Quantum Inf Process (2017) 16:294 DOI 10.1007/s11128-017-1747-z Dressed-state scheme for a fast CNOT gate Jin-Lei Wu1 · Xin Ji1 · Shou Zhang1 Received: 25 February 2017 / Accepted: 12 October 2017 © Springer Science+Business Media, LLC 2017 Abstract A dressed-state scheme, which aims to speed up the adiabatic population transfer, is applied for constructing a fast controlled-not (CNOT) gate in a cavity quantum electrodynamics system. Numerical simulations indicate that the average fidelity for constructing the CNOT gate is quite high and the gate operation time is relatively short. Moreover, the effects of the atomic spontaneous emissions and the photon leakages from the system on the average fidelity are discussed and the results show the scheme is robust against decoherence. Keywords CNOT gate · Shortcuts to adiabaticity · Dressed states 1 Introduction Quantum logic gates are key elements of a quantum computer which possesses stronger computational power and faster operational speed than a classical computer [1]. As we all know, all gate operations in quantum computation can be decomposed into a series of elementary one-qubit unitary gates and two-qubit conditional gates which are universal for quantum computation [2,3]. Many schemes have been proposed to implement quantum gate operations in various physical systems, such as ion-trap systems [4], linear optical systems [5,6], cavity QED systems [7–10], nuclear magnetic resonance systems [11] and superconducting systems [12,13]. The CNOT gate is an important two-qubit universal gate which can be used to construct multiqubit gates combined with single-qubit gates. Up to present, many schemes have been proposed B 1 Xin Ji jixin@ybu.edu.cn Department of Physics, College of Science, Yanbian University, Yanji, Jilin 133002, People’s Republic of China 123 294 Page 2 of 15 J.-L. Wu et al. to construct CNOT gates [14–18]. For example, Franson et al. realized a CNOT gate through the quantum Zeno effect of two optical qubits by two-photon absorbtion [15]; Sangouard et al. implemented a CNOT gate by adiabatic passage with an optical cavity [16]; Shao et al. proposed two schemes to perform CNOT gates via quantum Zeno dynamics (QZD) and stimulated Raman adiabatic passage (STIRAP), respectively [18]. As we can see from the references [14–18], QZD and STIRAP are two techniques widely used to perform population transfer because of their robustness against decoherence in proper conditions. However, either QZD or STIRAP has its own unavoidable defects. QZD is sensitive to the atomic spontaneous emission and variation in operation time. STIRAP usually requires a relatively long interaction time which will accumulate decoherence and destroy the desired dynamics. Therefore, many researchers have paid more attention to speeding up the adiabatic population transfer (i.e., shortcuts to adiabaticity) [19–31]. Besides, some remarkable achievements for shortcuts to adiabaticity have been implemented in experiment [32–36]. There are two methods, transitionless quantum driving and Lewis–Riesenfeld invariant, widely used. Also, many schemes have been proposed for constructing quantum gates based on the two methods [37–40]. However, the two methods suffer from some experimental obstacles in many cases. On one hand, transitionless quantum driving requires either a direct coupling between the initial and target states [21,41,42] or an unavailable coupling in the original Hamiltonian [32]. On the other hand, Lewis–Riesenfeld invariant usually leads to