External field-induced dynamics of a charged particle on a closed helix
EExternal field-induced dynamics of charged particles on a closed helix
Ansgar Siemens ∗ and Peter Schmelcher
1, 2, † Zentrum f¨ur Optische Quantentechnologien, Fachbereich Physik,Universit¨at Hamburg, Luruper Chaussee 149, 22761 Hamburg Germany Hamburg Center for Ultrafast Imaging, Universit¨at Hamburg,Luruper Chaussee 149, 22761 Hamburg Germany (Dated: February 8, 2021)We investigate the dynamics of a charged particle confined to move on a toroidal helix while beingdriven by an external time-dependent electric field. The underlying phase space is analyzed forlinearly and circularly polarized fields. For low driving amplitudes and a linearly polarized field, wefind a split-up of the chaotic part of the phase space which prevents the particle from inverting itsdirection of motion. This effectively allows for directed transport without breaking the well-knownsymmetries for the commonly investigated directed transport. Within our chosen normalized units,the resulting average transport velocity is constant and does not change significantly with thedriving amplitude. A very similar effect is found in case of the circularly polarized field and lowdriving amplitudes. Furthermore, when driving with a circularly polarized field, we unravel a secondmechanism of the split-up of the chaotic phase space region for very large driving amplitudes. Thereexists a wide range of parameter values for which trajectories may travel between the two chaoticregions by crossing a permeable cantorus. The limitations of these phenomena, as well as theirimplication on manipulating directed transport in helical geometries are discussed.
I. INTRODUCTION
Helical structures and patterns can be frequently foundin nature, with systems ranging from molecules like DNAor amino acids to self-assembled configurations of par-ticles in nanotubes [1]. Especially for quasi 1D struc-tures, the helical geometry can offer advantages such asincreased stability and resistance to deformations [2, 3].In the last decades great progress was made in attemptsto synthesize artificial 1D nanostructures, such as helicalCNT’s [4], with hopes for applications in nano electroniccircuits [5–8]. Therefore, there is a great interest in un-derstanding how the electronic properties of 1D struc-tures are affected by helical geometries.Already in minimal models, intriguing phenomena canresult from the geometric confinement to a 1D helix. Itwas demonstrated that, due to the geometry, ballisticlong-range Coulomb interacting particles on a 1D helicalpath can form bound states [9, 10] and can even build1D lattice structures [10–12]. Novel physics resultingfrom this behavior has been reported in several worksdiscussing relevant setups [13–20]. Effects range frommechanical properties like an unusual electrostatic bend-ing response [12], to intriguing nonlinear dynamics, suchas the scattering of bound states at an inhomogeneity inthe 1D path [13] or the tuning of the dispersion relationof a 1D chain of bound particles by varying the helix ra-dius [14, 15]. In the latter example, a degeneracy of theband structure for a specific helix radius was identifiedwhich prevents excitations from dispersing through thesystem. ∗ [email protected] † [email protected] In helical systems, the novel effects typically emergedue to the fact that the acting forces are partially com-pensated by confining forces of the helix, and are there-fore not limited to Coulomb interactions. Effects ofdipole-dipole interactions [16–18], as well as external elec-tric fields [11, 20] have been explored. Previous investi-gations of external electric fields considered adiabaticallyvarying forces and demonstrated the possibility of us-ing an external electric field for controlled state transfer[20], and inducing crystalline lattice ordering of particles[11]. For crystalline particles on a closed helix exposedto a static electric field, an unconventional pinned-to-sliding transition has been observed [19]. Investigatingthe dynamics