Controlling vortical motion of particles in two-dimensional driven superlattices
CControlling vortical motion of particles in two-dimensional driven superlattices
Aritra K. Mukhopadhyay ∗ and Peter Schmelcher
1, 2, † Zentrum f¨ur Optische Quantentechnologien, Fachbereich Physik,Universit¨at Hamburg, Luruper Chaussee 149, 22761 Hamburg, Germany The Hamburg Centre for Ultrafast Imaging, Universit¨at Hamburg,Luruper Chaussee 149, 22761 Hamburg, Germany (Dated: June 1, 2020)We demonstrate the control of vortical motion of neutral classical particles in driven superlat-tices. Our superlattice consists of a superposition of individual lattices whose potential depths aremodulated periodically in time but with different phases. This driving scheme breaks the spatialreflection symmetries and allows an ensemble of particles to rotate with an average angular veloc-ity. An analysis of the underlying dynamical attractors provides an efficient method to control theangular velocities of the particles by changing the driving amplitude. As a result, spatially peri-odic patterns of particles showing different vortical motion can be created. Possible experimentalrealizations include holographic optical lattice based setups for colloids or cold atoms.
Introduction.—
Due to their experimental controlla-bility, driven lattice potentials have become an importanttest bed for the exploration of non-equilibrium physicalphenomena [1–3]. The inherent non-linearity and tunablesymmetries in these systems allow us to realize differ-ent non-equilibrium transport phenomena, the ‘ratcheteffect’ being one of them [4–15]. A ratchet rectifies ran-dom particle motion into unidirectional particle trans-port in an unbiased non-equilibrium environment. Cer-tain spatio-temporal symmetries of the system need tobe broken in order to realize it [16–18]. This leads to nu-merous applications across different disciplines, such ascontrolling the transport of atomic ensembles in ac-drivenoptical lattices [19, 20] both in the ultracold quantum [1]and classical regimes [2, 12], colloidal transport in drivenholographic optical lattices [21], particle separation basedon physical properties [22–24] and motion of vortices intype-II superconductors [25–27]. Due to the widespreadapplicability of such directed transport, there has beenextensive research to control the strength and direction ofthe ratchet current. Setups using one dimensional (1D)driven lattices have been shown to effectively accelerate,slow down or even completely reverse the direction oftransport [18, 28, 29]. Two dimensional (2D) driven lat-tices on the other hand offer a higher variability in termsof transport direction and for particles to be transportedparallel, orthogonal or at any arbitrary angle with respectto the direction of the driving force [21, 30, 31].In contrast to 1D, the 2D ratchet setups also allow forthe possibility to convert random particle motion intorotational or vortical motion leading to non-zero angularvelocity of the particles. This is particularly interestingsince it provides a method to realize rotational motionof neutral particles analogous to the motion of chargedparticles in a magnetic field. In fact, similar mecha-nisms have been used to generate artificial magnetic fieldsfor exploring topological quantum states with cold neu-tral atoms in periodically modulated lattices [32, 33].However, the extensive research on symmetry-breaking induced directed transport in the classical regime hasmostly focused on translational currents and the controlof rotational currents has remained largely unexplored.The few existing setups either lead to a diffusive vorticalmotion over an extended space [34] or requires speciallytailored potentials [35, 36] and temporally correlated col-ored noise [37, 38]. Furthermore, due to the lack of spa-tial tunability of the underlying lattice potential, thesesetups do not allow patterns of multiple vortices in spaceanalogous to the different spatial configurations of artifi-cial magnetic fluxes in the quantum regime [39].In this work, we address these key limitations andpresent a setup to realize controllable rotational motionof classical particles along closed spatial paths in drivensuperlattices. The individual lattices are modeled by aperiodic arrangement of Gaussian potential wells whosedepths can be individually modulated in a time-periodicmanner. We show that modulating different wells withthe same driving amplitude but different driving phasesallow us to break the relevant symmetries and generatenon-zero average angular velocities for an ensemble ofparticles. The angular velocities of individual trajectoriescan be controlled by varying the driving amplitude. Ad-ditionally, we demonstrate periodic spatial arrangementsof different types of rotational motion by modulating thedifferent potential wells with different driving amplitudesand phases.
