Analog of photon-assisted tunneling originated from dark Floquet state in periodically modulated waveguide arrays

1.2 MB·

by  Liping Li

Analog ofphoton-assistedtunneling originated from dark Floquetstateinperiodically modulated
waveguidearrays
Liping Li1, Xiaobing Luo2,∗ Xiaoxue Yang1, Xin-You Lu¨1,† and Ying Wu1‡
1 Wuhan National Laboratory for Optoelectronics and School of Physics,
Huazhong University of Science and Technology, Wuhan, 430074, P. R. China and
2 Department of Physics, Jinggangshan University, Ji′an 343009, P. R. China
(Dated:January13,2016)
We theoretically report an analog of photon-assisted tunneling (PAT) in a periodically driven lattice array
without a static biased potential by studying a three-channel waveguide system. This analog of PAT can be
achieved by only periodically modulating the top waveguide and adjusting the distance between the bottom
and its adjacent waveguide. It is numerically shown that the PAT resonances also exist in the five-channel
6
waveguidesystemandprobablyexistinthewaveguidearrayswithotheroddnumbersofwaveguides,butthey
1
0 will become weak as the number of waveguides increases. With origin different from traditional PAT, this
2 typeofPATfoundinourworkiscloselylinkedtotheexistenceofthezero-energy(dark)Floquetstates. Itis
readilyobservableunder currentlyaccessibleexperimentalconditionsandmaybeusefulforcontrollinglight
n
propagationinwaveguidearrays.
a
J
PACSnumbers:42.65.Wi,42.25.Hz
2
1
I. INTRODUCTION ofanintegernumberofphotonswiththeoscillatingfield[9].
]
The static tilt (dcbias potential)leadsto suppressionof tun-
h
p nelingwhichisrelatedtolocalizedWannier-Starkstates[10].
Controllingquantumtunnelingandtransportthroughape-
- Whenamultipleofthedrivingfrequencyofacfieldmatches
nt riodic driving field has been a subject of intense studies in theenergydifferencebetweenadjacentrungsoftheWannier-
the last decades, for its relevance to fundamental physics
a Starkladder,thesystemisabletoabsorboremitphotonswith
u tests as well as to great potential application in nanoscale sufficient energy to bridge the energy difference created by
q devices[1,2]. Amongthemostintriguingaspectsofthesub-
the dc bias potential, throughwhich tunneling is (partly) re-
[ ject,coherentdestructionoftunneling(CDT)[3]andphoton-
stored (PAT). So far, PAT has been experimentally observed
assisted tunneling (PAT)[4] represent two seminal results.
1 in Josephson junctions[11], coupled quantum dots[12, 13],
v CDT is a resonant effect discovered in the pioneering work, semiconductorsuperlattices [14, 15] and Bose-Einstein con-
9 inwhichthecoherenttunnelingbetweenstatesisalmostcom-
densatesinopticallattices[16].
2 pletely suppressedwhenthe system parametersare carefully
8 chosenat the isolated degeneracypointof quasi-energies[3]. In this article, we have studied the tunneling dynamics in
2 It has so far generated great interests and has recently been lattice arrayswith controllableboundary. Owing to the sim-
0 observed experimentally in different physical systems[5, 6]. plicity and flexibility offered by optical settings, the engi-
.
