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Journal of Physics A Mathematical and General

Free field-matter commutation relations in operator radiation reaction theory

B. R. Mollow

Published Nov 1, 1975• 32 citations
Publisher

Abstract

Resonant light scattering is considered in the Heisenberg picture by making use of operator radiation reaction theory. A crucial commutator between matter and free-field variables is shown to vanish, and the validity of the quantum fluctuation-regression theorem, like that of the optical Bloch equations which govern the reduced atomic density matrix, is firmly established, apart from renormalization difficulties, within the rotating wave approximation.

DOI
10.1088/0305-4470/8/11/003

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Resonant light scattering is considered in the Heisenberg picture by making use of operator radiation reaction theory. A crucial commutator between matter and free-field variables is shown to vanish, and the validity of the quantum fluctuation-regression theorem, like that of the optical Bloch equations which govern the reduced atomic density matrix, is firmly established, apart from renormalization difficulties, within the rotating wave approximation.
A fully quantum-mechanical treatment of resonant light scattering is presented. The incident field is assumed to be described by a coherent state, and is allowed to be intense enough to cause saturation. Complete solutions are obtained for the correlated atom-field pure state vector, including multiphoton contributions of arbitrary order. The frequency spectrum of the scattered field is evaluated and is found to agree exactly with the result previously obtained by means of the quantum fluctuation-regression theorem. A derivation of the fluctuation-regression theorem and of the optical Bloch equations is given which is fully quantum mechanical and which relies upon no assumption of statistical factorization of atom and field states. The accuracy of the result found for the scattered - field spectrum is thus shown to be limited only by the assumption of the smallness of the saturated linewidth compared to the (optical) atomic resonance frequency. The one-photon approximation is analyzed in some detail. The method of adding an imaginary term to the upper-atomic-state energy is clarified, and it is shown how the vacuum and one-photon amplitudes thereby obtained may be used, within a simple and plausible iteration scheme, to construct the complete multiphoton spectrum. A variety of commonly used injection schemes and methods of representing atomic relaxation are discussed, and comparisons are made with results found by other authors. The entire analysis is performed with the aid of a canonical transformation which replaces the applied field by a ccc number. It is thus proved quite rigorously and generally that the use of a ccc-number applied field is a fully quantum-mechanical procedure, provided only that radiation-reaction terms are retained.
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.
Abstract The atom-field interaction is treated in semiclassical radiation theory, starting from the transformation from the minimal coupling Hamiltonian to the electric dipole form used extensively in quantum optics. The Heisenberg and density matrix approaches are developed and applied to two-state atoms, Bloch equations, Rabi oscillations, Maxwell-Bloch equations, and transition rates for absorption and emission. Einstein’s theory of blackbody radiation based on momentum fluctuations and dissipation is reviewed. The Einstein fluctuation formula is derived and used to introduce wave-particle duality, Hanbury Brown-Twiss correlations, and photon bunching.
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.
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.
In many problems in quantum optics, damping plays an important role. These include, for example, the decay of an atom in an excited state to a lower state and the decay of the radiation field inside a cavity with partially transparent mirrors. In general, damping of a system is described by its interaction with a reservoir with a large number of degrees of freedom. We are interested, however, in the evolution of the variables associated with the system only. This requires us to obtain the equations of motion for the system of interest only after tracing over the reservoir variables. There are several different approaches to deal with this problem. In this chapter, we present a theory of damping based on the density operator in which the reservoir variables are eliminated by using the reduced density operator for the system in the Schrödinger (or interaction) picture. We also present a ‘quantum jump’ approach to damping. In the next chapter, the damping of the system will be considered using the noise operator method in the Heisenberg picture. An insight into the damping mechanism is obtained by considering the decay of an atom in an excited state inside a cavity. The atom may be considered as a single system coupled to the radiation field inside the cavity. Even in the absence of photons in the cavity, there are quantum fluctuations associated with the vacuum state. As discussed in Chapter 1, the field may be visualized as a large number of harmonic oscillators, one for each mode of the cavity.