of confined particles being driven by time-dependent external forces is therefore a natural next step.Periodic driving is at the core of many intriguing phe-nomena, such as resonances and chaos. In driven sys-tems, already simple models can often yield quite com-plex dynamics and give valuable insight into real physi-cal systems. For example, the model of a driven morse-oscillator can give insight into the (vibrational) stabilityof molecules [21]. In the same spirit, particles in drivendouble well potentials have been studied to explain thetunneling dynamics (or the suppression thereof) througha potential barrier [22–28]. Studies of driven Hamilto-nian systems, i.e. particle dynamics in time dependentperiodic potentials, often possesses a focus on the ma-nipulation of transport phenomena due to the choice ofthe driving potential. There, the transport of diffusivetrajectories is usually induced by breaking certain spatio-temporal symmetries [29–35]. However, other manipula-tion techniques, like the possibility of switching betweenballistic and diffusive motion by introducing localized dis-order [36], have been demonstrated. Furthermore, thepresence of spatially varying forces has been linked toa variety of intriguing phenomena [37–41], such as the a r X i v : . [ n li n . C D ] F e b formation of density waves [40]. Based on this under-standing of driven systems, a plethora of applications,including velocity filters [42, 43], spectrometers [44–46],or batteries extracting energy from thermal fluctuations[34, 47–54], have been proposed.Motivated by the complexity arising when particles areconfined to curved space, we investigate in this work theinfluence of time periodic forces on particles in helicalconfinement. As a prototype, we consider a single par-ticle confined to a toroidal helix. The combination ofdriving and confining forces leads to spatially and tem-porally varying effective forces. We analyze the phasespace and identify two different scenarios by which thechaotic phase space region can be split. We explain howthese splits are induced by the different scales of oscilla-tions in the driving potential, and how they influence thecorresponding transport phenomena.Our manuscript is structured as follows. Sec. II con-tains the parametrization of the toroidal helix, a discus-sion of the Lagrangian, and the general equations of mo-tion for our setup. We further discuss the considereddriving laws. In Sec. III and IV we investigate and an-alyze the dynamics int the presence of driving with alinearly polarized and a circularly polarized electric fieldrespectively. Finally, in Sec. V we provide our conclu-sions. II. PARTICLES IN HELICAL GEOMETRIESWITH EXTERNAL DRIVING
We consider a single particle with charge q confined tomove along a toroidal helix (see Fig. 1(a) for a visual-ization). The parametrization of the particles positionsis then given by the following equation r ( u ) := ( R + r cos( u )) cos( u/M )( R + r cos( u )) sin( u/M ) r sin( u ) , u ∈ [0 , πM ](1)where R is the torus radius determining how stronglyour helix is bent, r is the radius of the helix, and M arethe total number of helical windings. Since the path isclosed we have r ( u ) = r ( u + 2 πM ), and the parametersobey the following restriction R = M h/ π , where h isthe pitch of the helix. When u changes by an amountof 2 π , the particle moves the distance of one winding onthe helix. When u changes by an amount of 2 πM , theparticle circles once around the torus and is exactly atthe same position it started in.The driving force is assumed to be caused by an exter-nal electric field E . The potential energy V ( u, t ) of theparticle is then given by V ( u, t ) = q E ( t ) · r ( u ) (2)Our system is then described by the following Lagrangian L = m (cid:18) d r ( u ) dt (cid:19) − q E · r ( u ) (3) FIG. 1. (a) A 3D illustration of the parametric function r ( u ),for M = 10, r = 0 .
8, and R = 2 .
5. (b) The potential V x ( u )created by a static field in the x-direction shown for toroidalhelices with M = 10, R = 2 .
