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

Phase-dependent line shapes in pump-probe interactions: A Floquet approach

Manoj Kumar, M. Ll. Pons, Joseph H. Eberly

Published Aug 1, 1991• 19 citations
Publisher

Abstract

We present analytical and numerical results for the line shapes for a two-level atom interacting with pump and probe fields. We focus our attention on a time regime which is small enough that relaxation effects could be neglected, but sufficiently large that one could integrate the signal over a large number of population oscillations between the excited and ground states. As in our earlier work [J. H. Eberly and V. D. Popov, Phys. Rev. A 37, 2012 (1988)], we consider situations where the signal is determined by the atomic inversion, and identify the transient line shape with the time-averaged inversion considered as a function of the probe frequency. We present a formulation, based on the Floquet approach, that allows us to obtain formal expressions for the transient line shape in terms of infinite continued fractions, which can be evaluated numerically. We also present line shapes computed by directly solving the equations of motion numerically and integrating the inversion numerically over long times to compare with our continued-fraction results. We discuss the dependence on initial conditions, the relative pump-probe phase, and the non-Lorentzian character of these line shapes.

DOI
10.1103/physreva.44.1995

Related papers

Phase-dependent pump-probe line-shape formulas

Joseph H. Eberly, V. D. Popov

Pump-probe interactions underlie the physics of four-wave and other wave-mixing effects. We present analytic formulas for probe response in a special category of pump-probe experiments. We consider a two-level system excited by two distinct fields, the stronger field with fixed frequency and the weaker with a tunable frequency. We focus attention on the spectral regions near the Rabi sidebands of the strong field and its various subharmonics. We consider times sufficiently short that relaxation effects can be ignored, but sufficiently long to encompass many cycles of population oscillation between the ground and excited states. Because there is no damping, both fields are in this sense strong, and we report here the time average of the response. Our results are obtained by making a suitable second-order rotating wave approximation and are shown to involve the initial phase relation between the two fields as well as the initial angles of orientation of the two-level system's Bloch vector. Depending on these initial phase and orientation angles, the absorption line shape may be either positive or negative and either symmetric or antisymmetric.

Physical Review A•Mar 1, 1988•40 citations

Transient lineshapes in nonlinear mixing of pump and probe fields

Manoj Kumar, Joseph H. Eberly

We present results on transient lineshapes in signals generated coherently via the nonlinear mixing of pump and probe fields in a model two-level system. We confine ourselves to times short compared to atomic relaxation times and hence treat both fields nonperturbatively. However, the probe is assumed to be much weaker than the pump. Transient lineshapes have been studied earlier in the case of atomic relaxation times and hence treat both fields nonperturbatively. However, the probe is assumed to be much weaker than the pump. Transient lineshapes have been studies earlier in the case of atomic inversion. Here, we identify the lineshape with the intensity of the light scattered in a particular direction and around a particular central frequency. The lineshapes are found to exhibit, in the neighbourhood of subharmonic resonances, highly non-lorentzian profiles.

Annual Meeting Optical Society of America•Jan 1, 1990•0 citations

Transient-absorption phases with strong probe and pump pulses

Becquet, V., Cavaletto, S.

The quantum dynamics of a system of Rb atoms, modeled by a V-type three-level system interacting with intense probe and pump pulses, are studied. The time-delay-dependent transient-absorption spectrum of an intense probe pulse is thus predicted, when this is preceded or followed by a strong pump pulse. Numerical results are interpreted in terms of an analytical model, which allows us to quantify the oscillating features of the resulting transient-absorption spectra in terms of the atomic populations and phases generated by the intense pulses. Strong-field-induced phases and their influence on the resulting transient-absorption spectra are thereby investigated for different values of pump and probe intensities and frequencies, focusing on the atomic properties which are encoded in the absorption line shapes for positive and negative time delays.

