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Physical Review A

Influence of non-Markovian relaxation processes on self-induced transparency: Memory function theory for optical solitons

Guram T. Adamashvili, David J. Kaup, A. Knorr, C. Weber

Published Jul 28, 2008• 12 citations
Publisher

Abstract

The influence of transverse relaxation on the resonance soliton ($2\ensuremath{\pi}$ hyperbolic-secant pulse) in solids is considered. It is shown that memory effects can be considered in terms of generalized non-Markovian optical Bloch equations. The equations of self-induced transparency with a memory function are presented. Explicit forms for the first-order effects of this relaxation on the frequency shift and the pulse height of a resonant $2\ensuremath{\pi}$ soliton in solids are given and discussed. It is shown that memory effects do lead to a qualitative change in the dynamics of the influence of transverse relaxations on a $2\ensuremath{\pi}$ pulse in comparison to the McCall-Hahn theory and others. The dynamics of the changes of the inverse width of the pulse are obtained in both the Markovian and the non-Markovian cases with realistic parameters that can be achieved in current experiments.

DOI
10.1103/physreva.78.013840

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Above a critical power threshold for a given pulse width, a short pulse of coherent traveling-wave optical radiation is observed to propagate with anomalously low energy loss while at resonance with a two-quantum-level system of absorbers. The line shape of the resonant system is determined by inhomogeneous broadening, and the pulse width is short compared to dissipative relaxation times. A new mechanism of self-induced transparency, which accounts for the low energy loss, is analyzed in the ideal limit of a plane wave which excites a resonant medium with no damping present. The stable condition of transparency results after the traversal of the pulse through a few classical absorption lengths into the medium. This condition exists when the initial pulse has evolved into a symmetric hyperbolic-secant pulse function of time and distance, and has the area characteristic of a "$2\ensuremath{\pi}$ pulse." Ideal transparency then persists when coherent induced absorption of pulse energy during the first half of the pulse is followed by coherent induced emission of the same amount of energy back into the beam direction during the second half of the pulse. The effects of dissipative relaxation times upon pulse energy, pulse area, and pulse delay time are analyzed to first order in the ratio of short pulse width to long damping time. The analysis shows that the $2\ensuremath{\pi}$ pulse condition can be maintained if losses caused by damping are compensated by beam focusing. In an amplifying inhomogeneously broadened medium an analytic "$\ensuremath{\pi}$ pulse area" solution is presented in the limit of a sharp leading edge of the pulse. The dynamics of self-induced transparency are studied for the particular effects of Doppler velocities upon a resonant gas. The analysis of transparency for random orientations of dipole moments associated with degenerate rotational states yields modified forms of self-induced transparency behavior, which indicates a finite pulse energy loss as a function of distance in some cases. The effect of self-induced transparency on the photon echo is considered. Experimental observations of self-induced transparency have been made in a ruby sample at resonance with a pulsed ruby-laser beam. Single and multiple $2\ensuremath{\pi}$ pulse outputs have been observed, and pulse areas measured in the range of $2\ensuremath{\pi}$. The experimental results are compared with the predictions of the ideal-plane-wave theory. Deviations from the ideal-plane-wave theory are discussed. An analysis is made of the effect of a transverse mode of the propagating beam upon the transparency properties of the pulse.