1 Recently,CDThasbeenfoundtooccuroverawiderangeof neered photonic waveguides provide an ideal system for ex-
0 systemparametersinodd-N-statesystemswhereonestate is ploration of tunneling phenomena, in which spatial propa-
6
periodicallydrivenwithrespecttoothers[7]. Suchextension gation of light mimics the temporal dynamics of a quantum
1
ofdestructionoftunnelingtoafiniteparameterrange,referred particle in a lattice array[17, 18]. As illustrated in Fig.1, we
:
v toasdarkCDT,isattributedtotheexistenceoflocalizeddark present an optical implementationof our Hamiltonian in the
Xi Floquetstate withzeroquasi-energy[7,8]. AsthedarkCDT form of a linear array of tunneling-coupled optical waveg-
wasintroducedinthehigh-frequencyregimewherethedriv- uideswhichischaracterizedin:(i)thattherefractiveindexof
r
a ing frequency is larger than all energy scales of the system, the top boundarywaveguideis modulatedperiodicallyalong
itnaturallyleadstocertaininterestingquestions: whetherthe the propagation direction; and (ii) that the distance ωs be-
2
dark CDT and associated dark Floquet state still exist in the tweenthebottomboundarywaveguideanditsnearestneigh-
non-high-frequencyregimes which are accessible by adjust- borisdifferentfromotheridenticalnearest-neighboringspac-
ing, for example, either the coupling strength between the ingsωs1. Thus, throughadjustmentof the distance ωs2, the
neighboring states or the driving frequency? what new be- coupling strength between the bottom boundary waveguide
haviorswillemergeinthenon-high-frequencyregimes? and its nearestneighborcan be tunedto be sufficiently large
incomparisontothemodulationfrequency. Theroleofpho-
Photon-assistedtunneling(PAT)referstoaphenomenonin
tonisplayedbya periodicmodulationof thethe topbound-
which tunneling contact disabled by a static tilt (dc bias po-
ary waveguide with a certain modulation frequency. Gener-
tential) can be restored when the system exchanges energy
ally, PAT occurs in a system with a static biased potential
which strongly suppresses usual Josephson oscillations. But
herewereportananalogofphoton-assistedtunnelinginape-
riodically driven lattice array without static tilt (dc bias po-
∗Electronicaddress:[email protected]
†Electronicaddress:[email protected] tential)bycomprehensivelystudyinga three-channelwaveg-
‡Electronicaddress:[email protected] uidesystem. OurnumericalanalysisdiscoversthatdarkCDT
2
(strongsuppressionof tunneling)and darkFloquetstate still
existinthethree-channelwaveguidesystemeveninthenon-
da
high-frequency regimes where the modulation frequency of i 1 = σ(z)a +Ω a ,
1 1 2
theperiodicallymodulatedtopwaveguideisroughlyequalto dz
oitrssamdjaalcleerntth(amnidthdelec)owuapvliengguisdtree.nHgtohwbeevtewr,ewenhethneinbtoetgtoemrmaunld- iddazj = Ω1aj−1+Ω1aj+1, (j=2,3,...,N−2)
tiplesofthemodulationfrequencyapproximatelyequaltothe da
i N−1 = Ω a +Ω a ,
couplingstrength between the bottom and its adjacent(mid- 1 N−2 2 N
dz
dle) waveguide, the light tunneling from the top waveguide
da
totheothersisrestoredasaclearsignatureofphoton-assisted i N = Ω2aN−1, (1)
dz
tunneling.DifferentfromthePATobservedintheearlierstud-
ieswhichusuallyrequiresastaticbiasedpotentialtoinitialize where a is field modeamplitude in the j-th waveguide, z
j
the systemin a self-trappedstate, thistypeof PAT isclosely the propagation distance, Ω the coupling strength between
1
linked to a dark Floquet state with zero quasi-energy. Our neighboring waveguides with spacing ωs , Ω the coupling
1 2
results are applicable for the five-channelwaveguide system strength between the bottom boundary waveguide (j = N)
andalsoextendabletowaveguidearrayswithanoddnumber and its adjacent waveguide (j = N − 1), and σ(z) the nor-
ofwaveguides. malized difference between the propagationconstants of the
topboundarywaveguideandthe otherwaveguidesof the ar-
ray. Weconsideraharmonicmodulationofthelinearrefrac-
x (a) x (b) tiveindexofthetopboundarywaveguidealongthepropaga-
Modulated waveguide
Light tion direction with σ(z) = Asin(ωz), where A is the relative
ωs
1 Modulated waveguide depth of the harmoniclongitudinalmodulation, and ω is the
Light
… spatial modulation frequency. In a different perspective, the
ωs
ωs 1 aboveequation(1) can be regardedas describingthe system
1
z z ofaquantumwaveinaperiodicallydrivenlatticearrayifzis
viewedastimet.
ωs
… Movable waveguide 2 As is well known, the periodic time-dependent equa-

ωs tion (1) admits solutions in the form of Floquet states
2
(a ,a ,...,a )T = (a˜ ,a˜ ,...,a˜ )Texp(−iεz), where ε is the
Movable waveguide 1 2 N 1 2 N
quasi-energyand the amplitudes(a˜ ,a˜ ,...,a˜ )T are periodic
1 2 N
withmodulationperiodT = 2π/ω. Ouranalysisisbasedon
FIG.1: (Coloronline)(a)Schematicofatunneling-coupledoptical theFloquettheorywhichoffersapowerfultoolforthetreat-
waveguidearraywithcontrollableboundary thatrealizesananalog mentoftheperiodicallydrivensystem.