pulse schemes that either need an infinite energy gap to be perfect [23], or do not smoothly turn on or off [23,43]. Recently, Baksic et al. proposed a new method to speed up adiabatic population transfer by using dressed states [44]. It is more experimentally feasible than transitionless quantum driving and Lewis–Riesenfeld invariant, because it skillfully escapes from the experimental obstacles the latter two methods suffer from. By using the dressed-state method, we have proposed a scheme for fast two-atom quantum state transfer and entanglement generation in a cavity QED system [45]; Kang et al. [46] proposed a scheme for generating a three-qubit W state in a superconducting system; schemes for holonomic quantum computation [47] and quantum state conversion [48] are also proposed. In this work, we apply the dressed-state method to construct a fast CNOT gate in a cavity QED system which consists of two identical five-level atoms trapped, respectively, in two single-mode optical cavities connected by a fiber. 2 Physical model and general process The schematic setup for constructing the fast CNOT gate is shown in Fig. 1. Two identical five-level atoms A and B are trapped in two single-mode optical cavities, respectively. The short fiber limit (l)/(2π c) ≤ 1 ensures that only the resonant modes of the fibers interact with the cavity modes [49], where l and are the length of the fiber and the decay rate of the cavity field into a continuum of fiber modes, respectively. Each atom has an upper level |e and four lower levels |a, |g1 , |g2 and |g0 . The atomic transitions |eA(B) ↔ |g0 A(B) is resonantly coupled to the mode of the cavity A(B) with corresponding coupling constant gA(B) , and the transitions 123 Dressed-state scheme for a fast CNOT gate Page 3 of 15 294 Fig. 1 The diagrammatic sketch of cavity-atom combined system, atomic level configuration and related transitions |eB ↔ |aB , |eB ↔ |g1 B and |eB ↔ |g2 B are resonantly driven by classical fields with time-dependent Rabi frequencies i (t)(i = 11, 1a, 22, 21, 3a, 32) shown at the right side in Fig. 1 which will be described hereinafter in detail. The quantum information is encoded in the states |g1 g1 AB ≡ |00, |g1 g2 AB ≡ |01, |g0 g1 AB ≡ |10, |g0 g2 AB ≡ |11, (1) where |gi g j AB (i = 0, 1; j = 1, 2) denotes the atoms A and B in the states |gi A and |g j B , respectively. Assume the system is initially in the superposition state |0 = sin ε sin β|g1 g1 AB + sin ε cos β|g1 g2 AB + cos ε sin β|g0 g1 AB + cos ε cos β|g0 g2 AB , (2) where ε and β are chosen as two real parameters for convenience of simulations. After a CNOT gate operation on the initial state |0 , the outcome state becomes | = sin ε sin β|g1 g1 AB + sin ε cos β|g1 g2 AB + cos ε sin β|g0 g2 AB + cos ε cos β|g0 g1 AB . (3) Here, atom A acts as the control qubit, and atom B is the target qubit. Only three steps are needed to achieve such a CNOT gate operation. Firstly, we implement the complete population transfer |g0 g1 AB → |g0 aAB by using the laser pulses interacting with atom B to drive the atomic transitions |eB ↔ |g1 B and |eB ↔ |aB with the corresponding Rabi frequencies 11 (t) and 1a (t), respectively. Then the initial state becomes |1 = sin ε sin β|g1 g1 AB + sin ε cos β|g1 g2 AB + cos ε sin β|g0 aAB + cos ε cos β|g0 g2 AB . (4) Secondly, we implement the complete population transfer |g0 g2 AB → |g0 g1 AB by using the laser pulses interacting with atom B to drive the atomic transitions |eB ↔ 123 294 Page 4 of 15 J.