Physical Review•Jul 10, 1969•1,394 citations
The influence of transverse relaxation on the resonance soliton (2\ensuremathπ2\ensuremath{\pi}2\ensuremathπ hyperbolic-secant pulse) in solids is considered. It is shown that memory effects can be considered in terms of generalized non-Markovian optical Bloch equations. The equations of self-induced transparency with a memory function are presented. Explicit forms for the first-order effects of this relaxation on the frequency shift and the pulse height of a resonant 2\ensuremathπ2\ensuremath{\pi}2\ensuremathπ soliton in solids are given and discussed. It is shown that memory effects do lead to a qualitative change in the dynamics of the influence of transverse relaxations on a 2\ensuremathπ2\ensuremath{\pi}2\ensuremathπ pulse in comparison to the McCall-Hahn theory and others. The dynamics of the changes of the inverse width of the pulse are obtained in both the Markovian and the non-Markovian cases with realistic parameters that can be achieved in current experiments.
A theory of dispersive soliton of the self-induced transparency in a medium consisting of atoms or semiconductor quantum dots of two types is considered. A two-component medium is modeled by a set of two-level atoms of two types embedded into a conductive host material. These types of atoms correspond to passive atoms (attenuator atoms) and active atoms (amplifier atoms) with inverse population of the energetic levels. The complete solution is given of the Maxwell--Bloch equations for ensembles of two-type atoms with different parameters and different initial conditions by inverse scattering transform. The solutions of the Maxwell--Bloch equations for many-component atomic systems by inverse scattering transform are also discussed. The influence of the difference between dipole moments of atoms, the longitudinal and transverse relaxation times, pumping, and conductivity on the soliton is taken into account by means of perturbation theory. The memory effects are described in terms of generalized non-Markovian optical Bloch equations. The condition of a balance between the energy supplied and lost is obtained.
In a recent experiment [ DevoeR. G.BrewerR. G., Phys. Rev. Lett.50, 1269 ( 1983)], it was found that the optical Bloch equations could not satisfactorily explain the signal that was observed for free-induction decay in the impurity ion cyrstal Pr3+:LaF3. Several theories have been proposed to explain this failure of the Bloch equations. In this paper, the general validity conditions for the optical Bloch equations are examined within the limits of a Markovian relaxation model. The specific problem to be considered is the interaction of an optical field with two-level atoms. The atoms undergo relaxation as a result of coupling to a perturber bath that, itself, is negligibly affected by the relaxation process. First, a simple decay-parameter model is assumed for the relaxation of atomic density-matrix elements. Such a model is found to lead to a set of generalized Bloch equations of which the conventional Bloch equations form a subset. Subsequently, more-realistic models for relaxation in both vapors and solids are considered within the limits of the impact approximation (i.e., the duration of a fluctuation can be viewed as instantaneous with respect to all relevant time scales in the problem). It is found that, even in the impact (Markovian) approximation, the generalized Bloch equations cannot be expected to provide an adequate description of relaxation, owing to effects in which a fluctuation-induced change in the atomic transition frequency persists between fluctuations. In vapors this frequency shift is produced by velocity-changing collisions that change the atomic resonance frequency (as seen in the laboratory frame), whereas in solids it is produced by local-field fluctuations. The limiting conditions under which one can expect both the generalized and the conventional Bloch equations to retain their validity are explored.
A review is given of the current status of the self-induced transparency effect. The McCall–Hahn theory (2π pulses) is considered and its generalization to the case of phase modulation of radiation pulses, degeneracy of transitions, and two-photon resonance is discussed. The relationship with the theory of nonlinear waves, obeying the Korteweg–de Vries equation, is pointed out. The possibility of similar coherent effects in direct interband transitions in semiconductors is discussed. A summary is given of the experimental results demonstrating a strong fall in the absorption and a considerable slowing down of light pulses, typical of the self-induced transparency. Possible physical applications of the effect, based on these properties, are discussed.
A more accurate description of the 2 pi pulse of self-induced transparency is obtained. This is an exact solution of an approximate form of the Maxwell-Bloch equations in which only backscattering is neglected; the equations are valid at densities less than about 10 18 atoms cm -3 . At such densities the 2 pi sech pulse of McCall and Hahn (1967, 1969) remains a good approximate well into the picosecond range. The more accurate solution is chirped by a factor proportional to the square of the ratio of the spectral width of the pulse to its carrier wave frequency. The stability of this pulse solution and other N soliton solutions of the approximate Maxwell-Bloch equations is demonstrated.
Above a critical power threshold for a given pulse width, a short pulse of coherent traveling-wave optical radiation is observed to propagate with anomalously low energy loss while at resonance with a two-quantum-level system of absorbers. The line shape of the resonant system is determined by inhomogeneous broadening, and the pulse width is short compared to dissipative relaxation times. A new mechanism of self-induced transparency, which accounts for the low energy loss, is analyzed in the ideal limit of a plane wave which excites a resonant medium with no damping present. The stable condition of transparency results after the traversal of the pulse through a few classical absorption lengths into the medium. This condition exists when the initial pulse has evolved into a symmetric hyperbolic-secant pulse function of time and distance, and has the area characteristic of a "$2\ensuremath{\pi}$ pulse." Ideal transparency then persists when coherent induced absorption of pulse energy during the first half of the pulse is followed by coherent induced emission of the same amount of energy back into the beam direction during the second half of the pulse. The effects of dissipative relaxation times upon pulse energy, pulse area, and pulse delay time are analyzed to first order in the ratio of short pulse width to long damping time. The analysis shows that the $2\ensuremath{\pi}$ pulse condition can be maintained if losses caused by damping are compensated by beam focusing. In an amplifying inhomogeneously broadened medium an analytic "$\ensuremath{\pi}$ pulse area" solution is presented in the limit of a sharp leading edge of the pulse. The dynamics of self-induced transparency are studied for the particular effects of Doppler velocities upon a resonant gas. The analysis of transparency for random orientations of dipole moments associated with degenerate rotational states yields modified forms of self-induced transparency behavior, which indicates a finite pulse energy loss as a function of distance in some cases. The effect of self-induced transparency on the photon echo is considered. Experimental observations of self-induced transparency have been made in a ruby sample at resonance with a pulsed ruby-laser beam. Single and multiple $2\ensuremath{\pi}$ pulse outputs have been observed, and pulse areas measured in the range of $2\ensuremath{\pi}$. The experimental results are compared with the predictions of the ideal-plane-wave theory. Deviations from the ideal-plane-wave theory are discussed. An analysis is made of the effect of a transverse mode of the propagating beam upon the transparency properties of the pulse.