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Physical Review A

Carrier-Frequency Distance Dependence of a Pulse Propagating in a Two-Level System

J.-C. Diels, Erwin Louis Hahn

Published Aug 1, 1973• 56 citations
Publisher

Abstract

A study is made of the distance dependence of carrier phase and average frequency of an electromagnetic pulse propagating through a quantum two-level system. Phenomenological differential equations are introduced to describe the distance rate of change of first- and second-moment deviations of the pulse carrier frequency from its original input frequency. In equilibrium the equations predict characteristics of pulse propagation in steady state. For both absorber and amplifier cases, computer plots are presented of the distance dependence of pulse shape, carrier phase, Fourier amplitude, and pulse energy for selected types of homogeneous and inhomogeneous line broadening and for specific input-pulse conditions. Off-resonance formation of a single $2\ensuremath{\pi}$ hyperbolic secant pulse of self-induced transparency may evolve from input-pulse areas less than $\ensuremath{\pi}$, as well as slightly greater than $\ensuremath{\pi}$, accompanied by frequency modulation and pulse breakup. The amplification of pulsed carrier radiation is accompanied by frequency pulling toward resonance. With and without phase modulation, stability conditions for stable pulse propagation are obtained. Either frequency pulling toward or frequency pushing away from resonance is dominant for an absorber, according to the dominance, respectively, of homogeneous or inhomogeneous line broadening. In the slowly-varying-pulse-envelope approximation, a steady-state solution is found for a symmetrically-phase-modulated pulse propagating in an amplifier with scattering losses. The mean carrier frequency for the pulse is displaced off resonance by one linewidth above the resonance transition frequency of a homogeneously broadened system.

DOI
10.1103/physreva.8.1084

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Physical Review A•Jun 8, 2006•23 citations
A study is made of the distance dependence of carrier phase and average frequency of an electromagnetic pulse propagating through a quantum two-level system. Phenomenological differential equations are introduced to describe the distance rate of change of first- and second-moment deviations of the pulse carrier frequency from its original input frequency. In equilibrium the equations predict characteristics of pulse propagation in steady state. For both absorber and amplifier cases, computer plots are presented of the distance dependence of pulse shape, carrier phase, Fourier amplitude, and pulse energy for selected types of homogeneous and inhomogeneous line broadening and for specific input-pulse conditions. Off-resonance formation of a single 2\ensuremathπ2\ensuremath{\pi}2\ensuremathπ hyperbolic secant pulse of self-induced transparency may evolve from input-pulse areas less than \ensuremathπ\ensuremath{\pi}\ensuremathπ, as well as slightly greater than \ensuremathπ\ensuremath{\pi}\ensuremathπ, accompanied by frequency modulation and pulse breakup. The amplification of pulsed carrier radiation is accompanied by frequency pulling toward resonance. With and without phase modulation, stability conditions for stable pulse propagation are obtained. Either frequency pulling toward or frequency pushing away from resonance is dominant for an absorber, according to the dominance, respectively, of homogeneous or inhomogeneous line broadening. In the slowly-varying-pulse-envelope approximation, a steady-state solution is found for a symmetrically-phase-modulated pulse propagating in an amplifier with scattering losses. The mean carrier frequency for the pulse is displaced off resonance by one linewidth above the resonance transition frequency of a homogeneously broadened system.
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.
We present a numerical method for solving the Maxwell-Bloch equations describing pulse propagation for a two-level medium. The method is accurate, efficient, stable, and well suited for this type of simultaneous equations. By applying the numerical scheme we investigate the evolutions of pulse area, pulse propagation, pulse velocity, and spectral shapes under both homogeneous and inhomogeneous broadening conditions. The results show that the area evolution and pulse-reshaping procedure are significantly influenced by detuning and inhomogeneous line shape, which also impact the oscillation tail and pulse peak. In addition, the pulse-peak traces indicated the pulse velocity always increases with greater deviation in pulse-area value from the value 2\ensuremath{\pi}. We also demonstrate the pulse velocity increased for a larger detuning or a wider inhomogeneous line shape. Furthermore, the spectral feature shows that pulse spectra evolve into an oscillating shape.
An analytic solution is obtained for the self-consistent equations describing propagation of an ultrashort pulse in a resonant two-level medium. The first approximation in respect of the dispersion is used to allow for phase modulation and for the dependence of the group velocity on the intensity and on the running value of the frequency. The conditions for the formation of a shock wave are established. It is shown that asymmetric broadening of the spectrum of a pulse results in coherent absorption, as a result of which the propagation of this pulse in such a medium leaves behind a polarization at the atomic transition frequency responsible for the radiation represented by the pulse.
Propagation of single- and two-color hyperbolic secant femtosecond laser pulses in a three-level $\ensuremath{\Lambda}$-type quantum system is investigated by solving the Maxwell and density matrix equations with the finite-difference time-domain and Runge-Kutta methods. As a first study of our modeling, we simulate pulse self-induced transparency (SIT) in two-level systems and see how this phenomenon can be controlled by manipulating the initial relative phase between the SIT pulse and a second control pulse, provided the ratio between both pulse frequencies obeys the relation ${\ensuremath{\omega}}_{1}∕{\ensuremath{\omega}}_{2}=3$. We then examine frequency down-conversion processes that are observed with single- and two-color pulses the envelope area of which is equal to or a multiple of $2\ensuremath{\pi}$, for pulse frequencies close to resonance with the transitions of a three-level $\ensuremath{\Lambda}$ medium. Also, phase-sensitive phenomena are discussed in the case of two-color $\ensuremath{\omega}\text{\ensuremath{-}}3\ensuremath{\omega}$ pulses propagating resonantly in the three-level system. In particular, possibilities for such coherent control are found for frequency down-conversion processes when the ratio of the frequencies of optical transitions is ${\ensuremath{\omega}}_{13}∕{\ensuremath{\omega}}_{12}=3$. The conditions for quantum control of four-wave mixing processes are also examined when the pulse frequencies of two-color $\ensuremath{\omega}\text{\ensuremath{-}}3\ensuremath{\omega}$ pulses are far from any resonance of the three-level system. We demonstrate the possibility to cancel the phase sensitivity of the four-wave coupling in a $\ensuremath{\Lambda}$-type system by competition effects between optical transitions.