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Journal of Systems Engineering and Electronics

Establishing formal state space models via quantization for quantum control systems

Daoyi Dong, Zonghai Chen

Published Jun 1, 2005• 4 citations
OpenAlex

Abstract

Formal state space models of quantum control systems are deduced and a scheme to establish formal state space models via quantization could been obtained for quantum control systems is proposed. State evolution of quantum control systems must accord with Schrodinger equations, so it is foremost to obtain Hamiltonian operators of systems. There are corresponding relations between operators of quantum systems and corresponding physical quantities of classical systems, such as momentum, energy and Hamiltonian, so Schrodinger equation models of corresponding quantum control systems via quantization could been obtained from classical control systems, and then establish formal state space models through the suitable transformation from Schrodinger equations for these quantum control systems. This method provides a new kind of path for modeling in quantum control.

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Formal state space models of quantum control systems are deduced and a scheme to establish formal state space models via quantization could been obtained for quantum control systems is proposed. State evolution of quantum control systems must accord with Schrodinger equations, so it is foremost to obtain Hamiltonian operators of systems. There are corresponding relations between operators of quantum systems and corresponding physical quantities of classical systems, such as momentum, energy and Hamiltonian, so Schrodinger equation models of corresponding quantum control systems via quantization could been obtained from classical control systems, and then establish formal state space models through the suitable transformation from Schrodinger equations for these quantum control systems. This method provides a new kind of path for modeling in quantum control.
Formal state space models of quantum control systems are deduced and a scheme to establish formal state space models via quantization could been obtained for quantum control systems is proposed. State evolution of quantum control systems must accord with Schrodinger equations, so it is foremost to obtain Hamiltonian operators of systems. There are corresponding relations between operators of quantum systems and corresponding physical quantities of classical systems,such as momentum, energy and Hamiltonian, so Schrodinger equation models of corresponding quantum control systems via quantization could been obtained from classical control systems, and then establish formal state space models through the suitable transformation from Schrodinger equations for these quantum control systems. This method provides a new kind of path for modeling in quantum control.
Quantum control theory is a new interdiscipline, which mainly concerns how to control quantum states of microsystems. To get better control results, it is necessary to study the modeling and simulation of the systems. With the consideration of the characteristics of quantum control, direct mechanism modeling and quantization modeling for quantum control systems are studied, and simulation of quantum control systems is also analyzed. Direct mechanism modeling uses the mechanism information of systems in quantum physics and suitably models some quantum control systems with the known mechanism. Quantization modeling makes the most of the corresponding relations between quantum operators and classical physical quantities, and uses the information of models in classical control to establish the corresponding models in quantum control.
Courses about Quantum Mechanics are generally developed at the level of the mathematical formalism, its syntactic level, seldom treating the interpretation of the theory, its semantic level. In the majority of cases, this first formal approach is not followed by any other course addressing the numerous interpretations of the theory. It has become apparent that this strategy leads to a deficit in the professional formation of the students, exceedingly driving them to technical areas of application of Quantum Mechanics. However, applications such as those on quantum computing and entanglement, as they become more and more encrusted in the deepest foundations of the theory, are asking more and more for the development of this kind of ability. In this paper we develop an approach to deal with second quantization in the realm of Quantum Mechanics. We show that the second quantization Schrodinger equation can be mathematically derived from a classical Hamiltonian written in the phase space (Q,P)(Q,P)(Q,P), obtained by a canonical transformation upon the original phase space (q,p)(q,p)(q,p), using an axiomatic quantization procedure developed elsewhere. We apply this quantization process to a bosonic system, modelled by the harmonic oscillator problem. We then make reverse engineering to show that Schwinger's second quantization approach to fermionic systems furnishes the path to derive a Schrodinger equation explicitly written in terms of the usual momenta and positions operators (q^,p^)(\hat{q},\hat{p})(q^​,p^​) for such systems. Finally, we use these mathematical developments to address the commonly accepted statement that 'the spin has no classical analog'.
A model of quantum mechanical systems and its ensemble are established from control theory viewpoint. The state vector of wave function and density matrix of quantum mechanical systems are transformed into evolution matrix state of geometry space. The manipulation problem of an abstract physical concept is changed into a control design problem on a state of geometry space, which is easy to deal with mathematically and suitable for control. (Finally,) five basic problems about quantum sysetms control, which may be solved based on the model, are (presented.)
When applied quantization to classical control system, quantum mechanics opens completely new perspectives; by exploiting quantum mechanical features, such as, time evolution of quantum state and costate (generalized momenta) of the Hamiltonian of the system, one can formulate and solve the optimal control problems of the quantum mechanical system more efficiently than the classical system. The physical perspective of quantization of the optimal feedback system is outlined by exploiting the formulation of the optimization problem of the system within the context of Hamilton-Jacobi theory and Pontryagin maximum/minimum principle in abstract Hubert space. One of the most striking example is the optimal control problem of synthesizing the feedback law of quantum diffusion process. The optimal diffusion process is shown to satisfy the Schrödinger wave equation of energy with the feedback law as potential function.
We develop a general `classicalization' procedure that links Hilbert-space and phase-space operators, using Weyl's operator. Then we transform the time-dependent Schrödinger equation into a phase-space picture using free parameters. They include position Q and momentum P . We expand the phase-space Hamiltonian in an ℏ-Taylor series and fix parameters with the condition that coefficients of ℏ 0 , -iℏ 1 ∂/∂ Q and iℏ 1 ∂/∂ P vanish. This condition results in generalized Hamilton equations and a natural link between classical and quantum dynamics, while the quantum motion-equation remains exact . In this picture, the Schrödinger equation reduces in the classical limit to a generalized Liouville equation for the quantum-mechanical system state. We modify Glauber's coherent states with a suitable phase factor S ( Q , P , t ) and use them to obtain phase-space representations of quantum dynamics and quantum-mechanical quantities.