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Journal of Physics B Atomic Molecular and Optical Physics

Coherence effects in frequency-modulated laser spectroscopy

Gian Bartolo Picotto, Vladimir Wataghin

Published Jun 14, 1992• 1 citation
Publisher

Abstract

The purpose of this work was to investigate coherence effects in the interaction of two counterpropagating electromagnetic waves with the atoms of a gas, in the case in which one of the waves is weak and frequency modulated, while the other is sufficiently intense to cause saturation. Solving the equations for the density matrix in first-order perturbation theory, the authors obtained in analytic form the real part of the susceptibility integrated over the atom's velocity, expressed in terms of the rate equation model (REM) predictions plus the coherence corrections. The relaxation constants of the levels involved in the transition were kept different in order to allow a wider range of applications of the results. Calculated line shapes related either to the frequency-modulated (FM) sideband or to the ordinary dispersion techniques are presented. The REM approximation and the coherence corrections are shown in detail for an optical line which is of interest for frequency standard investigations.

DOI
10.1088/0953-4075/25/11/006

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The purpose of this work was to investigate coherence effects in the interaction of two counterpropagating electromagnetic waves with the atoms of a gas, in the case in which one of the waves is weak and frequency modulated, while the other is sufficiently intense to cause saturation. Solving the equations for the density matrix in first-order perturbation theory, the authors obtained in analytic form the real part of the susceptibility integrated over the atom's velocity, expressed in terms of the rate equation model (REM) predictions plus the coherence corrections. The relaxation constants of the levels involved in the transition were kept different in order to allow a wider range of applications of the results. Calculated line shapes related either to the frequency-modulated (FM) sideband or to the ordinary dispersion techniques are presented. The REM approximation and the coherence corrections are shown in detail for an optical line which is of interest for frequency standard investigations.
We develop a field-nonlinear theory of sub-Doppler spectroscopy in a gas of two-level atoms, based on a self-consistent solution of the Maxwell-Bloch equations in the mean field and single-atom density matrix approximations. This makes it possible to correctly take into account the effects caused by the free motion of atoms in a gas, which lead to a nonlinear dependence of the spectroscopic signal on the atomic density even in the absent of a direct interatomic interaction (e.g., dipole-dipole interaction). Within the framework of this approach, analytical expressions for the light field were obtained for an arbitrary number of resonant waves and arbitrary optical thickness of a gas medium. Sub-Doppler spectroscopy in the transmission signal for two counterpropagating and co-propagating waves has been studied in detail. A previously unknown red shift of a narrow sub-Doppler resonance is predicted in a counterpropagating waves scheme, when the frequency of one wave is fixed and the frequency of the other wave is varied. The magnitude of this shift depends on the atomic density and can be more than an order of magnitude greater than the known shift from the interatomic dipole-dipole interaction (Lorentz-Lorenz shift). The found effects, caused by the free motion of atoms, require a significant revision of the existing picture of spectroscopic effects depending on the density of atoms in a gas. Apart of fundamental aspect, obtained results are important for precision laser spectroscopy and optical atomic clocks.
We develop a field-nonlinear theory of sub-Doppler spectroscopy in a gas of two-level atoms, based on a self-consistent solution of the Maxwell-Bloch equations in the mean-field and single-atom density-matrix approximations. This makes it possible to correctly take into account the effects caused by the free motion of atoms in a gas, which lead to a nonlinear dependence of the spectroscopic signal on the atomic density even in the absence of a direct interatomic interaction (e.g., dipole-dipole interaction). Within the framework of this approach, analytical expressions for the light field were obtained for an arbitrary number of resonant waves and arbitrary optical thickness of a gas medium. Sub-Doppler spectroscopy in the transmission signal for two counterpropagating and copropagating waves has been studied in detail. A previously unknown redshift of a narrow sub-Doppler resonance is predicted in a counterpropagating waves scheme, when the frequency of one wave is fixed and the frequency of the other wave is varied. The magnitude of this shift depends on the atomic density and can be more than an order of magnitude greater than the known shift from the interatomic dipole-dipole interaction (Lorentz-Lorenz shift). The found effects, caused by the free motion of atoms, require a significant revision of the existing picture of spectroscopic effects depending on the density of atoms in a gas. Apart from the fundamental aspect, obtained results are important for precision laser spectroscopy and optical atomic clocks.