-L. Wu et al. Fig. 2 Schematic representation of the three steps for constructing a CNOT gate |g2 B and |eB ↔ |g1 B with the corresponding Rabi frequencies 22 (t) and 21 (t), respectively. Then the state of the system becomes |2 = sin ε sin β|g1 g1 AB + sin ε cos β|g1 g2 AB + cos ε sin β|g0 aAB + cos ε cos β|g0 g1 AB . (5) Finally, we implement the complete population transfer |g0 aAB → |g0 g2 AB by using the laser pulses interacting with atom B to drive the atomic transitions |eB ↔ |aB and |eB ↔ |g2 B with the corresponding Rabi frequencies 3a (t) and 32 (t), respectively. And thus the state of the system becomes the expected outcome state |. The process above is equivalent to a CNOT gate operation, which indicates we can construct a CNOT gate by such three steps shown in Fig. 2. 3 Dressed-state scheme for constructing the CNOT gate For the first step, the time-dependent interaction Hamiltonian of the whole system is written as (setting h̄ = 1) H1 (t) = Hal (t) + Hacf , Hal (t) = 11 (t)|eB g1 | + 1a (t)|eB a| + H.c., (gk ak |ek g0 | + νbak† ) + H.c., Hacf = (6) k=A,B where aA(B) is the annihilation operator of the mode of the cavity A(B), b is the annihilation operator of the fiber mode, and ν is the coupling strength between the cavities modes and the fiber mode. For simplicity, we assume gA = gB = g. Then with the initial state |0 in Eq. (2), dominated by the Hamiltonian (6), the whole system evolves in the Hilbert subspace spanned by |φ1 = |g1 g1 AB |000, |φ2 = |g1 g2 AB |000, |φ3 = |g0 g1 AB |000, |φ4 = |g0 g2 AB |000, |φ5 = |g1 eAB |000, |φ6 = |g0 eAB |000, 123 Dressed-state scheme for a fast CNOT gate Page 5 of 15 294 |φ7 = |g1 aAB |000, |φ8 = |g1 g0 AB |001, |φ9 = |g0 aAB |000, |φ10 = |g0 g0 AB |001, |φ11 = |g1 g0 AB |010, |φ12 = |g0 g0 AB |010, |φ13 = |g1 g0 AB |100, |φ14 = |g0 g0 AB |100, |φ15 = |eg0 AB |000, (7) in which the unsubscripted ket |i jk (i, j, k = 0, 1) denotes i, j and k photon in the cavity A, fiber and cavity B, respectively. Then according to the theory of quantum Zeno dynamics [50,51], the eigenstates of the atom–cavity–fiber interaction Hamiltonian Hacf can be split into several eigenspaces (i.e., quantum Zeno subspaces), and the eigenstates in the same Zeno subspace are with the same eigenvalue. It is easy to know that the differences between the eigenvalues of Hacf are positively related to the values of g and v. The atom–laser interaction Hamiltonian Hal (t) brings a coupling between different eigenspaces with a coupling strength positively related to 11 (t) and 1a (t). By choosing the quantum Zeno limit condition 11 (t), 1a (t) g, v, the coupling between different subspaces can be ignored, which means that the evolution of the system will be limited in the subspace where the initial state exists in. Because the initial state |0 is the dark state of Hacf (i.e., Hacf |0 = 0), the whole system will approximatively evolve in the Zeno subspace consisting of dark states of Hacf H P = {|φ1 , |φ2 , |φ3 , |φ4 , |φ7 , |φ9 , |φd } , (8) corresponding to the projections P α = |αα|, (|α ∈ H P ). (9) |φd = (ν|φ6 − g|φ12 + ν|φ15 )/ 2ν 2 + g 2 . (10) Here, Therefore, by setting v = g, we can rewrite the system Hamiltonian as the following form [18,52,53] P α Hal (t)P α = 1 (t)|φ3 φd | + 2 (t)|φ9 φd | + H.c., (11) H (t) α √ √ in which 1 (t) = 11 (t)/ 3 and 2 (t) = 1a (t)/ 3. Except |g0 g1 AB , obviously, |g1 g1 AB , |g1 g2 AB and |g0 g2 AB do not participate in the system evolution governed by the Hamiltonian (11). By choosing 1 (t) = −(t) sin θ (t), 2 (t) = (t) cos θ (t), (12) with (t) = 1 (t)2 + 2 (t)2 and θ (t) = − arctan[1 (t)/ 2 (t)], we can easily obtain the time-dependent eigenstates of H (t) |ϕd (t) = cos θ (t)|φ3 + sin θ (t)|φ9 , 123 294 Page 6 of 15 J.