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Chapman & Hall/CRC eBooks

Modeling of Quantum Control Systems; Examples

Domenico D’Alessandro

Published Jul 13, 2021• 0 citations
Publisher

Abstract

Quantum systems whose dynamics depends on one or more control functions are called quantum control systems. This chapter describes elements of the theory of field-particle interaction. It starts with the classical theory of electrodynamics and shows how to obtain the quantum mechanical description through a process called canonical quantization. The chapter presents physical examples of quantum control systems of interest in applications and highlights the fact that such systems have a common mathematical structure. In the external system, the motion of the charges, are fixed and independent of the field. Canonical quantization is the process which associates quantities in classical mechanics with operators in quantum mechanics.

DOI
10.1201/9781003051268-2

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Quantum systems whose dynamics depends on one or more control functions are called quantum control systems. This chapter describes elements of the theory of field-particle interaction. It starts with the classical theory of electrodynamics and shows how to obtain the quantum mechanical description through a process called canonical quantization. The chapter presents physical examples of quantum control systems of interest in applications and highlights the fact that such systems have a common mathematical structure. In the external system, the motion of the charges, are fixed and independent of the field. Canonical quantization is the process which associates quantities in classical mechanics with operators in quantum mechanics.
Quantum systems whose dynamics depends on one or more control functions will be named quantum control systems. In a typical situation the objects of control are charged particles while the control functions are appropriately shaped electromagnetic fields. Therefore, to derive the models, in this chapter we describe elements of the theory of field-particle interaction. We start with the classical theory of electrodynamics and show how to obtain the quantum mechanical description through a process called canonical quantization. We present physical examples of (finite dimensional) quantum control systems of interest in applications and we highlight the fact that such systems have a common mathematical structure, as bilinear systems whose evolution is determined by an element of the Lie group of unitary matrices. In a parallel to classical control theory, we identify a state, an input, and an output in our description of quantum control systems.
In this chapter we review the main ideas of the canonical quantization in nonrelativistic quantum mechanics and optics, which we suppose the reader learned in previous courses. We first consider the least action principle and the action functional for classical many-body systems and also for the Schrödinger wavefunction, interpreted as the complex field of a classical field theory. Then, after having analyzed the Legendre transformation from the Lagrangian to the Hamiltonian formulation, we illustrate the canonical quantization of particles, and finally we extend the approach to the canonical quantization of bosonic and fermionic fields, considering also the role of temperature.
Quantum control refers to our ability to manipulate quantum systems. This tutorial-style chapter focuses on the use of classical electromagnetic fields to steer the system dynamics. In this approach, the quantum nature of the control stems solely from the underlying dynamics, through the exploitation of destructive and constructive interference to reach the control target. We first discuss two basic control principles -- coherent control which uses manipulation in frequency or time to design these interferences, and adiabatic following where access to the control target is enabled by tracking the time-dependent ground state. For complex control targets and system dynamics that exceed the scope of these basic principles, optimal control theory provides a powerful suite of tools to design the necessary protocols. A key consideration for the successful application of optimal control theory is a proper choice of the optimization functional. All concepts are illustrated using recent work from my research group, with a focus on controlling atoms and superconducting qubits. The chapter concludes with an outlook on integrating coherent control with engineered dissipation and a discussion of open questions in the field.
This chapter contains a brief compilation of information about various fields and contractions. The field formalism is fully applicable also to the ordinary quantum mechanics of systems with a finite number of degrees of freedom. The simplest example is that of a particle in a given external field. Preliminary separation of the δ function is necessary, for example, when calculating the amplitude by the stationary-phase method (the semiclassical approximation), since the conservation laws expressed by the δ functions automatically make the kernel of the quadratic form of the second variation of the action degenerate in the neighborhood of a classical trajectory. Nonrelativistic field theory is the quantum mechanics of systems with an arbitrary number of identical particles in the second-quantized formalism. In the classical theory such systems are described by a complex field ψ, ψ + which is an ordinary or anticommuting function.
This chapter reviews the classical description of radiation. It describes how classical electromagnetic radiation interacts with a quantum mechanical particle. The general rule is that, even though the wave functions and the electromagnetic potentials change with the gauge, all physically observable quantities are independent of the gauge. The chapter also describes the process of the absorption of light by a system in terms of the effect of an external classical electromagnetic field on the system. The total emission rate is the sum of the induced and the spontaneous rates. Spontaneous emission is just the quantum mechanical version of the classical phenomenon of radiation from an accelerating charge. A very important consequence of the fact that the electromagnetic field is quantized is that there are vacuum fluctuations of the field, analogous to the zero point motion of a harmonic oscillator. Einstein gave a simple statistical argument to determine the rate at which spontaneous emission takes place.
This introductory chapter sets out a general structure of physical theories which is applicable to both classical and quantum mechanics. We start with measurable properties of a given physical system, be it classical or quantum. These properties are called observables . We then introduce a definition of the state of a physical system in terms of measured values of a sufficiently large set of observables. A theory to describe the system should consists of four basic components: Basic mathematical framework This comprises a set of elements endowed with some specific mathematical structure and properties. In mathematics such a set is generally known as a space . Description of states States are described by elements of the space in the chosen mathematical framework. For this reason the space is called the state space of the system. Description of observables Observables are to be described by quantities defined on the state space. The description should yield all possible values of observables. The relationship between observables and states should be explicitly stated. The following two cases are of particular interest: 2 For a deterministic theory like classical mechanics a state should determine the values of all observables. For a probabilistic theory like quantum mechanics a state should determine the probability distribution of the values of all observables. Description of time evolution (dynamics).