Abstract

Quantum logic gate is indispensable to quantum computation. One of the important qubit operations is the quantum controlled-not (CNOT) gate that performs a NOT operation on a target qubit depending on the state of the control qubit. In this paper we present a scheme to realize the quantum CNOT gate between two spatially separated atoms via shortcuts to adiabatic passage. The influence of various decoherence processes on the fidelity is discussed. The strict numerical simulation results show that the fidelity for the CNOT gate is relatively high.

© 2015 Optical Society of America

1. Introduction

Quantum computers have the potential to perform certain computational tasks more efficiently than classical ones. And quantum computation requires quantum logic gates that use the interaction within pairs of qubits to perform conditional operations. As we know that a universal set of quantum operations can be decomposed into a series of single- and two-qubit gates [1, 2]. So many schemes have been proposed [3–12] both on the theory and experiments, and there are many different kinds of physical models can be used to implement gate operations, such as cavity quantum electrodynamics (QED) [4], nuclear magnetic resonance (NMR) system [12], linear optics [3], ion trap and superconducting devices [6–9]. The controlled-not (CNOT) gate is one of the universal gates which can be used to construct any multiqubit gates combined with single-qubit gates and also allows to prepare entangled states from factorizable superposition states. While entanglement is at the heart of quantum information processing, so it confers to the CNOT gate a broad application prospect. Recently, lots of theoretical schemes [13–16] have been proposed to implement two-qubit logic gates with trapped atoms or molecular systems. For instance, Zheng implemented a phase gate through the adiabatic evolution [13] and Sangouard et al implemented a CNOT gate by adiabatic passage with an optical cavity [15]. Adiabatic evolutionin have been used to improve the fidelity and avoid the errors in these two schemes.

However, the adiabatic condition usually limits the speed of the evolution of the system. That is adiabatic passage technique usually requires a relatively long interaction time. If the required evolution time is too long, the speed of the system evolution will be slowed down, that the dissipation caused by decoherence, noise, and losses would destroy the expected dynamics finally. Therefore, finding shortcuts to adiabaticity is of great significance to cut down the interaction time of a method and has drawn much more attentions than ever before. Thus, a variety of schemes have been proposed to construct shortcuts to adiabatic passage in both theories and experiment [17–28]. The shortcuts to adiabatic passage can also be used to implement the logical gate operations. Recently, Chen et al [29] proposed a scheme of shortcuts to adiabatic passage for performing a π phase gate and we proposed a scheme of constructing the multiqubit controlled phase gate via shortcut to adiabatic passage [30].

In this paper, by combining Lewis-Riesenfeld invariants with quantum Zeno dynamics (QZD), we propose an effective scheme to implement the CNOT gates between two spatially separated atoms via constructing the shortcuts to adiabatic passage. The CNOT gate can be achieved in a much shorter interaction time via shortcuts to adiabatic passage than that based on adiabatic passage and it is very robustness to the decoherence caused by the atomic spontaneous emission, cavity decay as well as the fiber decay.

The paper is organized as follows. In Sec. 2, we recall the principles of QZD and invariants Lewis-Riesenfeld. In Sec. 3 we effectively construct the shortcuts to adiabatic passage in a cavity-fibre-cavity system by combining the Lewis-Riesenfeld invariants and QZD, and show how to use the shortcut to implement the CNOT gates. We show the numerical simulation results and give feasibility analysis in Sec. 4. The conclusion appears in Sec. 5.

2. Preliminary theory

2.1. Quantum Zeno dynamics

Quantum Zeno effect is an interesting phenomenon in quantum mechanics. Recent studies [31–33] show that a quantum Zeno evolution will evolve away from its initial state, but it remains in the Zeno subspace defined by the measurements [31] via frequently projecting onto a multidimensional subspace. This is known as QZD. We consider a system which is governed by the Hamiltonian

HK=Hobs+KHmeas,
where Hobs is the Hamiltonian of the investigated quantum system and the Hmeas is the interaction Hamiltonian performing the measurement. K is a coupling constant, and when it satisfies K → ∞, the whole system is governed by the evolution operator
U(t)=exp[itn(KλnPn+PnHobsPn)],
where Pn is one of the eigenprojections of Hmeas with eigenvalues λn(Hmeas = ∑nλnPn).

2.2. Lewis-Riesenfeld invariants

A brief introduction of the Lewis-Riesenfeld invariants theory [34, 35] is given in this section. Considering a system which is governed by a time-dependent Hamiltonian H(t), and we can seek the time-dependent Hermitian invariants I(t) that is related to the original Hamiltonian H(t) and satisfies

ih¯I(t)t=[H(t),I(t)].
Obviously, its expectation values remain constant all the time, and drives the system state evolve along the initial eigenstate of I(t). For the time-dependent Schrödinger equation ih¯|Ψ(t)t=H(t)|Ψ(t), the solution can be expressed by the superposition of dynamical modes |Φn(t)〉 of the invariants I(t)
|Ψ(t)=nCneiθn|Φn(t),
where n = 0, 1,..., and Cn is one of the time-independent amplitudes, θn is the Lewis-Riesenfeld phase. |Φn(t)〉 is one of the orthonormal eigenvectors of the invariant I(t) with the corresponding real eigenvalue λn, satisfying I(t)|Φn(t)〉 = λnn(t)〉. And the Lewis-Riesenfeld phase satisfies
θn(t)=1h¯0tdtΦn(t)|ih¯tH(t)|Φn(t).

3. Shortcuts to adiabatic passage for the CNOT gate

The schematic setup for implementing the CNOT gate between two spatially separated atoms is shown in Fig. 1. We consider a cavity-fibre-cavity system, in which two atoms are trapped in the corresponding optical cavities connected by a fiber. Under the short fiber limit (lv)/(2πc) ≪ 1, only the resonant mode of the fiber will interact with the cavity mode [36], where l is the length of the fiber and v is the decay rate of the cavity field into a continuum of fiber modes. Atoms A and B possess four ground states |g0〉, |g1〉, |g2〉, |a〉 and one excited state |e〉. The B atom transitions |g1〉 ↔ |e〉, |g2〉 ↔ |e〉 and |a〉 ↔ |e〉 are coupled to the laser pulses, with the corresponding Rabi frequencies Ω1(t), Ω2(t) and Ωa(t), respectively. The transition |g0〉 ↔ |e〉 of atom A (B) is strongly coupled to its single mode cavity A (B) with the coupling constant λA(λB).

 figure: Fig. 1

Fig. 1 The schematic setup of CNOT gate implementation. The two atoms are trapped in two spatially separated optical cavities connected by a fiber, and each atom possesses five atomic levels.

Download Full Size | PPT Slide | PDF

We encode the quantum information in the states |g1g1AB ≡ |00〉, |g1g2AB ≡ |01〉, |g0g1AB ≡ |10〉 and |g0g2AB ≡ |11〉. For a general input state

|Ψ0=α1|g1g1AB+α2|g1g2AB+α3|g0g1AB+α4|g0g2AB,
after implementing the CNOT gate operation, the outcome becomes
|Ψ=α1|g1g1AB+α2|g1g2AB+α3|g0g2AB+α4|g0g1AB,
where |gigjAB(i = 0, 1; j = 1, 2) denotes the state of atom A and B, α1,2,3,4 are complex coefficients and satisfy the normalization condition. Here atom A acts as the control qubit, and atom B is the target qubit.

In order to achieve the CNOT gate operation, four steps are needed. Firstly, we transfer the population of |g0g1AB to −|g0aAB completely with the help of laser pulses resonant with B atomic transitions |g1B ↔ |eB and |aB ↔ |eB with the corresponding Rabi frequencies Ω11(t) and Ωa1(t), respectively. After the interaction, the initial state becomes

|Ψ1=α1|g1g1AB+α2|g1g2ABα3|g0aAB+α4|g0g2AB.

Secondly, the population of |g0g2AB is completely transferred to −|g0g1AB with the similar method as the first step, and the corresponding laser pulses resonant with B atomic transitions |g2B ↔ |eB and |g1B ↔ |eB with the corresponding Rabi frequencies Ω21(t) and Ω12(t), respectively. And then the state of the system becomes

|Ψ2=α1|g1g1AB+α2|g1g2ABα3|g0aABα4|g0g1AB.

Then the population of |g0aAB is completely transferred to −|g0g2AB by the similar method as the first step, and the corresponding laser pulses resonant with B atomic transitions |aB ↔ |eB and |g2B ↔ |eB with the corresponding Rabi frequencies Ωa2(t) and Ω22(t), respectively. Now, the state of the system reads

|Ψ3=α1|g1g1AB+α2|g1g2AB+α3|g0g2ABα4|g0g1AB.

Finally the state |g0g1AB induces a π phase, then turns into −|g0g1AB and the corresponding laser pulses are resonant with B atomic transitions |g1B ↔ |eB and |aB) ↔ |eB with the corresponding Rabi frequencies Ω′1(t) and Ω′a(t), respectively. As a result, the state of the system turns into

|Ψ4=α1|g1g1AB+α2|g1g2AB+α3|g0g2AB+α4|g0g1AB,
which is equal to a CNOT gate operation. Figure 2 represents these four steps for constructing the CNOT gate.

 figure: Fig. 2

Fig. 2 Schematic representation of the four steps of the construction of the CNOT gate. The initial state is denoted by an empty circle and the final state is represented by a full black circle.

Download Full Size | PPT Slide | PDF

In the following we will explain how to construct the shortcuts to adiabatic passage for implementing the CNOT gate in detail. For the first step, the Hamiltonian for the cavity-fiber-cavity system as shown in Fig. 1 in the interaction picture can be written as ( = 1)

H1=Ha-l+Ha-c-f,
Ha-l=Ω11(t)|eBg1|+Ωa1(t)|eBa|+H.c.,
Ha-c-f=λAaA|eAg0|+λBaB|eBg0|+ηb(aA+aB)+H.c.,
where η is the coupling strength between cavity mode and the fiber mode, b is the annihilation operator for the fiber mode, aA(B) is the annihilation operator for the corresponding cavity field, and λA(B) is the coupling strength between the corresponding cavity mode and the trapped atom. For convenience, we assume λA = λB = λ, and aA = aB = a. For the initial state |ϕ1〉 = |g1g1AB|000〉AfB (here |000〉AfB denotes the cavities A and B as well as the fiber are vacuum state), the system evolves in the subspace which can be spanned by the vectors
|ϕ1=|g1g1AB|000AfB,|ϕ2=|g1eAB|000AfB,|ϕ3=|g1aAB|000AfB,|ϕ4=|g1g0AB|001AfB,|ϕ5=|g1g0AB|010AfB,|ϕ6=|g1g0AB|100AfB.
Setting Ω11(t), Ωa1(t) ≪ η, λA(B), the Hilbert subspace can be divided into five invariant Zeno subspaces [32, 33]:
ΓP0={|ϕ1,|ϕ3},ΓP1={|ψ1},ΓP2={|ψ2},ΓP3={|ψ3},ΓP4={|ψ4},
with the eigenvalues E0 = 0, E1=(AB)/2, E2=(AB)/2, E3=(A+B)/2, and E4=(A+B)/2, where A = λ2 + 2η2, B=λ4+4η4, and the corresponding projection Piα=|αα|, (|α〉 ∈ ΓPi). Here
|ψ1=(C+B)AB22λη2|ϕ2+λ2B2η2|ϕ4AB2η|ϕ5+|ϕ6,|ψ2=(C+B)AB22λη2|ϕ2+λ2B2η2|ϕ4+AB2η|ϕ5+|ϕ6,|ψ3=(C+B)A+B22λη2|ϕ2+λ2+B2η2|ϕ4A+B2η|ϕ5+|ϕ6,|ψ4=(C+B)A+B22λη2|ϕ2+λ2+B2η2|ϕ4+A+B2η|ϕ5+|ϕ6,
C = 2η2λ. Under the above conditions, the system Hamiltonian can be rewritten as the following form [33]:
H1i,α,β(EiPiα+PiαHalPiβ)=E1|ψ1ψ1|+E2|ψ2ψ2|+E3|ψ3ψ3|+E4|ψ4ψ4|,
which indicates the transition between |ϕ1〉 ↔ |ϕ2〉 can be inhibited.

