Fast coherent manipulation of quantum states in open systems

Jie Song; Zi-Jing Zhang; Yan Xia; Xiu-Dong Sun; Yong-Yuan Jiang

doi:10.1364/OE.24.021674

1. Introduction

The techniques based on adiabatic dynamics have been widely developed for manipulating quantum states because the adiabatic process is robust against the fluctuations of external control fields [1–4]. However, the required evolution time should be sufficiently long in the process due to the fact that the adiabatic approximation needs to be satisfied [5]. Therefore, much of the theoretical effort has been focused on speeding up adiabatic passage. For example, different authors apply the method of transitionless quantum driving to engineer fast evolution of quantum system [6–10]. In these protocols, the system in a given initial state is driven to a prescribed final state on a short time scale without losing the robustness property. By combining shortcuts to adiabaticity and optimal control, Campbell et. al. develop a hybrid method to achieve quantum state control in the Lipkin-Meshkov-Glick model [11]. Wei et. al. propose a modified shortcut scheme beyond rotating wave approximation (RWA) by adding a composite pulse in the absence of noise [12]. Ibáñes et. al design time-dependent pulses to perform quantum state transfer without RWA [13]. In addition, the method to shortcut of adiabatic evolution has been applied to the generation of quantum states and the implementation of quantum gate in cavity QED systems [14–16].

On the other hand, there is considerable interest in achieving the shortcuts for non-Hermitian systems. These methods based on non-Hermitian Hamiltonian make it possible to improve the speed of evolution by controlling the imaginary parts of non-Hermitian Hamiltonian [17–23]. In many systems, the actual implementation of shortcut process is hindered by many difficulties, the most prominent of which being the removal of quantum decoherence. Unfortunately, the inevitable coupling between a system and its surrounding environment is the cause of decoherence phenomena [24]. Thus it is necessary to investigate the dynamics evolution for a wide range of parameters in open systems.

In this paper, we discuss how to implement fast quantum state manipulation in a two-level system. We propose to introduce an additional Hamiltonian allowing for significant speedup when RWA is not valid in open systems. Because the additional Hamiltonian only depends on the real part of counter-rotating wave (CRW) terms, the influence of the imaginary part may be omitted during the evolution process. With the increasing of noisy parameters, the fast evolution passage cannot be achieved successfully. Using quantum feedback control, the noise is suppressed efficiently by showing that a high-fidelity quantum state is generated on short time scales. The feedback parameters and the noise strength need not be accurately controlled during the whole evolution process. The organization of this paper is as follows. In Sec. 2, we present the model that we shall study. Sec. 3 describes the mechanism of implementing the shortcuts in a noisy environment. In Sec. 4, a detailed analysis of the dynamics behaviors for various parameters is given and the experimental feasibility is discussed. Finally, we conclude with a summary of the paper in Sec. 5.

2. The model

We consider an open two-level system coupled to a general boson environment. The master equation in Lindblad form is given by

\dot{ρ} = - i [H_{s}, ρ] - \frac{τ}{2} [(L_{s}^{†} L_{s} ρ - 2 L_{s} ρ L_{s}^{†} + ρ L_{s}^{†} L_{s}),

where L_s = |0_s〉〈1_s| and τ is the spontaneous emission rate of atom. The subscript s denotes the system (atom). The system Hamiltonian without RWA is written as

H_{s} = Δ (t) | 1_{s} 〉 〈 1_{s} | + \frac{Ω (t) (1 + e^{- 2 i ω_{l} t})}{2} | 0_{s} 〉 〈 1_{s} | + H . c .,

where ∆(t) = ω(t) + ω₀ − ω_l is the effective detuning. ω₀ is the atomic transition frequency and ω_l is the frequency of driving field. The control of time-dependent parameter ω(t) is realized with an off-resonant Stark laser pulse. This is so-called RWA with neglecting the CRW term

Ω (t) e^{- 2 i ω_{1} t}

if the coupling strengths are much less than the effective detunning. However, the effect of CRW terms on the dynamics evolution cannot be neglected in open systems. To analyze the decoherence dynamics, we use the effective Hamiltonian approach [25,26] to solve master equation. Then the environment may be modelled as an ancilla which has the same dimension of Hilbert space as the atom. Let |m_s〉and |n_e〉 be the basis for the system and the ancilla, respectively. Here the subscript e denotes the ancilla. In the master equation, the density matrix has four elements. We assume that the matrix elements of system are mapped to the pure state of composite system

| Ψ (t) 〉 = \sum_{m, n = 0}^{1} ρ_{m n} (t) | m_{s} 〉 | n_{e} 〉 .

