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This chapter formulates the Hartree-Fock approximation with density matrices, and discusses the properties of the Hartree-Fock equations together with the energy functional for the ground state energy. A generalization to finite thermal excitations is performed by applying the variational principle to the grand canonical ensemble. Finally, the equations for time-dependent Hartree-Fock (TDHF) are described both at zero and at finite temperature.

This chapter provides an account of atomic resonance fluorescence and its depolarization by external magnetic fields, also known as the Hanle effect. Experiments in mercury vapour and their theoretical description are explained both by classical and quantum mechanical density matrix methods. Radiative lifetimes of alkali atoms and Group IIB atoms are reported. Optical and radio-frequency coherence, and experiments using pulsed and intensity modulated radiation are discussed.

This chapter develops the fundamental notions and concepts of the probabilistic and statistical interpretation of quantum mechanics. It begins with a brief review of the mathematical structure given by a Hilbert space of state vectors and an algebra of observables represented by selfadjoint operators. Special emphasis is laid on the connection between the spectral representation of a selfadjoint operator and a corresponding random variable that describes the possible measurement outcomes. The chapter further introduces the concepts of a composite quantum system, of the reduced density matrix, and of entangled quantum states. A section on quantum entropies is included, as well as the mathematical formulation of the generalized theory of quantum measurements in terms of completely positive quantum operations and effects.

This chapter characterizes the statistical properties of a quantum mechanical ensemble in terms of a density matrix. However, if selective measurements of one or several observables are carried out on the ensemble, it will split into a number of sub-ensembles, each sub-ensemble being conditioned on a particular outcome of the measurements. The mathematical description of the collection of sub-ensembles thus created leads to probability distributions on projective Hilbert space. The chapter develops an appropriate mathematical framework which enables the general formulation of such a distribution, and leads to the concepts of stochastic state vectors and stochastic density matrices. These concepts are required in later chapters to construct appropriate stochastic differential equations describing the continuous monitoring of open quantum systems.

This chapter explains the excitation of atoms by polarized resonance radiation and their interaction with static and radio-frequency fields, known as optical double-resonance. The Brossel-Bitter experiment on mercury vapour is discussed and classical and quantum mechanical theories of the effect are developed. The phenomena of radiation trapping and coherence narrowing are explained, and the effect of collision broadening is examined. Experiments involving intensity-modulated light are reported, and the density matrix formulation of optical double-resonance experiments is developed.

Chapter 6 begins by transforming the optical wave itself into a quantized form, and the full implications of the wave‐particle duality of Chapter 2 become clear. The electric field may be pictured as an infinite ladder of discrete states and this structure allows optical interactions to be described photon by photon. The origin of spontaneous emission emerges from this treatment as a transition process that is stimulated by electric field fluctuations of the vacuum, and Weisskopf–Wigner theory explains why it is irreversible. The subject of coherent states is introduced and intriguing possibilities outlined regarding the noise properties of components of these states that form different quadratures in terms of the optical phase. Next, statistical analysis of light fields is considered to serve as a basis for more quantitative comparisons of fields and their noise properties. Methods of calculating the first and second degrees of coherence are presented, and the ranges of values corresponding to “classical” versus “quantum” light sources are discussed. The reduced density matrix is developed to justify analysis that focuses on only certain aspects of optical dynamics while ignoring others, as an important tool for simplifying complex problems. Finally, in view of its historical importance, the problem of calculating the fluorescence spectrum of resonantly excited two‐level atoms is considered using both a method of moments and dressed atom theory.

Nuclear magnetic resonance (NMR) exploits the interaction of nuclei with magnetic fields. A strong static field is applied to polarise the nuclear magnetic moments, time-dependent magnetic radio frequency fields are used to stimulate the spectroscopic response, and magnetic-field gradients are needed to obtain spatial resolution. Following the description of the different magnetic fields used in NMR spectroscopy and imaging, the behaviour of magnetic nuclei exposed to these fields is treated first in terms of the classic vector model, and then the density-matrix concept is introduced. The latter is required to describe the couplings among nuclei, for example, the dipole-dipole interaction which dominates the ¹H NMR spectrum of most solid materials. In addition, knowledge of the density matrix is helpful to understand multi-quantum coherences as well as the imaging methods developed for investigations of solid materials.

This chapter focuses on coherence and decoherence, starting from the basis of quantum mechanics and including the classical paradox of Schrödinger’s cat and entanglement. Decoherence and relaxation are simultaneously accounted for by an evolution equation for the density matrix, which is analysed in the case of spin tunnelling and simplifies when decoherence is complete. The final section discusses the possible exploitation of coherence in quantum computing.

This chapter reminds, without entering into details, the main mathematical concepts and results relevant for finite system quantum mechanics. The basic postulates single out a Hilbert space of wave functions with self-adjoint linear operators corresponding to observables as was originally discovered by von Neumann. The chapter connects the contemporary terminology of linear Hilbert space operators with quantum physics. Important concepts like linear operators, measures, self-adjointness, spectral measures, density matrices, and tensor products are discussed and illustrated in the light of observables, probability for quantum systems and composite systems. A first example of a useful algebra of observables, the Weyl algebra, is described in detail and linked to the classical phase space of a point particle.

This chapter applies the theoretical tolls developed in Chapter 3 to macroscopic systems in thermal equilibrium, wherein the Gibbs variational principle is understood as defining the equilibrium state.

This chapter analyses time evolution in macroscopic systems, and critiques the standard equation of motion for the density matrix. The practical difference between microscopic and macroscopic equations of motion and the necessary relation to external sources is recognized explicitly.

