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Physical Review A

Limitations of steady state solutions to a two-state model of population oscillations and hole burning

M. G. Payne, L. Deng, Kaijun Jiang

Published Oct 12, 2006• 4 citations
Publisher

Abstract

We consider a two-state system driven by an on-resonance, continuous wave pump laser and a much weaker pulsed probe laser that is slightly detuned from the pump laser frequency (usually this detuning is about ${\ensuremath{\omega}}_{p}\ensuremath{-}{\ensuremath{\omega}}_{P}=\ensuremath{\Delta}\ensuremath{\approx}1\phantom{\rule{0.3em}{0ex}}\mathrm{kHz}$). The upper state population is assumed to be slowly decaying, but the off-diagonal element of the density matrix decays rapidly due to homogeneous broadening. This model has been solved by others in rare-earth-element-doped fibers and crystals in a usual steady state approximation for slow optical wave propagation. We show that in general the usual steady state approximation does not apply unless either $\ensuremath{\Delta}\ensuremath{\tau}⪢1$ or $(2S+1){\ensuremath{\gamma}}_{2}\ensuremath{\tau}⪢1$ where ${\ensuremath{\gamma}}_{2}$ is the decay rate of the excited state population, $\ensuremath{\tau}$ is the pulse length of the probe field, and $2S$ is the saturation parameter. Both conditions, however, are not satisfied in many population-oscillation- and corresponding group-velocity-reduction-related studies. Our theory and corresponding numerical simulations have indicated that for probe pulses that are much shorter than the lifetime of the upper state, there is no analytical theory for the amplitude, pulse shape, and group velocity of the probe field. In addition, there is no reason to assume that the group velocity remains small when ${\ensuremath{\gamma}}_{2}\ensuremath{\tau}⪡1$ and there is no reason to believe that many pulse length decays can be obtained for such short pulses.

