Hybrid Impulsive Control for Closed Quantum Systems

Abstract

The state transfer problem of a class of nonideal quantum systems is investigated. It is known that traditional Lyapunov methods may fail to guarantee convergence for the nonideal case. Hence, a hybrid impulsive control is proposed to accomplish a more accurate convergence. In particular, the largest invariant sets are explicitly characterized, and the convergence of quantum impulsive control systems is analyzed accordingly. Numerical simulation is also presented to demonstrate the improvement of the control performance.

1. Introduction

One of major concerns in quantum control is how to steer quantum states to a desired target state precisely and efficiently. A solution to this quantum state transfer problem will help us to advance some promising applications such as quantum computation and quantum chemistry. The main difficulty in quantum control is due to the limitations on the application of observation and feedback in quantum systems. Open-loop control has therefore been a commonly adopted approach in quantum control, where recorded control signals obtained from numerical simulations are implemented to real quantum systems. Among existing open-loop control design methods, the Lyapunov method could be the most popular one and has been tested in real applications [1–6]. Despite great advances have been made in Lyapunov methods, they may fail to achieve the control goal if the internal Hamiltonian is not strong regular [7]. This nonideal case means that distances between the eigenvalues of the internal Hamiltonian are not distinct. It is worth pointing out that this nonideal case does exist in many practical quantum systems such as coupled spin systems and harmonic oscillator systems [3, 4].

In particular, this paper will study the state transfer for closed quantum systems modeled as the following Schrödinger equation: 𝑖𝜓̇(𝑡)=𝐻₀𝜓(𝑡), (1) where 𝐻₀ is the internal Hamiltonian. For quantum systems, the control is implemented to the system through electromagnetic fields. Our basic problem is to transfer a quantum state from an initial state to a desired target state. The difficulty for the Lyapunov control of nonideal quantum system mainly comes from the fact that the system could be driven to undesired limit points in the invariant set. There exist a few results in the recent literature handling such a nonideal case. In [7], the complete controllability of the quantum systems with twofold degeneracy was investigated, and the basic idea is to apply a weak constant field to eliminate the degeneracy. In [4], the implicit Lyapunov method was used to deal with such a nonideal case. However, it is difficult to characterize invariant sets which are critical for the following convergence analysis. Therefore, we propose a new hybrid impulsive control strategy for closed quantum systems under the nonideal case. Nowadays, the impulsive control has proved to be an effective method to accomplish good performance for classical systems [8–12]. This motivates us to apply such a control idea to quantum systems.

The basic idea of the hybrid impulsive control is to divide the control into a piecewise continuous open-loop coherent control 𝑢₁(𝑡) and an impulsive control 𝑢₂. 𝑢₁(𝑡) design is similar to the traditional Lyapunov control which drives states to invariant sets. Specifically, the system under the piecewise continuous control can be described by 𝑖𝜓̇(𝑡)=𝐻₀+^𝑟_𝑙=1𝐻_𝑙𝑢_1𝑙(𝑡)𝜓(𝑡),𝑡≠𝑡_𝑘, (2) where 𝐻_𝑙 is the control Hamiltonian and 𝑢_1𝑙(𝑡) is real-valued control function (𝑙∈𝐽={1,2,…,𝑟}). The continuous-time coherent control 𝑢₁(𝑡) is implemented through the control Hamiltonians 𝐻_𝑙 when 𝑡≠𝑡_𝑘. Due to the nonideal quantum system, this control cannot guarantee the convergence to the desired target state. After a certain instant 𝑡_𝑘, the controlled state would sufficiently approach an undesired limit point. Hence, the control could fail to drive the system state. Then we need to “kick” the state out of undesired limit points and drive it re-converge to new, hopefully desired, state. At 𝑡=𝑡_𝑘, the piecewise continuous control 𝑢₁(𝑡) is switched off, and the “kicking” effect is accomplished by the impulsive control 𝑢₂(𝑡), by which the controlled system at the instant 𝑡_𝑘 becomes 𝑖𝜓̇(𝑡)=𝐻₀+𝐵_𝑘𝛿𝑠−𝑡_𝑘𝜓(𝑡),𝑡≠𝑡_𝑘, (3) where the introduced Hermitian operator 𝐵_𝑘 is the impulsive control Hamiltonian to be designed, 𝛿(⋅) is the Dirac impulse function with 𝑢₂(𝑡)=0, 𝑡≠𝑡_𝑘, and 𝑡_𝑘 is the impulsive instant at which the impulsive control is implemented, satisfying lim_𝑘→∞𝑡_𝑘=∞. With the impulsive control 𝑢₂(𝑡), the state satisfies |𝜓(𝑡_𝑘+ℎ)⟩=𝑒^{−𝑖∫𝑡𝑘𝑡𝑘+ℎ[𝐻0+𝐵𝑘𝛿(𝑠−𝑡𝑘)]𝑑𝑠}|𝜓(𝑡_𝑘)⟩, for sufficiently small ℎ>0. As ℎ→0⁺, we have lim_ℎ→0⁺|𝜓(𝑡_𝑘+ℎ)⟩=|𝜓(𝑡⁺_𝑘)⟩=𝑒^{−𝑖𝐵𝑘}|𝜓(𝑡_𝑘)⟩ and define |𝜓(𝑡_𝑘)⟩=|𝜓(𝑡⁻_𝑘)⟩=lim_ℎ→0⁺|𝜓(𝑡_𝑘−ℎ)⟩. In the control process, in order to keep the coherence of the controlled state, the proposed control strategy is the open-loop coherent control.