For another initial state |ϕ′1〉 = |g0g1AB|000〉AfB, the subspace can be spanned by the vectors

|ϕ1=|g0g1AB|000AfB,|ϕ2=|g0eAB|000AfB,|ϕ3=|g0aAB|000AfB,|ϕ4=|g0g0AB|001AfB,|ϕ5=|g0g0AB|010AfB,|ϕ6=|g0g0AB|100AfB,|ϕ7=|eg0AB|000AfB.
For the condition Ω11(t), Ωa1(t) ≪ η, λA(B), the Hilbert subspace in this system can be divided into five invariant Zeno subspaces
ΓP0={|ϕ1,|ψ0,|ϕ3},ΓP1={|ψ1},ΓP2={|ψ2},ΓP3={|ψ3},ΓP4={|ψ4},
with the eigenvalues E′0 = 0, E′1 = −λ, E′2 = λ, E3=A, and E4=A. Here
|ψ0=ηA(|ϕ2λη|ϕ5+|ϕ7),|ψ1=12(|ϕ2+|ϕ4|ϕ6+|ϕ7),|ψ2=12(|ϕ2|ϕ4+|ϕ6+|ϕ7),|ψ3=λ2A(|ϕ2Aλ|ϕ4+2ηλ|ϕ5Aλ|ϕ6+|ϕ7),|ψ4=λ2A(|ϕ2+Aλ|ϕ4+2ηλ|ϕ5+Aλ|ϕ6+|ϕ7),
and the corresponding projection Piα=|αα|, (|α ∈ Γ′Pi). With the above conditions, the effective Hamiltonian in this system can be rewritten as
Heff=ηA(Ω11(t)|ψ0ϕ1|+Ωa1(t)|ψ0ϕ3|+H.c.).
In order to construct the shortcuts for the gate performing by the dynamics of invariant based inverse engineering, we need to find out the Hermitian invariant operator I(t), which satisfies ih¯I(t)t=[Heff(t),I(t)]. Since Heff(t) possesses SU(2) dynamical symmetry, I(t) can be easily given by [37, 38]
I(t)=χ(cosνsinβ|ψ0ϕ1|+cosνcosβ|ψ0ϕ3|+isinν|ϕ3ϕ1|+H.c.),
where χ is an arbitrary constant with units of frequency to keep I(t) with dimensions of energy, ν and β are time-dependent auxiliary parameters which satisfy the equations
ν˙=ηA[Ω11(t)cosβΩa1(t)sinβ],β˙=ηAtanν[Ωa1(t)cosβ+Ω11(t)sinβ].
Then we can derive the expressions of Ω11(t) and Ωa1(t) easily as follows:
Ω11(t)=Aη(β˙cotνsinβ+ν˙cosβ),Ωa1(t)=Aη(β˙cotνcosβν˙sinβ).
The solution of Shrödinger equation ih̄∂|Ψ(t)〉/∂t = Heff(t)|Ψ(t)〉 with respect to the instantaneous eigenstates of I(t) can be written as |Ψ(t)〉 = ∑n=0,± Cnenn(t)〉, where θn(t) is the Lewis-Riesenfeld phase in Eq. (5), Cn = 〈Φn(0)|ϕ′1〉, and |Φn(t)〉 is the eigenstate of the invariant I(t)
|Φ0(t)=cosνcosβ|ϕ1isinν|ψ0cosνsinβ|ϕ3,|Φ±(t)=12[(sinνcosβ±isinβ)|ϕ1+icosν|ψ0(sinνsinβicosβ)|ϕ3].
In order to transfer the population from state |ϕ′1〉 to −|ϕ′3〉, we choose the parameters as
ν(t)=ε,β(t)=πt2tf,
where ε is a time-independent small value and tf is the total pulse duration. After the precise calculation, we can easily obtain
Ω11(t)=Aπη2tfcotεsinπt2tf,Ωa1(t)=Aπη2tfcotεcosπt2tf.
When t = tf,
|Ψ(tf)=sinεsinθ|ϕ1+(isinεcosε+isinεcosεcosθ)|ψ0+(cos2εsin2εcosθ)|ϕ3,
where θ = π/(2sinε) = |θ±| (θ± are the Lewis-Riesenfeld phases). We let θ satisfy the condition θ = 2 (N = 1, 2, 3...), then |Ψ(tf)〉 =−|ϕ′3〉 = −|g0aAB|000〉AfB can be obtained. Thus we can obtain |Ψ1〉 = α1|g1g1AB +α2|g1g2ABα3|g0aAB + α4|g0g2AB successfully. Since the lasers have no effect on the transition |g2B ↔ |eB in this step, the states |g0g2AB and |g1g2AB〉 will not change any more.

The second step is similar to the first step. The Hamiltonian in the interaction picture in this step can be written as ( = 1)

H2=Ω21(t)|eBg2|+Ω12(t)|eB|g1+λAaA|eAg0|+λBaB|eBg0|+ηb(aA+aB)+H.c..
By using the same method as the first step, we can obtain
Ω21(t)=Aπη2tfcotεsinπt2tf,Ω12(t)=Aπη2tfcotεcosπt2tf.
When the conditions t = tf and θ = 2 (N = 1, 2, 3...) are met, the population of state |g0g2AB is completely transferred to −|g0g1AB in this step. Thus |Ψ2〉 = α1|g1g1AB + α2|g1g2ABα3|g0aABα4|g0g1AB is achieved.

In the third step, the Hamiltonian in interaction picture reads ( = 1)

H3=Ωa2(t)|eBa|+Ω22(t)|eBg2|+λAaA|eAg0|+λBaB|eBg0|+ηb(aA+aB)+H.c..
We can obtain
Ωa2(t)=Aπη2tfcotεsinπt2tf,Ω22(t)=Aπη2tfcotεcosπt2tf.
Similarly, when the conditions t = tf and θ = 2 (N = 1, 2, 3...) are satisfied, the population of state |g0aAB is completely transferred to −|g0g2AB in this step. Then |Ψ3〉 = α1|g1g1AB + α2|g1g2AB +α3|g0g2ABα4|g0g1AB is achieved.

The fourth step is a little different from the other three steps. In this step, the Hamiltonian in interaction picture can be written as ( = 1)

H4=Ω1(t)|eBg1|+Ωa(t)|eBa|+λAaA|eAg0|+λBaB|eBg0|+ηb(aA+aB)+H.c..
At first, by using the same way as the three steps above we can obtain
Ω1(t)=Aπη2tfcotεsinπt2tf,Ωa(t)=Aπη2tfcotεsinπt2tf.
Here we choose t = 2tf and θ = 2 (N = 1, 2, 3...) in this step, then the population of state |g0g1AB is completely transferred to −|g0g1AB. That the state |Ψ4〉 = α1|g1g1AB + α2|g1g2AB + α3|g0g2AB + α4|g0g1AB is obtained. By now, we implement the CNOT gate successfully.

4. Numerical simulations and feasibility analysis

In the following, we present the numerical validation of the mechanism proposed for the construction of the CNOT gate. Figure 3(a) shows the time-dependence laser pulse Ωi(t)/λ as a function of λt for a fixed value ε = 0.25, λA = λB = λ and tf = 15/λ. The temporal evolution of the initial qubit states |g0g1AB and |g0g2AB have been given in Fig 3(b) and (c), respectively. From Fig. 3(b) and (c) we can see that, when the control qubit A is |g0A, the target qubit B will change its state completely, that is the function of a CNOT gate. Figure 4 shows the fidelity F = 〈Ψ0|Ψ(tf)〉 as a function of ε and λtf for the initial state |Ψ0〉. From Fig. 4 we can see that, the effect of tf on fidelity can be ignored, and we choose tf = 15/λ in our scheme. Figure 4 shows that the fidelity oscillates with respect to ε. In our scheme, we choose ε = 0.25 and the fidelity can be higher than 99%. Whether a scheme is available largely depends on the robustness to the loss and decoherence. It is indispensable to examine the robustness of our shortcuts schemes against decoherence mechanisms including the cavity decay, the fiber decay, and the atomic spontaneous emission. In the following, we take the effect of decoherence into account on this model. The corresponding master equation for the whole system density matrix ρ(t) has the following form:

ρ˙=i[Htotal,ρ]κf2[bbρ2bρb+ρbb]j=A,Bκj2[ajajρ2ajρaj+ρajaj]j=A,Bk=g0,g1,g2,aγj2[σej,ejρ2σkj,ejρσej,kj+ρσej,ej],
where Htotal = H1 + H2 + H3 + H4. κA(B) is the cavity A (B) decay rate, γA(B) is A (B) atomic spontaneous emission rate from the excited state |eA(B) to the ground state |kA(B)(k = g0, g1, g2, a), respectively. σej,kj = |ej〉 〈kj (j = A, B). For simplicity, we assume κA = κB = κf = κ, γA = γB = γ/5. The initial condition ρ(0) = |Ψ0〉 〈Ψ0|. Figure 5 shows the fidelity of the CNOT gate versus the dimensionless parameter γ/λ with different values of κ by numerically solving the master Eq. (36). From Fig. 5 we can see that the fidelity of the CNOT gate is higher than 98.4% even when the values of γ and κ are comparable to γ = 0.1λ and κ = 0.1λ. It shows that the CNOT gate in our scheme is robust against decoherence due to cavity decay, the fiber decay as well as the atomic spontaneous emission.

 figure: Fig. 3

Fig. 3 (a) Temporal profile of the time dependence Rabi frequencies Ωi(t)/λ versus λt with Ωi(t) = Ω11(t) (dash blue line), Ωa1(t) (solid blue line), Ω21(t) (dash red line), Ω12(t) (solid red line), Ω22(t) (solid green line), Ωa2(t) (dash green line), Ω′a(t) (solid violet line) and Ω1(t) (dash violet line). (b) Time evolutions of the populations of the corresponding system states with the initial states |g0g1AB. (c) Time evolutions of the populations of the corresponding system states with the initial state |g0g2AB. The system parameters are set to be ε = 0.25, λA = λB = λ and tf = 15/λ.

Download Full Size | PPT Slide | PDF

 figure: Fig. 4

Fig. 4 The fidelity of the CNOT gate versus ε and λtf regardless of the decoherence.

Download Full Size | PPT Slide | PDF

 figure: Fig. 5

Fig. 5 The effect of atomic spontaneous emission γ on the fidelity of the CNOT gate with different values of the cavity decay κ.

Download Full Size | PPT Slide | PDF

Now we give a brief analysis of the feasibility in experiment for this scheme. We may employ a cesium atom for our proposal, the lowest state |a〉 corresponds to F = 3, m = 2 hyperfine state of 62S1/2 electronic ground state, |g1〉 corresponds to F = 3, m = 3 hyperfine state of 62S1/2 electronic ground state, |g2〉 corresponds to F = 4, m = 3 hyperfine state of 62S1/2 electronic ground state, |g0〉 corresponds to F = 4, m = 4 hyperfine state of 62S1/2 electronic ground state, and the excited state |e〉 corresponds to F = 4, m = 3 hyperfine state of 62P1/2 electronic ground state. In recent experiments [39], the suitable paramater of toroidal microcavities for strong-coupling cavity QED has been investigated and the relevant cavity QED parameter (λ, κ, γ) = 2π × (750, 3.5, 2.62) MHz is realizable. With these parameters the fidelity of the CNOT gate is close to 99%. So our scheme is robust against both the cavity decay, the fiber decay as well as the atomic spontaneous emission and could be very promising and useful for quantum information processing.

5. Conclusion

In conclusion, we have proposed a promising scheme to directly implement a CNOT gate in cavity-fiber-cavity systems which have advantages in long-distant quantum information processing and quantum computation. During the whole process the system keeps in a Zeno subspace without exciting the cavity field and the fiber, and all the atoms are in the ground states, thus the scheme is robust against the cavity, fiber and atomic decay. The shortcut scheme we propose is stable with respect to the systematic errors [21], and the control process in our scheme is robustness with respect to the pulse area, to the detuning, or to both parameters by a single-shot shaped pulse [40]. The numerical simulation results denote that our scheme is very robust against the decoherence caused by atomic spontaneous emission, cavity decay and fiber decay, so it can be a more reliable choice in experiment. The analysis in our scheme can also extend to other physical systems, such as ion trap, superconducting devices and quantum dot. Our scheme is promising and might enlighten applications of shortcuts to adiabatic passage for quantum CNOT gate in long-distant experimental systems. It is possible to extend this analysis to more complicated quantum gates, such as Toffoli gate and Multiqubit controlled unitary gate.

Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grant Nos. 11464046 and 61465013.

References and links

1. D. P. DiVincenzo, “Two-bit gates are universal for quantum computation,” Phys. Rev. A 51, 1015–1022 (1995). [CrossRef]   [PubMed]  

2. A. Barenco, C. H. Bennett, R. Cleve, D. P. DiVincenzo, N. Margolus, P. Shor, T. Sleator, J. Smolin, and H. Weinfurter, “Elementary gates for quantum computation,” Phys. Rev. A 52, 3457–3467 (1995). [CrossRef]   [PubMed]  

3. Y. F. Huang, X. F. Ren, Y. S. Zhang, L. M. Duan, and G. C. Guo, “Experimental Teleportation of a Quantum Controlled-NOT Gate,” Phys. Rev. Lett. 93, 240501 (2004). [CrossRef]  

4. S. B. Zheng, “Implementation of Toffoli gates with a single asymmetric Heisenberg XY interaction,” Phys. Rev. A 87, 042318 (2013). [CrossRef]  

5. B. Qiao, H. E. Ruda, and J. Wang, “Multiqubit computing and error-avoiding codes in subspace using quantum dots,” J. Appl. Phys. 91, 2524–2529 (2002). [CrossRef]  

6. J. I. Cirac and P. Zoller, “Quantum computations with cold trapped ions,” Phys. Rev. Lett. 74, 4091–4094 (1995). [CrossRef]   [PubMed]  

7. M. Šašura and V. Bužek, “Multiparticle entanglement with quantum logic networks: Application to cold trapped ions,” Phys. Rev. A 64, 012305 (2001). [CrossRef]  

8. C. P. Yang and S. Chun, “Possible realization of entanglement, logical gates, and quantum-information transfer with superconducting-quantum-interference-device qubits in cavity QED,” Phys. Rev. A 67, 042311 (2003). [CrossRef]  

9. C. P. Yang and S. Han, “Realization of an n-qubit controlled-U gate with superconducting quantum interference devices or atoms in cavity QED,” Phys. Rev. A 73, 032317 (2006). [CrossRef]  

10. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, 2000).

