Abstract
We construct shortcuts to adiabatic passage to achieve controllable and fast quantuminformation transfer (QIT) between arbitrary two distant nodes in a twodimensional (2D) quantum network. Through suitable designing of timedependent Rabi frequencies, we show that perfect QIT between arbitrary two distant nodes can be rapidly achieved. Numerical simulations demonstrate that the proposal is robust to the decoherence caused by atomic spontaneous emission and cavity photon leakage. Additionally, the proposed scheme is also insensitive to the variations of the experimental parameters. Thus, the proposed scheme provides a new perspective on robust quantum information processing in 2D quantum networks.
Introduction
In quantum information and quantum computation, one of the essential ingredients is the realization of controllable and fast quantuminformation transfer (QIT) between arbitrary remote nodes in a quantum network. In recent years, the QIT has been accomplished by several approaches, i.e., the resonant π pulses, composite pulses, stimulated Raman adiabatic passage (STIRAP), and their variants^{1,2,3}. Although the resonant π pulse technique can fast transfer quantum information^{1}, its highly sensitivity to the deviations of pulse areas restricts its extensive application in quantum information processing. The adiabatic passage techniques^{2,3} are robust versus variations of the experimental parameters while they usually need a long operation time. Thus the decoherence, which is one of the main obstacles in quantum information and quantum computation, would strongly affect the dynamics of the system, furthermore, may lead to the schemes become useless. The “shortcuts to adiabaticity technique”, which combines the advantages of resonant π pulses and adiabatic techniques, has been considered as a promising venue to achieve fast and highfidelity QIT, and has attracted much attention in recent years^{4,5,6,7,8,9,10,11,12,13,14,15}. In view of shortcuts to adiabaticity, Chen and Muga^{6,7} have successfully performed fast population transfer in threelevel atom systems via applying the opposite variation tendency in the timedependent laser pulse. After that, the shortcuts to adiabaticity technique has been extended from oneatom system to twoor multiatom system^{12,13,14,15}.
The scalability is still another obstacles in accomplishing the quantum information and quantum computation under current cavity quantum electrodynamics technology. The emergence of coupled cavity system^{16} which can overcome the scalability and meet the requirement of kinds of quantum tasks, i.e., simulation of quantum manybody phenomena^{17,18,19,20,21,22,23}, performing remote quantum information transfer^{24,25,26}, entanglement generation^{27,28,29,30,31,32,33,34,35} and quantum gate operations between two distant nodes^{36,37}. All such works typically focus on the cases of either twosite or onedimensional (1D) coupled cavity arrays. Extending such researches to more complex coupled cavity arrays (i.e., twodimensional (2D) or threedimensional (3D)) is more significance for quantum computation. There have been several studies considering the 2D coupled cavity arrays, which have respectively considered the realization of the fractional quantum Hall system^{38} and 2D oneway quantum computation^{39}. Recently, we have proposed protocols to realize the coherent coupling of multiple atoms^{40} and to realize twoqubits unconventional geometric phase gates in a 2D coupled cavity array^{41}.
The quantum Zeno effect is an interesting phenomenon in quantum mechanics and has been demonstrated in many experiments^{42,43,44,45}. It has been shown that a system can actually evolve away from its initial state, but still remain in the Zeno subspace defined by the measurements via frequently projecting onto a multidimensional subspace, which is known as quantum Zeno dynamics (QZD)^{46,47,48}. In general, if a system is governed by Hamiltonian H _{ K } = H _{obs} + KH _{meas}, where H _{obs} is the Hamiltonian of the subsystem to be investigated, H _{meas} is an additional interaction Hamiltonian performing the “measurement”, and K is a coupling constant. In a strong coupling limit K → ∞, the whole system will remain in the same Zeno subspace, and is governed by the evolution operator defined as \(U(t)=\exp (\,\,it{\sum }_{n}K{E}_{n}{P}_{n}+{P}_{n}{H}_{{\rm{obs}}}{P}_{n})\), with P _{ n } being the eigenvalue projection of H _{meas} with eigenvalues E _{ n } (H _{meas} = ∑_{ n } E _{ n } P _{ n }).
