Distributed atomic quantum information processing via optical fibers

The qudit system may offer great flexibilities for quantum information processing. We investigate the possibility of realizing elementary quantum gates between two high-dimensional atoms in distant cavities coupled by an optical fiber. We show that highly reliable special swap gate is achievable by different detuning. The numerical simulation shows that the proposed elementary gate is robust against the atomic spontaneous decay, photon leakage of cavities and optical fibers by choosing the experimental parameters appropriately.

is driven by the cavity mode with coupling constant g i 2 . The mode number of the fiber is on the order of ν π l c /2 , where l is the length of the fiber and ν is the decay rate of the cavity field. When ν π ≤ l c /2 1, there is only one fiber mode which essentially interacts with the cavity modes and the cavity-fiber coupling is described by the Hamiltonian as follows 29 where b is the annihilation operator for the fiber mode, † a j is the creation operator for the j-th cavity mode, and ν is the cavity-fiber coupling strength.
Assume that the classical field and cavity mode are detuned from the respective transition by ∆ i 1 and ∆ i 2 . In the interaction picture, the Hamiltonian describes the following atom-field interaction ∑ = Ω + + . .
Distributed qudit computation. It is well-known that the qubit rotations and two-qubit CNOT gate are universal for synthesizing multi-qubit circuit. In this case, one only needs to construct CNOT gate using the system in Fig. 1. In fact, for two three-level atomic systems, each of them has two ground states |e 1 〉, |e 2 〉, and one excite state |r〉. Let atomic transition ↔ e r 2 be driven by a classical laser field with Rabi frequency Ω, while the transition ↔ e r 1 be driven by the cavity mode with coupling constant g. Assume that the classical field and the cavity mode are detuned from respective transition by Δ 1 and Δ 2 . In the interaction picture, the Hamiltonian is simplified as Moreover, when µ π = + t k (2 1) and χ π = + t s (2 1/2) for some integers k and s, it reduces to the special SWAP gate This gate may be used to generate CNOT gate, as shown in Fig. 2.
Qudit case. Now, we consider the qudit-based quantum computation. From previous result 55 Since the qudit gate X d may be realized assisted by the classical fields 54 . In the follow, our consideration is to realize C 2 [X d ] with the proposed atomic systems in Fig. 1 S e e k ikk j . After a proper evolving time t (μt = (2k + 1)π and χ π = + t s (2 1/2) for some integers k and s), it leads to a special swapping gate as follows: ie e e e i e e ee id t , denotes the identity operation for all the other terms except to |e i e j 〉 and |e j e i 〉 of two atoms. From the circuit in Fig. 3(a), it easily follows that 1 with 0 k being a zero vector of k-dimension. The two-qudit gate X ij d may be used to realize controlled qudit gate C 2 [X d ]. From Fig. 3 may be decomposed into special two-qudit gates as follows while the other subspace is unchanged for the following evaluations. From the Hamiltonian H eff in Eq. (12), after a proper evolution time t (χ π = t k 2 ), it follows a two-qudit rotation ee ee e e e e e e ee ee id t (2 , , , 0): (10) and (18), it follows that www.nature.com/scientificreports/ 5 Scientific RepoRts | 7: 1234 | DOI: Similarly, one can get Two phase gates yield to  (10) and (21), it follows that From Eqs (10) and (22), it follows that . From Eqs (27) and (28), we obtain

