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Quantum transport with coupled cavities on the Apollonian network G. M. A. Almeida and A. M. C. Souza∗ Departamento de F´ısica, Universidade Federal de Sergipe, 49100-000 São Cristo´vão, Brazil (Dated: November1, 2012) We study the dynamics of atomic and photonic excitations in the Jaynes-Cummings-Hubbard (JCH) model when each cavity occupies the nodes of an Apollonian network (AN).Such structure showssmall-worldandscale-freeproperties,andhasbeeninvestigatedwithinmanyphysicalmodels. BynumericallydiagonalizingthesystemHamiltonianinthesingle-excitationsubspace,weevaluate the time evolution of an initial state prepared as an even superposition of both atomic and photonic modes fully localized at a specific node. For the large hopping regime we provide a detailed description of the quantum transport properties of the AN and show that the complex interplay 2 betweenthemanycavity-cavitydegreesoffreedominducesbothextendedandlocalizedstatesvary- 1 ing periodically. The excitation is most likely to be found in the initial node and its propagation 0 stronglydependson theinitialconditions. Wealso discussthestrongatom-cavitycouplingregime, 2 whereatomic andphotonicmodespropagates identically,andtheJCHregime, wherethesystemis t allowedtoroambetweenatomicandphotonicdegreesoffreedom,sincethehoppingparameterand c theatom-cavitycouplingstrengthareofthesameorder. Inthisregime,differentcavitiescontribute O eitherto the atomic or photonic component mostly, dependingon the initial conditions. 1 PACS numbers: 42.50.Pq, 42.50.Ex, 03.67.-a, 89.75.Da 3 ] I. INTRODUCTION calized field modes, they have derived the dynamics on h the single-excitationsubspaceforavarietyofinteraction p - Cavity quantum electrodynamics systems have been regimes focusing on the atomic state transfer and found nt widely considered as a suitable choice for implementa- that, by setting the right parameters, one can achieve a a tion of quantum information processing schemes [1–3]. high-fidelity state transfer without populating the field u Averysimple cavitysystemconsistsof atwo-stateatom mode, thus avoiding photon loss. Makin et al. [18] con- q coupled to a single mode of an electromagnetic field in- sidered a one-dimensional array of cavities and dealing [ sideanidealcavity. Suchsystemisdescribedbythe well withbothatomicandphotonicexcitationdynamicsthey 1 known Jaynes-Cummings (JC) model [4], which is ex- discussed the localized and delocalized behaviour of the v actly solvable and has been useful for investigating fun- system. Theyhavealsoshownthatbyparabolicallycou- 0 damental aspects of the interaction between light and pling the cavities,or encodingthe initialstate as a gaus- 5 matter. Further generalizations of the JC model include sianwave-packet,onecanavoidpulsedispersion. In[19], 4 multiple atoms, N-level atom, dissipation, and so forth Dong et al. discussed the binary transmission dynamics 8 . [5]. inachainofcoupledcavities,eachonecontainingatree- 0 Experimental advances in optical microcavities [6], levelatomandshowedthata certainclass ofcoding and 1 photoniccrystals[7],andsuperconductingdevices[8]has decoding protocol can improve the transmission fidelity. 2 1 brought interest to Hubbard-like models where photons Theseresultsindicatethateveninthesingle-excitation : can hop through an array of coupled-cavity systems. In subspace the JCH model presents a rich dynamics due v particular,theJaynes-Cummings-Hubbard(JCH)model to the interplay between atomic and photonic degrees i X [9,10]providesanewframeworkforstudyingmany-body of freedom. Thus it would be interesting to apply this r phenomenasuchasphasetransitionswherestronglycor- model to more complex structures. Lately, a class of a related photons play the role. Most of recent work has networks which are neither completely regular nor com- focused on the Mott insulator-superfluidquantum phase pletely random,namedsmall-world networks[20], where transition of polaritons (combined atomic and photonic the average length of the shortest path l between two excitations) [11–13]. nodes increases logarithmically with the network size