Setup.—
We consider N non-interacting classi-cal particles of mass m in a 2D potential land-scape V ( r ≡ ( x, y, , t )= (cid:80) + ∞ m,n = −∞ ˜ U mn ( t ) e − β ( r − r mn ) formed by a lattice of 2D Gaussian wells centeredat positions r mn = ( mL, nL, m, n ∈ Z . Thedepths of the wells are modulated periodically intime by the site-dependent driving law ˜ U mn ( t ) =˜ V mn (cos( ωt + φ mn ) −
1) with driving frequency ω , driv-ing amplitude ˜ V mn and a temporal phase shift φ mn . In-troducing dimensionless variables r (cid:48) = r L and t (cid:48) = ωt and dropping the primes for simplicity, the equation ofmotion for a single particle at position r = ( x, y,
0) with a r X i v : . [ n li n . C D ] M a y Figure 1. Schematic representation of the two superlatticesetups A and B formed by the superposition of four squaresublattices driven with an amplitude V but at different phases φ i = ( i − π , i = 1 , , ,
4. Each colored (red) circle denotesthe position of an individual Gaussian well. The thick dashedlines in black denote the boundary of the lattice unitcells.The spatial period of setup A is (2 , ,
0) whereas that of setupB is (3 , ,
0) due to the presence of empty sites without anywells. The blue and green regions in Fig. (a) denote plaquetteshaving clockwise and anti-clockwise chirality with respect tothe spatial orientation of the wells with driving phases φ i .Remaining parameters are: V = 0 . α = 3, γ = 0 . velocity ˙r = ( ˙ x, ˙ y,
0) reads¨ r + γ ˙ r = + ∞ (cid:88) m,n = −∞ αU mn ( t ) ( r − R mn ) e − α ( r − R mn ) + ξξξ ( t )(1)where U mn ( t ) = V mn (cos( t + φ mn ) −
1) is the effectivesite dependent driving law with time period T = 2 π and driving amplitude V mn = ˜ V mn mω L . R mn = ( m, n, γ = ˜ γmω is the effective dissipation coefficient and the parame-ter α = βL is a measure of the widths of the wells. ξξξ ( t ) = ( ξ x , ξ y ,
0) denotes thermal fluctuations modeledby Gaussian white noise of zero mean with the prop-erty (cid:104) ξ i ( t ) ξ j ( t (cid:48) ) (cid:105) = 2 Dδ ij δ ( t − t (cid:48) ) where i, j ∈ x, y and D = ˜ γk B T mω L is the dimensionless noise strength with T and k B denoting the temperature and Boltzmannconstant respectively. Unless mentioned otherwise, wechoose V mn = V for all the wells, α = 3 and γ = 0 . φ mn forms a sub-lattice of our system. Our setup is hence a driven su-perlattice formed by the superposition of different sub-lattices, each driven with a distinct driving phase φ mn .Possible experimental realizations of such a 2D potentialinclude holographic optical lattices [21, 40–43] or opticalsuperlattices [44] with the lattice depth modulated viastandard amplitude modulation techniques [45, 46]. Therotational dynamics of particles in such a setup could beobserved with colloidal particles or with cold atoms inthe classically describable regime of microkelvin temper-atures [12, 21]. Rotational current due to symmetry breaking.—
Theasymptotic dynamics of particles in our setup can be ei-ther confined within a lattice unitcell such as in linear os-cillatory motion or vortical motion along arbitrary closedspatial curves. There can also be unconfined diffusive orballistic motion throughout the lattice. Different parti-cles exhibiting vortical motion can, in general, possessdifferent angular velocities. Hence in order to distin-guish vortical motion of a trajectory from ballistic, diffu-sive and vortical dynamics of other trajectories, we usethe angular velocity Ω ( t ) = [ ˙r ( t ) × ¨r ( t )] / ˙r ( t ) which isequivalent to the definition of curvature of planar curvesmeasuring the speed of rotation of the velocity vectorabout the origin [34, 47]. Since the particle dynamicsis confined to the xy plane, the only possible non-zerocomponent of Ω ( t ) is along ˆz , the unit vector along the z direction. The mean angular velocity of a trajectory isdefined as ¯Ω = t lim t →∞ (cid:82) t Ω ( t (cid:48) ) dt (cid:48) . For trajectories rotat-ing along a closed spatial curve with period ηT , the meanangular velocity can be expressed as ¯Ω = πτηT ˆz = τη ˆz (since T = 2 π ), where 2 πτ denotes the total