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Journal of Physics A Mathematical and General

Theory of radiation reaction and atom self-energies: an operator reaction field

R Saunders, R. Bullough, F Ahmad

Published May 1, 1975• 23 citations
Publisher

Abstract

The field generalizes in a natural way the c-number theory of radiation reaction and frequency shifts due to Lorentz. By a trivial extension the operator field can be applied to the more complicated cases of a single multi-level atom and the super-radiant problem of N atoms on the same site where these are coupled either to the vacuum or the vacuum and additional imposed quantized fields. By further extension it also applies to the case of N atoms on different sites. Explicit expressions for positive and negative frequency parts of the total operator field are given in terms of arbitrary dipole operator sources and are apparently valid for times t>> omega s -1 >> omega max -1 where omega s is the atomic resonance and omega max a cut-off used systematically throughout the theory.

DOI
10.1088/0305-4470/8/5/011

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. 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.

Cambridge University Press eBooks•Sep 2, 1993•0 citations
The field generalizes in a natural way the c-number theory of radiation reaction and frequency shifts due to Lorentz. By a trivial extension the operator field can be applied to the more complicated cases of a single multi-level atom and the super-radiant problem of N atoms on the same site where these are coupled either to the vacuum or the vacuum and additional imposed quantized fields. By further extension it also applies to the case of N atoms on different sites. Explicit expressions for positive and negative frequency parts of the total operator field are given in terms of arbitrary dipole operator sources and are apparently valid for times t>> omega s -1 >> omega max -1 where omega s is the atomic resonance and omega max a cut-off used systematically throughout the theory.
A natural all-order decorrelation procedure is described for the interaction of a single two-level atom with all modes of the radiation field which replaces the semi-classical boson approximation for the atom by an exact fermion treatment. In consequence the vacuum Lamb shift appears as twice the Bethe shift for each level separately instead of the limiting neo-classical shift obtained in the Bose theory. Stimulated emission and absorption modify the Einstein A-coefficient and the vacuum shift is extended to include a field dependent shift. The all-order decorrelated perturbation theory yields an equation of motion for the single-atom dipole moment which includes those reported from operator radiation reaction theory recently; but it also generalizes these by including the effects of stimulated processes. The method is extended to a many-atom case in which it becomes necessary to modify the procedure slightly.
Abstract The atom-field interaction is formulated within the fully quantized-field theory, starting from a detailed analysis of the transformation from the fundamental minimal coupling interaction Hamiltonian to the electric dipole Hamiltonian used extensively in quantum optics. Spontaneous emission, radiative level shifts, and the natural radiative lineshape are treated in both the Schrodinger and Heisenberg pictures, with emphasis on the roles of vacuum field fluctuations, radiation reaction, and the fluctuation-dissipation relation between them. The shortcomings of semiclassical radiation theories are discussed.
We review the use of operator reaction field theory in quantum optics by considering the problem of one atom in an isotropic three-dimensional broad-band squeezed vacuum. In these terms we give a quantum electrodynamical analysis of this system exact up to all contributions of order one in the ratio γω 0 -1 , the ratio of the A -coefficient to the atomic resonance frequency. These terms include all of the counter-rotating terms at this order. At zero order, namely in the usual rotating wave approximation, we find the usual Einstein rate equation while we confirm that the fluorescence spectrum consists of two Lorentzian peaks. However, at order one we find an additional oscillating term in the rate equation and additional resonant dispersive terms correct for each of the two Lorentzian peaks in the fluorescence spectrum. Similar results are expected for broad-band correlated squeezed vacua of arbitrary, rather than isotropic, geometry.
In this paper, we dwell on three issues: (1) revisit the relation between vacuum fluctuations and radiation reaction in atom-field interactions, an old issue that began in the 1970s and settled in the 1990s with its resolution recorded in monographs; (2) the fluctuation–dissipation relation (FDR) of the system, pointing out the differences between the conventional form in linear response theory (LRT) assuming ultra-weak coupling between the system and the bath, and the FDR in an equilibrated final state, relaxed from the nonequilibrium evolution of an open quantum system; (3) quantum radiation from an atom interacting with a quantum field: We begin with vacuum fluctuations in the field acting on the internal degrees of freedom (idf) of an atom, adding to its dynamics a stochastic component which engenders quantum radiation whose backreaction causes quantum dissipation in the idf of the atom. We show explicitly how different terms representing these processes appear in the equations of motion. Then, using the example of a stationary atom, we show how the absence of radiation in this simple cases is a result of complex cancellations, at a far away observation point, of the interference between emitted radiation from the atom and the local fluctuations in the free field. In so doing we point out in Issue 1 that the entity which enters into the duality relation with vacuum fluctuations is not radiation reaction, which can exist as a classical entity, but quantum dissipation. Finally, regarding issue 2, we point out for systems with many atoms, the co-existence of a set of correlation-propagation relations (CPRs) describing how the correlations between the atoms are related to the propagation of their (retarded non-Markovian) mutual influence manifesting in the quantum field. The CPR is absolutely crucial in keeping the balance of energy flows between the constituents of the system, and between the system and its environment. Without the consideration of this additional relation in tether with the FDR, dynamical self-consistency cannot be sustained. A combination of these two sets of relations forms a generalized matrix FDR relation that captures the physical essence of the interaction between an atom and a quantum field at arbitrary coupling strength.
. 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.