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Journal of Physics B Atomic Molecular and Optical Physics

Analytical solutions of the susceptibility for Doppler-broadened three-level atomic systems

Heung‐Ryoul Noh, Han Seb Moon

Published Nov 23, 2012• 5 citations
Publisher

Abstract

We present an analytical calculation of the susceptibility in Doppler-broadened three-level atomic systems using a diagrammatic method. The optical coherences responsible for the susceptibility of the probe field are obtained from the optical Bloch equations for Ladder (lower or upper transition probed), V and Λ-type three-level atoms up to the first (arbitrary) order in the probe (coupling) field Rabi frequency, and are then averaged over a Maxwell–Boltzmann velocity distribution. Finally, we obtain analytical solutions of the susceptibility for four different three-level atomic systems. In particular, the general expressions of the susceptibility are obtained where the wavelengths of the probe and coupling fields are different from each other.

DOI
10.1088/0953-4075/45/24/245002

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Journal of Atomic Molecular Condensate and Nano Physics•Aug 22, 2016•1 citation
We present an analytical calculation of the susceptibility in Doppler-broadened three-level atomic systems using a diagrammatic method. The optical coherences responsible for the susceptibility of the probe field are obtained from the optical Bloch equations for Ladder (lower or upper transition probed), V and Λ-type three-level atoms up to the first (arbitrary) order in the probe (coupling) field Rabi frequency, and are then averaged over a Maxwell–Boltzmann velocity distribution. Finally, we obtain analytical solutions of the susceptibility for four different three-level atomic systems. In particular, the general expressions of the susceptibility are obtained where the wavelengths of the probe and coupling fields are different from each other.
We present an analytical study of susceptibility in pump–probe laser spectroscopy for an atomic medium composed of two-level atoms in the Doppler-broadened limit. We derive an accurate analytical formula for susceptibility up to the first (arbitrary) order at the Rabi frequency of the probe (pump) beam. This is expressed in a succinct single term rather than as a complicated summation, as presented in previous papers. We also derive analytical solutions for the linewidth and peak value of the absorption spectrum of the probe beam.
The propagation of a probe field through a four-level Y-type atomic system is described in the presence of two additional coherent radiation fields, namely, the control field and the coupling field. An expression for the probe response is derived analytically from the optical Bloch equations under steady state condition to study the absorptive properties of the system under probe field propagation through an ensemble of stationary atoms as well as in a Doppler broadened atomic vapor medium. The most striking result is the conversion of electromagnetically induced transparency (EIT) into electromagnetically induced absorption (EIA) as we start switching from weak probe regime to strong probe regime. The dependence of this conversion on residual Doppler averaging due to wavelength mismatch is also shown by choosing the coupling transition as a Rydberg transition.
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.
We investigate the effect of the residual Doppler averaging on the susceptibility and optical switching of a weak probe field propagating through a four-level inverted-Y type atomic system driven by two strong coherent fields, namely the pump field (control field) and the repump field. A theoretical model of the inverted-Y type system is developed by using density matrix method and a set of sixteen optical Bloch equations (OBEs) are solved analytically under steady state condition to determine the probe coherence term. The imaginary part of susceptibility shows electromagnetically induced transparency (EIT) under Doppler free condition as well as under Doppler broadening. The line parameters of 5S1/2 → 5P3/2 → 5D5/2 and 5S1/2 → 5P3/2 → 7S1/2 transitions executed by 87Rb atoms are used to simulate the susceptibility and group index related to probe field propagation in order to show the effect of residual Doppler broadening due to wavelength mismatch between the pump field and the probe field. Under Doppler free condition the probe field propagation is subluminal on resonance while that under Doppler broadened condition is superluminal on resonance for both the transitions.
A density matrix based complete analytical solution for the probe transmission through a three-level cascade type Doppler free atomic system is obtained. The probe beam couples the ground level to the energy level lying at the middle whereas the control field is acting between the level at the middle and the upper most energy level. The analysis is valid for comparable values of probe and control Rabi frequencies too. The probe response shows EIT window when Raman condition is satisfied. The variation of the EIT width with control Rabi frequency under Doppler free condition is studied using the parameters of 5S1/2→5P3/2→5D5/25S_{1/2}\rightarrow 5P_{3/2}\rightarrow 5D_{5/2}5S1/2​→5P3/2​→5D5/2​ transitions of 87^{87}87Rb. At high value of the probe field intensity compared to that of the control field, enhancement in probe absorption is noticed. The increase of the EIT width with control Rabi frequency under Doppler free situation is examined.