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Physical Review A

Line shape and frequency shift of Lamb dip and crossover-resonance dip in closely spaced transitions

Swapan Kumar Mandal, Pradip Narayan Ghosh

Published Apr 1, 1992• 29 citations
Publisher

Abstract

The line shape of a pair of Doppler-broadened closely spaced transitions sharing a common lower level and under the effect of a standing-wave monochromatic radiation field is studied. The eight steady-state optical Bloch equations are solved analytically to obtain the polarization by retaining terms up to the third power in the electric-field amplitude. The absorption coefficient derived for the standing-wave case consists of two Lamb dips for the two allowed transitions at ${\mathrm{\ensuremath{\omega}}}_{1}$ and ${\mathrm{\ensuremath{\omega}}}_{2}$ and a crossover resonance dip at the intermediate frequency ${\mathrm{\ensuremath{\omega}}}_{\mathit{c}}$=(${\mathrm{\ensuremath{\omega}}}_{1}$+${\mathrm{\ensuremath{\omega}}}_{2}$)/2. The Lamp-dip peak frequencies show intensity-dependent shifts. Numerical computations are presented for different values of Rabi frequencies and relaxation linewidth parameters, which are small compared to the frequency difference \ensuremath{\Delta}=${\mathrm{\ensuremath{\omega}}}_{2}$-${\mathrm{\ensuremath{\omega}}}_{1}$. The computed line shape and frequency shift are compared with the observed results in the Balmer ${\mathrm{H}}_{\mathrm{\ensuremath{\alpha}}}$ line of hydrogen and the infrared Zeeman spectra of methane.

DOI
10.1103/physreva.45.4990

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The line shape of a pair of Doppler-broadened closely spaced transitions sharing a common lower level and under the effect of a standing-wave monochromatic radiation field is studied. The eight steady-state optical Bloch equations are solved analytically to obtain the polarization by retaining terms up to the third power in the electric-field amplitude. The absorption coefficient derived for the standing-wave case consists of two Lamb dips for the two allowed transitions at \ensuremathω1{\mathrm{\ensuremath{\omega}}}_{1}\ensuremathω1​ and \ensuremathω2{\mathrm{\ensuremath{\omega}}}_{2}\ensuremathω2​ and a crossover resonance dip at the intermediate frequency \ensuremathωc{\mathrm{\ensuremath{\omega}}}_{\mathit{c}}\ensuremathωc​=(\ensuremathω1{\mathrm{\ensuremath{\omega}}}_{1}\ensuremathω1​+\ensuremathω2{\mathrm{\ensuremath{\omega}}}_{2}\ensuremathω2​)/2. The Lamp-dip peak frequencies show intensity-dependent shifts. Numerical computations are presented for different values of Rabi frequencies and relaxation linewidth parameters, which are small compared to the frequency difference \ensuremath{\Delta}=\ensuremathω2{\mathrm{\ensuremath{\omega}}}_{2}\ensuremathω2​-\ensuremathω1{\mathrm{\ensuremath{\omega}}}_{1}\ensuremathω1​. The computed line shape and frequency shift are compared with the observed results in the Balmer H\ensuremathα{\mathrm{H}}_{\mathrm{\ensuremath{\alpha}}}H\ensuremathα​ line of hydrogen and the infrared Zeeman spectra of methane.
A theory of line shape for a Doppler broadened probe in the presence of a strong infrared pump is presented. The optical analog of Bloch equations has been developed; the T1, T2 relaxations are introduced phenomenologically in these equations. Doppler broadening of both the pump and signal transitions are taken into account to obtain the absorption coefficient for the signal. This leads to a Doppler broadened Gaussian function weighted by a pump induced Lorentzian function whose position and linewidth are controlled by the detuning and power respectively of the pump radiation. The absorption coefficient is further modulated by a Lorentzian function containing the fourth power of the radiation. The off-resonance pumping leads to two symmetrically located ‘‘Lamb dips’’ or ‘‘peaks’’ around the Gaussian pedestal for co- and counter propagating laser radiations. A calculation of the line shape of the central sub-Doppler feature for the on-resonance pumping is in good agreement with the observed variation of linewidth with pressure. Numerical results are presented graphically to demonstrate the pump and signal power dependence of the central hole or peak for the bent and cascade-type double resonances. The results obtained confirm the expected behavior based on level populations.
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 semiclassical theory of two-photon Lamb dips is presented, when one of the radiations is in the radio-frequency region and the other is in the optical region. Perturbation solutions as well as an exact recurrence formula are discussed. For the latter case numerical results are given. The resonance line shape of the two-photon Lamb dip shows a variety of changes according to the experimental conditions. When both the optical and the rf fields are weak a line is of a dispersion form. When the rf field is increased, the line changes to a Lorentzian form. When the optical field is increased, the line is shifted and broadened. If the radio frequency is close to or less than the Doppler width of the gas, a two-photon Lamb dip is accompanied with satellite dips which correspond to molecules with nonzero velocity components along the propagation of the laser radiation. The intensity of a two-photon Lamb dip is expected to decrease rapidly with increasing radio frequency. Additional effects when the sample cell is placed inside the laser cavity are also discussed. Those results agree qualitatively with recent observations by Fruend, R\omheld, and Oka.
A semiclassical theory of two-photon Lamb dips is presented, when one of the radiations is in the radio-frequency region and the other is in the optical region. Perturbation solutions as well as an exact recurrence formula are discussed. For the latter case numerical results are given. The resonance line shape of the two-photon Lamb dip shows a variety of changes according to the experimental conditions. When both the optical and the rf fields are weak a line is of a dispersion form. When the rf field is increased, the line changes to a Lorentzian form. When the optical field is increased, the line is shifted and broadened. If the radio frequency is close to or less than the Doppler width of the gas, a two-photon Lamb dip is accompanied with satellite dips which correspond to molecules with nonzero velocity components along the propagation of the laser radiation. The intensity of a two-photon Lamb dip is expected to decrease rapidly with increasing radio frequency. Additional effects when the sample cell is placed inside the laser cavity are also discussed. Those results agree qualitatively with recent observations by Fruend, R\"omheld, and Oka.
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