Local equilibria and state transfer of charged classical particles on a helix in an electric field
J. Plettenberg, J. Stockhofe, A. V. Zampetaki, P. Schmelcher
LLocal equilibria and state transfer of charged classical particles on a helix in anelectric field
J. Plettenberg, ∗ J. Stockhofe, A. V. Zampetaki, and P. Schmelcher
1, 2, † Zentrum f¨ur Optische Quantentechnologien, Universit¨at Hamburg,Luruper Chaussee 149, 22761 Hamburg, Germany The Hamburg Centre for Ultrafast Imaging, Universit¨at Hamburg,Luruper Chaussee 149, 22761 Hamburg, Germany
We explore the effects of a homogeneous external electric field on the static properties and dynam-ical behavior of two charged particles confined to a helix. In contrast to the field-free setup whichprovides a separation of the center-of-mass and relative motion, the existence of an external forceperpendicular to the helix axis couples the center-of-mass to the relative degree of freedom leadingto equilibria with a localized center of mass. By tuning the external field various fixed points arecreated and/or annihilated through different bifurcation scenarios. We provide a detailed analysisof these bifurcations based on which we demonstrate a robust state transfer between essentiallyarbitrary equilibrium configurations of the two charges that can be induced by making the externalforce time-dependent.
PACS numbers: 45.20.D-,37.10.Ty,37.90.+j,05.45.-a
I. INTRODUCTION
One of the commonly emerging structures in nature isthat of the helix. Especially in substances which are in-extricably connected with life such as the DNA moleculeand particular proteins, the helical structure appears tobe crucial for their stability and functionality [1–4]. Inaddition, this structure constitutes a major factor for theexhibition of peculiar physical effects in the aforemen-tioned macromolecules such as a negative differential re-sistance [5], a proximity induced superconductivity [6]and the existence of topological states accompanied witha quantized current [7]. Even more, due to their heli-cal structure both the DNA and the helical proteins areexpected to act as efficient spin-filters [8–10] or as field-effect transistors [11–14] making them good candidatesfor the developing field of molecular electronics.Meanwhile, less complex inorganic systems such as car-bon nanotubes are shaped nowadays in different forms,including a helix [15–17]. Given a pre-established he-lical confinement, identical long-range interacting parti-cles such as dipoles and charges can display a plethoraof different intriguing phenomena [18–24]. Even on thelevel of classical mechanics a helical constraint restrictsthe motion of the particles inducing an oscillatory effec-tive two-body potential for repulsively interacting elec-tric charges [20, 21]. This potential supports a numberof bound states tunable by the geometrical parametersof the confining helix. If the helix is inhomogeneoussuch bound states can dissociate through the scatteringof the particles, or conversely be created out of the scat-tering continuum [22]. For the many-body system differ-ent crystalline configurations can be formed accompanied ∗ [email protected] † [email protected] with non-trivial vibrational band structures (tuned alsoby the geometry parameters of the helix) as a result of thecomplex potential landscape of the effective interactions[23, 24].Apart from the interaction potential also the externalpotential acting on the particles is crucially affected bythe presence of a helical constraint. Of particular interestis the case of a constant transverse (i.e. perpendicular tothe helix axis) electric field acting on helically confinedcharges. This is known to induce a superlattice potentialfor any charge carrier, whose parameters can be easilytuned by adjusting the external applied field [25, 26].Such a field is supposed to enhance the spin-polarizedtransport through DNA [27] or helical proteins [28]. Evenmore, for non-interacting electrons in a tight-binding ap-proach the rotation of a transverse electric field in thetransverse plane leads to an adiabatic charge pumpingwhose current is quantized, pointing to the existence oftopological states [7].In view of the above studies the natural question ariseswhat would be the combined effect of interactions andan external transverse electric field acting on helicallyconfined charges. Since the two potentials (interactionand external) introduce in general different characteris-tic lengths of the same scale, frustration phenomena canappear leading to a rich static and dynamical behavior.A typical system presenting such a complex behavior isthe well-known Frenkel-Kontorova model which consistsof harmonically interacting particles on top