Introduction

In 1963, E. T. Jaynes and F. W. Cummings published a paper entitled “Comparison of Quantum and Semiclassical Radiation Theories, with Application to the Beam Maser.” This paper was written at a time when the quantum theory of the laser had not yet been worked out, and represented an attempt at discussing the new behaviour that could be expected from treating the field quantum mechanically instead of classically. Jaynes and Cummings began by considering the interaction between a single two-level molecule and a single field mode, a simple situation which has become known as the Jaynes-Cummings model. Treating then the maser medium as a beam of two-level systems initially in a mixture of their upper and lower states characterized by a temperature T, they found that the field mode reached a thermal equilibrium with temperature Tf = ΩT/ω, where ω is the transition frequency of the two-level systems and Ω is the field mode frequency, provided that T > 0. For T < 0, the steady state field was found to contain an infinite amount of energy, an unphysical result resulting from the neglect of losses in the model. Jaynes and Cummings then proceeded to develop an alternative semi-classical theory of the beam maser, introducing their famous neo-classical theory of spontaneous emission.

An important follow-up paper was published by F. W. Cummings, who studied the long-time behavior of a two-level system interacting with a quantized single-mode field initially in a coherent state.

Ed Jaynes was born in Waterloo, Iowa, on 5 July 1922, the son of a surgeon. He attended Cornell College and the University of Iowa, receiving the B.A. degree in physics from the latter in 1942, and was then engaged in Doppler radar development at the Sperry Gyroscope Company in New York in 1942–43. He was subsequently appointed an Ensign in the U.S. Navy, and worked at the Naval Research Laboratory at Anacostia on microwave IFF equipment.

Lt(jg) Jaynes was discharged from the Navy in 1946 and spent that summer at Stanford working with W.W. Hansen on the design of the first linear electron accelerator. In the 1946–47 school year he was a graduate student at Berkeley, a student of J.R. Oppenheimer. When Oppenheimer left to take over the Institute for Advanced Study in Princeton in the summer of 1947, Jaynes also transferred to Princeton University. He received his Ph.D. in physics there in 1950, with a thesis in solid-state theory supervised by Eugene Wigner.

He then returned to Stanford, where he was Instructor, Assistant Professor, and Associate Professor during 1950–60. At this time he also consulted for Varian Associates on problems of magnetic resonance instrumentation. In 1960 he was appointed Professor of Physics at Washington University in St. Louis, and in 1975 became Wayman Crow professor of Physics.

ABSTRACT. Each transition dipole in a single atom has a radiation reaction field which acts on itself and on the other transition dipoles of the atom. The self interaction on a given transition gives rise to spontaneous emission and radiative decay on that transition. Here we emphasize that the mutual radiative interaction of different transition dipoles can sometimes have important consequences. The mutual radiative interaction can substantially alter radiative decay rates and inhibit the radiative decay of certain expectation values. The hydrogen atom and the harmonic oscillator are cited as examples and a possible experimental test of mutual radiation reaction involving a single electron in a Penning trap is discussed.

Introduction

Modern quantum mechanics began with Heisenberg's picture of a bound charge as a “virtual orchestra” of transition dipoles oscillating at different frequencies. It is known from modern quantum theory that each of these dipoles generates a radiation reaction field which, acting on the dipole itself, gives rise to spontaneous emission and radiative relaxation (Milonni, 1976). In this paper we consider the mutual interaction of different transition dipoles within a single “atom”. We begin by considering some simple cases of mutual radiation reaction in classical electrodynamics and proceed to the quantum theory of mutual radiation reaction (MRR), citing the hydrogen atom and the charged harmonic oscillator as examples. We point out that the equations describing MRR are not at all new, but can be traced to the earliest treatments of the interaction of atoms with the quantized electromagnetic field.

Let me first point out that my title is consciously ambiguous. It reflects the marvelous imprecision that is possible with language. Am I the independent thinker or is someone else? Whose recollections are these anyway? And is the “of” equivalent to “about” or to “by”? And who here is “independent,” of what, or of whom?

In apposition to this observation, let me quote one for whom ambiguity is anathema:

I will touch upon the work of several independent thinkers tonight, but what I have to say is mostly, of course, about E. T. Jaynes. Those in this roomful of independent thinkers surely recognize both his independence and his originality. He is a man who has marched to a different drummer upon a road less traveled by. Those of us gathered here tonight, and many others in the world of science and engineering, now find themselves following in his footsteps.

Where many of us have had one career in one field, Ed Jaynes has had several. The broad collection of expertise from a wide variety of different disciplines in this gathering reflects this diversity of his ideas and their applications.

