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Annual Meeting Optical Society of America

Bifurcations and chaos in pulse-train superradiance

Albert T. Rosenberger

Published Jan 1, 1988• 0 citations
Publisher

Abstract

The superradiant response of a collection of three-level atoms to a train of pump pulses has been investigated numerically. It is assumed that the size of the system is small enough that propagation may be neglected and that the pump is resonant. The atoms are considered to be homogeneously broadened, and all relaxation rates are taken to be equal. The period of the pump pulse train is of the order of the atomic relaxation time, and the width of individual pulses is <4% of the period. Coherence remaining from previous pulses results in qualitative changes in the superradiant pulse train as the pump intensity is increased. Three cases of increasing complexity have been studied: square pump pulses, nondegenerate energy levels; mode-locked pulses, non-degenerate levels; and mode-locked pulses, degenerate levels. The sequence of behavior patterns is somewhat different for different cases. Features of the superradiant output common to the three cases are inversion of the electric field, alternation of positive and negative pulses, and recovery of pulses of one sign after more complicated behavior such as period-doubling, chaos, or intermittency. The effects of varying pump period or pulse shape are found to be less straightforward than that of varying pump intensity. The feasibility of experimental observation is discussed.

DOI
10.1364/oam.1988.we7

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The superradiant response of a collection of three-level atoms to a train of pump pulses has been investigated numerically. It is assumed that the size of the system is small enough that propagation may be neglected and that the pump is resonant. The atoms are considered to be homogeneously broadened, and all relaxation rates are taken to be equal. The period of the pump pulse train is of the order of the atomic relaxation time, and the width of individual pulses is <4% of the period. Coherence remaining from previous pulses results in qualitative changes in the superradiant pulse train as the pump intensity is increased. Three cases of increasing complexity have been studied: square pump pulses, nondegenerate energy levels; mode-locked pulses, non-degenerate levels; and mode-locked pulses, degenerate levels. The sequence of behavior patterns is somewhat different for different cases. Features of the superradiant output common to the three cases are inversion of the electric field, alternation of positive and negative pulses, and recovery of pulses of one sign after more complicated behavior such as period-doubling, chaos, or intermittency. The effects of varying pump period or pulse shape are found to be less straightforward than that of varying pump intensity. The feasibility of experimental observation is discussed.
A theoretical model for the propagation of coherent light pulses in resonant media where a large number of atoms cooperate in spontaneous emission (superradiance) is discussed. The medium is supposedly free of inhomogeneous broadening, and no lossy relaxation mechanisms are taken into account. The conditions for the emission of all spontaneous radiation in the direction of pulse propagation are described. A semi-classical model is used and, at each point in the medium, the propagation field and the spontaneous damping field are considered separately. The equation governing the evolution of a superradiant system in the presence of an applied pulse is derived. A mechanism of energy transfer from the traveling pulse to the medium is described which includes superradiant spontaneous emission effects. As a result of the neglect of all lossy relaxation processes, all energy stored in the medium is eventually returned to the pulse by coherent spontaneous emission. Therefore, in this theory, all pulses travel through the medium without energy loss. Numerical results for square input pulses are discussed which show two possible types of propagation behavior: (a) Low input intensities produce a negative tail to the input (positive) pulse and sharpening of the tail with the advance in the medium; (b) higher input intensities produce, after sufficient penetration, a steady-state pulse envelope somewhat similar to the $2\ensuremath{\pi}$ pulses of self-induced transparency. The steady-state solutions of the equations of motion are found numerically and the pulse shapes are computed for different degrees of superradiance. In the limit of strong superradiance, very asymmetric pulse envelopes of large area are found. For negligible superradiance, approximately $2\ensuremath{\pi}$ hyperbolic-secant pulses are obtained. The effects of inhomogeneous broadening are briefly considered, and the consequences of important superradiant damping in photon-echo experiments are qualitatively discussed.
We numerically analyze superradiant dynamics in atomic ensembles that have different transition frequencies using a numerical model that can take account of the transient behavior of an unsynchronized superradiant state. The numerical results unveil that the superradiant emission of a periodic pulse train can be induced by means of collective multiple frequency generation, which we call superradiant mixing. This is, in fact, due to the superradiant coupling of unsynchronized atomic ensembles. We numerically investigate the superradiant mixing in detail, varying the collective decay rate, repumping rate, and the number of the individual atomic ensembles with detuned frequencies. This work broadens our understanding of the collective atomic behavior in a detuned system, and it also suggests a novel method for frequency generation without relying on the conventional Kerr nonlinear effect.