5, and helix radii of r = 0 . r = 1 (dotted gray).(c) The potential landscape V x ( u, t ) for a linearly polarized oscillating field in x-directionshown for t = 0 (orange) and t = π/ω (dashed gray). Theinset in the top right corner visualizes the driving direction ofthe field (red) for a top view of the setup. (d) The potentiallandscape V xy ( u, t ) for driving with a circularly polarized fieldin the xy-plane shown for the times t = [0 , π/ωM, π/ωM ].After the time t = 2 π/ωM the motion of the potential re-peats, being shifted by ∆ u = 2 π . Again, the inset in the topright corner visualizes the driving field (red) for a top view ofthe setup. Equation (3) already accounts for the confining forces ofthe setup by only allowing positions r ( u ) on the para-metric helical curve. Since r ( u ) is known, we can alreadyevaluate the derivative in the kinetic energy term andrewrite Eq. 3 as L = m (cid:18) d r ( u ) du (cid:19) (cid:18) dudt (cid:19) − q E · r ( u ) (4)where ( d r ( u ) /du ) = r + ( R + r cos( u )) /M . Fromthis, we obtain the following equations of motion for anarbitrary driving field E ( t ) (cid:18) d r ( u ) du (cid:19) d udt = d ( d r ( u ) /du ) du (cid:18) dudt (cid:19) − q E ( t ) m d r ( u ) du (5)Let us now take a closer look at the driving potential V ( u, t ) = q E ( t ) · r ( u ) created by the electric field. Evenin the static case, i.e. without time dependence, the po-tential can become quite complex and possesses multipleminima. This is shown in Fig. 1(b) for a static field par-allel to the x-axis. In this case the electric field is givenby E = E e x and the potential energy becomes V x ( u ) = qE ( R + r cos( u )) cos( u/M ) (6)This potential consists of two terms: The R cos( u/M )term creates a long wavelength cosine shaped potentialthat is maximal at the position that extends most intothe x-direction (for u = 0 or u = 2 πM ) and minimal forthe position extending most into the negative x-direction(for u = πM ). Since it is caused by the overall toroidalshape of the curve r ( u ) we will call this the torus in-duced potential (TIP). On top of that, there is a smallermodulation given by the r cos( u ) cos( u/M ) term. Sincethis modulation originates from the helix windings wewill call this the winding induced potential (WIP). Theamplitude of the WIP can be modulated via the helixradius r (shown in Fig. 1(b) for r = 0 . r = 1).Due to the cos( u/M ) dependence, the WIP oscillationalamplitude also changes with the position on the torus.The amplitude is largest for u ∈ [0 , πM, πM ] and van-ishes for u ∈ [ πM/ , πM/ M .In this work we focus on two different time depen-dent fields: Driving with a field oscillating parallel tothe x-axis, and driving with a field rotating in the xy-plane. In the first case, the driving field becomes E ( t ) = E cos( ωt ) e x . The resulting potential V x ( u, t ) is a stand-ing wave with the shape shown in Fig. 1(c) V x ( u, t ) = qE ( R + r cos( u )) cos( u/M ) cos( ωt ) (7)When we consider an electric field rotating in the xy-plane the driving becomes slightly more complex. Inthis case, the electric field can be written as E ( t ) = E { cos( ωt ) , sin( ωt ) , } and the potential landscape be-comes V xy ( u, t ) = − qE cos( ωt − u/M ) ( R + r cos( u )) (8)Figure 1(d) visualizes the time evolution of this poten-tial by showing the potential landscape at different times t . The three curves in the figure correspond to the cases t = 0 (orange), t = π/ωM (gray), and t = 2 π/ωM (blue).Due to the symmetries of the toroidal helix, we only needto consider the time ∆ t = 2 π/ωM needed to rotate byone winding to understand the driving, since the poten-tial movement repeats after this time; it is just shiftedby a distance of ∆ u = 2 π . The time evolution of thepotential landscape resembles a ‘crawling’ motion: Thelocal extrema of the potential oscillate between being apotential minimum and a potential maximum, with aconstant phase shift of 2 π/M between neighboring min-ima (or maxima). A video showing the time evolutionof V xy ( u, t ) can be found in the supplementary material.It should also be noted, that for V xy ( u, t ) the equationsof motion are not symmetric with regards to the spatio-temporal symmetries given by ( u → − u + ∆ u , t → t + τ )and ( u → u + ∆ u , t → − t + τ ); a necessary criterionfor directed transport within the chaotic sea [29, 31]. Incontrast, these symmetries are conserved for V x ( u, t ).We can eliminate redundant parameters by introduc-ing dimensionless units. Without loss of generality, wechoose to express distances in units of 2 h/π and time inunits of ω/ π . We also normalize the particle mass and charge to m = q = 1 (which is the same as ‘absorbing’both values in the driving amplitude). The remainingindependent system parameters are the winding number M , the helix radius r , and the driving amplitude E . III. PARTICLE DYNAMICS FOR A LINEARLYPOLARIZED FIELD
In this section we will analyze the dynamics when thesystem is driven by an electric field oscillating parallel tothe x-axis. For this we will examine the phase space of thesystem and understand how it is decomposed for differentparameter regimes. The dimensions of the phase spaceare made up of the three parameters: position u , momen-tum p , and time t . Since our Lagrangian is periodic intime, we can use a Poincar´e surface of sections (PSOS)to visualize the phase space in a two dimensional stro-poscopic u ( p ) dependence. Note, that our momentum p refers to the canonical momentum given by p = du/dtm ( r + a ( R + r cos( u )) ) (9)We start our investigation by considering a toroidalhelix with M = 10 and r = 0 .
