Quantum Discord (QD) is a useful indicator to measure quantum correlations in the composite bipartite system, which has been initially introduced by OLLIVIER H and ZUREK W H^[1]. Now it has been found that QD captures more quantum correlations and is more widespread than the entanglement. Especially in some quantum systems whose entanglement is zero, but QD is present. Due to its critical role in quantum information processing, QD has attracted huge attention. In recent years, QD was investigated quite intensively^[2-9]. For example, AIOBI H et al studied quantum entanglement and QD of a pair of two-level atoms with no direct interaction in the presence of a dissipative environment^[2]. WANG Bo et al investigated the dynamics of quantum discord using an exactly solvable model where each qubit independently interacts with its own zero-temperature reservoir^[3]. YI Xue-xi et al discussed the discord of a bipartite two-level system coupling to an XY spin-chain environment in a transverse field^[4]. Unfortunately, the evaluation of QD requires a potentially complex optimization procedure in general and analytical results can be obtained only in a few restricted case. In order to avoid the optimization procedure of QD, DAKIC B et al put forward a new measure of quantum correlations for a more general bipartite quantum system, namely Geometric Quantum Discord (GQD)^[10]. Its advantage is that it is able to give its analytical expression easily. So far, some new research progresses have been made for the GQD in different quantum systems^[11-16]. For example, in Ref.[11] the GQD for a class of two-qubit non-X states is investigated. CHEN Hong et al discussed the level surfaces of GQD under some typical kinds of decoherence channels for a class non-X-structured state^[12].

Cavity Quantum Electrodynamics (CQED), which concerns the interaction between photons and atoms within highly reflective cavities, provides an ideal physical implementation flat roofs for distributed quantum information processing. In the experiment, two atoms have been successfully prepared in an entangled state of the Einstein-Podolsky-Rosen, and a quantum phase gate operating on quantum bits has been realized by CQED technology, and so on^[17-19]. In order to achieve distributed quantum computation, in 1997 PELLIZZARI T firstly introduced the coupled-cavity model and proposed a scheme for reliable transfer of quantum information between two atoms^[20]. Because the coupled-cavity model can avoid the difficulty of manipulating multiple atoms in a single cavity, it has attracted much attention. In recently years, researchers have studied the possibility of quantum information processing realized via coupled-cavity systems^[21-27]. For example, OGDEN C D et al discussed the dynamics of a system consisting of two cavities^[21]. ZHANG Ye-qi et al proposed a scheme of entanglement transfer through arrays of cavities coupled by optical fibers^[22]. HUANG Xiao-bin et al put forward an efficient scheme to generate three-atom W states in coupled cavities via transitionless quantum driving^[23]. With the development of the CQED technique and the deepening of research, researchers have extended two coupled cavities to multiple coupled cavities, even to two-dimensional coupled cavities and coupled cavities networks^[28-32]. For example, HUANG Xiao-bin et al analyzed the dynamics of a two-dimensional coupled cavity system under one-excitation condition^[28]. The coherent transport behavior of a single-photon in T-type coupled cavity arrays has been investigated^[30]. Motivated by previous researches, we have extended the T-type coupled-cavity model to a new coupled-cavity model with a center. The dynamics of GQD under one-excitation condition is discussed. Because the coupled-cavity system with a center has a higher symmetry than T-type coupled-cavity systems, it may have important applications in quantum information processing and quantum computation. Based on this system, we have proposed a one-step method to realize an n-qubit controlled phase gate with one cat-state qubit simultaneously controlling n-1 target cat-state qubits, and one of its advantages is that the gate operation time is independent of the number of the cat-state qubits^[33]. We hope that this work has some reference values for quantum information processing.