In practical implementations, the 𝛿 function in the impulsive control can be substituted with pulses of finite duration if the duration is sufficiently short compared with the time scale of quantum systems [13]. In recent years, the impulsive control idea has been used in the control of open quantum systems to suppress decoherence, for example, bang-bang pulses [13–18], and the minimal time control of spin systems [19, 20]. It is realized by a sequence of unitary operations, characteristic of instantaneous pulses. The feasibility of this impulsive control was supported by physical experiments; see [14–17] and the references therein. The so called hard pulses in NMR are analogous to this picture.

With this understanding, this paper will focus on the development of hybrid impulsive control design itself to achieve more accurate convergence under the nonideal cases. Moreover, the invariant set of the controlled system is characterized explicitly, which is shown to be strictly smaller than that obtained using classical Lyapunov methods. The convergence analysis is then obtained via an extending LaSalle invariance principle for impulsive systems in [21]. Simulation studies show improved control performance.

The rest of this paper is organized as follows. In Section 2, we design the hybrid impulsive control for nonideal systems by the Lyapunov function based on the state distance. Properties of the controlled system are discussed, and the convergence is analyzed by explicit characterization of the LaSalle invariant set. In Section 3, using the Lyapunov function based on the state error, the hybrid impulsive control of quantum systems is investigated. Section 4 includes numerical simulation to demonstrate the effectiveness and advantage of the proposed methods. Finally, some conclusions are drawn in Section 5.

2. Hybrid Impulsive Control Based on the State Distance

In practice, we do not have much freedom to choose the control Hamiltonian due to the structure limitations of the control fields [3, 4]. The impulsive control Hamiltonian cannot be chosen arbitrarily to achieve the state transfer instantaneously. Thus in this paper, 𝐻_𝑙 is fixed and assumed to be known beforehand. Denote 𝐵_𝑘=𝑒^{−𝑖𝐵𝑘}, which is unitary. With the hybrid impulsive control fields 𝑢(𝑡), system (1) becomes a closed quantum impulsive control system as follows: 𝑖𝜓̇(𝑡)=𝐻₀+^𝑟_𝑙=1𝐻_𝑙𝑢_1𝑙(𝑡)𝜓(𝑡),𝑡≠𝑡_𝑘,𝜓𝑡⁺_𝑘=𝐵_𝑘𝜓𝑡_𝑘. (4) In the following, we denote 𝐻=𝐻₀+∑^𝑟_𝑙=1𝐻_𝑙𝑢_1𝑙(𝑡).

2.1. Hybrid Impulsive Control Design and Dynamical Properties of Controlled Systems

In quantum control, the goal state |𝜓_𝑓⟩ is usually chosen to be an eigenstate of 𝐻₀, that is, 𝐻₀|𝜓_𝑓⟩=𝜆_𝑓|𝜓_𝑓⟩. We select a Lyapunov function based on the Hilbert-Schmidt distance between |𝜓⟩ and |𝜓_𝑓⟩, that is, 𝑉₁(𝜓(𝑡))=𝑉₁(𝑡)=(1/2)(1−|⟨𝜓_𝑓|𝜓⟩|²). When 𝑡≠𝑡_𝑘, the time derivative of 𝑉₁ is given by 𝑉̇₁(𝑡)=−^𝑟_𝑙=1𝑢_1𝑙ℑ𝜓𝜓_𝑓𝜓_𝑓𝐻_𝑙𝜓, (5) where ℑ(⋅) and ℜ(⋅) denote the imaginary part and real part of a complex number, respectively. When 𝑡=𝑡_𝑘, the Dini derivative of 𝑉₁ is given by 𝐷⁻𝑉₁(𝑡_𝑘)≜lim_ℎ→0⁺((𝑉₁(𝑡_𝑘)−𝑉₁(𝑡_𝑘−ℎ))/ℎ). The difference of 𝑉₁ is described as Δ𝑉₁(𝑡_𝑘)=𝑉₁(𝑡⁺_𝑘)−𝑉₁(𝑡_𝑘).

In order that the designed control can work in the case of the initial state being orthogonal to the goal state, we rewrite (5) as 𝑉̇₁(𝑡)=−∑^𝑟_𝑙=1𝑢_1𝑙|⟨𝜓|𝜓_𝑓⟩|ℑ[𝑒^{𝑖∠⟨𝜓|𝜓𝑓⟩}⟨𝜓_𝑓|𝐻_𝑙|𝜓⟩]. We need to design the control law such that 𝑉̇₁(𝑡)≤0, 𝑡≠𝑡_𝑘 and Δ𝑉₁(𝑡_𝑘)≤0. Choose the piecewise continuous control law as follows: 𝑢_1𝑙(𝑡)=𝐾_𝑙𝑓_𝑙ℑ𝑒^{𝑖∠⟨𝜓|𝜓𝑓⟩}𝜓_𝑓𝐻_𝑙𝜓,𝑙∈𝐽,𝑡≠𝑡_𝑘, (6) where 𝐾_𝑙>0 is the control gain and the function 𝑓_𝑙(⋅) passes through the origin of plane 𝑥_𝑙−𝑦_𝑙 monotonically satisfying 𝑓_𝑙(𝑥_𝑙)𝑥_𝑙≥0 with 𝑥_𝑙=ℑ[⟨𝜓_𝑓|𝐻_𝑙|𝜓⟩]. To avoid the confusion, we define ∠⟨𝜓|𝜓_𝑓⟩=0^∘ if ⟨𝜓|𝜓_𝑓⟩=0. In addition, the impulsive control matrix 𝐵_𝑘 should be chosen to satisfy Δ𝑉₁(𝑡_𝑘)≤0; that is, 𝐵^∗_𝑘𝜌_𝑓𝐵_𝑘−𝜌_𝑓≥0, (7) where 𝜌_𝑓=|𝜓_𝑓⟩⟨𝜓_𝑓| is the density matrix of target state |𝜓_𝑓⟩. Inequality (7) holds at least for the unitary matrix 𝐵_𝑘 which commutes with 𝜌_𝑓. The control law satisfying (6) and (7) is the designed control law. In the following, the properties of system (4) will be studied to show that 𝑢(𝑡) can make the system leave the initial state even if |𝜓(0)⟩ is an eigenstate of 𝐻₀.

