Quantum Information Processing in An Array of Fiber Coupled Cavities

Transcription

1 Commun. Theor. Phys. (Beijing, China) 53 (010) pp c Chinese Physical Society and IOP Publishing Ltd Vol. 53, No., April 15, 010 Quantum Information Processing in An Array of Fiber Coupled Cavities LI Jian (Ó ), ZOU Jian (Õ ), and SHAO Bin (ÅÉ) Department of Physics, Beijing Institute of Technology, Beijing , China (Received May 5, 009) Abstract We consider a fiber coupled cavity array. Each cavity is doped with a single two-level atom. By treating the atom-cavity systems as combined polaritonic qubits, we can transform it into a polaritonic qubit-qubit array in the dispersive regime. We show that the four fiber coupled cavity open chain and ring can both generate the four qubit W state and cluster state, and can both transfer one and two qubit arbitrary states. We also discuss the dynamical behaviors of the four fiber coupled cavity array with unequal couplings. PACS numbers:.8.et, Hk, Mn Key words: fiber coupled cavity array, state transfer, multipartite entanglement 1 Introduction Entanglement is a remarkable feature of quantum mechanics and is the key resource of quantum computation and quantum information processing. 1] Not only the entanglement between two subsystem has been paid much attention, the multipartite entanglement can also be used to implement the quantum information processing. Different from the bipartite entanglement, the multipartite entanglement is not so fragile to be destroyed. People have found two types of inequivalent multipartite entangled states, the Greenberger Horne Zeilinger (GHZ) state, and the W state, which cannot be converted to each other under stochastic local operations and classical communication. ] An important property of the W state is that it is robust against loss of qubits. From recent studies, a modified W-class state can also finish the task of teleportation and the superdense coding. 35] Considering the importance of W states, there have been various schemes to prepare W states proposed theoretically and demonstrated experimentally. 68] Cluster state is a highly entangled state, 910] and it can be used to perform universal one way quantum computation, which realized by a series of one qubit measurements on the state at the same time. 10] Due to the important applications, various schemes have been proposed to generate cluster states in ion-trap system, resonant microwave cavities, and solid state system. 111] In optical systems, a four qubit cluster state has been prepared and applied to the Grover search algorithm. 1518] One key concentration of the quantum physical system in quantum information processing is scalability. There are several prospective candidates for scalable quantum computation. One of them is the coupled cavity array system. Experimentally, it has been developed in nanocavities in photonic crystals and the Josephson junction arrays, and it has been shown that this system can be used in many remarkable quantum information processing. 193] Moreover, another candidate, cavity QEDs coupled by optical fibers, has also been discussed, 5] and it has been shown that it would be able to construct optical networks more conveniently. And also, this fiber coupled cavity array has individual lattice sites which can be easily addressed by the optical lasers. In this paper, we concentrate on the fiber coupled cavity array system. Our array contains a series of cavities which are coupled by the fibers, and each cavity is doped with one two-level atom. We find that a two qubit universal gate can be realized in such a system. We investigate the four fiber coupled cavity open chain and ring, and find that both of them can be used to transfer one qubit arbitrary state and two qubit arbitrary state. In addition, the W state and cluster state can also be realized. We also investigate the quantum information processing of the four fiber coupled cavity array with unequal couplings. This paper is organized as follows. In Sec., we introduce the model, and give the effective Hamiltonian