2. Figure 2 shows thePSOS of the system for electric field amplitudes E = 80, E = 20, and E = 4. In Fig. 2(a), for E = 80, we ob-serve a mixed phase space that mainly allows three dif-ferent kinds of trajectories: Chaotic trajectories, and twotypes of (quasi-) periodic trajectories which we will referto as Type-I and
Type-II trajectories. Type-I trajecto-ries (marked I and I in the figure) are invariant spanningcurves [55, 56] for which the particle momentum is toolarge to be significantly affected by the driving. The driv-ing results only in a weak modulation of their dynamics.Towards smaller momenta the Type-I trajectories bor-der on a ‘sea’ of chaotic trajectories which contains twolarge regular islands. These regular islands correspond tothe Type-II trajectories and describe motion around thetorus in phase with the driving period, i.e. after one driv-ing period the particle on a Type-II trajectory has circledthe torus exactly once. Both regular islands describe thesame kind of motion, but in opposite directions.As one might expect, the size of the chaotic portionof phase space decreases when the driving amplitude isdecreased. This can be seen in Fig. 2(b) where E = 20.The Type-I trajectories, as well as the two main fixedpoints we identified in the previous figure are still present.However, the chaotic region now occupies a much smallermomentum range of the phase space. In addition, atthe center of the chaotic region around p ≈ u, p ] ≈ [10 . , . , [15 . , .
8] and [20 . , . u and is hardly affected by thedriving. The reason for their appearance is as follows:When the driving amplitude decreases, so does the accel-eration of the particle. Below a certain threshold the par- FIG. 2. Poincar´e surfaces of sections (PSOS) for a particle on the toroidal helix, driven by a linearly polarized oscillating fieldfor (a) E = 80, (b) E = 20, and (c) E = 4. Different colors are assigned to the trajectories for easier differentiation. EachPSOS features between 45 and 75 trajectories, each simulated for 2000 driving periods. (Quasi-) periodic trajectories betweenthe two chaotic regions (around p = 0) in (c) are only shown in the inset (top left of (c)) to emphasize the splitting of thechaotic sea into two parts in the main figure. The inset in (b) visualizes the particle motion on V x ( u ) for the three differenttypes of trajectories (I-III) during a driving period in the range u ∈ [ − πM/ , πM/ | p | ≈ . ticle has hardly moved before the driving field acceleratesthe particle in the opposite direction. With decreasingdriving amplitude an increasing amount of trajectorieswith initial conditions around p = 0 will exhibit this be-havior. The effect on the phase space can be seen in Fig.2(c) for E = 4 (note the adjusted range of p values). Herethe driving amplitude is sufficiently small, such that forevery u there is a (quasi-) periodic trajectory (picturedonly in the inset of (c)) close to p = 0 that is hardlyaffected by the driving and mostly stays in place. Aninteresting result is, that the appearance of these tra-jectories is splitting the chaotic sea into two parts: Onewith p > p < p > p = 0. The sameis of course true for trajectories starting in the chaoticregion with p <
0. In other words: Due to the (quasi-)periodic trajectories appearing around p = 0, we havetwo (symmetric) chaotic seas, both with non-zero aver-age momentum. In a way, this split-up of the chaotic seaallows for directed transport in each of the two chaotic re-gions, without the presence of broken symmetries usuallyassociated with transport in chaotic phase space regions.In Fig. 3 we take a closer look at this split-up of thechaotic sea. A close up of the split appearing in the PSOS is shown in Fig. 3(a-c). For clarity, the PSOS’s of Fig.3(a-c) only contain initial conditions from the chaotic re-gion with p <
