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Physical Review A

Absorption and Emission Line-Shape Functions for Driven Atoms

B. R. Mollow

Published Mar 1, 1972• 145 citations
Publisher

Abstract

The effect of a driving or pump field on the emission and absorption line-shape functions for an atom is discussed. The pump field is assumed to oscillate at a frequency near the resonance frequency for transitions between a single pair of atomic states, and transitions between one of these states and any other state of the atom are analyzed. The pump field is treated classically, and atomic relaxation is treated in general terms. The emission and absorption line-shape function (the latter is defined as the rate of absorption of energy from a weak signal field, applied in addition to the pump field, as a function of the signal-field frequency) are found by evaluating the relevant two-time atomic correlation functions in the Markoff approximation. In the limit of high pump-field intensity, both the absorption and the emission spectra are doubly peaked at frequencies which differ from the usual resonance frequency by $\ifmmode\pm\else\textpm\fi{}\frac{1}{2} \ensuremath{\Omega}$, where $\ensuremath{\Omega}$ is the frequency of the pump-field-induced oscillations in the populations of the two strongly coupled states. The absorption and the emission spectra are represented by essentially the same function in the limit of high pump-field intensity, as they are also in the limit of vanishing pump-field intensity. For intermediate pump-field intensities, however, the two functions are quite different in form, and no simple proportionality exists between them.

DOI
10.1103/physreva.5.1522

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Stimulated Emission and Absorption near Resonance for Driven Systems

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The rate of absorption of energy from a weak signal field by an atom driven by a strong pump field is evaluated. The pump field and the signal field are assumed to induce transitions between the same pair of states, and their frequencies are both assumed to lie near the atomic resonance frequency for the transition in question. We find that the signal-field absorption line-shape function takes on negative values, representing stimulated emission rather than absorption, even though population inversion does not occur. This amplification of the signal field, which is most pronounced at high pump intensities, is shown to occur primarily at the expense of the pump field, which suffers an increased rate of attenuation. The results are discussed in the context of a theorem which expresses the absorption line-shape function for general atomic systems in terms of a suitable atomic correlation function.

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Strong amplification of sidebands in a strongly driven three-level atomic system

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We have considered the excitation spectra for a three-level atom interacting simultaneously with a strong pump field and a weak signal field. The atom consists of an upper excited state ‖2〉 and two lower states ‖1〉 and ‖3〉 where the transition ‖1〉↔‖3〉 is electric-dipole forbidden from parity considerations. The pump field, whose frequency mode ωb is initially populated, operates near resonance between the states ‖2〉 and ‖3〉, ‖2〉↔‖3〉, while the signal field describes the radiative decays ‖2〉→‖3〉 and ‖2〉→‖1〉, respectively. Using the Green function method, the spectral functions for the signal field have been calculated in the limit of high photon densities of the pump field describing the following processes: (i) one-photon process; the spectra consist of a main peak at the signal frequency ω=ω21 and a pair of sidebands at ω=ω21±Ω−(ω23−ωb)/2, where Ω2=Ω2b/2+(ω23−ωb)2/4 with Ωb and (ω23−ωb) being the Rabi frequency and the detuning of the pump field, respectively. The intensity of the main peak at ω=ω21 is positive indicating signal-field absorption while those of the sidebands are always negative implying amplification (stimulated emission) of the signal field. The sum of the intensities (in absolute value) of the sidebands is twice that of the main peak at resonance while in the off-resonance case it depends on the value of the ratio (ω23−ωb)/2Ω. (ii) Stimulated three-photon process; the spectra consists of a pair of sidebands peaked at ω=ω21−2ωb±Ω−(ω23−ωb)/2 whose intensity is negative indicating amplification. At resonance the sidebands have equal intensities but at finite detuning asymmetry arises enhancing one peak while diminishing the intensity of the other. (iii) Stimulated two-photon process; the stimulated Raman spectra consist of a peak at ω=ω31, which has a delta function distribution, and a pair of strongly amplified sidebands at ω=ω31±Ω+(ω23−ωb)/2. The computed spectra for the three processes are presented graphically for different values of Rabi frequencies and detunings.