ofphoton-assistedtunneling.(b)Atypicaltripletwaveguidesystem. Inthe case ofΩ = Ω , the equation(1) isreducedto the
1 2
The refractive index of the top boundary waveguide is modulated onediscussedinRefs.[7,8],whereasignificantsuppression
periodically along the propagation direction. The space separation
oftunneling(theso-calleddarkCDT)existsforawiderange
between the bottom boundary waveguide and its nearest neighbor
ofsystemparametersinallodd-N-statesystemswithidentical
ωs isadjustablebymovingthebottomboundarywaveguidetowards
2 couplingconstantsbecause of occurrenceof the zero-energy
theotherwaveguides,whilethespacingsωs betweenothernearest-
1 (dark) Floquetstate. The case of Ω , Ω has yet been ex-
neighboringwaveguidesarefixed. 1 2
plored. Inthiswork,wewillfocusonthatcaseandgivetwo
example models (three- and five-waveguide systems) to ex-
plorenewphysicsinthesesystems.
A.PATinthree-guidesystem
We start our consideration for the three-guidesystem, the
minimal one for odd-N-state systems. In this case, the dy-
namicalequationsareoftheform
II. LIGHTTUNNELINGDYNAMICSINMODULATED
PHOTONICLATTICESWITHCONTROLLABLE
da
BOUNDARY i 1 = Asin(ωz)a +Ω a ,
1 1 2
dz
da
i 2 = Ω a +Ω a ,
Wenowdiscusslightdynamicsinaphotonicstructurecom- dz 1 1 2 3
posed of an array of waveguides, in which the refractive in- da
i 3 = Ω a . (2)
dexofthetopboundarywaveguideismodulatedperiodically dz 2 2
alongthepropagationdirectionandthedistancebetweenthe
other boundary waveguide and its nearest neighbor is ad- To study the system’s beam dynamics, we solve numeri-
justable, as schematically illustrated in Fig.1. The coupled- callythecoupled-modeequation(2)withthelightinitiallylo-
modeequationsdescribingthebeamdynamicsinsuchstruc- calizedin the1-thwaveguide(thetopboundarywaveguide).
turescanbewrittenas Withthenumericalsolution,wecomputetheintensityoflight
3
stayingin theinitialwaveguideby P (z) = |a (z)|2 andmea- methodisinvalid.Theseresultswillgreatlyenrichourunder-
1 1
sure the minimum value of P (z) over a long-enoughpropa- standingofdarkFloquetstateanddarkCDT.
1
gation distance. When Min(P ) is not zero, the tunneling is
1
suppressed as the light is not allowed to be fully transferred 1(a) 1(d) 1(g)
fFriogm.2(tah)e,w1-ethdimspoldayeM(giuni(dPe))tovetrhseusotthheercomuopdliensg(sgtureidnegst)h.ΩIn Min P10.5 Min P10.5 Min P10.5
1 2
atthefixedparametersA = 6.6,ω = 3,Ω = 1. ForΩ = 0, 0 0 0
1 2 2(b) 1(e) 0.5(h)
the system is in fact a two-guide system in which the con-
ǫ 0 ǫ 0 ǫ 0
ventionalCDThappensonlyattheisolateddegeneracypoint
ofthequasi-energies,andconsequentlyMin(P )takesazero −2 ǫ1 ǫ2 ǫ3 −1 ǫ1 ǫ2 ǫ3 −0.5 ǫ1 ǫ2 ǫ3
ivsaolluaetebdecdaeugseenoeurarcdyripvoiningtp.aWrahmeenteΩri2sissetintocrbeeas1selidghfrtolymoffzetrhoe, 〈〉Pj0.15(c) 〈〉Pj0.15(f) 〈〈〈PPP123〉〉〉 〈〉Pj0.15(i) 〈〈〈PPP123〉〉〉
thevalueofMin(P1)becomesrelativelylargeexceptatase- 00 3 6 9Ω12 15 1820 00 3.8A/ω 7 10 00 A5/.ω1 8.4 10
ries of very sharp dips. In general, periodic modulation of 2
the top boundarywaveguidewill yield a significantsuppres-
FIG.2: (Coloronline) PATinthree-guideoptical system. Theleft
sionofthelighttunnelinginthethree-guidesystemevenwith
column: (a)theminimumvalueof intensityof lightattheinitially
Ω1 , Ω2,asshowninFig.2(a). However,atparticularvalues populated guide-1, Min(P1) versusΩ2, with A = 6.6,ω = 3,Ω1 =
of the coupling strength Ω2, Ω2 ≈ nω with n being integer, 1;(b)thecorrespondingquasi-energyǫ versusΩ2 and(c)thetime-
the valueof Min(P ) exhibitsa series of sharp dips, in anal- averagedpopulationhP ibelongingtothezero-energy(dark)Floquet
1 j
ogyto the n-photon-likeresonanceswhich destroythe effect stateversusΩ2.Themiddlecolumn:(d)Min(P1)versusA/ωforthe
of suppression of tunneling. It also can be observed that as 1-photon resonance ω = 3,Ω2 = 3,Ω1 = 1; (e)thecorresponding
the couplingstrengthΩ is increased,the higherphoton-like quasi-energyǫversusA/ωand(f)thetime-averagedpopulationhPji
2
belonging tothezero-energy (dark)Floquet stateversus A/ω. The
resonancesbecomeveryweakandthusarealmostnotvisible.