11. D. P. DiVincenzo, S. L. Braunstein, and H. K. Lo, Scalable Quantum Computers (Wiley- VCH, Berlin, 2001).

12. N. A. Gershenfeld and I. L. Chuang, “Bulk spin-resonance quantum computation,” Science 275, 350–356 (1997). [CrossRef]   [PubMed]  

13. S. B. Zheng, “Nongeometric conditional phase shift via adiabatic evolution of dark eigenstates: a new approach to quantum computation,” Phys. Rev. Lett. 95, 080502 (2005). [CrossRef]   [PubMed]  

14. D. Sugny, L. Bomble, T. Ribeyre, O. Dulieu, and M. Desouter-Lecomte, “Rovibrational controlled-NOT gates using optimized stimulated Raman adiabatic passage techniques and optimal control theory,” Phys. Rev. A 80, 042325 (2009). [CrossRef]  

15. N. Sangouard, X. Lacour, S. Guérin, and H. R. Jauslin, “CNOT gate by adiabatic passage with an optical cavity,” Eur. Phys. J. D 37, 451–456 (2006). [CrossRef]  

16. X. Q. Shao, L. Chen, S. Zhang, and K. H. Yeon, “Fast CNOT gate via quantum Zeno dynamics,” J. Phys. B: At. Mol. Opt. Phys. 42, 165507 (2009). [CrossRef]  

17. M. Demirplak and S. A. Rice, “Adiabatic population transfer with control fields,” J. Phys. Chem. A 107, 9937–9945 (2003). [CrossRef]  

18. M. Demirplak and S. A. Rice, “Assisted adiabatic passage revisited,” J. Phys. Chem. B 109, 6838–6844 (2005). [CrossRef]  

19. M. Demirplak and S. A. Rice, “On the consistency, extremal, and global properties of counterdiabatic fields,” J. Chem. Phys. 109, 154111 (2008). [CrossRef]  

20. E. Torrontegui, S. Ibáñez, S. Martínez-Garaot, M. Modugno, A. del Campo, D. Guery-Odelin, A. Ruschhaupt, X. Chen, and J. G. Muga, “Shortcuts to adiabaticity,” Adv. At. Mol. Opt. Phys , 62, 117–169 (2013). [CrossRef]  

21. A. Ruschhaupt, X. Chen, D. Alonso, and J. G. Muga, “Optimally robust shortcuts to population inversion in two-level quantum systems,” New J. Phys. 14, 093040 (2012). [CrossRef]  

22. X. Chen, I. Lizuain, A. Ruschhaupt, D. Guéry-Odelin, and J. G. Muga, “Shortcut to Adiabatic Passage in Two- and Three-Level Atoms,” Phys. Rev. Lett. 105, 123003 (2010). [CrossRef]   [PubMed]  

23. K. H. Hoffmann, P. Salamon, Y. Rezek, and R. Kosloff, “Time-optimal controls for frictionless cooling in harmonic traps,” Europhys. Lett. 96, 60015 (2011). [CrossRef]  

24. A. del Campo, “Shortcuts to adiabaticity by counter-diabatic driving,” Phys. Rev. Lett. 111, 100502 (2013). [CrossRef]  

25. M. Lu, Y. Xia, L. T. Shen, J. Song, and N. B. An, “Shortcuts to adiabatic passage for population transfer and maximum entanglement creation between two atoms in a cavity,” Phys. Rev. A 89, 012326 (2014). [CrossRef]  

26. Y. H. Chen, Y. Xia, Q. Q. Chen, and J. Song, “Efficient shortcuts to adiabatic passage for fast population transfer in multiparticle systems,” Phys. Rev. A 89, 033856 (2014). [CrossRef]  

27. A. Walther, F. Ziesel, T. Ruster, S. T. Dawkins, K. Ott, M. Hettrich, K. Singer, F. Schmidt-Kaler, and U. Poschinger, “Controlling fast transport of cold trapped ions,” Phys. Rev. Lett. 109, 080501 (2012). [CrossRef]   [PubMed]  

28. J. F. Schaff, X. L. Song, P. Capuzzi, P. Vignolo, and G. Labeyrie, “Shortcut to adiabaticity for an interacting Bose-Einstein condensate,” Europhys. Lett. 93, 23001 (2011). [CrossRef]  

29. Y. H. Cheng, Y. Xia, Q. Q. Chen, and J. Song, “Fast and noise-resistant implementation of quantum phase gates and creation of quantum entangled states,” Phys. Rev. A 91, 012325 (2015). [CrossRef]  

30. Y. Liang, Q. C. Wu, S. L. Su, X. Ji, and S. Zhang, “Shortcuts to adiabatic passage for multiqubit controlled-phase gate,” Phys. Rev. A 91, 032304 (2015). [CrossRef]  

31. P. Facchi, V. Gorini, G. Marmo, S. Pascazio, and E. C. G. Sudarshan, “Quantum Zeno dynamics,” Phys. Lett. A 275, 12–19 (2000). [CrossRef]  

32. P. Facchi and S. Pascazio, “Quantum Zeno Subspaces,” Phys. Rev. Lett. 89, 080401 (2002). [CrossRef]   [PubMed]  

33. P. Facchi, G. Marmo, and S. Pascazio, “Quantum Zeno dynamics and quantum Zeno subspaces,” J. Phys: Conf. Ser. 196, 012017 (2009).

34. H. R. Lewis and W. B. Riesenfeld, “An exact quantum theory of the time-dependent harmonic oscillator and of a charged particle in a time-dependent electromagnetic field,” J. Math. Phys. 10, 1458 (1969). [CrossRef]  

35. M. A. Lohe, “Exact time dependence of solutions to the time-dependent Schrodinger equation,” J. Phys. A: Math. and Theor. 42, 035307 (2009). [CrossRef]  

36. A. Serafini, S. Mancini, and S. Bose, “Distributed quantum computation via optical fibers,” Phys. Rev. Lett. 96, 010503 (2006). [CrossRef]   [PubMed]  

37. A. Mostafazadeh, Dynamical Invariants, Adiabatic Approximation, and the Geometric Phase (Nova Science Pub Incorporated2001).

38. X. Chen, E. Torrontegui, and J. G. Muga, “Lewis-Riesenfeld invariants and transitionless quantum driving,” Phys. Rev. A 83, 062116 (2011). [CrossRef]  

39. S. M. Spillane, T. J. Kippenberg, K. J. Vahala, K. W. Goh, E. Wilcut, and H. J. Kimble, “Ultrahigh-Q toroidal microresonators for cavity quantum electrodynamics,” Phys. Rev. A 71, 013817 (2005). [CrossRef]  

40. D. Daems, A. Ruschhaupt, D. Sugny, and S. Guérin, “Robust quantum control by a single-shot shaped pulse,” Phys. Rev. Lett. 111, 050404 (2013). [CrossRef]   [PubMed]  

References

  • View by:

  1. D. P. DiVincenzo, “Two-bit gates are universal for quantum computation,” Phys. Rev. A 51, 1015–1022 (1995).
    [Crossref] [PubMed]
  2. A. Barenco, C. H. Bennett, R. Cleve, D. P. DiVincenzo, N. Margolus, P. Shor, T. Sleator, J. Smolin, and H. Weinfurter, “Elementary gates for quantum computation,” Phys. Rev. A 52, 3457–3467 (1995).
    [Crossref] [PubMed]
  3. Y. F. Huang, X. F. Ren, Y. S. Zhang, L. M. Duan, and G. C. Guo, “Experimental Teleportation of a Quantum Controlled-NOT Gate,” Phys. Rev. Lett. 93, 240501 (2004).
    [Crossref]
  4. S. B. Zheng, “Implementation of Toffoli gates with a single asymmetric Heisenberg XY interaction,” Phys. Rev. A 87, 042318 (2013).
    [Crossref]
  5. B. Qiao, H. E. Ruda, and J. Wang, “Multiqubit computing and error-avoiding codes in subspace using quantum dots,” J. Appl. Phys. 91, 2524–2529 (2002).
    [Crossref]
  6. J. I. Cirac and P. Zoller, “Quantum computations with cold trapped ions,” Phys. Rev. Lett. 74, 4091–4094 (1995).
    [Crossref] [PubMed]
  7. M. Šašura and V. Bužek, “Multiparticle entanglement with quantum logic networks: Application to cold trapped ions,” Phys. Rev. A 64, 012305 (2001).
    [Crossref]
  8. C. P. Yang and S. Chun, “Possible realization of entanglement, logical gates, and quantum-information transfer with superconducting-quantum-interference-device qubits in cavity QED,” Phys. Rev. A 67, 042311 (2003).
    [Crossref]
  9. C. P. Yang and S. Han, “Realization of an n-qubit controlled-U gate with superconducting quantum interference devices or atoms in cavity QED,” Phys. Rev. A 73, 032317 (2006).
    [Crossref]
  10. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, 2000).
  11. D. P. DiVincenzo, S. L. Braunstein, and H. K. Lo, Scalable Quantum Computers (Wiley- VCH, Berlin, 2001).
  12. N. A. Gershenfeld and I. L. Chuang, “Bulk spin-resonance quantum computation,” Science 275, 350–356 (1997).
    [Crossref] [PubMed]
  13. S. B. Zheng, “Nongeometric conditional phase shift via adiabatic evolution of dark eigenstates: a new approach to quantum computation,” Phys. Rev. Lett. 95, 080502 (2005).
    [Crossref] [PubMed]
  14. D. Sugny, L. Bomble, T. Ribeyre, O. Dulieu, and M. Desouter-Lecomte, “Rovibrational controlled-NOT gates using optimized stimulated Raman adiabatic passage techniques and optimal control theory,” Phys. Rev. A 80, 042325 (2009).
    [Crossref]
  15. N. Sangouard, X. Lacour, S. Guérin, and H. R. Jauslin, “CNOT gate by adiabatic passage with an optical cavity,” Eur. Phys. J. D 37, 451–456 (2006).
    [Crossref]
  16. X. Q. Shao, L. Chen, S. Zhang, and K. H. Yeon, “Fast CNOT gate via quantum Zeno dynamics,” J. Phys. B: At. Mol. Opt. Phys. 42, 165507 (2009).
    [Crossref]
  17. M. Demirplak and S. A. Rice, “Adiabatic population transfer with control fields,” J. Phys. Chem. A 107, 9937–9945 (2003).
    [Crossref]
  18. M. Demirplak and S. A. Rice, “Assisted adiabatic passage revisited,” J. Phys. Chem. B 109, 6838–6844 (2005).
    [Crossref]
  19. M. Demirplak and S. A. Rice, “On the consistency, extremal, and global properties of counterdiabatic fields,” J. Chem. Phys. 109, 154111 (2008).
    [Crossref]
  20. E. Torrontegui, S. Ibáñez, S. Martínez-Garaot, M. Modugno, A. del Campo, D. Guery-Odelin, A. Ruschhaupt, X. Chen, and J. G. Muga, “Shortcuts to adiabaticity,” Adv. At. Mol. Opt. Phys,  62, 117–169 (2013).
    [Crossref]
  21. A. Ruschhaupt, X. Chen, D. Alonso, and J. G. Muga, “Optimally robust shortcuts to population inversion in two-level quantum systems,” New J. Phys. 14, 093040 (2012).
    [Crossref]
  22. X. Chen, I. Lizuain, A. Ruschhaupt, D. Guéry-Odelin, and J. G. Muga, “Shortcut to Adiabatic Passage in Two- and Three-Level Atoms,” Phys. Rev. Lett. 105, 123003 (2010).
    [Crossref] [PubMed]
  23. K. H. Hoffmann, P. Salamon, Y. Rezek, and R. Kosloff, “Time-optimal controls for frictionless cooling in harmonic traps,” Europhys. Lett. 96, 60015 (2011).
    [Crossref]
  24. A. del Campo, “Shortcuts to adiabaticity by counter-diabatic driving,” Phys. Rev. Lett. 111, 100502 (2013).
    [Crossref]
  25. M. Lu, Y. Xia, L. T. Shen, J. Song, and N. B. An, “Shortcuts to adiabatic passage for population transfer and maximum entanglement creation between two atoms in a cavity,” Phys. Rev. A 89, 012326 (2014).
    [Crossref]
  26. Y. H. Chen, Y. Xia, Q. Q. Chen, and J. Song, “Efficient shortcuts to adiabatic passage for fast population transfer in multiparticle systems,” Phys. Rev. A 89, 033856 (2014).
    [Crossref]
  27. A. Walther, F. Ziesel, T. Ruster, S. T. Dawkins, K. Ott, M. Hettrich, K. Singer, F. Schmidt-Kaler, and U. Poschinger, “Controlling fast transport of cold trapped ions,” Phys. Rev. Lett. 109, 080501 (2012).
    [Crossref] [PubMed]
  28. J. F. Schaff, X. L. Song, P. Capuzzi, P. Vignolo, and G. Labeyrie, “Shortcut to adiabaticity for an interacting Bose-Einstein condensate,” Europhys. Lett. 93, 23001 (2011).
    [Crossref]
  29. Y. H. Cheng, Y. Xia, Q. Q. Chen, and J. Song, “Fast and noise-resistant implementation of quantum phase gates and creation of quantum entangled states,” Phys. Rev. A 91, 012325 (2015).
    [Crossref]
  30. Y. Liang, Q. C. Wu, S. L. Su, X. Ji, and S. Zhang, “Shortcuts to adiabatic passage for multiqubit controlled-phase gate,” Phys. Rev. A 91, 032304 (2015).
    [Crossref]
  31. P. Facchi, V. Gorini, G. Marmo, S. Pascazio, and E. C. G. Sudarshan, “Quantum Zeno dynamics,” Phys. Lett. A 275, 12–19 (2000).
    [Crossref]
  32. P. Facchi and S. Pascazio, “Quantum Zeno Subspaces,” Phys. Rev. Lett. 89, 080401 (2002).
    [Crossref] [PubMed]
  33. P. Facchi, G. Marmo, and S. Pascazio, “Quantum Zeno dynamics and quantum Zeno subspaces,” J. Phys: Conf. Ser. 196, 012017 (2009).
  34. H. R. Lewis and W. B. Riesenfeld, “An exact quantum theory of the time-dependent harmonic oscillator and of a charged particle in a time-dependent electromagnetic field,” J. Math. Phys. 10, 1458 (1969).
    [Crossref]
  35. M. A. Lohe, “Exact time dependence of solutions to the time-dependent Schrodinger equation,” J. Phys. A: Math. and Theor. 42, 035307 (2009).
    [Crossref]
  36. A. Serafini, S. Mancini, and S. Bose, “Distributed quantum computation via optical fibers,” Phys. Rev. Lett. 96, 010503 (2006).
    [Crossref] [PubMed]
  37. A. Mostafazadeh, Dynamical Invariants, Adiabatic Approximation, and the Geometric Phase (Nova Science Pub Incorporated2001).
  38. X. Chen, E. Torrontegui, and J. G. Muga, “Lewis-Riesenfeld invariants and transitionless quantum driving,” Phys. Rev. A 83, 062116 (2011).
    [Crossref]
  39. S. M. Spillane, T. J. Kippenberg, K. J. Vahala, K. W. Goh, E. Wilcut, and H. J. Kimble, “Ultrahigh-Q toroidal microresonators for cavity quantum electrodynamics,” Phys. Rev. A 71, 013817 (2005).
    [Crossref]
  40. D. Daems, A. Ruschhaupt, D. Sugny, and S. Guérin, “Robust quantum control by a single-shot shaped pulse,” Phys. Rev. Lett. 111, 050404 (2013).
    [Crossref] [PubMed]

