Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical Review A

Effects of the dynamic Lorentz shift on four-wave parametric interactions in a strongly driven two-level system

Richard M. Friedberg, Sven R. Hartmann, Jamal T. Manassah

Published Jul 1, 1990• 20 citations
Publisher

Abstract

We calculate the effect of the local-field correction (LFC) on the local response at ${\mathrm{\ensuremath{\omega}}}_{\mathit{s}}$ and 2${\mathrm{\ensuremath{\omega}}}_{\mathit{p}}$-${\mathrm{\ensuremath{\omega}}}_{\mathit{s}}$ of a medium consisting of two-level systems when subject to a pump of arbitrary intensity at ${\mathrm{\ensuremath{\omega}}}_{\mathit{p}}$ and a weak probe at ${\mathrm{\ensuremath{\omega}}}_{\mathit{s}}$. The pump is considered strong if its reduced intensity I=${\mathrm{\ensuremath{\omega}}}_{\mathit{R}}^{2}$/${\ensuremath{\gamma}}_{1}$${\ensuremath{\gamma}}_{2}$ is large compared to unity. Here ${\mathrm{\ensuremath{\omega}}}_{\mathit{R}}$ is the Rabi frequency and ${\ensuremath{\gamma}}_{1}$ and ${\ensuremath{\gamma}}_{2}$ are the longitudinal and transverse relaxation rates. In the weak pump case the LFC manifests itself as a fixed or static frequency shift. When the pump is strong, the response is independent of the LFC. At intermediate pump intensities the effect of LFC is dependent on the intensity of the pump. Both line-shift and line-shape distortions are present. The LFC thus gives rise to a shift in the resonance response that varies according to the intensity and frequency of the applied radiation. We call this effect the dynamic Lorentz shift.

DOI
10.1103/physreva.42.494

Related papers

Effect of local-field correction on a strongly pumped resonance

Richard M. Friedberg, Sven R. Hartmann, Jamal T. Manassah

We present a general formalism that incorporates local-field corrections into Bloch's equations and use it to calculate the strong-field susceptibility of a pressure-broadened gas consisting of two-level atoms. The major effect is to introduce a dynamic Lorentz shift that varies throughout the resonance line shape. We also calculate the susceptibility associated with a weak field (probe) in the presence of a strong field (pump) when the pump and probe are at the same frequency. Here the effect is to introduce both a variable shift and a greater distortion of the line shape.

Physical Review A•Sep 1, 1989•70 citations

Line shape, frequency shift, Rabi splitting, and two-photon resonances in four-level double-resonance spectroscopy with closely spaced intermediate levels

Swapan Kumar Mandal, Pradip Narayan Ghosh

The interaction of a four-level atomic system with two electromagnetic radiations has been studied. The two intermediate levels are closely spaced such that a single radiation frequency can induce transitions from the ground state to both of them. In a cascade-type double resonance, the pump electric-field amplitude is much stronger than the amplitude of the signal field, which can induce transitions from both of the intermediate levels to the upper state. The 15 optical Bloch equations obtained from the Liouville equations are solved analytically to obtain the Doppler-free signal line shape. The computed signals, exhibiting two Lorentzian line profiles, are presented for different pump frequencies. In addition to these one-photon transitions, we obtain a weak two-photon resonance peak. The two-photon resonance becomes weaker when the detuning of the pump frequency from a resonance frequency is increased. When the pump frequency ${\mathrm{\ensuremath{\Omega}}}_{\mathit{p}}$ is intermediate between the pump resonance frequencies ${\mathrm{\ensuremath{\omega}}}_{1}$ and ${\mathrm{\ensuremath{\omega}}}_{2}$, the two-photon resonance is very weak and almost vanishes when ${\mathrm{\ensuremath{\Omega}}}_{\mathit{p}}$=(${\mathrm{\ensuremath{\omega}}}_{1}$+${\mathrm{\ensuremath{\omega}}}_{2}$)/2. When the pump frequency is on resonance with one of the transition frequencies, the Rabi splitting of the signal shows asymmetry because of the presence of a close-lying transition which is pumped off resonance. The intensity-dependent shifts of the transitions are such that they repel each other more and more as the intensity of the pump field is increased. These shifts arise from the nonresonant interaction between the atom and the field.