of a sinu-soidal lattice potential [29]. The more involved form ofboth the external and the interaction potential in ourcase of helically confined charges makes even the two-body problem highly non-trivial, thereby motivating itsstudy. We find that the total potential landscape of sucha system is altered significantly by tuning the externalelectric field, giving rise to various kinds of local equilib-ria which emerge through different bifurcation scenarios.One of the observed effects is the merging of two neigh- a r X i v : . [ phy s i c s . c l a ss - ph ] S e p boring potential wells into one for high enough externalfields. Making use of this field-driven merging of the wellswe can achieve through particular quench protocols a ro-bust state transfer which involves the charges at differentinterparticle separations.This paper is organized as follows. In Sec. II wepresent our setup of the two charges confined to the he-lix in the presence of a constant electric field and dis-cuss briefly the features of the non-interacting case. Sec-tion III contains the study of the static problem for dif-ferent values of the external force, including a full dis-cussion of the existing critical points and the differentbifurcations. Using our knowledge of the static problemwe present in Sec. IV some applications of our system tostate transfer. Finally, Sec. V contains our conclusions. II. SETUP
We consider a system of two identical classical par-ticles of charge q and mass m each which interact viaCoulomb interaction. The particles are constrained tomove along a helix curve and exposed to a homogeneouselectric field E , thus experiencing a force F = q E . Dueto the constraint, their position vectors r i (where i = 1 , ϕ i ∈ R via r i = r ( ϕ i ) = (cid:0) R sin ϕ i , R cos ϕ i , Bϕ i (cid:1) , (1)where R denotes the helix radius and 2 πB the pitch (seeFig. 1). FIG. 1. (Color online) Illustration of the setup: Two identi-cal classical charges are constrained to move on a helix curveof radius R and pitch 2 πB . Additionally, they are subject toa homogeneous force field F (pointing along the x -axis in thissketch). Then the classical Lagrangian including the kinetic en-ergy, the external potential induced by the force F andthe Coulomb repulsion (with coupling constant g >
0) is given by L ( ϕ , ϕ , ˙ ϕ , ˙ ϕ )= (cid:88) i =1 (cid:16) m | ∂ ϕ i r ( ϕ i ) | ˙ ϕ i + F · r ( ϕ i ) (cid:17) − g | r ( ϕ ) − r ( ϕ ) | = (cid:88) i =1 (cid:16) m R + B ) ˙ ϕ i + F x R sin ϕ i + F y R cos ϕ i + F z Bϕ i (cid:17) − g (cid:112) R [1 − cos( ϕ − ϕ )] + B ( ϕ − ϕ ) . (2)Note the appearance of a geometry factor R + B inthe kinetic term here, such that when switching to arc-length coordinates s i = ϕ i √ R + B the canonical form m ˙ s i would be restored. The Coulomb repulsion po-tential is governed by the Euclidean interparticle dis-tance in the full three-dimensional space, which gives riseto an intricate non-monotonic effective potential in theone-dimensional angular coordinates. By virtue of thisgeometry-induced deformation of the interaction poten-tial metastable bound states can be formed for chargedparticles constrained to a helix, despite the underlyingCoulomb force being purely repulsive [21]. In a simi-lar fashion, the presence of the constraint turns the ho-mogeneous three-dimensional force field into a nontrivialpotential landscape when viewed in the angular coordi-nates. At a given position along the helix curve, a par-ticle only feels the component of the force that is locallytangential to the curve which gives rise to an oscillatorystructure in the effective one-dimensional potential.Let us start by briefly discussing the case of a singleparticle on the helix. From Eq. (2), the effective poten-tial reads V ( ϕ ) = − F x R sin ϕ − F z Bϕ , (3)where we have set F y = 0 and consider F x > F z (the force componentparallel to the helix axis) induces a linear decrease of thepotential energy, thus causing constant acceleration. Incontrast, F x (the force component perpendicular to thehelix axis), gives rise to a spatially oscillating contribu-tion to the effective potential. By itself, this contributionhas minima at ϕ = π + 2 πk , k ∈ Z , i.e. when the par-ticle is maximally pushed to the far end of the helix bythe perpendicular force, and correspondingly maxima at ϕ = − π + 2 πk , k ∈ Z . Whether there are fixed pointsin the presence of both F x and F z depends on their ratioas well as on the geometry of the helix. Specifically, if RB F x | F z | ≥ , (4)there is a spatially periodic sequence of maxima and min-ima, while otherwise no fixed points exist and there is nobound motion. III. CRITICAL POINTS AND BIFURCATIONS
