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Physical Review A

Coherence properties of the N -atom-radiation interaction and the Holstein-Primakoff transformation

Franco Persico, G. Vetri

Published Nov 1, 1975• 33 citations
Publisher

Abstract

We consider a system of $N$ two-level atoms initially in the ground state and interacting with a small [$〈n(t=0)〉\ensuremath{\ll}N$] coherent radiation field. The field is assumed to be monochromatic, and its wavelength is taken to be much larger than the dimensions of the region of the $N$ atoms. The statistical properties of both the atomic system and the radiation field are studied as functions of time. Using the Holstein-Primakoff transformation, it is shown that the initial coherence of the system tends to disappear after the energy of the system has been exchanged between atoms and field a number of times approximately given by $\frac{N}{〈n(t=0)〉}$.

DOI
10.1103/physreva.12.2083

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We consider a system of NNN two-level atoms initially in the ground state and interacting with a small [〈n(t=0)〉\ensuremath≪N〈n(t=0)〉\ensuremath{\ll}N〈n(t=0)〉\ensuremath≪N] coherent radiation field. The field is assumed to be monochromatic, and its wavelength is taken to be much larger than the dimensions of the region of the NNN atoms. The statistical properties of both the atomic system and the radiation field are studied as functions of time. Using the Holstein-Primakoff transformation, it is shown that the initial coherence of the system tends to disappear after the energy of the system has been exchanged between atoms and field a number of times approximately given by N〈n(t=0)〉\frac{N}{〈n(t=0)〉}〈n(t=0)〉N​.
The time development of NNN two-level atoms linearly coupled to a monochromatic, long-wavelength radiation field is studied. Both systems are assumed to be initially in coherent states, the radiation mode being strongly populated with an average number nnn of photons larger than NNN. The decay of atomic coherence can be followed up to a time t∗{t}^{*}t∗, after which the perturbation expansion in powers of ${n}^{\ensuremath{-}\frac{1}{2}}$ used to solve the problem breaks down. We show that it increases proportional to t2{t}^{2}t2 at large times, as measured from the variances of the components of the total angular momentum of the atoms. The results are briefly discussed in connection with the problem of conversion of incoherent into coherent radiation.
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.
We investigate properties of the cooperative emission of radiation from NNN identical two-level atoms by using, for the collective atomic operators, the Holstein-Primakoff transformation to boson operators. In the basis of the Glauber states, a weight function is associated with the atomic density operator. The equation of motion for the weight function is derived from the master equation and approximate solutions are given in the two limit cases of superradiance and diffusion. The moments of the radiation intensity are calculated and are found to be in close agreement with previous works.
A method utilising the coherent state variables for the electromagnetic field and appropriate spin variables for two-level atomsis developed for obtaining the dynamical evolution of the field and the atoms separately. The case of a single radiation mode interacting with two two-level atoms is studied. The time evolution of the photon number, starting from n photons and both atoms at their higher levels, is given. Furthermore, when the initial state consists of n photons and one of the atoms in its excited state with the other in its ground state, the temporal development of the photon number, as well as the motion of each atom, are obtained. In the latter case it becomes apparent that energy exchange takes place among the atomic systems, while the field acts as a transfer agent.
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.
In an earlier paper, we considered a Hamiltonian trilinear in boson operators, which described several physical systems of interest such as interaction of NNN two-level atoms with a single-mode resonant radiation field, parametric amplification, frequency conversion, Raman and Brillouin scattering, etc. We found that the dynamics of the system can almost exactly be determined if either one of the three modes is strong for all times or any two modes are separately strong in two complementary time domains. In the present paper, we determine the statistical behavior of the system in each case. We give explicit results for a number of initial states (e.g., coherent, chaotic, a mixture of coherent and chaotic, number state, etc.) of the modes 1, 2, and 3.