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Physical Review A

Susceptibility of strongly driven two-level atoms: A non-Markovian analysis

J. R. Brinati, Salomon Sylvain Mizrahi, Gilberto Aparecido Prataviera

Published Jul 1, 1997• 6 citations
Publisher

Abstract

The absorptive part of the susceptibility of a system constituted by a strongly driven two-level atom (interacting with a reservoir) is analyzed when probed by a weak field. Memory effects are taken into account (the correlation time of the reservoir variables is finite) and we verify that they lead to several distinct features for the absorption and stimulated emission of radiation, when compared with those obtained in the Markovian approximation. Moreover, we verify that memory effects lead to a signature, namely, the sensitive dependence of the maxima of the absorption lines on the Rabi frequency at off-resonance pumping. Other characteristics of the spectrum are enhanced, even in the presence of a small but non-null correlation time of the reservoir, for instance, the maxima of the dispersionlike absorption curves at on-resonance pumping. In this work several situations are analyzed for different values of the parameters of the problem: the off-resonance frequencies of the driving field, the Rabi frequency, the memory time, and temperature of the reservoir.

DOI
10.1103/physreva.56.322

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Physical Review A•Oct 1, 1995•7 citations
The absorptive part of the susceptibility of a system constituted by a strongly driven two-level atom (interacting with a reservoir) is analyzed when probed by a weak field. Memory effects are taken into account (the correlation time of the reservoir variables is finite) and we verify that they lead to several distinct features for the absorption and stimulated emission of radiation, when compared with those obtained in the Markovian approximation. Moreover, we verify that memory effects lead to a signature, namely, the sensitive dependence of the maxima of the absorption lines on the Rabi frequency at off-resonance pumping. Other characteristics of the spectrum are enhanced, even in the presence of a small but non-null correlation time of the reservoir, for instance, the maxima of the dispersionlike absorption curves at on-resonance pumping. In this work several situations are analyzed for different values of the parameters of the problem: the off-resonance frequencies of the driving field, the Rabi frequency, the memory time, and temperature of the reservoir.
We analyze the effects of a reservoir having a generic spectrum (colored noise) on the index of refraction of a system of strongly driven two-level atoms (gas) probed by a weak field. We show that for high Rabi frequencies, $\ensuremath{\Omega},$ a simple analytic expression results for the susceptibility function (with respect to the probe) when $\ensuremath{\nu}\ensuremath{\approx}\ensuremath{\Omega},$ where $\ensuremath{\nu}$ is the detuning between driving and probe fields frequencies. Several distinct features are revealed in that function when compared to the one resulting from a reservoir with a white-noise spectrum (atom-vacuum coupling). We also show that the Mach-Zehnder interferometry could permit distinguishing between the two kinds of spectra and we point out the sensitive dependence of the index of refraction of the atomic gas on $\ensuremath{\Omega}$ and find a signature for the non-Markovian evolution.
