Fast CNOT gate between two spatially separated atoms via shortcuts to adiabatic passage

Yan Liang; Chong Song; Xin Ji; Shou Zhang

doi:10.1364/OE.23.023798

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

H_{K} = H_{obs} + K H_{meas},

where H_obs is the Hamiltonian of the investigated quantum system and the H_meas 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 [- i t \sum_{n} (K λ_{n} P_{n} + P_{n} H_{obs} P_{n})],

where P_n is one of the eigenprojections of H_meas with eigenvalues λ_n(H_meas = ∑_nλ_nP_n).

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

i \bar{h} \frac{\partial I (t)}{\partial 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

i \bar{h} \frac{\partial | Ψ (t) 〉}{\partial t} = H (t) | Ψ (t) 〉

, the solution can be expressed by the superposition of dynamical modes |Φ_n(t)〉 of the invariants I(t)

| Ψ (t) 〉 = \sum_{n} C_{n} e^{i θ_{n}} | Φ_{n} (t) 〉,

where n = 0, 1,..., and C_n is one of the time-independent amplitudes, θ_n is the Lewis-Riesenfeld phase. |Φ_n(t)〉 is one of the orthonormal eigenvectors of the invariant I(t) with the corresponding real eigenvalue λ_n, satisfying I(t)|Φ_n(t)〉 = λ_n|Φ_n(t)〉. And the Lewis-Riesenfeld phase satisfies

θ_{n} (t) = \frac{1}{\bar{h}} \int_{0}^{t} d t^{'} 〈 Φ_{n} (t^{'}) | i \bar{h} \frac{\partial}{\partial t^{'}} - H (t^{'}) | Φ_{n} (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 Ω_a(t), respectively. The transition |g₀〉 ↔ |e〉 of atom A (B) is strongly coupled to its single mode cavity A (B) with the coupling constant λ_A(λ_B).

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.

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| 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},

after implementing the CNOT gate operation, the outcome becomes

| 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},

where |g_ig_j〉_AB(i = 0, 1; j = 1, 2) denotes the state of atom A and B, α_1,2,3,4 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 |g₀g₁〉_AB to −|g₀a〉_AB completely with the help of laser pulses resonant with B atomic transitions |g₁〉_B ↔ |e〉_B and |a〉_B ↔ |e〉_B with the corresponding Rabi frequencies Ω₁₁(t) and Ω_a1(t), respectively. After the interaction, the initial state becomes

| 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} .

Secondly, the population of |g₀g₂〉_AB is completely transferred to −|g₀g₁〉_AB with the similar method as the first step, and the corresponding laser pulses resonant with B atomic transitions |g₂〉_B ↔ |e〉_B and |g₁〉_B ↔ |e〉_B with the corresponding Rabi frequencies Ω₂₁(t) and Ω₁₂(t), respectively. And then the state of the system becomes

| 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} .

Then the population of |g₀a〉_AB is completely transferred to −|g₀g₂〉_AB by the similar method as the first step, and the corresponding laser pulses resonant with B atomic transitions |a〉_B ↔ |e〉_B and |g₂〉_B ↔ |e〉_B with the corresponding Rabi frequencies Ω_a2(t) and Ω₂₂(t), respectively. Now, the state of the system reads

| 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} .

Finally the state |g₀g₁〉_AB induces a π phase, then turns into −|g₀g₁〉_AB and the corresponding laser pulses are resonant with B atomic transitions |g₁〉_B ↔ |e〉_B and |a〉_B) ↔ |e〉_B with the corresponding Rabi frequencies Ω′₁(t) and Ω′_a(t), respectively. As a result, the state of the system turns into

| 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},

which is equal to a CNOT gate operation. Figure 2 represents these four steps for constructing the CNOT gate.

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.