right column: (g) Min(P ) versus A/ω for the 2-photon resonance
1
Foradeepinsightintothetunnelingdynamicsobtainedin ω=3,Ω =6,Ω =1;(h)thecorrespondingquasi-energyǫversus
2 1
Fig.2(a),wenumericallycomputethequasi-energiesandFlo- A/ωand(i)thetime-averagedpopulationhP ibelongingtothezero-
j
quetstatesofthissystemasshowninFig.2(b)-(c). Asshown energy(dark)FloquetstateversusA/ω.
inFig.2(b),thisthree-statesystemalwayspossessesaFloquet
statewithzeroquasi-energyregardlessofthevalueofΩ .The
2 InFig.2(d),weshowhowthevalueofMin(P )variesunder
1
other two quasi-energies make a set of close approaches to
conditionsthat the modulationamplitude is increased, while
eachotherasΩ2isincreased.Atthepointsofcloseapproach, its frequency is held constant at ω = 3 and the coupling
namely,atΩ2 ≈nω,thevalueofMin(P1)displayssharpdips strengthheldΩ = ω,Ω = 1. Thiscorrespondstothen = 1
2 1
and the tunneling is significantly restored. We also plot the photonresonance.ForA=0,thesystemisself-trappedinthe
time-averaged population distribution hPji = ( 0Tdz|aj|2)/T 1-th waveguide due to the existence of an imbalanced dark
forthezero-energyFloquetstate(a ,a ,a )T inFRig.2(c).Con- state (−Ω /Ω ,0,1)T with Ω /Ω > 1, andthusthevalueof
1 2 3 2 1 2 1
sideringthatthedynamicsisdeterminedbytheFloquetstate, Min(P )isnonzero. Whentheperiodicdrivingisapplied,as
1
self-trapping(suppressionoftunneling)oflightintensityini- Aisincreasedfromzero,thevalueofMin(P ) rapidlydrops
1
tially populating at the 1-th mode (guide) will take place if tozero,whichindicatesthatthephotonresonancedestroysthe
hP i>0.5holds.AsseeninFig.2(c),thezero-energyFloquet self-trappingeffect. When A/ωisincreasedfurther,Min(P )
1 1
state has negligible population at the central mode (guide) takes extremely low values about zero except at a sequence
whilethepopulationhP iismuchlargerthan0.5forallvalues of very narrow peaks. These peaks are precisely centeredat
1
of Ω exceptthose in the vicinity of Ω ≈ nω. Correspond- A/ω= 3.83,7.01,...,thezerosof J (A/ω). InFig.2(e)-(f),we
2 2 1
ingly, suppression of tunneling (CDT) occurs for all values plotthequasi-energiesandthepopulationdistributionsofthe
ofΩ exceptthelocationsofphotonresonances,asshownin darkFloquetstateforthefirstphotonresonance(n = 1). Ap-
2
Fig.2(a). The Floquet state with zero quasi-energyis essen- parently,thequasi-energiesaredegeneratewhenJ (A/ω)=0,
1
tiallythedarkFloquetstate,notonlyforitszeroquasi-energy and when away from A = 0 the dark Floquetstate has aver-
butalsoforitsnegligiblepopulationatthecentralwaveguide; agedpopulationatthe1-thmode(guide)wellbelowthevalue
thesuppressionoftunneling(CDT)isofthedarkCDTasitis of 0.5. Therefore it can be concluded that the well-defined
causedbythedarkFloquetstateratherthanleveldegenaracy. quasienergycrossingsinsteadofthedarkFloquetstatearethe
In fact, the CDT-PAT transition found in Fig.2(b) is closely originoftheextremelysharppeaksseeninFig.2(d).
