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

M. G. Payne; L. Deng; K. J. Jiang

doi:10.1103/PhysRevA.74.043810

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

M. G. Payne, L. Deng, K. J. Jiang

Emeriti, Retired, & In Memoriam (COSM)

National Institute of Standards and Technology

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Abstract

<div class="line" id="line-25"> 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  ω p − ω P = Δ ≈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  Δτ &GreaterGreater; 1  or  (2 S +1) γ 2 τ &GreaterGreater; 1  where  γ 2  is the decay rate of the excited state population,  τ  is the pulse length of the probe field, and  2 S  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  γ 2 τ &LessLess; 1  and there is no reason to believe that many pulse length decays can be obtained for such short pulses. </div>

Original language	American English
Journal	Physical Review A
Volume	74
DOIs	https://doi.org/10.1103/PhysRevA.74.043810
State	Published - Oct 16 2006

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Physical Sciences and Mathematics
Physics

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@article{b8b61caec916405eab5edb2ab595ac4a,

title = "Limitations of steady state solutions to a two-state model of population oscillations and hole burning",

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  ω p − ω P = Δ ≈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  Δτ &GreaterGreater; 1  or  (2 S +1) γ 2 τ &GreaterGreater; 1  where  γ 2  is the decay rate of the excited state population,  τ  is the pulse length of the probe field, and  2 S  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  γ 2 τ &LessLess; 1  and there is no reason to believe that many pulse length decays can be obtained for such short pulses. ",

author = "Payne, {M. G.} and L. Deng and Jiang, {K. J.}",

note = "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.",

year = "2006",

month = oct,

day = "16",

doi = "10.1103/PhysRevA.74.043810",

language = "American English",

volume = "74",

journal = "Physical Review A",

issn = "2469-9926",

publisher = "American Physical Society",

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TY - JOUR

T1 - Limitations of steady state solutions to a two-state model of population oscillations and hole burning

AU - Payne, M. G.

AU - Deng, L.

AU - Jiang, K. J.

N1 - 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.

PY - 2006/10/16

Y1 - 2006/10/16

N2 - 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  ω p − ω P = Δ ≈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  Δτ &GreaterGreater; 1  or  (2 S +1) γ 2 τ &GreaterGreater; 1  where  γ 2  is the decay rate of the excited state population,  τ  is the pulse length of the probe field, and  2 S  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  γ 2 τ &LessLess; 1  and there is no reason to believe that many pulse length decays can be obtained for such short pulses.

AB - 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  ω p − ω P = Δ ≈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  Δτ &GreaterGreater; 1  or  (2 S +1) γ 2 τ &GreaterGreater; 1  where  γ 2  is the decay rate of the excited state population,  τ  is the pulse length of the probe field, and  2 S  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  γ 2 τ &LessLess; 1  and there is no reason to believe that many pulse length decays can be obtained for such short pulses.

UR - https://doi.org/10.1103/PhysRevA.74.043810

U2 - 10.1103/PhysRevA.74.043810

DO - 10.1103/PhysRevA.74.043810

M3 - Article

SN - 2469-9926

VL - 74

JO - Physical Review A

JF - Physical Review A

ER -

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

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