•Jan 8, 2018•3 citations

Transient-absorption phases with strong probe and pump pulses

Vadim Becquet, Stefano M. Cavaletto

Abstract The quantum dynamics of a system of Rb atoms, modeled by a V -type three-level system interacting with intense probe and pump pulses, are studied. The time-delay-dependent transient-absorption spectrum of an intense probe pulse is thus predicted, simulating pump-probe experiments in which this is preceded or followed by a strong pump pulse. Numerical results are interpreted in terms of an analytical model based on interaction operators , which quantify the transformation undergone by the system under the action of an intense pulse. The oscillating features of the resulting transient-absorption spectra, due to the coupling of several excited states, are thus interpreted in terms of the atomic population and phase changes imposed by the pump and probe pulses. Strong-field-induced phases and their influence on the resulting transient-absorption spectra are thereby investigated for different values of pump and probe intensities and frequencies, focusing on the atomic properties which are encoded in the absorption line shapes for positive and negative time delays.

arXiv:1703.09182•Journal of Physics B Atomic Molecular and Optical Physics•Sep 22, 2017•6 citations

Floquet analysis of extended Rabi models based on high-frequency expansion

Yibo Liu, Lijun Mao, Yunbo Zhang

The extended quantum Rabi models make a significant contribution to understanding of the quantum nature of the atom-light interaction. We transform two kinds of extended quantum Rabi model, the anisotropic Rabi model and the asymmetric Rabi model, into the rotating frame and regard them as periodically driven quantum systems. The analytical solutions of the quasienergy spectrum as well as the Floquet modes for both models are constructed by applying the Floquet theory and the high-frequency expansion, which is applied to the nonstroboscopic dynamics of physical observables such as atomic inversion, transverse magnetization, and atom-field correlation. For the anisotropic Rabi model, the quasienergy fits well with the numerical results even when the rotating-wave coupling is in the deep strong-coupling regime $g\ensuremath{\simeq}\ensuremath{\hbar}\ensuremath{\omega}$ if the counter-rotating term is small enough compared to the driving frequency. Avoided level crossing may occur for quasienergy with the same parity when the positive branch spectrum lines for the total excitation number $N$ cross the negative branch lines for $N+2$, while the high-frequency expansion fails to predict this due to the conservation of the total excitation number. Furthermore, we present analytical and numerical studies of the long-time evolution of population and conclude that the analytical method is credible for the population dynamics. For the asymmetric Rabi model, we find that the external bias field which breaks the parity symmetry of the total excitation number tends to cluster the upper and lower branches into two bundles, and the detuning induced gap in the first temporal Brillouin zone shows a quadratic dependence on the bias. Fourier analysis is applied to extract the frequency composition and two-frequency driving behavior is revealed. Varying the bias strength will change the time-averaged value of the oscillation, which shows how the bias competes with the detuning and atom-field coupling in the driving dynamics. Both models prove that treating the Hamiltonian in the rotating frame by Floquet theory gives an alternative tool in the study of interaction between atom and light.

arXiv:2202.09751•Physical Review A•May 20, 2022•5 citations

Absorption and Emission Line-Shape Functions for Driven Atoms

B. R. Mollow

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.

