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Physical Review A

Self-induced transparency of excitons and the dispersion law of steady-state exciton-photon pulses

J. Goll, Hermann Haken

Published Nov 1, 1978• 45 citations
Publisher

Abstract

We determine the dispersion law of the carrier wave of an exciton-photon pulse which propagates as a solitary wave. For short pulses, four branches $\ensuremath{\Omega}=\ensuremath{\Omega}$ ($\stackrel{\ensuremath{\rightarrow}}{\mathrm{K}}$) appear: a branch which goes through the intersection point $\ensuremath{\hbar}cK={E}_{o}+\frac{{\ensuremath{\hbar}}^{2}{K}^{2}}{2M}$ of the uncoupled modes of the free exciton and the free light field, two polaritonlike branches and an excitonlike branch. The polaritonlike branches have a normal dispersion, whereas the branch which goes through the intersection point of the free modes has an anomalous dispersion for sufficiently short pulse width. This branch contains the solution of McCall and Hahn as a special case. For very short pulses the branch with anomalous dispersion continues into the two polaritonlike branches. For increasing pulse width a forbidden energy gap appears at $\stackrel{\ensuremath{\rightarrow}}{\mathrm{K}}=0$. For long pulses, only the two polaritonlike solutions exist.

DOI
10.1103/physreva.18.2241

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We determine the dispersion law of the carrier wave of an exciton-photon pulse which propagates as a solitary wave. For short pulses, four branches \ensuremathΩ=\ensuremathΩ\ensuremath{\Omega}=\ensuremath{\Omega}\ensuremathΩ=\ensuremathΩ (K\ensuremath→\stackrel{\ensuremath{\rightarrow}}{\mathrm{K}}K\ensuremath→​) appear: a branch which goes through the intersection point \ensuremathℏcK=Eo+\ensuremathℏ2K22M\ensuremath{\hbar}cK={E}_{o}+\frac{{\ensuremath{\hbar}}^{2}{K}^{2}}{2M}\ensuremathℏcK=Eo​+2M\ensuremathℏ2K2​ of the uncoupled modes of the free exciton and the free light field, two polaritonlike branches and an excitonlike branch. The polaritonlike branches have a normal dispersion, whereas the branch which goes through the intersection point of the free modes has an anomalous dispersion for sufficiently short pulse width. This branch contains the solution of McCall and Hahn as a special case. For very short pulses the branch with anomalous dispersion continues into the two polaritonlike branches. For increasing pulse width a forbidden energy gap appears at K\ensuremath→=0\stackrel{\ensuremath{\rightarrow}}{\mathrm{K}}=0K\ensuremath→​=0. For long pulses, only the two polaritonlike solutions exist.
The phenomenon of self-induced transparency (SIT) in the exciton range of the spectrum consists of the propagation through the crystal of solitary wave packets of coherent excitons and photons with anomalous small losses. In the running frame of reference at the site occupied by the soliton a single aperiodic transformation of photons into excitons and a reverse motion of the energy into the field takes place. The solutions of the SIT equations have been received in the slowly-varying amplitude approximation. Taking into account the phase modulation, the authors have shown the appearance of the branch of the carrier wave frequency with anomalous dispersion not only at the given pulse width but at the given pulse amplitude. They have also found the exact solution of the above mentioned equations taking into account the effect of saturation of the dipole momentum transition from the crystal ground state to the exciton state. It leads to the increasing of the soliton time width, to the decreasing of its amplitude and worsens the SIT observation.
We derive the properties of self-induced transparency (SIT) solitary waves in a one-dimensional periodic structure doped uniformly with resonance two-level atoms. In our model, the electromagnetic field is treated classically and the dopant atoms are described quantum mechanically. The resulting solitary waves take the form of ultrashort (picosecond) laser pulses which propagate near the band edge of the nonlinear photonic band gap (PBG) material doped with rare-earth atoms such as erbium. Solitary wave formation involves the combined effects of group velocity dispersion (GVD), nonresonant Kerr nonlinearity, and resonant interaction with dopant atoms. We derive the general Maxwell-Bloch equations for a nonlinear PBG system and then demonstrate the existence of elementary solitary wave solutions for frequencies far outside the gap where GVD effects are negligible and for frequencies near the photonic band edge where GVD effects are crucial. We find two distinct new types of propagating SIT solitary wave pulses. Far from Bragg resonance, we recapture the usual McCall-Hahn soliton with hyperbolic secant profile when the nonlinear Kerr coefficient ${\ensuremath{\chi}}^{(3)}=0.