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Physical Review A

Resonant nearly degenerate backward four-wave mixing in open and closed systems

Duncan G. Steel, J. T. Remillard

Published Nov 1, 1987• 37 citations
Publisher

Abstract

This paper presents perturbation calculations using the density matrix to describe the nearly degenerate four-wave-mixing spectral response in resonant systems subject to state-specific reservoir coupling. The calculations provide physical insight into the origin of narrow resonances observed in recent experimental work in atomic sodium and ultranarrow resonances in crystals. The calculations are for systems which are homogeneously broadened as well as systems which are inhomogeneously broadened due to velocity effects or random crystal fields. The calculations show that it is possible to eliminate the effects of either kind of broadening and measure the decay rate of the dipole coherence as well as the decay rates of both states of the transition. Furthermore, the calculations indicate that the method of cross-correlated optical fields which was used to eliminate the effects of laser jitter in longitudinal relaxation rate measurements could be extended to the measurements of slow dephasing rates in crystals.

DOI
10.1103/physreva.36.4330

Related papers

Effects of state-specific reservoir coupling on four-wave-mixing spectroscopy.

Jing Liu

The purpose of this research is to study the dynamic behavior of a resonant system using four-wave-mixing spectroscopy (FWM). From this study, we have obtained rich information about the FWM spectral response of systems which have different state-specific reservoir couplings. Atomic sodium (Na) is used as our model system. It is shown that the FWM spectral response depends strongly on how each level couples to the vacuum radiation field (through spontaneous emission). This effect is illustrated in the line shape obtained from a nearly degenerate FWM (NDFWM) experiment for a simple two-level system in Na. A narrow spike on top of a wide resonance is observed at zero pump-probe detuning ($\delta$ = 0) when the upper-state decay rate $\gamma\sb2$ to the reservoir is larger than the lower-state decay rate $\gamma\sb1$. The width of the spike is 2$\gamma\sb1$ and the width of the wide resonance is determined by the total upper-level decay rate (2$\gamma\sbsp{2}{\rm t}$). From the NDFWM spectrum, all the decay rates characterizing a two-level system may be measured directly. If magnetic substate degeneracy is included, the NDFWM line shape has contributions arising from orientation and alignment gratings, as well as from population gratings. For a two-level system closed with respect to population, the orientation or alignment gratings may contribute a narrow resonance at $\delta$ = 0 in the NDFWM response. These features are demonstrated for the first time by using NDFWM. Reservoir coupling can also occur through collisions with externally added gas. Collisions can cause a closed two-level system to become open, resulting in NDFWM spectra that show a narrow resonance at $\delta$ = 0 with a width below the residual Doppler width (Dicke narrowing). This is also the first time that Dicke narrowing has been observed in population terms. Finally, we present a calculation of the NDFWM response for strong degenerate pump fields and a weak probe field. We compare the differences between NDFWM spectra of closed two- and three-level systems in the presence of strong pumps.

•Jan 1, 1989•0 citations

Decay induced narrow resonances in backward nearly degenerate four-wave mixing spectroscopy

Jing Liu, J. T. Remillard, Duncan G. Steel

We present results showing narrow resonant behavior in a collisionless four-wave mixing interaction in atomic sodium. In a two-level system the excited state radiatively decays to the ground state at rate ${\ensuremath{\gamma}}_{\mathrm{sp}}$, and the two states decay to the reservoir at ${\ensuremath{\gamma}}_{1}$ and ${\ensuremath{\gamma}}_{2}$. If ${\ensuremath{\gamma}}_{1}$ and ${\ensuremath{\gamma}}_{2}$ equal the transit time, the resonance width is determined by the total decay rate out of state 2. In the presence of additional excited-state decay to another state, a narrow resonance with width ${\ensuremath{\gamma}}_{1}$ is oberved as a result of the effect of the additional excited-state decay on the magnitude of the ground-state probability amplitude.

