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Journal of Applied Physics

Spontaneous Emission from an Inverted Spin System

Amnon Yariv

Published Apr 1, 1960• 14 citations
Publisher

Abstract

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.

DOI
10.1063/1.1735687

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Several experiments [1,2] have noted recently that when an inverted two-level spin system was permitted to radiate spontaneously, the resulting oscillation was characterized by an appreciable amplitude modulation. The phenomenon was first believed to be the result of interference of different spin packets in an inhomogeneously broadened spectrum [1]. A theoretical analysis (which will be reported separately) shows that this is not the case. The spins are not independent but are coupled together by means of their radiation field. This explanation has since been by its original authors.

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A NONLINEAR MODEL OF THE COLLECTIVE SPONTANEOUS EMISSION BY A SPIN SYSTEM

Salomon Sylvain Mizrahi, V. V. Dodonov, D. Otero

We propose a nonlinear quantum master equation, which results in the nonlinear Bloch-like equations, similar to those which are used usually to describe the collective spontaneous emission by a system of two-level atoms or a spin system in a magnetic field. On the one hand, the new equation is the unique simplest homogeneous and nonsingular generalization of the linear master equations for quantum systems in the finite-dimensional Hilbert space. On the other hand, it corresponds to the mean-field approximation, and in this form it can be used for a larger class of systems. Our equations describe both triggered and pure spin emission processes (when the population is completely inverted), and their solutions show the occurrence of secondary emission peaks, as observed in experiments.

Modern Physics Letters B•Nov 20, 1996•1 citation
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.
Several experiments [1,2] have noted recently that when an inverted two-level spin system was permitted to radiate spontaneously, the resulting oscillation was characterized by an appreciable amplitude modulation. The phenomenon was first believed to be the result of interference of different spin packets in an inhomogeneously broadened spectrum [1]. A theoretical analysis (which will be reported separately) shows that this is not the case. The spins are not independent but are coupled together by means of their radiation field. This explanation has since been by its original authors.
A theoretical model is developed for the behavior of an optical maser medium in a static magnetic field of arbitrary strength in the ZZZ direction and an electromagnetic field which is composed of traveling waves with both states of polarization and with both +z+z+z and $\ensuremath{-}z$ directions. An integral formulation of a truncated phenomenological density matrix is introduced to treat the degeneracy of the atomic energy levels within the natural width. This integral is an exact solution of the phenomenological differential equation for the density matrix which was introduced by Wilcox and Lamb. The inherent degeneracy in the electromagnetic field is included by treating the field in a classical manner. The integral formulation is iterated to yield the first- and third-order contributions to the electric polarization vector P in a spatial element drd\mathrm{r}dr, and the electric field and the electric polarization are made self-consistent in this spatial element by using coefficients in Maxwell's equations which are slowly varying functions of position and time. Atomic motion, hyperfine, isotope, and other line-broadening effects are included in the polarization vector P. The general linear-amplifier, linear-oscillator, and closed-path-oscillator or photon-rate-gyroscope problems are formulated and in the simplest case require the analysis of eight nonlinear equations. These reduce to four equations for the linear oscillator with an axial magnetic field, and these examples are discussed in detail and compared with experiment. An anomalous situation arises for the ja=jb{j}_{a}={j}_{b}ja​=jb​ or Fa=Fb{F}_{a}={F}_{b}Fa​=Fb​ linear amplifiers or linear-oscillator masers, and the development indicates that these systems are stable near line center only for either right or left circular polarization in zero or weak magnetic fields. A magnetic field "dip" near line center is discussed. Special examples which require four or fewer nonlinear equations are discussed.
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.
In an electron-spin two-level system a bichromatic radiation field, consisting of a transverse microwave field and a radio frequency field oriented parallel to the static magnetic field, can induce multiple photon transitions of the type ${\ensuremath{\sigma}}_{\text{mw}}^{+}+k{\ensuremath{\pi}}_{\text{rf}}$, with the photons ${\ensuremath{\sigma}}_{\text{mw}}^{+}$ and ${\ensuremath{\pi}}_{\text{rf}}$ absorbed from the microwave and radio frequency fields, and kkk being an integer. The two-level spin system may become transparent under such a bichromatic radiation field. This phenomenon directly depends on the presence of the $\ensuremath{\pi}$ photons of the longitudinal radio frequency field. An analytical description of the multiple photon processes and the resulting transparency effects, based on a toggling-frame approach, is given. The effective field amplitudes of the multiple photon transitions are found to have a Bessel-function-like dependence on the radio frequency amplitude. The behavior of the magnetization vector during a bichromatic pulse is illustrated by using classical equations of motion. The theoretically predicted effects are verified experimentally. As an example for an application of the investigated type of electromagnetically induced transparency, a one-pulse echo experiment is described, where the free evolution period is created artificially.
Using the formalism developed earlier, we treat spontaneous emission from a pair of identical two-level atoms A1{A}_{1}A1​, A2{A}_{2}A2​, whose separation r21{r}_{21}r21​ can be comparable to the wavelength $\ensuremath{\lambda}$. We obtain expressions for time-dependent intensities and damping rates with the initial conditions (a) both atoms inverted, (b) prior excitation by a short $\frac{1}{2}\ensuremath{\pi}$ pulse, and (c) only A1{A}_{1}A1​ inverted. The results in (a) are compared with those obtained for a model consisting of two initially excited harmonic oscillators O1{O}_{1}O1​, O2{O}_{2}O2​. The atoms exhibit superradiant behavior, whereas O1{O}_{1}O1​, O2{O}_{2}O2​ tend to trap radiation. In (a), the intensity pattern $\mathcal{R}(\ensuremath{\theta},t)$ develops lobes in different directions at different times, so that the spatial distribution of photons at time $t\ensuremath{\rightarrow}\ensuremath{\infty}$ is the same as in the independent-atom case ${r}_{21}\ensuremath{\gg}\ensuremath{\lambda}$. For the oscillators, the lobes of $\mathcal{R}(\ensuremath{\theta},t)$ do not change direction, but only become more pronounced as time increases. In (b) and (c), the lobes oscillate back and forth at frequency $2{\ensuremath{\Omega}}_{12}$ corresponding to the shifts $\ifmmode\pm\else\textpm\fi{}{\ensuremath{\Omega}}_{12}$ of the triplet and singlet states due to the ${A}_{1}\ensuremath{-}{A}_{2}$ interaction. The intensity can therefore have a sinusoidal component. Field correlation functions calculated for (a) and (c) show that A2{A}_{2}A2​ and A2{A}_{2}A2​ radiate simultaneously around the frequencies $\ensuremath{\epsilon}\ifmmode\pm\else\textpm\fi{}{\ensuremath{\Omega}}_{12}$, where $\ensuremath{\epsilon}$ is the single-atom resonant frequency. The spectrum is calculated for case (c), and shows the effects of coherent linewidth enhancement in addition to the frequency shifts.
We propose a nonlinear quantum master equation, which results in the nonlinear Bloch-like equations, similar to those which are used usually to describe the collective spontaneous emission by a system of two-level atoms or a spin system in a magnetic field. On the one hand, the new equation is the unique simplest homogeneous and nonsingular generalization of the linear master equations for quantum systems in the finite-dimensional Hilbert space. On the other hand, it corresponds to the mean-field approximation, and in this form it can be used for a larger class of systems. Our equations describe both triggered and pure spin emission processes (when the population is completely inverted), and their solutions show the occurrence of secondary emission peaks, as observed in experiments.