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Physical Review

Coherence in a Model of Interacting Radiation and Matter

Richard H. Picard, C. R. Willis

Published Jul 5, 1965• 16 citations
Publisher

Abstract

Some preliminary results of a theoretical investigation of the origin of laser coherence properties are presented. A model consisting of $N$ two-level atoms in a perfect cavity interacting through their dipole moments with a single mode of the radiation field is adopted, and the resulting many-body problem is made solvable through the introduction of the self-consistent-field approximation (SCFA). It is shown that the SCFA is consistent with Glauber's suggestion that an ideal laser might be in a "coherent state," or pure eigenstate of the positive-frequency part of the electric-field operator. A conservation law is derived which assures the positive definiteness of the field-density operator, and it is shown that Glauber's coherent state is that solution of the SCFA equations which minimizes the corresponding constant of the motion. When the SCFA is modified by the introduction of particle-field correlations, the conservation law is destroyed, and the coherent state is no longer and allowed solution.

DOI
10.1103/physrev.139.a10

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Some preliminary results of a theoretical investigation of the origin of laser coherence properties are presented. A model consisting of NNN two-level atoms in a perfect cavity interacting through their dipole moments with a single mode of the radiation field is adopted, and the resulting many-body problem is made solvable through the introduction of the self-consistent-field approximation (SCFA). It is shown that the SCFA is consistent with Glauber's suggestion that an ideal laser might be in a "coherent state," or pure eigenstate of the positive-frequency part of the electric-field operator. A conservation law is derived which assures the positive definiteness of the field-density operator, and it is shown that Glauber's coherent state is that solution of the SCFA equations which minimizes the corresponding constant of the motion. When the SCFA is modified by the introduction of particle-field correlations, the conservation law is destroyed, and the coherent state is no longer and allowed solution.
Coherence properties of the single-mode electromagnetic radiation emitted by a system of identical two-level atoms which is initially in the superradiant state are determined approximately. The approximation is valid only for short times when the state of the atomic system does not differ appreciably from the superradiant state. The electromagnetic field radiated spontaneously possesses full coherence in the sense of Glauber's definition, its density operator having the form of that for the ideal laser radiation. This coherence disappears when, because of external radiation, stimulated and spontaneous emission both take place.
Recently Glauber has described the properties of coherent radiation fields, and has constructed the density matrix of the field in two simple cases: (1) The radiating system is a classical radiator and no reaction is considered; (2) the central-limit theorem applies to a collection of radiators. This paper investigates other simple "almost" exactly soluble problems, in which a quantum-mechanical two-level system interacts with a quantized electromagnetic field originally in a pure coherent state in a single mode. The first-order correlation function ${G}^{(1)}=〈{E}^{\ensuremath{-}}{E}^{+}〉$ is compared with $〈{E}^{\ensuremath{-}}〉〈{E}^{+}〉$ at resonance when the stimulating field is initially a pure coherent state and the two-level system is initially in its excited state. The corresponding quantities are also computed for a field whose initial density matrix is a Gaussian superposition of coherent states (e.g., blackbody radiation), as well as for a field which is initially described as having a given number of photons.
We investigate the long-time behavior of a model consisting of N two-level atoms in a lossless cavity. The Hamiltonian of our system contains the radiation oscillators in addition to the matter Hamiltonian and the usual ∫j·Adv interaction term. In order to treat the system perturbatively, it would be necessary to remove the tremendous degeneracy of the system. Since this is prohibitively difficult, and since we are interested in the long-time behavior of the system, we solve the quantum mechanical Liouville equation directly for a wide class of physically important initial distribution functions. We show the effective expansion parameter is γ̃Nγ̃ where γ̃ is a dimensionless atomic dipole moment and N is the number of atoms. In the lowest order we find the self-consistent field approximation. In the next order, particle-field correlations appear. We explicitly solve the equations of motion for the particle-field correlations in terms of the average quantities that appear in the self-consistent field approximation. We show the self-consistent field approximation consists of five first-order differential equations. Next we show the equations of motion for the density matrix of the system correct to order (γ̃Nγ̃)2 are equivalent to eight first-order differential equations. The three additional equations are needed to describe the three second moments of the density matrix of the electromagnetic field that appear in second order. Our lowest-order microscopic equations are equivalent to semiphenomenological theories and our higher-order equations contain only the measurable second-order moments of the electromagnetic field in addition to the variables that appear in semi-phenomenological theories.
Coherent spontaneous emission from a system of NNN two-level atoms interacting with a quantized radiation field is treated for the case in which Dicke's "cooperation number" rrr is macroscopically large. It is shown how to modify the quasiclassical approach to this problem to incorporate quantum effects that are lost in the self-consistent-field approximation. The statistics of the emitted radiation are found to vary markedly with the initial state of the system of atoms. The photon statistics tend to that typical of blackbody radiation when the initial state of the atomic system is that which would result from incoherent pumping ($m\ensuremath{\sim}r$). When, on the other hand, the atoms are initially in a superradiant state ($m\ensuremath{\ll}r$), the emitted radiation may be represented approximately by a coherent state.
Principles of coherent radiation are reviewed as they apply to the coherent interaction between two quantum Ievel atomic oscillators and classical radiation fields.The equivalence between electric dipole and magnetic dipole interactions is outlined for macroscopic systems.The power mechanism for formation of coherent signals is briefly calculated.The self-induced transparency mechanism and area-theorem connected with this mechanism serves as a means of predicting properties of photon echoes.It is stressed that when the classical field is a self-consistent solution to both the equations of the density matrix and Maxwell's equations a reaction field need not be added.
We consider the description of the quantum dynamics of a two-level system interacting with a single Glauber coherent mode of the radiation field. An exact solution (within the rotating wave and electric dipole approximations) is provided for both the time-independent and time-dependent Green’s functions for the coupled matter–radiation system. The atom (molecular) transition probabilities are shown to be Poisson averages of the corresponding semiclassical problem when the latter is likewise treated in the rotating wave approximation. At low radiation intensities, the quantum and semiclassical results differ, but at high intensities, they become the same when the average photon number n0 in the field satisfies the condition n01/2≫1. A similar behavior is displayed for the expectation value of the electric field of the coherent mode in the presence of the atom. The spectrum of the radiation, resonantly scattered by the atom, is shown to coincide with the results of Mollow.