Quantum state transmission in a cavity array via two-photon exchange

Yu-Li Dong, Shi-Qun Zhu, and Wen-Long YouSchool of Physical Science and Technology, Soochow University, Suzhou, Jiangsu 215006, P. R. China

The dynamical behavior of a coupled cavity array is investigated when each cavity contains a three-level atom. For the uniform and staggered intercavity hopping, the whole system Hamiltonian can beanalytically diagonalized in the subspace of single-atom excitation. The quantum state transfer alongthe cavities is analyzed in detail for distinct regimes of parameters, and some interesting phenomenaincluding binary transmission, selective localization of the excitation population are revealed. Wedemonstrate that the uniform coupling is more suitable for the quantum state transfer. It is shownthat the initial state of polariton located in the first cavity is crucial to the transmission fidelity, andthe local entanglement depresses the state transfer probability. Exploiting the metastable state, thedistance of the quantum state transfer can be much longer than that of Jaynes-Cummings-Hubbardmodel. A higher transmission probability and longer distance can be achieved by employing a classof initial encodings and final decodings.

PACS number: 42.50.Pq, 05.60.Gg, 03.67.Hk

I. INTRODUCTION

It is an important task in quantum information pro-cessing when a quantum state is transferred from one lo-cation to another. It refers to not only quantum commu-nication [1], but also large-scale quantum computing [2].Different protocols are proposed to realize quantum statetransfer (QST). For example, based on measurement andreconstruction, an unknown quantum state can be faith-fully transferred to an arbitrarily distant location viaquantum teleportation [3]. During the quantum telepor-tation, the measured result instead of the quantum stateis transferred by a classical communication channel. Al-ternatively, a quantum state can be sent through a chan-nel directly. There are two types of such state transferwhich depends on the transfer distance. For long distancecommunication, the information encoded in photons canbe transferred by an optical fiber [4, 5], which has beenwidely used in quantum communication and cryptogra-phy applications [6, 7]. Compared with the long distancecommunication, a kind of short distance communicationwhich can be used between adjacent quantum processorswas recently introduced [8]. Using an unmodulated spinchain as a channel, the state transmits from one end ofthe chain to another with some fidelity. Subsequently,many schemes are proposed to gain higher even perfecttransfer fidelity [9–15].

On account of the possibility of individual addressing,an array of coupled cavities is probably a promising can-didate for simulating spin chains. The coupled cavitiescan be realized in various physical systems [16], such asphotonic crystals [17], superconducting resonators [18],and cavity-fiber-cavity system [19]. So far, the Heisen-berg chains of spin 1/2 [20], of any high spin [21], andwith next-nearest-neighbor interactions [22] are simu-lated in the coupled cavities. In these simulation pro-cesses, the cavity field is ingeniously removed, and thesame procedure occurs when the QST is investigatedin Ref. [23]. However, the array of coupled cavities,a hybrid system combining the spinor atom and pho-

tons, would provide richer phenomena than the pure spinchains or Bose-Hubbard model [24–26]. For this reason,the temporal dynamics [27–29] and the binary transmis-sion [30, 31], i.e., the QST including both components inone-dimensional (1D) coupled cavities, are analyzed indetail.

The binary transmission mentioned above is restrictedin single-excitation subspace, i.e., only one photon trans-fers along the cavities in cascades, and usually the trans-fer distant extensively depends on the lifetime of atomicexcitation state. Quite recently, the two-photon propaga-tion in waveguide has been attracting increasing interestfrom many researchers, such as two-photon transport in1D waveguide coupled to a two-level emitter [32], a three-level emitter [33], and a nonlinear cavity [34]. Comparedwith one-photon electric-dipole E1 transition, the two-photon 2E1 transition that emits two photons simulta-neously is a second-order transition [35]. Therefore, themetastable state decaying via 2E1 transition has remark-able longer lifetime than the excited state decaying viaE1 transition. The typical example is the lifetime forthe 22S1/2 state in hydrogen is of the order of fractionsof a second, which is several orders longer than that ofsingle-photon excitation state. The long lifetime of themetastable state could provide sufficient operation timeto transfer information. More recently, Law et al. re-veal that the photons trend to bound together [36] in thetwo-excitation subspace when the ratio of vacuum Rabifrequency to the tunneling rate between cavities exceedsa critical value. Motivated by these aspects, it is de-sired to explore the short distance binary transmission ofthe model that bounding two photons as a quasiparticle[37, 38].