Let us now turn to the full two-body problem. As canbe seen from Eq. (2), the deformed interaction potentialdepends on the angle difference ϕ − ϕ only, which inthe absence of the external force leads to a separation ofthe center-of-mass (COM) and relative motion [21, 22].We thus transform to the relative coordinate ϕ = ϕ − ϕ and the COM coordinate Φ = ϕ + ϕ , such that theLagrangian turns into L ( ϕ, Φ , ˙ ϕ, ˙Φ) = m ( R + B ) ˙Φ + m R + B ) ˙ ϕ − g (cid:112) R (1 − cos ϕ ) + B ϕ + 2 F x R sin Φ cos ϕ F z B Φ , (5)where again we have restricted ourselves to F y = 0 and F x >
0. It can be seen that the F z component does notbreak the COM separation and only induces a constantacceleration of Φ. In contrast, F x couples ϕ and Φ andimpedes the decoupling. We are mostly interested in theinterplay of this F x -induced term and the geometricallymodified interaction and will therefore focus on F z = 0in the following.Let us first switch to dimensionless units according to˜ x = xR , ˜ t = t (cid:114) gmR , ˜ L = LRg , ˜ F x,y,z = F x,y,z R g , (6)where ˜ x and ˜ t denote rescaled length and time variables,˜ L the rescaled Lagrangian and ˜ F x,y,z the rescaled forcecomponents. The rescaled pitch parameter is given by b = BR . Effectively, this amounts to setting m = g = R = 1. We will omit the tildes in the following. Indimensionless units, the potential term in the Lagrangianof Eq. (5) reads (with F z = 0) V ( ϕ, Φ) = 1 (cid:112) − ϕ + b ϕ − F x sin Φ cos ϕ . (7)The rest of this section is devoted to a discussion of thestationary points of this effectively two-dimensional po-tential landscape, which correspond to the fixed pointsof the two-body problem, and their bifurcations as theexternal force F x is varied. Let us therefore start by re-capitulating the fully force-free case of F x = 0 [21] inwhich the COM moves freely. The interaction potentialis dominated by the oscillatory cosine term for bϕ (cid:28) bϕ (cid:29) b (i.e. onthe pitch-to-radius ratio of the helix), the oscillatory con-tribution to the interaction potential can induce a finitenumber of metastable bound states with respect to ϕ .For b = 0 . V ( ϕ ) possesses a minimum-maximum pair in every interval [2 πr − π, πr ] , r ∈ N , up ϕ/ π V ( ϕ ) FIG. 2. (Color online) Two-body interaction potential as afunction of the relative coordinate ϕ for b = 0 .
2. Triangularmarkers pointing upwards (downwards) indicate the locationsof local minima (maxima). Beyond a critical ϕ = 2 πr c , indi-cated by the thick vertical line, no further maxima or minimaexist ( r c = 4 here). to some critical integer r c which depends on b ( r c = 4 inthe example shown in the figure). We will refer to theinteger r as the “order” of the associated pair of criti-cal points. With increasing order r , the potential wellsaround the minima become shallower due to the impactof the Coulomb repulsion at larger ϕ . The integer r c that counts the total number of minima (and maxima)decreases with increasing b and eventually, for b (cid:38) . r c = 0.For F x = 0 the above discussion of the critical pointsapplies with respect to the relative coordinate ϕ , whilethe COM is free such that all fixed points of the full two-dimensional potential V ( ϕ, Φ) are neutrally stable withrespect to the Φ-direction. Switching on a small F x > π + kπ, k ∈ Z . The stability ofthe two-dimensional fixed points with respect to the Φ-direction is governed by the product sin Φ cos ϕ and thusdepends both on k and on the order index r with respectto the relative coordinate. This results in a fixed pointlandscape as schematically shown in Fig. 3(a). The fullconfiguration space can be divided into cells of length π in the Φ-direction and length 2 π in the ϕ -direction, whereeach cell (and the fixed points it contains) can be labeledby the index k and the order r introduced above. Fixedpoints with k + r even are unstable in the Φ-direction,thus each minimum-maximum pair with respect to ϕ turns into a two-dimensional saddle-maximum pair (cellsof type A marked by blue solid ellipses in Fig. 3) as aninfinitesimal F x is switched on. In contrast, for k + r oddthe fixed points are stable with respect to Φ and thusany minimum-maximum pair with respect to ϕ results ina two-dimensional minimum-saddle pair (cells of type Bmarked by green dashed ellipses in Fig. 3). While cellswith r > r c contain no fixed points in the force-free limit F x = 0, this changes already at extremely small values of F x : For each type-A cell with r > r c a saddle-maximumpair emerges in a saddle-node bifurcation, while in con-trast type-B cells with r > r c remain empty of fixedpoints for any F x . The fate of the fixed points when F x isfurther increased is radically different for the type-A cellsand the type-B cells: In the A