Physical Review Letters•Aug 17, 1987•8 citations

Effects of population pulsations on backward, nearly degenerate four-wave mixing spectroscopy and optical phase conjugation

Duncan G. Steel, Stephen Colby Rand, J. T. Remillard +1 more

We discuss the effects of population pulsations on the spectral response of backward nearly degenerate four-wave mixing (NDFWM). We include the effects of state-specific decay and spontaneous emission on both two-level and inverted-v-type three-level systems. We compare the results with experiments in both the gas phase and the condensed phase that show ultranarrow resonances. In addition, we examine the response in the presence of strong pump fields. We show that in a simple two-level system, the NDFWM response results in an ac-Stark splitting of the spectral response, whereas in a simple three-level system with decay to a metastable state, a single narrow resonance is observed, which broadens in the presence of strong pumps.

Journal of the Optical Society of America B•Jan 1, 1988•1 citation

Resonant degenerate four-wave mixing spectroscopy of transitions with degenerate energy levels: Saturation and polarization effects

Thomas A. Reichardt, Robert P. Lucht

The physics of the degenerate four-wave mixing process for resonant transitions between two degenerate energy levels is investigated by direct numerical integration of the time-dependent density matrix equations. The Zeeman structure of the upper and lower energy levels is included in a multistate formulation of the density matrix equations. The inclusion of the Zeeman structure enables the investigation of the degenerate four-wave mixing process for different polarization configurations of the forward pump, backward pump, and probe beams. Saturation curves and lineshapes are calculated for different polarization configurations and for numerous low-J transitions. At low laser intensity, the results of our calculations are in excellent agreement with perturbation theory in terms of the relative intensities of the degenerate four-wave mixing signal for linear polarization configurations. As the laser intensity increases and the resonance starts to saturate, we find in general that the relative degenerate four-wave mixing reflectivity increases for the crossed polarization configurations compared to the parallel polarization configuration because the saturation intensity is higher. However, for some resonance transitions, some of the crossed polarization configurations saturate at lower laser intensities than the parallel polarization configuration, even though the reflectivity for these crossed polarization configurations is much lower than for the parallel polarization configuration in the perturbative intensity limit. This result is explained in terms of the coupling of the various Zeeman states during the degenerate four-wave mixing interaction for specific polarization configurations. The effect of saturation on the resonance line shapes for the different polarization configurations is also investigated. Finally, a limited number of calculations are performed for resonances that are Doppler broadened as well as collision broadened. The effect of saturation on the reflectivity of the crossed polarization configurations compared to the parallel polarization configuration is even more significant for resonances with comparable Doppler and collisional broadening.

The Journal of Chemical Physics•Dec 8, 1999•22 citations

Interaction of closely spaced resonances in degenerate four-wave-mixing spectroscopy

Thomas A. Reichardt, Robert P. Lucht

The interaction of two closely spaced resonances in degenerate four-wave-mixing spectra is investigated theoretically. Degenerate four-wave-mixing spectra for the phase-conjugate geometry (counterpropagating pump beams) are calculated by integrating the time-dependent density-matrix equations at numerous grid points along the phase-matching axis and summing the polarization contribution from each of these grid points. Both purely homogeneously broadened resonances and resonances that are both collision and Doppler broadened are considered. The homogeneously broadened results are found to agree with the results of the widely used Abrams–Lind expression for the case of stationary absorbers exposed to a steady-state excitation. We examine the effects of power broadening, pressure broadening, transition dipole moment, and population on the spectra of two neighboring homogeneously broadened resonances. As the laser power is increased, the real parts of the resonance susceptibilities destructively interfere with each other in the region between the lines, resulting in a significant and persistent dip in reflectivity. When Doppler broadening is included in the calculations, similar interference effects are observed, although at first the interference dip begins to disappear until the homogeneous part of the line shapes is broadened significantly by saturation. Using this numerical approach, we obtain good agreement with experimental results of primarily Doppler-broadened neighboring resonances.

