Interaction Induced Directed Transport in AC-Driven Periodic Potentials

0.3 MB·English

by  Benno Liebchen

Interaction Induced Directed Transport in AC-Driven Periodic Potentials
Benno Liebchen1,∗ and Peter Schmelcher2,3,†
1SUPA, School of Physics and Astronomy, University of Edinburgh, Edinburgh EH9 3FD, United Kingdom
2Zentrum fu¨r Optische Quantentechnologien, Universita¨t Hamburg,
Luruper Chaussee 149, 22761 Hamburg, Germany
3The Hamburg Centre for Ultrafast Imaging, Universita¨t Hamburg,
Luruper Chaussee 149, 22761 Hamburg, Germany
(Dated: January 6, 2015)
We demonstrate that repulsive power law interactions can induce directed transport of particles
indissipative, ac-drivenperiodic potentials, inregimes wheretheunderlyingnoninteractingsystem
exhibits localized oscillations. Contrasting the well-established single particle ratchet mechanism,
this interaction induced transport is based on the collective behaviour of the interacting particles
5
yielding a spatiotemporal nonequilibrium pattern comprising persistent travelling excitations.
1
0
PACSnumbers: 05.60.Cd,05.45.Xt,05.45.Yv,05.45.-a
2

n
a Introduction The ratchet effect describes the emer- ratchet mechanism.
J
genceofadirectedparticletransportdue toaninterplay Setup We consider N repulsively power law interact-

3 of symmetry breaking[1–3] and nonlinearityin a system ing point particles in one dimension with coordinates

where all applied forces and gradients vanish after aver- x and mass m that are exposed to a frictional force
] j
S aging over time and space [2–7]. Originally inspired by F = γx˙ and a spatially periodic and laterally os-
R
−
P the desire of understanding how molecular motors con- cillating lattice potential V(x,t) = V cos2[kx f(t)].
0
. vertchemicalenergyintodirectionalmotion[4],ratchets Here L=π/k is the distance between two adjace−nt min-
n
i havenowadvancedtoawidespreadandtopicalparadigm imaandf(t)=a[cos(ω0t)+sin(2ω0t)]isthebiharmonic
l
n with applications in various branches of nonequilibrium drivinglaw,withfrequencyω andamplitudea. Thedy-
0
[ atomic [3, 8–11], condensed matter [7, 12–17] and bio- namicsofthismodelisdescribedbyN couplednonlinear,

1 physics [18–23]. Specifically, modern atomic cooling and non autonomous Newtonian equations of motion:
v trapping techniques [24, 25] allow for particularly clean
6 and versatile realizations of ratchets in optical lattices N
9 [8–11, 26, 27]. Since the established ratchet effect is of mx¨j +γx˙j +V0ksin{2k[xj −f(t)]}− X Fjk =0
5 single particle character that can survive interactions, k=1,k6=j
0 x x
1.0 mcleosrtegwimoreks(foforcruesveidewosnstehee[3n,o6n,-i7n,t1e1ra])ctoirngonsicnognletinpuaortuis- Fjk =α|xj −j −xk|kr+2 (1)
models with infinitely many degrees of freedom and ef-
0 This model might describes charged (r = 1) or dipolar
5 fectivelylinearorKuramoto-typeinteractions[6,28,29].
particles (r = 2) in an ac-driven (optical) lattice poten-
1 However, many relevant setups, such as driven dipolar
tial including a dissipative frictional force.
: atoms or molecules [30, 31], charged colloids [32] or mi-
v Tostudytransportphenomenaatafixedparticledensity
i croscopic ions [33] experience power-law (Coulomb and
X dipole-dipole) interactions, evoking the question of their weemployperiodicboundaryconditionsx+LN ∼=x. We
incorporate the corresponding Ewald sums for the inter-
r impact on directed currents. For the dilute Hamiltonian
a actionforcesF inoursimulations[35],butnotethatall
regime, it was already shown that these interactions can jk
results presented below remain qualitatively valid when
induceintriguingdynamicalreversalsofthetransportdi-
truncating the interaction beyond next neighbour ones.
rection [34].
