© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

Cambridge University Press eBooks

The evolution of interacting quantum systems

G. Grynberg, Alain Aspect, Claude Fabre, Claude N. Cohen-Tannoudji

Published Sep 2, 2010• 0 citations

Abstract

In this work we shall study the interaction of matter and light. In so doing we shall rely heavily on the description of such processes provided by quantum mechanics. This appears on a number of levels: firstly, a quantum description of matter is indispensable if one wants to understand on the microscopic scale the different kinds of interaction processes that can occur. Secondly, a quantum description of light often turns out to be useful, sometimes necessary, to better understand these processes. We shall study phenomena such as spontaneous emission, which can only be properly treated by a theory taking into account the quantum nature of both light and matter. In the following chapters we shall address, amongst others, the following question: ‘given an atom prepared at a given time in a particular state and subjected from this time onwards to electromagnetic radiation, what is the state of the atom and radiation at any later moment in time?’ In order to be able to answer this question it will be necessary for us to know how to calculate the evolution of a quantum system in a small number of typical situations. These methods we shall demonstrate in the first chapter. The evolution of the coupled atom–light system depends on the temporal dependence of the applied light field, which could, for example, be applied from a given moment and thereafter remain unchanged in intensity, or, perhaps, be appreciable only for a finite period of time (pulsed excitation).

DOI: 10.1017/cbo9780511778261.003

Related papers

An Overview Of Quantum Electrodynamics And Matter–Radiation Field Interaction

Abraham Nitzan

Many dynamical processes of interest are either initiated or probed by light, and their understanding requires some knowledge of this subject. This chapter is included in order to make this text self contained by providing an overview of subjects that are used in various applications later in the text. In particular, it aims to supplement the elementary view of radiation–matter interaction as a time-dependent perturbation in the Hamiltonian, by describing some aspects of the quantum nature of the radiation field. This is done on two levels: The main body of this chapter is an essentially qualitative overview that ends with a treatment of spontaneous emission as an example. The Appendix gives some more details on the mathematical structure of the theory. In elementary treatments of the interaction of atoms and molecules with light, the radiation field is taken as a classical phenomenon.

Oxford University Press eBooks•Apr 6, 2006•0 citations

WAVE MECHANICS

Murray, Iii Sargent

In this book we treat the interaction of radiation with matter with special application to gain (amplifying) media typically existing in lasers. Tradition ally the interaction of radiation with matter is treated in terms of the index of refraction and the absorption of light, utilizing the model of electrons on springs. In Chap. 3 we show that this model is reasonable for some linear absorbing systems but is very awkward, if not outright unworkable, for a microscopic description of gain media. The essential failure of the electron-spring model is its difficulty in yielding net stimulated emission (and therefore gain) for an ensemble of systems.t In contrast, a more physically sound, quantum treatment of matter provides for the concept of population inversion and explains stimulated emission in a natural way. Hence from the very start we are compelled to base our understanding of the interaction of radiation with matter on a quantum-mechanical description of atoms. In Chaps. 1-13 a classical field interacts with quantum-mechanical atoms; in the remainder of the book, both field and atoms are treated quantum mechani cally.

CRC Press eBooks•Mar 5, 2018•0 citations

Photons

P. K. Basu

The classical or semiclassical theory presented in Chapter 2 gives a satisfactory account of light-matter interactions in some cases. However, a detailed understanding of different processes requires a quantum mechanical theory of radiation. The quantum theory of light, like classical theory, is also based on Maxwell's equations. However, in quantum theory the electric field F and the magnetic field H are treated as operators. This chapter develops the correspondence between the two and outlines the method for converting field vectors F and H into quantum mechanical operators. Topics discussed include wave equation in a rectangular cavity, quantization of the electric field, time-dependent perturbation theory, interaction of an electron with the magnetic field, and second-order perturbation theory. Exercises are provided at the end of the chapter.

Oxford University Press eBooks•Jan 23, 2003•0 citations

Introduction

Lampros A. A. Nikolopoulos

Some general discussion is presented aiming to provide the physical context in conjunction with the theoretical formulations under which the present book was developed; namely, the interaction of isolated atomic systems with ultrashort laser fields (classically represented) using the non-relativistic Schrödinger or density matrix formulations for the time-evolution of a quantum system.