2015 (2)

Y. H. Cheng, Y. Xia, Q. Q. Chen, and J. Song, “Fast and noise-resistant implementation of quantum phase gates and creation of quantum entangled states,” Phys. Rev. A 91, 012325 (2015).
[Crossref]

Y. Liang, Q. C. Wu, S. L. Su, X. Ji, and S. Zhang, “Shortcuts to adiabatic passage for multiqubit controlled-phase gate,” Phys. Rev. A 91, 032304 (2015).
[Crossref]

2014 (2)

M. Lu, Y. Xia, L. T. Shen, J. Song, and N. B. An, “Shortcuts to adiabatic passage for population transfer and maximum entanglement creation between two atoms in a cavity,” Phys. Rev. A 89, 012326 (2014).
[Crossref]

Y. H. Chen, Y. Xia, Q. Q. Chen, and J. Song, “Efficient shortcuts to adiabatic passage for fast population transfer in multiparticle systems,” Phys. Rev. A 89, 033856 (2014).
[Crossref]

2013 (4)

E. Torrontegui, S. Ibáñez, S. Martínez-Garaot, M. Modugno, A. del Campo, D. Guery-Odelin, A. Ruschhaupt, X. Chen, and J. G. Muga, “Shortcuts to adiabaticity,” Adv. At. Mol. Opt. Phys,  62, 117–169 (2013).
[Crossref]

A. del Campo, “Shortcuts to adiabaticity by counter-diabatic driving,” Phys. Rev. Lett. 111, 100502 (2013).
[Crossref]

S. B. Zheng, “Implementation of Toffoli gates with a single asymmetric Heisenberg XY interaction,” Phys. Rev. A 87, 042318 (2013).
[Crossref]

D. Daems, A. Ruschhaupt, D. Sugny, and S. Guérin, “Robust quantum control by a single-shot shaped pulse,” Phys. Rev. Lett. 111, 050404 (2013).
[Crossref] [PubMed]

2012 (2)

A. Ruschhaupt, X. Chen, D. Alonso, and J. G. Muga, “Optimally robust shortcuts to population inversion in two-level quantum systems,” New J. Phys. 14, 093040 (2012).
[Crossref]

A. Walther, F. Ziesel, T. Ruster, S. T. Dawkins, K. Ott, M. Hettrich, K. Singer, F. Schmidt-Kaler, and U. Poschinger, “Controlling fast transport of cold trapped ions,” Phys. Rev. Lett. 109, 080501 (2012).
[Crossref] [PubMed]

2011 (3)

J. F. Schaff, X. L. Song, P. Capuzzi, P. Vignolo, and G. Labeyrie, “Shortcut to adiabaticity for an interacting Bose-Einstein condensate,” Europhys. Lett. 93, 23001 (2011).
[Crossref]

K. H. Hoffmann, P. Salamon, Y. Rezek, and R. Kosloff, “Time-optimal controls for frictionless cooling in harmonic traps,” Europhys. Lett. 96, 60015 (2011).
[Crossref]

X. Chen, E. Torrontegui, and J. G. Muga, “Lewis-Riesenfeld invariants and transitionless quantum driving,” Phys. Rev. A 83, 062116 (2011).
[Crossref]

2010 (1)

X. Chen, I. Lizuain, A. Ruschhaupt, D. Guéry-Odelin, and J. G. Muga, “Shortcut to Adiabatic Passage in Two- and Three-Level Atoms,” Phys. Rev. Lett. 105, 123003 (2010).
[Crossref] [PubMed]

2009 (4)

M. A. Lohe, “Exact time dependence of solutions to the time-dependent Schrodinger equation,” J. Phys. A: Math. and Theor. 42, 035307 (2009).
[Crossref]

P. Facchi, G. Marmo, and S. Pascazio, “Quantum Zeno dynamics and quantum Zeno subspaces,” J. Phys: Conf. Ser. 196, 012017 (2009).

D. Sugny, L. Bomble, T. Ribeyre, O. Dulieu, and M. Desouter-Lecomte, “Rovibrational controlled-NOT gates using optimized stimulated Raman adiabatic passage techniques and optimal control theory,” Phys. Rev. A 80, 042325 (2009).
[Crossref]

X. Q. Shao, L. Chen, S. Zhang, and K. H. Yeon, “Fast CNOT gate via quantum Zeno dynamics,” J. Phys. B: At. Mol. Opt. Phys. 42, 165507 (2009).
[Crossref]

2008 (1)

M. Demirplak and S. A. Rice, “On the consistency, extremal, and global properties of counterdiabatic fields,” J. Chem. Phys. 109, 154111 (2008).
[Crossref]

2006 (3)

N. Sangouard, X. Lacour, S. Guérin, and H. R. Jauslin, “CNOT gate by adiabatic passage with an optical cavity,” Eur. Phys. J. D 37, 451–456 (2006).
[Crossref]

C. P. Yang and S. Han, “Realization of an n-qubit controlled-U gate with superconducting quantum interference devices or atoms in cavity QED,” Phys. Rev. A 73, 032317 (2006).
[Crossref]

A. Serafini, S. Mancini, and S. Bose, “Distributed quantum computation via optical fibers,” Phys. Rev. Lett. 96, 010503 (2006).
[Crossref] [PubMed]

2005 (3)

S. M. Spillane, T. J. Kippenberg, K. J. Vahala, K. W. Goh, E. Wilcut, and H. J. Kimble, “Ultrahigh-Q toroidal microresonators for cavity quantum electrodynamics,” Phys. Rev. A 71, 013817 (2005).
[Crossref]

S. B. Zheng, “Nongeometric conditional phase shift via adiabatic evolution of dark eigenstates: a new approach to quantum computation,” Phys. Rev. Lett. 95, 080502 (2005).
[Crossref] [PubMed]

M. Demirplak and S. A. Rice, “Assisted adiabatic passage revisited,” J. Phys. Chem. B 109, 6838–6844 (2005).
[Crossref]

2004 (1)

Y. F. Huang, X. F. Ren, Y. S. Zhang, L. M. Duan, and G. C. Guo, “Experimental Teleportation of a Quantum Controlled-NOT Gate,” Phys. Rev. Lett. 93, 240501 (2004).
[Crossref]

2003 (2)

M. Demirplak and S. A. Rice, “Adiabatic population transfer with control fields,” J. Phys. Chem. A 107, 9937–9945 (2003).
[Crossref]

C. P. Yang and S. Chun, “Possible realization of entanglement, logical gates, and quantum-information transfer with superconducting-quantum-interference-device qubits in cavity QED,” Phys. Rev. A 67, 042311 (2003).
[Crossref]

2002 (2)

B. Qiao, H. E. Ruda, and J. Wang, “Multiqubit computing and error-avoiding codes in subspace using quantum dots,” J. Appl. Phys. 91, 2524–2529 (2002).
[Crossref]

P. Facchi and S. Pascazio, “Quantum Zeno Subspaces,” Phys. Rev. Lett. 89, 080401 (2002).
[Crossref] [PubMed]

2001 (1)

M. Šašura and V. Bužek, “Multiparticle entanglement with quantum logic networks: Application to cold trapped ions,” Phys. Rev. A 64, 012305 (2001).
[Crossref]

2000 (1)

P. Facchi, V. Gorini, G. Marmo, S. Pascazio, and E. C. G. Sudarshan, “Quantum Zeno dynamics,” Phys. Lett. A 275, 12–19 (2000).
[Crossref]

1997 (1)

N. A. Gershenfeld and I. L. Chuang, “Bulk spin-resonance quantum computation,” Science 275, 350–356 (1997).
[Crossref] [PubMed]

1995 (3)

J. I. Cirac and P. Zoller, “Quantum computations with cold trapped ions,” Phys. Rev. Lett. 74, 4091–4094 (1995).
[Crossref] [PubMed]

D. P. DiVincenzo, “Two-bit gates are universal for quantum computation,” Phys. Rev. A 51, 1015–1022 (1995).
[Crossref] [PubMed]

A. Barenco, C. H. Bennett, R. Cleve, D. P. DiVincenzo, N. Margolus, P. Shor, T. Sleator, J. Smolin, and H. Weinfurter, “Elementary gates for quantum computation,” Phys. Rev. A 52, 3457–3467 (1995).
[Crossref] [PubMed]

1969 (1)

H. R. Lewis and W. B. Riesenfeld, “An exact quantum theory of the time-dependent harmonic oscillator and of a charged particle in a time-dependent electromagnetic field,” J. Math. Phys. 10, 1458 (1969).
[Crossref]

Alonso, D.

A. Ruschhaupt, X. Chen, D. Alonso, and J. G. Muga, “Optimally robust shortcuts to population inversion in two-level quantum systems,” New J. Phys. 14, 093040 (2012).
[Crossref]

An, N. B.

M. Lu, Y. Xia, L. T. Shen, J. Song, and N. B. An, “Shortcuts to adiabatic passage for population transfer and maximum entanglement creation between two atoms in a cavity,” Phys. Rev. A 89, 012326 (2014).
[Crossref]

Barenco, A.

A. Barenco, C. H. Bennett, R. Cleve, D. P. DiVincenzo, N. Margolus, P. Shor, T. Sleator, J. Smolin, and H. Weinfurter, “Elementary gates for quantum computation,” Phys. Rev. A 52, 3457–3467 (1995).
[Crossref] [PubMed]

Bennett, C. H.

A. Barenco, C. H. Bennett, R. Cleve, D. P. DiVincenzo, N. Margolus, P. Shor, T. Sleator, J. Smolin, and H. Weinfurter, “Elementary gates for quantum computation,” Phys. Rev. A 52, 3457–3467 (1995).
[Crossref] [PubMed]

Bomble, L.

D. Sugny, L. Bomble, T. Ribeyre, O. Dulieu, and M. Desouter-Lecomte, “Rovibrational controlled-NOT gates using optimized stimulated Raman adiabatic passage techniques and optimal control theory,” Phys. Rev. A 80, 042325 (2009).
[Crossref]

Bose, S.

A. Serafini, S. Mancini, and S. Bose, “Distributed quantum computation via optical fibers,” Phys. Rev. Lett. 96, 010503 (2006).
[Crossref] [PubMed]

Braunstein, S. L.