related to the sharp localization-delocalization transition of In Fig.2(g), we show the values of Min(P ) in the three-
1
populationhP iforthezero-energy(dark)Floquetstate. Note guide optical system for the n = 2 resonances. As is ex-
1
thatthedarkFloquetoriginallydiscoveredanddefinedinthe pected,thevaluesofMin(P ) exhibita numberofextremely
1
high-frequencylimit can be reduced to the well-known dark sharppeakscenteredonthezerosofJ (A/ω)wherethequasi-
2
statebymeansofhigh-frequencyaveragingmethod[7,8]. In energieswillbedegenerate;seeFig.2(h). Similartothecase
thisconsideredmodel(2),however,thedarkFloquetstateand ofn = 1resonance,thesharppeaksinthen = 2resonanceis
the associated CDT can still exist in the non-high-frequency alsocausedbytheleveldegeneracyratherthanthedarkFlo-
regimes where the coupling strength Ω is much larger than quet state, as it is shown in Fig.2(i) that the populationhP i
2 1
the modulation frequency and the high-frequency averaging correspondingto the dark Floquet state is well below 0.5 at
4
thepointsofquasi-energycrossings.
da
i 1 = Asin(ωz)a +Ω a ,
1 1 2
dz
da
Min P10.15(a)0 .330.4 0.5 0.6Ω7 =2 1(c) ΩΩΩ111===111..25 iidddazz4j == ΩΩ11aa3j++1+Ω2Ωa15a,j−1, j=2,3
010 (b) 1 2 ω 2 5 Min P10.5 18 iddaz5 = Ω2a4. (3)
P1 0.5 0.6 0.75
n 0.5 Thedynamicsforthefive-guidesystemareinvestigatedby
Mi
00 11.5 ω3 Ω2=35 00 3 6 9Ω212 15 1820 (d3ir)ewctitihnttehgerlaitgiohntionfittihalelytimloec-adliezpeednadtetnhteSgcuhirdo¨ed1in.gInerFeiqgu.4a,tiwone
showthevalueofMin(P )asafunctionofΩ whichexhibits
1 2
FIG.3: (Coloronline)(a)-(b)Theminimumvalueofpopulationdis- asequenceofPATresonanceswithsimilarbehaviorasthatof
tributionP1,Min(P1),asafunctionofthemodulationfrequencyω a three-guide system. The higher n-photon resonances with
withΩ2 = 2andΩ2 = 3respectively. Otherparametersarechosen n ≥ 3 become very weak, almost invisible, as illustrated in
asΩ1 = 1andA = 6.6. (c)Min(P1)versusΩ2 withdifferentvalues theinsetof Fig.4(a). By comparisonofFig.2(a)with Fig.4
ofΩ .Otherparametersarechosenasω=3andA=6.6.
1 (a), it is apparentthat the same order PAT resonance for the
five-guidesystem is much narrowerand weaker than for the
three-guidesystem. Like the case ofthe three-guidesystem,
thisfive-guidesystemalsopossessesadarkFloquetstatewith
zeroquasi-energyandnegligiblepopulationatalloftheeven
In order to observe the n-photon-like resonances from a
j-thguides(modes),asillustratedinFigs.4(b)and(c).Reason
different angle, we also plot Min(P ) as a function of the
1 fortheexistenceoftheanalogofPATresonancesinthefive-
modulation frequency ω for two fixed parameters Ω = 2
2 guide system lies in that populationdistribution hP i for the
and Ω = 3 in Fig.3 (a) and (b) respectively. We can read- 1
2 dark Floquet state simultaneously displays a series of sharp
ily observe that the n-photon-like resonances occur at com-
dipsatthepositionsofPATresonances[seeFigs.4(c)].