Lemma 1. For control law (6) and (7), if the initial state is an eigenstate of 𝐻₀ with 𝐻₀|𝜓(0)⟩=𝜆₀|𝜓(0)⟩ and ⟨𝜓(0)|𝜓_𝑓⟩=0, then the following conclusions hold: (i)if there exists 𝑙∈𝐽 such that ℑ⟨𝜓_𝑓|𝐻_𝑙|𝜓(0)⟩≠0, then ⟨𝜓(𝑡)|𝜓_𝑓⟩≠0(𝑡>0); (ii)if ℑ⟨𝜓_𝑓|𝐻_𝑙|𝜓(0)⟩=0, for all 𝑙∈𝐽, and there exists 𝑙∈𝐽 such that ℜ⟨𝜓_𝑓|𝐻_𝑙|𝜓(0)⟩≠0 and 𝜆₀≠0, then ⟨𝜓(𝑡)|𝜓_𝑓⟩≠0, 𝑡>𝑡^′>0, where 𝑡^′ is sufficiently small; otherwise, the designed control fields cannot achieve the state steering of the closed-loop system.

Proof. (i) For a sufficiently small 𝑑𝑡, as 𝑡≠𝑡_𝑘, we have 𝑖|𝜓̇(0)⟩=𝑖lim_𝑑𝑡→0[|𝜓(𝑑𝑡)⟩−|𝜓(0)⟩]/𝑑𝑡=𝐻|𝜓(0)⟩; that is, as 𝑑𝑡→0, |𝜓(𝑑𝑡)⟩=(𝐼−𝑖𝐻𝑑𝑡)|𝜓(0)⟩. Since ⟨𝜓(0)|𝜓_𝑓⟩=0, the inequality ⟨𝜓_𝑓|𝜓(𝑑𝑡)⟩≠0 is equivalent to ∑^𝑟_𝑙=1𝑢_1𝑙⟨𝜓_𝑓|𝐻_𝑙|𝜓(0)⟩=∑^𝑟_𝑙=1𝑢_1𝑙(𝑖ℑ⟨𝜓_𝑓|𝐻_𝑙|𝜓(0)⟩+ℜ⟨𝜓_𝑓|𝐻_𝑙|𝜓(0)⟩)≠0. By (6), we have 𝑢_1𝑙ℑ⟨𝜓_𝑓|𝐻_𝑙|𝜓(0)⟩≥0, for all 𝑙∈𝐽. It follows from the assumption in case (i) that there exists 𝑙∈𝐽 such that 𝑢_1𝑙ℑ⟨𝜓_𝑓|𝐻_𝑙|𝜓(0)⟩>0, and consequently, ⟨𝜓_𝑓|𝜓(𝑑𝑡)⟩≠0. Since 𝑉̇₁≤0, we obtain that ⟨𝜓(𝑡)|𝜓_𝑓⟩≠0, 𝑡∈(0,𝑡₁). By (7) we have |⟨𝜓_𝑓|𝜓(𝑡⁺₁)⟩|≥|⟨𝜓_𝑓|𝜓(𝑡₁)⟩|=|⟨𝜓_𝑓|𝜓(𝑡⁻₁)⟩|>0. Hence, we obtain that ⟨𝜓(𝑡)|𝜓_𝑓⟩≠0, (𝑡>0).
(ii) Initially, the system evolves freely because 𝑢_1𝑙(0)=0, 𝑙∈𝐽. For a sufficiently small 𝑡^∗<𝑡₁, one can obtain that |𝜓(𝑡^∗)⟩=𝑒^{−𝑖𝐻0𝑡∗}|𝜓(0)⟩=𝑒^{−𝑖𝜆0𝑡∗}|𝜓(0)⟩. Moreover, as 𝑑𝑡→0, we have ⟨𝜓_𝑓|𝜓(𝑡^∗+𝑑𝑡)⟩=⟨𝜓_𝑓|(𝐼−𝑖𝐻𝑑𝑡)|𝜓(𝑡^∗)⟩=𝑑𝑡[−𝑖cos(𝜆₀𝑡^∗)−sin(𝜆₀𝑡^∗)]∑^𝑟_𝑙=1⟨𝜓_𝑓|𝐻_𝑙𝑢_1𝑙(𝑡^∗)|𝜓(0)⟩. Noticing that there exists 𝑙∈𝐽 such that ⟨𝜓_𝑓|𝐻_𝑙|𝜓(0)⟩≠0, we have 𝑢_1𝑙(𝑡^∗)=𝐾_𝑙𝑓_𝑙[ℑ(𝑒^{−𝑖𝜆0𝑡∗}⟨𝜓_𝑓|𝐻_𝑙|𝜓(0)⟩)]=𝐾_𝑙𝑓_𝑙[−sin(𝜆₀𝑡^∗)⟨𝜓_𝑓|𝐻_𝑙|𝜓(0)⟩]≠0. Similar to the discussion in case (i), we obtain that ⟨𝜓_𝑓|𝜓(𝑡^∗+𝑑𝑡)⟩≠0, and then ⟨𝜓(𝑡)|𝜓_𝑓⟩≠0, 𝑡>𝑡^∗. This completes the proof.