D. P. DiVincenzo, S. L. Braunstein, and H. K. Lo, Scalable Quantum Computers (Wiley- VCH, Berlin, 2001).

Bužek, V.

M. Šašura and V. Bužek, “Multiparticle entanglement with quantum logic networks: Application to cold trapped ions,” Phys. Rev. A 64, 012305 (2001).
[Crossref]

Capuzzi, P.

J. F. Schaff, X. L. Song, P. Capuzzi, P. Vignolo, and G. Labeyrie, “Shortcut to adiabaticity for an interacting Bose-Einstein condensate,” Europhys. Lett. 93, 23001 (2011).
[Crossref]

Chen, L.

X. Q. Shao, L. Chen, S. Zhang, and K. H. Yeon, “Fast CNOT gate via quantum Zeno dynamics,” J. Phys. B: At. Mol. Opt. Phys. 42, 165507 (2009).
[Crossref]

Chen, Q. Q.

Y. H. Cheng, Y. Xia, Q. Q. Chen, and J. Song, “Fast and noise-resistant implementation of quantum phase gates and creation of quantum entangled states,” Phys. Rev. A 91, 012325 (2015).
[Crossref]

Y. H. Chen, Y. Xia, Q. Q. Chen, and J. Song, “Efficient shortcuts to adiabatic passage for fast population transfer in multiparticle systems,” Phys. Rev. A 89, 033856 (2014).
[Crossref]

Chen, X.

E. Torrontegui, S. Ibáñez, S. Martínez-Garaot, M. Modugno, A. del Campo, D. Guery-Odelin, A. Ruschhaupt, X. Chen, and J. G. Muga, “Shortcuts to adiabaticity,” Adv. At. Mol. Opt. Phys,  62, 117–169 (2013).
[Crossref]

A. Ruschhaupt, X. Chen, D. Alonso, and J. G. Muga, “Optimally robust shortcuts to population inversion in two-level quantum systems,” New J. Phys. 14, 093040 (2012).
[Crossref]

X. Chen, E. Torrontegui, and J. G. Muga, “Lewis-Riesenfeld invariants and transitionless quantum driving,” Phys. Rev. A 83, 062116 (2011).
[Crossref]

X. Chen, I. Lizuain, A. Ruschhaupt, D. Guéry-Odelin, and J. G. Muga, “Shortcut to Adiabatic Passage in Two- and Three-Level Atoms,” Phys. Rev. Lett. 105, 123003 (2010).
[Crossref] [PubMed]

Chen, Y. H.

Y. H. Chen, Y. Xia, Q. Q. Chen, and J. Song, “Efficient shortcuts to adiabatic passage for fast population transfer in multiparticle systems,” Phys. Rev. A 89, 033856 (2014).
[Crossref]

Cheng, Y. H.

Y. H. Cheng, Y. Xia, Q. Q. Chen, and J. Song, “Fast and noise-resistant implementation of quantum phase gates and creation of quantum entangled states,” Phys. Rev. A 91, 012325 (2015).
[Crossref]

Chuang, I. L.

N. A. Gershenfeld and I. L. Chuang, “Bulk spin-resonance quantum computation,” Science 275, 350–356 (1997).
[Crossref] [PubMed]

M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, 2000).

Chun, S.

C. P. Yang and S. Chun, “Possible realization of entanglement, logical gates, and quantum-information transfer with superconducting-quantum-interference-device qubits in cavity QED,” Phys. Rev. A 67, 042311 (2003).
[Crossref]

Cirac, J. I.

J. I. Cirac and P. Zoller, “Quantum computations with cold trapped ions,” Phys. Rev. Lett. 74, 4091–4094 (1995).
[Crossref] [PubMed]

Cleve, R.

A. Barenco, C. H. Bennett, R. Cleve, D. P. DiVincenzo, N. Margolus, P. Shor, T. Sleator, J. Smolin, and H. Weinfurter, “Elementary gates for quantum computation,” Phys. Rev. A 52, 3457–3467 (1995).
[Crossref] [PubMed]

Daems, D.

D. Daems, A. Ruschhaupt, D. Sugny, and S. Guérin, “Robust quantum control by a single-shot shaped pulse,” Phys. Rev. Lett. 111, 050404 (2013).
[Crossref] [PubMed]

Dawkins, S. T.

A. Walther, F. Ziesel, T. Ruster, S. T. Dawkins, K. Ott, M. Hettrich, K. Singer, F. Schmidt-Kaler, and U. Poschinger, “Controlling fast transport of cold trapped ions,” Phys. Rev. Lett. 109, 080501 (2012).
[Crossref] [PubMed]

del Campo, A.

A. del Campo, “Shortcuts to adiabaticity by counter-diabatic driving,” Phys. Rev. Lett. 111, 100502 (2013).
[Crossref]

E. Torrontegui, S. Ibáñez, S. Martínez-Garaot, M. Modugno, A. del Campo, D. Guery-Odelin, A. Ruschhaupt, X. Chen, and J. G. Muga, “Shortcuts to adiabaticity,” Adv. At. Mol. Opt. Phys,  62, 117–169 (2013).
[Crossref]

Demirplak, M.

M. Demirplak and S. A. Rice, “On the consistency, extremal, and global properties of counterdiabatic fields,” J. Chem. Phys. 109, 154111 (2008).
[Crossref]

M. Demirplak and S. A. Rice, “Assisted adiabatic passage revisited,” J. Phys. Chem. B 109, 6838–6844 (2005).
[Crossref]

M. Demirplak and S. A. Rice, “Adiabatic population transfer with control fields,” J. Phys. Chem. A 107, 9937–9945 (2003).
[Crossref]

Desouter-Lecomte, M.

D. Sugny, L. Bomble, T. Ribeyre, O. Dulieu, and M. Desouter-Lecomte, “Rovibrational controlled-NOT gates using optimized stimulated Raman adiabatic passage techniques and optimal control theory,” Phys. Rev. A 80, 042325 (2009).
[Crossref]

DiVincenzo, D. P.

A. Barenco, C. H. Bennett, R. Cleve, D. P. DiVincenzo, N. Margolus, P. Shor, T. Sleator, J. Smolin, and H. Weinfurter, “Elementary gates for quantum computation,” Phys. Rev. A 52, 3457–3467 (1995).
[Crossref] [PubMed]

D. P. DiVincenzo, “Two-bit gates are universal for quantum computation,” Phys. Rev. A 51, 1015–1022 (1995).
[Crossref] [PubMed]

D. P. DiVincenzo, S. L. Braunstein, and H. K. Lo, Scalable Quantum Computers (Wiley- VCH, Berlin, 2001).

Duan, L. M.

Y. F. Huang, X. F. Ren, Y. S. Zhang, L. M. Duan, and G. C. Guo, “Experimental Teleportation of a Quantum Controlled-NOT Gate,” Phys. Rev. Lett. 93, 240501 (2004).
[Crossref]

Dulieu, O.

D. Sugny, L. Bomble, T. Ribeyre, O. Dulieu, and M. Desouter-Lecomte, “Rovibrational controlled-NOT gates using optimized stimulated Raman adiabatic passage techniques and optimal control theory,” Phys. Rev. A 80, 042325 (2009).
[Crossref]

Facchi, P.

P. Facchi, G. Marmo, and S. Pascazio, “Quantum Zeno dynamics and quantum Zeno subspaces,” J. Phys: Conf. Ser. 196, 012017 (2009).

P. Facchi and S. Pascazio, “Quantum Zeno Subspaces,” Phys. Rev. Lett. 89, 080401 (2002).
[Crossref] [PubMed]

P. Facchi, V. Gorini, G. Marmo, S. Pascazio, and E. C. G. Sudarshan, “Quantum Zeno dynamics,” Phys. Lett. A 275, 12–19 (2000).
[Crossref]

Gershenfeld, N. A.

N. A. Gershenfeld and I. L. Chuang, “Bulk spin-resonance quantum computation,” Science 275, 350–356 (1997).
[Crossref] [PubMed]

Goh, K. W.

S. M. Spillane, T. J. Kippenberg, K. J. Vahala, K. W. Goh, E. Wilcut, and H. J. Kimble, “Ultrahigh-Q toroidal microresonators for cavity quantum electrodynamics,” Phys. Rev. A 71, 013817 (2005).
[Crossref]

Gorini, V.

P. Facchi, V. Gorini, G. Marmo, S. Pascazio, and E. C. G. Sudarshan, “Quantum Zeno dynamics,” Phys. Lett. A 275, 12–19 (2000).
[Crossref]

Guérin, S.

D. Daems, A. Ruschhaupt, D. Sugny, and S. Guérin, “Robust quantum control by a single-shot shaped pulse,” Phys. Rev. Lett. 111, 050404 (2013).
[Crossref] [PubMed]

N. Sangouard, X. Lacour, S. Guérin, and H. R. Jauslin, “CNOT gate by adiabatic passage with an optical cavity,” Eur. Phys. J. D 37, 451–456 (2006).
[Crossref]

Guery-Odelin, D.

E. Torrontegui, S. Ibáñez, S. Martínez-Garaot, M. Modugno, A. del Campo, D. Guery-Odelin, A. Ruschhaupt, X. Chen, and J. G. Muga, “Shortcuts to adiabaticity,” Adv. At. Mol. Opt. Phys,  62, 117–169 (2013).
[Crossref]

Guéry-Odelin, D.

X. Chen, I. Lizuain, A. Ruschhaupt, D. Guéry-Odelin, and J. G. Muga, “Shortcut to Adiabatic Passage in Two- and Three-Level Atoms,” Phys. Rev. Lett. 105, 123003 (2010).
[Crossref] [PubMed]

Guo, G. C.

Y. F. Huang, X. F. Ren, Y. S. Zhang, L. M. Duan, and G. C. Guo, “Experimental Teleportation of a Quantum Controlled-NOT Gate,” Phys. Rev. Lett. 93, 240501 (2004).
[Crossref]

Han, S.

C. P. Yang and S. Han, “Realization of an n-qubit controlled-U gate with superconducting quantum interference devices or atoms in cavity QED,” Phys. Rev. A 73, 032317 (2006).
[Crossref]

Hettrich, M.

A. Walther, F. Ziesel, T. Ruster, S. T. Dawkins, K. Ott, M. Hettrich, K. Singer, F. Schmidt-Kaler, and U. Poschinger, “Controlling fast transport of cold trapped ions,” Phys. Rev. Lett. 109, 080501 (2012).
[Crossref] [PubMed]

Hoffmann, K. H.

K. H. Hoffmann, P. Salamon, Y. Rezek, and R. Kosloff, “Time-optimal controls for frictionless cooling in harmonic traps,” Europhys. Lett. 96, 60015 (2011).
[Crossref]

Huang, Y. F.

Y. F. Huang, X. F. Ren, Y. S. Zhang, L. M. Duan, and G. C. Guo, “Experimental Teleportation of a Quantum Controlled-NOT Gate,” Phys. Rev. Lett. 93, 240501 (2004).
[Crossref]

Ibáñez, S.

E. Torrontegui, S. Ibáñez, S. Martínez-Garaot, M. Modugno, A. del Campo, D. Guery-Odelin, A. Ruschhaupt, X. Chen, and J. G. Muga, “Shortcuts to adiabaticity,” Adv. At. Mol. Opt. Phys,  62, 117–169 (2013).
[Crossref]

Jauslin, H. R.

N. Sangouard, X. Lacour, S. Guérin, and H. R. Jauslin, “CNOT gate by adiabatic passage with an optical cavity,” Eur. Phys. J. D 37, 451–456 (2006).
[Crossref]

Ji, X.

Y. Liang, Q. C. Wu, S. L. Su, X. Ji, and S. Zhang, “Shortcuts to adiabatic passage for multiqubit controlled-phase gate,” Phys. Rev. A 91, 032304 (2015).
[Crossref]

Kimble, H. J.

S. M. Spillane, T. J. Kippenberg, K. J. Vahala, K. W. Goh, E. Wilcut, and H. J. Kimble, “Ultrahigh-Q toroidal microresonators for cavity quantum electrodynamics,” Phys. Rev. A 71, 013817 (2005).
[Crossref]

Kippenberg, T. J.

S. M. Spillane, T. J. Kippenberg, K. J. Vahala, K. W. Goh, E. Wilcut, and H. J. Kimble, “Ultrahigh-Q toroidal microresonators for cavity quantum electrodynamics,” Phys. Rev. A 71, 013817 (2005).
[Crossref]

Kosloff, R.

K. H. Hoffmann, P. Salamon, Y. Rezek, and R. Kosloff, “Time-optimal controls for frictionless cooling in harmonic traps,” Europhys. Lett. 96, 60015 (2011).
[Crossref]

Labeyrie, G.

J. F. Schaff, X. L. Song, P. Capuzzi, P. Vignolo, and G. Labeyrie, “Shortcut to adiabaticity for an interacting Bose-Einstein condensate,” Europhys. Lett. 93, 23001 (2011).
[Crossref]

Lacour, X.

N. Sangouard, X. Lacour, S. Guérin, and H. R. Jauslin, “CNOT gate by adiabatic passage with an optical cavity,” Eur. Phys. J. D 37, 451–456 (2006).
[Crossref]

Lewis, H. R.