-L. Wu et al. √ |ϕ± (t) = [sin θ (t)|φ3 ∓ |φd − cos θ (t)|φ9 ]/ 2, (13) with the eigenvalues E d = 0 and E ± = ±(t), respectively. Assume that the system evolution is started at t = ti and finished at t = t f . The state transfer from |φ3 to |φ9 can be achieved via√|ϕd (t) by setting θ (ti ) = 0 and θ (t f ) = π/2 under the adiabatic criterion |θ̇(t)| 2(t) [54,55], which requires a relatively long operation time. In the following, we use the dressed-state scheme to speed up the state transfer. For convenience, we transform the time-dependent Hamiltonian (11) to the timeindependent eigenstates frame by the unitary operator U (t) = j=d,± |ϕ j ϕ j (t)|. In the time-independent eigenstates frame the Hamiltonian (11) becomes Had (t) = (t)Mz + θ̇ (t)M y , (14) √ where Mz = |ϕ+ ϕ+ | − |ϕ− ϕ− | and M y = i(|ϕ+ + |ϕ− )ϕd |/ 2 + H.c.. In order to protect the system evolution from the second term of the Hamiltonian (14) which leads to an imperfect population transfer, we introduce the modified Hamiltonian Hmod (t) = H (t) + Hc (t) to govern a perfect population transfer with the addition of a correction Hamiltonian Hc (t). Hc (t) can be given by the general form Hc (t) = U † (t)[gx (t)Mx + gz (t)Mz ]U (t), (15) √ with Mx = (|ϕ− − |ϕ+ )ϕd |/ 2 + H.c.. gx (t) and gz (t) are two undetermined parameters. Thus the Hamiltonian (11) becomes Hmod (t) = H (t) + Hc (t) = 1 (t)|φ3 φd | + 2 (t)|φ9 φd | + H.c., (16) with the modified pulses 1 (t) = gx (t) cos θ (t) − [gz (t) + (t)] sin θ (t), 2 (t) = gx (t) sin θ (t) + [gz (t) + (t)] cos θ (t), (17) and then the Hamiltonian (6) becomes H1mod (t) = 11 (t)|eB g1 | + 1a (t)|e B a| + gk ak |ek g0 | + νbak† + H.c., (18) k=A,B √ √ with 11 (t) = 31 (t) and 1a (t) = 32 (t). With reference to Ref. [44], we introduce a set of dressed states |ϕ̃±,d (t) by the unitary operation |ϕ̃±,d (t) = V † (t)|ϕ±,d . We choose the unitary operator V (t) = exp[iμ(t)Mx ], 123 (19) Dressed-state scheme for a fast CNOT gate Page 7 of 15 294 with an Euler angle μ(t). After transforming the modified Hamiltonian (16) to the time-independent dressed-state frame defined by V (t), the Hamiltonian (16) becomes Hnew (t) = V (t)Had (t)V † (t) + V (t)U (t)Hc (t)U † (t)V † (t) + i = η(t)(|ϕ̃+ ϕ̃+ | − |ϕ̃− ϕ̃− |) + [ξ1 (t)|ϕ̃+ ϕ̃d | + ξ2 (t)|ϕ̃− ϕ̃d | + H.c.], d V (t) † V (t) dt (20) with |ϕ̃±,d = V (t)|ϕ̃±,d (t) and three time-dependent parameters η(t) = [gz (t) + (t)] cos μ(t) − θ̇ (t) sin μ(t), √ ξ1 (t) = {i[gz (t) + (t)] sin μ(t) + i θ̇ (t) cos μ(t) + μ̇(t) − gx (t)}/ 2, √ ξ2 (t) = {i[gz (t) + (t)] sin μ(t) + i θ̇ (t) cos μ(t) − μ̇(t) + gx (t)}/ 2. (21) After simple calculations, we choose gx (t) = μ̇(t), gz (t) = −(t) − θ̇(t) , tan μ(t) (22) to remove the second term of the Hamiltonian (20), which prevents the Hamiltonian (20) from driving transitions between different dressed states. Then back to the original frame, the dark dressed state, which corresponds to the zero eigenvalue of Hnew (t), is written as |ϕ0 (t) = U † (t)V † (t)|ϕ̃d = cos μ(t) [cos θ (t)|φ3 + sin θ (t)|φ9 ] − i sin μ(t)|φd . (23) If the parameters satisfy θ (ti ) = 0, θ (t1 f ) = π/2 and μ(ti ) = μ(t1 f ) = 0, where ti(1 f ) is the initial (final) time of the first step for constructing the CNOT gate, the desired population transfer |φ3 → |φ9 will be achieved by the system evolution along the dark dressed state |ϕ0 (t). Based