Here ρ_mn(t) = 〈m_s|ρ|n_s〉 and ρ is the density matrix of system in Eq. (1). The density matrix corresponds to the wave function which satisfies the following equation

i \partial t | Ψ (t) 〉 = i \sum_{m, n = 0}^{1} {\dot{ρ}}_{m n} (t) | m_{s} 〉 | n_{e} 〉 .

If one knows the wave function |Ψ(t)〉, the time evolution of system can be obtained by mapping the wave function back to the density matrix. Because the base vectors |m_s〉 and |n_e〉 are time-independent, the time derivative is given as

{\dot{ρ}}_{m n} = 〈 m_{s} | \dot{ρ} | n_{s} 〉

. Inserting the completeness relation and substituting Eq. (1) into Eq. (4), we get the following result

\begin{array}{l} i \partial_{t} | Ψ (t) 〉 = \sum_{m, n, p} 〈 m_{s} | H_{s, n o n} | p_{s} 〉 〈 p_{s} | ρ (t) | n_{s} 〉 | m_{s} 〉 | n_{e} 〉 \\ - \sum_{m, n, p} 〈 m_{s} | ρ (t) | p_{s} 〉 〈 p_{s} | H_{s, n o n}^{†} | n_{s} 〉 | m_{s} 〉 | n_{e} 〉 \\ + i τ \sum_{m, n, p, q} 〈 m_{s} | L_{s} | p_{s} 〉 〈 p_{s} | ρ (t) | q_{s} 〉 〈 q_{s} | L_{s}^{†} | n_{s} 〉 | m_{s} 〉 | n_{e} 〉 . \end{array}

Operators and Hamiltonian for the ancillary system satisfy the following relation [26]

〈 m_{e} | O_{e} | n_{e} 〉 = 〈 n_{s} | O_{s}^{†} | m_{s} 〉 .

Here the non-Hermitian Hamiltonian is

H_{s, n o n} = H_{s} - \frac{i π}{2} L_{s}^{†} L_{s}

. In Eq. (5), the second term and the third term are written as follows

\begin{array}{l} - \sum_{m, n, p} 〈 m_{s} | ρ (t) | p_{s} 〉 〈 p_{s} | H_{s, n o n}^{†} | n_{s} 〉 | m_{s} 〉 | n_{e} 〉 \\ \Rightarrow - \sum_{m, n, p} 〈 m_{s} | ρ (t) | p_{s} 〉 〈 n_{e} | H_{e, n o n} | p_{e} 〉 | m_{s} 〉 | n_{e} 〉, \end{array}

\begin{array}{l} i τ \sum_{m, n, p, q} 〈 m_{s} | L_{s} | p_{s} 〉 〈 p_{s} | ρ (t) | q_{s} 〉 〈 q_{s} | L_{s}^{†} | n_{s} 〉 | m_{s} 〉 | n_{e} 〉 \\ \Rightarrow i τ \sum_{m, n, p, q} 〈 m_{s} | L_{s} | p_{s} 〉 〈 p_{s} | ρ (t) | q_{s} 〉 〈 n_{e} | L_{e} | q_{e} 〉 | m_{s} 〉 | n_{e} 〉 . \end{array}

Then the master equation (1) may be rewritten in the following Schrödinger-like equation:

i \partial_{t} | Ψ (t) 〉 = \sum_{n} H_{s, n o n} ρ (t) | n_{s} 〉 | n_{e} 〉 - H_{e, n o n} ρ (t) | n_{s} 〉 | n_{e} 〉 + i τ L_{e} L_{s} ρ (t) | n_{s} 〉 | n_{e} 〉 .

Thus, we have

i \partial_{t} | Ψ (t) 〉 = H_{e f f} | Ψ (t) 〉 .

The effective Hamiltonian H_eff is given by

H_{e f f} = H_{s, n o n} - H_{e, n o n} + i τ L_{e} L_{s} .

The first two terms describe non-Hermitian evolution of the system and the ancilla, respectively. The third term which corresponds to quantum jump process [27] describes the coupling between atom and ancilla. In fact, the wave function equation (Eq. (10)) is equivalent to the markovian master equation. The excited state decays into ground state when the environmental noise is included. The evolution processes for ancilla and system are independent if the interaction between them is omitted (iτL_e L_s = 0). Then the states of system and ancilla can be written as a product state in the whole dynamics process. As a result, the non-Hermite Hamiltonian H_s,_non (H_e,_non) governs the evolution of system (ancilla).