paratively broad intervals around ω = Ω /n. The width of
2
suchphoton-assistedtunnelingresonancesismuchlargerthan
those of PAT resonancesobserved in the literature. We now 1(a) 1(d)
P1 0.71 P1
elaboratethe physicsunderlyingthis photon-assistedtunnel- Min 0.5 Min 0.5
ing resonances. The actual resonanceconditiondoes notre- 0.68
0 0
ferdirectlytoΩ2 = nωbutrathertothetunnelingfrequency 2(b) 2(e)
ofthe model(2) withoutperiodicmodulation. The unmodu-
ǫ 0 ǫ 0

latedthree-guideopticalsystem(2)admitsthreeenergylevel ǫ1 ǫ2 ǫ3 ǫ4 ǫ5 ǫǫ11 ǫǫ22 ǫǫ33 ǫǫ44 ǫǫ55
−2 −2
ilamesvbe0al,sl±ainsqcωeΩd021d=+arΩqk22sΩ,ta21atne+dwΩtih22th.ezIsneprasocueecnhoefargtswyysoretesnmueli,tgsthhibneoterhxienisgstueenpncpeerregosyf- 〈〉Pj0.0150 (c) 3 6 9 12 15 1〈〈〈〈〈8PPPPP12345〉〉〉〉〉2 0 〈〉Pj0.0150 (f) 3.8 7 〈〈〈〈〈PPPPP12345〉〉〉〉〉1 0
Ω A/ω
sion of tunneling when the periodic modulation is switched 2
off. The periodic modulation effectively creates “photons” FIG. 4: (Color online) PAT in five-guide optical system. The left
thatbridgetheenergygapbetweenneighboringenergylevels. column: (a)theminimumvalueof population distributionatguide
Thus, a photon-assisted tunneling resonance can occur at a 1,Min(P ),versusΩ withA = 6.6,ω = 3,Ω = 1; (b)thecorre-
1 2 1
modulationfrequencywhichsatisfiestheresonancecondition sponding quasi-energy ǫ versusΩ2 and(c)thetime-averagedpopu-
dωi0ff=erennωc.eWωheonfΩth2eiusncmonosdiudleartaebdlysylastregmerwthiallnbΩec1,otmheeepnrienrcgiy- lΩat2i.oTnhhePrjiigbhetlcoonlguimngn:to(dt)hMezine(rPo-1e)nveerrgsyus(dAa/rωk)fFolrotqhuee1t-spthatoetovnerrseuss-
pallycharact0erizedbyΩ andthereforetheresonancecondi- onanceω=3,Ω2 =2.8,Ω1 =1;(e)thecorrespondingquasi-energy
tion is approximatelygiv2en by Ω = nω. As clearly seen in ǫversusA/ωand(f)thetime-averagedpopulationhPjibelongingto
2 thezero-energy(dark)FloquetstateversusA/ω.
theinsetinFig.3(c),thepositionofn-photon-likeresonance
doesslightly shiftwith increasingΩ due to the dependence
1
oftheenergydifference(tunnelingfrequency)ω onΩ . InFig.4(d),weplotMin(P1)obtainedinthefive-guidesys-
0 1
tem as a function of the modulation parameter A/ω for the
1-photonresonanceΩ = 2.8,ω = 3,Ω ≈ ω. As discussed
2 2
B.PATinfive-guidesystem
before,wecanobservethatthevaluesofMin(P )arepeaked
1
atthezerosofJ (A/ω),atwhichCDToccurs,whilebetween
1
We now turn to the case of the five-guide optical system thepeaksMin(P )takeextremelylowvaluesasresultofPAT.
1
wherethedynamicalequationsare However,comparedwith the case of three-guidesystem, the
5
peaksinMin(P )areconsiderablylowerandbroader. Ascan III. POSSIBILITYOFEXPERIMENTALREALIZATION
1
be clearly seen from Figs.4(e) and (f), the peaks in Min(P )
1
are indeed centered at the points of closest approach of the Now,wediscusstheexperimentalpossibilityofobserving
quasi-energieswherethe darkFloquetstate hasa population ourtheoreticalpredictionsbasedon the coupled-modeequa-
hP1i > 0.5. Thenumericalresultsestablishagainafirmlink tions. Amorerigorousdynamicsforoursystemcanbesimu-
between PAT and dark Floquet state in our considered sys- latedbytheSchro¨dingerequationforthedimensionlessfield
tems. amplitudeE,whichdescribesthelightpropagationalongthe
zaxisofanarrayofN waveguides
C.Tunnelingdynamicsinthefour-andsix-guideoptical
systemsandbeyond
∂E 1∂2E
i =− −pR(x,z)E. (4)
Finally, we briefly discuss the case of the four- and six- ∂z 2 ∂2x
guidesystems. Thedynamicsfor N = 4and N = 6are pre-
sented in Fig.(5) on the basis of a full numericalanalysis of Herexandzarethenormalizedtransverseandlongitudinal
equation(1) with the lightinitially populatedin the guide1. coordinates, and p describes the refractive index contrast of
Ittells the existence of a sharp transitionfromCDT to com- theindividualwaveguide.Foroursystem,therefractiveindex
plete tunneling for both cases of N = 4 and N = 6 when of the first waveguide is harmonically modulated along the
thecouplingstrengthΩ isincreasedfromzero. Aclose ex- propagationdirection, while all other N −1 waveguides are
2
aminationof the tunnelingdynamicsatΩ = nω showsthat unmodulated.Thecorrespondingrefractiveindexdistribution
2
thevalueofMin(P )displaysnarrowpeaksnearlyatzerosof ofthiskindofwaveguidesystemisgivenby
1
J (A/ω) where a pair of quasi-energies become degenerate.