0. In Fig. 3(a), at E = 7, the emerging(quasi-) periodic regions around p = 0 are clearly visi-ble. However, changing the direction of motion is stillpossible and happens indeed frequently. The momentumevolution p ( t ) of a representative example trajectory isshown in Fig. 3(d)(blue curve). From this p ( t ) curvewe can see that already for E = 7 there are effectivelytwo momentum ranges the particle can be confined to.The particle frequently switches between having eitherpositive or negative momentum for extended periods oftime.When the driving amplitude is decreased further to E = 4 (see Fig. 3(b)), the two chaotic phase space re-gions are almost separated from each other. An inversionof the direction of movement now happens much less fre-quently. In the phase space this can be seen from thedecreasing density for p >
0. From the correspondingexample trajectory in Fig. 3(d)(red curve), we see thatthe momentum inversion now also takes a much longertime than for E = 7. It takes our example trajectoryalmost ∼ p < p > E = 3 (Fig. 3(c)), the two phase spaceregions are completely separated from each other. Noneof our trajectories cross into the phase space region with p >
0. In this regime, the dynamics of all simulated tra-jectories resemble that of our example trajectory in Fig.3(d)(yellow curve): The trajectories are chaotic while
FIG. 3. (a-c) PSOS created from ∼ trajectorieswith initial conditions in the chaotic sea ( u ∈ [0 ,
1] and p ∈ [ − . , − . p = 0 splits the chaotic sea into two seas, when the drivingamplitude E is decreased. (d) Representative example tra-jectories emanating in the chaotic sea for p < E = 3(yellow), E = 4 (orange), and E = 7 (blue). The two mo-mentum regimes the particles are confined to are clearly vis-ible. Transition between the two regimes is more likely forlarger driving amplitudes. (e) The average transport velocity v av and switch time t s as a function of the driving amplitude.Each data point was obtained from simulation numbers andtimes similar to those of (a-c). sustaining a strictly negative momentum.A better understanding of the Type-III trajectories canbe gained from statistical averages. We consider the av-erage velocity v av , as well as the mean switch time t s .For a set of trajectories u ( u i , p i , t ) with initial conditions u ( t = 0) = u i and p ( t = 0) = p i the average velocity isdetermined by averaging the mean velocities of all tra-jectories v av = 1 N T N (cid:88) i =1 (cid:90) T du ( u i , p i , t ) dt dt (10)where T is the simulation time of individual trajectories.We define the mean switch time as the average time aparticle spends with p > p <
0) before invertingthe direction of its motion. Note, that within our nu- merical simulations, there are limitations regarding thecalculation and accuracy of t s . We can only determine t s accurately from our simulations, if we (on average)observe at least one switch in the time T . Since eachtrajectory was simulated for T = 5000 driving periods,our value of t s is accurate for values below t s (cid:46) trajectories for 5000 time steps,count the total number of switches n in all simulations,and then calculate t s = 0 . /n .Figure 3(e) shows both v av and t s as a function of thedriving amplitude. For better insight into the dynamicsof the Type-III trajectories both curves were only ob-tained from trajectories with p ( t = 0) >
0. Until thesplit-up of the chaotic region at about E = 4 both quan-tities increase with decreasing driving amplitude. From t s we see that long before the two chaotic regions are sep-arated from each other, the particles perform very long’flights’ without inverting the direction of their motion.Even for E = 7, where the chaotic regions are still rea-sonably well connected in the phase space (see Fig. 3(a)),we have a mean switch time of t s >
600 driving periods.In the figure, our mean switch time exceeds the crit-ical value of t s = 2500 for driving amplitudes E < t s inthis regime of driving amplitudes because the change ofthe direction of motion happens too infrequently. Con-sequently, in this regime the choice of initial conditions( p ( t = 0) <