We examine the radiative properties of a two-level atom driven by a strong bichromatic field with frequencies ${\mathrm{\ensuremath{\omega}}}_{1}$=${\mathrm{\ensuremath{\omega}}}_{0}$-${\mathrm{\ensuremath{\delta}}}_{1}$ and ${\mathrm{\ensuremath{\omega}}}_{2}$=${\mathrm{\ensuremath{\omega}}}_{0}$+${\mathrm{\ensuremath{\delta}}}_{2}$, which can be asymmetrically placed about the atomic transition frequency ${\mathrm{\ensuremath{\omega}}}_{0}$ and can have different Rabi frequencies ${\mathrm{\ensuremath{\Omega}}}_{1}$ and ${\mathrm{\ensuremath{\Omega}}}_{2}$. Applying the optical Bloch equations for the bichromatic excitation, we derive an infinite set of equations of motion for the time evolution of the atomic variables. The equations are solved numerically by matrix inversion and Laplace transforms. Using the quantum regression theorem, we then solve for the steady-state total fluorescence intensity, and the resonance-fluorescence and absorption spectra. The spectra are found to depend on the frequency difference 2\ensuremath{\delta}=${\mathrm{\ensuremath{\delta}}}_{1}$+${\mathrm{\ensuremath{\delta}}}_{2}$, the average detuning \ensuremath{\Delta}=${\mathrm{\ensuremath{\omega}}}_{0}$-1/2(${\mathrm{\ensuremath{\omega}}}_{1}$+${\mathrm{\ensuremath{\omega}}}_{2}$), and the Rabi frequencies of the driving fields. For ${\mathrm{\ensuremath{\Omega}}}_{1}$=${\mathrm{\ensuremath{\Omega}}}_{2}$=\ensuremath{\Omega}, the total fluorescent intensity displays a series of maxima for \ensuremath{\Delta}=\ifmmode\pm\else\textpm\fi{}n\ensuremath{\delta}\ensuremath{\mp}${\mathrm{\ensuremath{\Omega}}}^{2}$/4\ensuremath{\delta}, where the n are the odd integers. The intensity-dependent shift from the resonances n\ensuremath{\delta} is explained as an analog of the generalized Bloch-Siegert shift. The resonance-fluorescence spectrum for \ensuremath{\Delta}\ensuremath{\ne}0 appears to contain more peaks than that for \ensuremath{\Delta}=0.This is due to the splitting of the central peak and the even sidebands into doublets. The absorption spectrum of a weak-probe beam for \ensuremath{\Delta}=0 and equal Rabi frequencies consists of a symmetric series of dispersionlike sidebands separated from ${\mathrm{\ensuremath{\omega}}}_{0}$ by integer multiples of \ensuremath{\delta}, together with a central absorption peak at ${\mathrm{\ensuremath{\omega}}}_{0}$, whose amplitude oscillates with \ensuremath{\Omega}. For \ensuremath{\Delta}\ensuremath{\ne}0 and/or ${\mathrm{\ensuremath{\Omega}}}_{1}$\ensuremath{\ne}${\mathrm{\ensuremath{\Omega}}}_{2}$, the odd sidebands remain dispersionlike, while the central peak and the even sidebands split into absorption-emission doublets. A simple physical interpretation of the spectral features is given in terms of the dressed-atom model.
Recent femtosecond processes [Phys. Rev. Lett. 66, 2464 (1991)] confirm the importance of non-Markovian effects since the dynamical characteristic times of the system become of the same time scale as the reservoir. We analyze here the memory effects in the density-matrix coherence for a driven two-level atom in the stationary regime. The coherence parameter \ensuremath{\eta} is written in terms of the detuning, the strength of the external driving force, and the temperature as the reservoir. We verify that the memory effects, present in the density matrix, are enhanced by the external driving force. We introduce the Shanon information entropy and we verify that, while in the Markovian approximation the entropy decreases monotonically as function of the detuning, in the non-Markovian approach it is characterized by a minimum, which enables us to differentiate between the two processes.
The absorption spectrum of a weak probe radiation by a two-level atom strongly driven by an intense radiation field is examined. Using a Green's function method developed for calculations of general multiphoton processes, the author obtains some earlier results in which the spectrum has regions (in the probe frequency domain) of amplification and regions of attenuation. Such a spectrum is shown explicitly to be due to a coherent interference effect arising from the interaction of the atom with the vacuum field. Furthermore, an equation that relates the area under the amplification part of the spectrum to that under the attenuation part is obtained. This indicates that amplification cannot occur if a very-wide-band probe beam is used instead of a narrow-band one. At high driving intensities the absorption spectrum is shown explicitly to be inversely proportional to the Rabi frequency and has a dispersion-like behaviour near the Rabi sidebands at driving resonance.
