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Mathematical Proceedings of the Cambridge Philosophical Society

On a quantum mechanical model for a maser I

Martin Hasler

Published Mar 1, 1976• 1 citation
Publisher

Abstract

Abstract The model is closely connected with a model by Lamb and Scully (10). Atoms described as two-level systems, initially in an incoherent superposition of the two levels, interact successively during a time T with an electromagnetic field of which only one mode is taken into consideration. In the limit as infinitely many atoms have interacted, it is shown that the field either approaches a thermal distribution or is excited to arbitrarily high Photon numbers according to whether or not the lower level of the atoms is initially more probable than the upper level. It is also shown that in any case the correlations between pure Photon number states converge to 0. If the atoms are initially in the upper level it is proved that the Photon number grows roughly as the square root of the number of atoms that have interacted. Throughout the discussion number-theoretical properties of T play a disturbing role. The last mentioned result in fact depends on a sharp (but arbitrary) value for T and is therefore disqualified for physical interpretation.

DOI
10.1017/s030500410005235x

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Abstract The model is closely connected with a model by Lamb and Scully (10). Atoms described as two-level systems, initially in an incoherent superposition of the two levels, interact successively during a time T with an electromagnetic field of which only one mode is taken into consideration. In the limit as infinitely many atoms have interacted, it is shown that the field either approaches a thermal distribution or is excited to arbitrarily high Photon numbers according to whether or not the lower level of the atoms is initially more probable than the upper level. It is also shown that in any case the correlations between pure Photon number states converge to 0. If the atoms are initially in the upper level it is proved that the Photon number grows roughly as the square root of the number of atoms that have interacted. Throughout the discussion number-theoretical properties of T play a disturbing role. The last mentioned result in fact depends on a sharp (but arbitrary) value for T and is therefore disqualified for physical interpretation.
We study the relaxation properties of the quantized electromagnetic field in a cavity under repeated interactions with single two-level atoms, so-called one-atom maser. We improve the ergodic results obtained in [BP] and prove that, whenever the atoms are initially distributed according to the canonical ensemble at temparature T>0, all the invariant states are mixing. Under some non-resonance condition this invariant state is known to be thermal equilibirum at some renormalized temperature T* and we prove that the mixing is then arbitrarily slow, in other words that there is no lower bound on the relaxation speed.
There are only fistful of systems in physics that can be described by an exact theory from first principles and, at the same time, can be investigated under experimental conditions approaching the idealized theoretical ones. The one-atom-maser or micromaser provides such an example, making a detailed study of the fundamental properties of the atom-field interaction possible. The situation realized in a micromaser is very close to the ideal case of a single two-level atom interacting with a single quantized mode of a cavity field. In the micromaser the atoms have a dual purpose of both pumping the field and also probing it via measurements of the outgoing atoms. Any other means of measuring the field inside the resonator has the detrimental effect of lowering its Q-factor and thereby the photon storage time. The long photon storage time of the resonator allows for the decay of the field to be negligible during the passage of an atom and its interaction with the field. The behavior of atoms in a cavity is governed by the oscillatory exchange of energy between the atoms and the field, which is called Rabi oscillation. For cavity fields in the vacuum and few photon number states (Fock states), Rabi oscillations have been measured in the past. To prevent thermal effects from polluting the pure number states these experiments were performed at low temperatures (below 1 K). However, the observed resolution of the Rabi oscillations measurements was quite disappointing (only about 2% of the predicted theoretical value) and precluded more ambitious experiments like atom-atom correlations or time resolved investigations of the maser field dynamics. The observed contrast of Rabi oscillations can be considered as a figure of merit for the experimental resolution of quantum features of a maser field. Therefore, a high resolution in the measurements of Rabi oscillations is crucial. This thesis describes the methods and improvements applied to enhance the resolution of Rabi oscillations by a factor of 8. The following upgrades and developments proved to be crucial in this work: 1) The base temperature of the cryogenic system was lowered to 0.3 K, i.e. one third of the original value. At the same time, the time during which the system remains at the base temperature was improved by almost of an order of magnitude (to more than 12 h). 