For the characterization of invariant sets, properties of the states such that 𝑉̇₁=0 are studied. Since the proof is similar to that of Proposition 4 in [1], we omit it here.

Lemma 2. If ⟨𝜓₀|𝜓_𝑓⟩=0 and the conditions (i) or (ii) in Lemma 1 hold, the following conclusions are equivalent: (i)𝑉̇₁(𝑡)=0, 𝑡≠𝑡_𝑘,(ii)𝑖|𝜓̇(𝑡)⟩=𝐻₀|𝜓(𝑡)⟩, 𝑡≠𝑡_𝑘,(iii)there exists 𝜆_𝑙∈ℝ such that ⟨𝜓_𝑓|(𝜆_𝑙𝐼−𝐻_𝑙)|𝜓(𝑡)⟩=0, 𝑡≠𝑡_𝑘, 𝑙∈𝐽.

Lemma 2 only characterizes the states guaranteeing that 𝑉̇₁=0 at specific instants. We need to characterize the states from which the system trajectories stay in the set 𝑉̇₁=0 and Δ𝑉₁=0. We first present the extensive LaSalle invariance principle for impulsive systems in [21].

Lemma 3. Consider the following differential impulsive system on an open set 𝒟: 𝑥̇(𝑡)=𝑓_𝑐(𝑥(𝑡)),𝑥(0)=𝑥₀,𝑡≠𝑡_𝑘,Δ𝑥(𝑡)=𝑓_𝑑(𝑥(𝑡)),𝑡=𝑡_𝑘. (8) If there exists a continuous function 𝑉 such that 𝑉^′(𝑥)𝑓_𝑐(𝑥)≤0,𝑡≠𝑡_𝑘 and Δ𝑉(𝑡_𝑘)≤0, then 𝑥(𝑡)→ℳ as 𝑡→∞, where ℳ is the largest invariant set contained in 𝔼≜{𝑥∶𝑉^′(𝑥)𝑓_𝑐(𝑥)=0}∩{𝑥∶Δ𝑉(𝑥)=0}.

2.2. Convergence Analysis

The following theorem presents the characterization of the invariant set for the nonideal systems under the hybrid impulsive control, by which the invariant set is smaller compared with that obtained by the conventional Lyapunov method. In the following, the unitary matrix 𝐵_𝑘 is designed such that it commutes with 𝐻₀, 𝑘∈ℤ⁺.

Theorem 4. Consider system (4) with the hybrid impulsive control satisfying (6) and (7). The largest invariant set is given by 𝐺=𝕊^2𝑛−1⋂𝐸₁⋂𝐸₂ with 𝐸₁={|𝜓⟩∶|𝜓⟩∈𝑀^𝑙_𝑘,forall𝑙∈𝐽,𝑘=1,2,…}, 𝐸₂={|𝜓⟩∶|𝜓⟩∈𝑁_𝑘,𝑘=1,2,…}, and 𝑀^𝑙₁∶=𝜓∶ℑ𝜓𝜓_𝑓𝜓_𝑓𝑋^𝑠𝑙_𝑙𝜓𝑠=0,_𝑙=1,2,…,𝑚_𝑙,𝑀^𝑙_𝑘∶=𝜓∶ℑ𝜓^𝑘−1_𝑗=1𝐵^∗_𝑗𝜓_𝑓𝜓_𝑓𝑋^𝑠𝑙_𝑙¹_{𝑗=𝑘−1}𝐵_𝑗𝜓=0,𝑠_𝑙=1,2,…,𝑚_𝑙𝑁,𝑘≥2,_𝑘∶=𝜓∶𝜓^𝑘−1_𝑗=1𝐵_𝑗𝜓_𝑓²−𝜓𝜓_𝑓²=0,𝑘≥1, (9) where 𝑋¹_𝑙,𝑋²_𝑙,…,𝑋^𝑚𝑙_𝑙 constitute the basis of the set {(𝑖)^𝑠[𝐻₀^(𝑠),𝐻_𝑙],𝑠=0,1,2,…}, 𝑙∈𝐽. Hence, system (4) converges to 𝐺 under the hybrid impulsive control.

