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Journal of Modern Optics

Long-term Behaviour of the Field in a Lossless Microscopic Maser

Henryk Paul

Published Apr 1, 1989• 8 citations
Publisher

Abstract

An asymptotic form of the density matrix for the field inside the cavity of a microscopic maser is derived. It is shown that it contains essentially the same information as that which has been inferred from numerical experiments in which the results of repeated measurements on the atoms as they leave the cavity have been taken into account. In particular, both approaches predict that the field, which is assumed to be initially of a thermal nature, evolves towards certain ‘trapping levels’. The analytic approach presented here is more general since it includes the evaluation of non-diagonal matrix elements.

DOI
10.1080/09500348914550581

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Physical Review A•Oct 1, 1986•545 citations
An asymptotic form of the density matrix for the field inside the cavity of a microscopic maser is derived. It is shown that it contains essentially the same information as that which has been inferred from numerical experiments in which the results of repeated measurements on the atoms as they leave the cavity have been taken into account. In particular, both approaches predict that the field, which is assumed to be initially of a thermal nature, evolves towards certain ‘trapping levels’. The analytic approach presented here is more general since it includes the evaluation of non-diagonal matrix elements.
We study the one-atom maser with a periodic-pump scheme for the injected atoms. The dynamics of the system is solved also in a regime where the losses of the field cannot be neglected while the atom is inside the cavity, as is the situation in an optical maser. The time development of the maser field is described by a general evolution operator, whose matrix representation in the damping basis is known analytically for zero temperature. The numerical evaluation of this matrix yields the stationary as well as the transient properties of the maser. Noise-producing mechanisms of the pump are included explicitly in the evolution operator for the field. These are a finite efficiency of the pump laser, a velocity spread in the atomic beam, and spontaneous transitions between the atomic levels and to the ground state. We find that features such as trapped states and sub-Poissonian photon statistics also occur in the high-loss regime like in an optical resonator. The observation of such effects will, as in the ordinary micromaser experiments, depend very sensitively on the reduction of the pump noise. For highly regular excitation, we find a pronounced oscillation of the field-correlation function for a wide range of the transit time, which leads to a distinct line splitting in the spectrum of the field. The amount of the splitting is determined by the periodicity of the pump. For less regular excitation, and in particular in the Poissonian limit, these oscillations disappear.
Trapping states in the one-atom maser or micromaser are predicted by maser theory and are a direct consequence of the quantization of the electromagnetic field. They have been found in our experiment when the inversion of atoms leaving the cavity is observed while the interaction time is scanned. They correspond to quantum states of the atom-cavity system and under certain conditions represent Fock states. The trapping states occur when the atom-field dynamics complete a full evolution (or multiples thereof). Owing to feedback through the maser dynamics, a stabilization of the photon number in the steady state field of the maser cavity is achieved.
The transient response of a three-level optical maser is calculated for a density of homogeneously broadened atoms coupled to one homogeneously broadened cavity mode. The atoms are treated in terms of the density matrix and the cavity mode is treated classically. Solutions are found analytically for the steady state and for the linearized form of two simplifications of the general equations. These special solutions have been previously obtained by Bloom and Statz and de Mars. Numerical solutions are found for a certain set of parameters. The response linewidth is found assuming one cavity mode coupled to a density of inhomogeneously broadened atoms.
We derive a master equation for the dynamics of a maser or laser with periodic injection of excited atoms. The model includes regular pumping with single atoms, but it also covers the case of periodic injection of groups of atoms, whose number may fluctuate. To obtain a quasicontinuous master equation for such a periodically kicked dynamics, we consider a macroscopic state of the field that is obtained from the true microscopic state by a time average over many periods. The master equation for that time-averaged state correctly describes both the stationary and the transient properties of the maser on a coarse, macroscopic scale. Numerical and analytical results based on this equation are compared with a microscopic, step-by-step resolution of the field's evolution along the periods. Deviations from the microscopic results only occur when frequencies of the order of the (mechanical) injection frequency of the pump become relevant. The recently predicted regularity-induced line splitting in the emission spectrum of a maser is an example for such a situation.
We present the theory of a truly microscopic maser consisting of a single-mode high-Q resonator in which a monoenergetic beam of excited two-level atoms is injected at such a low flux that at most one atom at a time is present inside the cavity. Both a microscopic theory and a heuristic Fokker-Planck approach are presented. We show that the micromaser exhibits a number of novel features that are averaged out in usual masers and lasers. First, the field is in general sub-Poissonian, which reflects the quantization of both the field and its sources. Second, the onset of maser oscillations may be followed by a succession of abrupt transitions in the state of the field. Finally, as the atomic flux through the resonator is increased, the maser threshold acquires characteristics of a continuous phase transition, whereas the subsequent changes in the field distribution become analogous to first-order phase transitions.