Physical Review A•Jun 1, 1993•18 citations

Four-wave mixing in strongly driven four-level systems

F. A. M. de Oliveira, Cid Bartolomeu de Araujo, J. R. Rios Leite

A nonperturbative theoretical study of four-wave-mixing processes in four-level systems is presented. Using the density-matrix formalism, numerical results are obtained for two arbitrarily intense pump fields with the third field a weak probe. The fourth wave, radiated by the system, is obtained analytically for some special cases. We find, for different frequency detunings, Rabi frequencies, and relaxation rates that the line shapes exhibit: (i) dynamic Stark splitting and shifting, and merging of resonances; (ii) power broadening with two-photon antiresonance behavior resulting from a destructive interference between different contributions to the optical polarization; and (iii) extra resonances induced by field intensities and saturation processes.

Physical Review A•Dec 1, 1988•17 citations

Optical Amplification due to Parametric Interaction of Four Waves with a Four-Level System

Motowo Tsukakoshi

Four-wave parametric interactions in a four-level system are treated using semiclassical theory. Line-shape functions for a weak probe field and three resonant pump fields are presented and are evaluated numerically in a model case of equalized interaction energies and relaxation rates. For sufficiently intense pump fields, the line shape splits into five peaks, and negative absorption, i.e. amplification, occurs in some frequency regions of the probe field.

Japanese Journal of Applied Physics•Jul 1, 1982•1 citation

Four-wave parametric interactions in a strongly driven two-level system

Robert W. Boyd, Michael G. Raymer, Paul Narum +1 more

We present an analysis of four-wave parametric amplification resulting from the nonlinear response of a two-level atomic system. The atomic dipole moment induced by weak optical fields at frequencies ${\ensuremath{\omega}}_{3}$ and ${\ensuremath{\omega}}_{4}$ in the presence of an optical field of arbitrary intensity at frequency ${\ensuremath{\omega}}_{1}$, where ${\ensuremath{\omega}}_{3}+{\ensuremath{\omega}}_{4}=2{\ensuremath{\omega}}_{1}$, is obtained by solving the density-matrix equations of motion with phenomenological damping constants. The resulting nonlinear polarization can induce loss or three-photon gain in either weak wave and can act to parametrically couple these waves. The spatial evolution of the weak-field amplitudes is obtained by finding approximate solutions to the Helmholtz wave equations using the nonlinear polarization for the source terms. These solutions predict large gain for either or both of the weak waves under experimentally attainable conditions. Depending on the circumstance, this gain can be attributed solely to the three-photon gain, solely to the parametric coupling, or to an interplay between these effects. In addition, the solutions show an enhancement in the gain when $|{\ensuremath{\omega}}_{3}\ensuremath{-}{\ensuremath{\omega}}_{1}|=|{\ensuremath{\omega}}_{4}\ensuremath{-}{\ensuremath{\omega}}_{1}|={\ensuremath{\Omega}}^{\ensuremath{'}}$, where ${\ensuremath{\Omega}}^{\ensuremath{'}}$ is the generalized Rabi frequency associated with the driving of the atoms by the wave at frequency ${\ensuremath{\omega}}_{1}$.