Physically, we can distinguish four different forces de-
In this work, we demonstrate that repulsive power law
termining the dynamics of a given particle ensemble:
interactionscanevenberesponsibleforthe emergenceof
(i) the force exerted by the instantaneous static lattice
adirectedparticletransport. Consideringionsordipoles
(F := V ksin(2kx) V k). If dominant, this force
in dissipative and laterally oscillating lattice potentials, s − 0 ∼ 0
drags the particles along the minima of the oscillating
interactioninducedcurrentsappearinparameterregimes
lattice. (ii) The time dependent pseudo force which acts
where frictional energy losses suppress any directed cur-
on a particle in the comoving coordinate system of the
rent in the corresponding noninteracting system. The
lattice (F := mf¨(t) γf˙(t) ( aω ) γ2+(mω )2).
physicaloriginofthesecurrentsisaninteractioninduced o − − ∼ − 0 p 0
(iii) The frictional forces F = γx˙ γω L slowing
collective behaviour of the particles which is represented f − ∼ 0
down the particles in favour to keep them at fixed po-
by a spatiotemporal nonequilibrium pattern comprising
sitions in the laboratory coordinate system. (iv) The
persistent travelling excitations and has to be carefully
repulsive interaction forces F α/Lr+1. In the fol-
distinguished from the well established single particle jk ∼
lowing parameters are chosensuch that F ,F ,F and in
s o f
2
many cases also Fjk are comparable for sufficiently long 〈〈 d 〉〉 20 〈〈 d 〉〉
times, i.e. none of these forces can be treated as a small
perturbation. To be in a regimewhere the drivingforces 300 10 (a) #particles
1000
andthe nonlinear components of Fs as well as the power 200 0 (b)
lawshapeofF playasignificantrole,ω ischosenclose 0 50 t/T
jk 0 0
x
to the resonance of the underlying linearised single par- 100 0 150 300
ticle problem(a damped drivenharmonic oscillator),i.e.
2ω0 ∼ p2Vk2/m. This way, we are far from the com- 00 200 400 600 800 t/T
paratively simple regimes of adiabatically slow and fast
driving resulting in effectively time independent prob- FIG. 1. Time evolution hhdii(t) for N = 6 (ochre: N = 20),
lems. averaged over 104 ensembles, for α = 0 (red, 2nd lowest);
In the following we investigate the impact of different α=1(green,4thlowest);α=2(blue,highest);α=5(black,
interaction strengths α > 0 on the transport proper- 2nd highest; ochre 3rd highest); α = 20 (cyan, 3rd lowest);
α=175(purple,lowest). Inset(a): Magnificationoftheshort
ties of spatially uniform initial states with x (t = 0) =
j
time behaviour of hhd(t)ii. Inset (b): Distribution of hdi for
L(j N/2+1/2)+a (N even and j =0,1...N 1) and
− − α = 5.0 and 10000 ensembles at t = 1000(2π/ω0). Parame-
random velocities v (t = 0), uniformly chosen in an in-
j ters: a = m = k = 1.0;V = 25;γ = 8.6;v0 ∈ (−250,250);
terval (−v0,v0). To avoidcapture by specific low-energy ω0=3.53605 (2ω0 =p2Vk2/m).
asymptotic states (attractors) we choose v to be large,
0
such that frictional forces dominate at short time scales.
Experimentally, different values of α can of course be
achieved indirectly by tuning other parameters such as t/T 104 [FT(v)]2 1.5
ω ,V or γ that are easier to control. 102
0 0
Results WeintegrateEq.(1)usingtheRunge-Kutta- 19900 100 1
10−2
Dormand-Prince integration scheme [36] and calculate P 0 1 2 3 ω
the time evolution of the ensemble averaged position 0.5
hd(t)i := (1/N)Pxj(t) (without setting x+NL back 19920 O PP
to x). Although noise is of course naturally present in
0
dissipativemanyparticlesystems due tothe fluctuation-
dissipation theorem [37, 38], we do not include it explic-
19940
itly in our simulations but simply note that our results −0.5
areindeedrobustwithrespecttoweak(Gaussian)noise.
Fig.1anditsinset(a)showthetimeevolutionof d(t) ,
hh ii 19960 −1
where . refers to the averaging over many N-particle
hhii
ensembles. For vanishing interactions (α = 0) as well as
for weak interactions (α=1.0) d(t) saturates after a −1.5
short initial growth at t/T 20hhand iwie observe that all 19980 1 3 5 j
∼
particlesasymptoticallyperformpureonsiteoscillations.