0
This closely resembles the case of the high-frequencymod- 1 x− jw 6
ulation ω ≫ max(Ω1,Ω2) where CDT is dominated by the R(x,z) = [1+ fj(z)]exp− w j! , (5)
zcearliozsatoiofnJc0(eAn/teωre).dnAesarslhyoawtnzeirnosFoigfsJ.0(5(A)/(ωb))iasnfdai(rely),stmheallloe-r f1(z) = µj=s−Xi(nN−(ω2)z), fj(z)=0(j,1), x 
for the four-guide system, but still generates high peaks for
thesix-guidesystem. withthepositionofeachwaveguidebeing jw ,thechannel
j
width w , the longitudinal modulation amplitude µ, and the
x
modulationfrequencyω. Thereinthesuper-Gaussianfunction
exp(−x6/w6)describestheprofileofasinglewaveguidewith
1 1 x
(a) (d) widthw .Inourdiscussion,allthewaveguidespacings[jw −
Min P10.5 Min P10.5 (j−1)wxj−1]areidenticalexceptthatthespacingbetweentjhe
0 0 bottomboundarywaveguideanditsneighborisvariable.
0 3 6 9Ω 12 15 1820 0 3 6 9Ω 12 15 1820
1 2 1 2 In what follows, we will illustrate our main results with
(b) (e)
Min P10.5 Min P10.5 tahetrifipeleldt wpraovpeagguaidtieonsyesqteumatio(Nn (=4)3w)itbhyrdeairleiscttilcyeixnpteegrrimatienng-
00 2.4 A/ω5.5 8.6 10 00 2.4 A/ω5.5 8.6 10 tal parameters. We set wx = 0.3,p = 2.78,µ = 0.2 and
1(c) 2(f) ω = 3.45 × (π/100). We characterize two distinct waveg-
ǫ 0 ǫ 0 uide spacings as ws and ws respectively, where ws = w
ǫ1 ǫ2 ǫ3 ǫ4 ǫ1 ǫ2 ǫ3 ǫ4 ǫ5 ǫ6 1 2 1 1
−10 2.4 5.5 8.6 10 −20 2.4 5.5 8.6 10 standsfor the separationbetween the top waveguideand the
A/ω A/ω middlewaveguideandws = w theseparationbetweenthe
2 −1
bottomwaveguideandthemiddlewaveguide. Furtherweset
FIG.5: (Coloronline)Theleftcolumn: thecharacteristicsoffour- ws = 3.2andchoosedifferentvaluesofws toobservePAT
guide optical system; The right column: the characteristics of six- 1 2
guideopticalsystem.(a)and(d):Min(P )versusΩ atA=6.6,ω= resonance. As in the experiments[19, 20], wx and wj are in
3,Ω =1;(b)and(e):Min(P )versusA/1ωatΩ =32,ω=3,Ω =1; units of 10 µm, and p = 2.78 corresponds to a real refrac-
(c)a1nd(f):quasi-energiesεv1ersusA/ωatΩ =23,ω=3,Ω =11. tiveindexof3.1×10−4. Inallsimulationsweexcitedthetop
2 1 channelatz=0,usingthefundamentallinearmodeoftheiso-
latedwaveguide. Itisinstructivetonormalizethemodulation
frequency to the beating frequency of the unmodulated lin-
Moreover, we have simulated multiwaveguide systems of eardual-corecouplerwith spacingws , Ω = 2Ω = 2π/T ,
1 b 1 b
other numbersof waveguides. The numericalresults, which whereT isabeatingperiodrepresentingtheshortestdistance
b
are not displayed here, show that the PAT resonances prob- forthelightreturningto theinputwaveguide. Foroursetof
ably occur in all the odd-N-guideopticalsystems, while the parametersonehasT =100andthusω=3.45Ω .
b 1
PAT resonancesbecomeweakerwiththeincreaseofnumber Thebeamdynamicsofathree-guideopticalsystemarevi-
ofguides. However,alltheeven-N-guideopticalsystemsex- sualizedin Fig.(6) forthreevaluesofws , whichfirmlyver-
2
hibit CDT to complete tunneling transition without appear- ifies the predictions from the coupled-modeequation (2). It
ance of n-photon-like resonance when the coupling strength can be readily observed from Figs.(6) (a) and (b) that the
Ω isincreasedfromzero,whichistotallydifferentfromthe lighttunnelingisalmostcompletelysuppressed,asthethree-
2
caseofodd-N-guidesystem. channelwaveguide system has equalchannelspacing ws =
1
6
modulatedperiodicallyalongthelightpropagationdirection.