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Spontaneous emission and absorption properties of a driven three-level system

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We investigate the steady-state spontaneous emission spectrum of a three-level atom driven by two coherent fields and the absorption spectrum of a weak probe passing through a collection of such driven atoms. We find that the fluorescence spectrum is strongly affected by the decay rates of all the levels involved in the atomic evolution and not just by the decay parameters of the specific transition whose emission spectrum is being monitored. In particular, the spectral components can acquire very different widths and peak heights relative to the case of the standard resonance fluorescence in which a two-level system is driven by a single near-resonant field. An external probe signal passing through the gas of three-level atoms may be absorbed or amplified, as in the standard two-level case, but under specific operating conditions, amplification (or absorption) occurs over the entire range where the atomic response is appreciable. In this case the existence of amplification or absorption is controlled solely by the population difference between two dressed states of the system. We provide numerical results in support of our arguments for arbitrary values of the atomic and field parameters and also develop an analytic description in the limit of strong driving fields that leads to explicit line-shape and linewidth formulas.

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We have considered the induced processes that occur in a driven two-level atom, where a laser photon is absorbed and emitted by the ground and by the excited states of the atom, respectively. In the low-intensity limit of the laser field, the induced spectra arising when a laser photon is absorbed by the ground state of the atom consist of two peaks describing induced-absorption and stimulated-emission processes, respectively, where the former prevails over the latter. Asymmetry of the spectral lines occurs at off-resonance and its extent depends on the detuning of the laser field. The physical, process where a laser photon is emitted by the excited state is the reverse of that arising from the absorption of a laser photon by the ground state of the atom. The former differs from the latter in that the emission of a laser photon by the excited state occurs in the low-frequency regime and that the stimulated-emission process prevails over that of the induced absorption. In this case, amplification of ultrashort pulses is likely to occur without the need of population inversion between the optical transitions. The computed spectra are graphically presented and discussed.

Canadian Journal of Physics•Jun 1, 1992•2 citations

Interference spectra in a two-level atom

Constantine Mavroyannis

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Canadian Journal of Physics•Dec 1, 1990•6 citations

Kinetics And Optical Properties Of The Strongly Driven Gas Medium Of Interacting Atoms

Андрій Сіжук, Philip Hemmer

This paper investigates stimulated emission and absorption near resonance for a driven system of interacting two-level atoms. Microscopic kinetic equations for the density matrix elements of N-atom states including atomic motion are built, taking into account atom-field and atom-atom interactions. Analytical solutions are given for the resulting macroscopic equations in different limits, for a system composed of a strong coherent “pump” field and a weak counter-propagating “probe” field. It was shown that the existence of a dipole-dipole (long-range) interaction between atoms separated by distance less than the pump wave-length can cause the formation of periodic polarization and population structures (gratings in time and space) in the pumped medium without a probe field. The magnitude of pump induced population grating can have a strong dependence on the relation between the pump field detuning and the polarization decay rate. The “interaction” between pump and probe induced polarization/population gratings through a dipole-dipole interaction mechanism causes the absorption line shape asymmetry. Under certain conditions, this asymmetry is revealed in increasing probe gain for the “red”-shifted (relative to pump) probe and suppressing the gain for the “blue”-shifted probe field when pump is “red”-shifted relative to the ensemble averaged resonant frequency. The theoretical results are consistent with experimental data for the probe gain or absorption as the function of frequency and the dependance of the gain on atomic density for sodium vapor when the pump laser is tuned near the D 2 line. Here the dependance of gain on particle density was explained in the terms of the long-range interaction between the atoms.

