Dynamical decoupling and dynamical isolation in temporally modulated coupled pendulums

0.23 MB·

by  Grazia Salerno

epl draft
Dynamical decoupling and dynamical isolation in temporally
modulated coupled pendulums
4 GRAZIA SALERNOandIACOPO CARUSOTTO
1
0 INO-CNRBECCenterandDipartimentodiFisica,Universita` diTrento,viaSommarive14,38123Povo,Italy
2

r
p
A PACS 45.20.D-–Newtonianmechanics
PACS 46.40.Ff–Resonance,damping,anddynamicstability

6 PACS 03.65.Vf–Phases:geometric;dynamicortopological
1
Abstract –We theoretically study the dynamics of a pair of coupled pendulums subject to a peri-

] odictemporalmodulationoftheiroscillationfrequency.Inspiredfromanalogousdevelopmentsin
h
quantummechanics,weanticipatedynamicallocalizationanddynamicalisolationeffects,aswell
p
- astheoccurrenceofnon-trivialcouplingphases.Perspectivesinthedirectionofstudyingsynthetic
s gaugefieldsinaclassicalmechanicscontextareoutlined.
s
a
l
c
.
s
c
i
s Introduction. – Dynamical localization is a surpris- atesyntheticgaugefieldforphotons[25,26].
y ing consequence of quantum mechanics applied to par- InthisLetter,wereportatheoreticalstudyofdynami-
h
ticles subject to a strong time-dependent external force. callocalization phenomena in a classicalmechanics con-
p
This effect was first observed as renormalization of the text.Thedynamicstabilizationoftheinvertedpendulum
[
magneticresponseofanatomilluminated byastrongrf whenitspivotpointismadetooscillateinspaceisawell

2
field [1]. In a solid state context, dynamical localization celebratedexampleofnon-trivialmechanicaleffectstem-
v
was proposed in [2–6] as a dramatic suppression of the mingfromatemporalmodulationofthesystemparame-
8
7 d.c.conductivityofametalinastronga.c.field. Another ters[27]. Hereweconsiderasystemoftwocoupledpen-
9 closelyrelatedeffectisthecoherentdestructionoftunnel- dulums,whoseoscillationfrequenciesareindependently
3 inginadouble-wellgeometry,firstpredictedin[7,8]and andperiodicallyvariedin time. Inanalogyto the coher-
.
1 extensivelycomparedtodynamicallocalizationin[10]. ent destruction of tunneling of a quantum particle in a
0 Whiletheexperimentalstudyoftheseeffectsinsolids double-well potential, we predict a dynamical decoupling
4 is made difficult by the unavoidable material imperfec- effect,whereexchangeofenergybetweenthependulums
1
tions and electronic decoherence, the robust coherence issuppressed. Whenthependulumsaredrivenbyanex-
:
v and the clean periodic potential experienced by atomic ternalforce,weanticipateanoveldynamicisolationeffect,
Xi matter waves in temporally modulated optical lattices wherethetemporalmodulationeffectivelydecouplesthe
has allowed for a clear observation of Bloch band sup- systemfromtheexternalforce.
r
a pression in a new atomic physics context [9]. Further
Thesystemandthetheoreticalmodel.– Thesystem
studies of dynamical matter wave localization effects
oftwoidenticalcoupledpendulumsismodelledasapair
were reported in [11,12] using Bose-condensed atomic
ofcoupledharmonicoscillatorsofequalmassesmfollow-
samples. Following the proposal [13,14], this research
ingthemotionequations:
line culminated in the observation of a dynamically-
inducedsuperfluidtoMott-insulatortransition[15]. mx˙ =p (1)
1 1
Veryexcitingfurtherdevelopmentsofthese ideasaim
p˙ =−mω2[1+ν (t)] x +k(x −x )−ξ x˙ (2)
at using more complex modulation schemes to gen- 1 0 1 1 2 1 1 1
erate non-trivial hopping phases between the lattice mx˙2 =p2 (3)
sites [16,17] and then synthetic gauge fields for neutral p˙ =−mω2[1+ν (t)] x +k(x −x )−ξ x˙ . (4)
2 0 2 2 1 2 2 2
atoms [18–23]. Correspondingly to these advances in
atomicphysics,thesameideasarebeingexploredinpho- The x variables indicate the spatial displacement of
1,2
tonics to observe dynamical localization of light in cou- the pendulums from the equilibrium position. The lin-
pledopticalwaveguides[24]and,veryrecently,togener- earised form of the motion equations is legitimate in
p-1
G.Salernoetal.
