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IRE Transactions on Electron Devices

Super-radiation and super-regeneration

Carl Greifinger, GEORGE L. BIRNBAUM

Published Jul 1, 1959• 3 citations
Publisher

Abstract

The transient behavior of a two-level spin system coupled to an electric circuit is investigated by using the equations of Bloembergen and Pound. The equations are solved, in the limit where the circuit ringing time is very short compared with all other characteristic times, for two cases: 1) the spin-lattice and spin-spin relaxation times both infinite, with an externally applied driving field, and 2) the spin-lattice relaxation time infinite but the spin-spin relaxation time finite, in the absence of an external field. In case 1), it is shown that the motion of an initially inverted magnetization under the action of an applied signal consists roughly of two stages: in the first stage, the effect of radiation damping is unimportant and the motion of the system is determined principally by the applied signal via the ordinary Bloch equations, whereas in the second stage, the motion is essentially the same as if the applied signal had been turned off and only radiation damping were present. In case 2), it is shown that a delayed peak in the emitted radiation should be observed under certain conditions. The delayed peak condition is identical with that derived by Bloom. Curves are presented showing the peak power and the time at which the delayed peak occurs as functions of the relevant parameters. In connection with the ordinary maser behavior of a two-level spin system, it is shown that for values of the parameters typical of steady-state maser amplification, the effects of radiation damping should be unimportant. Finally, systems are examined for which the radiation damping time is much shorter than all other characteristic times (super-regenerative systems). It is indicated how such systems might be operated as one-shot multivibrators or as linear amplifiers. For the latter type of operation, an expression for the gain is derived which is found to be similar to that encountered in ordinary circuit theory.