0) becomes apparent in the statistics of v av .While t s > v av increases withincreasing t s . When the two chaotic phase space regionssplits up at around E = 4, v av reaches a plateau (see v av in Fig. 3(e)). After the split-up, the Type-III tra-jectories have a consistent mean velocity of slightly lessthan v av ≈ π . This velocity corresponds to a positionchange of about one helix winding during each drivingperiod. More precisely, each driving period the chaoticType-III trajectories move between neighboring minimain the WIP. Therefore, the dynamics of Type-III trajec-tories are similar to the Type-II trajectories, except thatthey are mostly determined by the minima of the WIPwith the TIP being a perturbation that is mostly respon-sible for the chaos. In contrast, the Type-II trajectoriesare mostly determined by the minima of the TIP, withthe WIP acting as a perturbation. For even lower driv-ing amplitudes the perturbation due to the TIP becomessmall enough for the Type-III trajectories to stabilize intoa series of fixed points (similar to the ones shown in Fig.4(a) for a rotating driving field).Since the Type-III trajectories emerge due to the WIP,it is no surprise that the occurrence of the phase spacesplit depends on the helix radius r . For larger values of r , the (quasi-) periodic trajectories around the Type-IIIfixed points will already stabilize for larger values of E ,since the relative strength of the ‘perturbation’ due tothe TIP decreases. For a large enough r , it is possible forthe Type-III fixed points to stabilize before the chaoticregion is splitting up. In extreme cases this may evenprevent the occurrence of chaotic Type-III trajectories.The only independent system parameter did not dis-cuss so far is the winding number M . Changing M doesnot significantly affect the overall dynamics. However,due to the relation R = M h/ π and our choice of units(thereby setting h = π/ M will change thetorus radius R , thereby changing the momentum of theType-II trajectories. This, in turn, changes e.g. the driv-ing amplitude required for a mixed phase space as shownin Fig. 2(a). This also changes the ratio of r/R and maycause the periodic Type-III fixed points to stabilize atdifferent driving amplitudes. Increasing M also increasesthe number of extrema in the WIP, leading to more fixedpoints in the (quasi-) periodic Type-III trajectories oncethey stabilize. Besides this, however, the split-up of thechaotic phase space region is mostly unaffected. IV. PARTICLE DYNAMICS IN THE PRESENCEOF A CIRCULARLY POLARIZED FIELD
Another intriguing split-up in the phase space can beobserved when driving with a circularly polarized field inthe xy-plane. In this case, the driving law is character-ized by the time dependent potential landscape V xy ( u, t )given in Eq. (8). In this section, we will encounter tra-jectories that are very similar to the Type-I-III trajecto-ries that were classified in Sec. III. We will again referto them as Type-I, II, and III trajectories. Type-I tra-jectories are again invariant spanning curves that limitthe momentum of chaotic trajectories and are hardly af-fected by the driving. Type-II trajectories move aroundthe torus in phase with the driving. This time, how-ever, the potential V xy describes a running wave, andthe Type-II trajectories correspond to particles that aretrapped in one of the moving potential wells. Type-IIItrajectories refer to trajectories that are unable to inverttheir direction of movement and move between succes-sive minima of V xy during each driving period with anaverage velocity of v av = 2 π .An overview of the phase space for M = 10 and r = 0 . E . The chaotic regionis surrounded by Type-I trajectories. Also, the r sin( u )dependence of V av leads to the presence of Type-III tra-jectories for very low driving amplitudes which, due toperturbations in form of the R dependent term in V av ,can be chaotic and lead to a splitting of the chaotic seasimilar to the one discussed in Sec. III. At the same time,however, there are major differences. Since our drivinglaw breaks parity and time inversion symmetries in theequations of motion, the resulting phase space is not sym-metric anymore. Instead of two fixed points with Type-IItrajectories like in Fig. 2, there is now only one that cor- FIG. 4. Poincare Surfaces of Sections (PSOS) for a parti-cle on a toroidal helix driven by a circularly polarized fieldin the xy-plane. The inset in (c) visualizes the dynamics ofdifferent trajectories (I, II, and III) on V xy ( u, t ) in the range u ∈ [ − πM/ , πM/ V xy and move aroundthe torus once during every driving period. (III) Chaotic andregular trajectories that (after the chaotic sea has split) moveone helical winding during each driving period. The PSOSare shown for driving amplitudes of (a) E = 3, (b) E = 10,(c) E = 40, (d) E = 400, and (e) E = 1000. In (e) the color-ing was changed to emphasize the split of the chaotic region;except for one highlighted periodic trajectory (black) all datapoints are colored blue. The two chaotic regions correspondto chaotic motion that is trapped in the moving potential, andchaotic motion that is (on average) slower than the movingpotential. The yellow curve in (f) shows the motion of thehighlighted (black) trajectory of (e) in the moving potential. responds to motion around the torus with the same di-rection as the rotation of the driving field . Furthermore,the emergence of Type-III trajectories with decreasingdriving amplitude is not symmetric anymore. For ourexample parameters (quasi-) periodic Type-III trajecto-ries with p > E = 500.The plotted trajectory u ( t ) + 2 πM t has a negative (orpositive) slope if the particle is moving faster (or slower)around the torus than the rotating driving field. Theparticle in the figure starts in the chaotic region outsideof the cantorus barrier (i.e. it is not a Type-II trajec-tory). In this region it will (on average) be too slow tomove in phase with the driving field. Once it crosses thecantorus, the dynamics become that of a chaotic Type-IItrajectory. This is highlighted by the inset, which zoomsinto a small region of the trajectory during which theparticle crosses the cantorus, briefly becomes a chaoticType-II trajectory, and then crosses the cantorus againinto the other chaotic region. The times particles spendas chaotic Type-II trajectories follow a power law with acritical exponent that depends on the driving amplitudeand the permeability of the cantorus.The permeability of the cantorus does not simply de-crease with the driving amplitude until the two chaoticregions are separated from each other. It switches mul-tiple times between being more or less permeable beforethe driving amplitude is large enough to separate the twochaotic regions. This is demonstrated in Figs. 5(b-d).They each show the PSOS of six trajectories (with thesame initial conditions [ u, p ] used for each figure) for var- FIG. 5. (a) Example trajectory in a comoving referenceframe that moves in phase with the driving field. When theparticle crosses the cantorus and becomes a chaotic Type-IItrajectory, u ( t ) − πMt will become constant, which is demon-strated in the inset. (b-d) Each figure shows a PSOS for sixtrajectories (with the same initial conditions for each figure)for (b) E = 300, (c) E = 400, and (d) E = 500. In (b)and (c) only one of the six trajectories manages to cross thecantorus, whereas in (c) all trajectories frequently switch be-tween the two types of chaotic motion. (e) Average velocityfor particles started in the chaotic sea with initial conditionschosen close to [ u, p ] ≈ [15 π, trajectories, each simulated for 10 driving periods.The chaotic Type-II trajectories are faster than those from theother chaotic region, so the velocity decreases with decreasingpermeability of the cantorus. ious driving amplitudes. For E = 300 and E = 400, thetwo regions are almost separated from each other and inboth cases only one of the trajectories manages to crossthe cantorus. Despite the vast difference in driving am-plitudes, there is very little difference in the permeabilityof the cantorus. In contrast, for E = 500 all of the trajec-tories switch frequently between the two regions. In thiscase, the presence of a cantorus is not even obvious fromthe phase space alone. Only when looking at the individ-ual trajectories (such as the one shown in Fig. 5(a)), wecan distinguish between the different chaotic dynamics ofthe two chaotic regions.The average velocity is different for both chaotic re-gions, and we shall use this to analyze the split-up of thechaotic region. This is shown in Fig. 5(e). It shows theaverage velocity v av as a function of the driving ampli-tude. Each data point was obtained from 10 trajecto-ries started in the chaotic region around [ u, p ] ≈ [15 π, driving periods for eachtrajectory. Note, that for very low E , when the Type-III trajectories for p > p < p <