Increasing the strength of the repulsive interactions, one FIG. 2. Asymptotic trajectories xj(t) (color/grey scale de-
might expect that d(t) saturates even earlier as re- notes the value of xj(t)−L/2 modulo L = π) over particle
hh ii number j = 1,2...6 and time t/T. Inset: Low frequency ex-
pulsive interactions restrict the mobility of the particles
tract of the squared Fourier spectrum of a typical vj(t) :=
dynamics even further thereby impeding the emergence
x˙j(t) trajectory. Parameters as in Fig. 1 (N =6,α=5).
ofanysingleparticlebasedtransportinthesystem. How-
ever, for α = 2;5 d(t) grows linerly, i.e. we observe
hh ii
apermanentdirectedcurrentofthe underlyingionsthat
emergesspontaneouslywhentherepulsiveinteractionex- It is illustrative to further resolve this effect on a trajec-
ceeds a certain threshold. Fig. 1, inset (b) shows that tory level (Fig. 2). While for all non transporting cases
apart from a few ensembles that stay close to their ini- (α = 0;1;20;175) we asymptotically obtain exclusively
tial positions, after 1000 oscillations of the lattice d(t) on site oscillations (not shown), for α = 5.0 the trans-
h i
is very similar for most ensembles which propagate with port is represented by a spatiotemporal pattern. The
similar velocities through the lattice. For stronger in- latterconsistsofphasesofon-siteoscillations(exemplar-
teractions (α = 20;175) d(t) saturates quickly as for ilyencircledandmarkedwithan’O’)thatareinterrupted
hh ii
α = 0, i.e. the directed current dies out after a short bypropagationprocessesfromonelatticesitetoitsright
initialdriftandeachoftheunderlyingionsistrappedon neighbour (’P’) and by propagationprocesses oftwo lat-
itslatticesite. Hence,weobserveadirectedcurrentthat tice sites (’PP’). The spatiotemporal pattern of the on-
is interactioninduced but does not survivefor strongin- siteoscillationsandpropagationprocessesisreminescent
teractions. of a situation where all particles oscillate approximately
3
4 〈〈v 〉〉 (a) 〈〈E 〉〉 tractors exist as long as the energy transfer from the os-
2 d kin 20 cillating lattice to the particles can periodically balance
0 10 the frictional energy loss, which often happens within a
γ singleperiodofthedriving. Theslowestpossibledirected
−2 0
00 22 44 66 88 1100 1122 1144 motion which allows for such a balancing is typically a
(b) motionofadistanceLforadrivingperiodT. Insynchro-
nizationtheorythisisa1:1(orak :k,k N)phaselock-
∈
ing resonance between the particle motion and the lat-
(c) ticeoscillation[39,40]. Aslowermotionofe.g. onlyL/2
within T would correspond to a particle motion which
alternatingly gains and looses energy to the lattice and
E
(d)
∆ does not allow for a balancing of the energy. Note that
20 the existence of such a minimal transport velocity is in
strong contrast to both the Hamiltonian regime (γ = 0)
00 0.1 0.2 0.3 0.4 0.5 t/T and the regime of strong noise (high temperature/low
particlemass)wherethedirectedtransportemergesasan
asymmetric(chaoticdeterministicorBrownian)diffusion
FIG.3. (a): Asymptoticsingleparticle (α=0) transportve-
which can generally become arbitrarily slow [6]. In the
locity hhv ii (blue, upper curve) and kinetic energy per par-
d
ticle hhEkinii = (m/2)hhx˙2ii (green, lower curve) as a func- considered dissipative system, the minimal asymptotic,
tion of γ, respectively averaged over 104 single particle en- time-averaged transport velocity of one particle in the
sembles and the time interval t/T ∈ (500,1000). Remaining lattice that is typically possible is v = ω /(2k) 1.77,
d 0
parameters as in Fig. 1. (b): Projection of the asymptotic corresponding to the long plateau in (Fig. 3a; bl≈ue) be-
quasiperiodic dynamics of representative, exemplary trajec-
tween γ 2.1 and γ 7.3. When γ becomes sufficiently
toriesinthe(x,v,t)-singleparticlephasespaceontothex−v ≈ ≈
large, the transporting attractors (Fig. 3b; black) disap-
plane (x taken modulo L = π) for γ = 8.0,α = 0 (black,
pearandwehaveexclusivelynontransportingattractors
highest saturation), γ = 8.6,α = 0 (blue), γ = 8.6,α = 5
(green, lowest saturation). (c): Asymptotic kinetic ener- (Fig. 3b; blue) in the underlying phase space. Values for
gies per particle hhEkinii/N for N = 6 and α = 0 (red); vd that are smaller than ω0/(2k) in Fig. 3a are facil-
hh ii
α = 1 (green); α = 2 (blue); α = 5 (black); α = 20 (cyan); itated by non transporting attractors that coexist with
α = 175 (purple). The red dot is lying on top of the green transporting ones. Some initial conditions then lead to
and the cyan one. (d): Time evolution of the kinetic en-
on-site oscillations while others lead to transport with
ergy E∆ := (m/2)∆˙2(t) for α=0 (red, curves without dip), v = ω /(2k). Larger transport velocities are possible
α = 5.0 (blue), ∆(t = 0) = 2.25 and different ∆˙(t = 0) < 0 h di 0
due to higher resonances such as the peak at γ 1.1
(line styles). m=γ =1.0. ≈
representing particles that move 4L within 2T,i.e. a 4:2
phase locking.