cells, there is a pitchforkbifurcation scenario leading to a quartet of two saddles,a maximum and a minimum at large F x , while in the Bcells all fixed points are annihilated in saddle-node eventsas F x is increased. Ultimately, at large values F x , thisleads to a checkerboard pattern of fixed points as illus-trated in Fig. 3(b), which is governed by the F x -term inEq. (7). In the following subsections, we provide a moredetailed analysis of the bifurcations in the two types ofconfiguration space cells. AABBB AABBBBBAAA BBAAA Φ π ϕ π (a) r = 1 r = 2 r = 3 r = 4 r = 52 r c k = 1 k = 2 k = 3 k = 4 AABBB AABBBBBAAA BBAAA Φ π ϕ π (b) minimummaximumsaddle FIG. 3. (Color online) Schematic view of the fixed-pointlandscape (not to scale) for (a) small values of F x and (b) largevalues of F x . Fixed points are indicated by markers, wherecircles stand for saddles, asterisks for minima and pluses formaxima, respectively. Different types of configuration spacecells (A and B, see text) are marked by solid blue and dashedgreen ellipses, respectively. In the type-A cells of r = 2, thefixed points emerging in the subcritical pitchfork bifurcationare indicated in red. This bifurcation results in two close-lyingminima at the same Φ with r = 1 and r = 2, respectively,separated by a saddle, as highlighted by the red shaded areas.Protocols for controlled transfer between these minima arediscussed in Sec. IV. A. Subcritical pitchfork bifurcations
Let us first focus on the fixed points in the cells of typeA, with k + r even, marked by blue ellipses in Fig. 3.For small values of F x , these cells contain a saddle anda maximum each. For r ≤ r c , this is true even in theforce-free limit, and for r > r c such a pair forms at verysmall values of F x (see below for a more quantitative estimate). Let us note first that all type-A configurationspace cells of the same order r are fully equivalent, sincethe potential V ( ϕ, Φ) is invariant under Φ → Φ + 2 π . Incontrast, there is no periodicity in ϕ due to the Coulombterm.With increasing F x , the relative distances between theparticles in the equilibrium configurations change whilethe COM coordinate remains localized at Φ = π + πk .Thus, in the two-dimensional landscape as sketched inFig. 3 the fixed points in the type-A cells move alongthe ϕ -axis. Specifically, in each of these cells the initialsaddle moves towards smaller ϕ and eventually crosses ϕ = (2 r − π , i.e. it moves into the lower half of the cell.This can be seen to be accompanied by a stability change:The initial saddle turns into a minimum. In parallel, twoadditional saddles emerge which subsequently remain at ϕ = (2 r − π but separate along the Φ-axis (as indicatedin red in the A cells of order r = 2 in Fig. 3).Figs. 4(a,b) illustrate the three-dimensional configu-rations of the charges on the helix that correspond tothese fixed points (for r = 1 and b = 0 . F x . In con-trast, in the initial saddle configuration (circle markersin Fig. 4(a)) the charges are located a bit more thanhalf a winding apart and pushed towards the same edgeof the helix by the increasing force, which reduces theirangular distance. When the distance drops below a half-winding, the configuration becomes fully stable (also withrespect to translating its center of mass). This is accom-panied by the emergence of two new equilibria in whichthe charges are separated by exactly half a winding, butwith a displaced center of mass. Inspecting the COMcoordinates of the fixed points vs. F x reveals the typ-ical subcritical pitchfork bifurcation form, in which thesymmetric equilibrium (with respect to reflection aboutthe initial Φ = π + πk ) gains stability while two unstablesymmetry-broken solutions emerge, see Fig. 4(c). Panels(d,e) of Fig. 4 show the corresponding deformation of thetwo-dimensional potential landscape V ( ϕ, Φ).Quantitatively, the critical force at which the subcrit-ical pitchfork bifurcation occurs in the type-A cells oforder r is found to be F PF, rx = b π (2 r − b π (2 r − ] / , (8)which at the same time provides an upper bound on thecritical force at which the saddle-maximum pair emergesin the initially empty cells with r > r c . While for large r the critical force F PF, rx always goes to zero, its overalldependence on r is an intricate one and is also controlledby b in a non-monotonic way, see Fig. 5. F x < F PF , x (a) maximumsaddleCOM max./saddlecritical points F x > F PF , x (b) maximumminimumCOM max./min.saddle 1COM saddle 1saddle 2COM saddle 2 ∆ Φ / π F x F PF , x (c) saddleminimumsaddle 1saddle 2 (d) Φ / π ϕ / π −1 0 1 2 311.051.1 (e) Φ / π ϕ / π −1 0 1 2 30.811.2 FIG. 4. (Color online) Subcritical pitchfork bifurcation inconfiguration cells of type A, r = 1, b = 0 .