Withthestandardcoupled-modetheory,thesystemcanbede-
scribedbyadriventhree-statediscretemodelwithtwodistinct
couplingstrengths Ω and Ω , where Ω stands for the cou-
1 2 1
plingstrengthbetweenstates1and2,andΩ betweenstates
2
2 and 3. In studyingthe three-state discrete model, we have
foundthat(i)astrongsuppression(CDT)associatedwiththe
zero-energy(dark)Floquetstatepersistseveninthenon-high-
frequency modulation regimes where ω ≤ max(Ω ,Ω ) ex-
1 2
ceptataseriesofresonancepositions;(ii)atparticularvalues
of the coupling strength Ω , Ω ≈ nω with n being integer,
2 2
the tunnelingdynamicsis (partly)restored, analogousto the
FIG.6: (Coloronline)Lightpropagationinthree-guideopticalsys- n-photon-like resonances which overcome the effect of sup-
tems with different spacings between the bottom and the middle
pression of tunneling. The numerical calculations illustrate
waveguide fortheinput beamcentered atthetopwaveguide. First
thatthePATresonancesexistinthefive-statesystemandalso
row:(a)and(b):therefractiveindexdistributionR(x,z)andthelight
propagation |E(x,z)|2 for a three-guide system with equal channel probably exist in the systems with arbitrary odd number of
spacingws =ws =3.2;Secondrow:(c)and(d):therefractivein- coupledstates. Inparticular,thePATresonanceswillbecome
1 2
dexdistributionR(x,z)andthelightpropagation|E(x,z)|2forathree- weaker with the increase of number of states (modes). This
guidesystemwithunequalchannelspacingws = 3.2,ws = 2.22; typeofPATfoundinourworkhasadifferentoriginfromtra-
1 2
Thirdrow: (e) and (f): therefractive index distributionR(x,z) and ditionalPAT. Itiscloselyrelatedto theexistenceofthe dark
thelightpropagation|E(x,z)|2forathree-guidesystemwithunequal Floquetstate. Themainresultsaredemonstratedbythedirect
channelspacingws1=3.2,ws2 =1.2. numericalsimulationsofpropagationdynamicsbasedonthe
fullcontinuousmodelwithrealisticexperimentalparameters,
whichindicatesthatthePATfoundinourworkcanbereadily
ws = 3.2. At ws = 2.22, the light coupling between the
2 2 testedinthecurrentexperimentalsetup.
waveguidechannelsisrestored[seeFig.(6)(c)and(d)]. The
revivaloflighttunnelingisasignatureofPATresonancepre-
dicted by the coupled-mode theory. In fact, our numerical
simulation(notshownhere)revealsthatthebeatingperiodof Acknowledgments
anunmodulatedlineardual-corecouplerwithachannelspac-
ing 2.22 is about100/3.45. As such, we have Ω2 ≈ 3.45Ω1 The work is supportedin partby the NationalFundamen-
and Ω2 ≈ ω, which is in fact the position of the first pho- talResearchProgramofChina(GrantNo. 2012CB922103),
ton resonance. As the channel spacing ws2 is reduced fur- the National Science Foundation (NSF) of China (Grant
ther, it is expectableto observeagain the strong suppression Nos. 11375067,11275074,11374116,11204096,11405061
oflighttunneling[seeFig.(6)(e)and(f)]. Theseresultsarein and 11574104), the Fundamental Research Funds for the
goodqualitativeagreementwiththoseinFig.2(a)basedonthe Central Universities, HUST (Grant No. 2014QN193). X.
coupled-modeequation. Luo is also partially supported by the NSF of China un-
der Grants11465009,11165009,the ProgramforNew Cen-
tury Excellent Talents in University of Ministry of Educa-
IV. CONCLUSION tionof China (NCET-13-0836),Scientific andTechnological
Research Fund of Jiangxi Provincial Education Department
In summary, we have theoretically reported an analog of under Grant No. GJJ14566, and the financial support pro-
PATinathree-channelwaveguidesystem,inwhichthespace vided by the Key Subject of Atomic and Molecular Physics
separationbetweenthebottomandthemiddlewaveguidesis in Jiangxi Province. X. Luo thanks Prof. Biao Wu for his
adjustable and the refractive index of the top waveguide is supportsovertheyears.
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