N Coupled-cavitysystemsallowthecontrolandmeasure- (l ∝ lnN), has been widely studied within many fields, ment of individual nodes. Therefore, it also provides a from social and biological systems [20, 21] to classical reliable setup for distributed quantum information pro- and quantum transport dynamics [22–24]. In particular, cessing [14], whichrequiresentanglementgeneration[15] a class of complex networks called Apollonian networks and quantum state transfer between distant nodes in a (AN) [25], which are simultaneously small-world, scale- network [16]. Recently, Ogden et al. [17] studied quan- free (that displays a power-lawdegree distribution), and tum state transfer between two coupled cavities, each can be embedded in Euclidean space, has brought some one containing a single two-levelatom. In terms of delo- attention. The free-electron gas [26], magnetic models [27],tight-bidingmodels[28],correlatedelectronsystems [29],andquantumwalks[30]insuchnetworkshavebeen investigated. AN’sarisefromtheproblemofspace-filling ∗ Electronicaddress: [email protected] packing of spheres [31]. It can be deterministic or ran- 2 In what follows,Sec. II, we introduce the JCH model. In Sec. III we present our numerical results for the time evolutionatdifferentenergyregimes. Finally,inSec. IV, we draw our conclusions. II. THE JAYNES-CUMMINGS-HUBBARD MODEL Letusconsiderasystemofcoupledcavitieswhereeach oneoccupiesthenodesofanApolloniannetwork. Atwo- level system g , e coupled to a single field mode can {| i | i} describedbytheJCHamiltonian[4](intherotatingwave approximation), ω hJC = aσ +ω a†a +β(σ+a +σ−a†), (1) j 2 z f j j j j j j FIG. 1. The second generation (n = 2) Apollonian packing (solid lines) and thecorresponding AN (dotted lines). whereσz|gi=−|giandσz|ei=|ei,σj+(σj−)anda†j (aj) are,respectively,theatomicandphotonicraising(lower- ing)operatorsforcavityj, β isthe atom-cavitycoupling dom,anddependsontheinitialconfiguration. Themost strength,ωa theatomictransitionfrequency,ωf thefield commonpackingstartswiththreemutuallytouchingcir- mode frequency, and we set ~ = 1 for convenience. If cles. The first generation, n = 1, is achieved by insert- we include photon evanescent hopping between nearest- ing a maximal circle in the interstice bounded by the neighbourcavities(tight-bindingapproximation),wecan three initial circles. Further generations are obtained represent the whole system by the JCH Hamiltonian by repeating this process for every empty space. This packing can be mapped to the AN as shown in Fig. 1. N(n) N(n) At a given generation, the number of nodes is given by H = hJjC − κAi(jn)a†iaj, (2) N(n) = (3n +5)/2, and the number of connections by Xj=1 iX,j=1 C(n)=(3n+1+3)/2(n=0,1,2,...). Ineachgenerationa newgroupofnodessharingthesamedegreeγ iscreated. where κ is the cavity-cavity coupling strength, N(n) Moreprecisely,thereare3n−1,3n−2,3n−3,...,32,3,1,and number of nodes for a fiven AN generation n, and A(n) ij 3nodes with degreeγ =3,3 2,3 22,...,3 2n−2,3 2n−1, the adjacency matrix elements. We now restrict the sys- and2n+1,respectively,wher·ethe·lastnum·berrepr·esents tem basis to the single-excitation subspace. These exci- the three corner nodes. The central node, often referred tations can be photonic or atomic, that is Q g,1 or | i| i as the hub node, has the largest degree. Q e,0 , respectively, where Q 1,...,N(n) speci- | i| i ∈ { } In this paper, we study the dynamics of the JCH fies the cavity. Then the whole system state is written model defined on the AN. We numerically diagonalize as the tensor product of every single-cavity state, where the Hamiltonian in the single-excitation subspace and one excitation must be conserved(the other non-excited and solve Schrödinger equation thus obtaining the time cavities are in the g,0 state), so the Hilbert space has evolution of atomic and photonic occupation probabil- D =2N(n)=3n+|5 diimensions. In such basis, the ma- ity distribution for fully localized initial states prepared trixformoftheHamiltonian(2)canbesimplyexpressed in an even superposition of atomic and photonic modes. by [18] We consider three different energy regimes. For large