Because we are interested in the system, only the dynamics of system is considered in this case. The angle β is defined by tan(2β) = (Ω(t) + Ω(t) cos(2ω_lt))/(∆(t) − iτ/2). The eigenstates of this Hamiltonian H_s,_non are

\begin{array}{l} | +_{s} 〉 = \sin (β) | 0_{s} 〉 + \cos (β) | 1_{s} 〉, \\ | -_{s} 〉 = \cos (β) | 0_{s} 〉 + \sin (β) | 1_{s} 〉 . \end{array}

The partners are

\begin{array}{l} | +_{s}^{~} 〉 = \sin (β *) | 0_{s} 〉 + \cos (β *) | 1_{s} 〉, \\ | -_{s}^{~} 〉 = \cos (β *) | 0_{s} 〉 + \sin (β *) | 1_{s} 〉 . \end{array}

Using this method [17], the additional Hamiltonian reads

H_{a} = i \sum_{χ = +, -} | \partial_{t} χ_{s} 〉 〈 {\tilde{χ}}_{s} | - 〈 {\tilde{χ}}_{s} | \partial_{t} χ_{s} 〉 | χ_{s} 〉 〈 {\tilde{χ}}_{s} | .

Substituting Eqs. (12)–(13) into Eq. (14), we have

H_{a} = (\begin{matrix} 0 & i \dot{β} \\ - i \dot{β} & 0 \end{matrix})

with

\dot{β} = \frac{[\dot{Ω} (t) (1 + \cos (2 ω_{l} t)) - 2 ω_{l} Ω (t) \sin (2 ω_{l} t)] ν - Ω (t) (1 + \cos (2 ω_{l} t)) \dot{Δ} (t)}{2 ν^{2} + 2 Ω {(t)}^{2} {(1 + \cos (2 ω_{l} t))}^{2}},

where ν = ∆(t) − ikτ/2 and H_a is the additional Hamiltonian corresponding to the system. In order to evaluate the role of τ in constructing H_a, τ has been replaced by kτ (k is a constant). When k = 1 or k = 0, it means that the noise term is included or not in the additional Hamiltonian. Correspondingly, if the parameter k is not zero in β, the real part in the additional pulse should be considered. The imaginary parts in H_s are included, while they are neglected in the additional Hamiltonian because the population transfer is perfect without considering them. We use the Allen-Eberly pulse with

Ω (t) = Ω_{0} sech (\frac{π t}{2 t_{0}})

and

Δ (t) = \frac{2 ϵ^{2} t_{0}}{π} \tanh (\frac{π t}{2 t_{0}})

. Without loss of generality, the initial state is chosen as |1〉_s. In Fig. 1(a), one notices that the population of state |0_s〉 is about 0.99 when the system reaches the steady state (t > 2.2×10⁻¹⁰s). The population transfer is completed perfectly, which shows that the effect of the imaginary parts of H_s may be neglected in constructing H_a. In Fig. 1(b), when the parameter k is not zero and the spontaneous emission rate is large enough, population transfer is freezed. With the choice of k = 0, the transfer happens and the population of |0〉 approaches 1. It shows that imaginary parts in the parameter

\dot{β}

might destroy the shortcut process as the spontaneous emission rate is large. Thus, the parameter k in Eq. (16) should be chosen as zero in the presence of noise.

Fig. 1 (a) Time evolution of populations of states |1_s〉 (green line) and |0_s〉 (blue line) for τ = 0; (b) Contour plots for the population of state |0_s〉 as a function of k and t with the parameter τ = 4Ω₀. The other parameters are Ω₀ = 2πMHz, ω_l = 10³ Ω₀, ϵ = 40Ω₀, and t₀ = 1 ns.