The dynamics of the pulse shape and spectra of superfluorescence, stimulated superradiance, and coherent amplification of ultrashort light pulses is studied for a system of four-level atoms under incoherent cw pumping taking into account homogeneous and inhomogeneous broadening of the luminescence lines. It is found that incoherent pumping at a high level of excitation may significantly change the shape of the emission pulse in cooperative coherent processes. Also, it is found that, depending on the ratio of parameters of the pump, nonradiative transitions, and relaxation, different regimes of superfluorescence and stimulated superradiance may be realized.
Up to now, the problem of the influence of level degeneracy on superradiance has been neglected. We describe here a first approach to this problem, in the frame of "small system" quantum model, generalized to the case of a collection of atoms with two degenerate levels of angular momenta jjj and ${j}^{\ensuremath{'}}$ connected by an electric or magnetic dipole transition. The differences between this case and the well-known case of two (nondegenerate) level atoms are of two types. The first type is due to the competition between different transitions sharing a common state: superradiant emission on a transition inhibits or initiates superradiance on transitions with smaller transition probability, depending upon which state, upper or lower, is the common one. This effect does not come from the equality of the different wavelengths and would also appear with the same states even if the degeneracy were removed; its qualitative influence on the features of the superradiant pulse can be easily estimated. The second type of differences is quite specific to level degeneracy and consists in interferences between the transitions of the collective system having the same frequency and polarization. In analogy with the description of collections of two-level atoms as spin-\textonehalf{} systems, a group-theoretical description of the considered collection of atoms is found suitable; the problem of the determination of a basis of collective states and its group-theoretical implications are discussed; in particular, it is found that the isotropic character of spontaneous emission allows us to use tensorial formalism and simplifies the formal description of the problem.
A theoretical model for superradiant damping effect in inhomogeneously broadened media and its application to photon-echo problems are discussed. Only optically thin systems are considered, and all nonradiative relaxation mechanisms are neglected. Numerical calculations are performed for a very wide square inhomogeneous line shape and square applied pulses. It is shown that excitation of the system becomes increasingly difficult as the superradiant lifetime ${\ensuremath{\tau}}_{s}$ becomes comparable with T2∗{T}_{2}^{*}T2∗​. The energy reradiated after the end of the exciting pulse is considerable for ${\ensuremath{\tau}}_{s}\ensuremath{\approx}{T}_{2}^{*}$ and for some applied pulse areas. Two-pulse sequences produce multiple echoes as the microscopic polarizations spontaneously rephase. The first-echo amplitude maximum is found to occur for first-applied-pulse areas increasingly above $\frac{1}{2}\ensuremath{\pi}$ as ${\ensuremath{\tau}}_{s}$ decreases. For a given first-pulse area, the echo intensities increase initially as ${\ensuremath{\tau}}_{s}$ decreases (negligible damping) and finally decrease and vanish (important damping). The ratio of the second- to first-echo amplitudes is found to be proportional to $\frac{{T}_{2}^{*}}{{\ensuremath{\tau}}_{s}}$ for small superradiance levels and to decrease when ${\ensuremath{\tau}}_{s}$ approaches T2∗{T}_{2}^{*}T2∗​. An approximate estimate of this ratio is derived for the region of linearity in $\frac{{T}_{2}^{*}}{{\ensuremath{\tau}}_{s}}$.
The dynamical evolution of an excited-state population in a macroscopic volume of three-level molecules is considered where the population is driven optically between the initial ground state and the excited states by an externally applied (ccc-number) coherent pump. The intermediate and ground states are taken as nonradiatively coupled whereas the excited-to-intermediate-state transition is coupled to the radiation field which is treated quantum mechanically. It is found that stimulated Raman processes can significantly influence the dynamical evolution of population inversion and macroscopic polarization by producing coherence effects among the populations. We examine the evolution of collective relaxation between the excited and intermediate levels in the time regime of the pump pulse duration. It is shown, using the mean-field approximation, that for uniform pumping and conditions such that the pump Rabi rate ${\ensuremath{\omega}}_{R}$, the characteristic collective radiation time ${\ensuremath{\tau}}_{R}$, and the pump pulse duration ${\ensuremath{\tau}}_{P}$ satisfy the inequalities, ${\ensuremath{\omega}}_{R}>\frac{1}{{\ensuremath{\tau}}_{R}}>\ensuremath{\gamma}$ and ${\ensuremath{\tau}}_{P}\ensuremath{\gtrsim}{\ensuremath{\tau}}_{R}$ ($\ensuremath{\gamma}$ is a characteristic internal dephasing rate for the molecules), the system is left in a state with classical transverse polarization when the pump pulse terminates. The system can evolve collectively from such a state only as superradiant (classical) evolution. For superfluorescent evolution which requires a state of preparation of complete inversion (zero transverse polarization), it is shown that the pump pulse must be effectively of area $\ensuremath{\pi}$ and that ${\ensuremath{\tau}}_{P}<{\ensuremath{\tau}}_{R}$. Our results show that when the former conditions are satisfied the delay time ${\ensuremath{\tau}}_{D}$ for collective free pulse evolution is a function of both ${\ensuremath{\tau}}_{P}$ and ${\ensuremath{\tau}}_{R}$. It is shown further that the pump pulse shape as well as temporal duration significantly affect the final state of preparation. The results of this work are interpreted in connection with recent reported results of experiments in superfluorescence and superradiance.