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Zeitschrift für Naturforschung A

Phase Structures in the Micromaser Photon Statistics

János A. Bergou, Pál Bogár

Published Jan 1, 1999• 1 citation
Publisher

Abstract

Abstract In the photon number distribution p(θ,k) of the micromaser, distinct structures are formed by the ridges connecting the peaks in the θ-k space. We refer to these structures as phases. By a simple condition we can distinguish between "semiclassical" and "quantum" regimes of operation near and far above threshold, respectively. In the semiclassical regime, the equations of state of the phases θ= θ(k) are monotonous and coincide with the steady state solutions of the semiclassical theory. They reflect typical features of the micromaser dynamics. The transition jumps, e. g., between the phases decrease with increasing pumping parameter θ, signifying the onset of the Jaynes-Cummings collapse. On increasing 9 further, the phases first disintegrate and then restructure into new kinds of phases in the quantum regime. The equations of state are no longer monotonous. Large single peaks, the quantum island states (QIS), develop in the neighborhoods of minima. The system undergoes oscillations between two kinds of quantum island states, QIS- and QIS+ , as a function of θ. The disintegration and transformation of phases recur periodically as θ is varied, and the phases in the semiclassical region are followed by consecutive phase structures in the quantum regime. The subsequent collapses and revivals of phases are directly connected to the Jaynes-Cummings collapse and revival. The observation of these phase structures and the accompanying QIS is experimentally feasible.

DOI
10.1515/zna-1999-0106

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Abstract In the photon number distribution p(θ,k) of the micromaser, distinct structures are formed by the ridges connecting the peaks in the θ-k space. We refer to these structures as phases. By a simple condition we can distinguish between "semiclassical" and "quantum" regimes of operation near and far above threshold, respectively. In the semiclassical regime, the equations of state of the phases θ= θ(k) are monotonous and coincide with the steady state solutions of the semiclassical theory. They reflect typical features of the micromaser dynamics. The transition jumps, e. g., between the phases decrease with increasing pumping parameter θ, signifying the onset of the Jaynes-Cummings collapse. On increasing 9 further, the phases first disintegrate and then restructure into new kinds of phases in the quantum regime. The equations of state are no longer monotonous. Large single peaks, the quantum island states (QIS), develop in the neighborhoods of minima. The system undergoes oscillations between two kinds of quantum island states, QIS- and QIS+ , as a function of θ. The disintegration and transformation of phases recur periodically as θ is varied, and the phases in the semiclassical region are followed by consecutive phase structures in the quantum regime. The subsequent collapses and revivals of phases are directly connected to the Jaynes-Cummings collapse and revival. The observation of these phase structures and the accompanying QIS is experimentally feasible.
A distinctive signature of the granulated nature of the electromagnetic field is the revivals known to occur in the dynamics of a two-level atom interacting with a single-field mode (Jaynes-Cummings model). The development of high-Q single-mode cavities combined with progress in Rydberg atom spectroscopy has led to a possibility of verifying these predictions. Revival experiments using micromasers have been performed by Walther et al. Here the cavity mode is first allowed to reach a steady state, and then probe atoms are injected into the resonator. By measuring their state as they exit the cavity, one can infer the state of the field. We show that the micromaser experiences Jaynes-Cummingslike revivals when the pump parameter θ is varied. However, in contrast to the conventional Jaynes-Cummings situation, which considers initially coherent fields, the micromaser photon statistics are usually not Poissonian and depend sensitively on θ. The micromaser revivals cannot be explained by a semiclassical theory or linearized quantum fluctuations about the semiclassical solution. Rather, an exact quantum treatment, considering fully the granularity of the cavity mode, is required.
We study the photon statistics of a micromaser with a nonzero thermal background for a large range of the pumping parameter \ensuremath{\theta}. The photon statistics are examined in a two-dimensional space, parametrized by \ensuremath{\theta} and k, where k is the photon number. For values of \ensuremath{\theta} which are not too large, there are prominent peaks in the photon-number distribution. The locations of these peaks are determined by well-defined structures in \ensuremath{\theta}-k space. These structures lie along curves in this space. As \ensuremath{\theta} is increased, these structures disappear and the number distribution becomes diffuse. When \ensuremath{\theta} becomes sufficiently large, however, new structures which are well localized in \ensuremath{\theta} and k appear. We call these states ``island states.'' These states exhibit a strong squeezing of the photon number and are fairly insensitive to the thermal background. Their noise increases slowly due to fluctuations of the atomic beam as long as the spread in values of the pump parameter \ensuremath{\theta} is smaller than the distance between the islands. Bistable behavior can be induced by fluctuations that overlap adjacent island states. We also present the photon statistics of a micromaser pumped by atoms in a mixture (incoherent superposition) of their upper and lower levels. It is shown that the noise in island states can be drastically reduced by an optimum amount of injected absorption. These features render these states experimentally feasible, so that they could be used for generating squeezed quantum states of the radiation field that are well localized in \ensuremath{\theta} and k.
Two micromasers are coupled by sharing a common pump beam. A stream of excited atoms first passes through one micromaser and then enters a second one. The steady state of the second micromaser exhibits a second threshold at θ2= 2. The second threshold is accompanied with a jump of π in the phase of the second micromaser. For higher values of the pump parameter θ new branches of the steady-state intensity keep appearing. In certain parameter regions the phase of the second micromaser locks to that of the first and exhibits correlated emission laser-like noise quenching. In other regions the intensity fluctuations exhibit antibunching. The approach to steady state is also highly nonclassical. There are sharp jumps of magnitude π in the phase of the second micromaser and the intensity tends to overshoot before they settle to their steady-state value. In this transient regime the system exhibits phase noise squeezing.
We present a semiclassical method for determining the phase distribution and the average photon number of a steady-state micromaser field pumped by a stream of resonant two-level atoms in a superposition of their upper and lower states. Assuming the field to have a photon number distribution with a single sharp peak, we find an equation for the amplitude and the phase of the steady-state field, which has two solutions. The stability analysis shows which one of the two is physically realizable. Using the very simple time evolution of particular atomic states, valid in the semiclassical limit, we find an expression for the phase distribution of the field that contains the mean photon number as a parameter. That can be determined from the stable solution.
The statistics of the beam of excited atoms that pumps a micromaser influence the photon number distribution of the cavity field. Of the cases studied so far, regular injection statistics minimize the number fluctuations and Poisson statistics maximize them. Here we extend these studies in two directions. Previous master equations describing sub-Poissonian injection statistics were derived by using standard reservoir theory. Recently, corrections to these equations have been found that involve the commutator of the gain and loss operators. Describing the system by a discrete map, we can avoid the approximations inherent in reservoir theory. The results are compared to those of the master equation. Differences are found to be small for large photon number but increasingly important for decreasing photon number and pump beam intensity, indicating the breakdown of reservoir theory in this regime. We give an approximate expression for the photon number distribution for weak damping and discuss its connection with semiclassical theory. Finally, we generalize our treatment for super-Poissonian pump fluctuations.
We present the quantum theory of a microscopic maser operating on a degenerate two-photon transition between levels of the same parity. We derive both a master equation and a Fokker-Planck equation for this system, and show that quantum effects may have a substantial influence on the behavior of the maser. They modify the oscillation threshold and make external triggering of this maser unnecessary, whereas, according to semiclassical theory, such a triggering is required to start up the maser oscillation. We derive the phase-diffusion properties of the field and show that the diffusion coefficient is complex in this case, its imaginary part being associated with a frequency shift of the field inside the cavity. We show that, in steady state, the photon-number statistics is sub-Poissonian for a wide range of pumping rates.