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IEEE Journal of Quantum Electronics

Density matrix solutions for the susceptibilities of a three-level system with arbitrary relaxation rates and field strengths

J. Ryan, N. M. Lawandy

Published Nov 1, 1986• 12 citations
Publisher

Abstract

The susceptibilities for a three-level system with arbitrary pump and signal field strengths are derived for the case of arbitrary longitudinal and transverse relaxation rates.
DOI
10.1109/jqe.1986.1072911

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It is well known that the presence of a real atomic level which is nearly resonant with the pump field can greatly enhance the Raman emission cross section. In order to accurately calculate the Raman gain in systems where resonance enhancement plays a dominant role, expressions for the pump and signal susceptibilities must be derived. These expressions should be valid for arbitrary field strengths in order to allow for pump and signal saturation. In addition, the theory should allow for arbitrary longitudinal and transverse relaxation rates. This latter point is extremely vital for three level atomic systems such as the alkali earth metals since they do not have population reservoirs and can have widely varying spontaneous lifetimes on the three pertinent transitions. Moreover, the dephasing rates are strong functions of electron states and are therefore also different for the three coupled pairs of levels. These considerations are not as important when molecular systems are concerned since the large reservoir of rotational states serve to produce essentially equal longitudinal recovery rates for the population of the three levels. The three level system with three arbitrary longitudinal and transverse relaxation rates was solved. There is no need for setting either pair of rates equal and the expressions are valid for arbitrarily strong fields.
This work focused on understanding the effects of arbitrary transverse and longitudinal relaxation rates on the susceptibilities of coherently driven three-level systems. The approximation of a single relaxation rate often made in previous work is strongly invalidated by the variation in the spontaneous emission lifetime between various atomic level pairs in systems such as cesium. It is of great importance to the problem of nonlinear infrared generation to determine the dependence of both real and imaginary susceptibility on relaxation rates. The imaginary susceptibility on the pump transition determines the absorption of pump photons and the imaginary susceptibility on the laser transition determines the spectral dependence of the gain. This is of particular importance for pure Raman emission (i.e., absorption at linecenter of the gain transition) as it determines the tunability characteristics we are aiming to predict. The real susceptibility is important when cavities are used at the signal field as this will determine the loaded resonance of the Raman oscillator. Researchers show that in some cases which result from having different relaxation rates mode splitting may result, allowing more than one frequency to have the same Raman wavelength, possibly resulting in a temporal instability.
Nuclear slowing-down factors account for the sharing of angular momentum between the electron and the nucleus in the effective Bloch description of alkali-metal spin dynamics in the spin-exchange-relaxation-free (SERF) regime. For collinear optical pumping and magnetic field, we find that longitudinal and transverse spin dynamics are characterized by different polarization-dependent slowing-down factors. Transverse dynamics are governed by the conventional factor q(p)q(p)q(p), with ppp the electron spin polarization, whereas longitudinal relaxation is governed by q(p)+p dq(p)/dpq(p)+p\,dq(p)/dpq(p)+pdq(p)/dp. The two factors coincide at zero polarization but differ substantially at high polarization, where the single-factor description overestimates the underlying pumping and relaxation rates inferred from longitudinal transients by factors approaching two to four, depending on the nuclear spin. We further derive closed-form expressions, valid at arbitrary polarization, for the residual spin-exchange relaxation of the transverse spin components at finite magnetic field along the pumping axis. This contribution is quadratic in the magnetic field and can become comparable to the zero-field linewidth at fields well within the SERF regime. The results are obtained by perturbation theory on the density-matrix dynamics linearized around the stationary state and are verified by numerical solutions of the microscopic density-matrix equation. These findings refine the effective Bloch description of alkali-metal spin dynamics and are directly relevant to atomic magnetometers and alkali-metal--noble-gas comagnetometers, including those used in precision searches for physics beyond the Standard Model.