fortheα complexvariablesdefinedas:
i=1,2
mω 1
0
α = x +i p . (7)
i r 2 i r2mω i
0
Complex conjugate equations hold for the α∗ . Physi-
1,2
cally the square modulus |α |2 is the instantaneous en-
i
ergyof the i-thpendulumandthe argumentof α isthe
i
oscillation phase. In experiments, both can be extracted
F (t)
ex N bymeasuringtheinstantaneousposition andvelocityof
NN
SS thependulums.
S
Inorderforthe coupling betweenthe oscillatorsto be
i(t) i(t) effective, the strong coupling condition, Ω ≫ γ, will be
1 2
assumed.Amongthemanyformsofthemodulationcon-
sideredintheliterature[16,17],inthefollowingweshall
concentrateourattentiononsinusoidalones:
Figure 1: Sketch of the physical system under considera-
tion. Each pendulum contains a magnet and the modulation vi(t)=(−1)iI0sin(wt). (8)
of its natural oscillation frequency is controlled by the time-
dependentcurrenti1,2(t)flowinginthecorrespondingcoil.The As we shall see better in the following of this Letter, the
firstpendulummaybeexternallydrivenbyatime-dependent mostconvenientregimewheretoobtainandobservethe
forceFex(t). dynamicallocalizationphysicsischaracterizedbythein-
equalitychain:
ω ≫w≫Ω. (9)
0
thesmalloscillationregimewherethedisplacementsare
muchsmallerthanthelengthLofthependulums: inthis The latter inequality is essential to describe the system
regime,thenaturaloscillationfrequencyofeachisolated in terms of an effective, time-averaged coupling. The
pendulumistheusualω = g/L.Thefrictionconstants formerhelpstoreducethoseparametricinstabilitiesthat
0
of the two pendulums havepthe samevalue ξ = ξ = ξ. mightotherwiseoccurforlargevaluesofthemodulation
1 2
Thecouplingbetweenthependulumsoccursviaaspring amplitudeI0[28,29].
ofconstantk. Thekeyelementtoachievethedynamical
Dynamicaldecouplingoftheoscillators.– Asapre-
decouplingandisolation effectsisthe temporalmodula-
liminary step, we have numerically integrated Eq. (6)
tionofthesystem,whichisincludedinEq.(2)andEq.(4)
in the vanishing friction case, γ = 0. The integration
asatemporalmodulationoftherestoringforcestrengths
methodisastandardfourthorderRunge-Kutta. InFig.2
ofrelativeamplitudeν (t). Oneofthepossibleconcrete
1,2 wehaveplottedtheevolutionofthe|α |sampledatthe
1,2
realizationsofthismodelissketchedinFig.1: eachpen-
frequency w of the modulation. At these stroboscopic
dulumcontains amagnet which feelsthe magnetic field
times t = 2πj/w, the system is described by a time-
j
generatedbyacoillocatedbelowitsaxis. Inthisway,the
independentdiscreteevolutionlaw. Thedifferentpanels
gravitationalrestoringforcefeltbyeachpendulumissup-
ofFig.2correspondtothesamemodulationfrequencyw
plemented by a contribution of magnetic origin, which
butdifferentvaluesoftheamplitudeI .
0
canbecontrolledviathe(time-dependent)currenti (t)
1,2 In panel (a), there is no modulation, I = 0: the am-
0
flowing in the corresponding coil. The effective modu-
plitudes of the two oscillators exhibit the usual beating
lation of the natural oscillator frequencies then has the
effect,i.e. aperiodicexchange ofenergybetweenthe os-
form:
cillators at the coupling frequency Ω. As a result of the
ω2 (t)=ω2 [1+ν (t)], (5)
1,2 0 1,2 modulation vi(t), for increasing values of its amplitude
I ,thefrequencyofthebeatismodified: inparticular,for
where the ν (t) are proportional to the currents i (t). 0
1,2 1,2
theparametersofpanels(b)and(c),thebeatfrequencyis
Introducing, for notational simplicity, the rescaledquan-
moreandmorereduced.ForevenlargeramplitudesI ,a
tities: 0
non-monotonicbehaviouroftheeffectivebeatfrequency
ν1,2(t)ω0 k ξ isobserved(notshown).