2. (a) Equilib-ria below the critical force, F x < F PF,1 x . Plus and circlemarkers indicate the particle positions in the maximum andsaddle configuration, respectively, arrows show the directionsinto which the latter move with increasing F x . (b) Equilibriaabove the critical force, F x > F PF,1 x . The prior saddle hasturned into a minimum and new saddles have emerged whosecenter-of-mass coordinates drift with increasing F x . (c) Bi-furcation diagram showing the displacement of the COM co-ordinate ∆Φ = Φ − ( π + πk ) in the corresponding equilibriumconfigurations as in (a,b) versus F x . (d,e) Color-encoded pro-files of the potential landscape V ( ϕ, Φ) for F x < F PF,1 x and F x > F PF,1 x , respectively, with locations of the fixed pointsindicated by the same markers as in (c). Dashed lines denotethe edges of the configuration space cells. The minima inthe neighboring type-B cells are also shown as black asteriskmarkers. B. Saddle-node bifurcations
Let us now turn to the configuration space cells of typeB, with k + r odd (indicated by green ellipses in Fig. 3).Again, translational invariance in the Φ-direction ensuresthat such type-B cells of different k but identical r areequivalent. For r > r c , the type-B cells are empty offixed points in the force-free limit and remain empty forany F x . For r ≤ r c , there is a minimum and a saddlein each such cell for infinitesimal F x >
0. Increasing F x further, these approach each other along the ϕ -directionand eventually undergo saddle-node annihilations at acritical force F SN, rx , such that at large F x all type-B fixedpoints have disappeared.Fig. 6(a) shows the three-dimensional equilibrium con- b F P F , r x r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 FIG. 5. (Color online) Critical force at which the subcriticalpitchfork bifurcation takes place in the type-A configurationspace cells as a function of b and the order r of the cell. Notethe non-monotonic dependence both on r and b . For values of b for which the lines are dashed (beyond the cross markers),the involved fixed points do not exist in the force-free limit butare created in a saddle-node bifurcation at some F x < F PF, rx . figurations of the saddle and minimum configurations, re-spectively. The saddle configuration is characterized byboth charges sitting on the side of the helix that facesaway from the force, thus minimizing the field-inducedpotential energy, with a separation of around one windingbetween them. It changes only weakly with F x . In con-trast, the minimum configuration has the same COM co-ordinate, but a separation of only slightly more than halfa winding at small F x . As F x is increased, the charges arefurther pushed towards the far edge of the helix by theforce which increases their relative separation. Eventu-ally, the minimum and saddle equilibria become identicaland annihilate, see the bifurcation diagram in Fig. 6(b).Figs. 6(c-f) illustrate the corresponding changes in thepotential landscape V ( ϕ, Φ). Note in Fig. 6(e) how closethe annihilation point (at which the saddle-node bifur-cation occurs in the r = 1 B cell) is to the minimum ofthe r = 2 A cell. The saddle-node annihilation can beviewed as a merging of the potential wells around the twoinitial minima into one. This observation forms the basisof the state transfer applications to be discussed in thenext section.Finally, let us comment on the critical force F SN, rx atwhich the saddle-node annihilation occurs. This dependsboth on the geometry (via the pitch parameter b ) andthe order r of the configuration space cell under consid-eration. In contrast to the above pitchfork bifurcations,in this case there is no closed expression for the criticalforce as a function of b and r . Fig. 7(a) comprises ournumerical results for F SN, rx , which indicate a monotonicdecay both with b and r . Fig. 7(b) provides a comparisonto the corresponding F PF, r +1 x as will be relevant for thediscussion in Sec. IV. F x < F SN , x (a) minimumsaddleCOMannihilation points ϕ / π F x F SN , x (b) minimumsaddle (c) Φ / π ϕ / π (d) ϕ/ π V ( ϕ ) (e) Φ / π ϕ / π (f) ϕ/ π V ( ϕ ) FIG. 6. (Color online) Saddle-node bifurcation in configu-ration cells of type B, r = 1, b = 0 .