inter-cavityhopping,thephotoniccomponentpropagates H = ∆I Z+βI X κA I2+Z, (3) practically free, without significantly exchange energy 1exc 2 N(n)⊗ N(n)⊗ − n⊗ 2 with the atomic modes in short time scales. In this regime we analyse the photonic component transport where∆=ωf ωa isthedetuningbetweenphotonicand − throughthe AN when the initial state is preparedin dif- atomic modes, Im is the m m identity matrix, X and ferentnodesandshowhowitchangesbyvaryingthenet- Z are the usual Pauli matr×ices, and An the adjacency N(n) worksize. Whentheatom-cavitycouplingstrengthdom- matrix. Thegroundstate Q g,0 isnotincluded ⊗Q=1 | i| i inates,we havea similarpropagationdynamics but with in this subspace, that is, the system is already set with both components travelling simultaneously. For compa- the single-excitation. rable hopping and atom-cavity coupling, the dynamics The AN does not take geometrical deformations into turns outto be morecomplexsince the systemcanroam account so we are only interested in the connections be- between atomic and photonic degrees of freedom, thus tween the nodes. By considering uniform cavity-cavity leading to different dynamics. coupling, the adjacency matrix A for n = 2, for exam- n 3 modes (i.e., a single-cavity eigenstate for ∆ = 0 [5]), ψ (0) = k (g,1 + e,0 )/√2 at node (cavity) k, we 1 | k i | i⊗ | i | i 1 write 2 D 21 15 20 251424 36 1035 40 9 39 πℓpkh(t)=(cid:12)(cid:12)(cid:12)Xj e−iEjthψk(0)|EjihEj|ℓ,gi(cid:12)(cid:12)(cid:12) (5) 22 5 23 37 7 38 (cid:12)(cid:12) (cid:12)(cid:12) (cid:12) (cid:12) 17 1619 4 41 8 43 and 18 42 2 D 2 331314 32 6 28 2163 27 33 πℓakt(t)=(cid:12)(cid:12)(cid:12)Xj e−iEjthψk(0)|EjihEj|ℓ,ei(cid:12)(cid:12)(cid:12) (6) (cid:12) (cid:12) 29 31 (cid:12) (cid:12) 12 as the probabilit(cid:12)y of finding both photonic an(cid:12)d atomic 30 excitations, respectively, at node ℓ in time t, where we have denoted ℓ,g ℓ g,1 and ℓ,e ℓ e,0 . | i≡| i| i | i≡| i| i FIG. 2. (Color online) The Apollonian network for n = 4 (43 nodes). The first generation is made up by nodes 1 to A. Large hopping regime 4, the second one includes 5 to 7, the third, 8 to 16, and the fourth, 17 to 43. The corner nodes were drawn bigger for a First,weinvestigatethelargehoppingregime(κ β). clearervisualization. Thisisthenodelabelling wewilladopt ≫ In this case, the atom-cavity interaction slowly takes through the paper. place so that the photonic excitation propagates practi- callyfree,whilethe atomiconedoesnot“move”inshort time scales. Remember that only the photons can hop ple, can be written by while atoms remain stationary. The atomic component dynamicsisthencorrelatedtothephotonicone,although 0 1 1 1 1 0 1 not exactly the same, except when the atom-cavity cou- 1 0 1 1 0 1 1 pling β dominates the hopping strength as we will see 1 1 0 1 0 0 0   later. A2 =1 1 1 0 1 1 1 (4)   The AN has a 2π/3 rotational symmetry, hence there 1 0 0 1 0 0 1 0 1 0 1 0 0 1 are several subsets of equivalent nodes (see Fig. 2) like   1 1 0 1 1 1 0 1,2,3 , 5,6,7 , 8,10,11,13,14,16 , and so forth, in   { } { } { } which ψ (0) would lead to the same dynamics. Like- k | i (see Fig. 1). In order to discuss our results in next sec- wise, for the initial state set at the hub node (k = 4), tion, we need the appropriate identificationof eachnode the occupation probabilities πtype(t) are equal for each ℓ4 (cavity),thenFig. 2providesthe nodenumberingwhich subset. we are goingto consider herein,up to the 4th AN gener- Figs. 3 and 4 show the time evolution of the pho- ation. tonic occupation probability πph(t), for different initial ℓk single-cavity eigenstates ψ (0) , in the n=3 and n=4 k | i AN,respectively. Althoughexhibitingarathercomplex III. NUMERICAL RESULTS dynamics, we can identify some general aspects of the propagation. The most remarkable feature is periodic Inthissectionwedisplaytheresultsfortime-evolution transition between localized and extended states. The oftheJHCmodelintheANobtainedbynumericaldiag- return probability πkpkh(t) can reachthe highest values so onalizationoftheHamiltonian(2)inthesingle-excitation that the excitation is most likely to be found in the ini- subspace thus finding its eigenvalues E and the corre- tialnode. Thislocalizationisstrongerfork =4,thehub j sponding normalized eigenstates E . We discuss the node. Also, it seems that πph(t) oscillates faster for the j kk | i system dynamics for different energy regimes and con- initial state set atnodes with a higher degree of freedom sider no detuning, ∆ = 0, between photonic and atomic γ. InFig. 