v (t)= , Ω= , γ = ,
1,2
2 mω 2m
0 Experimental proposal and dynamical isolation. –
the four equations of motion Eqs. (1)-(4) can be summa- In order to perform a quantitative study of the effect of
rizedinapairofcomplexequations: themodulationand,atthesametime,toproposeaviable
proceduretoexperimentallyobservethesephenomena,it
α˙ =−iω α −iv (t)(α +α∗)−γ(α −α∗) is useful to consider the realistic case of dissipative pen-
i 0 i i i i i i
Ω (6) dulumsdrivenbyanexternalforce,whichisassumedto
+i 2 (α3−i+α∗3−i−αi−α∗i). be monochromatic, Fex(t) = 2fexcosωext and to act on
p-2
Dynamicaldecouplingandisolationinmodulatedcoupledpendulums
αα| | / | (t=0) |10.015 (a) ωγ2 |A( ) | / fexex00..027.5505 (a)
0 1000 2000 ω0 t 3000 4000 5000 −0.01 −0.005 (ωex0−ω0)/ω0 0.005 0.01
αα| | / | (t=0) |10.15 (b) ωγ2 |A( ) | / fexex00..027.555 (b)
0 0
0 1000 2000 ω0 t 3000 4000 5000 −0.01 −0.005 (ωex0−ω0)/ω0 0.005 0.01
αα| | / | ( t=0) |10.15 (c) ωγ2 |A( ) | / fexex00..027.5505 0.0250 −0.01 −0.005(ωex−0ω0)/ω00.005 0.01 (c)
00 1000 2000 ω0 t 3000 4000 5000 −0.01 −0.005 (ωex0−ω0)/ω0 0.005 0.01
Figure 2: Numerical integration of the equations of motion Figure 3: Response spectra |Ai(ωex)| as a function of the fre-
Eq.(6). The blue (green)dotsindicate the modulus|α1| (|α2|) quency ωex of the external force. The stroboscopically sam-
pledresultsareshownforthefirst(second)pendulumasblue
oftheoscillationamplitudeofthefirst(second)pendulum,nor-
malized to the initial amplitude, while the lines are guidesto (green) dots, normalized to the peak amplitude fex/(2γ) of a
singleisolated pendulum in the small γ limit. The solidlines
the eyes. At the initial time t = 0, only the first oscillator is
show the result of an analytical calculation based on the ro-
excited,whileα2(t = 0) = 0. Theevolutionofthetwopendu-
tating wave approximation (see text). The different panels
lumsisstroboscopicallyfollowedatthemodulationfrequency
corresponds to different values of the modulation amplitude,
w. Thethreepanelsarefordifferentvaluesofthe modulation
amplitude,I0/w=0(a),I0/w=0.632(b),andI0/w=1.053(c). I0/w = 0 (a), I0/w = 1.21 (b), I0/w = 2.31 (c). The insetin
γSy/sωt0em=pfaerxa/mω0et=er0s.arew/ω0 =7.6×10−2, Ω/ω0 =0.68×10−2, (wfce)/x/ωsωh0o0=w=s74.a6n××e1n01l−0a−r2g.2e,mΩe/nωt0o=fth0e.6m8a×in1p0−lo2t,.γSy/ωst0em=p0a.1ra×m1e0te−r2s,:
the first pendulum only. Energy transfer to the second
pendulum is made possible by the spring that couples oscillate with the same (opposite) phase. At all frequen-
thetwopendulums. Thedriven-dissipativemotionequa- cies,theoscillationamplitudesofthetwopendulumsre-
tionsforαi=1,2thentaketheform: maincomparable.