2. (a) Equilibria belowthe critical force, F x < F SN,1 x . Asterisk and circle markersindicate the particle positions in the minimum and saddleconfiguration, respectively, arrows show the directions intowhich these move with increasing F x and eventually meet,annihilating the fixed points. (b) Bifurcation diagram show-ing the relative coordinate ϕ of the corresponding equilibriumconfigurations as in (a) versus F x . Their COM coordinateremains unchanged with increasing F x . (c,e) Color-encodedprofiles of the potential landscape V ( ϕ, Φ) for F x < F SN,1 x and F x ≈ F SN,1 x , respectively, with locations of the fixed points in-dicated by the same markers as in (a) and (d,f) slices alongΦ = π . The minimum in the type-A cell of order r = 2 isalso included here, indicated by a black asterisk. IV. APPLICATIONS TO STATE TRANSFER
Combining the above insights into the bifurcations ofthe fixed-point landscape makes it possible to devise pro-tocols for controlled transfer of the two-particle systembetween equilibria of different order r (i.e. of differentseparation between the charges). The key idea is to applyan external force F x ( t ) with a suitable time-dependence.For example, starting out in the minimum of a type-Bcell of order r , one can (near-adiabatically) increase F x until it crosses F SN, rx , such that this minimum is anni-hilated. In the subsequent dynamics, the system movesaway from the former equilibrium position which can bevisualized as the motion of an effective particle in thetwo-dimensional V ( ϕ, Φ) potential, having different ef-fective masses in the directions of ϕ and Φ, cf. Eq. (5)[22]. By symmetry, this effective particle will move alongthe ϕ -axis, and it will get trapped in the minimum of theneighboring type-A cell of order r + 1 (if this minimum b F S N , r x (a) r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 b F S N / P F , r x (b) r = 1 (SN) r = 2 (SN) r = 3 (SN) r = 2 (PF) r = 3 (PF) r = 4 (PF) FIG. 7. (Color online) (a) Critical force at which the respec-tive saddle-node bifurcation takes place in the type-B config-uration space cells as a function of b and the order r of thecell. (b) Comparison of the critical force for the saddle-nodebifurcation in type-B cells of order r to that of the pitchforkbifurcation in type-A cells of order r + 1. For values of b for which the former (dotted lines) is larger than the latter(solid/dashed lines as in Fig. 5), i.e. left of the crossings indi-cated by circular markers, a force-driven configuration trans-fer as described in Section IV is possible. already exists). This is schematically shown in Fig. 8. Inother words, the saddle-node bifurcation can be viewedas a merging of the type-B minimum of order r and thetype-A minimum of order r +1 into a single joint potentialwell in which the effective particle then oscillates. Slowlyramping down F x below F SN, rx again, the saddle-nodeannihilation is reversed and the order- r minimum andsaddle are re-established, but the system remains boundin the potential well of the order r + 1 minimum. In thisway, transfer between equilibrium configurations of dif-ferent ϕ , increasing the separation between the charges,has been achieved. We have checked the reliability of thistransfer protocol in direct numerical simulations, con-firming in particular its robustness against variations inthe F x ( t ) pulse shape.A necessary condition for this transfer scheme to beapplicable is that the minimum in the A-cell of order r + 1 forms at a smaller F x than the critical force forthe saddle-node annihilation in the B-cell of order r , i.e.that F SN, rx > F PF, r +1 x . As can be seen in Fig. 7(b) thisis fulfilled for a large range of values of b and r , particu-larly for b and/or r at which the involved potential wellsbecome reasonably deep and thus arguably are most rele-vant for applications. In cases in which F SN, rx > F PF, r +1 x does not hold, annihilating the minimum in the B-cell oforder r induces time evolution towards the saddle in theA-cell of order r + 1, but due to the lack of stability withrespect to Φ the system is prone to perturbations in thecenter-of-mass coordinate.The simple transfer protocol described so far is largelyinsensitive to the details of the force profile F x ( t ) as longas it slowly crosses F SN, rx twice as required. However, itis by construction limited to transferring the system fromthe B-cell minimum that is annihilated in the saddle-nodebifurcation to the A-cell minimum at increased ϕ . Trans-fer in both directions is possible by using a more rapidvariation of F x ( t ), temporarily driving the system further F SN , x t F x ( t ) V ( ϕ ) ϕ FIG. 8. (Color online) Transfer between minimum energyconfigurations by tuning the external force as a function oftime: The bottom panel shows a possible F x ( t ) protocol,crossing the critical force F SN,1 x twice. The top panels showslices through the instantaneous potential landscapes V ( ϕ )(at Φ = π fixed and b = 0 .