Journal of the Optical Society of America B•Oct 1, 1997•12 citations

Collision Effects on Zeeman Coherences Using Nearly Degenerate Four-Wave Mixing

Jenny Lam, Duncan G. Steel, R. A. McFarlane

Recently, we have shown that the nearly degenerate four-wave mixing (NDFWM) process can provide a direct and simultaneous measurement of the longitudinal (1/T1) and transverse (1/T2) relaxation rates of a resonant two-level atom. Furthermore, we observed that in the presence of buffer gases, the linewidth of the resonance describing the longitudinal relaxation process experiences a significant narrowing, belowo that of the natural width. Using a theoretical model, the narrowing was accounted for as arising from collisional effects in the ground-state population. 1 These observations were made for the case where the radiation fields are co-polarized, and we concluded that NDFWM offers a unique technique for the measurement of the ground-state velocity-changing cross section.2

Journal of the Optical Society of America B•Jan 1, 1984•0 citations
This paper presents perturbation calculations using the density matrix to describe the nearly degenerate four-wave-mixing spectral response in resonant systems subject to state-specific reservoir coupling. The calculations provide physical insight into the origin of narrow resonances observed in recent experimental work in atomic sodium and ultranarrow resonances in crystals. The calculations are for systems which are homogeneously broadened as well as systems which are inhomogeneously broadened due to velocity effects or random crystal fields. The calculations show that it is possible to eliminate the effects of either kind of broadening and measure the decay rate of the dipole coherence as well as the decay rates of both states of the transition. Furthermore, the calculations indicate that the method of cross-correlated optical fields which was used to eliminate the effects of laser jitter in longitudinal relaxation rate measurements could be extended to the measurements of slow dephasing rates in crystals.
The purpose of this research is to study the dynamic behavior of a resonant system using four-wave-mixing spectroscopy (FWM). From this study, we have obtained rich information about the FWM spectral response of systems which have different state-specific reservoir couplings. Atomic sodium (Na) is used as our model system. It is shown that the FWM spectral response depends strongly on how each level couples to the vacuum radiation field (through spontaneous emission). This effect is illustrated in the line shape obtained from a nearly degenerate FWM (NDFWM) experiment for a simple two-level system in Na. A narrow spike on top of a wide resonance is observed at zero pump-probe detuning (δ\deltaδ = 0) when the upper-state decay rate $\gamma\sb2$ to the reservoir is larger than the lower-state decay rate $\gamma\sb1$. The width of the spike is 2$\gamma\sb1$ and the width of the wide resonance is determined by the total upper-level decay rate (2$\gamma\sbsp{2}{\rm t}$). From the NDFWM spectrum, all the decay rates characterizing a two-level system may be measured directly. If magnetic substate degeneracy is included, the NDFWM line shape has contributions arising from orientation and alignment gratings, as well as from population gratings. For a two-level system closed with respect to population, the orientation or alignment gratings may contribute a narrow resonance at δ\deltaδ = 0 in the NDFWM response. These features are demonstrated for the first time by using NDFWM. Reservoir coupling can also occur through collisions with externally added gas. Collisions can cause a closed two-level system to become open, resulting in NDFWM spectra that show a narrow resonance at δ\deltaδ = 0 with a width below the residual Doppler width (Dicke narrowing). This is also the first time that Dicke narrowing has been observed in population terms. Finally, we present a calculation of the NDFWM response for strong degenerate pump fields and a weak probe field. We compare the differences between NDFWM spectra of closed two- and three-level systems in the presence of strong pumps.
We present results showing narrow resonant behavior in a collisionless four-wave mixing interaction in atomic sodium. In a two-level system the excited state radiatively decays to the ground state at rate ${\ensuremath{\gamma}}_{\mathrm{sp}}$, and the two states decay to the reservoir at ${\ensuremath{\gamma}}_{1}$ and ${\ensuremath{\gamma}}_{2}$. If ${\ensuremath{\gamma}}_{1}$ and ${\ensuremath{\gamma}}_{2}$ equal the transit time, the resonance width is determined by the total decay rate out of state 2. In the presence of additional excited-state decay to another state, a narrow resonance with width ${\ensuremath{\gamma}}_{1}$ is oberved as a result of the effect of the additional excited-state decay on the magnitude of the ground-state probability amplitude.
