Controllable and fast quantum-information transfer between distant nodes in two-dimensional networks

We construct shortcuts to adiabatic passage to achieve controllable and fast quantum-information transfer (QIT) between arbitrary two distant nodes in a two-dimensional (2D) quantum network. Through suitable designing of time-dependent 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.

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 i t KE P P H P ( ) exp( ) n n n n n obs , 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 time-dependent Rabi frequencies, we can controllably and fast transfer quantum-information between arbitrary two distant nodes in one-step. 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 Λ-type atom. The atoms have two ground states (labeled as g jk and f jk ) and one excited state (labeled as e jk ), as shown in Fig. 1(b). The ↔ g e jk jk transition of atom couples to the corresponding cavity mode with coupling rate g jk and detuning Δ jk . The ↔ f e jk 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)  The atom level scheme. The transition of the jkth atom |g〉 jk ↔ |e〉 jk is coupled to the cavity mode with detuning Δ jk , the corresponding coupling rate is g jk . The transition |f〉 jk ↔ |e〉 jk of the jkth atom is resonantly driven by a classical laser field, and the corresponding Rabi frequencies are Ω jk . (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  . 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 ω ∆ − 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 Φ . 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 Scientific RepoRts | 6: 8 | DOI:10.1038/s41598-016-0016-1 Next, we rewrite the Hamiltonian in Eq. (9) with the eigenvectors of H 2eff , 1eff and the Zeno condition K → ∞ are satisfied. Under the rotating-wave approximation, we have a new Hamiltonian Thus the Hilbert subspace splits into three invariant Zeno subspaces The system can be divided into three subsystems, 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 three-level 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 χ , with the corresponding eigenstates are . Thus, if the adiabatic condition θ χ / 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 quantum-information by using the dynamics of invariant based inverse engineering 7 NN NN 11 11 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 0 corresponding to the eigenvalues λ 0 = 0 and λ ± = ±1, respectively. Based on the Lewis-Riesenfeld theory 50 , the solution of the Schröodinger equation with respect to the instantaneous eigenstates of I t ( ) where C n is a time-independent amplitude and α n is the Lewis-Riesenfeld 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 Φ ′ t ( ) 0 , we suitably choose the feasible parameters γ(t) and β(t), 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 = |〈ψ|ψ(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, ξ = . arcsin(0 25) (the Zeno condition 22 . Thus, if the system is initially in one of these basics, the system will evolve in this subspace. In Fig. 2, the red-solid (green-solid) and blue-dashed (black-dashed) 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 Ω  g t 2 () jk (jk = 11, NN). In Fig. 3(b), we plot the time evolution of the populations of states |Φ 1 〉 = |f〉 11  Hamiltonian in Eq. (14). Figure 3(b) shows that a perfect and fast quantum-information 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 three-level single-atom model 7 , as the optimal value of ξ for the whole system faultlessly satisfy the condition. In Fig. 3(c) ) can be slightly populated, the whole system cannot be faultlessly considered as a three-level single-atom model, and the optimal value of ξ for the whole system will not faultlessly satisfy the condition ξ = − M (sin ) 1 4 . In order to get more insight to dynamic of the system governed by the total Hamiltonian, we plot the population of ( |Φ 〉 = | 〉 g f 0 pq 5 11 22 ) governed by the total Hamiltonian and effective Hamiltonian, respectively. 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 three-level 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 ψ ρ ψ = F t ( )  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.   Fig. 5(b)).

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, γ jk eg and γ 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), γ γ = jk eg 1 and γ γ = jk ef 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 solid-state qubit trapped in a 2D array of superconducting cavity system. In this system, the superconducting cavity can be strongly coupled to the solid-state 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 quantum-information, 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 large-scale 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 quantum-information 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 solid-state qubit trapped in a 2D array of superconducting cavity system or in large-scale arrays cavities in photonic crystals. Moreover, the proposed scheme can be extended to 3D coupled cavity system.