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Physical Review A

Linear response of a nonlinear stochastic oscillator

Leone Fronzoni, Paolo Grigolini A, R. Mannella, Bruno Zambon

Published Oct 1, 1986• 14 citations
Publisher

Abstract

A nonlinear oscillator driven by both a coherent external field and a (weak) fluctuation-dissipation process of standard type is studied. It is shown that in the extremely-low-friction regime an additional random process comes into play which makes the linear-response theory work in spite of the scarce amount of standard fluctuation. This additional random process depends on the interplay of the nonlinear character of the oscillator under study and the weak fluctuation-dissipation process of standard type, but it is of transverse type, i.e., it does not involve any energy-dissipation process. On the theoretical side, this result rests on the joint use of a procedure of elimination of ``irrelevant'' variables and a rotating-wave-approximation technique which makes the role played by the transverse relaxation-process transparent. These predictions are checked by means of an analog experiment. Furthermore, light is shed onto an incorrect conclusion on the range of validity of the linear-response theory arrived at in an earlier report. This incorrect conclusion is now traced back to an ambiguous feature of the experimental results. The frequency position of the absorption peak does not change significantly upon increase of the excitation field up to a certain value of the excitation field. After this initial plateau the frequency of the absorption peak exhibits a rapid increase with increasing the intensity of the radiation field. Then the increase of the linewidth of the absorption peak becomes slower again with a parallel slow increase of the linewidth of the absorption spectrum, the shape of which, furthermore, does not show any sign of saturation, thereby generating the false impression, when the experimental investigation is carried out in this region, of a complete breakdown of the linear-response theory.