Physical Review A•Jul 1, 1981•404 citations

Effects of strong driving fields in resonant four-wave mixing schemes with down-conversion

Sergei A Babin, Evgeniy V. Podivilov, David A. Shapiro +3 more

An explicit solution is obtained for the four-wave frequency mixing ${\ensuremath{\omega}}_{d}={\ensuremath{\omega}}_{a}\ensuremath{-}{\ensuremath{\omega}}_{b}+{\ensuremath{\omega}}_{c}$ of two strong fields a and c and two weak fields b and d in a four-level system with large Doppler broadening in collinear geometry, where the frequencies of weak fields are nearly equal, ${\ensuremath{\omega}}_{b}\ensuremath{\simeq}{\ensuremath{\omega}}_{d},$ and the medium is optically thin. Without weak fields there are two independent two-level systems. A pair of weak fields probes two other allowed transitions. A peak of the mixing coefficient as a function of intensity is found around an equal Rabi splitting of both two-level systems. The effect is based on a resonance between two closed cycles of four-wave mixing via different dressed states. Three, four, or six peaks are predicted in the dependence of the mixing coefficient on the frequency of the weak field; two of them are a consequence of averaging over velocities. The model allows an interpretation of the dependence of the output wave power on the intensity and detuning in recent experiments on frequency mixing in sodium vapor.