In this work, we investigate the binary transmissionin 1D array of coupled cavities, each of which containsa three-level atom. By adiabatically eliminating the in-termediate state, the individual cavity can be describedby the two-photon Jaynes-Cummings (JC) model with ametastable state and a ground state. In the single-atomexcitation subspace, the system Hamiltonian is able to bediagonalized exactly. It allows us to focus on the binary

arX

iv:1

202.

2147

v1 [

quan

t-ph

] 9

Feb

201

2

2

transmission with the explicit expression of the excita-tion population. We consider the uniform and staggeredintercavity hopping in different parameters regimes, es-pecially in the strong coupling and strong hopping lim-its. With the uniform hopping, we find that the initialstate of polariton located in the first cavity is essential tothe transmission fidelity. When the polariton is a pureatomic state, an oscillation behavior in the envelopmentof fidelity occurs and the state transfer probability in-creases obviously. On resonance, we reveal the optimaltransmission time is mainly governed by the intercav-ity hopping strength and the system size. We also showthat the exploiting of the metastable state overcomes thedeficit of short transfer distance during the lifetime ofexcitation state [28]. With the staggered hopping, theexcitation population trends to be localized in the firstseveral cavities and slows down the transfer speed. Thisphenomenon becomes more distinct as the distortion ofthe hopping strength enlarges. In the free evolution ofthe system, the transmission fidelity drops quickly as thesize of the array increases. How to enhance the fidelity ofthe transmission is important, since the fidelity of QST isexpected to be better than that by using straightforwardclassical communication. We finally demonstrate that aclass of initial encoding and final decoding process, whichis currently used in spin chains [13, 14], can also greatlyimprove the performance of binary transmission.

The paper is organized as follows. In Sec. II, we discussthe dynamics of an array of coupled cavities with theuniform coupling. In Sec. III, we consider the generalsituation with staggered coupling between the adjacentcavities. Section IV gives an effective method to improvethe fidelity of the quantum state transfer by encodingand decoding limited qubits in strong coupling regime.In Sec. V, we give a brief conclusion to the paper.

II. THEORY MODEL

We consider an array of coupled cavities, each of whichcontains a three-level atom in cascade configuration. Theindividual cavity is tuned to two-photon resonance withthe metastable state |e〉 and the ground state |g〉, andthese two states are coupled to an intermediate state |i〉with a single mode of the electromagnetic field with cou-pling strengths g1 and g2 respectively. The intermediatelevel is detuned by the amount δ from the average of theenergies of the |e〉 and |g〉. In the large detuning condition|δ| >> g1, g2, the intermediate state can be adiabaticallyeliminated, and the Hamiltonian of one individual cavitycan be written as a two-photon JC model [39, 40] (here~ = 1)

H(i) = ωaσ(i)ee + ωca

†iai + λ

(σ(i)eg a

2i + σ(i)

ge a†2i

), (1)

where ai (a†i ) is the photonic annihilation (creation) op-

erator, and σ(i)ge = |g〉(i)(i) 〈e| (σ(i)

eg = |e〉(i)(i) 〈g|) is atomictransition operator in the ith cavity, ωc is the resonance

frequency of the cavity and ωa is the effective energy ofatom. The metastable state |e〉 and the ground state|g〉 is coupled by a two-photon process with strength λ,which is given as λ = g1g2/δ. The two-photon JC modelhas been experimentally realized in high-Q superconduct-ing microwave cavity [41].

In Hamiltonian (1), the operator Q(i) = a†iai+

σ(i)ee − σ

(i)gg commutes with H(i), implying that the to-

tal number of atomic inversion and photonic excita-tion is conserved. The two-photon JC Hamiltonian canbe diagonalized analytically. Spanned by the subspace{|e, n− 2〉(i) , |g, n〉(i)

}(n ≥ 2), where n is the number

of photons, the Hamiltonian can be written by the po-laritonic state as

H(i)n =

(ωa + nωc − 2ωc λ

√n (n− 1)

λ√n (n− 1) nωc

). (2)

The eigenvalues and eigenfunctions ofH(i)n can be writ-

ten in terms of the dressed-state representation

|±, n〉(i) = χ±(θn) |g, n〉(i) ± χ∓(θn) |e, n− 2〉(i) , (3)

E(i)±,n =

∆

2+ nωc ± 2χn, (n > 2) (4)

where tan θn = 2λ√n(1 + n)/(∆ + 2χn), χ+ (x) = sinx,

χ− (x) = cosx, χn =√n(n− 1)λ2 + ∆2/4, and the de-

tuning ∆ = ωa − 2ωc.The N cavities linked via two-photon exchange are de-

scribed by the Hamiltonian

H =

N∑i=1

H(i) + ξ

N∑i,j=1

Aij

(a†2i a

2j + a2

i a†2j

), (5)

where ξ is the intercavity coupling of double photons andA is the adjacency matrix of the connectivity graph. Herethe photons only hop between the nearest-neighbor cav-ities, so the matrix element is given by Aij = δi,j±1.