α˙ =−i (ω¯ −iγ)α −iv (t)(α +α∗)+iδ F (t) Panel (b) shows a case where the coupling of the two
i 0 i i i i i,1 ex pendulums is dramatically suppressed: this effect is ap-
Ω Ω (10)
−i +iγ α∗+i (α +α∗ ) parent in the figure as the two peaks merge into a sin-
(cid:18)2 (cid:19) i 2 3−i 3−i glepeakandnosignificantexcitationistransferredtothe
second pendulum, which remains basically at rest with
wherewehavedefinedtheshort-handω¯ =ω +Ω/2and
0 0
a negligible oscillation amplitude. This behaviour is the
δ istheusualKroneckerdelta.
i,j
driven-dissipative manifestation of the dynamical decou-
For each ω , the equations of motion Eq. (10) have
ex
plingeffect,alreadyseeninthelowestpanelofFig.2for
beenintegrateduntilasteady-stateregime,showingreg-
vanishingfriction.
ular periodic oscillations, is achieved at long times t ≫
1/γ. For the stroboscopic sampling at t = 2πj/w, the Panel (c) shows a novel regime: while some effec-
j
steadyoscillationshavetheform: αi(t)≈Ai(ωex)e−iωext. tivecouplingofthetwopendulumsisstillpresent,their
ThecomplexamplitudesA areobtainedviaaFourier globalexcitationbytheexternalforceissuppressed. The
i
transformation of the stroboscopically sampled numeri- suppressedexcitationisvisibleasaverysmalloscillation
cal solutions. The moduli |A |, as a function of the fre- amplitudeofbothpendulums. Thepresenceof asignifi-
i
quency of the external force, give the response spectra cantcouplingisapparentintheinsetwheretheresponse
showninFig.3:eachpanelcorrespondstodifferentvalue ofthefirstpendulumisstillshowingadoublet.
ofthemodulationamplitudeI andillustratesadifferent These different regimes are illustrated in more detail
0
regime. inFig. 4. Response spectrahavebeennumericallycalcu-
The case of no modulation is shown in panel (a): the latedforanumber ofdifferentvaluesof themodulation
spectra are characterized by a pair of peaks, split by Ω amplitudeI . Foreachofthesevalues, weplotthemax-
0
andofequalwidthγ. Thelower(upper)frequencypeak imum of the amplitude |A (ω )| over the externaldrive
i ex
correspondstotheeigenmodewherethetwopendulums frequencyω , thatisthe resonantresponseatthe peaks.
ex
p-3
G.Salernoetal.

Numerical 1st osc. 1 Numerical
Numerical 2nd osc.
0.6 Theory 1st osc. Ω | 0.75 Theory
T| fheeff o/ r(y2 2 fnd) |osc. /eff 0.5 (a)
ex ex Ω
ex 0.5 | 0.25
ωγ)| ) / fex 0.4 00 0.5 1 1I.5/w 2 2.5 3
A( 0
2 | 0.3 6
x (
ma 0.2 ΩΩ/ )eff 4 (b)
0.1 arg( 2
0
0 0 0.5 1 1.5 2 2.5 3
0 0.5 1 1.5 2 2.5 3
I /w
I0 / w 0
Figure4: Blue(green)dotsshowthenormalizedmaximumof Figure5: Modulusoftheeffectivecouplingfrequency|Ω |/Ω
eff
the oscillation amplitude for the first (second) oscillator as a [upper(a)panel]anditsphasearg(Ω )(modulo2π)[lower(b)
eff
function of the modulation amplitude I0/w. The solid lines panel]. Dotsaretheresultsofthenumericalcalculationsasdis-
showtheresultofananalyticalcalculationbasedontherotating cussedinthetext,whilethesolidlinesareanalyticalpredictions
waveapproximation(seetext).SystemparametersasinFig.3. oftheRWapproximation.SystemparametersasinFig.3.
ofthiseffectwillbegiveninthenextsection.
The dynamicaldecoupling seen in Fig. 3(b) corresponds
hereto theminimum ofmax(|A |)thatisvisible around Analytical expressions within the rotating wave ap-
2
I /w ≃ 1.2. The dynamical isolation seen in Fig. 3(c) proximation.– Analytical insight in the physics of the
0
lies in the vicinity of the simultaneous minima of both modulated system can be obtained within the so-called
max(|A |)thatarevisiblearoundI /w ≃2.4. Inthefig- rotating wave (RW) approximation, well-known from
1,2 0
ure,notethefurtherdynamicaldecouplingpointaround quantum optics. Thisapproximationreliesonassuming
I0/w ≃ 2.7. Thesolidlinesaretheanalyticalpredictions that the natural oscillation frequency ω0 is much larger
oftherotatingwaveapproximationthatwillbepresented thanallotherinternalfrequenciesintheproblem,thatis
afterwards. ω0 ≫max(Ω,γ,w,|ωex−ω0|). Sincetheαvariablesrotate
at≈ ω andtheirconjugatevariablesα∗ rotateat≈ −ω ,
Further insight on the effective coupling of the two 0 0
theRWapproximationisstraightforwardlyimplemented
pendulums, resulting from the temporal modulation, is
by neglecting the α∗ terms in the motion Equations (10)
given in Fig. 5. In panel (a) we present the numerically
fortheαvariables.