Motivated by the space division of QZD, in this paper, we construct shortcuts to adiabatic passage to achieve controllable and fast QIT between arbitrary two nodes in a 2D quantum network. Through suitably designing the timedependent Rabi frequencies, we can controllably and fast transfer quantuminformation between arbitrary two distant nodes in onestep. The distinguished advantages of the proposal are: (i) information can be controllably transferred between arbitrary two nodes; (ii) the time to accomplish the task is shorter than that in conventional adiabatic passage technique; (iii) it is robust against the parameters fluctuations and the decoherence caused by atomic spontaneous emission and cavity photon leakage. Thus it provides a new perspective on robust quantum information processing in 2D quantum networks in the future.
The theoretical model and the construction of a shortcut passage
we consider a 2D (N × N) coupled cavity array, as shown in Fig. 1(a). Each cavity (denoted by jk) respectively couples to their neighboring ones through the x and y directions with intercavity photon hopping. Each cavity contains a Λtype atom. The atoms have two ground states (labeled as \({g\rangle }_{jk}\) and \({f\rangle }_{jk}\)) and one excited state (labeled as \({e\rangle }_{jk}\)), as shown in Fig. 1(b). The \({g\rangle }_{jk}\leftrightarrow {e\rangle }_{jk}\) transition of atom couples to the corresponding cavity mode with coupling rate g _{ jk } and detuning Δ_{ jk }. The \({f\rangle }_{jk}\leftrightarrow {e\rangle }_{jk}\) transition of atom is resonantly driven by a classical field with Rabi frequency Ω_{ jk } (j, k ∈ 1, …, N). In the interaction picture, the Hamiltonian for the system is (ħ = 1)
with
and
where a _{ jk } \(({a}_{jk}^{+})\) denotes the annihilation (creation) operator for the jkth cavity, and v is the hopping rate of photons between neighboring cavities. We adopt periodic boundary conditions a _{ j1} = a _{ jN } and a _{1k } = a _{ Nk } by introducing the nonlocal bosonic modes c _{ mn }, and diagonalize the Hamiltonian H _{2} via the Fourier transform: \({a}_{jk}=\frac{1}{N}{\sum }_{m,n}^{N}\exp [\,i(\frac{2\pi jm}{N}+\frac{2\pi kn}{N})]{c}_{mn}\). Thus we can rewrite the Hamiltonian H _{1} and H _{2} as
and
where \({\omega }_{mn}=2v(\cos \,\frac{2\pi n}{N}+\,\cos \,\frac{2\pi m}{N})\) (n = 0,1,2, …, N − 1, m = 0, 1, 2, …, N − 1). We now go into a new frame by defining H _{2} as a free Hamiltonian, and obtain the interaction Hamiltonian for the whole system as
In order to achieve the information transfer between arbitrary two distant nodes. We first assume that the atom to receive information is in the g〉_{ NN } state (in general, we assume that the atom in node NN), the atoms in other nodes are all in the state f _{ jk }〉 (jk ≠ NN). Assume that the information to be transferred is loaded in the node 11 (i.e., the atom in node 11 is initially in the f〉_{11} state), and the cavity modes are all in the vacuum state. Next, we apply two laser fields with Rabi frequency Ω_{ jk } to these two nodes (node 11 and NN). Thus, the Hamiltonian in Eq. (6) reduces to
Generally, the accurate dynamics evolution governed by the above Hamiltonian is complicated, but there still can be simplified in some regimes. If the atomic transition frequencies are set equal to one of the frequencies of nonlocal bosonic modes, i.e, appropriately adjusting the detuning to satisfy Δ_{11} = Δ_{ NN } = ω _{ pq }, the Hamiltonian in Eq. (7) becomes