DOI
10.1109/t-ed.1959.14549

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The Journal of Chemical Physics•Jul 2, 2013•5 citations
The transient behavior of a two-level spin system coupled to an electric circuit is investigated by using the equations of Bloembergen and Pound. The equations are solved, in the limit where the circuit ringing time is very short compared with all other characteristic times, for two cases: 1) the spin-lattice and spin-spin relaxation times both infinite, with an externally applied driving field, and 2) the spin-lattice relaxation time infinite but the spin-spin relaxation time finite, in the absence of an external field. In case 1), it is shown that the motion of an initially inverted magnetization under the action of an applied signal consists roughly of two stages: in the first stage, the effect of radiation damping is unimportant and the motion of the system is determined principally by the applied signal via the ordinary Bloch equations, whereas in the second stage, the motion is essentially the same as if the applied signal had been turned off and only radiation damping were present. In case 2), it is shown that a delayed peak in the emitted radiation should be observed under certain conditions. The delayed peak condition is identical with that derived by Bloom. Curves are presented showing the peak power and the time at which the delayed peak occurs as functions of the relevant parameters. In connection with the ordinary maser behavior of a two-level spin system, it is shown that for values of the parameters typical of steady-state maser amplification, the effects of radiation damping should be unimportant. Finally, systems are examined for which the radiation damping time is much shorter than all other characteristic times (super-regenerative systems). It is indicated how such systems might be operated as one-shot multivibrators or as linear amplifiers. For the latter type of operation, an expression for the gain is derived which is found to be similar to that encountered in ordinary circuit theory.
When a two-state quantum system, capable of making radiative transitions, is driven by a high-frequency field, there occur damping effects due to the reaction of the sample to its own radiated field. This damping, which becomes important for high-Q rf structures and for samples of large dc susceptibilities, is here analyzed classically for a simple prototype system—a magnetic spin system obeying the Bloch equations. These equations are augmented to include the radiation-damping torques and are then solved for the four cases of usual interest. Firstly, for steady-state slow passage, radiation damping broadens and lowers the absorption and dispersion signals. Secondly, for adiabatic fast passage, damping causes the dispersion signal to dip at the line center and causes the otherwise absent absorption signal to become finite. However, if the radiation damping field becomes larger than the rf driving field a complete adiabatic inversion is shown to be no longer possible. Thirdly, the conditions for the occurrence of delayed power peaks of the free-precession signal are discussed. Fourthly, for the case of the driven nonsteady state, the stimulated emission may, for large radiation damping, become quite large if the system is driven from an initially ``negative temperature'' condition.
The operation of a pulsed solid-state Maser consists essentially of alternate periods of spin inversion and stimulated spin emission. Inversion can be accomplished by a short driving pulse or an adiabatic rapid passage. In the emission phase a weak signal field causes the spin system to spiral back coherently and radiate. The emissive phase can be treated by equations which for finite relaxation times can not be solved in closed form, but which are amenable to machine computation. Such numerical solutions giving the transverse moment, output field, output power, and power gain as a function of time were computed for various relaxation times, input fields, and values of the oscillation parameter. The results are interpreted with respect to applications of both regenerative and superregenerative modes of operation.
This paper summarizes the results of calculations made in an attempt to explain the origin of the observed modulation effects in two-level maser experiments [1,2]. \n \nIn a spontaneously oscillating spin system any transverse magnetization, which couples rf power out of the spin system, which is due to the presence of a transverse rf field, which is in turn induced by the transverse magnetization. Any consideration of the dynamics of the spin system or the build-up of the radiation field must consequently treat the two systems simultaneously. \n \nOur analysis uses a model first formulated by Bloembergen and Pound [3], except that our problem is inherently nonlinear [4] and we cannot make the simplifying assumption Mz = const, which linearizes the the equations of motion, and are forced to have rescourse to an IBM 704.
The study of a two-level system (TLS) coupled to a loss mechanism (LM), the fluctuations of which are taken into account, is motivated by relating the problem to spin-lattice relaxation, radiation damping, and Brownian motion. A TLS of the electric dipole type (coupled to the LM through one variable only) is discussed first. The problem is formulated in terms of the Pauli spin matrices, and the Langevin equations for Brownian motion of a TLS are derived. Their solution---made possible by taking expectation values in LM space---contains the Weisskopf-Wigner exponential decay formula of radiation theory, as well as a second-order shift in frequency of the TLS produced by the (nondispersive) LM. A driving force is added to the problem and expressions are obtained for a driven lossy TLS of the electric dipole type. The same analysis is applied to a TLS of the magnetic dipole type and differential equations for the spin matrices---for which the expectation values in LM space has been taken---are obtained. Taking expectation values in TLS space converts these equations into the Bloch equations. Unlike the electric dipole case, no frequency shift is present.
The problem of the evolution in time of a system of two-level atoms that are coupled through their electromagnetic fields, is studied and solved in the framework of semiclassical radiation theory. The atoms may be in any initial states and the radiation reaction is fully taken into account. Both superradiance and time-dependent frequency shift or chirping effects are found. The assumption that all the atoms see the same field is shown to place severe constraints on the evolution of the system, in the form of constants of the motion. These constraints are most easily pictured in terms of the Bloch-vector representation of the atomic system, and they lead to explicit solutions for the time development of each atom. It is shown that the evolution of the complex system is describable by means of a collective super Bloch vector, whose behavior is similar to the behavior of the Bloch vector for a single isolated atom. The constraints on the motion also imply that the system cannot radiate all its energy coherently; some of it remains trapped, to be dissipated ultimately by incoherent processes. Some curves are presented to illustrate the behavior of the system in special cases.
We apply the methods of cavity quantum electrodynamics (CQED), to obtain a microscopic and fully quantum-mechanical picture of radiation damping in magnetic resonance, and the nascent formation of the free induction signal. Numerical solution of the Tavis-Cummings model - i.e., multiple spins 1∕2 coupled to a lossless single-mode cavity - shows in fine detail the transfer of Zeeman energy, via spin coherence, to excite the cavity - represented here by a quantized LC resonator. The case of a single spin is also solved analytically. Although the motion of the Bloch vector is non-classical, we nonetheless show that the quantum mechanical Rabi nutation frequency (as enhanced by cavity coupling and stimulated emission) gives realistic estimates of macroscopic signal strength and the radiation damping constant in nuclear magnetic resonance. We also show how to introduce dissipation: cavity losses by means of a master equation, and relaxation by the phenomenological method of Bloch. The failure to obtain the full Bloch equations (unless semi-classical conditions are imposed on the cavity) is discussed in light of similar issues arising in CQED (and in earlier work in magnetic resonance as well), as are certain problems relative to quantization of the electromagnetic near-field.