curvature of the curve with the turning number τ defined as thenumber of times the velocity vector winds about its ori-gin [48]. The net rotational current, defined as the meanangular velocity of an ensemble of particles with differ-ent initial conditions, is given by J Ω = (cid:104) ¯Ω (cid:105) where (cid:104) ... (cid:105) denotes the average over all trajectories. Since the onlypossible non-zero components of Ω ( t ) , ¯Ω and J Ω is along ˆz , we drop the symbol ˆz henceforth.The necessary condition for any setup to exhibit a netrotational current is to break the symmetries which keepsthe system invariant but changes the sign of the angularvelocity Ω ( t ) [34]. There are only two symmetry trans-formations which can change the sign of Ω ( t ): (i) time re-versal together with optional spatial inversion and space-time translations: S t : t −→ − t + t (cid:48) , r −→ ± r + δδδ and (ii)parity or reflection P about any plane perpendicular tothe xy plane with optional spatial rotation R in the xy plane and space-time translations: S p : r −→ R ( P r ) + δδδ , t −→ t + t (cid:48) . Since our setup is dissipative, S t is bro-ken independent of our choice of the lattice potential V ( r , t ). However, the superlattice potential allows usto preserve or break the symmetry S p by controllingthe driving phases of the underlying sublattices. In or-der to illustrate this, we consider two setups A and B(Figs. 1(a,b)) each consisting of four square sublatticeswith the same driving amplitude V = 0 .
41 but differ-ent phases φ i = ( i − π , i = 1 , , ,
4. The sublattices insetup A have lattice vectors (2 , ,
0) and (0 , , L A = (2 , , L B = (3 , ,
0) with thelattice vectors being (3 , ,
0) and (0 , , Figure 2. Typical trajectories exhibiting rotational motion in(a) setup A and (c) setup B respectively over one time pe-riod of rotation (in colorbars). The colored circles denote thepositions of individual Gaussian wells with different drivingphases φ i . Figures (b) and (d) show the fraction of parti-cles ρ ( ¯Ω ) possessing mean angular momentum ¯Ω for differentnoise strengths D in setup A and B respectively. The insetsshow the variation of the net rotational current J Ω with D .Remaining parameters are the same as in Fig. 1. ttes are characterized by clockwise or counter-clockwisearrangement of Gaussian wells with driving phase φ i , i.e.of opposite chirality. Since the parity transformation S p reverses chirality, each of these plaquettes break the S p symmetry. However since the unitcell has equal num-ber of plaquettes with opposite chirality (two clockwiseand two anti-clockwise), the unitcell and hence the entiresetup A is symmetric with respect to S p . This impliesthat although the setup A might allow trajectories withdifferent mean angular velocities ¯Ω , the net rotationalcurrent J Ω must be zero. In contrast, the entire unitcellof setup B has an anti-clockwise chirality which can bereversed by S p and hence the setup B breaks S p symme-try. As a result one can expect J Ω to be non-zero.In order to verify our symmetry analysis and explorethe behavior of rotational current in our system, we ini-tialize N = 10 particles randomly within a square re-gion x, y ∈ [ − , × [ − , v x , v y ∈ [ − . , . t f = 10 T by numerical integration of Eq. 1 for differ-ent noise strength D . In the deterministic limit D = 0,all the particles in setup A exhibit only rotational mo-tion along closed curves either with mean angular mo-mentum ¯Ω = (vortex) or − (antivortex). Fig. 2(a)shows a typical trajectory in this setup having ¯Ω = − . The velocity vector winds around its origin in clockwisedirection once during the period of rotation 2 T , hence τ = − η = 2. The vortical motion persists as thenoise strength is increased to D = 0 . ¯Ω = − and ¯Ω = signifying that the netrotational current J Ω = 0 (Fig. 2(b)), as predicted byour symmetry analysis. Even for higher noise strengthup to D = 0 . D > . ¯Ω = 0 and hence J Ω = 0. The particles insetup B also exhibit rotational motion, however unlikein setup A, all the particles in setup B possess a meanangular momentum ¯Ω = = 0 .