H. R. Lewis and W. B. Riesenfeld, “An exact quantum theory of the time-dependent harmonic oscillator and of a charged particle in a time-dependent electromagnetic field,” J. Math. Phys. 10, 1458 (1969).
[Crossref]

Liang, Y.

Y. Liang, Q. C. Wu, S. L. Su, X. Ji, and S. Zhang, “Shortcuts to adiabatic passage for multiqubit controlled-phase gate,” Phys. Rev. A 91, 032304 (2015).
[Crossref]

Lizuain, I.

X. Chen, I. Lizuain, A. Ruschhaupt, D. Guéry-Odelin, and J. G. Muga, “Shortcut to Adiabatic Passage in Two- and Three-Level Atoms,” Phys. Rev. Lett. 105, 123003 (2010).
[Crossref] [PubMed]

Lo, H. K.

D. P. DiVincenzo, S. L. Braunstein, and H. K. Lo, Scalable Quantum Computers (Wiley- VCH, Berlin, 2001).

Lohe, M. A.

M. A. Lohe, “Exact time dependence of solutions to the time-dependent Schrodinger equation,” J. Phys. A: Math. and Theor. 42, 035307 (2009).
[Crossref]

Lu, M.

M. Lu, Y. Xia, L. T. Shen, J. Song, and N. B. An, “Shortcuts to adiabatic passage for population transfer and maximum entanglement creation between two atoms in a cavity,” Phys. Rev. A 89, 012326 (2014).
[Crossref]

Mancini, S.

A. Serafini, S. Mancini, and S. Bose, “Distributed quantum computation via optical fibers,” Phys. Rev. Lett. 96, 010503 (2006).
[Crossref] [PubMed]

Margolus, N.

A. Barenco, C. H. Bennett, R. Cleve, D. P. DiVincenzo, N. Margolus, P. Shor, T. Sleator, J. Smolin, and H. Weinfurter, “Elementary gates for quantum computation,” Phys. Rev. A 52, 3457–3467 (1995).
[Crossref] [PubMed]

Marmo, G.

P. Facchi, G. Marmo, and S. Pascazio, “Quantum Zeno dynamics and quantum Zeno subspaces,” J. Phys: Conf. Ser. 196, 012017 (2009).

P. Facchi, V. Gorini, G. Marmo, S. Pascazio, and E. C. G. Sudarshan, “Quantum Zeno dynamics,” Phys. Lett. A 275, 12–19 (2000).
[Crossref]

Martínez-Garaot, S.

E. Torrontegui, S. Ibáñez, S. Martínez-Garaot, M. Modugno, A. del Campo, D. Guery-Odelin, A. Ruschhaupt, X. Chen, and J. G. Muga, “Shortcuts to adiabaticity,” Adv. At. Mol. Opt. Phys,  62, 117–169 (2013).
[Crossref]

Modugno, M.

E. Torrontegui, S. Ibáñez, S. Martínez-Garaot, M. Modugno, A. del Campo, D. Guery-Odelin, A. Ruschhaupt, X. Chen, and J. G. Muga, “Shortcuts to adiabaticity,” Adv. At. Mol. Opt. Phys,  62, 117–169 (2013).
[Crossref]

Mostafazadeh, A.

A. Mostafazadeh, Dynamical Invariants, Adiabatic Approximation, and the Geometric Phase (Nova Science Pub Incorporated2001).

Muga, J. G.

E. Torrontegui, S. Ibáñez, S. Martínez-Garaot, M. Modugno, A. del Campo, D. Guery-Odelin, A. Ruschhaupt, X. Chen, and J. G. Muga, “Shortcuts to adiabaticity,” Adv. At. Mol. Opt. Phys,  62, 117–169 (2013).
[Crossref]

A. Ruschhaupt, X. Chen, D. Alonso, and J. G. Muga, “Optimally robust shortcuts to population inversion in two-level quantum systems,” New J. Phys. 14, 093040 (2012).
[Crossref]

X. Chen, E. Torrontegui, and J. G. Muga, “Lewis-Riesenfeld invariants and transitionless quantum driving,” Phys. Rev. A 83, 062116 (2011).
[Crossref]

X. Chen, I. Lizuain, A. Ruschhaupt, D. Guéry-Odelin, and J. G. Muga, “Shortcut to Adiabatic Passage in Two- and Three-Level Atoms,” Phys. Rev. Lett. 105, 123003 (2010).
[Crossref] [PubMed]

Nielsen, M. A.

M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, 2000).

Ott, K.

A. Walther, F. Ziesel, T. Ruster, S. T. Dawkins, K. Ott, M. Hettrich, K. Singer, F. Schmidt-Kaler, and U. Poschinger, “Controlling fast transport of cold trapped ions,” Phys. Rev. Lett. 109, 080501 (2012).
[Crossref] [PubMed]

Pascazio, S.

P. Facchi, G. Marmo, and S. Pascazio, “Quantum Zeno dynamics and quantum Zeno subspaces,” J. Phys: Conf. Ser. 196, 012017 (2009).

P. Facchi and S. Pascazio, “Quantum Zeno Subspaces,” Phys. Rev. Lett. 89, 080401 (2002).
[Crossref] [PubMed]

P. Facchi, V. Gorini, G. Marmo, S. Pascazio, and E. C. G. Sudarshan, “Quantum Zeno dynamics,” Phys. Lett. A 275, 12–19 (2000).
[Crossref]

Poschinger, U.

A. Walther, F. Ziesel, T. Ruster, S. T. Dawkins, K. Ott, M. Hettrich, K. Singer, F. Schmidt-Kaler, and U. Poschinger, “Controlling fast transport of cold trapped ions,” Phys. Rev. Lett. 109, 080501 (2012).
[Crossref] [PubMed]

Qiao, B.

B. Qiao, H. E. Ruda, and J. Wang, “Multiqubit computing and error-avoiding codes in subspace using quantum dots,” J. Appl. Phys. 91, 2524–2529 (2002).
[Crossref]

Ren, X. F.

Y. F. Huang, X. F. Ren, Y. S. Zhang, L. M. Duan, and G. C. Guo, “Experimental Teleportation of a Quantum Controlled-NOT Gate,” Phys. Rev. Lett. 93, 240501 (2004).
[Crossref]

Rezek, Y.

K. H. Hoffmann, P. Salamon, Y. Rezek, and R. Kosloff, “Time-optimal controls for frictionless cooling in harmonic traps,” Europhys. Lett. 96, 60015 (2011).
[Crossref]

Ribeyre, T.

D. Sugny, L. Bomble, T. Ribeyre, O. Dulieu, and M. Desouter-Lecomte, “Rovibrational controlled-NOT gates using optimized stimulated Raman adiabatic passage techniques and optimal control theory,” Phys. Rev. A 80, 042325 (2009).
[Crossref]

Rice, S. A.

M. Demirplak and S. A. Rice, “On the consistency, extremal, and global properties of counterdiabatic fields,” J. Chem. Phys. 109, 154111 (2008).
[Crossref]

M. Demirplak and S. A. Rice, “Assisted adiabatic passage revisited,” J. Phys. Chem. B 109, 6838–6844 (2005).
[Crossref]

M. Demirplak and S. A. Rice, “Adiabatic population transfer with control fields,” J. Phys. Chem. A 107, 9937–9945 (2003).
[Crossref]

Riesenfeld, W. B.

H. R. Lewis and W. B. Riesenfeld, “An exact quantum theory of the time-dependent harmonic oscillator and of a charged particle in a time-dependent electromagnetic field,” J. Math. Phys. 10, 1458 (1969).
[Crossref]

Ruda, H. E.

B. Qiao, H. E. Ruda, and J. Wang, “Multiqubit computing and error-avoiding codes in subspace using quantum dots,” J. Appl. Phys. 91, 2524–2529 (2002).
[Crossref]

Ruschhaupt, A.

E. Torrontegui, S. Ibáñez, S. Martínez-Garaot, M. Modugno, A. del Campo, D. Guery-Odelin, A. Ruschhaupt, X. Chen, and J. G. Muga, “Shortcuts to adiabaticity,” Adv. At. Mol. Opt. Phys,  62, 117–169 (2013).
[Crossref]

D. Daems, A. Ruschhaupt, D. Sugny, and S. Guérin, “Robust quantum control by a single-shot shaped pulse,” Phys. Rev. Lett. 111, 050404 (2013).
[Crossref] [PubMed]

A. Ruschhaupt, X. Chen, D. Alonso, and J. G. Muga, “Optimally robust shortcuts to population inversion in two-level quantum systems,” New J. Phys. 14, 093040 (2012).
[Crossref]

X. Chen, I. Lizuain, A. Ruschhaupt, D. Guéry-Odelin, and J. G. Muga, “Shortcut to Adiabatic Passage in Two- and Three-Level Atoms,” Phys. Rev. Lett. 105, 123003 (2010).
[Crossref] [PubMed]

Ruster, T.

A. Walther, F. Ziesel, T. Ruster, S. T. Dawkins, K. Ott, M. Hettrich, K. Singer, F. Schmidt-Kaler, and U. Poschinger, “Controlling fast transport of cold trapped ions,” Phys. Rev. Lett. 109, 080501 (2012).
[Crossref] [PubMed]

Salamon, P.

K. H. Hoffmann, P. Salamon, Y. Rezek, and R. Kosloff, “Time-optimal controls for frictionless cooling in harmonic traps,” Europhys. Lett. 96, 60015 (2011).
[Crossref]

Sangouard, N.

N. Sangouard, X. Lacour, S. Guérin, and H. R. Jauslin, “CNOT gate by adiabatic passage with an optical cavity,” Eur. Phys. J. D 37, 451–456 (2006).
[Crossref]

Šašura, M.

M. Šašura and V. Bužek, “Multiparticle entanglement with quantum logic networks: Application to cold trapped ions,” Phys. Rev. A 64, 012305 (2001).
[Crossref]

Schaff, J. F.

J. F. Schaff, X. L. Song, P. Capuzzi, P. Vignolo, and G. Labeyrie, “Shortcut to adiabaticity for an interacting Bose-Einstein condensate,” Europhys. Lett. 93, 23001 (2011).
[Crossref]

Schmidt-Kaler, F.

A. Walther, F. Ziesel, T. Ruster, S. T. Dawkins, K. Ott, M. Hettrich, K. Singer, F. Schmidt-Kaler, and U. Poschinger, “Controlling fast transport of cold trapped ions,” Phys. Rev. Lett. 109, 080501 (2012).
[Crossref] [PubMed]

Serafini, A.

A. Serafini, S. Mancini, and S. Bose, “Distributed quantum computation via optical fibers,” Phys. Rev. Lett. 96, 010503 (2006).
[Crossref] [PubMed]

Shao, X. Q.

X. Q. Shao, L. Chen, S. Zhang, and K. H. Yeon, “Fast CNOT gate via quantum Zeno dynamics,” J. Phys. B: At. Mol. Opt. Phys. 42, 165507 (2009).
[Crossref]

Shen, L. T.

M. Lu, Y. Xia, L. T. Shen, J. Song, and N. B. An, “Shortcuts to adiabatic passage for population transfer and maximum entanglement creation between two atoms in a cavity,” Phys. Rev. A 89, 012326 (2014).
[Crossref]

Shor, P.

A. Barenco, C. H. Bennett, R. Cleve, D. P. DiVincenzo, N. Margolus, P. Shor, T. Sleator, J. Smolin, and H. Weinfurter, “Elementary gates for quantum computation,” Phys. Rev. A 52, 3457–3467 (1995).
[Crossref] [PubMed]

Singer, K.

A. Walther, F. Ziesel, T. Ruster, S. T. Dawkins, K. Ott, M. Hettrich, K. Singer, F. Schmidt-Kaler, and U. Poschinger, “Controlling fast transport of cold trapped ions,” Phys. Rev. Lett. 109, 080501 (2012).
[Crossref] [PubMed]

Sleator, T.

A. Barenco, C. H. Bennett, R. Cleve, D. P. DiVincenzo, N. Margolus, P. Shor, T. Sleator, J. Smolin, and H. Weinfurter, “Elementary gates for quantum computation,” Phys. Rev. A 52, 3457–3467 (1995).
[Crossref] [PubMed]

Smolin, J.

A. Barenco, C. H. Bennett, R. Cleve, D. P. DiVincenzo, N. Margolus, P. Shor, T. Sleator, J. Smolin, and H. Weinfurter, “Elementary gates for quantum computation,” Phys. Rev. A 52, 3457–3467 (1995).
[Crossref] [PubMed]

Song, J.

Y. H. Cheng, Y. Xia, Q. Q. Chen, and J. Song, “Fast and noise-resistant implementation of quantum phase gates and creation of quantum entangled states,” Phys. Rev. A 91, 012325 (2015).
[Crossref]

Y. H. Chen, Y. Xia, Q. Q. Chen, and J. Song, “Efficient shortcuts to adiabatic passage for fast population transfer in multiparticle systems,” Phys. Rev. A 89, 033856 (2014).
[Crossref]

M. Lu, Y. Xia, L. T. Shen, J. Song, and N. B. An, “Shortcuts to adiabatic passage for population transfer and maximum entanglement creation between two atoms in a cavity,” Phys. Rev. A 89, 012326 (2014).
[Crossref]

Song, X. L.