We analyze the fluorescence and absorption spectra of a two-level atom driven by a bichromatic field with frequencies ${\mathrm{\ensuremath{\omega}}}_{1}$ and ${\mathrm{\ensuremath{\omega}}}_{2}$, separated by ${\mathrm{\ensuremath{\omega}}}_{2}$-${\mathrm{\ensuremath{\omega}}}_{1}$=2\ensuremath{\delta}, and Rabi frequencies (at resonance) 2${\mathrm{\ensuremath{\Omega}}}_{1}$ and 2${\mathrm{\ensuremath{\Omega}}}_{2}$ such that their ratio \ensuremath{\alpha}=${\mathrm{\ensuremath{\Omega}}}_{2}$/${\mathrm{\ensuremath{\Omega}}}_{1}$1. We focus on the case of ${\mathrm{\ensuremath{\omega}}}_{1}$ close to the atomic frequency ${\mathrm{\ensuremath{\omega}}}_{0}$ and ${\mathrm{\ensuremath{\omega}}}_{2}$ near the Rabi sideband frequency ${\mathrm{\ensuremath{\omega}}}_{1}$+2${\mathrm{\ensuremath{\Omega}}}_{1}$; the detunings are denoted by ${\mathrm{\ensuremath{\Delta}}}_{1}$=${\mathrm{\ensuremath{\omega}}}_{0}$-${\mathrm{\ensuremath{\omega}}}_{1}$ and ${\mathrm{\ensuremath{\Delta}}}_{2}$=${\mathrm{\ensuremath{\omega}}}_{1}$+2${\mathrm{\ensuremath{\Omega}}}_{1}$-${\mathrm{\ensuremath{\omega}}}_{2}$. We find that the spectra depend critically on the detuning ${\mathrm{\ensuremath{\Delta}}}_{2}$: For large ${\mathrm{\ensuremath{\Delta}}}_{2}$, the fluorescence spectrum consists of the well known Mollow triplet, centered at ${\mathrm{\ensuremath{\omega}}}_{1}$; for smaller (but nonzero) ${\mathrm{\ensuremath{\Delta}}}_{2}$, the spectrum is composed of a triplet at ${\mathrm{\ensuremath{\omega}}}_{1}$ together with doublets near the sideband frequencies ${\mathrm{\ensuremath{\omega}}}_{1}$\ifmmode\pm\else\textpm\fi{}2${\mathrm{\ensuremath{\Omega}}}_{1}$. However, when ${\mathrm{\ensuremath{\Delta}}}_{2}$=0 (and \ensuremath{\alpha}\ensuremath{\ll}1), the spectrum consists of a doublet centered at ${\mathrm{\ensuremath{\omega}}}_{1}$ and triplets at ${\mathrm{\ensuremath{\omega}}}_{1}$\ifmmode\pm\else\textpm\fi{}2${\mathrm{\ensuremath{\Omega}}}_{1}$: there is then no fluorescence at ${\mathrm{\ensuremath{\omega}}}_{1}$. As \ensuremath{\alpha} increases, additional triplet structures appear in the spectrum at frequencies ${\mathrm{\ensuremath{\omega}}}_{1}$\ifmmode\pm\else\textpm\fi{}2n${\mathrm{\ensuremath{\Omega}}}_{1}$ with intensities proportional to ${\mathrm{\ensuremath{\alpha}}}^{2(\mathit{n}\mathrm{\ensuremath{-}}1)}$, n\ensuremath{\gtrsim}1, and a line reappears at ${\mathrm{\ensuremath{\omega}}}_{1}$, with intensity proportional to ${\mathrm{\ensuremath{\alpha}}}^{4}$. The absorption by the system of a weak probe beam is also strongly dependent on the detuning, and the spectrum is composed of emission-dispersion-absorption features located near ${\mathrm{\ensuremath{\omega}}}_{1}$ and ${\mathrm{\ensuremath{\omega}}}_{1}$\ifmmode\pm\else\textpm\fi{}2n${\mathrm{\ensuremath{\Omega}}}_{1}$. An analysis in the dressed-atom picture is presented which explains the physical origin of all these features, in both fluorescence and absorption. \textcopyright{} 1996 The American Physical Society.
The description of a non-Markovian process is presented and the asymptotic solution of the generalized master equation is obtained for a two-level atom pumped externally by a monochromatic time-dependent electric field. We give the explicit expression for the absorption line-shape function and discuss its features as a function of the temperature and the memory parameter. We verified the following interesting results: (a) At a critical temperature the line shape undergoes a phase transition, the single-bump profile splits into two bumps; (b) for small values of the memory parameter k the linewidths broaden, becoming larger than the ones calculated under the Markovian approximation, and after reaching some maximum value they narrow monotonically with increasing values of k, this behavior being standard for any finite temperature.