Proof. When 𝑡=𝑡₀, from (6), we obtain that 𝑉̇₁𝑡₀=0⟺𝜓𝑡₀𝜓_𝑓×ℑ𝑒^{𝑖∠⟨𝜓(𝑡0)|𝜓𝑓⟩}𝜓_𝑓𝐻_𝑙𝜓𝑡₀𝜓𝑡=0,𝑙∈𝐽⟺ℑ₀𝜓_𝑓𝜓_𝑓𝐻_𝑙𝜓𝑡₀=0. (10) The main idea of the proof is sketched as follows. The interval [𝑡_𝑘−1,𝑡_𝑘] is divided into 𝑛_𝑘 sufficiently small intervals with duration 𝑑𝑡. We apply the Taylor expansion on the system state and omit the high order terms of 𝑑𝑡. By Lemma 2, the requirements 𝑉̇₁(𝑡)=0(𝑡≠𝑡_𝑘) and Δ𝑉₁(𝑡_𝑘)=0 for the whole system trajectory will be transformed to the conditions on the initial state. By the Taylor expansion and commutativity between 𝐻₀ and 𝐵_𝑘, it yields that 𝑉̇₁𝑡₀𝜓𝑡+𝑑𝑡=0⟺ℑ₀+𝑑𝑡𝜓_𝑓𝜓_𝑓𝐻_𝑙𝜓𝑡₀𝜓𝑡+𝑑𝑡=0⟺ℑ₀𝐼+𝑖𝐻₀𝑑𝑡𝜓_𝑓×𝜓_𝑓𝐻_𝑙𝐼−𝑖𝐻₀𝑑𝑡𝜓𝑡₀𝑖𝜓𝑡=0⟺ℑ₀𝜓_𝑓𝜓_𝑓𝐻₀,𝐻_𝑙𝜓𝑡₀⋮𝐷=0,₋𝑉₁𝑡₁(𝑖)=0⟺ℑ^𝑛1𝜓𝑡₀𝜓_𝑓×𝜓_𝑓𝐻^(𝑛1)₀,𝐻_𝑙𝜓𝑡₀=0. (11) At 𝑡=𝑡_𝑘+𝑑𝑡, the free evolution of |𝜓(𝑡)⟩ is given by 𝜓𝑡_𝑘+𝑑𝑡=𝐼−𝑖𝐻₀𝑑𝑡𝜓𝑡⁺_𝑘=𝐵_𝑘𝐼−𝑖𝐻₀𝑑𝑡𝜓𝑡_𝑘,𝑘=1,2,…. (12) Similar to the previous deduction, it follows from (11) and (12) that 𝑉̇₁𝑡₁𝜓𝑡+𝑑𝑡=0⟺ℑ₁+𝑑𝑡𝜓_𝑓𝜓_𝑓𝐻_𝑙𝜓𝑡₁(𝑖)+𝑑𝑡=0⟺ℑ^𝑠𝜓𝑡⁺₁𝜓_𝑓×𝜓_𝑓𝐻₀^(𝑠),𝐻_𝑙𝜓𝑡⁺₁(𝑖)=0,𝑠=0,1⟺ℑ^𝑠𝜓𝑡₁−𝑑𝑡𝐵^∗₁𝜓_𝑓×𝜓_𝑓𝐻₀^(𝑠),𝐻_𝑙𝐵₁𝜓𝑡₁(𝑖)−𝑑𝑡=0,𝑠=0,1,2,⟺ℑ^𝑠𝜓𝑡₀𝐵^∗₁𝜓_𝑓×𝜓_𝑓𝐻₀^(𝑠),𝐻_𝑙𝐵₁𝜓𝑡₀=0,𝑠=0,1,…,𝑛₁𝐷+1,₋𝑉₁𝑡₂(𝑖)=0⟺ℑ^𝑠𝜓𝑡₀𝐵^∗₁𝜓_𝑓×𝜓_𝑓𝐻₀^(𝑠),𝐻_𝑙𝐵₁𝜓𝑡₀=0,𝑠=0,1,…,𝑛₁+𝑛₂. (13) Consequently, it can be obtained that 𝐷⁻𝑉₁𝑡_𝑘(𝑖)=0⟺ℑ^𝑠𝜓𝑡₀^𝑘−1_𝑗=1𝐵^∗_𝑗𝜓_𝑓×𝜓_𝑓𝐻₀^(𝑠),𝐻_𝑙¹_{𝑗=𝑘−1}𝐵_𝑗𝜓𝑡₀=0, (14) where 𝑠=0,1,…,∑^𝑘_𝑖=1𝑛_𝑖. Noticing that the set {(𝑖)^𝑠[𝐻₀^(𝑠),𝐻_𝑙],𝑠=0,1,…,∑^𝑘_𝑖=1𝑛_𝑖}, 𝑙∈𝐽 has finite dimension, we denote its basis to be 𝑋¹_𝑙,𝑋²_𝑙,…,𝑋^𝑚𝑙_𝑙, 𝑙∈𝐽. Since the division of the interval [𝑡_𝑘−1,𝑡_𝑘] is random, (14) can be rewritten as 𝐷⁻𝑉₁𝑡_𝑘𝜓𝑡=0⟺ℑ₀^𝑘−1_𝑗=1𝐵^∗_𝑗𝜓_𝑓×𝜓_𝑓𝑋^𝑠𝑙_𝑙¹_{𝑗=𝑘−1}𝐵_𝑗𝜓𝑡₀𝑠=0,_𝑙=1,…,𝑚_𝑙. (15) For convenience, the set of the states satisfying (15) is denoted as 𝑀^𝑙_𝑘 in (9), 𝑙∈𝐽, 𝑘≥2.
In the following, we will discuss the conditions on the initial states from which the trajectories stay in the set {|𝜓⟩∶Δ𝑉₁(𝑡_𝑘)=0,𝑘=1,2,…}: Δ𝑉₁𝑡₁⟺𝜓𝑡=0₁𝐵^∗₁𝜓_𝑓²−𝜓𝑡₁𝜓_𝑓²⟺𝜓𝑡=0₁−𝑑𝑡𝐵^∗₁𝜓_𝑓²−𝜓𝑡₁−𝑑𝑡𝜓_𝑓²⟺𝜓𝑡=0₀𝐵^∗₁𝜓_𝑓²−𝜓𝑡₀𝜓_𝑓²=0. (16) Applying the similar technique, it follows that Δ𝑉₁𝑡_𝑘=0⟺𝜓𝑡₀^𝑘_𝑗=1𝐵^∗_𝑗𝜓_𝑓²−𝜓𝑡₀𝜓_𝑓²=0. (17) We denote the set of the states guaranteeing (17) to be 𝑁_𝑘 in (9).
In conclusion, all the states which stay in the intersection 𝐸₁∩𝐸₂ constitute the largest invariant set of system (4). By Lemma 3, we complete the proof.