Since the hopping between cavities does not changethe number of photons, the operator Q =

∑iQ

(i) com-mutes with the Hamiltonian H. For simplicity, the anal-ysis is restricted to the case of Q = 2 −N , i.e., the sys-tem contains only single-atom excitation or two-photonexcitation. Here it is assumed that all the cavities areequal. Then we extract the site indexes from the atomicand field operators, and define them as a set of barebases which form a complete Hilbert space. The systembases thus can be written as |M〉⊗ |g, 2〉 and |M〉⊗ |e, 0〉(M = 1, 2, ..., N), which denote that there are two pho-tons or an atomic excitation at cavity M . In these re-stricted bases, the Hamiltonian (5) can be representedas

Hres =∆

2IN ⊗ Z +

√2λIN ⊗X + 2ξA⊗ I2 − Z

2, (6)

where Im is the m × m identity matrix and X, Z arethe pauli operators acting on the certain site of cavity.

3

FIG. 1: (Color online) The probability of the certain cavityas a function of time with the length of the chain N = 101and on resonance ∆ = 0 for the initial state β = π/4. Theupper two plots show the population of atom (a) and photon(b) in the strong J-C coupling regime (λ = 200ξ), and thelower two plots are the population of atom (c) and photon(d) in the strong hopping regime (λ = ξ/200). The inset of(c) shows the population of the atom in the first cavity slowlydecreases with time.

If the open chain of the coupled cavity is considered,i.e., the hard-wall boundary conditions are assumed,the eigenvectors of the adjacency matrix A are [9, 28]

|m〉 =√

2N+1

∑M sin

(mMπN+1

)|M〉, with corresponding

eigenvalues Em = −2 cos mπN+1 for all m = 1, 2, ..., N . For

a 1D chain of cavities, the Hamiltonian Hres can be ex-pressed in the bases {|m〉 ⊗ |e, 0〉 , |m〉 ⊗ |g, 2〉} as blockdiagonal matrixes, in which the mth block appears as

Hres(m) =

(∆2

√2λ√

2λ −∆2 − 4ξ cos mπ

N+1

). (7)

Then the block matrix can be diagonalized,

the eigenvalues are Em± = −2ξ cos(mπN+1

)±√[

∆2 + 2ξ cos

(mπN+1

)]2+ 2λ2, and the correspond-

ing eigenvectors are |m,±〉 = |m〉 ⊗ |±〉, wherethe dressed state |±〉 = [

(∆− 2Em∓

)|e, 0〉 +

2√

2λ |g, 2〉]/√(

∆− 2Em∓)2

+ 8λ2. With these ana-

lytical expressions, the time evolution of arbitrary Ncavities can be straightforwardly derived. In the follow-ing, the quantum state transfer is studied in such model.It should be emphasized that there exist two channels,either atomic or photonic, to transfer information, whichis different from state transfer in a pure spin chain. Inthis model, the sender and receiver can select whichqubit is encoded and decoded. The ground state of

the system |g, 0〉⊗N with a zero eigenenergy will notinvolve any evolution by the Hamiltonian H. So onlythe propagation of the excitation state for atom andphotons needs to be considered.

FIG. 2: (Color online) Dynamics of the transfer probabilitiesof atom and photon respectively in strong coupling limit. Thesystem parameters are ∆ = 0, λ = 10, ξ = 1, N = 100 for (a)and (b) β = π/2, (c) and (d) β = π/4. The insets of (a) and(b) show the oscillation behavior in the envelopment.

The influence of the initial state located in the firstcavity is investigated. Supposing that the initial state isan entangled state in the first cavity

|ϕ(t = 0)〉 = |M = 1〉 ⊗ (cosβ |g, 2〉+ sinβ |e, 0〉) , (8)

the state at time t is given by |ϕ(t)〉 = U(t) |ϕ(0)〉, whereU(t) = e−iHt is the evolution operator of the whole sys-tem. Hence one has

|ϕ(t)〉 =

N∑M,m=1

|M〉 ⊗[f+M,m (t) |+〉+ f−M,m (t) |−〉

],

(9)where the probability amplitudes are given by

f±M,m (t) =2

N + 1e−iE

m± tχ± (β − αm)

× sin

(mπ

N + 1

)sin

(mMπ

N + 1

), (10)

with tanαm = −{[(∆− 2Em+

)2+ 8λ2]/[

(∆− 2Em−

)2+

8λ2]}1/2. Thus, the probabilities of occupation for atomand photons at the sth cavity are given by

Patom/photon,s =

∣∣∣∣∣N∑m=1

[f±s,mχ

−(αm)± f∓s,mχ+ (αm)]∣∣∣∣∣

2

.