with \({g}_{p,q}=\frac{{g}_{jk}}{N}{e}^{i(\frac{2\pi jp}{N}+\frac{2\pi kq}{N})}\). The first part in the above Hamiltonian describes the resonant interaction between atoms and the nonlocal bosonic mode as well as the laser fields, while the second term represents the dispersive interaction between the atoms and the nonresonant normal modes. Under the condition \({{\rm{\Delta }}}_{jk}{\omega }_{mn}\gg \frac{{g}_{jk}}{N}{e}^{i(\frac{2\pi jm}{m}+\frac{2\pi kn}{n})}\), the interaction of the atoms with the nonresonant normal modes can be neglected, the Hamiltonian reads
where
and
The above Hamiltonian H _{eff} shows that, the atoms can resonantly interact with the nonlocal bosonic mode c _{ p,q }, it means that the atoms resonantly interact with all the cavities simultaneously. Assume the system is initially in the state Φ_{0}〉 = f _{11}〉g _{ NN }〉0〉 (i.e., atoms in the node 11 and node NN are in the states f〉 and g〉, respectively, and the bosonic mode C _{ pq } is in the vacuum state), the whole system evolves in the subspace spanned by \(\{{{\rm{\Phi }}}_{1}\rangle \) \(={f\rangle }_{11}{g\rangle }_{NN}0\rangle \), \({\Phi }_{2}\rangle ={e\rangle }_{11}{g\rangle }_{NN}0\rangle \), \({\Phi }_{3}\rangle ={g\rangle }_{11}{g\rangle }_{NN}1\rangle \), \({\Phi }_{4}\rangle ={g\rangle }_{11}{e\rangle }_{NN}0\rangle \), \({\Phi }_{5}\rangle ={g\rangle }_{11}{f\rangle }_{NN}0\rangle \}\). In the proposed model, the interaction between atoms and nonlocal bosonic mode plays the role of continuous measurements on the interaction between atoms and the classical fields. Thus, we concentrate on the dynamical evolution of three new computation bases. In light of QZD, the eigenvalues of H _{2eff} are E _{1} = 0, E _{2} = −g, E _{3} = −g (we here assume g _{ pq } = g). Thus, the corresponding eigenvectors of Hamiltonian H _{2eff} are
Next, we rewrite the Hamiltonian in Eq. (9) with the eigenvectors of H _{2eff},
where
and
It is obvious that there are four nonzero energy eigenvalues ±Ω_{11}(t) and ±Ω_{ NN }(t) for the Hamiltonian \({H}_{{\rm{2eff}}}^{^{\prime} }\). Defining \({H}_{{\rm{1eff}}}^{^{\prime} }\) as a free Hamiltonian, and performing the unitary transformation \(U={e}^{i{H}_{{{\rm{1eff}}}^{t}}^{^{\prime} }}\) under condition \({H}_{{\rm{2eff}}}^{^{\prime} }\gg {H}_{{\rm{1eff}}}^{^{\prime} }\), we obtain
Therefore, setting \(\sqrt{2}g\gg {{\rm{\Omega }}}_{11}(t),{{\rm{\Omega }}}_{NN}(t)\), the condition \({H}_{{\rm{2eff}}}^{^{\prime} }\gg {H}_{{\rm{1eff}}}^{^{\prime} }\) and the Zeno condition K → ∞ are satisfied. Under the rotatingwave approximation, we have a new Hamiltonian
Thus the Hilbert subspace splits into three invariant Zeno subspaces H _{p0} = {Ψ_{1}〉,Φ_{1}〉,Φ_{5}〉}, H _{p1} = {Ψ_{2}〉}, H _{p2} = {Ψ_{3}〉}. The system can be divided into three subsystems, S _{1} = {Ψ_{1}〉,Φ_{1}〉,Φ_{5}〉}, S _{2} = {Ψ_{2}〉,Φ_{1}〉,Φ_{5}〉}, S _{3} = {Ψ_{3}〉,Φ_{1}〉,Φ_{5}〉}. Note that the interaction between the states in subsystems S _{2} and S _{3} is far weaker than that in subsystem S _{1}, thus this weak interaction can be neglected. Then the system can be considered as a threelevel atom system with two ground states Φ_{1}〉,Φ_{5}〉 and an excited state Ψ_{1}〉. The Hamiltonian for STIRAP reads
The instantaneous eigenvalues are \({\chi }_{0}=\mathrm{0,}{\chi }_{\pm }=\pm \,\sqrt{{{\rm{\Omega }}}_{11}^{2}+{{\rm{\Omega }}}_{NN}^{2}}/\sqrt{2}\), with the corresponding eigenstates are
Here \(\theta =\arctan ({{\rm{\Omega }}}_{11}(t)/{{\rm{\Omega }}}_{NN}(t))\). Thus, if the adiabatic condition \(\dot{\theta }\ll \chi /\sqrt{2}\) is fulfilled, QIT from initial state Φ_{1}〉 to target state Φ_{5}〉 is achieved adiabatically along the dark state Φ_{0}〉. However, the time to accomplished this task is long.
