Lyapunov-based unified control method for closed quantum systems

Abstract

A Lyapunov-based unified control method for closed quantum systems is proposed in this paper for the state transfers from an arbitrary initial state to a desired target state. Especially, the proposed control method can be used in the cases that the free Hamiltonian of the system is not strongly regular, and (or) the control Hamiltonian is not fully connected. Three types of control laws are proposed simultaneously based on the implicit Lyapunov control theory, in which two implicit functions and one Lyapunov function with virtual mechanical quantity are well-designed. The largest invariant set of the quantum control system is investigated, and the convergence of the control system designed is proved. A numerical simulation is implemented in detail to instruct the design procedure and demonstrate the effectiveness of the control method proposed.

Introduction

The quantum state preparation, transfer and manipulation are the main objectives of quantum control. Any quantum control theory is usually a method to obtain the convergent control law so as to achieve the desired objective of the quantum control based on the dynamic evolution equation of quantum systems [1], [2]. A closed quantum system refers a quantum system in the absolute zero degree or without the interaction with the envirenment, and it is the basis of reseaching on the open quantum systems. People have developed many quantum control theories for controlling closed quantum systems in recent twenty years, such as geometric control [3], Lie group decomposition control [4], Bang-bang control [5], [6], sliding mode control [7], optimal control [8], [9], [10], [11], [12], robust learning control [13] and Lyapunov-based control [14]. Among these methods, the optimal quantum control theory was proposed by Rabitz in 1980’s [15] and has been sucessfully applied in many quantum control systems [16], [17]. The Lyapunov control theory was first introduced in quantum control system in 2003 [18], and many research achievements have been obtained [21], [19], [20], [22]. Eigenstate transfer in the Schrödinger equation was well studied [23], and the steering of superposition states and pure states were developed [24]. The following was target states being mixed or non-generic in Liouville-von Neumann equation [25]. People also studied the trajectory tracking by means of the Lyapunov-based control [26].

A finite dimensional closed quantum system can be described by the Liouville-von Neumann equation $\partial \rho /\partial t=-\mathrm{i}[H,\rho ],$ where H is the Hamiltonian of the quantum system, and ρ is the density operator. The quantum control dynamical equation is given by $\partial \rho /\partial t=-\mathrm{i}[H_0+{\sum }_{c=1}^m{H}_c{u}_c\left(t\right),\rho ],$ where H₀ is the free Hamiltonian of the quantum system, and H_c is the control Hamiltonian corresponding to the Lyapunov-based control law u_c(t). In order to guarantee that quantum system state can arrive the desired target state, one needs to design a convergent control law, and the existing Lyapunov-based control methods request the quantum control system to satisfy the following conditions [27]:

• H₀ should commute with the desired target state ρ_e.

• H₀ should be strongly regular, which means ${\omega }_{j_1,k_1}\ne {\omega }_{j_2,k_2}$ if $(j_1,k_1)\ne (j_2,k_2),$ where ${\omega }_{j,k}={\lambda }_j-{\lambda }_k$ is the transition frequencies of H₀, and λ_j is eigenvalue of H₀. If H₀ is strongly regular, the transition energies between two different levels are distinguishable.

• H_c should be fully connected, which means (H_c)_j,k ≠ 0 if j ≠ k. If H_c is fully connected, all the levels are directly coupled.

The contribution of this paper is to propose a unified Lyapunov-based control method for the state convergence of closed quantum systems under action of control laws by strict mathematical proof. The existence of the control laws and the stability of the (non-)degenerate and (not) ideal quatum systems have been proved. The largest invariant set of such quantum system have been analyzed, and the convergence of the quantum system, from an arbitrary initial state to any desired target state, have been made. Specifically, for a quantum system, if the free Hamiltonian is strongly regular, and the control Hamiltonian is fully connected, the quantum system is called non-degenerate, otherwise it is called degenerate. If the free Hamiltonian of a non-degenerate quantum system commutes with the desired target state, the quantum system is called ideal. In fact, most of the actual systems are degenerate or not ideal. For example, the desired target state may not be an equilibrium of the system, hence the free Hamiltonian may not commute with the desired target state. Other examples are the V-type and Λ-type quantum systems [28], [29], which are degenerate and often encountered in practice. All of these cases do not meet the conditions required by the existing Lyapunov-based control methods, which means that the existing Lyapunov-based control methods may not be used for these cases of quantum systems in practice. Thus, an applied quantum control method is very essential and important. This motivates us to study and propose a Lyapunov-based unified control method for general closed quantum systems in this paper: no matter what kind of Hamiltonian it has in a closed quantum system, the designed control system will converge to an arbitrary desired target state ρ_e from an arbitrary initial state ρ₀ under the proposed control laws. In other words, the unified control method for closed quantum systems proposed in this paper does not need special conditions or assumptions on the system structures.

In the following, we first describe the proposed unified method in general and main theorem to ensure the convergence of quantum control systems in Section 2. Then the detailed design procedures of the control laws are derived, and the lemmas needed for the main theorem is proved also in Section 2. The largest invariant set of the quantum control system is analyzed in detail, and the virtual mechanical quantity is deigned and discussed in Section 3. A numerical simulation is implemented to verify the effectiveness of the method proposed in Section 4. Finally, Section 5 is the conclusion.

Common Notations. We adopt the following notations in this paper. $\mathbb{C}\left(\mathbb{R}\right)$ denotes the complex (real) domain, ${\mathbb{C}}^n\left({\mathbb{R}}^n\right)$ denotes the set of the n-dimensional complex (real) column vectors, and ${\mathbb{C}}^{m\times n}$ denotes the set of m × n complex matrices. $C^1(-\infty ,\infty )$ denotes the set of all differentiable functions defined on $(-\infty ,\infty )$ whose derivative is continuous. For a complex number z, the notations Re(z), Im(z), |z| and z* stand for its real part, imaginary part, modulus and complex conjugate, respectively. δ( · ) is the Dirac delta function, and δ_j,k is the Kronecker delta. Uppercase and lowercase letters in bold font represent matrices and vectors, respectively. For a matrix M, M_j,k denotes its entry in row j, column k. The j-th element of a vector b is denoted by b_j. $\mathbf{M}=\text{diag}(a_1,\cdots ,a_n)$ means that M is an n × n diagonal matrix with diagonal entries $a_1,\cdots ,a_n$. The rank of M is denoted by rank(M). The exponential of M is denoted by exp(M). The transpose of M and b are denoted by M^T and b^T, respectively. The conjugate transpose M† of a matrix M is defined by $M_{j,k}^{†}=M_{k,j}^{*},$ and Tr(M) denotes the trace of matrix M. A dagger † is also used to indicate the Hilbert space adjoint A† of an operator A. The commutator [A, B] of two operators A, B, is defined as $AB-BA$. [A⁽ⁿ⁾, B] is defined recursively as: $[A^{\left(0\right)},B]=B,$ $[A^{\left(1\right)},B]=[A,B]$ and $[A^{\left(n\right)},B]=[A,[A^{(n-1)},B]],$ $n=1,2,\cdots$. U(n) is the Lie group of n × n unitary matrices, and O_u(n)(ρ) is the subset of U(n) whose elements share the same spectrum with ρ. Finally,  i denotes the imaginary unit which satisfies the equation ${\mathrm{i}}^2=-1$.