of the whole system. Then, in Sec. 3, we discuss how to construct a two qubit universal gate. After that, the dynamical behaviors of the four qubit open chain and ring are considered in Secs. and 5 respectively. And then, in Sec. 6, we consider the effect of unequal couplings in the four qubit chain. At last, we give the conclusions and discussions in Sec. 7. Model Fig. 1 The fiber coupled cavity array system. The system under consideration is shown in Fig. 1. It consists of N coupled atom-cavity systems, each two near- Supported by National Natural Science Foundation of China under Grant No Corresponding author, zoujian@bit.edu.cn

2 No. Quantum Information Processing in An Array of Fiber Coupled Cavities 765 est cavities are coupled by an optical fiber. The Hamiltonian for the array of fiber coupled cavities is described by H = H 0 + H ac + H cf, (1) where H 0 = H ac = H cf = N N ωi a e i e + ωi c a i a i + ω f i b i b i, () N g i (a i g i e + a i e i g ), (3) J i (a i + a i+1 )b i + (a i + a i+1 )b i ]. () Here, H 0 is the Hamiltonian for the free energy of the atoms, cavity fields, and fibers, H ac describes the Hamiltonian of the interaction between the atoms and their respective cavity fields, which can be seen as a rotating-wave approximation Jaynes Cummings model, and H cf represents the coupling between the cavity fields and the fiber modes. In Eqs. () (), g i and e i are respectively the ground and excited states of the i-th atom which has the transition frequency of ω a i, and a i (a i ) and b i (b i ) denote the creation (annihilation) operators of the i-th cavity field and fiber mode with frequency ω c i and ωf i respectively. g i in Eq. (3) is the coupling of the i-th atom and its cavity, and J i in Eq. () represents the coupling strength of the i-th fiber mode and its neighbour cavities. For this system, the Hamiltonian of the atoms, cavities and their interactions N ωa i e i e + ωi ca i a i + g i (a i g i e + a i e i g )] can be diagonalized in a basis of mixed photonic and atomic excitations which are called polaritons. This polariton regime has been well discussed and used for many quantum information processing in the directly coupled cavity array system. 190,3] As we know, the total excitation number is a conserved quantity for the Hamiltonian, it is easy to find that if the atom and the cavity are resonant, i.e., ωi a = ωc i ω i, the eigenstates of ωi a e i e + ωi ca i a i + g i (a i g i e + a i e i g ) are n, ± i = (1/ )( n, g i ± n 1, e i ) with eigenvalues E (n,±) i = ω i n±g i n. Here, n i denotes the n-photon Fock state of the i-th cavity. The polaritonic states n, ± i can be created by the operators P (n,±) i = n, ± i 0, g. Due to the blockade effect, once a site of polariton is excited to 1, i or 1, + i, no further excitation is possible. And if we create only the polaritons P (1,) i in the atomcavity system initially with energy ω i g i, the polaritons corresponding to P (1,+) i will never even be created. 3] So we can redefine the states and operators. The polariton can be treated as a two level qubit with the two states E i 1, i = (1/ )( 1, g i 0, e i ) and G i 0, g i. The creation operator of the one excitation polariton is P i + P (1,) i = 1, i 0, g = E i G, and Pi = (P i + ). Then, the Hamiltonian of Eq. (1) can be rewritten as H p = N (ω i g i ) E i E + ω f i b i b i + 1 J i b i (P i + + P i+1 + ) + b i (P i + Pi+1 )]. (5) It is clear that the Hamiltonian (5) describes the array of N polaritonic qubits coupled by N 1 fibers. The sketch is shown in Fig.. For simplicity, we suppose the atoms and cavities in the array are the same, i.e., ω i ω and g i g (i = 1,,..., N). Fig. The fiber coupled cavity array can be simplified to the array of N polaritonic qubits coupled by N 1 fibers. Now we focus on the system dynamics when the polaritonic qubits are far off-resonant from the fiber modes, i.e., δ i J i, where δ i (ω g) ω f i. In this dispersive regime, one can apply the canonical transformation defined by the unitary operator { Ji } U = exp b i (P + i + P + i+1 ) b i (P i + P i+1 )], (6) δi and expand it to second order in J i, then obtain N UHU = (ω g) E i E + + N ω f i b i b i + J i J ( i ) (b i b i+1 δ i δ + b i b i+1)( E i+1 E G i+1 G ) i+1 i b i b i ( E i E + E i+1 E ) b i b i( G i G + G i+1 G )] + i (P i + P i+1 + P i P i+1 + ), (7)