It should be noticed that the basis of the set {(𝑖)^𝑠[𝐻₀^(𝑠),𝐻_𝑙],𝑠=0,1,2,…,∑^𝑘_𝑖=1𝑛_𝑖} can be obtained in finite steps, 𝑙∈𝐽. If the first 𝑛² elements in the set are linearly independent, then they constitute the basis. If [𝐻^(𝑠𝑙₀⁺¹⁾,𝐻_𝑙] can be represented by the first 𝑠_𝑙 elements, it is easy to obtain that [𝐻₀^(𝑘),𝐻_𝑙] can be represented by the linear combination of 𝐻_𝑙,[𝐻₀,𝐻_𝑙],…,[𝐻^(𝑠𝑙)₀,𝐻_𝑙], for all 𝑘>𝑠_𝑙.

Corollary 5. Consider system (4) with control field (6) without the impulsive control, that is, 𝐵_𝑘=𝐼, 𝑘=1,2,…. The largest invariant set is 𝐸⋂𝕊^2𝑛−1, where 𝐸={|𝜓⟩∶|𝜓⟩∈𝑀^𝑙_𝑘,𝑙∈𝐽,𝑘=0,1,…}, 𝑀^𝑙_𝑘={|𝜓⟩∶ℑ(𝑖^𝑘⟨𝜓|𝜓_𝑓⟩⟨𝜓_𝑓|[𝐻₀^(𝑘),𝐻_𝑙]|𝜓⟩)=0}, 𝑘≥0, and 𝑙∈𝐽.

Remark 6. If 𝐻₀ is strong regular, then the result in Corollary 5 reduces to Theorem 2 in [1]. For the nonideal case, it is clear that 𝐺⊂𝐸. From the viewpoint of physics, this implies that the proposed hybrid impulsive control can achieve more accurate convergence under the nonideal case. In general, the matrix 𝐵_𝑘 can be chosen to guarantee that 𝐸₁ and 𝐸₂ contain finite sets 𝑀^𝑙_𝑘 and 𝑁_𝑘. It can be found that the invariant set 𝐺 depends on the choice of impulsive control matrix 𝐵_𝑘. The optimal determination of 𝐵_𝑘 and impulsive instants 𝑡_𝑘 to minimize the invariant set are under study.

3. Hybrid Impulsive Control Based on the State Error

It is known that different Lyapunov functions may have different control effects. The relations among them were studied in our previous work [4]. In this section, we consider the hybrid impulsive control of quantum systems based on the state error between the controlled state and the goal state. Let 𝑉₂(𝜓(𝑡))=𝑉₂(𝑡)=(1/2)⟨𝜓(𝑡)−𝜓_𝑓|𝜓(𝑡)−𝜓_𝑓⟩=1−ℜ⟨𝜓_𝑓|𝜓(𝑡)⟩. Similar to the hybrid control design in Section 1, we consider the following quantum impulsive control system which is different from (4): 𝑖𝜓̇(𝑡)=𝐻₀+𝑢₁𝐻₁+𝜔𝐼𝜓(𝑡),𝑡≠𝑡_𝑘,𝜓𝑡⁺_𝑘=𝐵_𝑘𝜓𝑡_𝑘, (18) where 𝜔 is a new real scalar control field. For the convenience of the computation, the introduced 𝜔 may be used to adjust the global phase without changing the physical quantities regarding |𝜓⟩. While in practical implementation, it is not necessary to be implemented to the system. Similar conclusion can be drawn if there exists more than one control Hamiltonian 𝐻_𝑙, 𝑙≥2. The time derivative of 𝑉₂ is 𝑉̇₂(𝑡)=−𝜆_𝑓+𝜔ℑ𝜓_𝑓𝜓−ℑ𝜓_𝑓𝐻₁𝜓𝑢₁,𝑡≠𝑡_𝑘. (19) Let 𝑢₀=𝜆_𝑓+𝜔. We design the following control to ensure 𝑉̇₂(𝑡)≤0, 𝑡≠𝑡_𝑘: 𝜆_𝑓+𝜔=𝑢₀=𝐾₀𝑓₀ℑ𝜓_𝑓𝜓,𝑢₁=𝐾₁𝑓₁ℑ𝜓_𝑓𝐻₁𝜓, (20) where 𝐾₁>0, and the function 𝑦₁=𝑓₁(⋅) is defined as that in (6).