5weplotπph(t)(solidline)andalsotheatomic 44 modes. We are mostly interested in the analysis of pho- component πat(t) (dotted line) time evolution in detail. 44 tonic and atomic excitation occupation probabilities in As we said before the atomic mode barely moves, ex- different cavities for a given initial state, namely ψ(0) , cept for smalloscillations(see insetof Fig. 5) due to the | i through the time. The state at time t is given by photonic component behaviour. Since the atom-cavity ψ(t) = U(t) ψ(0) , where U(t) = e−iHt is the quan- interaction process has much lower associated energies, | i | i tum time evolution operator. If the initial state is pre- only the cumulative effect of these interactions in longer pared in a equal superposition of photonic and atomic time scales may significantly vary πat(t). Therefore, at ℓk 4 13 k = 3 40 k = 5 30 9 20 5 10 0.5 1 0.5 13 k = 4 40 k = 15 mber 59 00..34 number123000 00..34 e nu131 k = 5 ode 40 k = 18 d 0.2 N30 0.2 No 9 20 5 0.1 10 0.1 113 k = 15 0 3400 k = 21 0 9 20 5 10 1 0 20 40 60 80 100 0 20 40 60 80 100 Time (in units of k -1) Time (in units of k -1) FIG. 3. (Color online) Time evolution of the photonic exci- FIG. 4. (Color online) Same as Fig. 3 but for n = 4 (43 tation πph(t) through the n=3 AN (16 nodes) for the large nodes) and k=5, 15, 18, and 21. ℓk hopping regime. The vertical axis represents the occupation probability of a particular node (cavity). The single-cavity initial state ψ (0) = k (g,1 + e,0 )/√2 is prepared at 0.8 k nodes k =3,|4, 5,iand|15i.⊗Sy|stemi pa|ramieters are κ/β =103 0.5002 photonic and ∆=0. Thenode numberingis available in Fig. 2. 0.5000 atomic 0.6 0.4998 0 5 10 15 20 the large hopping regime we shall focus on the photonic mode propagation dynamics thus aiming to explain the 0.4 quantum transport properties of the AN. type 44 Now let us turn our attention to the probability of p findingtheexcitationinnodesotherthanthe initialone. 0.2 In general, such probabilities are very small compared to π (t), although they can reach relevant values in kk some specific nodes. The overallprobability distribution 0 strongly depends on the AN generation and the initial 0 5 10 15 20 Time (in units of k -1) conditions. In Fig. 3, by examining π (t) for k =3, we ℓk see that a significant amount of probability flows to the other nodes belonging to the same degree group, that is FIG.5. (Color online) Photonic(solid line) andatomic (dot- 1 and 2 (check Fig. 2 for node numbering). The same ted line) return probabilities for the initial state prepared in applies to π65(t) and π75(t) when k=5. For k =15, the ψ4(0) at the n = 4 AN. The inset shows small oscillations nodes10and11evenbelongingtothesamedegreegroup i|nπ4a4t(it)inducedbythephotonicmodepropagation. System are barely populated. It suggests that sharing the same parameters are κ/β =103 and ∆=0. degree does not necessarily imply in privileged network- ing. It turns clearer in Fig. 4 for k = 18 and 21, where the excitationismostly spreadto afewnodes withinthe A better way to visualize how the excitationprobabil- subnetwork [32] generated by the new corner nodes 1, 2, ity distributes between the nodes for given initial con- and4. Whenk =5forn=4(Fig. 4),theexcitationstill ditions is by calculating the long time average of πph(t) ℓk roams between nodes 6 and 7, but now avoiding trans- defined as port to higher generation nodes. For k = 15 in Fig. 4, there is a relevant probability of finding the excitation 1 T χph = lim πph(t)dt at nodes 20, 21, and 22. These nodes belong to the sub- ℓk T→∞T Z0 ℓk network formed by nodes 1, 2, and 5, where the node 15 play the role of hub node. These observations indicate = δ(Ei Ej) ℓ,g Ei Ei ψk(0) − h | ih | i× X that these subnetworks has a fundamental part on the i,j