matter ,105–113 (2015). K. Pyka, J. Keller, H. Partner, R. Nigmatullin, T. Burgermeister, D. Meier,K. Kuhlmann, A. Retzker, M. B. Plenio, W. Zurek, et al. , “Topologicaldefect formation and spontaneous symmetry breaking in ion coulomb crys-tals,” Nature communications , 1–6 (2013). M. Johanning, “Isospaced linear ion strings,” Applied Physics B , 71(2016). J. Pedregosa-Gutierrez and M. Mukherjee, “Defect generation and dynam-ics during quenching in finite size homogeneous ion chains,” arXiv preprintarXiv:2001.09586 (2020). M. Toda, “Vibration of a chain with nonlinear interaction,” Journal of thePhysical Society of Japan , 431–436 (1967). G. L. Kotkin and V. G. Serbo,
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