2) at the marked times and indi-cate the dynamics of the effective particle (black circle), whichinitially resides in the minimum of an r = 1 type-B cell. As F x is slowly increased, the particle adiabatically follows thisminimum until the critical value for the saddle-node bifurca-tion in this cell is reached. Then the corresponding minimumvanishes and the effective particle starts to roll towards theclose-lying minimum in the neighboring type-A cell (of order r = 2 here). Slowly tuning F x back below the critical value,the r = 1 minimum is restored but the particle still oscillatesin the vicinity of the r = 2 minimum. away from equilibrium. For instance, we consider suddenquenches from an initial F x to a larger value F (cid:48) x , hold-ing it for a delay time ∆ t and quenching back to F x . Ifthe initial F x < F SN, rx and the intermediate F (cid:48) x > F SN, rx ,this corresponds to a sudden switch from a two-well po-tential (as in panel 1 of Fig. 8) to a single-well potential(as in panel 3). After the quench to F (cid:48) x , the effectiveparticle will thus perform oscillations in the joint welland depending on the time ∆ t at which one switchesback to F x it may end up trapped in either of the twowells. Evidently this transfer between the minima worksin both directions but requires more fine-tuned choices of F (cid:48) x and ∆ t . Fig. 9(a) illustrates the dynamics triggeredby a sudden quench to a force F (cid:48) x > F SN, rx . The effec-tive particle starts out at the position of one minimumof the pre-quench potential at F x (these former minimaare indicated by diamond markers) and oscillates along ϕ through the joint potential well. For the chosen valueof F (cid:48) x , the turning point of the trajectory lies close to theposition of the desired other pre-quench minimum (dia-mond marker), see also the phase space plot in Fig. 9(b).Switching back from F (cid:48) x to F x after a suitable delay time∆ t adapted to the oscillation period will thus leave thesystem close to the desired target minimum with littleexcess energy.The quench-based transfer protocol is relatively stablein terms of varying the initial conditions. Quantifyingthis for a special case, let us analyze the transfer from r = 1 to r = 2 at b = 0 . F x = 0 .
1, i.e. as in Fig. 9(a,b).Instead of just a single trajectory as in that figure, weconsider now an ensemble of effective particles, initial- (a) Φ / π ϕ / π −0.5 0 0.51.522.5 (b) ˙ ϕ/ π ϕ / π (c) Φ / π ϕ / π −0.1 0 0.13.7544.25 (d) ˙ ϕ/ π ϕ / π FIG. 9. (Color online) (a) Color-encoded profile of the po-tential landscape V ( ϕ, Φ) at F (cid:48) x = 1, b = 0 .
2. The diamondmarkers represent the r = 1 (initial state) and r = 2 (targetstate) minima before the quench ( F x = 0 . F (cid:48) x , the effective particle oscillates around thenew minimum indicated by the asterisk and follows the oscil-latory trajectory indicated as a red line whose turning pointlies close to the target state. This is also seen in the phasespace plot of the trajectory in (b). For these parameters, ahalf period of oscillation takes around t = 3 .
82 and choosing adelay time ∆ t for switching back to the initial F x close to thiswill result in the system being trapped near the target min-imum. (c,d) The quench-based transfer scheme also appliesfor transfer between the r = 2 and r = 3 minima, F (cid:48) x = 0 . ized in a circular neighborhood of the r = 1 pre-quenchminimum at F x . We quench to a force F (cid:48) x > F SN, rx andback to F x after a delay time ∆ t and then evaluate foreach trajectory whether it ends up being energeticallytrapped near the r = 2 target minimum or not. Forone combination of F (cid:48) x and ∆ t the result is shown inFig. 10(a), demonstrating that although the ensemble isinitially spread relatively widely, a large fraction of tra-jectories indeed ends up trapped near the target state. Asa quantitative measure of this, we consider the fraction F of successfully transferred trajectories in the ensem-ble. Even for the rather widespread ensembles we use,this adopts maximum values of around 90% and upondeviating from the optimal parameters F (cid:48) x and ∆ t it stillremains substantially large, see Figs. 10(b,c).Such a quench-based protocol thus allows robust bi-directional configuration transfer between the stableequilibria of type-B, order r and type-A, order r +1 with-out affecting the center of mass. To complete the trans-fer toolbox, it is desirable to have a means of moving thecenter-of-mass coordinate in a controlled way. As seenabove, Φ can in principle be accelerated and deceleratedby a force component F z along the helix axis. Alterna-tively, one can employ a force that is purely perpendic-ular to the helix axis and rotate it, i.e. in Cartesian co-ordinates F = F (cos α, sin α, V F ( ϕ, Φ) = − F sin (Φ + α ) cos ϕ , (9) (a) t t Φ / π ϕ / π
123 0.51 successfulfailsstate transfer: (b) ∆ t F (c) F ′ x F FIG. 10. (Color online) (a) Robustness of the quench-basedtransfer scheme: Colors encode the V ( ϕ, Φ) potential land-scape for F x = 0 . b = 0 .
