Quantum transport with coupled cavities on the Apollonian network

1.7 MB·English

by  Guilherme M. A. Almeida

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 S˜ao Cristo´v˜ao, 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 pho-
tonic 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 Schr¨odinger 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. (Color online) Comparison between atomic and photonic propagation dynamics from the large hopping regime κ/β =
103 to the strong atom-cavity coupling regime κ/β =10−3. In (a), the photonic return probability to the hub node, πph, and
44
in (b) the atomic counterpart. System parameters are n = 4 and ∆ = 0. The middle range of κ/β values is where the JCH
model plays its main role.
ACKNOWLEDGEMENTS
This work was supported by the CNPq (Brazilian
agency). A.M.C.S. wishes to thank the National Insti-
tute of Science and Technology for Complex Systems.
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