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3. Constructing shortcuts in open systems

The interaction between system and ancilla should be included in open systems (iτL_e L_s ≠ 0). Then the steady state will be a mixed state instead of pure state because the noise destroys the coherence of quantum state. The result is also easily understood based on the dynamics mechanism governed by the effective Hamiltonian. In the presence of the interaction between system and ancilla, the entanglement between them will be generated. After tracing over the degrees of freedom of ancilla, the system should be in a mixed state. If one wants to suppress the effect of noise, the term iτL_e L_s will be decreased in the effective Hamiltonian. Using quantum measurement theory [28], the general form of master equation and the form of quantum measurement might be put in one-to-one correspondence in the Markovian limit. The master equation may be expressed by

ρ (t + d t) = \sum_{ξ = 0, 1} Ω_{ξ} (t + d t) ρ (t) Ω_{ξ} {(t + d t)}^{†},

where Ω_ξ (t + dt) is an operator and dt is a very short time. ξ = {0, 1} corresponds to all possible results of quantum measurement. The time evolution of density matrix is described by the following equation

\begin{matrix} ρ (t + d t) = {\begin{matrix} L_{0} ρ (t) L_{0}^{†} \\ L_{0} ρ (t) L_{s}^{†} d t \end{matrix} & \begin{matrix} i f ξ = 0, \\ i f ξ = 1, \end{matrix} \end{matrix}

where L₀ = 1 – iH′ dt and

H^{'} = H_{s} - i τ L_{s}^{†} L_{s} / 2

. If we introduce the feedback control in open systems, the term

L_{s} ρ (t) L_{s}^{†} d t

is changed into

U_{s} L_{s} ρ (t) L_{s}^{†} U_{s}^{†} d t

. Thus the corresponding dissipative term in master equation is given by

L [ρ] = - \frac{τ}{2} (L_{s}^{†} L_{s} ρ - 2 U_{s} L_{s} ρ L_{s}^{†} U_{s}^{†} + ρ L_{s}^{†} L_{s}) .

The term

2 U_{s} L_{s} ρ L_{s}^{†} U_{s}^{†}

means that the feedback control is immediately performed on the system only when the spontaneous emission event is detected. After the term

2 L_{s} ρ L_{s}^{†}

is substituted for

2 U_{s} L_{s} ρ L_{s}^{†} U_{s}^{†}

in Eq. (1), the third term in Eq. (5) is given by

+ i τ \sum_{m, n, p, q} 〈 m_{s} | L_{s} U_{s} | p_{s} 〉 〈 p_{s} | ρ (t) | q_{s} 〉 〈 q_{s} | L_{s}^{†} U_{s}^{†} | n_{s} 〉 | m_{s} 〉 | n_{e} 〉 .

From the definition in Eq. (6), we find the following relation

〈 q_{s} | L_{s}^{†} U_{s}^{†} | n_{s} 〉 = 〈 n_{e} | U_{e} L_{e} | q_{e} 〉 .

Finally, we obtain the new effective Hamiltonian in the following form

H_{e f f}^{'} = H_{s, n o n} - H_{e, n o n} + i τ U_{e} U_{s} L_{e} L_{s} .

We can convert the master equation with dissipative term of Eq. (19) into an evolution equation derived from the effective Hamiltonian

H_{e f f}^{'}

.

Taking the noise into account, the effective Hamiltonian approach for solving the master equation will be helpful in interpreting further analysis. If the noise has no influence on the dynamics, the total density matrix may be written as ρ = ρ_s ⊗ ρ_e. Here ρ_s and ρ_e denote the density operators for system and ancilla, respectively. We define that tan(2β′) = (Ω(t) + Ω(t) cos(ω_lt))/∆(t). The adiabatic basis corresponding to system and ancilla is connected to the original basis |0〉 and |1〉 by

(\begin{matrix} | +_{l} 〉 \\ | -_{l} 〉 \end{matrix}) = m_{l} (\begin{matrix} | 0_{l} 〉 \\ | 1_{l} 〉 \end{matrix})

with

m_{l} = (\begin{matrix} \sin (β^{'}) & \cos (β^{'}) \\ \cos (β^{'}) & - \sin (β^{'}) \end{matrix}),

where l = {s, e}. The Schrödinger-like equation in the adiabatic basis takes the form

i \partial_{t} Ψ (t) = H^{T} Ψ (t),

where Ψ(t) = [s₁(t), s₂(t), s₃(t), s₄(t)]^T. s₁(t), s₂(t), s₃(t), and s₄(t) are the amplitudes of the states |+_s〉|+_e〉, |+_s〉|−_e〉, |−_s〉|+_e〉 and |−_s〉|−_e〉, respectively. The effective Hamiltonian in the adiabatic basis is expressed by

H^{T} = M (β^{'}) (H_{s} \otimes I) M {(β^{'})}^{- 1} - M (β^{'}) (I \otimes H_{e}) M {(β^{'})}^{- 1} - i M (β^{'}) \dot{M} {(β^{'})}^{- 1} + i τ M (β^{'}) U_{e} U_{s} L_{e} L_{s} M {(β^{'})}^{- 1} .