We discuss the effects of population pulsations on the spectral response of backward nearly degenerate four-wave mixing (NDFWM). We include the effects of state-specific decay and spontaneous emission on both two-level and inverted-v-type three-level systems. We compare the results with experiments in both the gas phase and the condensed phase that show ultranarrow resonances. In addition, we examine the response in the presence of strong pump fields. We show that in a simple two-level system, the NDFWM response results in an ac-Stark splitting of the spectral response, whereas in a simple three-level system with decay to a metastable state, a single narrow resonance is observed, which broadens in the presence of strong pumps.
The physics of the degenerate four-wave mixing process for resonant transitions between two degenerate energy levels is investigated by direct numerical integration of the time-dependent density matrix equations. The Zeeman structure of the upper and lower energy levels is included in a multistate formulation of the density matrix equations. The inclusion of the Zeeman structure enables the investigation of the degenerate four-wave mixing process for different polarization configurations of the forward pump, backward pump, and probe beams. Saturation curves and lineshapes are calculated for different polarization configurations and for numerous low-J transitions. At low laser intensity, the results of our calculations are in excellent agreement with perturbation theory in terms of the relative intensities of the degenerate four-wave mixing signal for linear polarization configurations. As the laser intensity increases and the resonance starts to saturate, we find in general that the relative degenerate four-wave mixing reflectivity increases for the crossed polarization configurations compared to the parallel polarization configuration because the saturation intensity is higher. However, for some resonance transitions, some of the crossed polarization configurations saturate at lower laser intensities than the parallel polarization configuration, even though the reflectivity for these crossed polarization configurations is much lower than for the parallel polarization configuration in the perturbative intensity limit. This result is explained in terms of the coupling of the various Zeeman states during the degenerate four-wave mixing interaction for specific polarization configurations. The effect of saturation on the resonance line shapes for the different polarization configurations is also investigated. Finally, a limited number of calculations are performed for resonances that are Doppler broadened as well as collision broadened. The effect of saturation on the reflectivity of the crossed polarization configurations compared to the parallel polarization configuration is even more significant for resonances with comparable Doppler and collisional broadening.
The interaction of two closely spaced resonances in degenerate four-wave-mixing spectra is investigated theoretically. Degenerate four-wave-mixing spectra for the phase-conjugate geometry (counterpropagating pump beams) are calculated by integrating the time-dependent density-matrix equations at numerous grid points along the phase-matching axis and summing the polarization contribution from each of these grid points. Both purely homogeneously broadened resonances and resonances that are both collision and Doppler broadened are considered. The homogeneously broadened results are found to agree with the results of the widely used Abrams–Lind expression for the case of stationary absorbers exposed to a steady-state excitation. We examine the effects of power broadening, pressure broadening, transition dipole moment, and population on the spectra of two neighboring homogeneously broadened resonances. As the laser power is increased, the real parts of the resonance susceptibilities destructively interfere with each other in the region between the lines, resulting in a significant and persistent dip in reflectivity. When Doppler broadening is included in the calculations, similar interference effects are observed, although at first the interference dip begins to disappear until the homogeneous part of the line shapes is broadened significantly by saturation. Using this numerical approach, we obtain good agreement with experimental results of primarily Doppler-broadened neighboring resonances.
Recently, we have shown that the nearly degenerate four-wave mixing (NDFWM) process can provide a direct and simultaneous measurement of the longitudinal (1/T1) and transverse (1/T2) relaxation rates of a resonant two-level atom. Furthermore, we observed that in the presence of buffer gases, the linewidth of the resonance describing the longitudinal relaxation process experiences a significant narrowing, belowo that of the natural width. Using a theoretical model, the narrowing was accounted for as arising from collisional effects in the ground-state population. 1 These observations were made for the case where the radiation fields are co-polarized, and we concluded that NDFWM offers a unique technique for the measurement of the ground-state velocity-changing cross section.2