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AbstractBell states play an important role in quantum information and quantum communication. In thispaper we propose an architecture for realizing the perfect state transfer of the Bellstates with asingle basic operation. This architecture, we employs flux qubits to induce the necessary interactionbetween cavities, and We explicitly show that for a resonant interaction case, Bell states can betransferred from two cavities to another two distant cavities.PACS numbers: I. INTRODUCTIONBell states are two-qubit maximally entangled states, which play an important role inquantum communication and quantum information processing (QIP). Recently cavity quan-tum electrodynamics (QED), linear optics devices , and super conducting qubits , etc. havebeen well studied and quantum communication through a spin chain also has been welldiscussed, The physical system, composed of cavities and qubits, has attracted much atten-tion for QIP. Over the past twenty years, a large number of theoretical and experimentalworks have been done for implementing quantum information transfer. However no onehas discussed such a case in which Bell states are transferred between distant parties basedon circuit QED, although Ref discussed transferring the single excited state between twolocated NVEs.On the other hand, quantum communication over short distances througha spin chain, in which adjacent qubits are equally coupled by equal has been studied indetails[12], Near perfect state transfer has been achieved for uniform couplings provided aspatially varying magnetic field is introduced[13].48093
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The propagation of quantum informationin rings has also been investigated[13]. But all previous works did not discuss how to realizethe remote transmission of Bell states between cavities.In the recent works, the coupling between LC circuits and flux qubits. the strong cou-pling between hybrid-solid quantum systems, as well as the quantum state transfer betweensolid quantum systems have been well studied. We here will propose an architecture fortransferring Bell states among cavities. In this architecture, flux qubits are used to inducethe necessary interaction between cavities, We explicitly show that for resonant interactioncases, then the high-fidelity Bell state transfer from two cavities to another two cavities canbe implemented. Besides we also discuss the Bell state transfer from two cavities to anyother two cavities in a cavity chain. This work is important in the transmission of quantumentanglement over a long distance qubits.The paper is organized as follows. In Sec. II, we consider a physical system compossedof six cavities and two flux qubits and then introduce a Hamiltonian for such a system thetwo cavities on the left to the two cavities on the right.A brief conclusion is given in sec III. II. MODEL AND BELL STATE TRANSFERLet us consider six cavities and two flux qubits , as illustrated in Fig. 1. Each cavity FIG. 1: (color online).The six cavities group(1,2...6) are pided into three groups each groupcontains two cavities. Each cavity is connected to a flux qubit via a capacitor.here has two states a vacuum state |0⟩ and single photon state |1⟩, Each flux qubit has twolevels i.e, the ground level |g⟩ and the excited level |e⟩, We pide the six cavities into threegroups. Group 1 contain cavities 1 and 2, Group 2 contain cavities 3 and 4,
Group 3 containcavities 5 and 6. We denote the coupling constant of group i(1,2,3) with flux qubit j(1,2)as gj;i, In addition, we consider the cavities are identical. The cavity frequency is denotedas we which the frequency of flux qubit j is labelled as !jq . In a resonant interaction case(!c = !jq) and in the interaction picture, the Hamiltonian of the whole system is given byHI =g1;1[ −1 (a†1 + a†2) + h:c] + g1;2[ −1 (a†3 + a†4) + h:c] (1)+ g2;2[ −2 (a†3 + a†4) + h:c] + g2;3[ −2 (a†5 + a†6) + h:c]Where a†iis the photon creation operator of cavity i (i=1,2,3,4,5,6) while j = |j⟩⟨e| isthe lowing operator of flux qubit j(j=1,2).Now we define the co-operator J†i;j = 1 √2(a†i+a†j), which satisfies the bosonic commutationrelations:[Ji;j ; J†i;j ] ≈ 1 (2)Then we can rewrite the Hamiltonian(1) as flowsHI = g′1;1[ −1 J†1;2 + h:c] + g′1;2[ −1 J†3;4 + h:c] + g′2;1[ −2 J†3;4 + h:c] + g′2;2[ −2 J†5;6 + h:c] (3)here we use g′i;j =√2gi;j .Assume that the left-hand two cavities are initially in the Bell state 1 √2(|01⟩ + |10⟩),the other four cavities are initially in the vacuum state, and each flux qubit is initially inthe group state |g⟩. For simplicity, we define the Bell state of cavities group j as |i⟩j = 1 √2(|01⟩ + |10⟩) and the vacuum state of cavities group j as e |0⟩j = |00⟩. Thus ,the initialstate of this system can be written as | (0)⟩ =e |1⟩1|g⟩1e |0⟩2|g⟩2e |0⟩3 The Bell state from theleft-hand two cavities to the right is described by e |1⟩1|g⟩1e |0⟩2|g⟩2e |0⟩3 →e |0⟩1|g⟩1e |0⟩2|g⟩2e |1⟩3subspace formed by the following states:| (0)⟩ =e |1⟩1|g⟩1e |0⟩2|g⟩2e |0⟩3 (4)| (1)⟩ =e |0⟩1|e⟩1e |0⟩2|g⟩2e |0⟩3| (2)⟩ =e |0⟩1|g⟩1e |1⟩2|g⟩2e |0⟩3| (3)⟩ =e |0⟩1|g⟩1e |0⟩2|e⟩2e |0⟩3| (4)⟩ =e |0⟩1|g⟩1e |0⟩2|g⟩2e |1⟩3The quantum state transfer in this chain flows asi~@| (t)⟩@t= HI | (t)⟩ (5)At any instant, the quantum state of the system is given by| (t)⟩ =4 ∑i=0Ci(t)| (i)⟩ (6)Where, for the initial conditions C0(0) = 1; andCk(k) = 0(k = 1; 2; 3; 4). By setting thecoupling strength as g1;1 : g1;2 : g2;3 : g2;4 = 2 :√6 :√6 : 2 [26], we can easily obtain theexpression of the time-dependent coefficients,C0(t) = 18cos 4t + 12cos 2t + 38(7a)C1(t) = −14i sin 4t − 12i sin 2t (7b)C2(t) =√68cos 4t −√68(7c)C3(t) = −14i sin 4t − 12i sin 2t (7d)C4(t) = 18cos 4t − 12cos 2t + 38(7e)From Eq. (7), we can see that when for t = 2 , we have C0 = 0, and C4 = 1. This resultmeans that the original quantum state | (0)⟩ evolves to the state | (4)⟩, i.e. the Bell stateof the left-hand cavities is transferred onto the right two cavities. The Bell states transferhere is also illustrated in Fig.2.