The propagation of a weakly modulated light beam through a nonlinear material is treated. The optical field representing such a beam consists of a strong carrier-frequency component and two weak, symmetrically displaced sidebands that combine to form a field, which may be purely amplitude modulated (AM), frequency modulated (FM), or some combination of these two modulation forms. It is shown that, for any optical nonlinearity, two modulation forms exist that have the property that the form of modulation is invariant under propagation of the beam. If a modulation form other than one of these natural modes is injected into the nonlinear medium, the modulation form will change as the beam propagates, asymptotically approaching the natural mode that experiences the lower attenuation. Explicit expressions for these natural modes are presented for the case in which the nonlinear medium can be modeled as a collection of two-level atoms. For the special case of an on-resonance pump beam, the natural modes correspond to pure amplitude modulation and pure frequency modulation. This formalism provides a general description of saturation spectroscopy for both AM and FM fields. Formulas are derived for the rate at which a pure FM beam is converted to an AM beam owing to its interaction with a two-level atomic medium. We also consider the variation that is due to propagation of the depth of modulation of an AM wave interacting resonantly with two-level atoms. The formalism predicts the existence of spectral features whose shape depends sensitively on the internal relaxation processes of the material. Experimental spectra are presented for ruby, alexandrite, and fluorescein in glass and are interpreted.
A novel phase-modulation technique which permits subkilohertz-laser stability and new levels of precision in laser spectroscopy was reported recently. For spectroscopy, the basic arrangement consists of a combination of an optical pump and a probe field which is phase modulated. The pump prepares the atomic sample by burning a narrow hole within the atom's inhomogeneous line shape, and the probe beam samples the prepared hole when its modulation sidebands are swept into resonance. Off resonance, the probe is balanced as pairs of sidebands produce heterodyne beat signals of opposite phase which just cancel. On resonance, the balance is upset and yields a nonvanishing beat signal with a Lorentzian absorption or dispersion line shape and with residual noise approaching the shot noise limit. Here we investigate the theory of phase-modulation spectroscopy. We treat the nonlinear response of an atomic two-level quantum system subject to an intense pump and a weak copropagating or counterpropagating phase-modulated probe beam. The density-matrix equations of motion are solved by a Laplace-transform method and by the novel use of a translation operator which allows the infinite hierarchy of coupled equations to close. A solution equivalent to the rate-equation result is developed and coherence corrections are found which predict new resonances that have just been detected in this laboratory. The delayed pump-probe technique encountered in solid-state laser spectroscopy is analyzed in this context for two- and three-level quantum systems. The response of a Fabry-Perot cavity to a phase-modulated light wave is examined also and reveals an unexpected absorption feature.
The nonlinear interaction of two intense optical waves with three energy levels of an atomic system is analyzed theoretically. The frequencies of the applied fields are assumed to be on or near two of the resonances involving the three levels. Particular attention is given to the generation of power at the difference frequency of the two applied fields where their frequency separation lies in the far-infrared range of the spectrum. In the limit of large input fields where atomic transitions are driven to saturation, the generated difference-frequency power is no longer proportional to the product of the two input powers. It is shown that, in this limit, the total rate of emission of photons at the difference frequency may, in principle, approach a value given by rNT\frac{\mathrm{rN}}{T}TrN​, where NNN is the total number of atoms, TTT is the average time of coherent interaction of atoms and the radiation field, and rrr is a factor which in favorable cases approaches unity at low temperatures. Also, a number of additional effects are encountered due to coherent double-quantum transitions among various levels which may be ignored only for weak input powers. These effects are highly dependent on the modes of relaxations of atomic levels and the details of the line shape. The results are applied to several cases, in which the resonance condition yields large efficiency in nonlinear frequency mixing.
The interaction of a three-level atom in the A configuration with a frequency-modulated (FM) field is analyzed theoretically for the first time. The two-frequency model for solving the problem of frequency modulation spectroscopy of coherent dark resonances is described and analyzed for a three-level A system. The effectiveness of the two-frequency model is demonstrated by comparing the results obtained using this model with the results of solving the exact problem of interaction of a A system with an FM field, which can be solved by the density matrix method. It is shown that the simple two-frequency model corresponds to the exact solution and is in qualitative agreement with experimental data.