Next we introduce how to construct shortcuts to fast transfer quantuminformation by using the dynamics of invariant based inverse engineering^{7}. Here, we first introduce an invariant Hermitian operator \({I}_{{s}_{2}}(t)\), which satisfies the Schröodinger equation \(i\partial {I}_{{s}_{2}}(t)/\partial t=[{H}_{{s}_{2}}(t),{I}_{{s}_{2}}(t)]\) ^{49}, for \({H}_{{s}_{2}}(t)\) possesses the SU(2) dynamical symmetry. And \({I}_{{s}_{2}}(t)\) is given by
Here γ and β are the timedependent auxiliary parameters and satisfy the following equations,
where the dot represents a time derivative. By inversely deriving from Eq. (17), the explicit expressions of Ω_{11}(t) and Ω_{ NN }(t) are as follows:
The eigenstates of the invariant I _{ s2}(t) are
corresponding to the eigenvalues λ _{0} = 0 and λ _{±} = ±1, respectively. Based on the LewisRiesenfeld theory^{50}, the solution of the Schröodinger equation with respect to the instantaneous eigenstates of \({I}_{{s}_{2}}(t)\) is a superposition of orthonormal dynamical modes, \({\rm{\Psi }}(t)\rangle ={\sum }_{n}{C}_{n}{e}^{i{\alpha }_{n}}{{\rm{\Phi }}}_{n}(t)\rangle \), where C _{ n } is a timeindependent amplitude and α _{ n } is the LewisRiesenfeld phase and obeys the form,
In the proposal, α _{0} = 0, and
In order to get the target state Φ_{5}〉 along the invariant eigenstate \({{\rm{\Phi }}}_{0}^{\text{'}}(t)\rangle \), we suitably choose the feasible parameters γ(t) and β(t),
where ξ is a small value, which satisfies (sinξ)^{−1} = 4 M(M = 1, 2, 3, …) for a high fidelity of the target state. And we obtain
Once the Rabi frequencies are specially designed, the fast QIT from initial state to the target state in subsystem S _{1} will be implemented.