In contrast to the single particle transport that is based
in phase while an additional travelling collective exci- ontheexistenceoftransportingattractorswithacertain
tation represents the propagation process leading to di- minimal velocity, the presence of interactions allows for
rected transport. the emergence of directed currents as a collective phe-
A Fourier analysis of the transporting pattern (inset nomenon. Specifically, while for α =0 the Fourier spec-
in Fig. 2) reveals that it is of quasi-periodic character trum consists of peaks at ω0 and its higher harmonics
persisting asymptotically. The low frequency part of 2ω0;3ω0... (not shown) for α = 5 the Fourier spectrum
the Fourier spectrum shows subharmonic frequencies of (inset in Fig. 2) shows subharmonic frequencies repre-
nω /48withn=1,2,3...and2ω = 2Vk2/mreflecting senting the possibility of a directed current of the inter-
0 0
p
the collective nature of the interaction induced directed acting ensemble without requiring that the underlying
transport. particlespropagateby(atleast)onelattice siteperdriv-
Analysis Wenowfocusonthemechanismunderlying ing period. Indeed, in Fig. 1 the transporting ensembles
theinteractioninduceddirectedcurrents. Sinceournon- propagate with vd = L/(8T) = ω0/(16k) < L/T =
h i
linearsystemisdrivenoutofequilibriumandbreaksboth ω0/(2k). On the level of the 2N +1 dimensional N par-
parity- and time-reversalsymmetry it fulfills the general ticle phase space, the corresponding emerging collective
criteriaforobservingadirectedcurrent[3]. Indeed,inthe current (Fig. 1) is represented by a transporting attrac-
single particle regime (α = 0 or N = 1) a directed cur- tor(Fig.3b;green). Forvisualizingthishighdimensional
rentcanbeobserved,aslongasγ issufficientlysmall,i.e. attractor we exploit the fact that all N particle trajec-
for weak dissipation (Fig. 3a; blue curve). This current toriesunderlyingthecorrespondingtransportingpattern
is represented by one or several coexisting transporting (Fig. 2) are identical except for certain phase shifts, i.e.
limit cycle attractors (Fig. 3b; black) in the underlying the single trajectory shown in Fig. 3b is representative
single particle phase space (x,v,t). Physically such at- for the complete N particle attractor.
4
Let us further illuminate the microscopic mechanisms 〈〈d〉〉
t/T 〈〈E 〉〉
underlying the interaction induced transport. The in- 1000 947.1 1 kin
teraction among the particles can, in the course of the 1.5
952.9 (a) (b)
dynamics, reduce the loss of kinetic energy due to fric- 0
958.7
tion, which in turn of course facilitates the emergence 500 1.25
964.5 −1
of directed currents. Consider the time evolution of the
kineticenergyE (t):=m∆˙2/2correspondingtotherel- 1 3 5 j 1
∆
0
ative motion ∆:=x2 x1 of a free frictious two-particle 0 200 400 600 800 t/T
−
system mx¨ +γx˙ α/[x x ]2 =0. While the fric-
1,2 1,2 1 2
tionalenergylossconc±erningth−ecenterofmassdynamics FIG.4. Timeevolutionhhdii(t)forN =6,averagedover104
E (t):= mx˙2/2 with x:= (x +x )/2 is of course inde- ensemblesforα=0.25;2;5(threeuppercurvesingreen,blue
x 1 2
pendentofthevalueofα, E (t)initiallydecreasesfaster and black, lying almost on top of each other); α=20 (cyan,
∆
second lowest curve); α = 175 (purple, lowest curve) and
for α > 0 as compared to α = 0 (see Fig. 3d) but then
α = 0 (covered by the purple curve). Inset (a): Like Fig. 2.