on the process above, we achieve the transfer |g0 g1 AB → |g0 aAB , but the states |g1 g1 AB , |g1 g2 AB and |g0 g2 AB remain unchanged. Therefore, the first step is achieved for constructing the CNOT gate. Besides, the evolution process is not necessarily slow, and there is not a direct coupling between the initial and target states. Similar to the first step, the modified Hamiltonian of the second step for constructing the CNOT gate is written as H2mod (t) = 22 (t)|eB g2 | + 21 (t)|eB g1 | + (gk ak |ek g0 | + νbak† ) + H.c., (24) k=A,B 123 294 Page 8 of 15 J.-L. Wu et al. in which 22 (t) = √ 3{gx (t − t1 f ) cos θ (t − t1 f ) − [gz (t − t1 f ) + (t − t1 f )] sin θ (t − t1 f )}, √ 21 (t) = 3{gx (t − t1 f ) sin θ (t − t1 f ) + [gz (t − t1 f ) + (t − t1 f )] cos θ (t − t1 f )}. (25) If the parameters satisfy θ (t1 f ) = 0, θ (t2 f ) = π/2 and μ(t1 f ) = μ(t2 f ) = 0, where t1 f (2 f ) is the initial (final) time of the second step for constructing the CNOT gate, the desired transfer |g0 g2 AB → |g0 g1 AB will be achieved. Similarly, the modified Hamiltonian of the third step for constructing the CNOT gate is written as H3mod (t) = 3a (t)|eB a| + 32 (t)|eB g2 | (gk ak |ek g0 | + νbak† ) + H.c., + (26) k=A,B in which 3a (t) = 32 (t) √ 3{gx (t − t2 f ) cos θ (t − t2 f ) − [gz (t − t2 f ) + (t − t2 f )] sin θ (t − t2 f )}, √ = 3{gx (t − t2 f ) sin θ (t − t2 f ) + [gz (t − t2 f ) + (t − t2 f )] cos θ (t − t2 f )}. (27) If the parameters satisfy θ (t2 f ) = 0, θ (t3 f ) = π/2 and μ(t2 f ) = μ(t3 f ) = 0, where t2 f (3 f ) is the initial (final) time of the third step for constructing the CNOT gate, the desired transfer |g0 aAB → |g0 g2 AB will be achieved. By now, the transform |0 → | is achieved and we implement the CNOT gate successfully. 4 Numerical simulations First of all, 1 (t) and 2 (t) can be chosen as the Gaussian pulses [54,55] 1 (t) = 0 exp[−(t − t f /2 − t0 )2 /τ 2 ], 2 (t) = 0 exp[−(t − t f /2 + t0 )2 /τ 2 ], (28) in which the related parameters are chosen as t f = 20/0 and τ = t0 = 0.1t f . Besides, there exist the relations t1 f = t f , t2 f = 2t f and t3 f = 3t f . μ(t) is defined by μ(t) = − arctan 123 θ̇ (t) , G(t)/τ + (t) (29) Dressed-state scheme for a fast CNOT gate θ(t) (a) 294 π/2 1 (b) μ(t) 2 Page 9 of 15 0 −0.2 −0.4 0 Step 1 Step 2 10 20 Step 3 30 40 50 60 t/Ω−1 0 Fig. 3 θ (t) and μ(t) versus t/ −1 0 . The parameters used here are {t f = 20/ 0 , τ = t0 = 0.1t f } where G(t) = sech(t/τ ) is chosen to regularize μ(t) such that it can meet the condition μ(ti ) = μ(t f ) = 0 and make sin2 μ(t), the population of |φd [see Eq. (23)], as small as possible. Then, in order to check whether or not the parameters we choose are suitable, we plot θ (t) and μ(t) versus t/−1 0 in Fig. 3. Obviously, Fig. 3 shows that the chosen parameters are in accord with all of the boundary conditions {θ (ti ) = 0, θ (t1 f ) = π/2, μ(ti ) = μ(t1 f ) = 0}, {θ (t1 f ) = 0, θ (t2 f ) = π/2, μ(t1 f ) = μ(t2 f ) = 0} and {θ (t2 f ) = 0, θ (t3 f ) = π/2, μ(t√ 2 f ) = μ(t3 f ) = 0} for constructing the CNOT gate. Based on the relations {11 (t) = 31 (t), 1a (t) = √ 32 (t), Eqs. (25), (27)}, we can determine the six time-dependent modified pulses 11 (t), 1a (t), 22 (t), 21 (t), 3a (t) and 32 (t). In Fig. 4a, we plot the six modified pulses versus t/−1 0 . And in Fig. 4b, we plot the time-dependent population conversions of the three steps for constructing the CNOT gate, respectively. Here, for the Zeno limit condition i (t) g, v, we choose g = v = 100 . In Fig. 4a, the six modified pulses