0, leading to some biasin the data for very low E . Note also, that the curvemay slightly change for different simulation times, if theswitch time for the cantorus crossing exceeds the simula-tion time.At first, for very low E , v av decreases with increasingdriving amplitude which is caused by a combination ofType-III trajectories disappearing with increasing E , anda ‘bias’ in our initial conditions (compare v av in Fig. 3(e)and discussion thereof). Then, v av will (mostly) increasewith increasing driving amplitude until E ≈ E ≈ v av . From then on, there are peaks in v av whenever the trajectories can frequently switch be-tween the two chaotic regions: (e.g. the plateau around E = 500). Around E ∼ ∼ qER cos( ωt − u/M )of the running wave, with the smaller oscillations ∼ qEr cos( u ) cos( ωt − u/M ) acting as a perturbation that(for a wide parameter range) prevents the Type-II tra-jectories from stabilizing and becoming periodic. Due tothe position dependence of the smaller oscillations, theperturbation is always stronger for trajectories that aretightly bound i.e. closer to the fixed point, than for thosewith greater variations of ˙ u ( t ) − v av . This ‘perturbation’increases with increasing the helix radius r and thereforea larger helix radius requires larger driving amplitudesfor the chaotic region to split-up. Similar to the discus-sion of Sec. III, the winding number M changes the ratioof r/R and the velocity of the Type-II trajectories. Thiscan influence the general parameter regimes in which thesplit-up of the chaotic phase space region is encountered,however, we did not observe any changes in the underly- ing physics when varying the winding number M . V. SUMMARY AND CONCLUSION
We have investigated the dynamics of a charged parti-cle confined to a toroidal helix which is exposed to exter-nal driving forces originating from a time-dependent elec-tric field. While analyzing the phase space of the systemwe observed two different mechanisms for the split-upof the chaotic phase space region - both with their owninteresting consequences for the dynamics. We showedthat for low driving amplitudes the two different spa-tial scales of oscillating potential lead to a split-up ofthe chaotic region around p = 0. This prevents chaotictrajectories to invert the direction of their motion andleads to a consistent average velocity of v av ≈ π . Es-pecially notable is that this split effectively allows fordirected transport of the particle, even in a case wherethe spatio-temporal symmetries that are usually associ-ated with chaotic transport are not broken by the drivingfield.Specifically for driving with a circularly polarized fieldin the xy-plane, we found another mechanism for thesplit-up of the chaotic sea - this time splitting off a chaoticregion in which particles are trapped in a valley of thedriving potential. Trajectories confined to this separateregion of the phase space move around the torus in phasewith the driving field and will have a consistent averagevelocity of v av = 2 πM . Before this region is completelyseparated from the remainder of the chaotic sea, thereis a very large range of driving amplitudes for which thetrajectories can switch between the two chaotic regionsby crossing a permeable cantorus. The probability ofcrossing the cantous fluctuates heavily with the drivingamplitude.The presented spit-ups of the chaotic phase space re-gion are not unique to setups with confining forces andmainly depend on the different scales of oscillations inthe driving potential. A realization of similar physics ina driven lattice with spatially varying forces, or with ul-tracold atoms in an optical lattice seem feasible. Further-more, recent experiments have demonstrated the possi-bility of confining neutral atoms to a helical path [57],however, in such setups, the realization of our drivingforces may be a challenge. ACKNOWLEDGMENTS
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