(11)It is assumed that the system is on resonant condition,i.e., ∆ = 0. If the coupling in each cavity dominatesthe evolution, i.e., in the so called strong coupling limitλ >> ξ, the Hamiltonian becomes Hres = ξA ⊗ I2which has ignored the fast-rotating terms in the interac-tion picture. Such Hamiltonian mimics two Heisenbergspin chains [42] in atomic and photonic channels respec-tively. Then the information transferred from sender to

4

FIG. 3: (Color online) The binary transmission probabilitiesverse the number of cavities with different β. From top tobottom, β = 0, β = π/6, and β = π/4, with the systemparameters are ∆ = 0, λ = 10, and ξ = 1.

receiver can be encoded in either channel with speed ξ,and the selectivity of channels may be useful in binarycommunication. Compared with the Ref. [28], due tothe bounding effect of photon discussed here, the timefor the quantum state transfer is reduced by half giventhe both system have the same hopping strength (seeFig. 1 (a) and (b)). In other limit, that the hoppingdominates the evolution, i.e., ξ >> λ, the Hamiltonianreduces to Hres = 2ξA ⊗ (I2 − Z) /2. Governed by suchHamiltonian, only the photon is delocalized on the wholecoupled cavity array with double speed 2ξ (Fig. 1 (d)),whereas the atomic population is trapped in the first cav-ity without dispersion (Fig. 1 (c) and the inset in it).

In the following of this section, we focus our intereston the binary transmission in the strong coupling limit.We find that the temporal evolution of the state vec-tor sensitively depends on the initial state of the sys-tem. When β = π/2, i.e., the initial localized excitationis purely atomic, Figs. 2 (a) and (b) show that bothprobabilities oscillate in each envelopment and have thesame transmission behavior. When β = π/4, the ini-tial state is the maximum entangled state localized inthe first cavity. Meanwhile αm ' −π/4 in the strongcoupling limit on resonance. Then the transfer prob-abilities can be expressed simply as Patom/photon,N '

2∣∣∣∑m e

−iEm+ t sin(mπN+1

)sin(mNπN+1

)∣∣∣2 / (N + 1)2. Figs.

2 (c) and (d) show that the evolution of system has asimilar behavior as that of β = π/2. However, due to thecoherence of the initial local state, the oscillation in eachenvelopment disappears. Furthermore, the maximum oftransition probability is 0.135 and 0.139 for atom andphoton at time 51.84 and 51.73 respectively, which areabout half of those for β = π/2.

The maximum of transmission probabilities for vari-ous cavity lengths with different β is numerically evalu-ated. Fig. 3 shows that the entangled state located in thefirst cavity does not provide any benefit to the transmis-

FIG. 4: (Color online) On resonance, the optimal time to ofbinary transmission as a function of (a) ξ, from top to bottom,N = 80 and N = 40 with λ = 80 and β = 0; (b) N , from topto bottom, ξ = 1 and λ = 10, ξ = 4 and λ = 40 with β = 0.

sion probability, and the transition probability of atomicchannel is slightly less than that of photonic channel forvarious values of β. Also, the maximum transmissionprobabilities decrease with the number of cavities N .

The binary-channel optimal time to can be exploredwhen the transmission probability achieves its maximum.It is found that the time to is not sensitive to the cavity-atom coupling λ in the strongly coupling limit, while it ishighly dependent on the number of the cavities N and in-tercavity hopping amplitude ξ. As shown in Fig. 4 (a), toof atomic and photonic modes is plotted with respect to1/ξ for N = 40 and N = 80 with λ = 80 when two pho-tons are initially located in the first cavity. The atomicoptimal time is nearly identical to that of the photonicmode. A linear dependence of both to on 1/ξ is depicted,and the ratio increases with respect to the length of thecavity array N , as is also confirmed in Fig. 4 (b). Thefarther the distance is, the more time the propagationneeds. The time to scales linearly with N for atomicand photonic modes. Therefore, the optimal time satis-fies to ∼ N/ξ. Thus, the optimal time can be tuned bycontrolling intercavity hopping strength and system size.