can be replaced by complete Gaussian pulses by curve fitting and can be smoothly turned on and off. According to the recent experiment [35], the values and plus or minus signs of the pulses can be easily controlled by modulating their amplitudes and phases, respectively. From Fig. 4a, we know that the modified pulses amplitude 0 ≈ 2.50 , and then g = v ≈ 40 . However, Fig. 4b shows that all of the expected population conversions are achieved near perfectly, which shows that the CNOT gate can also be constructed even when the condition i (t) g, v is not strictly satisfied. Next, we show the average fidelity of each step in Fig. 5a and that of the whole gate process in Fig. 5b, respectively. The average fidelity is defined by F= 1 (2π )2 2π 0 2π |ideal |(t)|2 dεdβ. (30) 0 Here, |ideal = |1 for the step 1; |ideal = |2 for the step 2; |ideal = | for the step 3 and the whole gate process. |(t) is the state of the system governed by the Hamiltonian (18) for the step 1, Hamiltonian (24) for the step 2, or Hamiltonian (16) for the step 3. As shown in Fig. 5a, the average fidelity of each step is near unit at each finial time. Correspondingly, in Fig. 5b, the average fidelity of the whole gate 123 J.-L. Wu et al. (a) 3 0 Page 10 of 15 1.5 Ω’22 Ω’11 Ω’3a 0 i Ω’ (t)/Ω 294 Ω’1a −1.5 −3 Populations (b) Step 1 1 Ω’21 Step 2 Ω’ 32 Step 3 |g a〉 0 |g0g1〉 |g g 〉 |g g 〉 |g0g2〉 |g a〉 0 2 0.5 0 0 0 1 10 20 30 t/Ω−1 0 0 40 50 60 Fig. 4 a The six time-dependent modified pulses versus t/ −1 0 ; b the time-dependent population conversions of the three steps for constructing the CNOT gate versus t/ −1 0 . The parameters used here are {t f = 20/ 0 , τ = t0 = 0.1t f , g = v = 100 } Fidelity (a) Fidelity (b) 1 Step 1 0.9 0.8 0.7 Step 2 Step 3 1 0.8 0.6 0.4 0 10 20 30 40 50 60 −1 0 t/Ω Fig. 5 a The average fidelity of each step versus t/ −1 0 ; b the average fidelity of the whole gate process versus t/ −1 0 . The parameters used here are the same as in Fig. 4 process is F = 0.994 at t = 60/0 , which indicates our scheme is highly feasible within a very short gate operation time. For a clearer illustration, in Fig. 6, we give a truth table of the CNOT gate constructed by the above dressed-state scheme, which also indicates our scheme is highly feasible. In the following, we take the effect of decoherence on the dressed-state scheme into account. The whole system is dominated by the master equation ρ̇(t) = −i[H mod (t), ρ(t)] γj σe j ,e j ρ(t) − 2σk j ,e j ρ(t)σe j ,k j + ρ(t)σe j ,e j − 2 j=A,B k=a,g1 ,g2 ,g0 κj † a j a j ρ(t) − 2a j ρ(t)a †j + ρ(t)a †j a j − 2 j=A,B κf † − b bρ(t) − 2bρ(t)b† + ρ(t)b† b , (31) 2 123 Dressed-state scheme for a fast CNOT gate Page 11 of 15 294 Fig. 6 The truth table of the CNOT gate. The parameters used here are the same as in Fig. 4 1 0.994 1.000 0.994 1.000 0.5 0 |g g 〉 1 1 |g1g2〉 |g g 〉 0 1 |g g 〉 1 2 |g g 〉 |g0g2〉 |g g 〉 0 2 |g0g1〉 1 1 0.99 Fidelity 1 0.98 0.98 0.97 0.96 0 0 0.05 0.05 κ/Ω0 0.1 0.1 γ/Ω 0 Fig. 7 The fidelity as a function of γ / 0 and κ/ 0 . The parameters used here are the same as in Fig. 4 where H mod (t) = Himod (t) for the ith step (i = 1, 2, 3). γA(B) is the spontaneous emission rate of atom A(B) from the excited state |eA(B) to the ground state |kA(B) (k = a, g1 , g2 , g0 ); κA(B) denotes the photon leakage rate from the cavity A(B); σe j ,k j = |e j k|. For simplicity, we assume γA = γB = γ /4 and κA = κB = κ f = κ. Then the average fidelity is rewritten as F= 1 (2π )2 0 2π 2π |ideal |ρ(t)|ideal |dεdβ. (32) 0 Based on the above master equation, we