Results
To confirm the validity of all our above derivation, we first numerically simulate the dynamics governed by the derived effective Hamiltonian in Eq. (14), and compare it to the dynamics governed by the total Hamiltonian in Eq. (1). Note that the numerical computation we performed using the python package Qutip^{51}. The validity of the model is numerically simulated by taking the evolution of the population P = 〈ψψ(t)〉^{2} of the proposed state ψ〉. We consider the case with N = 2, and set the parameters in the following way: v = 2.0 g, gt _{ f } = 50, \(\xi =\arcsin \mathrm{(0.25)}\) (the Zeno condition \(\sqrt{2}g\gg {{\rm{\Omega }}}_{jk}(t)\) can be satisfied very well). For the total Hamiltonian, a new subspace is spanned by \(\{{\psi }_{1}\rangle =f{\rangle }_{11}g{\rangle }_{22}0{\rangle }_{11}0{\rangle }_{12}0{\rangle }_{21}0{\rangle }_{22}\), \({\psi }_{2}\rangle =e{\rangle }_{11}g{\rangle }_{22}0{\rangle }_{11}0{\rangle }_{12}0{\rangle }_{21}0{\rangle }_{22}\), \({\psi }_{3}\rangle =g{\rangle }_{11}e{\rangle }_{22}0{\rangle }_{11}0{\rangle }_{12}0{\rangle }_{21}0{\rangle }_{22}\), \({\psi }_{4}\rangle =g{\rangle }_{11}g{\rangle }_{22}1{\rangle }_{11}0{\rangle }_{12}0{\rangle }_{21}0{\rangle }_{22}\), \({\psi }_{5}\rangle =g{\rangle }_{11}g{\rangle }_{22}0{\rangle }_{11}1{\rangle }_{12}0{\rangle }_{21}0{\rangle }_{22}\), \({\psi }_{6}\rangle =g{\rangle }_{11}g{\rangle }_{22}0{\rangle }_{11}0{\rangle }_{12}1{\rangle }_{21}0{\rangle }_{22}\), \({\psi }_{7}\rangle =g{\rangle }_{11}g{\rangle }_{22}0{\rangle }_{11}0{\rangle }_{12}0{\rangle }_{21}1{\rangle }_{22}\), \({\psi }_{8}\rangle =g{\rangle }_{11}f{\rangle }_{22}0{\rangle }_{11}0{\rangle }_{12}0{\rangle }_{21}0{\rangle }_{22}\}\). Thus, if the system is initially in one of these basics, the system will evolve in this subspace. In Fig. 2, the redsolid (greensolid) and bluedashed (blackdashed) lines describe the time evolution of the population of state ψ _{1}〉 = f _{11} g _{22} 0_{11} 0_{12} 0_{21} 0_{22}〉 (ψ _{8}〉 = f _{22} 0_{11} 0_{12} 0_{21} 0_{22}〉) and state Φ_{1}〉 = f _{11} g _{22} 0_{ pq }〉 (Φ_{5}〉 = g _{11} f _{22} 0_{ pq }〉) governed by the total Hamiltonian and effective Hamiltonian, respectively. It is obvious that the approximations adopted during the deriving of the effective Hamiltonian are valid, since the two curves described by the total Hamiltonian and effective Hamiltonian are nearly coincided, and their deviation is small enough as soon as the parameters are fixed.
Next, we show how the operation time is shorten when considering the shortcuts to adiabatic passage. We first numerically simulate the time dependence of the Rabi frequencies for the atoms in Fig. 3(a) when gt _{ f } = 10, the other parameters are set the same as those in Fig. 2. As seen from Fig. 3(a), the maximum value of Ω_{ jk }/g is 0.83, which satisfies the conditions mentioned above (the Zeno condition \(\sqrt{2}g\gg {{\rm{\Omega }}}_{jk}(t)\) (jk = 11, NN). In Fig. 3(b), we plot the time evolution of the populations of states Φ_{1}〉 = f〉_{11}g〉_{22}0〉_{ pq } (bluedash line), Φ_{5}〉 = g〉_{11}f〉_{22}0〉_{ pq } (reddash line), and \({{\rm{\Psi }}}_{1}\rangle =\mathrm{1/}\sqrt{2}{(e\rangle }_{11}g{\rangle }_{22}\mathrm{0}{\rangle }_{pq}+g{\rangle }_{11}e{\rangle }_{22}\mathrm{0}{\rangle }_{pq})\) (magentadash line) under effective Hamiltonian in Eq. (14). Figure 3(b) shows that a perfect and fast quantuminformation transfer from the initial state Φ_{1}〉 to the target state Φ_{5}〉 can be achieved after reselecting the optimal value of ξ. Notice that the population of excited state Ψ_{1}〉 is less