2. At t an ensemble of effectiveparticles is initialized in a neighborhood of the r = 1 min-imum (lower red asterisk). After a quench to F (cid:48) x = 1 eachof these particles is propagated for ∆ t = t − t = 3 .
82 suchthat the final positions lie near the r = 2 minimum (upperred asterisk). For each trajectory it is then evaluated if, whenquenching back to F x = 0 .
1, it would end up energeticallytrapped in the r = 2 minimum (blue vertical hatching, trans-fer successful) or untrapped (black cross hatching, transferfailed). (b,c) Fraction of successfully transferred trajectories F as a function of the delay time ∆ t (at fixed F (cid:48) x = 1) and F (cid:48) x (at fixed ∆ t = 3 . while the ϕ -dependent interaction potential is, of course,unaffected. This means that adiabatic tuning of the ro-tation angle α leads to transport of the center-of-masscoordinate without changing the relative position of thecharges. Specifically, one can start with a force along the x -axis (as done above) and rotate it by a multiple of 2 π .This admits a simple transfer between type-A (or type-B)equilibria of different k (i.e. horizontally in Fig. 3).Transfer from a type-A to a type-B cell of the same or-der r is also possible, for instance with the following pro-tocol: Start in a type-B minimum with a force along the x -axis and rotate by α = π . Then the COM of the chargepair has already been transferred to the desired position.After slowly decreasing F x to almost F PF ,rx (such thatthe minimum is not lost crossing the pitchfork bifurca-tion point) and then quenching back to α = 0, the con-figuration will be close to the desired type-B equilibriumand oscillate around it. An intermediate reduction of theabsolute value of the force is helpful here in order to sup-press the potential gradient along Φ, that may otherwiseinduce undesired COM motion after the second quench.It also serves to bring the ϕ -positions of the pre-quenchand target states closer together and thus reduce the ex- cess energy. Subsequently, F x can be increased back toits initial value to complete the transfer. The reversetransfer from a B-cell to an A-cell of identical r can es-sentially proceed along the same lines; since one does notstart in a type-A minimum here, one can even drop therestriction F x ( t ) > F PF ,rx for all times. V. CONCLUSIONS
We have investigated the local equilibria and classi-cal dynamics of two charged particles confined to a he-lix and subject to a constant external electric field. Inthe absence of the field the effective two-body interactionpotential exhibits multiple minima depending on the ge-ometry parameters. The existence of a finite transverseelectric field induces a coupling of the center-of-mass tothe relative coordinate and thus alters the total potentiallandscape by localizing the center-of-mass coordinate i.e.favoring particular values of it. By increasing the exter-nal field amplitude the potential landscape keeps alteringand various critical points (saddles, minima or maxima)emerge or annihilate through different bifurcation sce-narios, signifying the very rich nonlinear behavior of thissystem.Among the observed effects there exists a merging oftwo potential wells corresponding to different relativeconfigurations into one with increasing the magnitude ofthe external force. By providing the force with a certaintime dependence, we succeed in transferring the particlesfrom one state into another in which the charges are sep-arated by a different interparticle distance. Apart fromthese transfers being in general robust with respect to theparticular type of the time dependence, different quenchprotocols can be used to achieve transfer between vari-ous states including also states with different values ofthe center-of-mass coordinate. Therefore by choosing asuitable protocol even a transfer between arbitrary statesis possible, at the cost of acquiring in general much excessenergy which should be somehow drained from the parti-cles (e.g. through friction) in order to retain control overthe transfer. Such transfer mechanisms are reminiscentof the field-driven charge pumping investigated recentlyin [7] for a tight-binding system with a helical structureargued to be a suitable model for helical molecules.Concerning the experimental realization of such asetup certain advances have been made apart from thefield of nanofabrication [15–17] also in the field of ultra-cold atoms [30–32] where the realization of optical helicaltraps has recently been proposed. From the theory side,further studies could be dedicated to investigations of themany-body analogue of the present system. Given thevariety and the wealth of effects in much simpler modelswhich combine both an external and an interaction po-tential, an example being the Frenkel-Kontorova model[29], it is natural to expect this to hold as well for themany-body system of helically confined charges in thepresence of an external field where especially the effec-tive interaction potential is already complex. [1] C. R. Calladine, H. Drew, B. Luisi, and A. Travers,
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