system transport dynamics for higher AN generations. ψ (0)E E ℓ,g , (7) k j j ×h | ih | i 5 (a) 40 (b) (c) >0 .00.505 110 320 0.04 30 80 240 l 0.03 e od 20 50 160 N 0.02 10 80 20 0.01 0 10 20 30 40 20 50 80 110 80 160 240 320 Node k FIG.6. (Coloronline)Photoniclong-timeaverageoccupationprobabilitiesχph for(a)n=4(43nodes),(b)n=5(124nodes), ℓk and (c) n = 6 (367 nodes). System parameters are κ/β = 103 and ∆ = 0. The plot shape is symmetrical since χph = χph ℓk kℓ for every k and ℓ, and the highest values are χph, i.e., the average return probability. Although we have not provided the kk node numbering for nodes from generations higher than n = 4, the purpose here is to give an idea of how the subnetworks andnodesfromdifferentgenerationsareconnectedamongthemselves. Ineachgenerationabove,thedarkestbranchrepresents the average excitation transport probability between nodes belonging to the current generation and to the previous one. The square boundaries point out connections between nodes havingthesame degree. where δ(E E ) = 1 for E = E and δ(E E ) = 0 uating the participation ratio of eigenstates defined as i j i j i j − − else. Fig. 6showsthedistributionofthisaveragefordif- ferentANgenerations. Firstofall,noticethatχpℓkh =χpkhℓ ξj = 1 , (8) for every k and ℓ (since the evolution is unitary), hence N(n) i,sE 4 this quantity provides us a clearer view on the way the |h | ji| s=P{g,e} iP=1 nodes stablish communication channels with each other. As it must be, the higher averages are χph, however, by for a given AN generation n. This quantity can assume kk looking at the other connections we can identify several values within 1, for completely localized states, and D, zones (square boundaries)andbranches apartwhich, in- for extended states ( i,sE = 1/√D for all i). Fig. 7 j terestingly, present self-similarity as n increases. The showstherelationshiph be|tweienξ /N andtheprobability j branches connect generations with each other while the of finding the system in each eigenstate, E ψ (0) 2, j k zones connect nodes from the same generation,i.e., with |h | i| for a given initial condition ψ (0) [Fig. 7(a)], and the k same degree γ. The strongest(darkest)branchis always | i associated eigenvalues E [Fig. 7(b)], for the n = 4 AN. j the one connecting nodes from the current generationto When we prepare ψ (0) in the hub node k = 4 (with k the previous one. It mostly represents the links between | i degree γ =24), the two most localized eigenstates (with the correspondinghubnodes fromthe 3n−2 subnetworks ξ /D 0.05), one being related to atomic excitation j having7nodes,toitsthreenearest-neighbournodeswith ≈ 4,e (E /κ 0)andtheothertothephotonicexcitation j γ =3. As n increases,the previous branchesstill remain | i ≈ 4,g (the highest eigenvalue), become more populated but with lower intensities since the aforementioned hub | i thus strongly leading the system dynamics. Only a few nodes now have a higher γ. Also, nodes from “distant” other eigenstates, not much localized, gives a relevant generationsseemtobebarelyconnected. Thisisbecause, contribution to the dynamics. Since we are considering when preparing the initial state in nodes from a current thelarge-hoppingregime,theatom-cavityinteractionen- generation, the excitation is mostly roams between the ergiesareverylowsoweclearlyidentifythemiddlerange initial node, other nodes with same degree belonging to of eigenvalues 0 [see Fig. 7(b)] as being those belong- local subnetworks, and its corresponding hub node from ≈ ing mostly to the atomic component of the system. In the previous generation. Notice in Fig. 6 that connec- Fig. 7(a), as we set ψ (0) for nodes having lower de- k tions between nodes from the same generation are only | i grees of freedom, for instance, k = 3, 5, and 21, where relevant within subnetworks composed up to 16 nodes. γ = 17, 12, and 3, respectively, the contribution from For instance, at a given generation n there are 3n−m localized eigenstates decreases and then more extended subnetworks having (3m +5)/2 nodes. Needless to say, states start to participate on the dynamics. Also, the thesedegreeinterconnectionsalsobecomeweakerasgen- eigenstates