grows again and becomes larger than for α = 0 (where
Inset (b): Like Fig. 3c.) Upper dots: α = 0.25 (covered by
E (t) exp[ 2γt/m]). The zero in E (t) (see blue
∆ ∆ the blue dot), α = 2 (blue) and α = 5 (black). Lower dots:
∝ −
curvesinFig.3d)herebyindicatesthecloseencounterie. α = 20 (cyan) and α = 175 (purple, covering the α = 0 dot
collision of two particles. Accordingly, a particle travel- and partly also thecyan one)
ling from one lattice site to an adjacent one, can reach Parameters: ω0 = 4.0;a = m = 1.0;γ = 7.0;V = 25; v0 ∈
thelatterwithahigherkineticenergywhenencountering (−250,250).
a collision on its path. Physically this collision induced
reduction of frictional energy losses is based on the fact
that upon a collision kinetic energy is converted into in- uniquely determined by a fixed point attractor at δ = 0
teraction energy which is not subject to frictional losses. with a global basin of attraction. Then, the state where
In the case of the complete N particle system in the os- the twoconsideredions exhibitanidenticalphaselocked
cillating lattice, this mechanismis stillpresentandleads dynamics is stable (for arbitrary values of a,ω and γ)
0
to a substantial increase of the asymptotic kinetic ener- and any initial state sufficiently close to it asymptoti-
gies(Fig.3c)forα=2;5(transport)ascomparedtothe cally approaches it. This result can be generalized to
energies in the noninteracting α= 0 and non transport- the N particle case, yielding α > KV π3/(Qk) with
0
ing α= 1;20;175 cases. In these non transporting cases Q = 4;3;2;(5 √5)/2;1 for N = 2,3;...6 and Q > 0
−
the particles asymptotically exhibit almost equidistant for N < . In fact, for α > KV π3/(Qk) we numeri-
0
∞
on site oscillations. callyobservthattheN particleensemblediscussedinthe
We now discuss why the interaction induced directed contextofFig.1genericallycomesclosetoatranslation-
transport is present only in some regime of the inter- ally invariant state (after redistributing the nonuniform
action strength (e.g. for α = 2;α = 5) but vanishes initial kinetic energies) and we observe a breakdown of
for comparatively large values of α (α = 20;α = 175). the N particle current at the same value of γ as in the
Clearly, in the limit α of a stiff ion chain where all single particle case. The uniform (or phase locked) N
→∞
N particles exhibit an identical dynamics, the N parti- particle dynamics is then described by the equation of
cle transport breaks down at the same friction strength motion of its center of mass coordinate x:=(1/N) x
i
P
γ as in the single particle case (N = 1 or α = 0). Us- (Eq.2),representingabiharmonicversionofanac-driven
ing linear stability analysis, we show that the N particle damped physical pendulum which is known to exhibit a
dynamics reduces to that of a stiff ion chain already for period doubling route to chaos [41]. Note that the cri-
finite values of α, no matter how strong the driving is. terion α > KV π3/(Qk) represents only an upper limit
0
Consideratwoparticlesystem(Eq.(1)forN =2;r=1) for the stability of the spatially uniform dynamics; in
with periodic boundary conditions and only next neigh- practice stability can of course occur already for smaller
bour interactions. Transformingx x˜ :=2k[x f(t)] values of α.
i i i
→ −
and then to center of mass x:=(x˜1+x˜2)/2 and relative Irregular transport We finally demonstrate that the
∆:=x˜2 x˜1 coordinatesyieldsafterinserting∆=2π+δ interaction induced directed currents are by no means
−
(2π is the equilibrium distance) and truncating in linear restricted to the regular regime as represented by the
order δ: propagating spatiotemporal pattern in Fig. 2. Fig. 4
0=mx¨+γx˙ 2V0k2sin(x)+F(t) (2) shows hhd(t)ii in a regime where the dynamics of the
− underlying ions is irregular (Fig. 4, inset a). Here, a
0=mδ¨+γδ˙+2V k2δcos(2x)+8k3αδ/π3 (3)
0 current emerges already for comparatively weak interac-
F(t)=2k mf¨(t)+γf˙(t) tions (α = 0.25) and is accompanied, as before, by a
h i
significantincrease of the mean kinetic energy per parti-
As cos(2x) ( 1,1) it follows from the linear Eq. (3) clecomparedtothenoninteracting(α=0)andtheother
∈ −
that for α > V π3/(4k) the asymptotic state for δ(t) is nontransporting cases (α = 20; α = 175) (Fig. 4, inset
0
5
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