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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 (h̄ = 1)

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 .,

where η is the coupling strength between cavity mode and the fiber mode, b is the annihilation operator for the fiber mode, a_A(B) is the annihilation operator for the corresponding cavity field, and λ_A(B) is the coupling strength between the corresponding cavity mode and the trapped atom. For convenience, we assume λ_A = λ_B = λ, and a_A = a_B = a. For the initial state |ϕ₁〉 = |g₁g₁〉_AB|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

\begin{array}{l} | 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} . \end{array}

Setting Ω₁₁(t), Ω_a1(t) ≪ η, λ_A(B), the Hilbert subspace can be divided into five invariant Zeno subspaces [32, 33]:

\begin{array}{l} 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} 〉}, \end{array}

with the eigenvalues E₀ = 0,

E_{1} = - \sqrt{(A - B) / 2}

,

E_{2} = \sqrt{(A - B) / 2}

,

E_{3} = - \sqrt{(A + B) / 2}

, and

E_{4} = \sqrt{(A + B) / 2}

, where A = λ² + 2η²,

B = \sqrt{λ^{4} + 4 η^{4}}

, and the corresponding projection

P_{i}^{α} = | α 〉 〈 α |

, (|α〉 ∈ Γ_Pi). Here

\begin{array}{l} | ψ_{1} 〉 & = & \frac{(C + B) \sqrt{A - B}}{2 \sqrt{2} λ η^{2}} | ϕ_{2} 〉 + \frac{λ^{2} - B}{2 η^{2}} | ϕ_{4} 〉 - \frac{\sqrt{A - B}}{\sqrt{2} η} | ϕ_{5} 〉 + | ϕ_{6} 〉, \\ | ψ_{2} 〉 & = & - \frac{(C + B) \sqrt{A - B}}{2 \sqrt{2} λ η^{2}} | ϕ_{2} 〉 + \frac{λ^{2} - B}{2 η^{2}} | ϕ_{4} 〉 + \frac{\sqrt{A - B}}{\sqrt{2} η} | ϕ_{5} 〉 + | ϕ_{6} 〉, \\ | ψ_{3} 〉 & = & - \frac{(- C + B) \sqrt{A + B}}{2 \sqrt{2} λ η^{2}} | ϕ_{2} 〉 + \frac{λ^{2} + B}{2 η^{2}} | ϕ_{4} 〉 - \frac{\sqrt{A + B}}{\sqrt{2} η} | ϕ_{5} 〉 + | ϕ_{6} 〉, \\ | ψ_{4} 〉 & = & \frac{(- C + B) \sqrt{A + B}}{2 \sqrt{2} λ η^{2}} | ϕ_{2} 〉 + \frac{λ^{2} + B}{2 η^{2}} | ϕ_{4} 〉 + \frac{\sqrt{A + B}}{\sqrt{2} η} | ϕ_{5} 〉 + | ϕ_{6} 〉, \end{array}

C = 2η² − λ. Under the above conditions, the system Hamiltonian can be rewritten as the following form [33]:

\begin{array}{l} H_{1} & ≃ & \sum_{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} |, \end{array}

which indicates the transition between |ϕ₁〉 ↔ |ϕ₂〉 can be inhibited.

For another initial state |ϕ′₁〉 = |g₀g₁〉_AB|000〉_AfB, the subspace can be spanned by the vectors

\begin{array}{l} | 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} . \end{array}

For the condition Ω₁₁(t), Ω_a1(t) ≪ η, λ_A(B), the Hilbert subspace in this system can be divided into five invariant Zeno subspaces

\begin{array}{l} 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}^{'} 〉}, \end{array}

with the eigenvalues E′₀ = 0, E′₁ = −λ, E′₂ = λ,

E_{3}^{'} = - \sqrt{A}

, and

E_{4}^{'} = \sqrt{A}

. Here

\begin{array}{l} | ψ_{0}^{'} 〉 & = & \frac{η}{\sqrt{A}} (| ϕ_{2}^{'} 〉 - \frac{λ}{η} | ϕ_{5}^{'} 〉 + | ϕ_{7}^{'} 〉), \\ | ψ_{1}^{'} 〉 & = & \frac{1}{2} (- | ϕ_{2}^{'} 〉 + | ϕ_{4}^{'} 〉 - | ϕ_{6}^{'} 〉 + | ϕ_{7}^{'} 〉), \\ | ψ_{2}^{'} 〉 & = & \frac{1}{2} (- | ϕ_{2}^{'} 〉 - | ϕ_{4}^{'} 〉 + | ϕ_{6}^{'} 〉 + | ϕ_{7}^{'} 〉), \\ | ψ_{3}^{'} 〉 & = & \frac{λ}{2 \sqrt{A}} (| ϕ_{2}^{'} 〉 - \frac{\sqrt{A}}{λ} | ϕ_{4}^{'} 〉 + \frac{2 η}{λ} | ϕ_{5}^{'} 〉 - \frac{\sqrt{A}}{λ} | ϕ_{6}^{'} 〉 + | ϕ_{7}^{'} 〉), \\ | ψ_{4}^{'} 〉 & = & \frac{λ}{2 \sqrt{A}} (| ϕ_{2}^{'} 〉 + \frac{\sqrt{A}}{λ} | ϕ_{4}^{'} 〉 + \frac{2 η}{λ} | ϕ_{5}^{'} 〉 + \frac{\sqrt{A}}{λ} | ϕ_{6}^{'} 〉 + | ϕ_{7}^{'} 〉), \end{array}