than 0.25 during the interaction. Thus, we can draw a conclusion that the effective model can be considered as a threelevel singleatom model^{7}, as the optimal value of ξ for the whole system faultlessly satisfy the condition. In Fig. 3(c), we plot the time evolution of the population of states ψ _{1}〉 = f〉_{11}g〉_{22}0〉_{11}0〉_{12}0〉_{21}0〉_{22} (bluesolid line), ψ _{8}〉 = g〉_{11}f〉_{22}0〉_{11}0〉_{12}0〉_{21}0〉_{22} (redsolid line), and \(\psi \rangle =\mathrm{1/}\sqrt{2}{(g\rangle }_{11}e{\rangle }_{22}\mathrm{0}{\rangle }_{11}\mathrm{0}{\rangle }_{12}\mathrm{0}{\rangle }_{21}\mathrm{0}{\rangle }_{22}+e{\rangle }_{11}f{\rangle }_{22}\mathrm{0}{\rangle }_{11}\mathrm{0}{\rangle }_{12}\mathrm{0}{\rangle }_{21}\mathrm{0}{\rangle }_{22})\) (magentasolid line) under the total Hamiltonian in Eq. (1). Also, a perfect and fast QIT from the initial state ψ _{1}〉 to the target state ψ _{8}〉 can be achieved at time t _{ f }. Compared to the effective Hamiltonian model, the population of excited state governed by total Hamiltonian is larger than that governed by the effective Hamiltonian. The reason for this can be explained as follow: during the operation, the intermediate states (i.e., g〉_{11}g〉_{22}1〉_{11}0〉_{12}0〉_{21}0〉_{22}, g〉_{11}g〉_{22}0〉_{11}1〉_{12}0〉_{21}0〉_{22}, etc.) can be slightly populated, the whole system cannot be faultlessly considered as a threelevel singleatom model, and the optimal value of ξ for the whole system will not faultlessly satisfy the condition \({(\sin \xi )}^{}\mathrm{1=4}\,M\). In order to get more insight to dynamic of the system governed by the total Hamiltonian, we plot the population of states ψ _{2}〉 to ψ _{7}〉 versus time in Fig. 3(d). From Fig. 3(d), we can see that all the populations of these states are smaller than 0.25, especially, the probabilities to find a photon in nodes 12 and 21 are less than 0.012. We can draw a conclusion that the system can be approximately considered as a threelevel atom system, although the specific procedures has small differences between the two dynamics. Thus, the information can be fast and perfect transferred between arbitrary two distant nodes under current condition.
As shown in Fig. (3), the proposal can be nearly treated as an adiabatic process which is insensitive to the fluctuations of parameters, such as the amplitude of the laser pulses Ω_{ jk }, the coupling constant g and the parameter ξ. Thus, we can choose a sets of parameters to obtain high fidelity and fast QIT. In Fig. 4, we plot the fidelity of the target state ψ _{8}〉 versus the value of ξ and the interaction time gt _{ f } governed by the total Hamiltonian H _{ I } when v = 2.0 g. The fidelity for the target state is defined as \(F=\langle {\psi }_{8}\rho (t){\psi }_{8}\rangle \), where ρ(t) is the density operator of the system at the time t _{ f } by solving the equation \(\dot{\rho }=i[{H}_{I},\rho ]\). As seen from Fig. 4, when gt _{ f } = 10, the optimal value of ξ for the highest fidelity (fidelity \(\simeq 1\)) of the state ψ _{8}〉 is from 0.235 to 0.265. The reason for this can be expressed as: the proposal is a adiabatic passage, thus it is robust versus variations in the experimental parameters. However, when the parameters are no longer approximately satisfied by the condition \({(\sin \xi )}^{1}=4\,M\) (M = 1, 2, 3, …), the fidelity will show an extreme fluctuation. Figure 4 also shows that it is hardly to get high fidelity when gt _{ f } < 10. Thus, in the proposed scheme, the fastest time to get the target state is t _{ f } = 10/g. Therefore, it is much faster than the general adiabatic process.