DOI
10.1103/physreva.34.3293

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Physical Review A•Jun 1, 1990•17 citations
A nonlinear oscillator driven by both a coherent external field and a (weak) fluctuation-dissipation process of standard type is studied. It is shown that in the extremely-low-friction regime an additional random process comes into play which makes the linear-response theory work in spite of the scarce amount of standard fluctuation. This additional random process depends on the interplay of the nonlinear character of the oscillator under study and the weak fluctuation-dissipation process of standard type, but it is of transverse type, i.e., it does not involve any energy-dissipation process. On the theoretical side, this result rests on the joint use of a procedure of elimination of ``irrelevant'' variables and a rotating-wave-approximation technique which makes the role played by the transverse relaxation-process transparent. These predictions are checked by means of an analog experiment. Furthermore, light is shed onto an incorrect conclusion on the range of validity of the linear-response theory arrived at in an earlier report. This incorrect conclusion is now traced back to an ambiguous feature of the experimental results. The frequency position of the absorption peak does not change significantly upon increase of the excitation field up to a certain value of the excitation field. After this initial plateau the frequency of the absorption peak exhibits a rapid increase with increasing the intensity of the radiation field. Then the increase of the linewidth of the absorption peak becomes slower again with a parallel slow increase of the linewidth of the absorption spectrum, the shape of which, furthermore, does not show any sign of saturation, thereby generating the false impression, when the experimental investigation is carried out in this region, of a complete breakdown of the linear-response theory.
The resonance fluorescence spectrum of linear and nonlinear oscillators driven by a monochromatic field of frequency ${\ensuremath{\omega}}_{0}$, near the resonant frequency \ensuremath{\omega}, is calculated both classically and quantum mechanically. The result for the linear oscillator is the same in both calculations: a \ensuremath{\delta} function at ${\ensuremath{\omega}}_{0}$. For the nonlinear oscillator, however, the classical and quantum-mechanical spectra near resonance are qualitatively different: Classically, a \ensuremath{\delta} function at ${\ensuremath{\omega}}_{0}$ is obtained; quantum mechanically, a weak-field calculation leads to a \ensuremath{\delta} function at ${\ensuremath{\omega}}_{0}$ together with finite-width peaks nearby. An intuitive explanation for the difference is suggested by the examination of the quantum-mechanical energy-correlation function for the linear oscillator. This function describes energy oscillations with frequencies within a band centered at ${\ensuremath{\omega}}_{0}$-\ensuremath{\omega}. The oscillations can be interpreted as the beating of the driven oscillation with those produced by the vacuum fluctuations (the latter having a frequency spread of the order of the relaxation constant). Because of the nonlinearity, the energy oscillation modulates the effect of the driving field to produce sidebands. Higher-order processes produce additional peaks. A heuristic picture of vacuum fluctuations is developed.
The linear and phase insensitive absorption of a single quanta via coherent interactions with a saturable system, even a single ground state qubit, is sufficient to deterministically generate quantum non-Gaussian states in an oscillator, even stimulated merely by increasing thermal oscillator energy. However, the resultant states only approach Fock states and therefore do not exhibit quantum coherence. Here we overcome this limitation using a minimal step: a nonlinear phase-insensitive absorption process added to the linear one. The coherent addition of such individually passive processes allows coherence to emerge and increase in phase space without an external drive and with minimal interaction requirements. The coherence of quantum non-Gaussian states emerges because the linear and nonlinear absorption processes are not mutually passive. In the simplest case rotationally symmetric Wigner functions of the oscillator Fock states convert their many negative regions to an extremely complex asymmetric structure in sharp contrast to the rotational symmetry of those obtained by the individual interactions. We extend this case to include an unsaturable absorber (oscillator) and analyse switching between linear and nonlinear absorptions, suitable for broad classes of experiments.
The nonlinear dynamics of dissipative quantum systems in incoherent laser fields is studied in the framework of a master equation with the random telegraph model describing the laser noise and the Markovian approximation dealing with the system-bath couplings. Floquet theory and time-dependent perturbation methods are used to facilitate both analytical and numerical solutions. We develop a theoretical formalism that provides a powerful tool for the detailed analysis of the dissipative quantum dynamics of multilevel systems driven by intense stochastic laser fields. It is found that the system relaxes to a steady state from the effect of the laser phase and frequency noise and the kinetics of this relaxation increases with the addition of dissipative terms, introduced by the coupling to the reservoir. Amplitude fluctuations show a different behavior. Other results concerning the destruction of quantum coherence and the dynamical localization will be established and further relaxation mechanisms such as spontaneous emission and the ionization process will also be considered.
We study the field correlation functions and the natural laser spectrum for a slow-inversion laser with the thresholdless intensity fluctuations. In such a laser, the spontaneous-emission-driven relaxation oscillations are responsible for some unique features: The field correlation function oscillates with the frequency of relaxation oscillations ${\ensuremath{\omega}}_{r}$ and the laser line changes its profile. We show that appreciably above threshold the laser spectrum consists of three lines. In addition to the main line at the frequency of the lasing mode ${\ensuremath{\omega}}_{c}$ appear two lines at frequencies ${\ensuremath{\omega}}_{c}+{\ensuremath{\omega}}_{r}$ and ${\ensuremath{\omega}}_{c}\ensuremath{-}{\ensuremath{\omega}}_{r}$. The spontaneous emission noise in such lasers gives rise to nonlinear stochastic effects: Quantum noise affects the oscillation frequency ${\ensuremath{\omega}}_{r}$ and the decay rates of the field correlation function.
A method of calculation of multiple-time correlation functions of the resonance fluorescence light from a two-level atom driven by a realistic laser field with phase and amplitude fluctuations is developed. Both kinds of fluctuations are assumed to be Gaussian. The phase fluctuations are treated in the phase-diffusion model. The averaging over the amplitude fluctuations can be performed in a closed way when the relative mean-square amplitude fluctuation is small compared with the ratio of the amplitude correlation decay rate to the characteristic atomic relaxation rate. There is no restriction to Markovian amplitude fluctuations. Arbitrary values for the mean Rabi frequency are allowed. Results are presented for the intensity, the intensity correlation function, and the spectrum of the fluorescent light. It is shown that even in the case when the relative strength of the amplitude fluctuations is small, significant effects can occur due to the finite correlation length of the fluctuations. The results for fast amplitude correlation decay correspond to the situation without amplitude fluctuations. When the correlation time of the amplitude fluctuations becomes comparable with the atomic relaxation times, the Rabi oscillations of the intensity and the intensity correlation function are nearly washed out, and non-Lorentzian line shapes for the side peaks in the spectrum are observed. It is found that the ratio of the heights of the central peak to the side peak increases with increasing correlation time.
Strong laser fields, when tuned on resonance to an atomic (molecular) transition, give rise to the well-known triplet of resonance fluorescence. The laser field is normally assumed to be strong and monochromatic, even though the spectrum of real lasers is never a \ensuremath{\delta} function. A standard way to treat laser fluctuations has been to assume a well-defined nominal frequency and a randomly fluctuating phase. Recently we introduced the generalized jump model (GJM) for correlated phase fluctuations, where correlated phase jumps are considered, leading to Markovian as well as non-Markovian stochastic behavior. In the GJM, each phase jump may be correlated to the previous jump, and the degree of correlation, the typical jump size, and the mean time between jumps are the three parameters defining the stochastic character of the laser field. In this paper we apply the GJM model to the case of resonance fluorescence from an atom excited by a stochastic field. The ``standard'' Mollow triplet is obtained in the monochromatic limit of small jumps, and several new predictions are made in the different limits of the stochastic parameters. In the small-jump limit, a new ``field effect'' is predicted. The spectral line shape and the triplet separation deviate from the Mollow predictions for a monochromatic field in a substantial way, giving rise to observable differences.In the highly correlated small-jump limit, which is approximated by the Kubo oscillator model, the previous results are corrected to include nonzero line shifts and extended to the case of non-Lorentzian line shapes. In all regions, even input fields of the same line shape (which would be considered identical in the one-parameter linewidth description) give rise to very different output spectra, justifying the more complete stochastic description. Effective relaxation constants ${\mathrm{\ensuremath{\Gamma}}}_{1}^{\mathrm{*}}$ and ${\mathrm{\ensuremath{\Gamma}}}_{2}^{\mathrm{*}}$ that simplify the description of the spectrum are calculated in all appropriate cases, and ${\mathrm{\ensuremath{\Gamma}}}_{1}^{\mathrm{*}}$ is found to equal the input laser linewidth. In addition to predicting the resonance fluorescence spectra, the present analysis provides a way to extract the stochastic parameters from the nonlinear measurement in those cases where the linear characterization of the input field cannot do it. The analytic work is fully supported by extensive numerical simulations.