6. An example trajec-tory in setup B in the deterministic limit can be seen inFig. 2(c). The velocity vector makes four anti-clockwise(at the four corners of the curve) and one clockwise (cor-responding to one full rotation along the curve) windingaround its origin during one period of rotation 5 T , hence τ = 3 and η = 5. For D (cid:54) . ¯Ω = 0 . J Ω = 0 . D > . J Ω . Control of rotational current.—
The question thatnaturally arises is that once we design a driven super-lattice which breaks the S p symmetry, for e.g. our setupB, can we predict the value of J Ω apriori? Specifically,how does the mean angular momentum ¯Ω of the trajec-tories depend on the system parameters? For a drivendissipative non-linear system like the present one, thiscan be answered by analyzing the asymptotic t → ∞ particle dynamics in the deterministic limit D = 0.The asymptotic dynamics of the particles is governedby the set of attractors underlying the phase space ofthe system, which can be of two types: (i) regular at-tractors denoting ballistic, linear oscillatory and rota-tional motions (ii) chaotic attractors denoting diffusivemotion. In order to distinguish between attractors cor-responding to rotational motion as compared to the oth-ers, we introduce a slightly modified angular momen-tum vector Ω (cid:48) ( t ) = [ ˙r ( t ) × ¨r ( t )] / [ | ˙r ( t ) || ¨r ( t ) | ]. Note that Ω (cid:48) ( t ) = sin ϑ ( t ) ˆz where ϑ ( t ) denotes the instantaneousangle between the velocity and acceleration vectors ofthe particle. Ω (cid:48) ( t ) transforms under S p and S t in ex-actly the same way as Ω ( t ). However since the values of Ω (cid:48) ( t ) are bounded in the interval [ − , Ω ( t ) which becomes large for small values of ˙r ( t ), it is agood quantity to differentiate between chaotic and reg-ular rotational dynamics of particles. To illustrate this,we inspect the bifurcation diagram of Ω (cid:48) ( t ) in Fig. 3(a) Figure 3. (a) Bifurcation diagram of Ω (cid:48) ( t ) as a function ofthe driving amplitude V depicting the chaotic (broad bluebands) and regular (thin blue lines) attractors of the setupB (see Fig. 1(b)). (b) The mean angular momentum ¯Ω ofthe attractors in Fig. 3(a) as a function of V . The values of ¯Ω for the regular attractors denoting rotational motion andthe turning number τ of the corresponding closed curves arelabeled with arrows. Remaining parameters are the same asin Fig. 1(b). as a function of the driving amplitude V for our setup Bby initializing particles with random position and veloci-ties and stroboscopically monitoring Ω (cid:48) ( t ) after an initialtransient [49]. For certain ranges of values of V , all theparticles in the setup exhibit chaotic motion (broad bluebands in Fig. 3(a)) such that Ω (cid:48) ( t ) takes all possible val-ues in the range [ − , V , theyperform regular periodic motion resulting in only spe-cific values of Ω (cid:48) ( t ). Most of these periodic motions cor-respond to particles performing rotational motion withdifferent non-zero ¯Ω (except for 0 . (cid:46) V (cid:46) .