and the corresponding projection

P_{i}^{α^{'}} = | α^{'} 〉 〈 α^{'} |

, (|α〉^′ ∈ Γ′_Pi). With the above conditions, the effective Hamiltonian in this system can be rewritten as

H_{eff} = \frac{η}{\sqrt{A}} (Ω_{11} (t) | ψ_{0}^{'} 〉 〈 ϕ_{1}^{'} | + Ω_{a 1} (t) | ψ_{0}^{'} 〉 〈 ϕ_{3}^{'} | + 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

i \bar{h} \frac{\partial I (t)}{\partial t} = [H_{eff} (t), I (t)]

. Since H_eff(t) possesses SU(2) dynamical symmetry, I(t) can be easily given by [37, 38]

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 .),

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

\begin{array}{l} \dot{ν} & = & \frac{η}{\sqrt{A}} [Ω_{11} (t) cos β - Ω_{a 1} (t) sin β], \\ \dot{β} & = & \frac{η}{\sqrt{A}} tan ν [Ω_{a 1} (t) cos β + Ω_{11} (t) sin β] . \end{array}

Then we can derive the expressions of Ω₁₁(t) and Ω_a1(t) easily as follows:

\begin{array}{l} Ω_{11} (t) & = & \frac{\sqrt{A}}{η} (\dot{β} cot ν sin β + \dot{ν} cos β), \\ Ω_{a 1} (t) & = & \frac{\sqrt{A}}{η} (\dot{β} cot ν cos β - \dot{ν} sin β) . \end{array}

The solution of Shrödinger equation ih̄∂|Ψ(t)〉/∂t = H_eff(t)|Ψ(t)〉 with respect to the instantaneous eigenstates of I(t) can be written as |Ψ(t)〉 = ∑_n=0,± C_ne^iθ_n|Φ_n(t)〉, where θ_n(t) is the Lewis-Riesenfeld phase in Eq. (5), C_n = 〈Φ_n(0)|ϕ′₁〉, and |Φ_n(t)〉 is the eigenstate of the invariant I(t)

\begin{array}{l} | Φ_{0} (t) 〉 & = & cos ν cos β | ϕ_{1}^{'} 〉 - i sin ν | ψ_{0}^{'} 〉 - cos ν sin β | ϕ_{3}^{'} 〉, \\ | Φ_{\pm} (t) 〉 & = & \frac{1}{\sqrt{2}} [(sin ν cos β \pm i sin β) | ϕ_{1}^{'} 〉 + i cos ν | ψ_{0}^{'} 〉 \\ - (sin ν sin β \mp i cos β) | ϕ_{3}^{'} 〉] . \end{array}

In order to transfer the population from state |ϕ′₁〉 to −|ϕ′₃〉, we choose the parameters as

ν (t) = ε, β (t) = \frac{π t}{2 t_{f}},

where ε is a time-independent small value and t_f is the total pulse duration. After the precise calculation, we can easily obtain

\begin{array}{l} Ω_{11} (t) & = & \frac{\sqrt{A} π}{η 2 t_{f}} cot ε sin \frac{π t}{2 t_{f}}, \\ Ω_{a 1} (t) & = & \frac{\sqrt{A} π}{η 2 t_{f}} cot ε cos \frac{π t}{2 t_{f}} . \end{array}

When t = t_f,

\begin{array}{l} | 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}^{'} 〉, \end{array}