The coupled-cavity model with a center is shown in Fig. 1. It consists of N+1 single-mode cavities, each containing a two level atom. Cavity 1 is placed at the center of the system and coupled with other N cavities via N fibers. We consider the situation that each atom resonantly interacts with the cavity field via a one-photon hopping. Under the short fiber limit and applying the rotating wave approximation, the Hamiltonian of the whole system in the interaction picture can be written as (setting $ \hbar $=1)^[20]

1 $ {H_{\rm{I}}} = \sum\limits_{j = 1}^{N + 1} {{g_j}{a_j}s_j^ + } + \sum\limits_{i = 1}^N {{J_i}{b_i}\left( {a_{i + 1}^ + + a_1^ + } \right)} + {\rm{H}}.{\rm{C}} $

where a_j⁺(a_j) (j=1, 2, …, N, N+1) stand for creation (annihilation) operator for the mode of cavity j, b_i is the annihilation operator for the ith fiber mode, s_j⁺=|e〉_jj〈g|(j=1, 2, …, N, N+1) is the raising operator of the jth atom between its ground state |g〉_j and excited state |e〉_j, g_j is the atom-cavity coupling constant, J_j(j=1, 2, …, N) denotes the cavity-fiber coupling constant, H.C stands for the conjugate terms. For the sake of simplicity, we take g₁=g₂=…=g_N+1=g and J₁=J₂=…=J_N=J.

Initially, we consider that all cavities and fibers are in the vacuum state, atom 1 is prepared in anexcited state, and the other atoms are in the ground states. Thus, the combined initial state is |φ_a1〉, and the whole system evolves in the one-excitation subspace spanned by the basis state vectors

2 $ \left\{ \begin{array}{l} \left| {{\varphi _{{\rm{a}}i}}} \right\rangle = {\left| {{e_i}} \right\rangle _{\rm{a}}}|0{\rangle _{\rm{c}}}|0{\rangle _{\rm{f}}}\;\;\;\;i = 1,2, \cdots ,N,N + 1\\ \left| {{\varphi _{{\rm{c}}i}}} \right\rangle = |g{\rangle _{\rm{a}}}{\left| {{1_i}} \right\rangle _{\rm{c}}}|0{\rangle _{\rm{f}}}\quad i = 1,2, \cdots ,N,N + 1\\ \left| {{\varphi _{{\rm{f}}l}}} \right\rangle = |g{\rangle _{\rm{a}}}|0{\rangle _{\rm{c}}}{\left| {{1_l}} \right\rangle _\text{f}}\quad l = 1,2, \cdots ,N \end{array} \right. $

where the subscripts a, c and f represent the atomic state, the state of cavity and the state of fiber mode, respectively; |j〉(j=0, 1) denotes the Fock state with j photons; |e_i〉_a stand for the ith(i=1, 2, …, N, N+1) atom being in an excited state and other atoms being in ground states, |g〉_a stand for all atom being in ground states; |1_i〉_c means that the ith cavity is in one photon state, other cavities are in vacuum states; |0〉_c denotes all cavities are in vacuum states; |1_l〉_f is similar to |1_i〉_c, |0〉_f is similar to |0〉_c. Then, the time evolution of the whole state vector is given by

3 $ \left| {\varphi (t)} \right\rangle = \sum\limits_{i = 1}^{N + 1} {{A_i}} \left| {{\varphi _{ai}}} \right\rangle + \sum\limits_{i = 1}^{N + 1} {{B_i}} \left| {{\varphi _{ci}}} \right\rangle + \sum\limits_{l = 1}^N {{C_l}} \left| {{\varphi _{fl}}} \right\rangle $

where t denotes time, A_i, B_i and C_l are the expansion coefficients, and $ \sum\limits_{i=1}^{N+1}\left(\left|A_{i}\right|^{2}+\left|B_{i}\right|^{2}\right)+\sum\limits_{l=1}^{N}\left|C_{l}\right|^{2}=1 $. The state vector |φ(t)〉 obeys following the schrödinger equation

4 $ {\rm{i}}\frac{\partial }{{\partial t}}|\varphi (t)\rangle = {H_{\rm{I}}}|\varphi (t)\rangle $

using the initial condition, we obtain by solving the Eq. (4)

