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

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

Published Jun 1, 1993• 18 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.47.4934

Related papers

Doppler-free absorptive signal lineshape of a four-level double λ-type system: Rabi splitting and two-photon effects

Saswata Ghosh, Swapan Mandal

Two electromagnetic fields (the stronger called the pump and the weaker called the signal) are allowed to interact with a four-level double λ-type system with two pairs of closely spaced levels. The present four-level system could also be viewed as an example of a double resonant system. Under the Doppler-free condition and using a semiclassical formulation of atom–field interaction, we derive the optical Bloch equations involving the said four-level system coupled to two external fields. These coupled optical Bloch equations are unsolvable in closed analytical form. In order to obtain the absorptive lineshapes, we use the usual perturbation method for getting the approximate analytical solution to the coupled optical Bloch equations for the density matrix elements. Through the off-diagonal complex density matrix elements, the introduction of phase angles between the levels participating in dipole-allowed transitions is automatic. It is shown that these phases are appreciable if the applied electromagnetic field is strong. For on-resonance pump conditions, we obtain the usual Rabi splitting. The effects of two-photon resonances are clearly shown for off-resonance pump conditions.

Journal of Physics B Atomic Molecular and Optical Physics•Jun 24, 2009•4 citations

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

Hyperpolarizability, two-photon resonances and quadratic Stark effects in Doppler free spectra for a four level system

Monojit Kora, Piyasi Shit, Swapan Mandal

Using pump-probe spectroscopy, a four level system (FLS) with a double V-type configuration is investigated under Doppler-free conditions. The semiclassical description of the atom-field interaction is used to derive the optical Bloch equations (OBE). Under steady-state and rotating wave approximation, the coupled OBE reduces to a system of simultaneous equations. The analytical expressions for the signal absorption are parametrically computed using theoretical data. The signal lineshapes exhibit two perfectly symmetric Lorentzians corresponding to two signal resonance frequencies when the pump position is exactly in the middle of the two signal resonance frequencies. These two Lorentzians repel each other and, hence, the shifts of the signal peak frequencies. For on-resonant pump, the ac (dynamic) Stark splittings are obtained. The off-resonant pump produces the two-photon resonances. The nonlinear polarization and hence the hyperpolarizability involving the lower pair of levels exhibit interesting results. The hyperpolarizability is synonymous with those of the quadratic Stark effects.

Journal of Modern Optics•Jul 23, 2026•0 citations

One-photon-cascade spectrum in a two-photon-pumped three-level system

Z Chen, Helen S. Freedhoff

The top level of a three-level atom is populated by resonant two-photon optical pumping from the bottom level by a field intense enough to produce dynamic Stark splitting. The frequencies and intensities are calculated for the one-photon cascade emission to the ground state via a nearly resonant intermediate level. Two strong (shifted) doublets near the atomic frequencies, and a weaker triplet centred at the laser frequency are predicted.

Journal of Physics B Atomic Molecular and Optical Physics•Feb 14, 1990•0 citations

Theory of two strongly interacting resonances in multiphoton ionization

Burke Ritchie

The author presents a theory for $N$-photon ionization in which two intermediate states can be in resonance with the ground state over a narrow range of photon frequencies. The theory is applied to the three-photon $6F$ resonance-state spin-orbit doublet in the four-photon ionization of Cs. The amplitude for the process is the sum of two components, each containing a complex line-shape function. In the limit of zero splitting, the previously studied isolated resonance result, in which the line shape is Lorentzian is recovered. For the splitting in Cs ($6F$), the absorption profile is studied as a function of the splitting parameters of the state, the $6S\ensuremath{\rightarrow}6F$ three-photon Rabi rate, and the field-induced shift and width for the $6S\ensuremath{\rightarrow}6F$ transition. This profile depends in a complicated way on the relative magnitudes of the shift and the spacing between the fine-structure levels.

Physical Review A•Feb 1, 1980•22 citations

Two-photon Resonance Absorption in Four-level Systems

Tien D. Kieu, Bryan J. Dalton

An unconventional perturbation treatment of the theory of two-photon absorption in a four-level atomic system close to two-photon resonance is developed on the basis of the optical Bloch equations. The results for the longtime populations of the intermediate and upper states are valid in low- and medium-intensity regimes. In both regimes the upper state population shows a lorentzian resonance centred on zero two-photon detuning with a width equal to the natural linewidth of the upper state. The intermediate state population shows a similar behaviour in the intermediate regime only, with no significant dependence on two-photon detuning in the low-intensity regime. However, in both regimes the population of the virtual intermediate state is at least comparable to that of the resonant upper state. The perturbation theory results have been checked by direct numerical calculations based on the Bloch equations. Numerical calculations confirm the presence of a high-intensity regime characterized by power shifts, broadening and saturation effects.