estimated magnitude of the effective coupling between
Toobtainaneffective,temporallyaveragedformofthe
the pendulums: this quantity is observable as the beat
stroboscopic dynamics of the system, we introduce the
frequency in Fig. 2, or as the separation between peaks
in the response spectra of Fig. 3. Most remarkable fea- new variables: βi(t) = αi(t) exp[i 0tvi(t′)dt′]exp[iωext].
tures of Fig. 5(a) are the vanishing effective coupling at For the chosen sinusoidal form ofRthe modulation, the
I /w ≃ 1.2 and I /w ≃ 2.7, corresponding to the zeros phasefactorinvolvingthemodulationisexactlyequalto
0 0
ofonlymax(|A2|)inFig. 4. Inthe following, wewillsee 1atthestroboscopictimestj =2πj/wthatareconsidered
thatthesenumericalvaluesarerelatedtothezerosofthe inthefigures. Sinceweareinthefastmodulationregime
zeroth-orderBesselfunction. w ≫ Ω, effectiveequations, which no longer depend on
thetimet,canbeobtained[30]byaveragingtheRWform
Another crucial consequence of the temporal modula-
oftheequations(10)overamodulationperiodT =2π/w:
tionisshowninthelowerpanelFig.5(b).Whiletheeigen-
modesofthebaresystemcorrespondtoin-phaseandout- β˙ =−i (ω¯ −ω ) β −γβ +i(Ωeff/2)β +ifeff (11)
of-phase oscillations, the modulation allows to tune the 1 0 ex 1 1 12 2 ex
relativephaseoftheoscillationofthetwopendulumsto β˙2 =−i (ω¯0−ωex) β2−γβ2+i(Ωe2f1f/2)β1. (12)
anyvaluefrom0to2π. Inmathematicalterms,theeffec-
TheeffectivecouplingsinEqs.(11)-(12)aredefinedby:
tivecouplingdevelopsanon-trivialphase.Notehowthis
pgoheassethdriosupglahyszeπroju.mBepfsorwehpernoecveeedritnhge,eitffiesctwivoertchounpotliinngg Ωeijff ≡ TΩZ Tdt eiR0t[vi(t′)−vj(t′)]dt′
0
inFig.3asizeableglobalshiftoftheresonancecurvesto-
wardslowerfrequenciesforgrowingI . Anexplanation andtheyarecomplexconjugatetoeachother,Ωeff =Ωeff∗.
0 12 21
p-4
Dynamicaldecouplingandisolationinmodulatedcoupledpendulums
Forthespecificmodulationconsideredhere,theeffective andγ arenegligibleforthechosenparameters. Theshift
couplinghasthesimpleexpression: can be interpreted as a classical analog of the quantum
Bloch-Siegertshiftofnuclearmagneticresonance[31].
Ωeff ≡ Ωe1f2f =Ω e−2iI0/wJ0(2I0/w) (13) TheshiftEq.(17)canbetakenintoaccountinEqs.(15)-
(16) by replacing ω¯ with ω˜ = ω¯ + ∆ω . In this way,
0 0 0 0
intermsoftheJ zero-orderBesselfunction. 1 Themod-
0 an excellent agreement for the analytical spectra (solid
ulusandthephaseofthisquantityareplottedasasolid
lines) with the resultof the numericalsimulations (dots)
lineinFig.5. Inparticular,themodulus|Ω |showsase-
eff is found in Fig. 3. This suggests a way to extract an es-
riesofzeros,whichareindicativeofacompletedynamical timate of the effective coupling Ω from the numerical
eff
decouplingbetweenthetwopendulums.