which mostly contribute to πph(t) has lower eration increases. kk associated energy levels for smaller γ. It shows that the oscillationperiodofthereturnprobabilitydependsonγ, Before discuss other energy regimes it would be use- as we have already mentioned. ful to explainthe reasonfor why the state alwayscarries More details about localized and extended states on relevant information about its initial conditions by eval- the AN and its relationship with the structure charac- 6 (a) 0.5 nature. In this case, as the photonic excitation propa- x / D gates through the network it exchanges energy with the j 2>| 0.4 kk == 43 atotormoaicmmboedtweseeinnpsuhcohtoanwicaayntdhaattotmheicwdheoglreeessysotfefmreesdtaormts. (0) 0.3 k = 5 Eachcavityprovidesitscontributiontomostlyeitherone k k = 21 of these two components, depending on the system ini- yE | j0.2 tialconditionsandthecharacteristicsingle-cavityenergy < levelsinarathercomplexway. Sinceafullylocalizedini- | 0.1 tial state of the form ψ (0) mostly leads the excitation k | i to return to the initial node, we presume that the oscil- 0 lation rate of πtype will be primarily relevant for driving 15 30 45 60 75 kk thewholesystemamongtheatomicandphotonicdegrees Eigenstate label (j) of freedom. Fig. 8 gives an idea of how the initial state maysignificantlychangesthewaybothmodescontribute (b) to the system dynamics in such energy regime. In Fig. 4 8(a)weshowthe long-timeaverageoccupationprobabil- 2 itydistributionχtype when ψ (0) issetatthehubnode, 0 k =4. Noticethaℓtkχat canb|ekevaliuatedbythesameway kℓ kE / j--42 0.0020 χ|aℓsp,hgEiqv.a.Wl7ueebsyc(apinnrcoslejueedcttinhingagtttshhoeemheeuigbgernnosuotdpaeste)o,sfoontnho|deℓre,sseimh,aoivnsetslthyeiagcdohneor-f -6 ℓ4 tributeto χat,andafewpracticallycontributeswiththe -0.002 ℓ4 -8 20 30 40 50 60 70 same amount for both components. In Fig. 8(b) we plot -10 the time evolution of the probability of finding the pho- 15 30 45 60 75 tonic and atomic excitation at any cavity by summing Eigenstate label (j) πtype over ℓ. In this case, the system notoriously fluctu- ℓ4 atebetweenbothcomponents. Itresultsmainlyfromthe FIG. 7. (Color online) (a) The normalized participation rate fast oscillation rate of π4p4h (since the hub node has the of eigenstates ξj/D (solid line) and the probability of find- higher γ) combined with the strong localization proper- ikng=t4h,e3,sy5s,taenmdi2n1.eiSgyesntsetmatepa|rEajmi,ettehrastarise|nhE=j|4ψk(4(30)ni|o2d,efso)r, thieens.ceTrheperaevseenratignegvaalgureesatadreiffχerp4e4hn≈ce.0.F2i6gsa.n8d(cχ)a4a4tn≈d80(.1d4) κ/β = 103, and ∆ = 0. The eigenstates are labeled accord- show the same quantities but now for k = 18 (γ = 3). ing to increasing energy values and, within each degenerate The higher averages χtype does not differ much for both gerrgoyupv,abluyesin(cirneausninitgsvoafluκe−s1)o.f Tξjh.e(ben)eTrghyelceovrerlessparoonudnindgzeenro- types, thus resulting inℓ1s8mall variations of ℓπℓty18pe. is shown on theinset. Actually, the ones lying at the middle P In summary, at the large hopping regime, κ β, the range ( 0) are not all degenerate. These energy levels arise ≫ from th≈e atomic modes and since κ β, its energy scale is interplay between atomic and photonic degrees of free- very small compared to thephotonic≫counterpart. domisminimalsothatthe atomicexcitationonlysignif- icantlypropagatesinlargertimescales. Thusthecavity- cavity coupling parameter, κ, allows the control of the teristic spectrum and symmetry properties can be found atomic mode velocity. When the JCH regime (κ β) ≈ in Ref. [28], where they considereda simple tight-biding takesplace,atomicandphotonicdegreesoffreedomstart electronicmodel. Thetendencyoftheexcitationforstay- tointerferewitheachother,thusresultingindifferentdy- inginthe initialnodehasbeenfoundforthecontinuous- namics . We can keep on decreasing κ/β until reaching time quantum walk in the same structure [30], and also the strong atom-cavity coupling regime, where photonic inasmall-worldnetworkdefinedbyaddingrandomlinks and atomic components have exactly the same dynam- in a one-dimensional ring of nodes [24]. By increasing ics. This regime transition is clearly shown on Fig. 9 the number of additional bonds, they have shown that where we plot the time evolution of the photonic [Fig. this effect becomes stronger thus avoiding dispersion. 