Most physicists, most of the time, are engaged in what Thomas Kuhn calls “normal science,” the process of solving problems under a prevailing paradigm. Part of Edwin Jaynes' career has been profitably devoted to this enterprise, but his best known work may aptly be called “abnormal” in this sense. We organized the Symposium on Physics and Probability not only to recognize Jaynes' accomplishments, but also to celebrate the scientific style and integrity that have been an inspiration to so many of his students and colleagues.

The articles in this volume are based, with a few exceptions, on lectures given at the Symposium, which was held 15-16 May 1992 at the University of Wyoming. The occasion for the Symposium was Jaynes' official retirement from his position as Wayman Crow Professor of Physics at Washington University — as well as to commemorate his seventieth birthday. The authors are former graduate students of Professor Jaynes, or colleagues, or people whose work has been so directly influenced by Jaynes that it was deemed appropriate to invite them to speak at the Symposium.

Jaynes is best known for his work in statistical physics and quantum optics. His seminal papers on information theory and statistical mechanics formulated the latter (in the spirit of Gibbs) without an ergodic hypothesis, relying instead on the maximum entropy principle to assign probabilities in a manner least biased with respect to the available information.

ABSTRACT. We investigate the possibility of enhancing the refractive properties in a nonabsorbing medium via two fundamentally different schemes. First there is the coherent preparation of three-level atoms where absorption is cancelled due to destructive interference while the refractivity is not hampered in the same way. There also is the possibility of cancelling absorption via a mixture of absorbing and emitting two-level atoms without the need of a coherent preparation. One drawback here, however, is high sensitivity to Doppler broadening, collisions and number fluctuations which makes this scheme practically infeasible.

Introduction

The various application of atomic coherence in laser physics and quantum optics has recently attracted considerable interest. It has been shown that atomic coherence can lead to absorption cancellation (Alzetta, et al., 1976; and Gray, et al., 1979) and quenching of spontaneous emission noise (Scully, 1985). More recently, the notion of noninversion lasing has received attention, and it was shown that atomic coherence leading to cancellation of absorption does not necessarily influence emission (Harris, 1989; Scully, et al., 1989; and Kocharovskaya and Khanin, 1988). There has been extensive research on many schemes that involve coherence between an upper or lower level laser doublet due to various means: microwave or Raman coherent coupling and spontaneous and incoherent pumping coupling just to name the most important.

ABSTRACT. We survey briefly some fifty years of thinking about physics and probability with the aim of explaining: (1) What I did not know then, but know now; (2) What I have been trying to accomplish in science and education, and to what extent these efforts have succeeded; (3) What remains unfinished, but where I think the greatest future opportunities lie; and (4) What personal and professional advice I can now give to young people (and wish someone had given me fifty years ago).

Introduction

A meeting like this is an overwhelming experience! I was overwhelmed not only by the sheer number of people who came here from so far; but even more by the kind sentiments expressed. Of course, I had looked forward to seeing again many former students and colleagues and had expected to have chats with each one, to renew our friendship and bring us both up to date about the other's work. But what one can actually do in two days is so helplessly short of what one wants to do! Some of the participants had to leave without our being able to talk at all; there was simply no time for it. Then perhaps this reminiscence can serve as a substitute channel for conveying my thanks and appreciation to all of you.

Introduction

The algebraic Bethe Ansatz is presented in this chapter. This is an important generalization of the coordinate Bethe Ansatz presented in Part I, and is one of the essential achievements of QISM. The algebraic Bethe Ansatz is based on the idea of constructing eigenfunctions of the Hamiltonian via creation and annihilation operators acting on a pseudovacuum. The matrix elements of the monodromy matrix play the role of these operators. The transfer matrix (the sum of the diagonal elements of the monodromy matrix) commutes with the Hamiltonian; thus constructing eigenfunctions of τ(µ) determines the eigenfunctions of the Hamiltonian.

The basis of the algebraic Bethe Ansatz is stated in section 1. The commutation relations between matrix elements of the monodromy matrix are specified by the R-matrix. The explicit form of the commutation relations allows the construction of eigenfunctions of the transfer matrix (the trace of the monodromy matrix). (Recall that the Hamiltonian may also be obtained from the transfer matrix via the trace identities.) Further developments of the algebraic Bethe Ansatz necessary for the computation of correlation functions are given in section 2. The general scheme is illustrated with some examples in section 3. The NS model, the sine-Gordon model and spin models are considered in detail. The Pauli principle for interacting one-dimensional bosons plays an important role in constructing the ground state of the system and is discussed in section 4. The eigenvalues of the shift operator acting on the monodromy matrix are calculated in section 5.