Discussion
It is necessary to discuss the influence of decoherence caused by atomic spontaneous emission and cavity photon leakage of the system. In the current model, the master equation of the whole system can be expressed by the Lindblad form^{52},
where κ _{ ls } denotes the decay rate of cavity, \({\gamma }_{jk}^{eg}\) and \({\gamma }_{jk}^{ef}\) represent the atomic decay from level e _{ jk }〉 to g _{ jk }〉 and e _{ jk }〉 to f _{ jk }〉, respectively. For simplicity, we assume κ _{ ls } = κ (ls = 11, 12, 21, 22), \({\gamma }_{jk}^{eg}={\gamma }_{1}\) and \({\gamma }_{jk}^{ef}={\gamma }_{2}\) (jk = 11, 22). The fidelity of the target state versus the ratios κ and γ _{1} (κ and γ _{2}) is shown in Fig. 5(a,b) when ξ = 0.25 and gt _{ f } = 10. As seen from Fig. 5, the fidelity decreases slowly with the increasing of cavity decay and atomic spontaneous emission. Figure 5 shows that the fidelity is still about 79.8% (83.3%) when κ = γ _{1} = 0.1 g (κ = γ _{2} = 0.1 g). Therefore, we can draw a conclusion that the proposal is robust against the spontaneous emission and cavity photon leakage.
Finally, let us give a brief analysis of the experimental feasibility for this scheme. The proposal can be realized in solidstate qubit trapped in a 2D array of superconducting cavity system. In this system, the superconducting cavity can be strongly coupled to the solidstate qubits such as Cooper pair boxes (CPB), and the corresponding microwave photons have small loss rates. As reported in ref. 53, the coupling strength in the interaction between CPBs and the circuit cavities is g˜2π × 50 MHZ, the corresponding photon lifetime is T _{ c } ~ 20 × 10^{−6} s, the dephasing time of the spin state is T _{ a }˜1 × 10^{−6} s. Thus, the required time for transferring the quantuminformation, in principle, is T~3.2 × 10^{−8} s, which is much shorter than T _{ c } and T _{ a }. The proposed idea can also be used for largescale arrays cavities in photonic crystals, in which the achievable parameters are predicted to be (g,κ,γ) = 2π × (2.5 × 10^{3}, 0.4, 1.6) MHz^{54}. As shown above, the required time for achieving the task is smaller than photon coherence time and the atom dephasing time. In recent experiments, a set of cavity quantum electrodynamics parameters (g, κ, γ) = 2π × (7.6, 2.8, 3.0) MHz is available in an optical cavity^{55,56,57}. Thus, based on the recent cavity QED technique or the technique to be improved soon, the proposal might be realizable in the future.
In conclusion, we have proposed a promising scheme to construct shortcuts to adiabatic passage to achieve controllable and fast quantuminformation transfer between arbitrary two nodes in 2D quantum networks. The proposal has several advantages. The first one is that information can be controllably transferred between arbitrary two nodes, which is the basic of quantum computation. Secondly, the operation time is shorter than that in conventional adiabatic passage technique. Third, the proposed scheme is robust against the parameter fluctuations and the decoherence caused by atomic spontaneous emission and cavity photon leakage. These are very benefit to the suppression of decoherence effect. The scheme provides a new perspective on robust quantum information processing in 2D quantum networks. In principle, the proposal can be realized in solidstate qubit trapped in a 2D array of superconducting cavity system or in largescale arrays cavities in photonic crystals. Moreover, the proposed scheme can be extended to 3D coupled cavity system.
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Acknowledgements
Z.R.Z acknowledge support from the Natural Science Foundation of Fujian Province under Grant No. 2016J01018 and the Natural Science Foundation of China under Grant No. 11404061.
Author contributions
Z.R.Z. wrote the main manuscript text and prepared all the figures. The author reviewed the manuscript.
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The authors declare no competing financial interests.
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