J. F. Schaff, X. L. Song, P. Capuzzi, P. Vignolo, and G. Labeyrie, “Shortcut to adiabaticity for an interacting Bose-Einstein condensate,” Europhys. Lett. 93, 23001 (2011).
[Crossref]

Spillane, S. M.

S. M. Spillane, T. J. Kippenberg, K. J. Vahala, K. W. Goh, E. Wilcut, and H. J. Kimble, “Ultrahigh-Q toroidal microresonators for cavity quantum electrodynamics,” Phys. Rev. A 71, 013817 (2005).
[Crossref]

Su, S. L.

Y. Liang, Q. C. Wu, S. L. Su, X. Ji, and S. Zhang, “Shortcuts to adiabatic passage for multiqubit controlled-phase gate,” Phys. Rev. A 91, 032304 (2015).
[Crossref]

Sudarshan, E. C. G.

P. Facchi, V. Gorini, G. Marmo, S. Pascazio, and E. C. G. Sudarshan, “Quantum Zeno dynamics,” Phys. Lett. A 275, 12–19 (2000).
[Crossref]

Sugny, D.

D. Daems, A. Ruschhaupt, D. Sugny, and S. Guérin, “Robust quantum control by a single-shot shaped pulse,” Phys. Rev. Lett. 111, 050404 (2013).
[Crossref] [PubMed]

D. Sugny, L. Bomble, T. Ribeyre, O. Dulieu, and M. Desouter-Lecomte, “Rovibrational controlled-NOT gates using optimized stimulated Raman adiabatic passage techniques and optimal control theory,” Phys. Rev. A 80, 042325 (2009).
[Crossref]

Torrontegui, E.

E. Torrontegui, S. Ibáñez, S. Martínez-Garaot, M. Modugno, A. del Campo, D. Guery-Odelin, A. Ruschhaupt, X. Chen, and J. G. Muga, “Shortcuts to adiabaticity,” Adv. At. Mol. Opt. Phys,  62, 117–169 (2013).
[Crossref]

X. Chen, E. Torrontegui, and J. G. Muga, “Lewis-Riesenfeld invariants and transitionless quantum driving,” Phys. Rev. A 83, 062116 (2011).
[Crossref]

Vahala, K. J.

S. M. Spillane, T. J. Kippenberg, K. J. Vahala, K. W. Goh, E. Wilcut, and H. J. Kimble, “Ultrahigh-Q toroidal microresonators for cavity quantum electrodynamics,” Phys. Rev. A 71, 013817 (2005).
[Crossref]

Vignolo, P.

J. F. Schaff, X. L. Song, P. Capuzzi, P. Vignolo, and G. Labeyrie, “Shortcut to adiabaticity for an interacting Bose-Einstein condensate,” Europhys. Lett. 93, 23001 (2011).
[Crossref]

Walther, A.

A. Walther, F. Ziesel, T. Ruster, S. T. Dawkins, K. Ott, M. Hettrich, K. Singer, F. Schmidt-Kaler, and U. Poschinger, “Controlling fast transport of cold trapped ions,” Phys. Rev. Lett. 109, 080501 (2012).
[Crossref] [PubMed]

Wang, J.

B. Qiao, H. E. Ruda, and J. Wang, “Multiqubit computing and error-avoiding codes in subspace using quantum dots,” J. Appl. Phys. 91, 2524–2529 (2002).
[Crossref]

Weinfurter, H.

A. Barenco, C. H. Bennett, R. Cleve, D. P. DiVincenzo, N. Margolus, P. Shor, T. Sleator, J. Smolin, and H. Weinfurter, “Elementary gates for quantum computation,” Phys. Rev. A 52, 3457–3467 (1995).
[Crossref] [PubMed]

Wilcut, E.

S. M. Spillane, T. J. Kippenberg, K. J. Vahala, K. W. Goh, E. Wilcut, and H. J. Kimble, “Ultrahigh-Q toroidal microresonators for cavity quantum electrodynamics,” Phys. Rev. A 71, 013817 (2005).
[Crossref]

Wu, Q. C.

Y. Liang, Q. C. Wu, S. L. Su, X. Ji, and S. Zhang, “Shortcuts to adiabatic passage for multiqubit controlled-phase gate,” Phys. Rev. A 91, 032304 (2015).
[Crossref]

Xia, Y.

Y. H. Cheng, Y. Xia, Q. Q. Chen, and J. Song, “Fast and noise-resistant implementation of quantum phase gates and creation of quantum entangled states,” Phys. Rev. A 91, 012325 (2015).
[Crossref]

Y. H. Chen, Y. Xia, Q. Q. Chen, and J. Song, “Efficient shortcuts to adiabatic passage for fast population transfer in multiparticle systems,” Phys. Rev. A 89, 033856 (2014).
[Crossref]

M. Lu, Y. Xia, L. T. Shen, J. Song, and N. B. An, “Shortcuts to adiabatic passage for population transfer and maximum entanglement creation between two atoms in a cavity,” Phys. Rev. A 89, 012326 (2014).
[Crossref]

Yang, C. P.

C. P. Yang and S. Han, “Realization of an n-qubit controlled-U gate with superconducting quantum interference devices or atoms in cavity QED,” Phys. Rev. A 73, 032317 (2006).
[Crossref]

C. P. Yang and S. Chun, “Possible realization of entanglement, logical gates, and quantum-information transfer with superconducting-quantum-interference-device qubits in cavity QED,” Phys. Rev. A 67, 042311 (2003).
[Crossref]

Yeon, K. H.

X. Q. Shao, L. Chen, S. Zhang, and K. H. Yeon, “Fast CNOT gate via quantum Zeno dynamics,” J. Phys. B: At. Mol. Opt. Phys. 42, 165507 (2009).
[Crossref]

Zhang, S.

Y. Liang, Q. C. Wu, S. L. Su, X. Ji, and S. Zhang, “Shortcuts to adiabatic passage for multiqubit controlled-phase gate,” Phys. Rev. A 91, 032304 (2015).
[Crossref]

X. Q. Shao, L. Chen, S. Zhang, and K. H. Yeon, “Fast CNOT gate via quantum Zeno dynamics,” J. Phys. B: At. Mol. Opt. Phys. 42, 165507 (2009).
[Crossref]

Zhang, Y. S.

Y. F. Huang, X. F. Ren, Y. S. Zhang, L. M. Duan, and G. C. Guo, “Experimental Teleportation of a Quantum Controlled-NOT Gate,” Phys. Rev. Lett. 93, 240501 (2004).
[Crossref]

Zheng, S. B.

S. B. Zheng, “Implementation of Toffoli gates with a single asymmetric Heisenberg XY interaction,” Phys. Rev. A 87, 042318 (2013).
[Crossref]

S. B. Zheng, “Nongeometric conditional phase shift via adiabatic evolution of dark eigenstates: a new approach to quantum computation,” Phys. Rev. Lett. 95, 080502 (2005).
[Crossref] [PubMed]

Ziesel, F.

A. Walther, F. Ziesel, T. Ruster, S. T. Dawkins, K. Ott, M. Hettrich, K. Singer, F. Schmidt-Kaler, and U. Poschinger, “Controlling fast transport of cold trapped ions,” Phys. Rev. Lett. 109, 080501 (2012).
[Crossref] [PubMed]

Zoller, P.

J. I. Cirac and P. Zoller, “Quantum computations with cold trapped ions,” Phys. Rev. Lett. 74, 4091–4094 (1995).
[Crossref] [PubMed]

Adv. At. Mol. Opt. Phys (1)

E. Torrontegui, S. Ibáñez, S. Martínez-Garaot, M. Modugno, A. del Campo, D. Guery-Odelin, A. Ruschhaupt, X. Chen, and J. G. Muga, “Shortcuts to adiabaticity,” Adv. At. Mol. Opt. Phys,  62, 117–169 (2013).
[Crossref]

Eur. Phys. J. D (1)

N. Sangouard, X. Lacour, S. Guérin, and H. R. Jauslin, “CNOT gate by adiabatic passage with an optical cavity,” Eur. Phys. J. D 37, 451–456 (2006).
[Crossref]

Europhys. Lett. (2)

K. H. Hoffmann, P. Salamon, Y. Rezek, and R. Kosloff, “Time-optimal controls for frictionless cooling in harmonic traps,” Europhys. Lett. 96, 60015 (2011).
[Crossref]

J. F. Schaff, X. L. Song, P. Capuzzi, P. Vignolo, and G. Labeyrie, “Shortcut to adiabaticity for an interacting Bose-Einstein condensate,” Europhys. Lett. 93, 23001 (2011).
[Crossref]

J. Appl. Phys. (1)

B. Qiao, H. E. Ruda, and J. Wang, “Multiqubit computing and error-avoiding codes in subspace using quantum dots,” J. Appl. Phys. 91, 2524–2529 (2002).
[Crossref]

J. Chem. Phys. (1)

M. Demirplak and S. A. Rice, “On the consistency, extremal, and global properties of counterdiabatic fields,” J. Chem. Phys. 109, 154111 (2008).
[Crossref]

J. Math. Phys. (1)

H. R. Lewis and W. B. Riesenfeld, “An exact quantum theory of the time-dependent harmonic oscillator and of a charged particle in a time-dependent electromagnetic field,” J. Math. Phys. 10, 1458 (1969).
[Crossref]

J. Phys. A: Math. and Theor. (1)

M. A. Lohe, “Exact time dependence of solutions to the time-dependent Schrodinger equation,” J. Phys. A: Math. and Theor. 42, 035307 (2009).
[Crossref]

J. Phys. B: At. Mol. Opt. Phys. (1)

X. Q. Shao, L. Chen, S. Zhang, and K. H. Yeon, “Fast CNOT gate via quantum Zeno dynamics,” J. Phys. B: At. Mol. Opt. Phys. 42, 165507 (2009).
[Crossref]

J. Phys. Chem. A (1)

M. Demirplak and S. A. Rice, “Adiabatic population transfer with control fields,” J. Phys. Chem. A 107, 9937–9945 (2003).
[Crossref]

J. Phys. Chem. B (1)

M. Demirplak and S. A. Rice, “Assisted adiabatic passage revisited,” J. Phys. Chem. B 109, 6838–6844 (2005).
[Crossref]

J. Phys: Conf. Ser. (1)

P. Facchi, G. Marmo, and S. Pascazio, “Quantum Zeno dynamics and quantum Zeno subspaces,” J. Phys: Conf. Ser. 196, 012017 (2009).

New J. Phys. (1)

A. Ruschhaupt, X. Chen, D. Alonso, and J. G. Muga, “Optimally robust shortcuts to population inversion in two-level quantum systems,” New J. Phys. 14, 093040 (2012).
[Crossref]

Phys. Lett. A (1)

P. Facchi, V. Gorini, G. Marmo, S. Pascazio, and E. C. G. Sudarshan, “Quantum Zeno dynamics,” Phys. Lett. A 275, 12–19 (2000).
[Crossref]

Phys. Rev. A (13)

Y. H. Cheng, Y. Xia, Q. Q. Chen, and J. Song, “Fast and noise-resistant implementation of quantum phase gates and creation of quantum entangled states,” Phys. Rev. A 91, 012325 (2015).
[Crossref]

Y. Liang, Q. C. Wu, S. L. Su, X. Ji, and S. Zhang, “Shortcuts to adiabatic passage for multiqubit controlled-phase gate,” Phys. Rev. A 91, 032304 (2015).
[Crossref]

M. Lu, Y. Xia, L. T. Shen, J. Song, and N. B. An, “Shortcuts to adiabatic passage for population transfer and maximum entanglement creation between two atoms in a cavity,” Phys. Rev. A 89, 012326 (2014).
[Crossref]

Y. H. Chen, Y. Xia, Q. Q. Chen, and J. Song, “Efficient shortcuts to adiabatic passage for fast population transfer in multiparticle systems,” Phys. Rev. A 89, 033856 (2014).
[Crossref]

S. B. Zheng, “Implementation of Toffoli gates with a single asymmetric Heisenberg XY interaction,” Phys. Rev. A 87, 042318 (2013).
[Crossref]

D. Sugny, L. Bomble, T. Ribeyre, O. Dulieu, and M. Desouter-Lecomte, “Rovibrational controlled-NOT gates using optimized stimulated Raman adiabatic passage techniques and optimal control theory,” Phys. Rev. A 80, 042325 (2009).
[Crossref]

D. P. DiVincenzo, “Two-bit gates are universal for quantum computation,” Phys. Rev. A 51, 1015–1022 (1995).
[Crossref] [PubMed]

A. Barenco, C. H. Bennett, R. Cleve, D. P. DiVincenzo, N. Margolus, P. Shor, T. Sleator, J. Smolin, and H. Weinfurter, “Elementary gates for quantum computation,” Phys. Rev. A 52, 3457–3467 (1995).
[Crossref] [PubMed]

M. Šašura and V. Bužek, “Multiparticle entanglement with quantum logic networks: Application to cold trapped ions,” Phys. Rev. A 64, 012305 (2001).
[Crossref]

