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Journal of Modern Optics

Photon-echo Varieties in Three-level Systems Formed by Non-rectangular Exciting Pulses of Arbitrary Area

Igor V. Yevseyev, Yu.V. Men'shikova, V. N. Tsikunov

Published Jul 1, 1990• 1 citation

Abstract

A technique for the derivation of the electric field strength of varieties of photon-echo signals formed by exciting pulses with smooth envelopes is developed. This technique is based on joint solution of the d'Alembert and Schrödinger equations. The matrix of the evolution operator is constructed, which permits a description of the state of an individual atom (molecule) resulting from the action of an arbitrary exciting pulse sequence. The special form of exciting pulses with smooth envelope considered here is, together with the model of rectangular exciting pulses, one of the models that permits an analytical solution when the areas of the exciting pulses are not small. As an example, the technique is applied to the calculation of the electric field strength of signals of the modified three-level photon echo and the three-level photon echo formed by exciting pulses with smooth envelopes. The signal characteristics of photon-echo varieties formed by rectangular and non-rectangular exciting pulses are compared. It is shown that if the spectral lines of the resonant transitions are broad then the shapes and the peaks of the echo signals depend essentially on the shapes of the exciting pulses. The appearance of the fine structure of the modified three-level photon-echo and three-level photon-echo signal shapes is predicted. The necessity of setting up experiments aimed at observing the theoretically predicted special features of these photon-echo varieties is pointed out.

DOI: 10.1080/09500349014551271

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A technique for the derivation of the electric field strength of varieties of photon-echo signals formed by exciting pulses with smooth envelopes is developed. This technique is based on joint solution of the d'Alembert and Schrödinger equations. The matrix of the evolution operator is constructed, which permits a description of the state of an individual atom (molecule) resulting from the action of an arbitrary exciting pulse sequence. The special form of exciting pulses with smooth envelope considered here is, together with the model of rectangular exciting pulses, one of the models that permits an analytical solution when the areas of the exciting pulses are not small. As an example, the technique is applied to the calculation of the electric field strength of signals of the modified three-level photon echo and the three-level photon echo formed by exciting pulses with smooth envelopes. The signal characteristics of photon-echo varieties formed by rectangular and non-rectangular exciting pulses are compared. It is shown that if the spectral lines of the resonant transitions are broad then the shapes and the peaks of the echo signals depend essentially on the shapes of the exciting pulses. The appearance of the fine structure of the modified three-level photon-echo and three-level photon-echo signal shapes is predicted. The necessity of setting up experiments aimed at observing the theoretically predicted special features of these photon-echo varieties is pointed out.

An expression for the electric field intensity of the photon echo formed in a gaseous medium on transition between two degenerate levels under the action of two linearly polarized exciting pulses is derived. Using the obtained relationships the effect of photon-echo shape locking is analysed for a square shaped second (‘locking’) exciting pulse as well as for the case of a smooth envelope of the hyperbolic secant form. It is shown that in degenerate systems as well as in non-degenerate ones, photon-echo shape locking takes place if the shape of the ‘locking’ pulse is square and does not take place in the case of a smooth envelope.

Echo responses of a three-level medium formed by the λ-scheme of quantum transitions under exposure to optical pulses consisting of an arbitrary number of oscillations have been studied theoretically. The cases of pulses consisting of few optical oscillations (for which the concept of an envelope cannot be used) and combinations of such pulses and quasi-monochromatic resonance signals have been considered. The approach used can be reduced to the renormalization of dipole moments of allowed quantum transitions through their multiplication by coefficients depending on the shape and duration of pump pulses and having absolute values in the range from zero (for nonresonance pulses) to unity (for resonance quasi-monochromatic pulses and broadband pulses consisting of few oscillations, whose spectrum covers the quantum transitions). A general equation has been proposed for the pulse area. In the limit of a large number of oscillations, it transforms into the well-known definition of the area of a quasi-monochromatic signal. The characteristics of primary and longlived photon echoes have been analyzed in detail. It has been shown that, when a medium is exposed to only pulses consisting of a few oscillations, three echo responses of both types can be principally generated at each frequency of the λ-scheme. Introduction of quasi-monochromatic pulses in pump pulse series decreases the number of echoes, and their qualitative character has a non-commutative property with respect to pulse permutation in a series. The extension of the proposed approach to more complex schemes of quantum transitions with the large number of quantum levels faces no principal difficulties.

In the limit of small exciting light-pulse areas the amplitude and polarization of the tri-level photon echo in gases formed by the exciting pulses with arbitrary shapes are obtained. It is shown that the dependence of the tri-level photon echo signal on the mutual orientation of the exciting pulse polarization vectors may be used to determine the optical coherence-matrix relaxation characteristics for an optically forbidden transition. The conditions under which a tri-level photon echo pulse repeats the shape of each of the exciting pulses are defined. It is proved that the tri-level photon-echo polarization properties do not depend on the shapes of the exciting pulses, either for small light-pulse areas or for the tri-level echo formation on narrow spectral lines.

The excitation of photon echo by noise pulses that are formed by modulation of the carrying frequency with Gauss noise is modeled. The modeling is based on optical Bloch equations the solution of which for noise pulse realizations is constructed by their stepwise approximation. In terms of the formalism of state transfer matrices, the two- and three-pulse excitation modes are analyzed. The complex envelopes of the primary and stimulated echo responses are determined. In the linear (low-level-signal) mode, the shape of the two-pulse echo corresponds to that of the time delayed and inverted noise pulse. The boundary of the linear mode, upon exceeding of which distortions of the shape of the noise pulse become noticeable, is determined. The shape of the stimulated (three-pulse) echo in the linear mode corresponds to that of the autocorrelation function of the noise pulse realization. Upon passage beyond the boundary of the linear mode, the shape of the three-pulse echo corresponds either to the cross-correlation function of distorted noise pulses (with different intensities) or to the autocorrelation function of distorted pulses (with the same intensities). The modeled photon echo excitation modes can be used in photon echo processors to process signals in the light range.

A general expression for coherent transients, free induction decay and echoes, of an atomic or spin two-level system excited by resonant laser or rf field in the form of two square pulses is given by using the Fourier-transform method. This expression interprets the main characteristics of the transient signals well. An analytic expression for the shapes of two-pulse echoes is derived by assuming infinite inhomogeneity and the same Rabi frequencies for the two pulses. Two-pulse echo shapes were numerically calculated for various combinations of pulse areas by using the analytic expression. Nuclear-magnetic-resonance experiment was carried out on an inhomogeneously broadened proton spin system. Experimental two-pulse echo shapes are compared with the calculated ones. Agreement between theory and experiment is quite excellent.