results. Anexplicitexpressionofitintermsoftheoscilla-
Thesameproceduremustbeappliedtotheamplitude
tionamplitudesisobtainedbytakingtheratioofEq.(15)
of the externaldriving force in Eq. (11), which gives the
andEq.(16):
effectiveaverageddrivingforce:
Ω =2(ω˜ +iγ−ω )(β /β )∗. (18)
eff 0 ex 2 1
feexff ≡ fTex Z Tdt eiR0tvi(t′)dt′ TheresultofreplacinginEq.(18)the βi withthe numer-
0 icallycalculatedA isshown bythe dots inFig. 5and is
i
that,forthespecificmodulationinEq.(8),hastheform: compared with the analytical RW prediction of Eq. (13).
Theagreementappearstobeverygoodforboththemag-
feff =f e−iI0/wJ (I /w). (14) nitude and the phase of the coupling, in particular the
ex ex 0 0
position of the dynamical decoupling points at which
Thezerosoffeexff determinetheparametersatwhichcom- Ωeff = 0. The small discrepancies occur when both nu-
pletedynamicalisolationfromtheexternalforceoccurs. merator and denominator of Eq. (18) go to zero and the
Explicitformsfor the steadyoscillation regime canbe procedureismoresensitivetonumericalerrors.
derived by setting the time derivatives in Eqs. (11)-(12) A similar comparison for the effectivedriving force is
tozero. Inthisway,oneobtainsthefollowinganalytical performedinFig.4,wherethenumericalresultsarecom-
formoftheresonancecurvesasfunctionofωex: pared to RW prediction of Eq. (14). The agreement is
againverygood,inparticularforwhatconcernstheposi-
2feff(ω¯ −iγ−ω )
β (ω )= ex 0 ex (15) tionof the dynamicalisolation points for which feff = 0
1 ex 4(ω¯ −iγ−ω )2−|Ω |2 ex
0 ex eff and both max(|A |) = 0. Of course, the agreement
1,2
feffΩ∗ gets worse when the inequality chain of Eq. (9) is only
β (ω )= ex eff . (16)
2 ex 4(ω¯ −iγ−ω )2−|Ω |2 marginallysatisfied.
0 ex eff
Connection to the Bose-Hubbard model. – Before
Whileaqualitativeagreementwiththenumericalpredic-
concluding,itisworthtohighlightthedirectconnection
tionsshowninFig.3isalreadypresentatthislevel,there
of the RW description of the system of coupled pendu-
isstillanoverallglobalshiftoftheresonances. Thisshift
lumstoadriven-dissipativeversionoftheBose-Hubbard
iseasilyexplainedbyincludingtheleadingordercorrec-
(BH) model of quantum condensed-matter. In the pres-
tion to the RW approximation. We allow for the α to
i
ence of anexternalgauge field [32], the BH modelis de-
also have a small counter-rotating part evolving at fre-
quency −ωex by writing αi(t) = αriw e−iωext+δαnirweiωext. scribedbytheHamiltonian:
Since the α and the α∗ are coupled by the non-RW U
termsinEqs.(10)andtheircomplexconjugateequations, Hˆ =− J eiφijaˆ†iaˆj +h.c. + 2aˆ†iaˆ†iaˆiaˆi (19)
co-rotating contributions appear in the equation for α Xhijih i Xi
from the counter-rotating terms of α∗. A straightfor-
ward calculation gives, to the leading order: δαnrw ≃ where aˆi and aˆ†i are bosonic on-site operators. The first
i
−(αrw)∗v (t)/(2ω ). After substituting this expression termdescribeshoppingofthebosonicparticles: thehop-
i i 0 ping amplitude is J and the phase φ describes a non-
intoEq.(10)andisolatingthenon-rotatingterms,byaver- ij
trivial tunneling phase. Of course, non-rotating wave
agingoverthe periodof the consideredmodulation, the
terms do not appear in the standard condensed-matter
effectivefrequencyshiftis:
BH model Hamiltonian in Eq. (19) as they would cor-
∆ω =−hv (t)2i /(2ω )=−I2/(4ω ). (17) respond to processes where the total numbers of parti-
0 i T 0 0 0
cles is not conserved. If particles are instead injected
This is the principal effect of the counter-rotating terms fromacoherentsourceandlostfromthesystem,thethe-
beyond the RW, and it is more and more important for oretical description requires a driving term of the form
growingI0. Allothernon-RWcontributionsinvolvingΩ HˆF = i[fex,i(t)aˆ†i+h.c.]aswellasdampingterms,tobe
typicalPlyincludedattheleveloftheMasterEquation[33].