C. P. Yang and S. Chun, “Possible realization of entanglement, logical gates, and quantum-information transfer with superconducting-quantum-interference-device qubits in cavity QED,” Phys. Rev. A 67, 042311 (2003).
[Crossref]

C. P. Yang and S. Han, “Realization of an n-qubit controlled-U gate with superconducting quantum interference devices or atoms in cavity QED,” Phys. Rev. A 73, 032317 (2006).
[Crossref]

X. Chen, E. Torrontegui, and J. G. Muga, “Lewis-Riesenfeld invariants and transitionless quantum driving,” Phys. Rev. A 83, 062116 (2011).
[Crossref]

S. M. Spillane, T. J. Kippenberg, K. J. Vahala, K. W. Goh, E. Wilcut, and H. J. Kimble, “Ultrahigh-Q toroidal microresonators for cavity quantum electrodynamics,” Phys. Rev. A 71, 013817 (2005).
[Crossref]

Phys. Rev. Lett. (9)

D. Daems, A. Ruschhaupt, D. Sugny, and S. Guérin, “Robust quantum control by a single-shot shaped pulse,” Phys. Rev. Lett. 111, 050404 (2013).
[Crossref] [PubMed]

Y. F. Huang, X. F. Ren, Y. S. Zhang, L. M. Duan, and G. C. Guo, “Experimental Teleportation of a Quantum Controlled-NOT Gate,” Phys. Rev. Lett. 93, 240501 (2004).
[Crossref]

J. I. Cirac and P. Zoller, “Quantum computations with cold trapped ions,” Phys. Rev. Lett. 74, 4091–4094 (1995).
[Crossref] [PubMed]

S. B. Zheng, “Nongeometric conditional phase shift via adiabatic evolution of dark eigenstates: a new approach to quantum computation,” Phys. Rev. Lett. 95, 080502 (2005).
[Crossref] [PubMed]

A. Walther, F. Ziesel, T. Ruster, S. T. Dawkins, K. Ott, M. Hettrich, K. Singer, F. Schmidt-Kaler, and U. Poschinger, “Controlling fast transport of cold trapped ions,” Phys. Rev. Lett. 109, 080501 (2012).
[Crossref] [PubMed]

X. Chen, I. Lizuain, A. Ruschhaupt, D. Guéry-Odelin, and J. G. Muga, “Shortcut to Adiabatic Passage in Two- and Three-Level Atoms,” Phys. Rev. Lett. 105, 123003 (2010).
[Crossref] [PubMed]

A. del Campo, “Shortcuts to adiabaticity by counter-diabatic driving,” Phys. Rev. Lett. 111, 100502 (2013).
[Crossref]

P. Facchi and S. Pascazio, “Quantum Zeno Subspaces,” Phys. Rev. Lett. 89, 080401 (2002).
[Crossref] [PubMed]

A. Serafini, S. Mancini, and S. Bose, “Distributed quantum computation via optical fibers,” Phys. Rev. Lett. 96, 010503 (2006).
[Crossref] [PubMed]

Science (1)

N. A. Gershenfeld and I. L. Chuang, “Bulk spin-resonance quantum computation,” Science 275, 350–356 (1997).
[Crossref] [PubMed]

Other (3)

M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, 2000).

D. P. DiVincenzo, S. L. Braunstein, and H. K. Lo, Scalable Quantum Computers (Wiley- VCH, Berlin, 2001).

A. Mostafazadeh, Dynamical Invariants, Adiabatic Approximation, and the Geometric Phase (Nova Science Pub Incorporated2001).

Cited By

Optica participates in Crossref's Cited-By Linking service. Citing articles from Optica Publishing Group journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (5)

Fig. 1
Fig. 1 The schematic setup of CNOT gate implementation. The two atoms are trapped in two spatially separated optical cavities connected by a fiber, and each atom possesses five atomic levels.
Fig. 2
Fig. 2 Schematic representation of the four steps of the construction of the CNOT gate. The initial state is denoted by an empty circle and the final state is represented by a full black circle.
Fig. 3
Fig. 3 (a) Temporal profile of the time dependence Rabi frequencies Ω i (t)/λ versus λt with Ω i (t) = Ω11(t) (dash blue line), Ωa1(t) (solid blue line), Ω21(t) (dash red line), Ω12(t) (solid red line), Ω22(t) (solid green line), Ωa2(t) (dash green line), Ω′ a (t) (solid violet line) and Ω1(t) (dash violet line). (b) Time evolutions of the populations of the corresponding system states with the initial states |g0g1AB. (c) Time evolutions of the populations of the corresponding system states with the initial state |g0g2AB. The system parameters are set to be ε = 0.25, λA = λB = λ and tf = 15/λ.
Fig. 4
Fig. 4 The fidelity of the CNOT gate versus ε and λtf regardless of the decoherence.
Fig. 5
Fig. 5 The effect of atomic spontaneous emission γ on the fidelity of the CNOT gate with different values of the cavity decay κ.

Equations (36)

Equations on this page are rendered with MathJax. Learn more.

H K = H obs + K H meas ,
U ( t ) = exp [ i t n ( K λ n P n + P n H obs P n ) ] ,
i h ¯ I ( t ) t = [ H ( t ) , I ( t ) ] .
| Ψ ( t ) = n C n e i θ n | Φ n ( t ) ,
θ n ( t ) = 1 h ¯ 0 t d t Φ n ( t ) | i h ¯ t H ( t ) | Φ n ( t ) .
| Ψ 0 = α 1 | g 1 g 1 AB + α 2 | g 1 g 2 AB + α 3 | g 0 g 1 AB + α 4 | g 0 g 2 AB ,
| Ψ = α 1 | g 1 g 1 AB + α 2 | g 1 g 2 AB + α 3 | g 0 g 2 AB + α 4 | g 0 g 1 AB ,
| Ψ 1 = α 1 | g 1 g 1 AB + α 2 | g 1 g 2 AB α 3 | g 0 a AB + α 4 | g 0 g 2 AB .
| Ψ 2 = α 1 | g 1 g 1 AB + α 2 | g 1 g 2 AB α 3 | g 0 a AB α 4 | g 0 g 1 AB .
| Ψ 3 = α 1 | g 1 g 1 AB + α 2 | g 1 g 2 AB + α 3 | g 0 g 2 AB α 4 | g 0 g 1 AB .
| Ψ 4 = α 1 | g 1 g 1 AB + α 2 | g 1 g 2 AB + α 3 | g 0 g 2 AB + α 4 | g 0 g 1 AB ,
H 1 = H a-l + H a-c-f ,
H a-l = Ω 11 ( t ) | e B g 1 | + Ω a 1 ( t ) | e B a | + H . c . ,
H a-c-f = λ A a A | e A g 0 | + λ B a B | e B g 0 | + η b ( a A + a B ) + H . c . ,
| ϕ 1 = | g 1 g 1 AB | 000 AfB , | ϕ 2 = | g 1 e AB | 000 AfB , | ϕ 3 = | g 1 a AB | 000 AfB , | ϕ 4 = | g 1 g 0 AB | 001 AfB , | ϕ 5 = | g 1 g 0 AB | 010 AfB , | ϕ 6 = | g 1 g 0 AB | 100 AfB .
Γ P 0 = { | ϕ 1 , | ϕ 3 } , Γ P 1 = { | ψ 1 } , Γ P 2 = { | ψ 2 } , Γ P 3 = { | ψ 3 } , Γ P 4 = { | ψ 4 } ,
| ψ 1 = ( C + B ) A B 2 2 λ η 2 | ϕ 2 + λ 2 B 2 η 2 | ϕ 4 A B 2 η | ϕ 5 + | ϕ 6 , | ψ 2 = ( C + B ) A B 2 2 λ η 2 | ϕ 2 + λ 2 B 2 η 2 | ϕ 4 + A B 2 η | ϕ 5 + | ϕ 6 , | ψ 3 = ( C + B ) A + B 2 2 λ η 2 | ϕ 2 + λ 2 + B 2 η 2 | ϕ 4 A + B 2 η | ϕ 5 + | ϕ 6 , | ψ 4 = ( C + B ) A + B 2 2 λ η 2 | ϕ 2 + λ 2 + B 2 η 2 | ϕ 4 + A + B 2 η | ϕ 5 + | ϕ 6 ,
H 1 i , α , β ( E i P i α + P i α H a l P i β ) = E 1 | ψ 1 ψ 1 | + E 2 | ψ 2 ψ 2 | + E 3 | ψ 3 ψ 3 | + E 4 | ψ 4 ψ 4 | ,
| ϕ 1 = | g 0 g 1 AB | 000 AfB , | ϕ 2 = | g 0 e AB | 000 AfB , | ϕ 3 = | g 0 a AB | 000 AfB , | ϕ 4 = | g 0 g 0 AB | 001 AfB , | ϕ 5 = | g 0 g 0 AB | 010 AfB , | ϕ 6 = | g 0 g 0 AB | 100 AfB , | ϕ 7 = | e g 0 AB | 000 AfB .
Γ P 0 = { | ϕ 1 , | ψ 0 , | ϕ 3 } , Γ P 1 = { | ψ 1 } , Γ P 2 = { | ψ 2 } , Γ P 3 = { | ψ 3 } , Γ P 4 = { | ψ 4 } ,
| ψ 0 = η A ( | ϕ 2 λ η | ϕ 5 + | ϕ 7 ) , | ψ 1 = 1 2 ( | ϕ 2 + | ϕ 4 | ϕ 6 + | ϕ 7 ) , | ψ 2 = 1 2 ( | ϕ 2 | ϕ 4 + | ϕ 6 + | ϕ 7 ) , | ψ 3 = λ 2 A ( | ϕ 2 A λ | ϕ 4 + 2 η λ | ϕ 5 A λ | ϕ 6 + | ϕ 7 ) , | ψ 4 = λ 2 A ( | ϕ 2 + A λ | ϕ 4 + 2 η λ | ϕ 5 + A λ | ϕ 6 + | ϕ 7 ) ,
H eff = η A ( Ω 11 ( t ) | ψ 0 ϕ 1 | + Ω a 1 ( t ) | ψ 0 ϕ 3 | + H . c . ) .
I ( t ) = χ ( cos ν sin β | ψ 0 ϕ 1 | + cos ν cos β | ψ 0 ϕ 3 | + i sin ν | ϕ 3 ϕ 1 | + H . c . ) ,
ν ˙ = η A [ Ω 11 ( t ) cos β Ω a 1 ( t ) sin β ] , β ˙ = η A tan ν [ Ω a 1 ( t ) cos β + Ω 11 ( t ) sin β ] .
Ω 11 ( t ) = A η ( β ˙ cot ν sin β + ν ˙ cos β ) , Ω a 1 ( t ) = A η ( β ˙ cot ν cos β ν ˙ sin β ) .
| Φ 0 ( t ) = cos ν cos β | ϕ 1 i sin ν | ψ 0 cos ν sin β | ϕ 3 , | Φ ± ( t ) = 1 2 [ ( sin ν cos β ± i sin β ) | ϕ 1 + i cos ν | ψ 0 ( sin ν sin β i cos β ) | ϕ 3 ] .
ν ( t ) = ε , β ( t ) = π t 2 t f ,
Ω 11 ( t ) = A π η 2 t f cot ε sin π t 2 t f , Ω a 1 ( t ) = A π η 2 t f cot ε cos π t 2 t f .
| Ψ ( t f ) = sin ε sin θ | ϕ 1 + ( i sin ε cos ε + i sin ε cos ε cos θ ) | ψ 0 + ( cos 2 ε sin 2 ε cos θ ) | ϕ 3 ,
H 2 = Ω 21 ( t ) | e B g 2 | + Ω 12 ( t ) | e B | g 1 + λ A a A | e A g 0 | + λ B a B | e B g 0 | + η b ( a A + a B ) + H . c . .
Ω 21 ( t ) = A π η 2 t f cot ε sin π t 2 t f , Ω 12 ( t ) = A π η 2 t f cot ε cos π t 2 t f .
H 3 = Ω a 2 ( t ) | e B a | + Ω 22 ( t ) | e B g 2 | + λ A a A | e A g 0 | + λ B a B | e B g 0 | + η b ( a A + a B ) + H . c . .
Ω a 2 ( t ) = A π η 2 t f cot ε sin π t 2 t f , Ω 22 ( t ) = A π η 2 t f cot ε cos π t 2 t f .
H 4 = Ω 1 ( t ) | e B g 1 | + Ω a ( t ) | e B a | + λ A a A | e A g 0 | + λ B a B | e B g 0 | + η b ( a A + a B ) + H . c . .
Ω 1 ( t ) = A π η 2 t f cot ε sin π t 2 t f , Ω a ( t ) = A π η 2 t f cot ε sin π t 2 t f .
ρ ˙ = i [ H total , ρ ] κ f 2 [ b b ρ 2 b ρ b + ρ b b ] j = A , B κ j 2 [ a j a j ρ 2 a j ρ a j + ρ a j a j ] j = A , B k = g 0 , g 1 , g 2 , a γ j 2 [ σ e j , e j ρ 2 σ k j , e j ρ σ e j , k j + ρ σ e j , e j ] ,

Metrics