25) de-pending on the value of V as shown in Fig. 3(b). Thisprovides an efficient method to design and control theangular momentum of the trajectories in our setup bysimply choosing the desired driving amplitude V . Ourprevious results (see Figs. 2(c,d)) is such an example forthe setup B with V = 0 . Multiple vortices.—
The ability to control the angu-lar momentum of the particles with different driving am-plitude V allows us to design lattices with spatially pe-riodic arrangements of multiple vortices. In order to il-lustrate this, we consider a specific setup as shown inFig. 4(a). It is designed such that the unitcell consists ofa collection of four plaquettes D , D , D and D . Eachplaquette consists of four Gaussian wells driven at dif-ferent phases φ i = ( i − π , i = 1 , , ,
4. The plaquettes
Figure 4. (a) Schematic representation of one unitcell of oursetup consisting of four plaquettes D , D , D and D withthe thick dashed lines denoting the plaquette boundaries. Thecolor filled circles denote the positions of individual Gaus-sian wells driven with amplitudes V = 0 .
51 or V = 0 . φ i . D and D ( D and D ) have anti-clockwise(clockwise) chirality with respect to the spatial orientationof the wells with driving phases φ i . Trajectories of particlesexhibiting vortical motion for D = 0 with positive (red) andnegative (blue) ¯Ω have been superimposed on the unitcell.The trajectories in D , D , D and D have ¯Ω = −
1, 1, − and respectively. An extract of the spatial arrangementsof the trajectories exhibiting vortical motion within differentplaquettes for D = 10 − and D = 10 − is shown in (b) and(c) respectively. Remaining parameters are the same as inFig. 1. D and D possess an anti-clockwise chirality whereas D and D have clockwise chirality with respect to thespatial arrangement of the wells with driving phases φ i .Additionally, the wells in D and D are driven withamplitude V = 0 .
51 and those in D and D with V = 0 . N = 10 particles randomly in this setup within a square region x, y ∈ [ − , × [ − ,
50] with small random velocities v x , v y ∈ [ − . , .
1] and propagate the ensemble up totime t f = 10 T . For D = 0, the particles exhibit vorticalmotion at long timescales with their angular momentumbeing governed by the plaquette they are trapped withinas shown in Fig. 4(a). The particles in D and D rotatewith ¯Ω = − ¯Ω = respectively, as predicted byFig. 3(b). Note that the plaquettes D and D can beobtained by a spatial parity transformation on D and D respectively. Hence the mean angular momentum ofthe particles in D and D has an opposite sign as com-pared to the particles in D and D respectively. Evenfor D = 10 − , such rotational motion persists and weobtain a periodic arrangement of particles in space ro-tating with different angular momenta (Fig. 4(b)). Fora higher strength D = 10 − , the vortical motion of par-ticles with ¯Ω = ± is destroyed and only the ones with ¯Ω = ± D (cid:62) × − eventually de-stroy all the vortex trajectories. Conclusions.—
We have demonstrated that superlat-ices of periodically driven localized wells provide highlycontrollable setups to realize different patterns of rota-tional motion of particles. The spatial arrangement ofthe lattices is responsible for breaking the relevant sym-metries, thus allowing for the non-zero average angularmomentum of an ensemble of particles. Our analysis ofthe underlying non-linear dynamical attractors providean efficient method to control the angular momentumof the particles as well as create a variety of periodicarrangements of vortical motion with different angularmomenta. Future perspectives include investigation ofrotational dynamics of particles operating in the purelyHamiltonian regime without dissipation, as well as in thequantum regime with the possibility to realize spatiallyvarying artificial magnetic fluxes.A.K.M acknowledges a doctoral research grant (Fund-ing ID: 57129429) by the Deutscher Akademischer Aus-tauschdienst (DAAD) and thanks J. 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