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Optics and Spectroscopy

Rabi frequency and optical Bloch equations for a short femtosecond optical pulse

G. M. Genkin, S. E. Razov, A. G. Razov

Published Jul 1, 2001• 2 citations
Publisher

Abstract

We investigate the interaction of a multilevel atom with a short femtosecond optical pulse beyond the scope of the rotating-wave approximation. We consider the behavior of a three-level atom under the action of two optical pulses resonant with two allowed transitions and a two-level atom under the action of a resonant optical pulse. It is shown that the imaginary part of the off-diagonal elements of the density matrix oscillates with high frequency due to the counter-rotating terms. It is shown also that an effective time-dependent addition to the Rabi frequency exists, which is of importance for ultrashort optical pulses.

DOI
10.1134/1.1388317

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We investigate the interaction of a multilevel atom with a short femtosecond optical pulse beyond the scope of the rotating-wave approximation. We consider the behavior of a three-level atom under the action of two optical pulses resonant with two allowed transitions and a two-level atom under the action of a resonant optical pulse. It is shown that the imaginary part of the off-diagonal elements of the density matrix oscillates with high frequency due to the counter-rotating terms. It is shown also that an effective time-dependent addition to the Rabi frequency exists, which is of importance for ultrashort optical pulses.
We investigate the interaction of a two-level atom with a short femtosecond optical pulse without the rotating-wave approximation. We have derived the population inversion due to the counterrotating terms. It is shown that due to the counterrotating terms the population inversion oscillates with high frequency and there is the effective time-dependent addition to the Rabi frequency that varies inversely proportionally to the length of an optical pulse and this effect is actual for a short femtosecond optical pulse. It is shown that there is the transient cubic nonlinearity due to the counterrotating terms.
Some features of the excitation of multilevel quantum systems under the action of electromagnetic pulses which are shorter than the inverse frequency of interlevel transitions are considered. It is shown that the interaction is characterized by a specific type of selectivity which is not connected with the resonant absorption of radiation. The simplest three-level model displays the inverse population of upper levels. The effect of an ultrashort laser pulse on a multilevel molecule was regarded as an instant reception of the oscillation velocity by the oscillator and this approach showed an effective excitation and dissociation of the molecule. The estimations testify to the fact that these effects can be observed using modern femtosecond lasers.
One of the simplest models involving the atom-field interaction is the coupling of a single two-level atom with single-mode optical field. Under the rotating wave approximation, this problem is reduced to a form that can be solved exactly. But the approximation is only valid when the two levels are resonant or nearly resonant with the applied electromagnetic radiation. Here we present an analytical solution without the rotating wave approximation and applicable to general atom-field interaction far away from the resonance. We find that there exists remarkable influence of the initial phase of optical field on the Rabi oscillations and Rabi splitting, and this issue cannot be explored in the context of the rotating wave approximation. Due to the retention of the counter-rotating terms, higher-order harmonic appears during the Rabi splitting. The analytical solution suggests a way to regulate and control the quantum dynamics of a two-level atom and allows for exploring more essential features of the atom-field interaction.
We investigate the properties of a homogeneously broadened three-level system interacting with two optical fields, Eω1(x,t) and Eω2(x,t), with central frequencies ω1 and ω2 whose amplitudes are assumed to be slowly varying in time compared with dephasing times of the optical transitions. We calculate Raman susceptibilities allowing each transition to be induced by both fields and without employing the rotating wave approximation (RWA); this enables correct calculation of the optical response of the medium when the fields are on-resonance or arbitrarily far off-resonance with the Raman transition. In some frequency regions the RWA causes substantial errors. A distinction is drawn between two different types of RWA: (a) a neglect of off-resonant components of the polarization for direct optical transitions, and (b) neglect of off-resonant components of off-diagonal density matrix elements of order Eω1(x,t)Eω2(x,t)* in the field amplitudes. The first RWA is known to become inaccurate when the pump frequency is significantly off-resonance with the pump transition. We show that the second RWA becomes inaccurate when ‖ω1−ω2‖ is small or comparable to the active Raman mode frequency (even when the field are close to resonance). Calculations based upon our expressions show that when the pump frequency is close to resonance with the optical transition from the ground state, a nonlinear absorption at the Stokes shifted frequency occurs. This absorption can significantly decrease the Raman gain. Also, we find that Rabi splittings in a three-level system can occur even when the pulse durations of the incident fields are short compared with the T1 times of the system. We solve analytically for the optical response of the system to two fields of arbitrary strength and frequency. We analyze this solution and its perturbative expansion in the field amplitudes. We use the Maxwell–Bloch equations developed here to propagate time-dependent pulses.
We calculate analytically the error introduced by the rotating-wave approximation (RWA) to the description of a two-level-atom response to a resonant sufficiently intense femtosecond optical pulse of a few-cycle duration and any envelope. The analytically calculated non-RWA refinements to the RWA response are of the lowest order, i.e., linear in the small ratio of the Rabi to optical frequencies. One type of such refinements is small-amplitude rapid oscillations at nearly twice the optical frequency superimposed on the standard RWA population-inversion evolution. The other refinement is an extension of the spectrum of the induced atomic dipole moment in which the third harmonic of the transition frequency appears. The spectrum of this harmonic has an internal structure dependent on both strength and duration of the pulse. The analytically found non-RWA corrections are in qualitative agreement with the recent results of different numerical approaches by other authors and a recent experiment.
We consider the case of interaction of a three-level atom with a high intensity pulse electromagnetic field. Rabi frequencies are assumed to be of the order of the transition frequency or greater. We demonstrate that coherent population trapping exists even in such a case when the usual rotating-wave approximation is no longer valid. However, a certain condition on the phases of electromagnetic wave components has to be satisfied in order to make coherent population trapping possible. This condition is specific for the strong field case in contrast to the case where the rotating-wave approximation is valid. Also we point out some elements for which a model of an atom with a finite number of levels may still be valid while the rotating-wave approximation breaks down.