5 $ \left\{ \begin{array}{l} {A_1} = \frac{1}{{N + 1}}\cos (gt) + \frac{{N{J^2}}}{{{\alpha ^2}}} + \frac{{N{g^2}}}{{(N + 1){\alpha ^2}}}\cos (\alpha t)\\ {A_2} = {A_3} = {A_4} = {A_5} = {A_6} = \cdots = {A_{N + 1}} = - \frac{1}{{N + 1}}\cos (gt) + \frac{{{J^2}}}{{{\alpha ^2}}} + \frac{{{g^2}}}{{(N + 1){\alpha ^2}}}\cos (\alpha t)\\ {B_1} = - i\frac{1}{{N + 1}}\sin (gt) - i\frac{{Ng}}{{(N + 1)\alpha }}\sin (\alpha t)\\ {B_2} = {B_3} = {B_4} = {B_5} = {B_6} = \cdots = {B_{N + 1}} = i\frac{1}{{N + 1}}\sin (gt) - i\frac{g}{{(N + 1)\alpha }}\sin (\alpha t)\\ {C_1} = {C_2} = \cdots = {C_N} = - \frac{{Jg}}{{{\alpha ^2}}} + \frac{{Jg}}{{{\alpha ^2}}}\cos (\alpha t) \end{array} \right. $

here we use the notation $ \alpha=\sqrt{g^{2}+(N+1) J^{2}} $.

In order to quantify the quantum correlation between the two subsystems, in Ref.[10] DAKIC B et al introduced GQD. The analytical expression of GQD is

6 $ D\left( \rho \right) = \frac{1}{4}\left( {{{\left\| \mathit{\boldsymbol{a}} \right\|}^2} + {{\left\| \mathit{\boldsymbol{T}} \right\|}^2} - {k_{\max }}} \right) $

where

7 $ \left\{ \begin{array}{l} {a_i} = \text{Tr}\mathit{\boldsymbol{\rho }}\left( {{\sigma _i} \otimes \mathit{\boldsymbol{I}}} \right)\\ {\left\| \mathit{\boldsymbol{a}} \right\|^2} = \sum\limits_{i = 1}^3 {\mathit{\boldsymbol{a}}_i^2} \\ {\mathit{\boldsymbol{T}}_{ij}} = \text{Tr}\mathit{\boldsymbol{\rho }}\left( {{\mathit{\boldsymbol{\sigma }}_i} \otimes {\mathit{\boldsymbol{\sigma }}_j}} \right)\\ {\left\| \mathit{\boldsymbol{T}} \right\|^2} = \text{Tr}\left( {{\mathit{\boldsymbol{T}}^{\rm{T}}}\mathit{\boldsymbol{T}}} \right) \end{array} \right. $

here Tr denotes tracing, ρ is the density matrix, I is identity matrix, σ_i(i=x, y, z)is the usual Pauli matrix, a=(a₁, a₂, a₃)^T stands for a column vector, T={T_ij} is the matrix with elements T_ij, ‖T‖²=Tr(T^TT), k_max is the largest eigenvalue of the matrix K=aa^T+TT^T, superscript T denotes a transpose of vector or matrix.

In the present paper, we use GQD as a criterion of quantum correlation. Firstly, we discuss the GQD between atom1 and atom2. Using Eq. (3), in the atom-atom bases {|ee〉, |eg〉, |ge〉 and |gg〉}, we obtain the density matrix ρ₁₂ of atom1 and atom 2 subsystem

8 $ {\mathit{\boldsymbol{\rho }}_{12}} = \left[ {\begin{array}{*{20}{c}} 0&0&0&0\\ 0&{{{\left| {{A_1}} \right|}^2}}&{{A_1}A_2^ * }&0\\ 0&{{A_1}A_2^ * }&{{{\left| {{A_2}} \right|}^2}}&0\\ 0&0&0&{1 - {{\left| {{A_1}} \right|}^2} - {{\left| {{A_2}} \right|}^2}} \end{array}} \right] $