Bulletin of the American Physical Society•Oct 7, 2011•0 citations
The effect of a driving or pump field on the emission and absorption line-shape functions for an atom is discussed. The pump field is assumed to oscillate at a frequency near the resonance frequency for transitions between a single pair of atomic states, and transitions between one of these states and any other state of the atom are analyzed. The pump field is treated classically, and atomic relaxation is treated in general terms. The emission and absorption line-shape function (the latter is defined as the rate of absorption of energy from a weak signal field, applied in addition to the pump field, as a function of the signal-field frequency) are found by evaluating the relevant two-time atomic correlation functions in the Markoff approximation. In the limit of high pump-field intensity, both the absorption and the emission spectra are doubly peaked at frequencies which differ from the usual resonance frequency by \ifmmode±\else\textpm\fi12\ensuremathΩ\ifmmode\pm\else\textpm\fi{}\frac{1}{2} \ensuremath{\Omega}\ifmmode±\else\textpm\fi21​\ensuremathΩ, where \ensuremathΩ\ensuremath{\Omega}\ensuremathΩ is the frequency of the pump-field-induced oscillations in the populations of the two strongly coupled states. The absorption and the emission spectra are represented by essentially the same function in the limit of high pump-field intensity, as they are also in the limit of vanishing pump-field intensity. For intermediate pump-field intensities, however, the two functions are quite different in form, and no simple proportionality exists between them.
The rate of absorption of energy from a weak signal field by an atom driven by a strong pump field is evaluated. The pump field and the signal field are assumed to induce transitions between the same pair of states, and their frequencies are both assumed to lie near the atomic resonance frequency for the transition in question. We find that the signal-field absorption line-shape function takes on negative values, representing stimulated emission rather than absorption, even though population inversion does not occur. This amplification of the signal field, which is most pronounced at high pump intensities, is shown to occur primarily at the expense of the pump field, which suffers an increased rate of attenuation. The results are discussed in the context of a theorem which expresses the absorption line-shape function for general atomic systems in terms of a suitable atomic correlation function.
We have considered the excitation spectra for a three-level atom interacting simultaneously with a strong pump field and a weak signal field. The atom consists of an upper excited state ‖2〉 and two lower states ‖1〉 and ‖3〉 where the transition ‖1〉↔‖3〉 is electric-dipole forbidden from parity considerations. The pump field, whose frequency mode ωb is initially populated, operates near resonance between the states ‖2〉 and ‖3〉, ‖2〉↔‖3〉, while the signal field describes the radiative decays ‖2〉→‖3〉 and ‖2〉→‖1〉, respectively. Using the Green function method, the spectral functions for the signal field have been calculated in the limit of high photon densities of the pump field describing the following processes: (i) one-photon process; the spectra consist of a main peak at the signal frequency ω=ω21 and a pair of sidebands at ω=ω21±Ω−(ω23−ωb)/2, where Ω2=Ω2b/2+(ω23−ωb)2/4 with Ωb and (ω23−ωb) being the Rabi frequency and the detuning of the pump field, respectively. The intensity of the main peak at ω=ω21 is positive indicating signal-field absorption while those of the sidebands are always negative implying amplification (stimulated emission) of the signal field. The sum of the intensities (in absolute value) of the sidebands is twice that of the main peak at resonance while in the off-resonance case it depends on the value of the ratio (ω23−ωb)/2Ω. (ii) Stimulated three-photon process; the spectra consists of a pair of sidebands peaked at ω=ω21−2ωb±Ω−(ω23−ωb)/2 whose intensity is negative indicating amplification. At resonance the sidebands have equal intensities but at finite detuning asymmetry arises enhancing one peak while diminishing the intensity of the other. (iii) Stimulated two-photon process; the stimulated Raman spectra consist of a peak at ω=ω31, which has a delta function distribution, and a pair of strongly amplified sidebands at ω=ω31±Ω+(ω23−ωb)/2. The computed spectra for the three processes are presented graphically for different values of Rabi frequencies and detunings.