Finally, we give a comparison between this schemeand that based on the Jaynes-Cummings-Hubbard model[28]. We note that in contrast to single-photon JC cou-pling g used in that model, the two-photon couplingstrength λ ' g2/δ ∼ 0.1g (assuming g1 = g2 = g). Topreserve the strong coupling condition, λ = 10ξ for in-stance, the speed of the quantum transfer will reduceto 2ξ ' 0.02g considering the photon’s bounding effect,which is about 1/5 of that of the Jaynes-Cummings-Hubbard model. With the same parameters used in Figs.2 (c) and (d), the maximum excitation probabilities ofatom and field are 0.139 and 0.141 at time 10.41 and10.40 [28]. It is clear that the weakening of the trans-mission fidelity in our scheme is less than 0.01, which isnearly neglectable. However, due to the long lifetime ofmetastable states, the distance of the propagation canobviously increase during the lifetime of atom. For in-stance, the spontaneous emission lifetimes of the D5/2

metastable state and the P1/2 level of 40Ca+ ion trappedin high finesse optical cavity are about 1s [43] and 7ns

5

[44] respectively, which leads to an increase of propaga-tion distance with a seven order of magnitude when themetastable state is exploited in the scheme.

III. STAGGERED HOPPING

Compared with the uniform hopping, a parabolic hop-ping was firstly advanced in theory to achieve the per-fect quantum transfer, and the dynamic behavior of suchkind of hopping is elaborately investigated in 1D Jaynes-Cummings-Hubbard model [28]. Recently, there is an-other kind of hopping [29] called staggered hopping drawsmuch interest. Some models with the staggered next-nearest coupling have been widely studied in condensedmatter physics. For instance, a Peierls distorted chain[45] can be used as a data bus [46] to transfer the infor-mation. In this section, we consider the staggered pat-tern of hopping strengths, i.e., the hopping strength ofthe next-nearest cavities are different between odd-to-even and even-to-odd, which is controlled by the distor-tion of the hopping strength κ. To solve this model, weexploited the same process dealt with Eq. (5), whichfirst diagonalize the auxiliary Hilbert space spanned bythe site of the cavity {|M〉}. Here the matrix ele-ment of the adjacency matrix A in Hres is rewritten asAji = Aij = [1− κ(−1)i]δi+1,j , (i 6 j). Here, we assumethat the number of the cavities is odd, and hence thenumbers of strong bonds and weak bonds are equal. Theeigenvectors of this adjacency matrix are given as

|o〉 =2

κ− 1

√κ

τN+1 − 1

N+12∑

M=1

τM−1 |2M − 1〉 , (12)

|m〉± =

√2

N + 1

[ N−12∑

M=1

sin

(2mMπ

N + 1

)|2M〉

±

N+12∑

M=1

sin

(2mMπ

N + 1+ θm

)|2M − 1〉

], (13)

with corresponding eigenvalues

εo = 0, (14)

εm+= −εm− = 2

√cos2

mπ

N + 1+ κ2 sin2 mπ

N + 1,(15)

where the bond alternation parameter τ = κ+1κ−1 and the

angle θm satisfies

eiθm =1− κεm+

(e−i

2mπN+1 − τ

). m = 1, 2, ...,

N − 1

2(16)

Now we investigate the dynamical behavior of the sys-tem under this staggered coupling. In the υth block, theHamiltonian is given as

FIG. 5: (Color online) The probability of the certain cavity asa function of time with the staggered coupling. The upper twoplots show the population of atom (a) and photon (b) in thestrong coupling regime (λ = 200ξ), and the lower two plotsare the population of atom (c) and photon (d) in the stronghopping regime (λ = ξ/200). The parameters are β = π/4,∆ = 0, N = 101, κ = −0.2.

Hres(υ) =

(∆2

√2λ√

2λ −∆2 + 2ξευ

), υ = o,m± (17)

The eigenvectors of the υth block of the Hamiltonian are

|υ,±〉 = |υ〉⊗|±υ〉 = |υ〉⊗(∆− 2Eυ∓

)|e, 0〉+ 2

√2λ |g, 2〉√(

∆− 2Eυ∓)2

+ 8λ2

,

(18)with the corresponding eigenenergy

Eυ± = ξευ ±

√(∆

2− ξευ

)2

+ 2λ2. (19)

For the initial entangled state Eq. (8) in the first cav-ity, the excitation populations of the atom and field inthe sth cavity after time t are

Patom,s = |∑υ,m

cυc′υ[χ− (αυ)χ+ (β − αυ) e−iE

υ+t

+χ+(αυ)χ− (β − αυ) e−iEυ−t]|2. (20)