The transformation matrix M(β′) equals to m_s ⊗ m_e. Because the total density matrix can be written as a direct product, the Hamiltonian of system corresponding to the first three terms should be expressed by

H_{s}^{'} = (\begin{matrix} 2 \cos^{2} (β^{'}) & 0 \\ 0 & 2 \sin^{2} (β^{'}) \end{matrix}) \otimes I + (\begin{matrix} 0 & i {\dot{β}}^{'} \\ - i {\dot{β}}^{'} & 0 \end{matrix}) \otimes I,

where I is a 2×2 identity matrix. Therefore, when the additional Hamiltonian has the same form as that in Eq. (15), the non-diagonal matrix elements in Eq. (27) is eliminated. Then the evolution can be accelerated. In the last term, L_e L_s is

L_{e} L_{s} = (\begin{array}{l} 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}) .

The collective spontaneous emission in Eq. (28) drives the system into a mixed state. We choose the feedback operation as

\begin{matrix} U_{e} = (\begin{matrix} \cos (θ) & i \sin (θ) \\ i \sin (θ) & \cos (θ) \end{matrix}), & U_{s} = (\begin{matrix} \cos (θ) & - i \sin (θ) \\ - i \sin (θ) & \cos (θ) \end{matrix}), \end{matrix}

where 0 ≤ θ ≤ 2π. As a result, the last term is expressed as follows

i τ U_{e} U_{s} L_{e} L_{s} = i τ (\begin{array}{l} 0 & 0 & 0 & \cos^{2} (θ) \\ 0 & 0 & 0 & - i \cos (θ) \sin (θ) \\ 0 & 0 & 0 & i \cos (θ) \sin (θ) \\ 0 & 0 & 0 & \sin^{2} (θ) \end{array}) .

Here the operators U_e and U_s act on the ancilla and system, respectively. In the equation mentioned above, U_e and U_s is replaced by I ⊗ U_e and U_s ⊗ I, respectively. When the decay process of excited state is suppressed, the system can evolve along adiabatical basis. One can notice that the matrix elements in Eq. (30) depend on θ. When θ = π/2, the off-diagonal element which induces decoherence is changed into on-diagonal element. As long as θ ≠ 0, the noise term is decreased effectively due to the fact that the feedback control might drive the atom back to excited state once the decay process is detected. Thus the feedback scheme can be implemented by randomly choosing θ, which corresponds to the error operation in performing the feedback scheme. In this paper, θ is a random number between π/3 and π/2.

4. Results and discussions

To illustrate the suppression effect, we present the populations versus spontaneous emission rate without feedback control in Fig. 2(a), where one can find that the population transfer is almost perfect as the parameter τ is varied over a large range. As the strength of noise increases, the population of the wanted state decreases. Then a feedback operation is performed on the atom. Physically, the feedback control will compensate the energy loss in the noisy environment. We assume that the system starts in |1_s〉. The evolution of quantum state is illustrated by the path traced in the standard Bloch sphere representation where the three dimensional coordinates may be expressed by x = ρ₀₁ + ρ₁₀, y = i(ρ₀₁ − ρ₁₀), and z = ρ₀₀ − ρ₁₁, respectively (ρ_jk = 〈j_s|ρ|k_s〉, {j, k} = {0, 1}). The coherent evolution of qubit is of importance to implement quantum information processing. Figure 2(b) shows that the system is approximately confined to the pure state during the whole evolution (see solid line). In the absence of feedback control, the dashed-dotted line corresponding to evolution path lies inside the Bloch sphere, which shows the system state is a mixed state. In Fig. 2(c), the intended state is assuming to be $| ϕ 〉 = (| 0_{s} 〉 - | 1_{s} 〉) / \sqrt{2}$ . For the case, all the pulses will be turned off when the system is driven to the state |ϕ〉. We observe that the fidelity is higher than 90% if the feedback operation is implemented.

Fig. 2 (a) Time evolution of the populations of levels |0_s〉 and |1_s〉. The solid line, dashed line and dotted line correspond to the cases with τ = 5Ω₀ (dashed line), τ = 15Ω₀ (dotted line) and τ = 35Ω₀ (solid line), respectively. (b) The Bloch sphere of the density matrix. We have taken τ = 35Ω₀ and θ = π/2 in (b). (c) The fidelity of state |ϕ〉 versus time t with τ = 40Ω₀. θ is a random number (π/3 ≤ θ ≤ π/2). The other common parameters are the same as those in Fig. 1.