1Anon-zerophaseinthedriving(8),orequivalentlyatemporalshift
[in17t]h.estroboscopicsampling,willresultinanextraphasefactorinΩeff repUlancdeedrbthyeCclnaussmicbaelrasp,epqruoxaitmionatsioonfmwohteiorenoapnearlaotgoorussatroe
p-5
G.Salernoetal.
Eqs. (15)-(16) are found. In particular, the complex hop- SCHERMANNJ.P.,Phys.Rev.Lett.,24(1970)861.
pingamplitudeoftheBHmodelcorrespondstothecom- [2] DUNLAPD. H.and KENKREV. M.,Phys.Rev.B,34(1986)
plex coupling between the pendulums, J eiφij ↔ Ωeff: 3625.
ij
the non-trivial Peierls phase φ = ridr· A/~ describ- [3] IGNATOVA.A.andROMANOVY.A.,Phys.StatusSolidiB,
ij rj 73(1976)327.
ingtheeffectofanexternalvectorpoRtentialactingonthe
[4] ZHAOX.-G.,J.Phys.,6(1994)2751.
quantumparticles[32]correspondstoanon-trivialphase
[5] HOLTHAUSM.,Phys.Rev.Lett.,69(1992)351.
of the Ωeijff coupling. An on-site interaction term analo- [6] HOLTHAUSM.andHONED.,Phys.Rev.B,47(1993)6499.
goustotheU terminEq.(19)directlyappearswhenthe [7] GROSSMANNF.,DITTRICHT.,P.JUNGandHA¨NGGI,Phys.
anharmonicityofthependulumsistakenintoaccountbe- Rev.Lett.,67(1991)516.
yondthelinearizedmotionEquations(1)-(4). [8] GRIFONIM.andHA¨NGGIP.,Phys.Rep.,304(1998)229.
Themodulationschemethathasbeenenvisagedinthe [9] MADISON K. W., FISCHER M. C., DIENER R. B., NIU Q.
present Letter for coupled pendulums corresponds to a andRAIZENM.G.,Phys.Rev.Lett.,81(1998)5093.
temporal modulation of the on-site energies of the BH [10] KAYANUMA Y. and SAITO K., Phys. Rev. A, 77 (2008)
010101.
lattice. There is however one crucial difference worth
[11] LIGNIERH., SIAS C., CIAMPINID., SINGH Y., ZENESINI
noting: whiletheglobalshakingoftheopticallatticepo-
A., MORSCH O. and ARIMONDO E., Phys. Rev. Lett., 99
tentialthatistypicallyusedforthispurposeinultracold
(2007)220403.
atom experiments [9,11,15,16] can only provide modu-
[12] ECKARDT A., HOLTHAUSM., LIGNIERH., ZENESINIA.,
lationamplitudesthatarelinearlydependentonthesite CIAMPINID.,MORSCHO.andARIMONDOE.,Phys.Rev.A,
position[30],systemsofpendulumsallowforanindivid- 79(2009)013611.
ual addressing of each single pendulum. This freedom [13] ECKARDT A., WEISS C. and HOLTHAUS M., Phys. Rev.
willbeveryusefulinviewofgeneratingsyntheticgauge Lett.,95(2005)260404.
fieldconfigurationsinourmechanicalsystem. [14] ECKARDTA.andHOLTHAUSM.,Europhys.Lett.,80(2007)
50004.