Pphoton,s = |∑υ,m

cυc′υ[−χ+(αυ)χ+ (β − αυ) e−iE

υ+t

+χ−(αυ)χ− (β − αυ) e−iEυ−t]|2, (21)

where tanαυ = −{[(∆− 2Eυ+

)2+ 8λ2]/[

(∆− 2Eυ−

)2+

8λ2]}1/2 and the amplitudes c′υ are dependent on the site

6

FIG. 6: (Color online) The probability of the certain cavity asa function of time with the staggered coupling. The upper twoplots show the population of atom (a) and photon (b) in thestrong coupling regime (λ = 200ξ), and the lower two plotsare the population of atom (c) and photon (d) in the stronghopping regime (λ = ξ/200), with the system parameters areβ = π/4, ∆ = 0, N = 101, and κ = −0.8.

of the cavity, when s is odd

c′o =2

κ− 1

√κ

τN+1 − 1τs−12 , (22)

c′m+(m) = −c′m−

(m) =

√2

N + 1sin

[(s+ 1)mπ

N + 1+ θm

],

(23)

and when s is even

c′o = 0, (24)

c′m+(m) = c′m−

(m) =

√2

N + 1sin

(smπ

N + 1

). (25)

The amplitudes of the initial state cυ equal to c′υ withs = 1.

When the distortion κ → 0, it returns to the uniformcoupling discussed above. Here, the behavior of the sys-tem is studied in both strong coupling and hopping re-gion for different values of κ. Figs. 5 (a) and (b) showthe evolution of the excitation along the array in thestrong coupling limit when κ = −0.2. Compare with theuniform case, the maximum of excitation probability re-duces to 0.08, and the speed of the propagation slowsdown lightly. When κ = −0.8, Figs. 6 (a) and (b) showthat it further reduces the speed and considerably lessensthe excitation probability, and the population is almostlocalized in the first cavity. In the strong hopping limit,Fig. 5 (c) shows the atomic population distributes on thefirst several cavities, especially the first several odd cav-ities, while the amplitude of the even sites vanishes. Forthe photonic mode, the speed is double referring to thestrong coupling and the maximum of excitation probabil-ity also reaches 0.08 (see Fig. 5 (d)). When the hopping

dominates, as the distortion increases to −0.8, only thephoton transfers along the array with the maximum ex-citation probability 0.0026 (see Figs. 6 (c) and (d)).

IV. MULTIQUBIT ENCODING

It is well known that the highest fidelity for classi-cal transmission of a quantum state is 2/3 [47]. How-ever, Fig. 3 implies that even in the case of β = 0, thetransmission fidelity exceeds 2/3 in a very short distance.Therefore, the further improvement of the transfer prob-ability of the state is necessary. In spin system, there areseveral methods to improve the fidelity of state transfer,such as modulating the couplings [9], or turning on/offthe coupling on demand [11]. Alternatively, one can im-prove the communication by encoding the information inthe multiple spins [12–14] without engineering the cou-pling.

In Ref. [28], a Gaussian wave packet is exploited as theinitial state. However, encoding all the qubits throughthe coupled cavity array is difficult to achieve in real-ity. Here, an alternative k-qubit encoding is used, whichonly needs limited number of qubits to be encoded anddecoded by the sender and receiver. In Sec. III, we haveshown that the excitation populations are almost trappedin the first several cavities either in strong coupling orstrong hopping. So we will focus the quantum transferin the uniform coupling discussed in Sec. II rather thanthe staggered hopping, especially in the strong couplingregime. If a k-qubit encoding is used as the initial state

|ϕ(t = 0)〉k =∑k−1ν=0 (−1)

ν |M = 2ν + 1〉⊗ |e, 0〉 /√k, the

maximum value can be achieved for k = 2, which meansthat one cannot get a higher transmitting probabilityeven using more qubits to encode. To obtain a highertransmitting probability, the decoding process at the an-other site of the cavities must be considered. The ex-citation state of the first cavity would ideally propa-gate to the [N − 2 (r − 1)]th site of the cavities, whilethe excitation state of the last cavity of the encodingwould ideally propagate to the Nth site of the cavities,i.e., |ϕidea〉atom =

∑r−1q=0 (−1)

q |M = Mq〉⊗|e, 0〉 /√r and

|ϕidea〉photon =∑r−1q=0 (−1)

q |M = Mq〉⊗|g, 2〉 /√r, where

Mq = N − 2 (r − 1) + 2q. Thus the probabilities of exci-tation can be calculated as

Patom/photon =

∣∣∣∣∣ 1√r

N∑m=1

r−1∑q=0

µ±m,q

∣∣∣∣∣2

,

where µ±m,q = (−1)q[f±Mq,mχ− (αm) ± f∓Mq,m

χ+ (αm)],

with the probability amplitudes

f±M,m (t) =2

(N + 1)√k

k−1∑ν=0

(−1)νe−iE

m± tχ∓ (αm)

× sin

(mMπ

N + 1

)sin

[(2ν + 1)mπ

N + 1

], (26)

7

FIG. 7: (Color online) The binary transmission probabilitiesverse the number of cavities with different k. From top tobottom, k = 8, k = 4, and k = 2, with the system parametersare ∆ = 0, λ = 10, and ξ = 1.

