arXiv:1412.7137 · Physical Review A

Interference and nonlinear properties of four-wave-mixing resonances in thermal vapor: Analytical results and experimental verification

Michal P. Parniak, Wojciech Wasilewski

Published Feb 17, 2015• 34 citations

Abstract

We develop a model to calculate nonlinear polarization in a nondegenerate four-wave mixing in diamond configuration which includes the effects of hyperfine structure and Doppler broadening. We verify the model against the experiment with $5{\phantom{\rule{0.16em}{0ex}}}^{2}{S}_{1/2},5{\phantom{\rule{0.16em}{0ex}}}^{2}{P}_{3/2},5{\phantom{\rule{0.16em}{0ex}}}^{2}{D}_{3/2}$, and $5{\phantom{\rule{0.16em}{0ex}}}^{2}{P}_{1/2}$ levels of rubidium 85. Treating the multilevel atomic system as a combination of many four-level systems we are able to express the nonlinear susceptibility of a thermal ensemble in a low-intensity regime in terms of Voigt-type profiles and obtain an excellent conformity of theory and experiment within this complex system. The agreement is also satisfactory at high intensity and the analytical model correctly predicts the positions and shapes of resonances. Our results elucidate the physics of coherent interaction of light with atoms involving higher excited levels in vapors at room temperature, which is used in an increasing range of applications.

DOI: 10.1103/physreva.91.023418

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arXiv:1412.7137 · Physical Review A

Interference and nonlinear properties of four-wave-mixing resonances in thermal vapor: Analytical results and experimental verification

Michal P. Parniak, Wojciech Wasilewski

Published Feb 17, 2015• 34 citations

arXiv HTML PDF

Abstract

5{\phantom{\rule{0.16em}{0ex}}}^{2}{S}_{1/2},5{\phantom{\rule{0.16em}{0ex}}}^{2}{P}_{3/2},5{\phantom{\rule{0.16em}{0ex}}}^{2}{D}_{3/2}

, and

5{\phantom{\rule{0.16em}{0ex}}}^{2}{P}_{1/2}

levels of rubidium 85. Treating the multilevel atomic system as a combination of many four-level systems we are able to express the nonlinear susceptibility of a thermal ensemble in a low-intensity regime in terms of Voigt-type profiles and obtain an excellent conformity of theory and experiment within this complex system. The agreement is also satisfactory at high intensity and the analytical model correctly predicts the positions and shapes of resonances. Our results elucidate the physics of coherent interaction of light with atoms involving higher excited levels in vapors at room temperature, which is used in an increasing range of applications.

DOI: 10.1103/physreva.91.023418

Related papers

Nondegenerate four-wave mixing in rubidium vapor: The diamond configuration

F. E. Becerra, R. T. Willis, S. L. Rolston +1 more

We investigate experimentally and theoretically nondegenerate four-wave mixing in rubidium vapor using a diamond configuration ( $5{S}_{1/2}$ , $5{P}_{1/2}$ , $5{P}_{3/2}$ , and $6{S}_{1/2}$ levels). We model this process of three classical fields interacting with a four-level atomic system with the Schr\"odinger equation. The experimental results are in good agreement with theoretical predictions for the generated light as a function of detuning.We investigate experimentally and theoretically nondegenerate four-wave mixing in rubidium vapor using a diamond configuration ($5{S}_{1/2}$, $5{P}_{1/2}$, $5{P}_{3/2}$, and $6{S}_{1/2}$ levels). We model this process of three classical fields interacting with a four-level atomic system with the Schr\"odinger equation. The experimental results are in good agreement with theoretical predictions for the generated light as a function of detuning.

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Forward four-wave mixing experiments in hot and cold atomic systems