9(a)]andatomic[Fig. 9(b)]returnprobabilitiesπtype for 44 a large range of κ/β values from 103 to 10−3. At the κ β regime, the dynamics for both atomic and pho- B. The JCH and strong atom-cavity coupling ton≪ic modes behaves exactly like at the large hopping regimes regimebut now creatingnew single-cavityeigenstates(if theinitialstateispreparedin ψ (0) )aspassingbyother k | i In what follows we consider the situation where the cavities, and propagating at half the speed of the pho- coupling between cavities and the atom-cavity interac- tonic excitation at the κ β regime [18]. Due to the ≫ tion strength are of the the same order (κ β). That is fact that the atoms inside the cavities does not actually ≈ the energy regime where the JCH model displays its full move,thereisasmalldelay,namelyδ,whichtendstodis- 7 0 10 0.8 (a) photonic (b) photonic atomic atomic -1 type 104 type 04.6 -2 10 0.4 -3 10 0.2 0 10 20 30 40 0 10 20 30 -1 40 50 Node Time (in units of ) 0 10 0.8 (c) photonic (d) photonic atomic atomic -1 type 10 18 type 018 .6 -2 10 0.4 -3 10 0.2 0 10 20 30 40 0 10 20 30 -1 40 50 Node Time (in units of ) FIG.8. (Coloronline)Figures(a)and(c)showsthephotonicandatomiclong-timeaverageoccupationprobabilitydistribution for k = 4 and k = 18, respectively, and (b) and (d), the corresponding time-dependent probability of finding both excitation types at any cavity, P πtype. System parameters are n = 4, κ/β = 1, and ∆ = 0. The return average probabilities are ℓ ℓk χp44h 0.26, χa44t 0.14, χp18h18 0.103, and χa18t18 0.105. ≈ ≈ ≈ ≈ turbthesingle-cavityeigenstates,butδ 0asκ/β 0. withinlocalsubnetworks. Thesepropertiesarealsovalid → → Bythesameway,atthe largehoppingregime,δ as for any analogue tight-binding model in the same AN. →∞ κ/β thus freezing the atomic component. →∞ At the JCH regime, where the hopping term and the IV. CONCLUSIONS atom-cavity coupling strength are of the same order, we haveshownthateachcavitygivesitsowncontributionfor Evenatthesingle-excitationsubspace,theJCHmodel theatomicandphotonicmodesthusinducingthesystem provides a rich dynamics. In addition, the Apollonian to roam between these two degrees of freedom. When network has ubiquitous transport features. We studied the atom-cavitycoupling term dominates, the excitation thepropagationdynamicsbyconsideringtheinitialstate propagatesmaintainingtheevensuperpositionofatomic as an even superposition of atomic and photonic modes and photonic modes (the single-cavity eigenstate). fullylocalizedinsomespecificnode. Atthelargehopping regime, where practically only the photonic component moves,wefocusedtheanalysisonthequantumtransport Even though high-fidelity excitation transport pro- dynamics in the AN. In such structure, the excitation is tocols in an one-dimensional chain of coupled cavities most likely to be found at the initial node due to the has been proposed [18, 19], such regular structures have characteristic network spectrum properties. By evalu- strongdispersiveproperties,incontrasttotheAN,where ating the long time average photonic occupation prob- we have shown that the excitation actually avoids prop- ability for different AN generations, we have discussed agating to some specific nodes, even when it is a first- the interplay between the nodes and subnetworks thus neighbour node, without any coding-decoding protocol. showing their role on the system dynamics. At a given Infact, we haveonly dealtwith the mostsimple fully lo- generationn,thenodeswithγ =3aremostlyconnected calized entangled initial state. Obviously, there are sev- withitscorrespondenthubnodefromthepreviousgener- eral possible combinations of initial states that should ationandwithsomeothernodeshavingthe samedegree generate different and much more complex dynamics. 8 (a) (b) 20 20 >0 .05.5 ) 1 - kof 15 15 0.4 s t i n u 10 10 0.2 in ( e m 5 5 0.1 i T 0 0 0 3 0 -3 3 0 -3 log (/kb) log (/kb) 10 10 FIG. 9. 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