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Physical Review A

Simple vectorial model for the spectrum of a two-level atom in an intense low-frequency field

Pietro Paolo Corso, L. Lo Cascio, Franco Persico

Published Aug 1, 1998• 21 citations
Publisher

Abstract

An ansatz is proposed for the analytic solution of the Heisenberg equations of a two-level atom in a classical, linearly polarized and intense low-frequency field. According to this ansatz, the motion of the vector representing the state of the atom on the Bloch sphere consists of a fast precession around a direction, which in turn basculates slowly around an axis perpendicular to the field. The time evolution of the atomic state, which is obtained analytically by this vectorial model, is shown to compare quite satisfactorily with the exact numerical solution of the Heisenberg equations. The limits of validity of the ansatz are discussed. The spectrum of the light scattered by the two-level atom is evaluated analytically using the ansatz and it is shown to reproduce very precisely the exact spectrum evaluated numerically. It is shown that the slow component of the motion of the atomic Bloch vector yields the high-order harmonic part of the spectrum, whereas the fast precession gives rise to the hyper-Raman part. The appearance of the overall spectrum is discussed as a function of the various physical parameters of the model. The insight provided by the vectorial model is exploited to investigate the manifest universality of the main features of the overall spectrum and to suggest experiments to test the validity of the vectorial model.

DOI
10.1103/physreva.58.1549

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An ansatz is proposed for the analytic solution of the Heisenberg equations of a two-level atom in a classical, linearly polarized and intense low-frequency field. According to this ansatz, the motion of the vector representing the state of the atom on the Bloch sphere consists of a fast precession around a direction, which in turn basculates slowly around an axis perpendicular to the field. The time evolution of the atomic state, which is obtained analytically by this vectorial model, is shown to compare quite satisfactorily with the exact numerical solution of the Heisenberg equations. The limits of validity of the ansatz are discussed. The spectrum of the light scattered by the two-level atom is evaluated analytically using the ansatz and it is shown to reproduce very precisely the exact spectrum evaluated numerically. It is shown that the slow component of the motion of the atomic Bloch vector yields the high-order harmonic part of the spectrum, whereas the fast precession gives rise to the hyper-Raman part. The appearance of the overall spectrum is discussed as a function of the various physical parameters of the model. The insight provided by the vectorial model is exploited to investigate the manifest universality of the main features of the overall spectrum and to suggest experiments to test the validity of the vectorial model.
We propose an extension of the vectorial model for a linearly polarized field to the more general elliptically polarized field. In particular, we will show how the emitted spectra depend on the ellipticity of the laser, using both an analytic and a numerical approach. A further result deals with the introduction of damping terms into the system. In this case, we show that the main effects of the damping terms are the disappearance of the hyper-Raman lines and the independence of the dynamics from the initial conditions.
Numerical results are presented on the spectrum of the light scattered into the vacuum modes by a two-level atom driven by a strong ramped field of frequency smaller than the natural atomic frequency. The attention is focused on the role of the hyper-Raman part of the spectrum, which is shown to decrease dramatically with increasing ramp time. It is suggested that this may explain lack of experimental observation of the hyper-Raman lines. For zero ramp time, the numerical intensity of both high-order harmonics and hyper-Raman peaks is shown to agree satisfactorily with previous analytical predictions except in the region of overlap of the two series of peaks. This lends support for the suggestion that the plateau in the spectrum is related to quantum-mechanical interference between high-order harmonics and hyper-Raman processes. There is also good agreement between the numerical amplitude of the plateau and previous analytical predictions. A physical mechanism for the cut-off frequency which terminates the plateau, based on approximate energy conservation in hyper-Raman transitions, is presented and is shown to be in agreement with present numerical and previous analytical results.
Analytic solutions of the optical Bloch equations for a two-level atom interacting with a strong polychromatic field whose frequencies are symmetrically positioned with respect to the atomic frequency are used to obtain the polarization spectrum of the atom. The spectrum is found to consist of a series of discrete peaks or dips superimposed on the continuous part of the spectrum. Physical interpretation of resonances exhibited in the continuous part of the spectrum is given using a semiclassical dressed-atom approach.
We propose and develop a method for theoretical description of circularly (elliptically) polarized optical pulse resonant coherent interactions with two-level atoms. The method is based on the time-evolution equations of a two-level quantum system in the presence of a time-dependent dipole perturbation for electric dipole transitions between states with total angular-momentum projection difference $(\ensuremath{\Delta}{J}_{z}=\ifmmode\pm\else\textpm\fi{}1)$ excited by a circularly polarized electromagnetic field [Feynman et al., J. Appl. Phys. 28, 49 (1957)]. The adopted real-vector representation approach allows for coupling with the vectorial Maxwell's equations for the optical wave propagation and thus the resulting Maxwell pseudospin equations can be numerically solved in the time domain without any approximations. The model permits a more exact study of the ultrafast coherent pulse propagation effects taking into account the vector nature of the electromagnetic field and hence the polarization state of the optical excitation. We demonstrate self-induced transparency effects and formation of polarized solitons. The model represents a qualitative extension of the well-known optical Maxwell-Bloch equations valid for linearly polarized light and a tool for studying coherent quantum control mechanisms.
We present a theory of harmonic generation by a strongly driven atom within the framework of non-relativistic QED. The atom is assumed to possess a finite number N of levels and ionization is neglected. This atom is dressed by the driving field using a unitary time-dependent transformation. Within the quasistatic approximation this transformation is shown to lead from the bare to the Floquet atom. The interaction of the Floquet atom with the vacuum modes is derived and it is shown to give rise both to the high-order harmonic and to the hyper-Raman spectrum. In the case N=2 the spectrum of the light scattered into the vacuum modes can be evaluated analytically and it is shown to consist of two parts. Neglecting interference between these two parts, the first of them is the high-order harmonic spectrum displaying peaks at the odd harmonics of the driving field. The second part is the hyper-Raman spectrum which has finite bandwidth and which displays peaks at the even harmonics of the laser frequency, centred at the Stark-shifted natural atomic frequency. These two spectra partially overlap and they are discussed in various ranges of intensity of the driving field, both in the minimal coupling and in the multipolar scheme. The results support recent suggestions that the presence of the plateau is related to interference between the two spectra. The dependence of the high-order harmonic spectrum from the driving field intensity is found to be related to that of the hyper-Raman spectrum in such a way that when the former increases the latter decreases and vice versa. It is argued that this might help to explain the non-visibility of the hyper-Raman lines when the high-order harmonics are visible.
The wave function of a two-level atom driven by a low-frequency laser pulse is obtained in analytical form and is used to determine the power spectrum of the dipole moment of the atom for different initial atomic states. The solution goes over some known approximate solutions and gives the energy of the two dressed states, the position of the hyper-Raman lines, the amplitudes both of the hyper-Raman lines and of the odd harmonics, and the position of the cutoff. The form of the hyper-Raman spectrum results independently of the initial atomic state and shows an onset and a cutoff of the emission.