Introduction

Impenetrable bosons in one space dimension are a special case (at coupling constant c → ∞) of the one-dimensional Bose gas (NS model) considered in detail in Chapter I. The results obtained there for the NS model can be specialized to the case of impenetrable bosons. It is to be said that all the formulæe are considerably simplified in this case. From the point of view of physics this is due to the fact that the δ-function potentials in the N-particle Hamiltonian (I.1.11) are now infinitely strong and the bosons cannot penetrate one another, so that the wave function should be equal to zero if the bosons' coordinates coincide. In this respect impenetrable bosons are rather similar to free fermions (see Appendix I.I). In this chapter correlation functions of impenetrable bosons are considered and their representations as Fredholm determinants of linear integral operators are given. The (very essential) simplification with respect to the general case is due to the fact that now all the auxiliary quantum fields can be set equal to zero (see IX.6), so that the kernels of these operators do not contain quantum fields. The determinant representation for the time-dependent correlator is easy to obtain in this case. Temperature correlation functions are considered as well as zero temperature ones.

Introduction

The quantum inverse scattering method (QSIM) appears as the quantized form of the classical inverse scattering method. It allows us to reproduce the results of the Bethe Ansatz and to move ahead. QISM is now a well developed branch of mathematical physics. In this chapter the fundamentals of QISM are given and illustrated by concrete examples.

In section 1 the general scheme of QISM, which allows the calculation of commutation relations between elements of the transfer matrix (necessary to construct eigenfunctions of the Hamiltonian in Chapter VII) is presented, and the quantum R-matrix is introduced. As in the classical case, the existence of an R-matrix and trace identities ensures that a Lax representation for the model exists. Thus, there are infinitely many conservation laws.

The Yang-Baxter equation, which is satisfied by the R-matrix, is discussed in section 2. Some important features of the R-matrix are also mentioned. The trace identities for the quantum nonlinear Schrödinger equation are proved in section 3. The general scheme of QISM is applied to the quantum sine-Gordon and Zhiber-Shabat-Mikhailov models in section 4. Spin models of quantum statistical physics are discussed in section 5. It is shown that a fundamental spin model can be constructed with the help of any given R-matrix.

The connection between classical statistical models on a two-dimensional lattice and QISM is established in section 6. QISM is the generalization of the classical inverse scattering method.

A method of solution of a number of quantum field theory and statistical mechanics models in two space-time dimensions is presented in this Part. This method was first suggested by H. Bethe in 1931 and is traditionally called the Bethe Ansatz. Later on the method was developed by Hulthen, Yang and Yang, Lieb, Sutherland, Baxter, Gaudin and others (see, and).

We begin the presentation with the coordinate Bethe Ansatz not only due to historical reasons but also because of its simplicity and clarity. The multi-particle scattering matrix appears to be equal to the product of two-particle matrices for integrable models. This property of two-particle reducibility is of primary importance when constructing the Bethe wave function. The important feature of integrable models is that there is no mass-shell multiple particle production. This property is closely connected to the existence of an infinite number of conservation laws in such models; this will be clear from Part II.

Four main models, namely the one-dimensional Bose gas, the Heisenberg magnet, the massive Thirring model and the Hubbard model, are considered in Part I. Eigenfunctions of the Hamiltonians of these models are constructed. Imposing periodic boundary conditions leads to a system of equations for the permitted values of momenta. These are known as the Bethe equations. This system can also be derived from a certain variational principle, the corresponding action being called the Yang-Yang action.

Introduction

The modern way to solve partial differential equations is called the classical inverse scattering method. (One can think of it as a nonlinear generalization of the Fourier transform.)

Nowadays, the classical inverse scattering method (CISM) is a well developed branch of mathematical physics (see Preface references. In this chapter, we shall give only the information necessary for the quantization which will be performed in the next chapter. The concepts of the Lax representation, the transition matrix and the trace identities are stated in section 1. Classical completely integrable partial differential equations will appear once more in this book. In Chapters XIV and XV we shall derive them for quantum correlation functions. In those chapters we shall study completely integrable differential equations from a different point of view. We shall apply the Riemann-Hilbert problem in order to evaluate the asymptotics. The classical r-matrix, which enables calculation of the Poisson brackets between matrix elements of the transition matrix (and also construction of the action-angle variables) is introduced in section 2. As explained there, the existence of the r-matrix guarantees the existence of the Lax representation. The r-matrix satisfies a certain bilinear relation (the classical Yang-Baxter relation). The existence of the r-matrix also guarantees the existence of an infinite number of conservation laws which restrict in an essential way the dynamics of the system. In the next chapter, the notion of the r-matrix will be generalized to the quantum case. In the first two sections of this chapter, general statements are demonstrated by example using the nonlinear Schrödinger equation which is the simplest dynamical model (it should be mentioned that in the classical case this name is more natural than the one-dimensional Bose gas).