We investigate the steady-state spontaneous emission spectrum of a three-level atom driven by two coherent fields and the absorption spectrum of a weak probe passing through a collection of such driven atoms. We find that the fluorescence spectrum is strongly affected by the decay rates of all the levels involved in the atomic evolution and not just by the decay parameters of the specific transition whose emission spectrum is being monitored. In particular, the spectral components can acquire very different widths and peak heights relative to the case of the standard resonance fluorescence in which a two-level system is driven by a single near-resonant field. An external probe signal passing through the gas of three-level atoms may be absorbed or amplified, as in the standard two-level case, but under specific operating conditions, amplification (or absorption) occurs over the entire range where the atomic response is appreciable. In this case the existence of amplification or absorption is controlled solely by the population difference between two dressed states of the system. We provide numerical results in support of our arguments for arbitrary values of the atomic and field parameters and also develop an analytic description in the limit of strong driving fields that leads to explicit line-shape and linewidth formulas.
We have considered the induced processes that occur in a driven two-level atom, where a laser photon is absorbed and emitted by the ground and by the excited states of the atom, respectively. In the low-intensity limit of the laser field, the induced spectra arising when a laser photon is absorbed by the ground state of the atom consist of two peaks describing induced-absorption and stimulated-emission processes, respectively, where the former prevails over the latter. Asymmetry of the spectral lines occurs at off-resonance and its extent depends on the detuning of the laser field. The physical, process where a laser photon is emitted by the excited state is the reverse of that arising from the absorption of a laser photon by the ground state of the atom. The former differs from the latter in that the emission of a laser photon by the excited state occurs in the low-frequency regime and that the stimulated-emission process prevails over that of the induced absorption. In this case, amplification of ultrashort pulses is likely to occur without the need of population inversion between the optical transitions. The computed spectra are graphically presented and discussed.
We have considered the interference spectra arising from the competition between a spontaneous process and one induced by a laser field in a two-level atom. Expressions for the spectral functions have been derived describing the spectra of the excited and ground states of the atom in the low- and high-intensity limit of the laser field. For the excited-state spectra in the low-intensity limit, the frequency profiles of the two peaks, which arise from the spontaneous and the induced processes, cancel each other out completely near the center of the line, while for the ground state the induced process dominates. For finite values of the detuning, the spectra of the excited state consist of two peaks, which have positive and negative frequency profiles, respectively. The computed spectra have been graphically presented and discussed. In the high-intensity limit, the dynamic Stark effect dominates the spectra of the excited and ground states of the atom. Expressions for the correlation functions have been derived that describe the emission or the absorption of a laser photon at two different times. The derived expressions for the corresponding delay functions in the low- and high-intensity limits have been found to be identical to those recently proposed in the literature. The laser field has been treated as a classical as well as a quantized entity.
This paper investigates stimulated emission and absorption near resonance for a driven system of interacting two-level atoms. Microscopic kinetic equations for the density matrix elements of N-atom states including atomic motion are built, taking into account atom-field and atom-atom interactions. Analytical solutions are given for the resulting macroscopic equations in different limits, for a system composed of a strong coherent “pump” field and a weak counter-propagating “probe” field. It was shown that the existence of a dipole-dipole (long-range) interaction between atoms separated by distance less than the pump wave-length can cause the formation of periodic polarization and population structures (gratings in time and space) in the pumped medium without a probe field. The magnitude of pump induced population grating can have a strong dependence on the relation between the pump field detuning and the polarization decay rate. The “interaction” between pump and probe induced polarization/population gratings through a dipole-dipole interaction mechanism causes the absorption line shape asymmetry. Under certain conditions, this asymmetry is revealed in increasing probe gain for the “red”-shifted (relative to pump) probe and suppressing the gain for the “blue”-shifted probe field when pump is “red”-shifted relative to the ensemble averaged resonant frequency. The theoretical results are consistent with experimental data for the probe gain or absorption as the function of frequency and the dependance of the gain on atomic density for sodium vapor when the pump laser is tuned near the D 2 line. Here the dependance of gain on particle density was explained in the terms of the long-range interaction between the atoms.