3 766 LI Jian, ZOU Jian, and SHAO Bin Vol. 53 where i = J i /(δ i). Assuming that the fiber modes are initially in the vacuum states, we can write the effective Hamiltonian in the interaction picture as follows H eff = i ( E i E + E i+1 E ) + i (P + i P i+1 + P i P + i+1 ). (8) In Eq. (8), the first term on the right hand describes the Stark shifts due to the dispersive interaction with the fiber vacuum, while the last term describes the dipole-dipole coupling between the nearest two polaritons induced by the fiber mode through the exchange of virtual fiber photons. Note that the strength of the Stark shifts and the effective interaction between the qubits are both i. 3 Two Qubit Quantum Universal Gate In this section, we will show how to construct a two qubit universal gate. The Hamiltonian of the two polaritonic array is H p = ( E 1 E + E E ) + (P 1 + P + P 1 P + ). (9) It is easy to obtain the time evolution operator U p (t) = e ihpt = e it At t = π/, we obtain a two-qubit gate U p (π/) = (1) e it cost i sint 0 0 i sint cost e it (10). (11) This is not a regular quantum gate, we can treat it as the combined quantum gate of SWAP gate and control-phase gate (CP), i.e., U p (π/) U SCP, where the subscript means the combined gate. This is a universal two quantum gate that the essential CNOT gate can be sufficient constructed with two U SCP gates and some single-qubit rotations 67] U CNOT = e i(π/) σx e i(π/) σ z USCP e i(π/) σx 1 e i(π/) σ z 1 e i(π/) σ z USCP e i(π/) σz, (1) where σ x i = E i G + G i E, and σ z i = E i E G i G (i = 1, ). In the similar way, we can also construct the control-phase gate (CP) and SWAP gate by a few single qubit rotation U CP = e i(π/) σy UCNOT e i(π/) σy, USWAP = U SCP U CP. (13) Four Fiber Coupled Cavity Open Chain In contrast to the bipartite quantum system, multipartite quantum system offers a much richer structure and various types of entanglement. The efficient and scalable preparation of multipartite entangled states is a key ingredient for the further characterization and experimental study of multipartite entanglement. In this section, we concentrate on the four polaritonic qubit open chain (Fig. 3(a)), and investigate the dynamical behaviors of the four polaritonic qubit open chain. Assume that the couplings between polaritons and the fiber modes are the same 1 = = 3 =. So the Hamiltonian becomes H p = ( E 1 E + E E + E 3 E + E E ) + (P 1 + P + P 1 P + + P + P 3 + P P P 3 + P + P 3 P + ). (1) First, we consider that there is only one excitation of the whole system. By solving the Schrödinger equation with the initial state EGGG, where EGGG = E 1 G G 3 G, we obtain ψ 1 (t) = 1 { 1 + e it 1 + cos ( t ) + i sin ( t )]} EGGG + 1 { 1 + e it 1 i sin ( t )]} GEGG + 1 {1 eit 1 + i sin ( t )]} GGEG + 1 { 1 + e it 1 + cos ( t ) + i sin ( t )]} GGGE. (15) We find that when t = 15.63, the excitation will approximately be transferred to the fourth qubit. In Fig., the solid line represents the occupation probability of E, which can achieve maximally. The dashed line is the occupation probability of E obtained directly from Hamiltonian (5). From Fig. we can see that Hamiltonian (5)

4 No. Quantum Information Processing in An Array of Fiber Coupled Cavities 767 is a good approximation of Hamiltonian (5). From numerical calculations we find that the occupation probabilities for all the fiber modes are less than 0.01, which means that the fiber modes are greatly suppressed in the dispersive regime. As we know, the zero excited state GGGG is the eigenstate of H p, so the arbitrary one qubit state (α G 1 + β E 1 ) GGG 3 will be transferred to the last qubit GGG 13 (α G + β E ) at t = 15.63, where α and β should be arbitrary complex numbers that satisfy the relation α + β = 1. Also, from Eq. (15) we find that at t = 11.05, we can obtain