DOI
10.1103/physreva.74.043810

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We consider a two-state system driven by an on-resonance, continuous wave pump laser and a much weaker pulsed probe laser that is slightly detuned from the pump laser frequency (usually this detuning is about \ensuremathωp\ensuremath−\ensuremathωP=\ensuremathΔ\ensuremath≈1kHz{\ensuremath{\omega}}_{p}\ensuremath{-}{\ensuremath{\omega}}_{P}=\ensuremath{\Delta}\ensuremath{\approx}1\phantom{\rule{0.3em}{0ex}}\mathrm{kHz}\ensuremathωp​\ensuremath−\ensuremathωP​=\ensuremathΔ\ensuremath≈1kHz). The upper state population is assumed to be slowly decaying, but the off-diagonal element of the density matrix decays rapidly due to homogeneous broadening. This model has been solved by others in rare-earth-element-doped fibers and crystals in a usual steady state approximation for slow optical wave propagation. We show that in general the usual steady state approximation does not apply unless either \ensuremathΔ\ensuremathτ⪢1\ensuremath{\Delta}\ensuremath{\tau}⪢1\ensuremathΔ\ensuremathτ⪢1 or (2S+1)\ensuremathγ2\ensuremathτ⪢1(2S+1){\ensuremath{\gamma}}_{2}\ensuremath{\tau}⪢1(2S+1)\ensuremathγ2​\ensuremathτ⪢1 where \ensuremathγ2{\ensuremath{\gamma}}_{2}\ensuremathγ2​ is the decay rate of the excited state population, \ensuremathτ\ensuremath{\tau}\ensuremathτ is the pulse length of the probe field, and 2S2S2S is the saturation parameter. Both conditions, however, are not satisfied in many population-oscillation- and corresponding group-velocity-reduction-related studies. Our theory and corresponding numerical simulations have indicated that for probe pulses that are much shorter than the lifetime of the upper state, there is no analytical theory for the amplitude, pulse shape, and group velocity of the probe field. In addition, there is no reason to assume that the group velocity remains small when \ensuremathγ2\ensuremathτ⪡1{\ensuremath{\gamma}}_{2}\ensuremath{\tau}⪡1\ensuremathγ2​\ensuremathτ⪡1 and there is no reason to believe that many pulse length decays can be obtained for such short pulses.
We consider the propagation of a Gaussian probe pulse in an absorptive, optically dense, two-level medium if a deep, persistent hole is created in advance by another pump field in the inhomogeneously broadened absorption spectrum of this medium. Both fields are well separated in time and the lifetime of the hole is assumed to be long with respect to the delay time between the pump and probe pulses. We show that the group velocity of the Gaussian probe pulse reduces several orders in magnitude, similarly to the reduction of the group velocity for the probe field in electromagnetically induced transparency (EIT) phenomena. In contrast to EIT, the width of the transparency window can be made very wide because of the saturation broadening of the hole.
Propagation of a short pulse in an amplifier with excited-state absorption (ESA) is considered. As is shown both analytically and numerically, if the stimulated emission cross section exceeds the ESA cross section, a steady-state pulse of a dissipative soliton type can exist. The parameters of the soliton, i.e., its width, amplitude, and velocity, are calculated. In the opposite case, amplification is inhibited unless ESA is suppressed using an additional control field applied to a transition adjacent to the ESA one [Phys. Rev. A 70, 043801 (2004)]. Propagation and amplification dynamics of a probe pulse in the presence of a strong control field is analyzed. In the limit of a continuous wave control field, a dissipative soliton solution forms, which is a pair of pulses similar to adiabatons. In the case of a pulsed control field, amplification of the probe pulse is transient. Conditions for optimal amplification are derived. Estimates of soliton parameters are made for rare-earth and transition-metal ion-activated laser crystals.
The problem of a light pulse propagating in a nonlinear laser medium is investigated. The electromagnetic field is treated classically and the active medium consists of thermally moving atoms which have two electronic states with independent decay constants ${\ensuremath{\gamma}}_{a}$ and ${\ensuremath{\gamma}}_{b}$ in addition to the decay constant ${\ensuremath{\gamma}}_{\mathrm{ab}}$ describing the phase memory. The self-consistency requirement that the field sustained by the polarized medium be equal to the field inducing the polarization leads to coupled equations of motion for the density matrix, and equations of propagation for the electromagnetic field. Although the theory is developed for a Doppler-broadened gaseous medium, it may also be applied to a solid medium with inhomogeneous broadening. A unified treatment is given encompassing a wide range of pulse durations from cw signals to psec pulses. Continuous pumping is allowed, as well as any amount of detuning of the carrier frequency of the pulse from the atomic resonance frequency. The three independent decay constants ${\ensuremath{\gamma}}_{a}$, ${\ensuremath{\gamma}}_{b}$, and ${\ensuremath{\gamma}}_{\mathrm{ab}}$ provide greater flexibility than that obtained by using 1T1\frac{1}{{T}_{1}}T1​1​ and 1T2\frac{1}{{T}_{2}}T2​1​. The equations are solved analytically in a few specialized cases and numerically in the general case. Flow charts for accomplishing the numerical integration are given. Among the special problems considered is the apparent paradox of pulses propagating faster than the velocity of light under circumstances described by Basov et al. It is shown that this contradiction with relativity arises from the use of an unphysical initial condition.
This paper presents an analytic study of the properties of a three-state system coupled to two external laser fields on two-photon resonance. The intermediate state is generally off single-photon resonance and can decay irreversibly out of the system. The two fields are assumed to have the same pulse-shaped time dependence but may have different strengths. Various general properties for arbitrary pulse shapes, detunings and decay rates are derived and illustrated with an exactly solvable analytic model. They include the conditions for complete population transfer and depletion, complete population loss or no loss, various limitations on the values of the populations, the effect of overdamping at large decay rates and others.
We present a detailed semiclassical study of the propagation of a pair of optical fields in resonant media with and without the adiabatic approximation. In the case of near- and on-resonance excitation, we show detailed calculations, both analytical and numerical, of the extremely slowly propagating probe pulse and the subsequent regeneration of a pulse via a coupling laser. Further discussions of the adiabatic approximation provide much subtle understanding of the process, including the effect on the bandwidth of the regenerated optical field. We show that the adiabatic condition does not generally predict that the regenerated field will have the same bandwidth as that of the original probe pulse. Indeed, we show analytically that for a set of coupling pulses well satisfying the adiabatic conditions, the regenerated field usually has a different bandwidth from the original probe pulse. For very-far-off resonance excitations, we show that the analytical solution is nearly detuning independent, providing a simple condition is satisfied. This surprising result is vigorously tested and compared to numerical calculations with very good agreement.
An exact analysis has been carried out for general analytical expressions for the second threshold of a single-mode homogeneously broadened laser and for the initial pulsation frequency at the second threshold, for arbitary physical values of the relaxation rates, and at an arbitrary detuning between the cavity frequency and the atomic resonance frequency. These expressions also give correspondingly exact forms for asymptotic cases that have been previously studied with some approximations. Earlier approximate results are partly confirmed and partly improved by these more general expressions. The physical status of various expressions and approximations is reconsidered and specified more clearly, including an analysis of what reasonably can be attained in lasers or masers. A general analytical proof is given of the fact that, for a larger detuning of the laser cavity from resonance, a higher value of the laser excitation is required to destabilize the steady-state solution (the second threshold). We also present results for the minimum value of the second threshold at fixed detuning as a function of the other parameters of the system and for the dependence of the ratio of the second threshold to the first threshold as a function of detuning. Minima of the second threshold and of the threshold ratio occur only if the population relaxation rate is equal to zero. The minima of the threshold ratio are shown to be bounded from above as well as from below (as functions of the relaxation rates, so long as the second threshold exists). The upper bound on the minima ratio is equal to 17. The variation of the second threshold in the semi-infinite parameter space of the decay rates is shown at various detunings in plots with a finite domain by normalizing the material relaxation rates to the cavity decay rate.