Effects of spontaneous decay and photon leakage.
In this section, we study the influence of atomic spontaneous decay and photon leakage of the cavities and fibers. For convenience, we rewrite the interaction Hamiltonian under the dipole and rotating wave approximation. The master equation for the density matrices of the system is expressed as due to the equal probability transition of ↔ r e i . In the follow, we will discuss the parameter conditions and experimental feasibility of the present scheme. With the choice of a scaling g, all the parameters can be reduced to the dimensionless units related to g.
To realize various rotations in Eqs (9) and (15), the rotation parameters χ and μ could achieve various values. In detail, consider the parameters of Δ 1 = 4g, Δ 2 = 4g + δ, ν = g and Ω = g 3 . The rotation parameters χ and μ are shown in Fig. 4(a,b) respectively. It follows that μ may be changed largely while χ is negative. The ratio of μ and χ is changed from −110 to −20 in Fig. 5(a). Moreover, if another set of parameters Δ 1 = 9g, Δ 2 = 9g + δ, ν = 4g and Ω = g 3 are considered, the rotation parameters χ and μ are shown in Fig. 4(c,d) respectively. In this case, both of them are positive where their ratio is shown in Fig. 5(b).
For the first set of parameters shown in Fig. 4(a), all the adiabatic conditions  v 0 are approximatively satisfied when g and δ/g are increased, as shown in Fig. 6(a-d). Here, v 2 < 0 should be avoided by choosing proper g and δ. If the second set of parameters shown in Fig. 4(c) are considered, the corresponding adiabatic conditions  v 0 i of are greatly improved and shown in Fig. 6(f-h). Specially, in this case, all the v i > 0 for all g > 2 and δ/g > 2. It means that the adiabatic conditions may be satisfied under the weak coupling g < 5.
In order to complete the quantum applications, proper quantum gates should be realized using special phases φ = μt and ψ χ = t with proper evolution times. The phases ratio φ ψ / of all the gates including the iSWAP gate Here, Δ 1 = 9g, Δ 2 = 9g + δ, ν = 4g, Ω = g 3 . Here, χ and μ are positive. and inverse iSWAP gate are shown in Fig. 7(a,b). Combined with Fig. 5(a), these gates may be efficiently realized. Moreover, if another set of parameters Δ 1 = 9g, δ ∆ = + g 9 2 , ν = g 4 , and Ω = g 3 are considered, the rotation parameters χ and μ are shown in Fig. 4(c,d) respectively. In this case, both of them are positive, and their ratio is shown in Fig. 5(b). The corresponding adiabatic conditions are improved and shown in Fig. 6(e,f). The phases ratio φ ψ / of different gates are shown in Fig. 7(c,d), which mean that the iSWAP gate and inverse iSWAP gate may be realized.
To consider atomic spontaneous emission and the decay of the Bosonic modes, let κ γ Γ = = = . g 0 01 , where Γ, κ, and γ are the decay rates for the atomic excited state, the cavity modes, and the fiber mode, respectively. The probability that the atoms undergo a transition to the excited state due to the off-resonant interaction with the classical fields is = Γ ∆ < . P / 001 1 1 2 for both cases. Meanwhile, the probability that the three modes c i are excited due to non-resonant coupling with the classical modes is   Fig. 4. (e-h) Denote the second case in Fig. 4.
The P 2 is shown in Fig. 8 for two groups of parameters. The effective decoherence rates due to the atomic spontaneous emission and the decay of the Bosonic modes are Γ′ = Γ < − P g 10 1 4 and κ κ ′ = < . × − P g 0 35 10 2 3 , respectively.
The fidelity of the iSWAP gate is defined by  Fig. 9. For the small ≈ . g 5 275, the fidelity may be reached to 0.982 after the evolution time ≈ . t 19 575, see Fig. 9(a). For the large ≈ . g 18 4, the fidelity may be reached to 0.994 after the evolution time ≈ . t 6 375. The ideal iSWAP gate is achieved after eight Rabi-like oscillations, see Fig. 10. In the regime ν  g i the fidelities of the gates have been consistently found to be essentially unaffected by fiber losses. In general, moreover, the direct effect of spontaneous emission proves to be more relevant than the indirect   Figure 10. The fidelity of the iSWAP gate vias g and evolution time. ∆ = g 9 1 , δ ∆ = + g 9 2 , ν = 4g, Ω = g 2 . (a) The diamonds refer to ≈ . g 5 275, the squares and the circle refer, respectively, to a variation of −0.025 and +0.025 of g. (b) The diamonds refer to ≈ . g 18 4, the squares and the circle refer, respectively, to a variation of −0.05 and +0.05 of g. defined by the hyperfine atomic level |F = 1, m = 0〉 of 5 2 P 1/2 . Each atom can be made localized at a fixed position in each cavity with high Q for long time 56 . Recent experiment 57 has achieved the parameters g/2π ≈ 750 MHz, κ/2π ≈ 2.62 MHz, and γ/2π ≈ 3.5 MHz in an ultrahigh-Q toroidal microresonators with the wavelength in the region 630~850 nm is predicatively achievable with the optical fiber decay rate 0.152 MHz 58 . By setting Ω = Ω = . g 0 35 1 2 , Δ 1 = 2.3g, Δ 2 = 2.4g, and ν = 0.8g, we can obtain a iSWAP gate the fidelity about 9.21% with κ ≈ . g 0 0035 and γ ≈ . g 0 0046 .

Conclusion
In conclusion, we have investigated the implementation of high-dimensional quantum computation for atoms trapped in distant cavities coupled by an optical fiber. The chosen ground states of each atom are coupled via the cavity mode and different classical fields in the Raman process. All the atoms do not undergo the real Raman transitions due to the large detuning while the atomic system is decoupled from the cavity modes and fiber modes. In the short fiber regime, reliable elementary gates could be reasonable even if imperfections (atomic spontaneous decay and photon leakage of the cavities and fibers) are considered. Let us also mention that, in the considered system, not only entangling and swap gates, but also perfect quantum state transfer is possible. Moreover, the proposed setup would also allow for entanglement preparation schemes between distributed atoms, and could useful in one-way quantum computation. These schemes would be useful for constructing large-scale and long-distance quantum computation or quantum communication networks.