A simple model of a maser or laser consisting of a single-mode field coupled to $N$ identical two-level atoms is considered. The semiclassical approximation is equivalent to neglecting the statistical correlations between the atoms and the field. The accuracy of this approximation is investigated by writing equations of motion for the correlations. To obtain a complete, self-consistent set of equations, it is necessary to include correlations of the field with itself (related to the coherence of the field) and correlations between different atoms. Relaxations and an energy source are introduced phenomenologically into the equations of motion. They are then solved for the case of steady-state oscillation. It is found that the correlations are smaller than the terms kept in the semiclassical theory by the order of 1 over the number of photons present in the field, larger if thermal photons abound, but smaller if the field relaxation is dominant. Expressions are also found for the amplitude and phase fluctuations of the field. The latter yield a linewidth for the maser oscillator in agreement with earlier calculations, but obtained by a different method.A simple model of a maser or laser consisting of a single-mode field coupled to $N$ identical two-level atoms is considered. The semiclassical approximation is equivalent to neglecting the statistical correlations between the atoms and the field. The accuracy of this approximation is investigated by writing equations of motion for the correlations. To obtain a complete, self-consistent set of equations, it is necessary to include correlations of the field with itself (related to the coherence of the field) and correlations between different atoms. Relaxations and an energy source are introduced phenomenologically into the equations of motion. They are then solved for the case of steady-state oscillation. It is found that the correlations are smaller than the terms kept in the semiclassical theory by the order of 1 over the number of photons present in the field, larger if thermal photons abound, but smaller if the field relaxation is dominant. Expressions are also found for the amplitude and phase fluctuations of the field. The latter yield a linewidth for the maser oscillator in agreement with earlier calculations, but obtained by a different method.

Physical ReviewMay 10, 196919 citations

Number-Conserving Approximations for the Theory of the Pairing Interaction in Nuclei. II

G. Do Dang, Abraham Klein

A previous study of the pairing interaction Hamiltonian by means of the equations of motion is extended. In the prior note, ground-state and single-quasiparticle properties were treated by an approximation method which was strictly particle conserving. According to the general theory, further improvement of the ground-state energies and single-particle occupation probabilities requires information about excited states of seniority zero and unity. A nonlinear, number-conserving random-phase approximation is developed for this purpose. Numerical calculations are carried out in a linearized version which is almost equivalent to the usual quasi-boson approximation. As part of the approximation, the conservation-of-particles condition is relaxed for the excited states. Results of the numerical study establish the consistency of the assumptions made and yield improved ground-state and single-particle properties. By comparison with the few models for which exact numerical results are available, the results of this paper are seen to be comparable with or superior to those of other improved approximation methods so far reported, especially so as the number of particles increases. Finally, it is pointed out that just as the results of the previous paper provide the input data for the present work, the results reported here can serve in like manner for more elaborate approximations, which are proposed.A previous study of the pairing interaction Hamiltonian by means of the equations of motion is extended. In the prior note, ground-state and single-quasiparticle properties were treated by an approximation method which was strictly particle conserving. According to the general theory, further improvement of the ground-state energies and single-particle occupation probabilities requires information about excited states of seniority zero and unity. A nonlinear, number-conserving random-phase approximation is developed for this purpose. Numerical calculations are carried out in a linearized version which is almost equivalent to the usual quasi-boson approximation. As part of the approximation, the conservation-of-particles condition is relaxed for the excited states. Results of the numerical study establish the consistency of the assumptions made and yield improved ground-state and single-particle properties. By comparison with the few models for which exact numerical results are available, the results of this paper are seen to be comparable with or superior to those of other improved approximation methods so far reported, especially so as the number of particles increases. Finally, it is pointed out that just as the results of the previous paper provide the input data for the present work, the results reported here can serve in like manner for more elaborate approximations, which are proposed.

Physical ReviewJul 22, 196623 citations

On the applicability of the quasiclassical approximation

P. V. Elyutin, D. V. Krivchenkov

Theoretical and Mathematical PhysicsMay 1, 19740 citations

Brief Review of Relaxation and Transport Theories

Jochen Gemmer B, Mathias Michel A, G. Mahler

Lecture notes in physicsJan 1, 20090 citations

Spin-boson systems: equivalence between the dilute-blip and the Born approximations

Claude Aslangul, No�lle Pottier, Daniel Saint-James

We demonstrate the full equivalence between the dilute-blip approximation introduced by Chakravarty and Leggett, and the second-order Born approximation of the general relaxation theory. Besides its simplicity, the latter scheme allows us to derive the explicit dynamics in a direct way, invoking only standard analysis based on Laplace transformation. As an example, the case of the symmetric double-well potential is revisited. In addition, explicit expressions of the coordinate of the particle are derived for some values of the coupling constant a, which serve as illustrations of the three cases : exponentially damped oscillatory relaxation ( a < 1/2 ) , non-exponential and non-oscillatory relaxation ( 1/2 < α < 1) and finally, dynamics above the critical value ( a > 1) .We demonstrate the full equivalence between the dilute-blip approximation introduced by Chakravarty and Leggett, and the second-order Born approximation of the general relaxation theory. Besides its simplicity, the latter scheme allows us to derive the explicit dynamics in a direct way, invoking only standard analysis based on Laplace transformation. As an example, the case of the symmetric double-well potential is revisited. In addition, explicit expressions of the coordinate of the particle are derived for some values of the coupling constant a, which serve as illustrations of the three cases : exponentially damped oscillatory relaxation ( a < 1/2 ) , non-exponential and non-oscillatory relaxation ( 1/2 < α < 1) and finally, dynamics above the critical value ( a > 1) .

Journal de physiqueJan 1, 1986102 citations

A simple model of a maser or laser consisting of a single-mode field coupled to

N

identical two-level atoms is considered. The semiclassical approximation is equivalent to neglecting the statistical correlations between the atoms and the field. The accuracy of this approximation is investigated by writing equations of motion for the correlations. To obtain a complete, self-consistent set of equations, it is necessary to include correlations of the field with itself (related to the coherence of the field) and correlations between different atoms. Relaxations and an energy source are introduced phenomenologically into the equations of motion. They are then solved for the case of steady-state oscillation. It is found that the correlations are smaller than the terms kept in the semiclassical theory by the order of 1 over the number of photons present in the field, larger if thermal photons abound, but smaller if the field relaxation is dominant. Expressions are also found for the amplitude and phase fluctuations of the field. The latter yield a linewidth for the maser oscillator in agreement with earlier calculations, but obtained by a different method.