a W state (1/)( EGGG GEGG + GGEG + GGGE ) with the fidelity If we apply one qubit σ z operation on the second qubit, it will become the general symmetric W state. Fig. 3 The four fiber coupled cavity array. (a) Open chain. (b) Ring. Fig. The occupation probability of E. The solid line is obtained from Hamiltonian (1), and the dashed is obtained from Hamiltonian (5). Here we choose δ = 0, ω f = 1. Now we assume that the first two qubits are initially in the maximal entangled state and the other two are in the ground state, i.e., (1/ )( EG 1 + GE 1 ) GG 3, then we obtain ( )] ( ) ( )] ψ (t) = 1 + cos t EGGG cos t i sin t GEGG ( ) ( )] ( )] cos t i sin t GGEG cos t GGGE. (16) Here and below, we discard the common phase factor. It is easy to see that at t = π/, the state will be GG 1 (1/ )( EG 3 + GE 3 ), which means that the maximal entangled state between the first two qubits has successfully transferred to the last two. Now we assume that the initial state is EEGG, then ψ 3 (t) = 1 cos ( t ) + cos ( t ) cos ( t ) + sin(t)sin( t )] EEGG i cos(t)sin( t ) ] + sin(t) EGEG + i sin(t)cos ( t ) eit sin ( t )] EGGE i sin(t)cos ( t ) eit sin ( t )] GEEG i cos(t)sin( t ) ] sin(t) GEGE

5 768 LI Jian, ZOU Jian, and SHAO Bin Vol cos(t) cos(t)cos ( t ) sin(t)sin( t )] GGEE. (17) From Eq. (17) we find that at t = 11.05, the state will become (1/)( EGEG + EGGE GEEG GEGE ) with a fidelity of It is noted that this is a product state of two maximal -qubit entangled state (1/ )( EG 1 GE 1 ) (1/ )( EG 3 + GE 3 ). Then, if we tune off the fibers 1 and 3, and let qubits and 3 evolve for t = π/, which means applying a local USCP 3 gate on qubits and 3, the state of the system will become a four qubit cluster state Ψ C = 1 ( EEGG + EGGE + GEEG GGEE ). (18) Now we consider the transfer of two qubit arbitrary state. Assume that the arbitrary state is initially in the first two polariton qubits, i.e., (α GG 1 + β GE 1 + γ EG 1 + δ EE 1 ) GG 3. Here, α, β, γ, and δ are arbitrary complex numbers that satisfy the relation α + β + γ + δ = 1. From Eqs. (15) and (17), it is easy to obtain that at τ = e ihpτ GEGG GGEG, e ihpτ EGGG GGGE, e ihpτ EEGG GGEE, (19) with high fidelities. Obviously, GGGG is the zero-energy eigenstate of H p, and it does not evolve with time. This means that at τ the original state evolves into: e ihpτ (α GG 1 + β GE 1 + γ EG 1 + δ EE 1 ) GG 3 GG 1 (α GG 3 β EG 3 γ GE 3 δ EE 3 ), (0) then cut off the fiber between qubit and qubit 3, and apply a local operation of σ 3 σ z U z SCP 3, we will obtain the arbitrary state α GG 3 + β GE 3 + γ EG 3 + δ EE 3 which means that the arbitrary two qubit state is transferred from the first two qubits to the last two qubits. 5 Four Fiber Coupled Cavity Ring In this section, we discuss the dynamical behaviors of the four polaritonic qubit ring (Fig. 3(b)) with the identical fiber couplings 1 = = 3 =. The Hamiltonian is H p-ring = ( E 1 E + E E + E 3 E + E E ) + (P 1 + P + P 1 P + + P + P 3 + P P P 3 + P + P 3 P + + P + P 1 + P P 1 + ), (1) where the periodic boundary conditions are considered. By choosing the same one excited initial state EGGG as that in Sec., and obtain ψ 1 (t) ring = cos(t)cos(t) EGGG i sin(t)cos(t) GEGG sin(t) sin(t) GGEG i sin(t) cos(t) GGGE. () So, when t = π/, we will obtain the W state (1/)( EGGG i GEGG GGEG i GGGE ) with a fidelity of 1. And also, if we choose a maximal entangled state (1/ )( EG 1 + GE 1 ) GG 3 between qubits 1 and, the entanglement will be perfectly transferred to qubits 3 and at t = π/ in the four polaritonic qubit ring. If we choose the two excited state EEGG as the initial state, then ψ 3 (t) ring = 1 ( )] i 3 + cos t EEGG sin( t ) EGEG 1 ( )] 1 cos t EGGE 1 ( )] i 1 cos t GEEG sin( t ) EGEG 1 ( )] 1 cos t GGEE. (3) At t = π/, we obtain (1/)( EEGG EGGE GEEG GGEE ), which is a four qubit cluster state. It is noted that the four polaritonic qubit ring can generate the cluster state directly without any operation. Considering the open chain and the ring schemes, both of them can implement the generation of the multipartite entangled state, but the ring is more suitable for the quantum information processing for its more symmetrical structure. 6 Four Polaritonic Qubit Chain with Unequal Couplings In this section, we consider the open chain of unequal couplings between polaritons and fiber modes. The whole system is governed by Hamiltonian (8) when N =. We suppose 1 = 3, but 1. Assuming the initial state is EGGG, we obtain ψ(t) = 1 cos( 1 t) i sin( 1 t) + i sin ( t) e it + cos ( 1 + t) e it] EGGG i 1 sin ( t) e it] GEGG

