Lyapunov-based unified control method for closed quantum systems
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 where H is the Hamiltonian of the quantum system, and ρ is the density operator. The quantum control dynamical equation is given by where H0 is the free Hamiltonian of the quantum system, and Hc is the control Hamiltonian corresponding to the Lyapunov-based control law uc(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]:
- 1.
H0 should commute with the desired target state ρe.
- 2.
H0 should be strongly regular, which means if where is the transition frequencies of H0, and λj is eigenvalue of H0. If H0 is strongly regular, the transition energies between two different levels are distinguishable.
- 3.
Hc should be fully connected, which means (Hc)j,k ≠ 0 if j ≠ k. If Hc 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 ρe 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. denotes the complex (real) domain, denotes the set of the n-dimensional complex (real) column vectors, and denotes the set of m × n complex matrices. denotes the set of all differentiable functions defined on 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 δj,k is the Kronecker delta. Uppercase and lowercase letters in bold font represent matrices and vectors, respectively. For a matrix M, Mj,k denotes its entry in row j, column k. The j-th element of a vector b is denoted by bj. means that M is an n × n diagonal matrix with diagonal entries . 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 MT and bT, respectively. The conjugate transpose M† of a matrix M is defined by 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 . [A(n), B] is defined recursively as: and . U(n) is the Lie group of n × n unitary matrices, and Ou(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 .
Section snippets
Main Result
Consider an N-level closed quantum system which is controllable. The state of the quantum system is represented by a density operator ρ. The evolution of ρ is described by the following Liouville-von Neumann equation :where the real-valued control law ut may be divided into three parts, time dependent control laws uq, vq, and constant valued control laws wq, . At least one uq ≠ 0. H0 is the free Hamiltonian of the quantum system, and
Analysis of the largest invariant set and design of virtual mechanical quantity
Let S be the set of all states such that which is composed of equilibrium points, then according to LaSalle’s invariance principle [31], every solution of the quantum system will approach to the largest invariant set contained in S. So, to achieve the goal that make the quantum system converge to the desired target state, the largest invariant set should be investigated first, and make sure it contain the desired target state.
With the control laws uq and vq represented by Eqs. (21) and
Numerical Simulation
The method proposed in this paper is effective for many state control problems of actual quantum systems. Among these quantum systems, the 3-level quantum systems, and corresponding actual quantum systems, were discussed by many researchers [33], [34], [35], [36], [37], [38]. In this section, the numerical simulation of a 3-level quantum system is implemented to illustrate the effectiveness of the method proposed. To demonstrate the universality of the method proposed, in the simulation, the
Conclusion
A Lyapunov-based unified control method of closed quantum systems, which can be used in the quantum systems of non-degenerate and non-ideal, was proposed in this paper. The convergence of the control system was proved based on the Lyapunov stability theorem and LaSalle’s invariance principle. Numerical simulation experiment of a three dimensional quantum system demonstrated the effectiveness of the proposed control method. The proposed method expanded the possibility of practical experiment of
Declaration of Competing Interest
None.
Acknowledgment
This work was supported by National Natural Science Foundation of China under Grant 61973290.
References (39)
- et al.
Sliding mode control of two-level quantum systems
Automatica
(2012) - et al.
Optimal control of two-level quantum systems
IEEE T. Automat. Contr.
(2001) - et al.
Quantum speed limits: from heisenberg’s uncertainty principle to optimal quantum control
J. Phys. A: Math. Theor.
(2017) - et al.
Optimal control of quantum-mechanical systems: Existence, numerical approximation, and applications
Phys. Rev. A.
(1988) - et al.
Krotov method for optimal control of closed quantum systems
RUSSIAN MATHEMATICAL SURVEYS
(2019) - et al.
Lyapunov control methods of closed quantum systems
Automatica
(2008) - et al.
Analysis of lyapunov method for control of quantum states
IEEE T. Automat. Contr.
(2010) - et al.
Superadiabatic driving of a three-level quantum system
Phys. Rev. A.
(2017) - et al.
Analysis of the quantum zeno effect for quantum control and computation
J. Phys. A: Math. Theor.
(2013) - et al.
Control of a two-level quantum system in a coherent feedback scheme
J. Phys. A: Math. Theor.
(2013)
Geometric algebra and the causal approach to multip@article quantum mechanics
J. Math. Phys.
Quantum control by decompositions of su(2)
Phys. Rev. A.
Bang-bang control design for quantum state transfer based on hyperspherical coordinates and optimal time-energy control
J. Phys. A: Math. Theor.
Fast state transfer in closed quantum system based on the improved bang-bang control
OPTIK
Time-optimal control of a two level quantum system via interaction with an auxiliary system
IEEE T. Automat. Contr.
Optimal control theory for quantum-classical systems: Ehrenfest molecular dynamics based on time-dependent density-functional theory
J. Phys. A: Math. Theor.
Time optimal simultaneous control of two level quantum systems
Automatica
Robust learning control design for quantum unitary transformations
IEEE T. Cybernetics
Lyapunov control of bilinear schrȵdinger equations
Automatica
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