Morgan & Claypool Publishers eBooks•Mar 1, 2019•0 citations

Interaction of an atom with the quantized electromagnetic field

G. Grynberg, Alain Aspect, Claude Fabre +1 more

In this chapter, we discuss a purely quantum approach to the interaction between an atom and the electromagnetic field. In this treatment, the atom and electromagnetic field form a single quantum system, whose evolution is handled globally within a unified formalism. This will thus have the merit of being fully consistent from the theoretical standpoint. But the main advantage in an entirely quantum approach is that it can treat the full range of matter–radiation interaction phenomena. In particular, it provides a rigorous description of the spontaneous emission of light by an excited atom , something that falls outside the scope of the semi-classical framework applied in Chapters 2 and 3, where the lifetime of an excited atomic state had to be fed in phenomenologically. It can describe other phenomena of the same type, such as parametric fluorescence by a nonlinear crystal subjected to pumping radiation (see Chapter 7), which underlies many recent developments in quantum optics. The fully quantum approach also has the merit of allowing a simple interpretation in terms of photons for the various matter–radiation interaction processes, such as absorption, stimulated emission, scattering, and also the basic processes of nonlinear optics. Indeed, it provides a unified framework for both stimulated and spontaneous processes. Finally, it can be used to tackle completely new situations where matter and radiation interact, which lie outside the scope of any semi-classical description, such as cavity quantum electrodynamics or the production of single-photon or entangled-photons states .

Cambridge University Press eBooks•Sep 2, 2010•0 citations

Light-Matter Interactions

Davıd L. Andrews, David S. Bradshaw

5.1 Foundations Up until now, we have only considered light propagating essentially in vacuum; this has been used as the basis to describe the effects of reflection, refraction, and transmission in a medium. These essentially passive phenomena, alone, serve as a reminder that nothing is achieved with light, or learned about it, without material interaction. However, a more extensive realm of science centers upon light actively coupling with matter - and many of the systems, in which the photonic aspects of light are most clearly manifest, concern matter in the form of atoms, molecules, or nanoparticles. A quantum analysis of light–matter interactions typically begins with the system Hamiltonian H, the energy operator of the system. In basic quantum mechanics, the "system" simply comprises matter, and the state wavefunctions are secured as eigenstates of the corresponding Hamiltonian Hmat. A quantum formulation of the radiation requires the additional inclusion of a radiation Hamiltonian Hrad. Here, it becomes appropriate to use the representation of quantum electrodynamics (QED): however, because the charges that comprise matter move at significantly sub-luminal speeds in almost all optics and photonics applications, the full relativistic form of QED is unnecessary. To complete the full Hamiltonian for systems in which light and matter are not isolated from each other, an interaction Hamiltonian Hint is also required. The system Hamiltonian can thus be decomposed into three parts: [41]

SPIE eBooks•Aug 1, 2018•0 citations

In this work we shall study the interaction of matter and light. In so doing we shall rely heavily on the description of such processes provided by quantum mechanics. This appears on a number of levels: firstly, a quantum description of matter is indispensable if one wants to understand on the microscopic scale the different kinds of interaction processes that can occur. Secondly, a quantum description of light often turns out to be useful, sometimes necessary, to better understand these processes. We shall study phenomena such as spontaneous emission, which can only be properly treated by a theory taking into account the quantum nature of both light and matter. In the following chapters we shall address, amongst others, the following question: ‘given an atom prepared at a given time in a particular state and subjected from this time onwards to electromagnetic radiation, what is the state of the atom and radiation at any later moment in time?’ In order to be able to answer this question it will be necessary for us to know how to calculate the evolution of a quantum system in a small number of typical situations. These methods we shall demonstrate in the first chapter. The evolution of the coupled atom–light system depends on the temporal dependence of the applied light field, which could, for example, be applied from a given moment and thereafter remain unchanged in intensity, or, perhaps, be appreciable only for a finite period of time (pulsed excitation).

Many dynamical processes of interest are either initiated or probed by light, and their understanding requires some knowledge of this subject. This chapter is included in order to make this text self contained by providing an overview of subjects that are used in various applications later in the text. In particular, it aims to supplement the elementary view of radiation–matter interaction as a time-dependent perturbation in the Hamiltonian, by describing some aspects of the quantum nature of the radiation field. This is done on two levels: The main body of this chapter is an essentially qualitative overview that ends with a treatment of spontaneous emission as an example. The Appendix gives some more details on the mathematical structure of the theory. In elementary treatments of the interaction of atoms and molecules with light, the radiation field is taken as a classical phenomenon.