When the number of encoding qubits equals to that ofdecoding qubits, the maximum values of the excitationprobabilities can be obtained. In Fig. 7, the maximumtransfer probability is plotted as a function of N for theclass of encoding states. The curves show that the propa-gation fidelity increases rapidly as the number of encod-ing qubits k increases. Clearly, the transmission prob-ability decreases with respect to the cavity number N ,and the decay becomes slower by encoding and decod-ing more qubits. The difference of excitation probabili-ties between atomic and photonic mode decreases as thenumber of encoding qubits increases. As is displayed inFig. 7, the maximum probabilities of the binary trans-mission are higher than 0.86 for k = 8, even up to the200 cavities.

V. CONCLUSION

To conclude, the dynamics of a cavity array is studiedwhen each cavity contains a three-level atom. Adiabat-ically eliminating the intermediate state, the individualcavity can be described by the two-photon JC model witha metastable state and a ground state. By exploiting

the metastable state, the transfer distance can be muchlonger than that of Jaynes-Cummings-Hubbard model atthe expense of longer transmission times. When the ad-jacent cavities are coupled by two-photon hopping, thewhole system Hamiltonian can be exactly diagonalizedin the subspace of single-atom excitation. For the uni-form and staggered intercavity hopping, we analyze thedynamics of the system in distinct regimes of parame-ters. With the staggered hopping, the excitation popu-lation trends to be localized in the first several cavitiesand slows down the transfer speed. This phenomenonbecomes more distinct as the distortion of the hoppingstrength increases. Compared with the staggered hop-ping, the uniform hopping is more suitable for the quan-tum state transfer. In the strong hopping limit, only thephoton is delocalized on the whole array of the coupledcavity with double speed. While in the strong couplinglimit, the initial state in the first cavity plays an im-portant role in the transmission fidelity for the binarytransmission. Due to the strong coupling between atomand quantum field in individual cavity, the difference ofmaximum probabilities between the binary transmissionis not very large. Finally, it is shown that a class ofinitial encodings and final decoding process used in spinsystem can also greatly improve the performance of bi-nary transmission. The analysis of the dynamics in thehigh-Q optical cavities can also be exploited in analo-gous systems, such as circuit quantum electrodynamicsand coupled photonic crystal cavities. The results of thiswork provide a step for studying the quantum state trans-fer in coupled cavities.

VI. ACKNOWLEDGMENTS

This work is supported by the Specialized ResearchFund for the Doctoral Program of Higher Education(Grant No. 20103201120002), Special Funds of the Na-tional Natural Science Foundation of China (Grant No.11047168), the National Natural Science Foundation ofChina (Grant Nos. 11074184 and 11004144), the Natu-ral Science Foundation of Jiangsu Province (Grant No.10KJB140010), and the PAPD.

[1] A. Zeilinger, A. Ekert, and D. Bouwmeester (editors),The Physics of Quantum Information: Quantum Cryp-tography, Quantum Teleportation, Quantum Computa-tion (Springer, Berlin, 2000).

[2] S. Lloyd, Science 261, 1569 (1993).[3] C. H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A.

Peres, and W. K. Wootters, Phys. Rev. Lett. 70, 1895(1993).

[4] J. I. Cirac, P. Zoller, H. J. Kimble, and H. Mabuchi,Phys. Rev. Lett. 78, 3221 (1997).

[5] A. D. Boozer, A. Boca, R. Miller, T. E. Northup, and H.

J. Kimble, Phys. Rev. Lett. 98, 193601 (2007).[6] A. Muller, H. Zbinden, and N. Gisin, Europhys. Lett. 33,

335 (1996).[7] R. Ursin, F. Tiefenbacher, T. Schmitt-Manderbach, H.

Weier, et al., Nat. Phys. 3, 481 (2007).[8] S. Bose, Phys. Rev. Lett. 91, 207901 (2003).[9] M. Christandl, N. Datta, A. Ekert, and A. J. Landahl,

Phys. Rev. Lett. 92, 187902 (2004).[10] M. Christandl, N. Datta, T. C. Dorlas, A. Ekert, A. Kay,

and A. J. Landahl, Phys. Rev. A 71, 032312 (2005).[11] S. G. Schirmer and P. J. Pemberton-Ross, Phys. Rev. A

8

80, 030301(R) (2009).[12] T. J. Osborne and N. Linden, Phys. Rev. A 69, 052315

(2004).[13] Z. M. Wang, C. Allen Bishop, M. S. Byrd, B. Shao, and

J. Zou, Phys. Rev. A 80, 022330 (2009).[14] C. A. Bishop, Y. C. Ou, Z. M. Wang, and M. S. Byrd,

Phys. Rev. A 81, 042313 (2010).[15] N. Y. Yao, L. Jiang, A. V. Gorshkov, Z. -X. Gong, A.