2) For the promotion of the atoms to the Rydberg regime new three-step diode laser setup was constructed. The excitation efficiency compared to the old dye laser was increased by two orders. Using top-of-fringe stabilization with new spectroscopic methods, specially designed error detection and feedback schemes as well as state-of-the-art PID regulators the continuous locking of lasers is about 8 h (compared to about 20 min with the old dye laser system). 3) The flexibility of using the three step excitation allows to excite different maser states and for the first time to investigate pure maser transition. 4) Various improvements on the atomic oven and the channeltron detection unit led to the reliable and stable production and detection of the atomic beam. Observed distribution of the atomic beam statistics reaches to within 6% the Poisson limit which is expected in the absence of experimental imperfections. 5) New developed techniques in magnetic field compensation allow to perform such measurements with two orders of magnitude better accuracy compared to the previous experiments. To demonstrate the new capabilities of the apparatus and to apply its improved resolution of quantum fields we performed a previously impossible in-depth analysis of mean photon number and the atomic inversion produced by a micromaser for various detunings between the cavity- and atomic resonance frequency.
We discuss the quantum theory of a simple model of a maser oscillator, consisting of one radiation-field mode interacting with a large number of stationary three-level atoms. The field and the atoms also interact with separate heat reservoirs which represent dissipation mechanisms and an incoherent pumping mechanism. The model is sufficiently simple that some analytical progress can be made with the nonlinear quantal equations before further approximation is necessary. We start from the quantal equation of motion of the field-atom density operator. We make immediate use of a diagonal coherent-state expansion for the field part of the density operator and a somewhat similar expansion for the atom part. This yields an exact equation of the Fokker-Planck form for a ccc-number weight distribution, which retains all the significance of the original operator equation, and which has the semiclassical equation for the same model as a first, fluctuation-free approximation. We make use of our basic Fokker-Planck equation in a variety of ways. We discuss the reduction of the equation under conditions that the atomic decay constants are large (large atomic linewidth), arriving finally at an equation of motion for a field-only weight function which serves to demonstrate the basic coherence properties of a maser. We derive and discuss the equation of motion of the generalized Wigner density for the maser model. The generalized Wigner density is a smoothed version (a convolution) of our basic weight distribution, and from it we derive an equivalent classical model including noise sources. Finally, we discuss other useful weight distributions and the number representation for the field. The equations we derive in these discussions make contact with the rate equations of Shimoda, Takahasi, and Townes, as well as with the more recent work of Lax and Louisell, Lax, and of Scully and Lamb.
Introduction In 1963, E. T. Jaynes and F. W. Cummings published a paper entitled “Comparison of Quantum and Semiclassical Radiation Theories, with Application to the Beam Maser.” This paper was written at a time when the quantum theory of the laser had not yet been worked out, and represented an attempt at discussing the new behaviour that could be expected from treating the field quantum mechanically instead of classically. Jaynes and Cummings began by considering the interaction between a single two-level molecule and a single field mode, a simple situation which has become known as the Jaynes-Cummings model. Treating then the maser medium as a beam of two-level systems initially in a mixture of their upper and lower states characterized by a temperature T , they found that the field mode reached a thermal equilibrium with temperature T f = Ω T /ω, where ω is the transition frequency of the two-level systems and Ω is the field mode frequency, provided that T > 0. For T < 0, the steady state field was found to contain an infinite amount of energy, an unphysical result resulting from the neglect of losses in the model. Jaynes and Cummings then proceeded to develop an alternative semi-classical theory of the beam maser, introducing their famous neo-classical theory of spontaneous emission. An important follow-up paper was published by F. W. Cummings, who studied the long-time behavior of a two-level system interacting with a quantized single-mode field initially in a coherent state.
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 study the resonant interaction of A initially unexcited two-level atoms with single-mode Fock, coherent and thermal cavity fields of arbitrary intensity. Two mechanisms lead to a spread of the eigenfrequencies: one is due to the photon-number distribution in the initial state and the other reflects the cooperativity of the system. Both affect the evolution of the photon number and photon statistics. We present approximate expressions for the eigenvalues and eigenvectors of the system. Its regular dynamical regions are found and the difference in shape of the collapses and revivals connected with the above mechanisms is discussed.