6 No. Quantum Information Processing in An Array of Fiber Coupled Cavities i 1 i sin( 1 t) sin ( t) e it] GGEG cos( 1 t) + i sin ( t) e it + cos ( 1 + t) e it] GGGE, () and if the initial state is EEGG, we obtain ψ(t) = 1 cos( t) + cos( 1 t)cos ( 1 + t) 1 + sin( t)sin ( 1 + t)] EEGG i sin( t) + cos( 1 t)sin ( t)] EGEG sin( 1 t)cos ( 1 + t) 1 cos( t)sin ( 1 + t) + i i + sin( t)sin ( 1 + t)] EGGE i sin( 1 t)cos ( 1 + t) 1 cos( t)sin ( 1 + t) i sin( t)sin ( 1 + t)] GEEG i + 1 sin( t) + cos( t)sin ( 1 + t)] GEGE cos( t) + cos( 1 t)cos ( 1 + t) + 1 sin( t)sin ( 1 + t)] GGEE. (5) From Eqs. () and (5), we can see that at time t 1 which satisfies cos( 1 t 1 ) = 1, cos( t 1 ) = 1, cos( 1 + t 1) = 1, and cos( 1 t 1 )cos( t 1 ) cos( 1 + t 1) = 1, EGGG GGGE, GEGG GGEG and EEGG GGEE. By applying the similar operation as that in Sec., the two qubit arbitrary state can also be transferred from qubit 1 and qubit to qubit 3 and qubit in the unequal coupling chains. In Eq. (), at time t which satisfies sin( 1 t ) = 1, cos( t ) = 1, and cos( 1 + t ) = 1, we can establish a four qubit W state. So in the unequal coupling chain of 1 = 3, the quantum information processing realized in the identical coupling chain can also be achieved. Now we give a further study of the unequal coupling chain. From Eq. (), at time t 3 which satisfies cos( 1 t 3 ) = 1, sin( t 3 )t = 1, and cos( 1 + t 3) = 1, there will be a maximal entangled state between qubit 1 and qubit. When 1, if cos( 1 t ) = 1, cos( t ) = 1, and sin( 1 + t ) = 1, from Eq. () we will also obtain a maximal entangled state between qubit 1 and at time t. So by choosing the unequal couplings properly, we can achieve the remote maximal entanglement establishment. 7 Conclusions and Discussions In conclusion, we have discussed the quantum information processing in fiber coupled cavity array. We have treated the atom-cavity as a combined polaritonic qubit, and this array system can be effective approximate to an array of qubit-qubit interaction in the dispersive regime. For this system, we have realized a universal two qubit gate U SCP, which can be used to construct the CNOT gate, CP gate, and SWAP gate. The four fiber coupled cavity open chain and ring have also been discussed. Both of them can be used to realize the four qubit W state and entanglement transfer. But because of the qubit ring has more symmetrical structure, it can accomplish the quantum information processing more perfectly. It is noted that we can also realize the cluster state through the four fiber coupled cavity open chain and ring. The qubit ring can directly generate the four qubit cluster state, while the qubit open chain needs a two qubit gate operation to realize the same process. In the four qubit open chain, we have presented a protocol to transfer the two arbitrary qubit state, and it just needs a few local operations to recover the arbitrary state. In addition, we have found these quantum information processing can also be achieved in the open chain of unequal couplings, in which, the remote maximal entangled state can be achieved. Compared with the similar proposal of quantum computation, the fiber coupled cavity array is less complex in terms of realizing quantum information processing, and each node for this array can be scaled up and addressed very conveniently. References 1] M.A. Nielsen and I.L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, Cambridge (000). ] W. Dür, G. Vidal, and J.I. Cirac, Phys. Rev. A 6 (000) ] P. Agrawal and A. Pati, Phys. Rev. A 7 (006) 0630.

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