Zhai, L. -M. Duan, and M. D. Lukin, Phys. Rev. Lett.106, 040505 (2011).

[16] H. J. Kimble, Nature (London) 453, 1023 (2008).[17] K. Hennessy, A. Badolato, M. Winger, D. Gerace, M.

Atatue, S. Gulde, S. Falt, E. L. Hu, and A. Imamoglu,Nature (London) 445, 896 (2007).

[18] R. J. Schoelkopf and S. M. Girvin, Nature (London) 451,664 (2008).

[19] A. Serafini, S. Mancini, and S. Bose, Phys. Rev. Lett. 96,010503 (2006).

[20] M. J. Hartmann, F. G. S. L. Brandao, and M. B. Plenio,Phys. Rev. Lett. 99, 160501 (2007).

[21] J. Cho, D. G. Angelakis, and S. Bose, Phys. Rev. A 78,062338 (2008).

[22] Z. X. Chen, Z. W. Zhou, X. Zhou, X. F. Zhou, and G.C. Guo, Phys. Rev. A 81, 022303 (2010).

[23] P. B. Li, Y. Gu, Q. H. Gong, and G. C. Guo, Phys. Rev.A 79, 042339 (2009).

[24] M. J. Hartmann, F. G. S. L. Brandao, and M. B. Plenio,Nat. Phys. 2, 849 (2006).

[25] A. D. Greentree, C. Tahan, J. H. Cole, and L. C. L.Hollenberg, Nat. Phys. 2, 856 (2006).

[26] M. I. Makin, J. H. Cole, C. Tahan, L. C. L. Hollenberg,and A. D. Greentree, Phys. Rev. A 77, 053819 (2008).

[27] C. D. Ogden, E. K. Irish, and M. S. Kim, Phys. Rev. A78, 063805 (2008).

[28] M. I. Makin, J. H. Cole, C. D. Hill, A. D. Greentree, and

L. C. L. Hollenberg, Phys. Rev. A 80, 043842 (2009).[29] F. Ciccarello, Phys. Rev. A 83, 043802 (2011).[30] K. Zhang and Z. Y. Li, Phys. Rev. A 81, 033843 (2010).[31] W. L. Yang, Z. Q. Yin, Z. Y. Xu, M. Feng, and C. H.

Oh, Phys. Rev. A 84, 043849 (2011).[32] J. T. Shen and S. Fan, Phys. Rev. Lett. 98, 153003

(2007).[33] D. Roy, Phys. Rev. Lett. 106, 053601 (2011).[34] J. Q. Liao and C. K. Law, Phys. Rev. A 82, 053836

(2010).[35] P. H. Mokler and R. W. Dunford, Physica Scripta. 69,

C1 (2004).[36] M. T. C. Wong and C. K. Law, Phys. Rev. A 83, 055802

(2011).[37] M. Alexanian, Phys. Rev. A 81, 015805 (2010).[38] M. Alexanian, Phys. Rev. A 83, 023814 (2011).[39] V. Buzek and B. Hladky, J. Mod. Opt. 40, 1309 (1993).[40] M. Alexanian, S. K. Bose, and L. Chow, J. Mod. Opt.

45, 2519 (1998).[41] M. Brune, J. M. Raimond, P. Goy, L. Davidovich, and S.

Haroche, Phys. Rev. Lett. 59, 1899 (1987).[42] A. Kay and D. G. Angelakis, Europhys. Lett. 84, 20001

(2008).[43] A. Kreuter, C. Becher, G. P. T. Lancaster, A. B. Mundt,

C. Russo, H. Haffner, C. Roos, J. Eschner, F. Schmidt-Kaler, and R. Blatt, Phys. Rev. Lett. 92, 203002 (2004).

[44] M. Sasura and V. Buzek, J. Mod. Optics. 49, 1593(2002).

[45] R. E. Peierls, Quantum Theory of Solids (Oxford Uni-versity Press, London, 1955).

[46] M. X. Huo, Y. Li, Z. Song, and C. P. Sun, Europhys.Lett. 84, 30004 (2008).

[47] M. Horodecki, P. Horodecki, and R. Horodecki, Phys.Rev. A 60, 1888 (1999).