Alexandre Mendes de [UNESP] Almeida Junior

In this work, we present a set of experiments involving four-wave mixing processes in rubidium atoms. We employ several experimental configurations exploring two- and three-level systems in cascade or Λ configurations, all of which are in a copropagating geometry, to investigate different physical phenomena present in each scenario. In the first experiment, we combine a cw diode laser and a 1 GHz femtosecond pulse train to generate coherent blue light through the four-wave mixing process in hot rubidium vapor. The generated coherent signal is analyzed by scanning either the diode frequency or the repetition rate of the frequency comb. In both cases, the cw laser induces an Autler- Townes splitting with a wide separation in the doublet structure due to the configuration of copropagating fields. To model these results, we employ the density matrix formalism to write the Bloch equations and solve them numerically. The solutions allow us to explore other cases beyond those involving our experimental setup, comparing the response of the upper-level population and the generated signal coherence for homogeneously and non-homogeneously broadened systems. Moreover, in this same experiment, we also observed an unexpected interference effect between excitation routes of the four-wave mixing process when the repetition rate of the pulse train is scanned. The theoretical model demonstrates that the interfering effect appears only when the repetition rate is carefully chosen to produce a pair of modes of the frequency comb resonant with both one- and two-photon transitions for the same atomic velocity group. In the second experiment, still with a hot rubidium vapor, we explore two symmetrical degenerate four- wave mixing signals induced by two almost copropagating laser beams, with~ka and~kb wave-vectors, and detected simultaneously in the 2~ka −~kb and 2~kb −~ka directions. Each spectrum has a single peak with a short frequency separation between them, which our model indicates is due to the nonlinear behavior of the refractive indexes, a consequence of the high intensity of both input fields. Finally, we use the same experimental setup for the two symmetrical signals to study the field intensity fluctuations of the degenerate four-wave mixing signals and the transmission signals. In this case, the sample is an ensemble of cold rubidium atoms in a magneto-optical trap. There is an intensity- intensity cross-correlation between the signals due to the resonant phase-noise to amplitude-noise conversion. In this type of cold system, we can neglect the Doppler- broadening, allowing a straightforward study of the dependency of the correlations on the laser detuning. In these measurements, we observe a type of oscillation in the correlation curves that the theoretical model indicates to be connected with Rabi oscillations, revealed by the fluctuations long after the transient period.In this work, we present a set of experiments involving four-wave mixing processes in rubidium atoms. We employ several experimental configurations exploring two- and three-level systems in cascade or Λ configurations, all of which are in a copropagating geometry, to investigate different physical phenomena present in each scenario. In the first experiment, we combine a cw diode laser and a 1 GHz femtosecond pulse train to generate coherent blue light through the four-wave mixing process in hot rubidium vapor. The generated coherent signal is analyzed by scanning either the diode frequency or the repetition rate of the frequency comb. In both cases, the cw laser induces an Autler- Townes splitting with a wide separation in the doublet structure due to the configuration of copropagating fields. To model these results, we employ the density matrix formalism to write the Bloch equations and solve them numerically. The solutions allow us to explore other cases beyond those involving our experimental setup, comparing the response of the upper-level population and the generated signal coherence for homogeneously and non-homogeneously broadened systems. Moreover, in this same experiment, we also observed an unexpected interference effect between excitation routes of the four-wave mixing process when the repetition rate of the pulse train is scanned. The theoretical model demonstrates that the interfering effect appears only when the repetition rate is carefully chosen to produce a pair of modes of the frequency comb resonant with both one- and two-photon transitions for the same atomic velocity group. In the second experiment, still with a hot rubidium vapor, we explore two symmetrical degenerate four- wave mixing signals induced by two almost copropagating laser beams, with~ka and~kb wave-vectors, and detected simultaneously in the 2~ka −~kb and 2~kb −~ka directions. Each spectrum has a single peak with a short frequency separation between them, which our model indicates is due to the nonlinear behavior of the refractive indexes, a consequence of the high intensity of both input fields. Finally, we use the same experimental setup for the two symmetrical signals to study the field intensity fluctuations of the degenerate four-wave mixing signals and the transmission signals. In this case, the sample is an ensemble of cold rubidium atoms in a magneto-optical trap. There is an intensity- intensity cross-correlation between the signals due to the resonant phase-noise to amplitude-noise conversion. In this type of cold system, we can neglect the Doppler- broadening, allowing a straightforward study of the dependency of the correlations on the laser detuning. In these measurements, we observe a type of oscillation in the correlation curves that the theoretical model indicates to be connected with Rabi oscillations, revealed by the fluctuations long after the transient period.

Feb 28, 20230 citations

Resonant multiwave-mixing spectra of gas-phase sodium: Nonperturbative calculations