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

\begin{array}{l} 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 . . \end{array}

By using the same method as the first step, we can obtain

\begin{array}{l} Ω_{21} (t) & = & \frac{\sqrt{A} π}{η 2 t_{f}} cot ε sin \frac{π t}{2 t_{f}}, \\ Ω_{12} (t) & = & \frac{\sqrt{A} π}{η 2 t_{f}} cot ε cos \frac{π t}{2 t_{f}} . \end{array}

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

\begin{array}{l} 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 . . \end{array}

We can obtain

\begin{array}{l} Ω_{a 2} (t) & = & \frac{\sqrt{A} π}{η 2 t_{f}} cot ε sin \frac{π t}{2 t_{f}}, \\ Ω_{22} (t) & = & \frac{\sqrt{A} π}{η 2 t_{f}} cot ε cos \frac{π t}{2 t_{f}} . \end{array}

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

\begin{array}{l} 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 . . \end{array}

At first, by using the same way as the three steps above we can obtain

\begin{array}{l} Ω_{1}^{'} (t) & = & \frac{\sqrt{A} π}{η 2 t_{f}} cot ε sin \frac{π t}{2 t_{f}}, \\ Ω_{a}^{'} (t) & = & \frac{\sqrt{A} π}{η 2 t_{f}} cot ε sin \frac{π t}{2 t_{f}} . \end{array}

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 Ω_i(t)/λ as a function of λt for a fixed value ε = 0.25, λ_A = λ_B = λ and t_f = 15/λ. The temporal evolution of the initial qubit states |g₀g₁〉_AB and |g₀g₂〉_AB 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 |g₀〉_A, 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(t_f)〉 as a function of ε and λt_f for the initial state |Ψ₀〉. From Fig. 4 we can see that, the effect of t_f on fidelity can be ignored, and we choose t_f = 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:

\begin{array}{l} \dot{ρ} & = & - i [H_{total}, ρ] - \frac{κ_{f}}{2} [b^{†} b ρ - 2 b ρ b^{†} + ρ b^{†} b] \\ - \sum_{j = A, B} \frac{κ_{j}}{2} [a_{j}^{†} a_{j} ρ - 2 a_{j} ρ a_{j}^{†} + ρ a_{j}^{†} a_{j}] \\ - \sum_{j = A, B} \sum_{k = g_{0}, g_{1}, g_{2}, a} \frac{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}}], \end{array}

where H_total = H₁ + H₂ + H₃ + H₄. κ_A(B) is the cavity A (B) decay rate, γ_A(B) is A (B) atomic spontaneous emission rate from the excited state |e〉_A(B) to the ground state |k〉_A(B)(k = g₀, g₁, g₂, a), respectively. σ_{e_j,k_j} = |e_j〉〈k_j (j = A, B). For simplicity, we assume κ_A = κ_B = κ_f = κ, γ_A = γ_B = 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.

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), Ω_a1(t) (solid blue line), Ω₂₁(t) (dash red line), Ω₁₂(t) (solid red line), Ω₂₂(t) (solid green line), Ω_a2(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 |g₀g₁〉_AB. (c) Time evolutions of the populations of the corresponding system states with the initial state |g₀g₂〉_AB. The system parameters are set to be ε = 0.25, λ_A = λ_B = λ and t_f = 15/λ.

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

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

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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 6²S_1/2 electronic ground state, |g₁〉 corresponds to F = 3, m = 3 hyperfine state of 6²S_1/2 electronic ground state, |g₂〉 corresponds to F = 4, m = 3 hyperfine state of 6²S_1/2 electronic ground state, |g₀〉 corresponds to F = 4, m = 4 hyperfine state of 6²S_1/2 electronic ground state, and the excited state |e〉 corresponds to F = 4, m = 3 hyperfine state of 6²P_1/2 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.

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Fast CNOT gate between two spatially separated atoms via shortcuts to adiabatic passage

Abstract

1. Introduction

2. Preliminary theory

2.1. Quantum Zeno dynamics

2.2. Lewis-Riesenfeld invariants

3. Shortcuts to adiabatic passage for the CNOT gate

4. Numerical simulations and feasibility analysis

5. Conclusion

Acknowledgments

References and links

Cited By

Figures (5)

Equations (36)

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