Theorem 7. Consider system (18) with control fields (7) and (20). The largest invariant set is given by 𝐾=𝐹₁⋂𝐹₂⋂𝕊^2𝑛−1, where 𝐹₁={|𝜓⟩∶|𝜓⟩∈𝑈_𝑘,𝑘=1,2,…}, 𝐹₂={|𝜓⟩∶|𝜓⟩∈𝑊_𝑘,𝑘=1,2,…}, and 𝑈₁∶={|𝜓⟩∶ℑ(⟨𝜓_𝑓|𝑋^𝑠|𝜓⟩)=0,𝑠=1,…,𝑚₁}, 𝑈_𝑘∶=𝜓∶ℑ𝜓_𝑓𝑋^𝑠1_{𝑗=𝑘−1}𝐵_𝑗𝜓=0,𝑠=1,…,𝑚₁,𝑊𝑘≥2,_𝑘∶=𝜓∶ℜ𝜓_𝑓𝜓−ℜ𝜓_𝑓^𝑘_𝑗=1𝐵_𝑗𝜓=0,𝑘≥1, (21) where 𝑋¹,𝑋²,…,𝑋^𝑚1 are the basis of the set {𝐼,(𝑖)^𝑠[𝐻₀^(𝑠),𝐻₁],𝑠=0,1,2,…}. Therefore, system (18) converges to 𝐾 with the hybrid impulsive control satisfying (7) and (20).

Proof. Let 𝜔=−𝜆_𝑓. When the system satisfies 𝑉̇₂=0, that is, 𝑢₁=0, the evolution of system (18) becomes 𝑖𝜓̇(𝑡)=𝐻₀−𝜆_𝑓𝐼𝜓(𝑡),𝑡≠𝑡_𝑘,𝜓𝑡⁺_𝑘=𝐵_𝑘𝜓𝑡_𝑘. (22) It follows from (22) that 𝜓𝑡_𝑘−1=𝜓𝑡+𝑑𝑡_𝑘−1+𝜓̇𝑡_𝑘−1=𝐻𝑑𝑡𝐼−𝑖₀−𝜆_𝑓𝐼𝑑𝑡𝜓𝑡_𝑘−1. (23) From (20), we obtain the following relation: 𝑉̇₂𝑡₀=0⟺ℑ𝜓_𝑓𝜓𝑡₀ℑ𝜓=0,_𝑓𝐻₁𝜓𝑡₀=0. (24) Similarly, we divide the interval [𝑡_𝑘−1,𝑡_𝑘] into 𝑛_𝑘 sufficiently small intervals. From (22)–(24), we have 𝐷⁻𝑉₂(𝑡₁)=0⇔ℑ(𝑖^𝑛1⟨𝜓_𝑓|[𝐻^(𝑛1)₀,𝐻₁]|𝜓(𝑡₀)⟩)=0. According to the similar method in the proof of Theorem 4, when 𝑡=𝑡_𝑘, it yields that 𝐷⁻𝑉₂𝑡_𝑘𝜓=0⟺ℑ_𝑓¹_{𝑗=𝑘−1}𝐵_𝑗𝜓𝑡₀ℑ𝑖=0,_𝑠𝜓_𝑓𝐻₀^(𝑠),𝐻₁¹_{𝑗=𝑘−1}𝐵_𝑗𝜓𝑡₀=0, (25) where 𝑠=0,…,∑^𝑘_𝑖=1𝑛_𝑖. Denote the basis of the set {𝐼,(𝑖)^𝑠[𝐻₀^(𝑠),𝐻₁],𝑠=0,1,…,∑^𝑘_𝑖=1𝑛_𝑖} to be 𝑋¹,𝑋²,…,𝑋^𝑚1. Equation (25) can be rewritten as 𝐷⁻𝑉₂(𝑡_𝑘)=0⇔ℑ⟨𝜓_𝑓|𝑋^𝑠∏¹_{𝑗=𝑘−1}𝐵_𝑗|𝜓(𝑡₀)⟩=0,𝑠=1,2,…,𝑚₁. This equality is denoted as 𝑈_𝑘 in (21).
Next, we characterize the initial states from which the system trajectories stay in {|𝜓⟩∶Δ𝑉₂(𝑡_𝑘)=0,𝑘=1,2,…}. From the definition of 𝑉₂, we have Δ𝑉₂(𝑡₁)=ℜ⟨𝜓_𝑓|𝜓(𝑡⁺₁)⟩−ℜ⟨𝜓_𝑓|𝜓(𝑡₁)⟩=ℜ⟨𝜓_𝑓|𝐵₁|𝜓(𝑡₁)⟩−ℜ⟨𝜓_𝑓|𝜓(𝑡₁)⟩. The following relations can be obtained: Δ𝑉₂𝑡₁𝜓=0⟺ℜ_𝑓𝜓𝑡₁−ℜ𝜓_𝑓𝐵₁𝜓𝑡₁𝜓=0⟺ℜ_𝑓𝐼−𝐵₁𝐼−𝑖𝐻₀−𝜆_𝑓𝐼×𝜓𝑡𝑑𝑡₁𝜓−𝑑𝑡=0⟺ℜ_𝑓𝜓𝑡₀−ℜ𝜓_𝑓𝐵₁𝜓𝑡₀=0. (26) By similar deduction, it yields that Δ𝑉₂(𝑡_𝑘)=0⇔ℜ⟨𝜓_𝑓|𝜓(𝑡₀)⟩−ℜ⟨𝜓_𝑓|∏^𝑘_𝑗=1𝐵_𝑗|𝜓(𝑡₀)⟩=0, which can be denoted as 𝑊_𝑘 in (21).
In conclusion, all the states which remain in the intersection 𝐹₁∩𝐹₂ constitute the largest invariant set of controlled system (18). By Lemma 3, the proof is completed.