Optica Acta International Journal of Optics•Nov 1, 1984•2 citations
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 \ensuremathΩp{\mathrm{\ensuremath{\Omega}}}_{\mathit{p}}\ensuremathΩp​ is intermediate between the pump resonance frequencies \ensuremathω1{\mathrm{\ensuremath{\omega}}}_{1}\ensuremathω1​ and \ensuremathω2{\mathrm{\ensuremath{\omega}}}_{2}\ensuremathω2​, the two-photon resonance is very weak and almost vanishes when \ensuremathΩp{\mathrm{\ensuremath{\Omega}}}_{\mathit{p}}\ensuremathΩp​=(\ensuremathω1{\mathrm{\ensuremath{\omega}}}_{1}\ensuremathω1​+\ensuremathω2{\mathrm{\ensuremath{\omega}}}_{2}\ensuremathω2​)/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.
Two electromagnetic fields (the stronger called the pump and the weaker called the signal) are allowed to interact with a four-level double λ-type system with two pairs of closely spaced levels. The present four-level system could also be viewed as an example of a double resonant system. Under the Doppler-free condition and using a semiclassical formulation of atom–field interaction, we derive the optical Bloch equations involving the said four-level system coupled to two external fields. These coupled optical Bloch equations are unsolvable in closed analytical form. In order to obtain the absorptive lineshapes, we use the usual perturbation method for getting the approximate analytical solution to the coupled optical Bloch equations for the density matrix elements. Through the off-diagonal complex density matrix elements, the introduction of phase angles between the levels participating in dipole-allowed transitions is automatic. It is shown that these phases are appreciable if the applied electromagnetic field is strong. For on-resonance pump conditions, we obtain the usual Rabi splitting. The effects of two-photon resonances are clearly shown for off-resonance pump conditions.
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.
Using pump-probe spectroscopy, a four level system (FLS) with a double V-type configuration is investigated under Doppler-free conditions. The semiclassical description of the atom-field interaction is used to derive the optical Bloch equations (OBE). Under steady-state and rotating wave approximation, the coupled OBE reduces to a system of simultaneous equations. The analytical expressions for the signal absorption are parametrically computed using theoretical data. The signal lineshapes exhibit two perfectly symmetric Lorentzians corresponding to two signal resonance frequencies when the pump position is exactly in the middle of the two signal resonance frequencies. These two Lorentzians repel each other and, hence, the shifts of the signal peak frequencies. For on-resonant pump, the ac (dynamic) Stark splittings are obtained. The off-resonant pump produces the two-photon resonances. The nonlinear polarization and hence the hyperpolarizability involving the lower pair of levels exhibit interesting results. The hyperpolarizability is synonymous with those of the quadratic Stark effects.
The author presents a theory for NNN-photon ionization in which two intermediate states can be in resonance with the ground state over a narrow range of photon frequencies. The theory is applied to the three-photon 6F6F6F resonance-state spin-orbit doublet in the four-photon ionization of Cs. The amplitude for the process is the sum of two components, each containing a complex line-shape function. In the limit of zero splitting, the previously studied isolated resonance result, in which the line shape is Lorentzian is recovered. For the splitting in Cs (6F6F6F), the absorption profile is studied as a function of the splitting parameters of the state, the $6S\ensuremath{\rightarrow}6F$ three-photon Rabi rate, and the field-induced shift and width for the $6S\ensuremath{\rightarrow}6F$ transition. This profile depends in a complicated way on the relative magnitudes of the shift and the spacing between the fine-structure levels.
An unconventional perturbation treatment of the theory of two-photon absorption in a four-level atomic system close to two-photon resonance is developed on the basis of the optical Bloch equations. The results for the longtime populations of the intermediate and upper states are valid in low- and medium-intensity regimes. In both regimes the upper state population shows a lorentzian resonance centred on zero two-photon detuning with a width equal to the natural linewidth of the upper state. The intermediate state population shows a similar behaviour in the intermediate regime only, with no significant dependence on two-photon detuning in the low-intensity regime. However, in both regimes the population of the virtual intermediate state is at least comparable to that of the resonant upper state. The perturbation theory results have been checked by direct numerical calculations based on the Bloch equations. Numerical calculations confirm the presence of a high-intensity regime characterized by power shifts, broadening and saturation effects.