Physical Review A•Feb 1, 1999•20 citations
We calculate the effect of the local-field correction (LFC) on the local response at \ensuremathωs{\mathrm{\ensuremath{\omega}}}_{\mathit{s}}\ensuremathωs​ and 2\ensuremathωp{\mathrm{\ensuremath{\omega}}}_{\mathit{p}}\ensuremathωp​-\ensuremathωs{\mathrm{\ensuremath{\omega}}}_{\mathit{s}}\ensuremathωs​ of a medium consisting of two-level systems when subject to a pump of arbitrary intensity at \ensuremathωp{\mathrm{\ensuremath{\omega}}}_{\mathit{p}}\ensuremathωp​ and a weak probe at \ensuremathωs{\mathrm{\ensuremath{\omega}}}_{\mathit{s}}\ensuremathωs​. The pump is considered strong if its reduced intensity I=\ensuremathωR2{\mathrm{\ensuremath{\omega}}}_{\mathit{R}}^{2}\ensuremathωR2​/{\ensuremath{\gamma}}_{1}$${\ensuremath{\gamma}}_{2} is large compared to unity. Here \ensuremathωR{\mathrm{\ensuremath{\omega}}}_{\mathit{R}}\ensuremathωR​ is the Rabi frequency and \ensuremathγ1{\ensuremath{\gamma}}_{1}\ensuremathγ1​ and \ensuremathγ2{\ensuremath{\gamma}}_{2}\ensuremathγ2​ are the longitudinal and transverse relaxation rates. In the weak pump case the LFC manifests itself as a fixed or static frequency shift. When the pump is strong, the response is independent of the LFC. At intermediate pump intensities the effect of LFC is dependent on the intensity of the pump. Both line-shift and line-shape distortions are present. The LFC thus gives rise to a shift in the resonance response that varies according to the intensity and frequency of the applied radiation. We call this effect the dynamic Lorentz shift.
We present a general formalism that incorporates local-field corrections into Bloch's equations and use it to calculate the strong-field susceptibility of a pressure-broadened gas consisting of two-level atoms. The major effect is to introduce a dynamic Lorentz shift that varies throughout the resonance line shape. We also calculate the susceptibility associated with a weak field (probe) in the presence of a strong field (pump) when the pump and probe are at the same frequency. Here the effect is to introduce both a variable shift and a greater distortion of the line shape.
The interaction of a four-level atomic system with two electromagnetic radiations has been studied. The two intermediate levels are closely spaced such that a single radiation frequency can induce transitions from the ground state to both of them. In a cascade-type double resonance, the pump electric-field amplitude is much stronger than the amplitude of the signal field, which can induce transitions from both of the intermediate levels to the upper state. The 15 optical Bloch equations obtained from the Liouville equations are solved analytically to obtain the Doppler-free signal line shape. The computed signals, exhibiting two Lorentzian line profiles, are presented for different pump frequencies. In addition to these one-photon transitions, we obtain a weak two-photon resonance peak. The two-photon resonance becomes weaker when the detuning of the pump frequency from a resonance frequency is increased. When the pump frequency ${\mathrm{\ensuremath{\Omega}}}_{\mathit{p}}$ is intermediate between the pump resonance frequencies ${\mathrm{\ensuremath{\omega}}}_{1}$ and ${\mathrm{\ensuremath{\omega}}}_{2}$, the two-photon resonance is very weak and almost vanishes when ${\mathrm{\ensuremath{\Omega}}}_{\mathit{p}}$=(${\mathrm{\ensuremath{\omega}}}_{1}$+${\mathrm{\ensuremath{\omega}}}_{2}$)/2. When the pump frequency is on resonance with one of the transition frequencies, the Rabi splitting of the signal shows asymmetry because of the presence of a close-lying transition which is pumped off resonance. The intensity-dependent shifts of the transitions are such that they repel each other more and more as the intensity of the pump field is increased. These shifts arise from the nonresonant interaction between the atom and the field.
A nonperturbative theoretical study of four-wave-mixing processes in four-level systems is presented. Using the density-matrix formalism, numerical results are obtained for two arbitrarily intense pump fields with the third field a weak probe. The fourth wave, radiated by the system, is obtained analytically for some special cases. We find, for different frequency detunings, Rabi frequencies, and relaxation rates that the line shapes exhibit: (i) dynamic Stark splitting and shifting, and merging of resonances; (ii) power broadening with two-photon antiresonance behavior resulting from a destructive interference between different contributions to the optical polarization; and (iii) extra resonances induced by field intensities and saturation processes.
We present an analysis of four-wave parametric amplification resulting from the nonlinear response of a two-level atomic system. The atomic dipole moment induced by weak optical fields at frequencies ${\ensuremath{\omega}}_{3}$ and ${\ensuremath{\omega}}_{4}$ in the presence of an optical field of arbitrary intensity at frequency ${\ensuremath{\omega}}_{1}$, where ${\ensuremath{\omega}}_{3}+{\ensuremath{\omega}}_{4}=2{\ensuremath{\omega}}_{1}$, is obtained by solving the density-matrix equations of motion with phenomenological damping constants. The resulting nonlinear polarization can induce loss or three-photon gain in either weak wave and can act to parametrically couple these waves. The spatial evolution of the weak-field amplitudes is obtained by finding approximate solutions to the Helmholtz wave equations using the nonlinear polarization for the source terms. These solutions predict large gain for either or both of the weak waves under experimentally attainable conditions. Depending on the circumstance, this gain can be attributed solely to the three-photon gain, solely to the parametric coupling, or to an interplay between these effects. In addition, the solutions show an enhancement in the gain when $|{\ensuremath{\omega}}_{3}\ensuremath{-}{\ensuremath{\omega}}_{1}|=|{\ensuremath{\omega}}_{4}\ensuremath{-}{\ensuremath{\omega}}_{1}|={\ensuremath{\Omega}}^{\ensuremath{'}}$, where ${\ensuremath{\Omega}}^{\ensuremath{'}}$ is the generalized Rabi frequency associated with the driving of the atoms by the wave at frequency ${\ensuremath{\omega}}_{1}$.
An explicit solution is obtained for the four-wave frequency mixing ${\ensuremath{\omega}}_{d}={\ensuremath{\omega}}_{a}\ensuremath{-}{\ensuremath{\omega}}_{b}+{\ensuremath{\omega}}_{c}$ of two strong fields a and c and two weak fields b and d in a four-level system with large Doppler broadening in collinear geometry, where the frequencies of weak fields are nearly equal, ${\ensuremath{\omega}}_{b}\ensuremath{\simeq}{\ensuremath{\omega}}_{d},$ and the medium is optically thin. Without weak fields there are two independent two-level systems. A pair of weak fields probes two other allowed transitions. A peak of the mixing coefficient as a function of intensity is found around an equal Rabi splitting of both two-level systems. The effect is based on a resonance between two closed cycles of four-wave mixing via different dressed states. Three, four, or six peaks are predicted in the dependence of the mixing coefficient on the frequency of the weak field; two of them are a consequence of averaging over velocities. The model allows an interpretation of the dependence of the output wave power on the intensity and detuning in recent experiments on frequency mixing in sodium vapor.