Robert P. Lucht, Rick P. Trebino, Larry A. Rahn

We perform nonperturbative calculations of resonant multiwave mixing of two high-intensity laser beams of different frequencies both for a general three-level system and for the 3 $^{2}$ ${\mathit{S}}_{1/2}$ --3 $^{2}$ ${\mathit{P}}_{1/2}$ electronic resonance of the sodium atom. Our calculations proceed by direct numerical integration of the density-matrix equations and subsequent Fourier analysis of the calculated time-dependent density-matrix elements. We examine the case where two nearly degenerate input beams are each detuned many collisional linewidths from electronic resonance, and the frequency difference between the input beams is tuned through a ground-state resonance. Unlike previous high-intensity analyses, each input beam can be resonant with numerous transitions simultaneously; for a three-level system, we refer to this as an ``M-type'' system, in analogy with \ensuremath{\Lambda}- and V-type systems. Calculated four-, six-, and eight-wave-mixing spectra exhibit subharmonic resonances of the type observed experimentally by Trebino and Rahn [Opt. Lett. 12, 912 (1987)]. The three-level system is studied to gain insight into the physics of these subharmonic resonances. For more quantitative comparison with experiment, we use a model that includes the 16 Zeeman and hyperfine states in the 3 $^{2}$ ${\mathit{S}}_{1/2}$ and the 3 $^{2}$ ${\mathit{P}}_{1/2}$ levels, and accounts for arbitrary linear polarization of the input laser radiation. The relative intensities of the resonance features in the calculated multiwave-mixing spectra show good agreement with experiment. These resonance features broaden and shift as laser intensity increases, as predicted for four-wave-mixing processes in previous, less general high-intensity analyses. Both theory and experiment also show additional resonant features in the eight-wave-mixing spectra that appear at very high intensity.We perform nonperturbative calculations of resonant multiwave mixing of two high-intensity laser beams of different frequencies both for a general three-level system and for the 3 $^{2}$${\mathit{S}}_{1/2}$--3 $^{2}$${\mathit{P}}_{1/2}$ electronic resonance of the sodium atom. Our calculations proceed by direct numerical integration of the density-matrix equations and subsequent Fourier analysis of the calculated time-dependent density-matrix elements. We examine the case where two nearly degenerate input beams are each detuned many collisional linewidths from electronic resonance, and the frequency difference between the input beams is tuned through a ground-state resonance. Unlike previous high-intensity analyses, each input beam can be resonant with numerous transitions simultaneously; for a three-level system, we refer to this as an ``M-type'' system, in analogy with \ensuremath{\Lambda}- and V-type systems. Calculated four-, six-, and eight-wave-mixing spectra exhibit subharmonic resonances of the type observed experimentally by Trebino and Rahn [Opt. Lett. 12, 912 (1987)]. The three-level system is studied to gain insight into the physics of these subharmonic resonances. For more quantitative comparison with experiment, we use a model that includes the 16 Zeeman and hyperfine states in the 3 $^{2}$${\mathit{S}}_{1/2}$ and the 3 $^{2}$${\mathit{P}}_{1/2}$ levels, and accounts for arbitrary linear polarization of the input laser radiation. The relative intensities of the resonance features in the calculated multiwave-mixing spectra show good agreement with experiment. These resonance features broaden and shift as laser intensity increases, as predicted for four-wave-mixing processes in previous, less general high-intensity analyses. Both theory and experiment also show additional resonant features in the eight-wave-mixing spectra that appear at very high intensity.

Physical Review AJun 1, 199223 citations

Four-wave mixing in a diamond configuration: Experiments with rubidium vapor

R. T. Willis, F. E. Becerra, Luis A. Orozco +1 more

Quantum Electronics and Laser Science ConferenceMay 1, 20070 citations

5{\phantom{\rule{0.16em}{0ex}}}^{2}{S}_{1/2},5{\phantom{\rule{0.16em}{0ex}}}^{2}{P}_{3/2},5{\phantom{\rule{0.16em}{0ex}}}^{2}{D}_{3/2}

, and

5{\phantom{\rule{0.16em}{0ex}}}^{2}{P}_{1/2}

We investigate experimentally and theoretically nondegenerate four-wave mixing in rubidium vapor using a diamond configuration (

5{S}_{1/2}

5{P}_{1/2}

5{P}_{3/2}

, and

6{S}_{1/2}

levels). We model this process of three classical fields interacting with a four-level atomic system with the Schr\"odinger equation. The experimental results are in good agreement with theoretical predictions for the generated light as a function of detuning.

5{S}_{1/2}

5{P}_{1/2}

5{P}_{3/2}

, and

6{S}_{1/2}

). The atomic polarization comes from all the contributions of different velocity classes of atoms in the ensemble modifying dramatically the total transient behavior of the light from FWM.

We perform nonperturbative calculations of resonant multiwave mixing of two high-intensity laser beams of different frequencies both for a general three-level system and for the 3

^{2}

{\mathit{S}}_{1/2}

--3

^{2}

{\mathit{P}}_{1/2}

electronic resonance of the sodium atom. Our calculations proceed by direct numerical integration of the density-matrix equations and subsequent Fourier analysis of the calculated time-dependent density-matrix elements. We examine the case where two nearly degenerate input beams are each detuned many collisional linewidths from electronic resonance, and the frequency difference between the input beams is tuned through a ground-state resonance. Unlike previous high-intensity analyses, each input beam can be resonant with numerous transitions simultaneously; for a three-level system, we refer to this as an ``M-type'' system, in analogy with \ensuremath{\Lambda}- and V-type systems. Calculated four-, six-, and eight-wave-mixing spectra exhibit subharmonic resonances of the type observed experimentally by Trebino and Rahn [Opt. Lett. 12, 912 (1987)]. The three-level system is studied to gain insight into the physics of these subharmonic resonances. For more quantitative comparison with experiment, we use a model that includes the 16 Zeeman and hyperfine states in the 3

^{2}

{\mathit{S}}_{1/2}

and the 3

^{2}

{\mathit{P}}_{1/2}

levels, and accounts for arbitrary linear polarization of the input laser radiation. The relative intensities of the resonance features in the calculated multiwave-mixing spectra show good agreement with experiment. These resonance features broaden and shift as laser intensity increases, as predicted for four-wave-mixing processes in previous, less general high-intensity analyses. Both theory and experiment also show additional resonant features in the eight-wave-mixing spectra that appear at very high intensity.