Similar to the discussion in Corollary 5, Theorem 7 can be reduced to Theorem 8 in [1] if there is no impulsive control, and 𝐻₀ is strong regular. We can see that our result reduces the invariant set for the nonideal case. This implies that the proposed hybrid impulsive control scheme can accomplish more accurate state transfer.

4. Numerical Simulation

Example 1. Consider the five-level system with the internal Hamiltonian and impulsive control Hamiltonian given by 𝐻₀=diag{1.0,1.2,1.2,2.0,2.15} and 𝐵_𝑘=diag{0,0,−𝜋,−𝜋,0}, respectively. The unitary operation 𝐵_𝑘=𝑒^{−𝑖𝐵𝑘}=diag{1,1,−1,−1,1} can be realized by performing the planar rotation on system states. It can be found that the system is a nonideal system. The control Hamiltonians are given by 𝐻₁=^{00𝑖𝑖000𝑖0𝑖−𝑖−𝑖000}_{−𝑖00000−𝑖000}, 𝐻₂=^{0𝑖000−𝑖00𝑖0000𝑖𝑖}_{0−𝑖−𝑖0𝑖00−𝑖−𝑖0}. Let the target state be |𝜓_𝑓⟩=00100^⊤, let the initial state be |𝜓₀⟩=01000^⊤, and let the controlled state be |𝜓⟩=[𝑐₁,𝑐₂,…,𝑐₅]^⊤. Take the control function to be 𝑓_𝑖(𝑥)=𝑥,𝑖=1,2. Choose the impulsive instant to be 𝑡_𝑘=3𝑘−1,𝑘∈ℤ⁺ and 𝐾₁=𝐾₂=0.2. Using the hybrid impulsive control based on the state distance, simple computation yields that the invariant set 𝐺={|3⟩} (without regard to the global phase), which implies that under the hybrid impulsive control the system converges to |𝜓_𝑓⟩. The populations of the controlled system are illustrated in Figure 1.
Now we compare performance of the hybrid impulsive control with that of classical Lyapunov control. If the impulsive control is not applied to the system, then the hybrid impulsive control is reduced to the classical Lyapunov control, by which the performance of the controlled system is shown in Figure 2. Hence, the proposed hybrid impulsive control improves the control performance.

Figure 1 

The population of the system state by the hybrid impulsive control based on the state distance.

Figure 2 

The population of the system state by the Lyapunov control without impulsive control.

Example 2. Consider the five-level system with the same internal Hamiltonian as the previous example. Let the target state and the initial state be |𝜓_𝑓⟩=00001^⊤ and |𝜓₀⟩=10000^⊤, respectively, and the impulsive control Hamiltonian 𝐵_𝑘=diag{−𝜋,0,0,−𝜋,0}. The unitary operation is chosen as 𝐵_𝑘=𝑒^{−𝑖𝐵𝑘}=diag{−1,1,1,−1,1}. The system is a nonideal system. The control Hamiltonian is given by 𝐻₁=^{0𝑖00𝑖−𝑖00𝑖𝑖0000𝑖}_{0−𝑖00𝑖−𝑖−𝑖−𝑖−𝑖0}. Let the control function be 𝑓₁(𝑥)=𝑥, and 𝐾₀=0.1, 𝐾₁=0.2. Choose the impulsive instant as 𝑡_𝑘=3𝑘−1, 𝑘∈ℤ⁺. Using the hybrid impulsive control based on the state error, simple computation yields that the invariant set 𝐾={|5⟩} (without regard to the global phase), which implies that under the hybrid impulsive control the system converges to |𝜓_𝑓⟩. Simulation results are illustrated in Figure 3.
When the hybrid impulsive control is reduced to the classical Lyapunov control, the trajectory of the controlled system is plotted in Figure 4. Moreover, if the implicit Lyapunov control strategy in [4] is employed with the same parameters, it fails to drive the system, as illustrated in Figure 5. Therefore, the proposed hybrid impulsive control improves the control performance.

Figure 3 

The population of the system state by the hybrid impulsive control based on the state error.

Figure 4 

The population of the system state by the Lyapunov control without impulsive control.

Figure 5 

The population of the system state by the implicit Lyapunov control in [4].

5. Conclusion

In this paper, the coherent hybrid impulsive control for closed quantum systems has been investigated for the nonideal case that 𝐻₀ is not strong regular. The dynamical properties of the resulted quantum impulsive control system have been discussed to facilitate the convergence analysis. Based on two kinds of Lyapunov functions, the largest invariant sets have been characterized explicitly. Consequently, more accurate convergence of the controlled system has been achieved by the extensive LaSalle invariance principle. Compared with some existing results, the improved control performance has been shown for the nonideal case. Since the practical implementation of impulsive control has been studied in known literature, we believe that it is feasible. The optimal determination of the impulsive control Hamiltonian and impulsive instants is worth to be explored in the future work.

Acknowledgment

This work is supported by the NNSF of China under Grant nos. 61174039, 61203128, the Connotative Construction Project of Shanghai University of Engineering Science (nhky-2012-13), the Fundamental Research Funds for the Central Universities of China, the “Chen Guang” Project supported by Shanghai Municipal Education Commission and Shanghai Education Development Foundation (no. 12CG65), and the Research Funds of Shanghai University of Engineering Science no. 2012td19.