This chapter outlines the quantum ideas that underpin magnetic resonance and explains their link to classical physics. It discusses some of the essential concepts governing the quantum behaviour of ensembles of nuclear spins, including the nature of quantum states, the eigenvalue equation and observables, the measurement process, dynamics, and the Schroedinger equation. A detailed description of angular momentum is given along with discussion of the role of symmetry, rotation operators, and both product and total angular momentum representations. Statistical ensembles of spins are introduced and the density matrix and Liouville formalism outlined, along with the various tensor bases for the description of spin systems, both independent and coupled. The spin Hamiltonian is described in detail and the evolution of spin systems under various Hamiltonian terms is discussed using the density matrix description, along with suitable recession diagrams.

This chapter provides a detailed description of the numerical algorithms used to study open and closed quantum systems out of equilibrium. In a first step, the physical formulas to be evaluated numerically, and the mathematical framework for all these problems, are established. Quasi-exact simulations in the full Hilbert space for relatively small system sizes are discussed. Next, it is shown that in some cases it is possible to abandon the costly treatment of the full Hilbert space while identifying subspaces that can be handled numerically and still contain all of the essential physics. Related algorithms, which are essentially effective in one spatial dimension, are discussed under the heading of “matrix product state (MPS) simulations” (or “density-matrix renormalization group (DMRG) simulations”). It is shown that this strategy works very well on short timescales, but encounters fundamental issues of quantum physics as time evolves.

Quantum statistical mechanics governs metals, semiconductors, and neutron stars. Statistical mechanics spawned Planck’s invention of the quantum, and explains Bose condensation, superfluids, and superconductors. This chapter briefly describes these systems using mixed states, or more formally density matrices, and introducing the properties of bosons and fermions. We discuss in unusual detail how useful descriptions of metals and superfluids can be derived by ignoring the seemingly important interactions between their constituent electrons and atoms. Exercises explore how gregarious bosons lead to superfluids and lasers, how unsociable fermions explain transitions between white dwarfs, neutron stars, and black holes, how one calculates materials properties in semiconductors, insulators, and metals, and how statistical mechanics can explain the collapse of the quantum wavefunction during measurement.

This chapter presents the basic elements of quantum theory, including the Dirac bra and ket notation for quantum states, Hermitian operators and their eigenstates, commutators and the Schrödinger and Heisenberg equations of motion. Simple model atoms are described as an effective spin and their properties related to the familiar Pauli spin-half operators. The density matrix is introduced to describe incoherent processes. The electric-dipole coupling between and atom and the electromagnetic field is treated leading to coupled equations for the probability amplitudes. Finally, quantized field modes are introduced and their elementary properties described leading to the fundamental Jaynes–Cummings model of a single two-state atom interacting with a quantized field mode.

This chapter introduces quantum spin chains for two different purposes: prototypes of infinite quantum systems discretized in space, and general models of stochastic processes in discrete time on a discrete quantum state space. The algebra of local observables is constructed. The probability functionals, i.e. the states, are defined in terms of reduced density matrices with their compatibility relations. Several examples of shift-invariant states are discussed: product states, classical states, and limiting Gibbs states. The chapter concludes with the construction of nearest-neighbour dynamics and the need for an appropriate algebra of quasi-local observables.

Unlike other optoelectronic devices, quantum cascade lasers (QCLs) cannot be described by simple drift-diffusion equation. Whereas in interband semiconductor lasers the assumption of separate quasi-Fermi levels in the conduction and valence bands allows the transport problem to be reduced to a quasi-equilibrium situation, such approximations completely break down in QCLs. Because of the subpicosecond long intersubband lifetime, the transport scattering time is never much shorter than the intersubband transition time, preventing the use of such approximation. For this reason, the development of predictive transport models requires a more sophisticated approach. This chapter discusses rate equation models, density matrix, full density matrix models, Monte-Carlo techniques, and non-equilibrium Green's function.

One of the most powerful tools for calculating quantum device performance is based on the density matrix operator. The operator is unique because it is time dependent in the Schrödinger picture. The approach is quite general, but in the systems of interest here, the Hilbert space of the operator includes both the quantum system such as a nano-vibrator or two-level system and the quantized vacuum radiation field. The equation of motion follows from the time dependent Schrödinger equation. It is possible, as we show, to include the generation of spontaneous emission in this system and then, because observables of interest do not depend on the vacuum field, trace over the vacuum field to create a new density matrix called the reduced density matrix. The resulting equations of motion are the Bloch equations. We use these equations to analyze several problems involving two- and three-level systems.

Chapter 3 discusses the interaction Hamiltonian, which determines the way that light interacts with matter. Simple perturbative analysis is applied to see if basic dynamics of atoms can be explained. The partial successes of perturbation theory are compared with predictions of an “exact” method of calculating the occupation probabilities of various atomic states. The “exact” method is shown to fail, establishing the need for improved approaches that yield correct results in later chapters. The density matrix is introduced as a tool for describing not only the populations of atomic energy levels but also the coherence that can be created and lost during the dynamic evolution of atoms in time. A vector model based on the Bloch vector is presented as a useful way of picturing coherent atom–field interactions in an optical “spin” space, which proves to be particularly useful in understanding multiple-pulse interactions. Mechanisms are described that cause line broadening in optical spectroscopy, such as the Doppler effect. In preparation for the extensive use in later chapters of models based on only two or three energy levels, it is also shown that multi-level real atoms can experimentally be converted into two-level systems for strict comparisons with theory.