Combining Eq.(7)and Eq.(8), we get

9 $ \left\{ {\begin{array}{*{20}{l}} {{a_1} = {a_2} = 0}\\ {{a_3} = 2{{\left| {{A_1}} \right|}^2} - 1}\\ {{T_{11}} = 2{A_1}{A_2} = {T_{22}}}\\ {{T_{33}} = 1 - 2\left( {{{\left| {{A_1}} \right|}^2} + {{\left| {{A_2}} \right|}^2}} \right)} \end{array}} \right. $

other element T_ij equals zero. From Eq.(6) and Eq.(9), the GQD between atom1 and atom2 can be easily computed as

10 $ {D_{{\rm{a}}12}} = \frac{1}{4}\left[ {a_3^2 + 2T_{11}^2 + T_{33}^2 - \max \left( {T_{11}^2,T_{33}^2 + a_3^2} \right)} \right] $

In order to discuss the influence of the number of coupled cavities on GQD between two atoms, we set g=J. When N equals 4, 8, 12 and 16, respectively, D_a12 as a function of the scale time Jt is plotted in Fig. 2. It is shown that D_a12 exhibits irregular oscillation, and the peaks of curves decrease as N increases, and so do the averages of D_a12. This shows that the GQD between atom1 and atom2 is weakened with the increasing of N. Similarly, in the case of the same parameter as before, the GQD between atom2 and atom3 is exhibited in Fig. 3. We can clearly see from Fig. 3 that GQD between atom2 and atom3 is also weakened with the increasing of N. This can be qualitatively explained as follows: according to the symmetry of the system, the probabilities of atom2, atom 3, …, atom N+1 sharing the excitation number of system are the same, then the probability of atom2 sharing the excitation number decreases as N increases. Therefore, the GQD between two atoms is weakened with the increasing of N. For example, in the[0, 50] region, the calculation results of the probabilities average of atom2 sharing the excitation number A₂² are: when N equals 4, A₂² equals 0.048 95; when N equals 8, A₂² equals 0.016 32; when N equals 12, A₂² equals 0.008 12; and when N equals 16, A₂² equals 0.004 85.

Assuming that N=8, the atom-cavity coupling constant g equals 0.1J, 0.5J, J, 2.0J, 5.0J and 10.0J, respectively, we plot the evolution of D_a12 with the scaled time Jt in Fig. 4. As shown in Fig. 4, as the coefficient g increases, D_a12 exhibits a quasi-periodic oscillation in the beginning; then it displays an irregular oscillation, finally it displays a quasi-periodic oscillation when g is larger than a certain value. From Eq. (9) and Eq. (10), we know that D_a12 depends on expansion coefficients A₁ and A₂. They are all the superposition of cosine functions with the angular frequency α or g. As g$ \ll $J, $ {A_1} \to \frac{1}{{N + 1}}\cos \left( {gt} \right) + \frac{{N{J^2}}}{{{\alpha ^2}}} $, $ {A_2} \to - \frac{1}{{N + 1}}\cos \left( {gt} \right) + \frac{{{J^2}}}{{{\alpha ^2}}} $; As g$ \gg $J, α→g. Therefore, in both cases either A₁ or A₂ is transformed into a cosine function of a single angular frequency. This leads that the evolution of D_a12 shows a quasi-periodic oscillation. On the other hand, the averages of curves decrease with the increasing of coupling constant g. When g equals 0.5J, J, 2.0J and 5.0J, the average D_a12 equals 0.021 28, 0.015 29, 0.009 12 and 0.005 82, respectively. This shows that GQD between two atoms is weakened with the increasing of the atom-cavity coupling constant. Physically, this result can be understood as follow: the probabilities of the cavity modes being excited increase as the coupling coefficient g increases. This leads that the probabilities of atoms sharing the excitation number of system decrease. Thus, the GQD between atoms is weakened.