In this book we treat the interaction of radiation with matter with special application to gain (amplifying) media typically existing in lasers. Tradition ally the interaction of radiation with matter is treated in terms of the index of refraction and the absorption of light, utilizing the model of electrons on springs. In Chap. 3 we show that this model is reasonable for some linear absorbing systems but is very awkward, if not outright unworkable, for a microscopic description of gain media. The essential failure of the electron-spring model is its difficulty in yielding net stimulated emission (and therefore gain) for an ensemble of systems.t In contrast, a more physically sound, quantum treatment of matter provides for the concept of population inversion and explains stimulated emission in a natural way. Hence from the very start we are compelled to base our understanding of the interaction of radiation with matter on a quantum-mechanical description of atoms. In Chaps. 1-13 a classical field interacts with quantum-mechanical atoms; in the remainder of the book, both field and atoms are treated quantum mechani cally.

The classical or semiclassical theory presented in Chapter 2 gives a satisfactory account of light-matter interactions in some cases. However, a detailed understanding of different processes requires a quantum mechanical theory of radiation. The quantum theory of light, like classical theory, is also based on Maxwell's equations. However, in quantum theory the electric field F and the magnetic field H are treated as operators. This chapter develops the correspondence between the two and outlines the method for converting field vectors F and H into quantum mechanical operators. Topics discussed include wave equation in a rectangular cavity, quantization of the electric field, time-dependent perturbation theory, interaction of an electron with the magnetic field, and second-order perturbation theory. Exercises are provided at the end of the chapter.

In this chapter, we discuss a purely quantum approach to the interaction between an atom and the electromagnetic field. In this treatment, the atom and electromagnetic field form a single quantum system, whose evolution is handled globally within a unified formalism. This will thus have the merit of being fully consistent from the theoretical standpoint. But the main advantage in an entirely quantum approach is that it can treat the full range of matter–radiation interaction phenomena. In particular, it provides a rigorous description of the spontaneous emission of light by an excited atom , something that falls outside the scope of the semi-classical framework applied in Chapters 2 and 3, where the lifetime of an excited atomic state had to be fed in phenomenologically. It can describe other phenomena of the same type, such as parametric fluorescence by a nonlinear crystal subjected to pumping radiation (see Chapter 7), which underlies many recent developments in quantum optics. The fully quantum approach also has the merit of allowing a simple interpretation in terms of photons for the various matter–radiation interaction processes, such as absorption, stimulated emission, scattering, and also the basic processes of nonlinear optics. Indeed, it provides a unified framework for both stimulated and spontaneous processes. Finally, it can be used to tackle completely new situations where matter and radiation interact, which lie outside the scope of any semi-classical description, such as cavity quantum electrodynamics or the production of single-photon or entangled-photons states .

5.1 Foundations Up until now, we have only considered light propagating essentially in vacuum; this has been used as the basis to describe the effects of reflection, refraction, and transmission in a medium. These essentially passive phenomena, alone, serve as a reminder that nothing is achieved with light, or learned about it, without material interaction. However, a more extensive realm of science centers upon light actively coupling with matter - and many of the systems, in which the photonic aspects of light are most clearly manifest, concern matter in the form of atoms, molecules, or nanoparticles. A quantum analysis of light–matter interactions typically begins with the system Hamiltonian H, the energy operator of the system. In basic quantum mechanics, the "system" simply comprises matter, and the state wavefunctions are secured as eigenstates of the corresponding Hamiltonian Hmat. A quantum formulation of the radiation requires the additional inclusion of a radiation Hamiltonian Hrad. Here, it becomes appropriate to use the representation of quantum electrodynamics (QED): however, because the charges that comprise matter move at significantly sub-luminal speeds in almost all optics and photonics applications, the full relativistic form of QED is unnecessary. To complete the full Hamiltonian for systems in which light and matter are not isolated from each other, an interaction Hamiltonian Hint is also required. The system Hamiltonian can thus be decomposed into three parts: [41]