$ However, when the host nonresonant Kerr coefficient is nonzero, we obtain the first new type of soliton. In this case, the optical soliton envelope function deviates from the hyperbolic secant profile and pulse propagation requires nontrivial phase modulation (chirping). We derive the dependence of the solitary wave structure on the Kerr coefficient ${\ensuremath{\chi}}^{(3)},$ the resonance impurity atom density, and the detuning of the average laser frequency from the atomic transition. When the laser frequency and the atomic transition frequencies are near the photonic band edge we obtain the second type of soliton. To illustrate the second type of soliton we consider two special cases. In the first case, GVD facilitates the propagation of an unchirped SIT-gap soliton moving at a velocity fixed by the material's parameters. The soliton structure changes dramatically as the laser frequency is tuned through the atomic resonance. In the second illustrative case we set the Kerr coefficient ${\ensuremath{\chi}}^{(3)}=0.$ In this case, the solution is a chirped pulse which arises from the balance between GVD and the resonance interaction with the dopant atoms. Finally, we show that under certain circumstances, these solitary wave solutions may persist even in the presence of (subpicosecond) dipolar dephasing of the dopant atoms and absorption losses of the host PBG material, provided that the system is incoherently pumped. These results may be relevant to the application of PBG materials as optical devices in fiber-optic networks.
For pt.I see ibid., vol.10, p.425 (1977). The effect of spatial dispersion or exciton formation on the steady propagation of a coherent light pulse in a dielectric medium is studied. The optical Bloch equation for a lattice of atomic dipoles coupled to one another by an exchange-type interaction is solved simultaneously with the Maxwell equation by the method of power-series expansion developed in the preceding paper. Two kinds of pulse solutions are obtained in general: one propagating by the medium of the radiation field and the other by excitation transfer between atomic dipoles. The former solution is SIT-like in the short pulse limit, but a polariton-soliton or a standing wave of non-linear polariton if the pulse width is long. The latter is an exciton-soliton containing little photon component. When the pulse width is extremely long, there appears a polariton-soliton of a new type in which the two propagation mechanisms are mixed. It is found that the spatial dispersion prevents the polariton-soliton solutions from existing in a certain frequency range. This effect is explained on the basis of the functional behavior of the dispersion relation of the non-linear polariton.
We have found a steady-state solution for a pulse propagating in imhogeneously broadened resonant two-level atoms embedded in a dispersive and nonlinear Kerr lossless host medium. The pulse is of an unchirped hyperbolic-secant form with a linear spatial phase delay. There are several salient features in this solution which are different from the results obtained previously. First, for the given parameters of the medium and carrier frequency, the group velocity, the amplitude, and the pulse width are uniquely determined. Second, the group velocity is independent of the density of the resonant atoms. Third, the solution exists in the positive dispersion region, which is different from solitons in fibers.
We examine theoretically the transparency of electromagnetic pulses through an infinite one-dimensional array of coupled optical microcavities uniformly filled with superconducting qubits-one per cavity. Two types of hybrid matter-light waves, i.e., polaritons and self-induced transparency solitons, govern the transparency of electromagnetic radiation in these media. The spectrum of linear excitations, i.e., polaritons, consists of two branches separated by a relatively wide forbidden band. In the nonlinear regime, the dispersion relation of the carrier wave is determined by soliton width that is controlled by the reciprocal qubit frequency. The separate dispersion curves lie within the polariton forbidden band. Soliton transparency requires that the carrier wave frequency exceeds a threshold value; the latter depends strongly on the pulse width. We find that for pulses with widths ranging from ultrashort to an intermediate limit, the threshold is of the order of the gap frequency value in the photon spectrum. For wider pulses, the threshold frequency gradually decreases to values that are toward the edge of the polariton lower band, provided the soliton width is larger than a critical value. In the overcritical regime, the bandgap appears in the spectrum of the soliton carrier wave, while a twin transparency window appears in the soliton pulse dispersion law. A possible experimental observation of the predicted effects within the proposed setup would be of interest in understanding the properties of self-induced transparency in general and applications in the design of quantum technological devices.