It is similar to the calculation of GQD between two atoms in Sec.2. The density matrix of cavity1 and cavity2 subsystem can be written as

11 $ {\mathit{\boldsymbol{\rho }}_{{\rm{c}}12}} = \left[ {\begin{array}{*{20}{c}} 0&0&0&0\\ 0&{{{\left| {{B_1}} \right|}^2}}&{{B_1}B_2^ * }&0\\ 0&{{B_2}B_1^ * }&{{{\left| {{B_2}} \right|}^2}}&0\\ 0&0&0&{1 - {{\left| {{B_1}} \right|}^2} - {{\left| {{B_2}} \right|}^2}} \end{array}} \right] $

Combining Eq. (7) and Eq. (11), we derive

12 $ \left\{ \begin{array}{l} {a_{{\rm{c}}1}} = {a_{{\rm{c}}2}} = 0\\ {a_{{\rm{c}}3}} = 2{\left| {{B_1}} \right|^2} - 1\\ {T_{{\rm{cl}}1}} = 2{B_1}{B_2} = {T_{{\rm{c}}22}}\\ {T_{{\rm{c}}33}} = 1 - 2\left( {{{\left| {{B_1}} \right|}^2} + {{\left| {{B_2}} \right|}^2}} \right) \end{array} \right. $

other element T_cij equals zero. Similar to the calculation of Eq. (10), the GQD between cavity1 and cavity2 can be derived as

13 $ {D_{{\rm{cl}}2}} = \frac{1}{4}\left[ {a_{{\rm{c}}3}^2 + 2T_{{\rm{c}}11}^2 + T_{{\rm{c}}33}^2 - \max \left( {T_{{\rm{c11}}}^2,T_{{\rm{c}}33}^2 + a_{{\rm{c}}3}^2} \right)} \right] $

Again, set g=J, the evolutions of D_c12 versus the scaled time Jt for different N are plotted in Fig. 5. From Fig. 5, we can see that the evolution of D_c12 is similar to that of D_a12, the peaks and the averages of curves decrease as N increases. This means that the GQD between two cavities is weakened with the increasing of N. The reason is as follows: the probabilities of cavity2, cavity 3, …, cavity N+1 sharing the excitation number of system are the same, then the probability of cavity2 sharing the excitation number decreases as N increases. This causes the GQD between two cavities to decrease.

When N equals 8 and the atom-cavity coupling constant g equals some given values, the evolutions of GQD (D_c12) between cavity1 and cavity2 versus the scaled time Jt are plotted in Fig. 6. As seen from Fig. 6, as g increases, the curves show a transition from quasi-periodic oscillation to irregular oscillation and then to quasi-periodic oscillation. The reason is similar to that in section 2.2, i.e., when g$ \ll $J or g$ \gg $J, B₁ and B₂ are all transformed into cosine functions of a single angular frequency. But the peaks of the curves increase as g increases, and so do the averages of D_c12. This shows that GQD between two cavities is strengthened with the increasing of g.

In conclusion, we introduce the coupled-cavity model with the center. It consists of N+1 single-mode cavities, and each cavity contains a two level atom. Cavity1 is at the center of the system, and is coupled to the remaining N cavities through N fibers. We consider the situation that atom resonantly interacts with cavity field via a one-photon hopping. Under one-excitation condition, the evolution of state vector of the system is derived. Using the numerical calculation, we plot the evolution curves of GQD between two atoms and between two cavities. The calculation results show: firstly, the peaks of curves GQD between two atoms and that between two cavities all decrease with the increasing of the number of coupled cavities, and so do the averages of curves; secondly, the averages of curves of GQD between two atoms decrease with the increasing of atom-cavity coupling constant, but the averages and the peaks of curves of GQD between two cavities increase with the increasing of atom-cavity coupling constant. These results show that GQD between two atoms and that between two cavities are weakened with the number of coupled cavities. On the other hand, as the atom-cavity coupling constant increases, GQD between two atoms is weakened, but GQD between two cavities is strengthened.