Finalremarksandconclusions.– Inspiredbyanalo- [15] ZENESINIA.,LIGNIERH.,CIAMPINID.,MORSCHO.and
gousquantumeffects,inthisLetterwehavetheoretically ARIMONDOE.,Phys.Rev.Lett.,102(2009)100403.
studied dynamical localization and dynamical isolation [16] STRUCK J., O¨LSCHLA¨GER C., WEINBERG M., HAUKE P.,
effects in a classical system of two coupled pendulums SIMONET J., ECKARDT A., LEWENSTEIN M., SENGSTOCK
whenatemporallyperiodicmodulationoftheoscillation K. and WINDPASSINGER P., Phys. Rev. Lett., 108 (2012)
frequencies is applied to them. The dynamical decou- 225304.
plingeffectcanbeusedinmechanicalengineeringtosup- [17] CREFFIELD C. E. and SOLS F., Phys. Rev. Lett., 100 (2008)
250402.
press the coupling between the two pendulums, while
[18] KOLOVSKYA.R.,Europhys.Lett.,93(2011)20003.
thedynamicalisolationallowstoisolatethesystemfrom
[19] CREFFIELD C. E. and SOLS F., Europhys. Lett., 101 (2013)
external forces. Of course, our results are valid for cou-
40001.
pledoscillatorsofanyphysicalnaturewithatimedepen- [20] HAUKEP.,TIELEMANO.,CELIA.,O¨LSCHLA¨NGERC.,SI-
dent frequency, for instance temporally modulated RLC MONETJ.,STRUCKJ., WEINBERGM.,WINDPASSINGERP.,
electriccircuits. SENGSTOCK K., LEWENSTEIN M. and ECKARDT A., Phys.
From the point of view of fundamental physics, the Rev.Lett.,109(2012)145301.
non-trivial coupling phase between the pendulums is [21] STRUCK J., WEINBERG M., O¨LSCHLA¨NGER C., WIND-
analogous to the Peierls phase of Bose-Hubbard mod- PASSINGER P, SIMONET J., SENGSTOCK K., HO¨PPNER R.,
els of quantum condensed-matter physics. In analogy HAUKEP.,ECKARDTA.,LEWENSTEINM.andMATHEYL.,
Nat.Phys.,9(2013)738.
toorbital magnetism and topological insulation, newin-
[22] AIDELSBURGER M., ATALA M., LOHSE M., BARREIRO J.
triguing phenomena are expected to appear in multi-
T., PAREDES B. and BLOCH I., Phys. Rev. Lett., 111 (2013)
dimensionallatticesofmanytemporallymodulatedpen-
185301.
dulums. Further exciting developments in nonlinear
[23] MIYAKEH.,SIVILOGLOUG.A.,KENNEDYC.J.,BURTON
physics are expected to arise as a result of the intrinsic
W.C.andKETTERLEW.,Phys.Rev.Lett.,111(2013)185302.
anharmonicityofpendulums. [24] LONGHIS., MARANGONIM., LOBINO M., RAMPONIR.,
LAPORTAP., CIANCIE.and FOGLIETTIV.,Phys.Rev.Lett.,
∗∗∗ 96(2006)243901.
[25] RECHTSMANM. C., ZEUNER J. M., PLOTNIKY., LUMER
We are grateful to Nicola Pugno, Andre´ Eckardt and Y., PODOLSKY D., DREISOW F., NOLTE S., SEGEVM. and
Jean Dalibard for stimulating discussions. We acknowl- SZAMEITA.,Nat.,496(2013)196.
edge partial financial support from ERC via the QGBE [26] HAFEZIM.,MITTALS.,FANJ.,MIGDALLA.andTAYLOR
J.M.,Nat.Phot.,7(2013)1001.
grantandfromProvinciaAutonomadiTrento.
[27] BUTIKOVE.I.,Am.J.Phys.,69(2001)755.
[28] ARNOLD V.I., Mathematical Methods of Classical Mechanics
References (Springer-Verlag,NewYork)1989.
[29] BERTHETR., PETROSYANA.andROMANB.,Am.J.Phys.,
[1] HAROCHE S., COHEN-TANNOUDJI C., AUDOIN C. and 70(2002)774.
p-6
Dynamicaldecouplingandisolationinmodulatedcoupledpendulums
[30] DALIBARDJ.,LectureNotesatColle`gedeFrance,A.Y.2012-13
www.phys.ens.fr/˜dalibard/CdF/Cours_2013.pdf
[31] BLOCHF.andSIEGERTA.,Phys.Rev.,57(1940)522.
[32] DALIBARD J., GERBIER F., JUZELIU¯NASG. and O¨HBERG
P.,Rev.Mod.Phys.,83(2011)1523.
[33] CARUSOTTOI.andCIUTIC.,Rev.Mod.Phys.,85(2013)299.
p-7