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The effects of the CRW terms on the population transfer are shown in Fig. 3(a). The blue (green) line corresponds to state |0_s〉 (|1_s〉). The populations vary with the increasing of driving field frequency ω_l when the CRW terms have not been included in additional Hamiltonian and the feedback control is absent. It shows that the influence of the CR terms should not be neglected in open systems. With the increasing of ω_l, the effects of the CRW terms will be decreased. In addition, the noise which is not large may be neglected when the frequency is changed within a large range. In this case, the additional Hamiltonian plays a key role during the whole evolution process. For example, without including the CRW terms in H_a, the population of state |ϕ〉 is about 0.95 when ω_l = 2π × 10²GHz in Fig. 3(a). However, if both CRW terms and noise terms paly a key role during the evolution, shortcut process may be performed perfectly by combining quantum feedback and the additional Hamiltonian. In Fig. 3(b), by using feedback control, the required interaction time is changed with the decreasing of the pulse duration t₀. One observes that the total time t is almost equal at t₀ = 1ns and t₀ = 0.1ns. In addition, the time t is decreased with choosing t₀ = 0.01ns. The phenomenon can be understood by dynamics process. The coupling constant in H_a is determined by the classical fields. With the choice of t₀ = 1ns and t₀ = 0.1ns, the pulses in H_a have a similar shape. As t₀ decreases, the width of peak in time domain becomes narrow. As a result, the evolution time is short with t₀ = 0.01ns.

Fig. 3 Plot of the populations for (a) ω_l = 2πGHz (dashed line), 2π × 10GHz (dotted line), and π × 10²GHz (solid line); (b) t₀ = 1ns (solid line), 0.1ns (dashed line), and 0.01ns (dotted line). The parameter is chosen as τ = 10Ω₀ and the other parameters are the same as in Fig. 1.

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Experimental implementation of the shortcut scheme without RWA will be feasible in different physical systems [29–31]. Our study shows that the CRW effect cannot be omitted when the ratio between ω_l and Ω is large. However, the imaginary parts of CRW terms may be neglected in noisy environment. Thus, the imaginary parts need not be considered in the pulse design corresponding to the additional Hamiltonian. In addition, the feedback operation is simple because only single qubit operation is performed on the qubit as the decay is detected. The implementation of the shortcuts to adiabaticity in open systems for the other model may be accomplished, i.e., in the Landau-Zener model, the pulse is expressed by Ω(t) = Ω₀(exp(−2iω_lt) + 1) and ∆(t) = ϵ²t. When we choose Ω₀ = 40MHz, ϵ = 400MHz, τ = 40MHz, and ω_l = 4GHz, the interaction time is about 1 × 10⁻⁷s and the fidelity of population transfer is 0.99 using the additional Hamiltonian and perfect feedback control.

We have studied the fast quantum state manipulation without RWA in the presence of noise. In open systems, we introduce an additional Hamiltonian to eliminate the effect of CRW terms on the shortcuts to adiabatic passage. The quantum state may be obtained with a high fidelity on a short time scale. The contributions of this paper can be summarized as follows: (1) The real part of CRW terms need be considered in designing the additional Hamiltonian, while the imaginary part can be omitted. (2) The noise terms need not been introduced in the additional Hamiltonian if the noise strength is large. (3) The effect of noise is suppressed when the additional Hamiltonian plays a main role during the evolution. (4) Quantum feedback control increases the fidelity under the condition that the noise cannot be neglected. The coherence evolution is realized during the whole process, which is important for quantum state manipulation.

Funding

National Natural Science Foundation of China (11675046; 11205037; 51176041; 11105030; 61575055), Doctoral Fund of Ministry of Education of China (20122302120003), PIRS OF HIT A201412, the Fundamental Research Funds for the Central Universities (AUGA5710056414), Natural Science Foundation of Heilongjiang Province of China (A201303), and Postdoctoral Scientific Research Developmental Fund of Heilongjiang Province (LBH-Q15060).

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Fast coherent manipulation of quantum states in open systems

Abstract

1. Introduction

2. The model

3. Constructing shortcuts in open systems

4. Results and discussions

Funding

References and links

Cited By

Figures (3)

Equations (30)

Optics Express