Physical Review A•Mar 1, 1972•145 citations
We present analytical and numerical results for the line shapes for a two-level atom interacting with pump and probe fields. We focus our attention on a time regime which is small enough that relaxation effects could be neglected, but sufficiently large that one could integrate the signal over a large number of population oscillations between the excited and ground states. As in our earlier work [J. H. Eberly and V. D. Popov, Phys. Rev. A 37, 2012 (1988)], we consider situations where the signal is determined by the atomic inversion, and identify the transient line shape with the time-averaged inversion considered as a function of the probe frequency. We present a formulation, based on the Floquet approach, that allows us to obtain formal expressions for the transient line shape in terms of infinite continued fractions, which can be evaluated numerically. We also present line shapes computed by directly solving the equations of motion numerically and integrating the inversion numerically over long times to compare with our continued-fraction results. We discuss the dependence on initial conditions, the relative pump-probe phase, and the non-Lorentzian character of these line shapes.
Pump-probe interactions underlie the physics of four-wave and other wave-mixing effects. We present analytic formulas for probe response in a special category of pump-probe experiments. We consider a two-level system excited by two distinct fields, the stronger field with fixed frequency and the weaker with a tunable frequency. We focus attention on the spectral regions near the Rabi sidebands of the strong field and its various subharmonics. We consider times sufficiently short that relaxation effects can be ignored, but sufficiently long to encompass many cycles of population oscillation between the ground and excited states. Because there is no damping, both fields are in this sense strong, and we report here the time average of the response. Our results are obtained by making a suitable second-order rotating wave approximation and are shown to involve the initial phase relation between the two fields as well as the initial angles of orientation of the two-level system's Bloch vector. Depending on these initial phase and orientation angles, the absorption line shape may be either positive or negative and either symmetric or antisymmetric.
We present results on transient lineshapes in signals generated coherently via the nonlinear mixing of pump and probe fields in a model two-level system. We confine ourselves to times short compared to atomic relaxation times and hence treat both fields nonperturbatively. However, the probe is assumed to be much weaker than the pump. Transient lineshapes have been studied earlier in the case of atomic relaxation times and hence treat both fields nonperturbatively. However, the probe is assumed to be much weaker than the pump. Transient lineshapes have been studies earlier in the case of atomic inversion. Here, we identify the lineshape with the intensity of the light scattered in a particular direction and around a particular central frequency. The lineshapes are found to exhibit, in the neighbourhood of subharmonic resonances, highly non-lorentzian profiles.
The quantum dynamics of a system of Rb atoms, modeled by a V-type three-level system interacting with intense probe and pump pulses, are studied. The time-delay-dependent transient-absorption spectrum of an intense probe pulse is thus predicted, when this is preceded or followed by a strong pump pulse. Numerical results are interpreted in terms of an analytical model, which allows us to quantify the oscillating features of the resulting transient-absorption spectra in terms of the atomic populations and phases generated by the intense pulses. Strong-field-induced phases and their influence on the resulting transient-absorption spectra are thereby investigated for different values of pump and probe intensities and frequencies, focusing on the atomic properties which are encoded in the absorption line shapes for positive and negative time delays.
Abstract The quantum dynamics of a system of Rb atoms, modeled by a V -type three-level system interacting with intense probe and pump pulses, are studied. The time-delay-dependent transient-absorption spectrum of an intense probe pulse is thus predicted, simulating pump-probe experiments in which this is preceded or followed by a strong pump pulse. Numerical results are interpreted in terms of an analytical model based on interaction operators , which quantify the transformation undergone by the system under the action of an intense pulse. The oscillating features of the resulting transient-absorption spectra, due to the coupling of several excited states, are thus interpreted in terms of the atomic population and phase changes imposed by the pump and probe pulses. Strong-field-induced phases and their influence on the resulting transient-absorption spectra are thereby investigated for different values of pump and probe intensities and frequencies, focusing on the atomic properties which are encoded in the absorption line shapes for positive and negative time delays.
The extended quantum Rabi models make a significant contribution to understanding of the quantum nature of the atom-light interaction. We transform two kinds of extended quantum Rabi model, the anisotropic Rabi model and the asymmetric Rabi model, into the rotating frame and regard them as periodically driven quantum systems. The analytical solutions of the quasienergy spectrum as well as the Floquet modes for both models are constructed by applying the Floquet theory and the high-frequency expansion, which is applied to the nonstroboscopic dynamics of physical observables such as atomic inversion, transverse magnetization, and atom-field correlation. For the anisotropic Rabi model, the quasienergy fits well with the numerical results even when the rotating-wave coupling is in the deep strong-coupling regime $g\ensuremath{\simeq}\ensuremath{\hbar}\ensuremath{\omega}$ if the counter-rotating term is small enough compared to the driving frequency. Avoided level crossing may occur for quasienergy with the same parity when the positive branch spectrum lines for the total excitation number NNN cross the negative branch lines for N+2N+2N+2, while the high-frequency expansion fails to predict this due to the conservation of the total excitation number. Furthermore, we present analytical and numerical studies of the long-time evolution of population and conclude that the analytical method is credible for the population dynamics. For the asymmetric Rabi model, we find that the external bias field which breaks the parity symmetry of the total excitation number tends to cluster the upper and lower branches into two bundles, and the detuning induced gap in the first temporal Brillouin zone shows a quadratic dependence on the bias. Fourier analysis is applied to extract the frequency composition and two-frequency driving behavior is revealed. Varying the bias strength will change the time-averaged value of the oscillation, which shows how the bias competes with the detuning and atom-field coupling in the driving dynamics. Both models prove that treating the Hamiltonian in the rotating frame by Floquet theory gives an alternative tool in the study of interaction between atom and light.
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.