plot the effect of decoherence on the average fidelity of the whole process for constructing the CNOT gate in Fig. 7. As we can see from Fig. 7, we learn that the influence of the photon leakages from the cavities on the fidelity is obviously greater than that of the atomic spontaneous emissions. It is not difficult to understand it. As we all know, QZD can effectively restrain the influence of the photon leakages from the cavities if the Zeno limit condition is met. But in our scheme, the amplitude of the six modified pulses is near 2.50 (see Fig. 4a) and the chosen parameters g = v = 100 do not satisfy the Zeno limit condition i g, v strictly. Even so, however, the fidelity of the whole process for constructing the CNOT gate is over 0.96 even when κ = γ = 0.10 . Therefore, the dressed-state scheme 123 294 (a) Page 12 of 15 J.-L. Wu et al. (b) 1 1 0.99 Fidelity 0.95 0.98 γ=κ=0 γ=κ=0.05Ω0 0.97 γ=κ=0.1Ω0 (γ,κ)=(2.62,3.5)×2π MHz Ω0=75×2π MHz 0.96 0.95 0 0.9 20 40 60 g/Ω 0 80 100 0 50 100 t /Ω−1 f 0 150 0.85 200 Fig. 8 a The fidelity as a function of g/ 0 with t f = 20/ 0 and different values of γ and κ; b the fidelity as a function of t f / −1 0 with g = 100 and different values of γ and κ. Other parameters used here are the same as in Fig. 4 for constructing the CNOT gate is robust against the decoherence induced by the atomic spontaneous emissions and the photon leakages from the cavity-fiber system. Considering the current experimental conditions, by using cesium atoms [56] and a set of predicted cavity QED parameters (g, κ, γ )/2π = (750, 3.3, 2.62) MHz [57], the CNOT gate can be constructed with a fidelity F = 0.99. Hence, even in the presence of decoherence, the dressed-state scheme for constructing a fast CNOT gate is also highly feasible. Finally, in order to explore the effect of the two main parameters g and t f on constructing the CNOT gate with different values of γ and κ, we plot the average fidelities of the whole process versus g/0 and t f /−1 0 in Figs. 8a, b, respectively. In Fig. 8a, with one pair of the same (zero or non-zero) values of γ and κ, a larger g will lead to a higher average fidelity, because a larger g can satisfy the Zeno limit condition better. In Fig. 8b, in the absence of decoherence (blue dashed line), the average fidelity will continuously rise up to near 1 with the increase of t f . For nonzero γ and κ, however, when t f is over a certain value (t f = 20/0 ), the average fidelities will be lower and lower as t f increases. The reason is that a larger t f leads to a greater decoherence accumulation which definitely destroys the desired dynamics. For the predicted cavity QED parameters (g, κ, γ )/2π = (750, 3.3, 2.62) MHz (red solid lines with 0 = g/10), g ≥ 100 and 10/0 < t f < 100/0 can guarantee a high average fidelity (F > 0.95). 5 Conclusion The dressed-state scheme does not require a direct coupling between the target and initial states, and all of the driving pulses are smoothly turned on or off, which ensures the high feasibility of the scheme in experiment. During the whole process of constructing the CNOT gate, the adiabatic condition need not be satisfied, and thus the gate operation is fast. Besides, the results of the numerical simulations indicate that 123 Dressed-state scheme for a fast CNOT gate Page 13 of 15 294 the construction of the CNOT gate is robust against the decoherence induced by the atomic spontaneous emissions and the photon leakages from the system. 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