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The Journal of Chemical Physics

Four-wave mixing in two-level atoms with stochastically modulated transition frequencies

Jennifer C. Green, Marvin D. Kemple, Gautam Vemuri

Published Oct 22, 2001• 3 citations
Publisher

Abstract

We investigate nonlinear wave mixing in two-level atoms wherein, using a dressed state formalism, it is shown that the signal arises as a consequence of quantum interference between relevant probability amplitudes for photon emission. We then report a theoretical study of the modifications that are induced in the four-wave mixing signals when the transition frequency of the atoms is stochastically modulated. Such modulations, which arise from perturbations in the immediate environment of the atoms, are shown to cause significant alterations in the nonlinear response of the atoms. We also describe how one can, via an investigation of the four-wave mixing signal, differentiate between Markov and non-Markov fluctuations in the transition frequency of the two-level system. The calculations rely on a recently developed Monte Carlo procedure that can be used to obtain exact solutions to a wide variety of spectroscopic problems, with minimal restrictions on atom, field, and noise parameters.

DOI
10.1063/1.1407841

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We investigate nonlinear wave mixing in two-level atoms wherein, using a dressed state formalism, it is shown that the signal arises as a consequence of quantum interference between relevant probability amplitudes for photon emission. We then report a theoretical study of the modifications that are induced in the four-wave mixing signals when the transition frequency of the atoms is stochastically modulated. Such modulations, which arise from perturbations in the immediate environment of the atoms, are shown to cause significant alterations in the nonlinear response of the atoms. We also describe how one can, via an investigation of the four-wave mixing signal, differentiate between Markov and non-Markov fluctuations in the transition frequency of the two-level system. The calculations rely on a recently developed Monte Carlo procedure that can be used to obtain exact solutions to a wide variety of spectroscopic problems, with minimal restrictions on atom, field, and noise parameters.
The effect of environmental perturbations on a system of two-level atoms is to alter the coupling between the atoms and their surroundings, which manifests itself as a change in the relaxation rates of the atoms. One of the ways in which environmental fluctuations can affect an atom is to modulate its transition frequency in a stochastic manner. We study the response of such two-level atoms to irradiation by a monochromatic laser, and report the results of numerical experiments on the spectrum of scattered light. It is observed that this spectrum is a sensitive indicator of the parameters that describe the stochastic modulation. In addition, the spectra exhibit substantially different features depending on whether the stochastic process follows a diffusion or a jump mechanism. The numerical results are based on a Monte Carlo procedure, which can be applied to stochastic processes in a wide variety of biological, chemical, and physical processes.
We investigate the interplay between six-wave mixing (SWM) and four-wave mixing (FWM) resulting from atomic coherence and polarization beat in a four-level atomic system. The dressed FWM evolution and competition pathways can be controlled by the coupling field to exhibit two FWM and SWM turning points, FWM+SWM, and FWM+FWM interference regions. Quantum interference between two FWM or one FWM and one SWM channels leads to nonlinear signal enhancement and suppression under different conditions. The fifth-order nonlinear response can be obtained by the phase control of the polarization beat between the FWM and SWM signals.
A theory of four-wave mixing is developed using an amplitude approach in the Schr\"odinger picture. It is shown that the four-wave mixing signal produced when three fields are incident on an ensemble of two-level atoms can be viewed as arising from constructive interference of the signals emitted at the various atomic sites. Fourteen distinct amplitudes contribute to the four-wave mixing signal. Some of these amplitudes involve final states in which one or two atoms are excited, and some involve final states in which photons are emitted into modes other than a phase-matched mode of interest. The amplitude approach has the advantage of clearly showing the atom-field correlations. It helps one to understand the underlying physical mechanisms responsible for creation of the four-wave mixing signal, as well as features in the line shape resulting from collisions and effects relating to atomic recoil on the absorption or emission of radiation. These so-called collision- and recoil-induced resonances are calculated explicitly using the amplitude approach, and an interpretation of the results in terms of the interference between different amplitudes is given. Our results, calculated for a finite interaction time between the atoms and the fields, are compared with analogous calculations in which an effective ground-state decay time is assumed. It is shown that the results differ if the atom-field interaction time is not the longest relevant time parameter in the problem.
We investigate the influence of the distribution of atom velocities in a hot rubidium sample on the correlation between field-intensity fluctuations of two independently generated four-wave mixing signals and between the transmission signals. The nonlinear process is driven by a single continuous-wave (cw) laser in a pure two-level system due to the forward geometry with circular and parallel polarization of the input fields. The intensity cross correlations of the four-wave mixing signals and the transmission signals present an oscillatory behavior with a clear dependence on the power of the incident fields, which indicates a connection with Rabi oscillations. A two-level theoretical model using stochastic differential equations to account for the mechanism of conversion of phase noise into amplitude noise shows good agreement with our experimental results. Moreover, we show how the response of the system is affected by the different atomic velocity groups.
The coupling between a chromophore and its surroundings can, in the event of environmental perturbations, result in a stochastic modulation of the chromophore’s transition frequency, which in turn can cause fluctuations in the population of the excited state. We demonstrate that by using these fluctuations in population inversion as the spectroscopic variable, one can extract information on whether the random modulation of the chromophore follows a diffusion process or a jump process. The population fluctuations also discriminate between Markov and non-Markov fluctuations of the atomic frequency. In addition, we show that this observable allows, in certain regimes, to unambiguously determine whether the population fluctuations are due to atomic frequency noise, or due to noise in the irradiating laser field. Finally, through a dressed state analysis, we demonstrate that the response of a randomly modulated atom is similar to that of an unmodulated atom driven by a noisy field that has amplitude fluctuations in only one quadrature.
We study phase-conjugate degenerate four-wave mixing (DFWM) in an optically pumped, two-level atomic system (the 3s2S1∕23s\phantom{\rule{0.2em}{0ex}}^{2}S_{1∕2}3s2S1∕2​, F=2F=2F=2, ${m}_{F}=2\ensuremath{\rightarrow}3p\phantom{\rule{0.2em}{0ex}}^{2}P_{3∕2}$, F=3F=3F=3, mF=3{m}_{F}=3mF​=3 transition in a diffuse, collision-free, thermal beam of sodium) using phase-modulated laser beams. In this investigation, we study the detailed line shape of the four-wave mixing signal as a function of the temporal delay between the probe field and the pump fields at different intensities of the cw pump waves and modulation frequencies. We show that phase modulation of the pump and probe beams leads to strong amplitude modulation of the phase-conjugate beam and report excellent agreement between our experimental observations of the DFWM spectra and the results of numerical calculations. Our numerical results suggest an enhancement of the ``local'' DFWM signal when the sidebands of the phase-modulated laser are resonant with dressed-state transitions, but that this resonance is less obvious in the total signal generated. We also show that the complex line shapes depend sensitively upon the delay of the probe beam, with the peak DFWM signal occurring when the probe beam is advanced compared to the pump beam.