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Physical Review A

Phase-dependent pump-probe line-shape formulas

Joseph H. Eberly, V. D. Popov

Published Mar 1, 1988• 40 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.37.2012

Related papers

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

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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.

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Theory of pump-probe and four-wave mixing spectroscopy

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In this paper we present theoretical calculations relating to pump-probe and four-wave mixing spectroscopy. In pump-probe spectroscopy, one measures the absorption profile for the situation when an arbitrary strong pump field drives a given transition and a probe (weak) field drives the same transition. We have studied the influence of level degeneracy and field polarization on pump-probe and four-wave mixing signals. Our results indicate that for some closed systems (no population loss to the reservoir) it is possible to observe resonances that are characterized by the ground state width. Previous calculations indicate that for a nondegenerate system this resonance can appear only for an open system, the so-called "pressure induced resonances.”1 Our result, which contradicts this conclusion, can be understood by noting the fact that there are in general three irreducible tensor2 (in general) contributions to the line shape (K = 0 population, K = 1 orientation, and K = 2 alignment). Thus, even if the system is closed for the population it may still remain open for orientation or alignment; this effect is responsible for the narrow resonance. Specific calculations are presented for J = 1/2 → 1/2 and J = 1→2 transitions.

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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

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.

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Analytical studies on pump-induced optical resonances in anM-typesix-level system

Saswata Ghosh, Swapan Mandal

In the domain of semiclassical formulation and for the Doppler-free atom–field interaction, we construct the optical Bloch equations involving an M-type six-level system coupled to two pump fields and a probe field. The response of the system is probed for different pump-induced transitions in double and triple-resonance situations. In order to obtain the coherent lineshapes (absorptive and dispersive), we use the usual perturbation method for obtaining the approximate analytical solutions to these coupled optical Bloch equations for the density matrix elements. The interferences between the probability amplitudes for different energy levels (dipole allowed and dipole forbidden) are taken care of. For off-resonance pump positions, the linewidths of the three probe transitions are insensitive to the pump Rabi frequencies. On the other hand, the shifts of the three resonance peaks are extremely sensitive to the pump Rabi frequencies. However, for on-resonance pump conditions, the sensitivities of pump Rabi frequencies on the linewidths of the resonance peaks and on the shifts of the resonance peak positions are opposite to those of their off-resonance counterparts. In particular, we have shown the asymmetric and symmetric Rabi splittings under different physical conditions, for non-zero and near-zero probe detuning, respectively. The Rabi splitting under triple-resonance conditions, significantly, modifies the dispersive lineshape at the centre of the absorption line. The two- and three-photon absorptions are also reported for different off-resonant pump positions.

Journal of Physics B Atomic Molecular and Optical Physics•Dec 1, 2010•8 citations
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 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.
When a two-level system interacts resonantly with a saturating pump laser, the absorption profile of a probe wave in the Rabi sidebands resembles two separate dispersion profiles. At sufficiently high probe intensity this effect is accompanied by a dispersive profile in the pump-wave absorption, considered as a function of the probe-wave frequency, and also by a dip in the population of the upper state of the two-level system. The relative stabilization of the ground state is interpreted in terms of two-laser-photon processes where absorption of a pump (probe) photon is followed by stimulated emission of a probe (pump) photon. The dispersive absorption profile is attributed to the competition between three- and four-photon scattering processes.
In this paper we present theoretical calculations relating to pump-probe and four-wave mixing spectroscopy. In pump-probe spectroscopy, one measures the absorption profile for the situation when an arbitrary strong pump field drives a given transition and a probe (weak) field drives the same transition. We have studied the influence of level degeneracy and field polarization on pump-probe and four-wave mixing signals. Our results indicate that for some closed systems (no population loss to the reservoir) it is possible to observe resonances that are characterized by the ground state width. Previous calculations indicate that for a nondegenerate system this resonance can appear only for an open system, the so-called "pressure induced resonances.”1 Our result, which contradicts this conclusion, can be understood by noting the fact that there are in general three irreducible tensor2 (in general) contributions to the line shape (K = 0 population, K = 1 orientation, and K = 2 alignment). Thus, even if the system is closed for the population it may still remain open for orientation or alignment; this effect is responsible for the narrow resonance. Specific calculations are presented for J = 1/2 → 1/2 and J = 1→2 transitions.
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 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.
In the domain of semiclassical formulation and for the Doppler-free atom–field interaction, we construct the optical Bloch equations involving an M-type six-level system coupled to two pump fields and a probe field. The response of the system is probed for different pump-induced transitions in double and triple-resonance situations. In order to obtain the coherent lineshapes (absorptive and dispersive), we use the usual perturbation method for obtaining the approximate analytical solutions to these coupled optical Bloch equations for the density matrix elements. The interferences between the probability amplitudes for different energy levels (dipole allowed and dipole forbidden) are taken care of. For off-resonance pump positions, the linewidths of the three probe transitions are insensitive to the pump Rabi frequencies. On the other hand, the shifts of the three resonance peaks are extremely sensitive to the pump Rabi frequencies. However, for on-resonance pump conditions, the sensitivities of pump Rabi frequencies on the linewidths of the resonance peaks and on the shifts of the resonance peak positions are opposite to those of their off-resonance counterparts. In particular, we have shown the asymmetric and symmetric Rabi splittings under different physical conditions, for non-zero and near-zero probe detuning, respectively. The Rabi splitting under triple-resonance conditions, significantly, modifies the dispersive lineshape at the centre of the absorption line. The two- and three-photon absorptions are also reported for different off-resonant pump positions.