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physica status solidi (b)

Inductions and Echoes in Systems with Unequidistant Spectrum

V. R. Nagibarov, N. K. Solovarov

Published Jan 1, 1970• 9 citations
Publisher

Abstract

Abstract A general method is developed of calculation of response signals of N‐particle systems with an unequidistant spectrum to pulse action of coherent physical fields resonant for any transition of the system. The case of a particle system where each particle has three energy levels E1 < E2 < E3 is analysed. It is shown that if such a system is subjected to pulse action of two generators exciting transitions E1 ⟷ E2 and E2 ⟷ E3 at initial time and at time τ to pulse action of a generator with frequency ω31 = h−1 (E3 — E1), the system generates induction and echo signals with power proportional to N2 at all possible transitions. Echo signals arise at times (1 + ϵ) τ, 2 τ, and (1 + 1/ϵ) τ after the first pulse, where ϵ = E2 — E1/E3 — E2. The observed responses may have a physical nature which is both identical with the excitation and different from it. Supplements are given by which the responses of particle systems may be calculated to a series of pulses different from those considered here. When ϵ →1 the results transform to the case of equidistant spectrum.

DOI
10.1002/pssb.19700370241

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Abstract A general method is developed of calculation of response signals of N‐particle systems with an unequidistant spectrum to pulse action of coherent physical fields resonant for any transition of the system. The case of a particle system where each particle has three energy levels E1 < E2 < E3 is analysed. It is shown that if such a system is subjected to pulse action of two generators exciting transitions E1 ⟷ E2 and E2 ⟷ E3 at initial time and at time τ to pulse action of a generator with frequency ω31 = h−1 (E3 — E1), the system generates induction and echo signals with power proportional to N2 at all possible transitions. Echo signals arise at times (1 + ϵ) τ, 2 τ, and (1 + 1/ϵ) τ after the first pulse, where ϵ = E2 — E1/E3 — E2. The observed responses may have a physical nature which is both identical with the excitation and different from it. Supplements are given by which the responses of particle systems may be calculated to a series of pulses different from those considered here. When ϵ →1 the results transform to the case of equidistant spectrum.
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.
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.
An analytical study of the photon echoes associated with the three-level system interacting resonantly with optical pulses possessing two different frequencies is performed. It is predicted that photon echoes arise at anomalous times corresponding to the correlation between the inhomogeneous broadenings for the different spectral lines as well as at the normal time expected in a simple two-level system. This new effect is caused by the coupling of the two resonant transitions through the common energy level. Furthermore, if the particle does not have inversion symmetry, the echoes with the sum or difference frequency of the exciting optical pulses can be observed in such a system. The quantitative analysis is also given for intensities, direction of propagation and polarizations of the anomalous echoes which depend upon the exciting pulses with different frequency.
The equations of motion of macroscopic quantities characterizing the nonequilibrium state of a many-particle system with a discrete energy spectrnm and subjected to the strong action of a pulsed generator are deduced by applying the Schroedinger equation for the system. Simple algorithms for the operator transformations related to impluse excitation are derived. The general properties of pulsed generators and of physical systems leading to the appearance of oscillating and stationary response signals of the excited particle system are established. As an illustration of application of the general relations, the nonequilibrium states of electron and nuclear spin systems produced by cross magneto-acoustic excitation are investigated. The prospects of developing pulse techniques for investigating electric and orbital magnetic moment systems are discussed. (auth)
The possibility of a class of optical echo-producing mechanisms is suggested that does not invoke the phase-reversal action of the second pulse. The mechanisms act selectively on the radiating atomic dipole moments, or on their radiation fields, with a periodic dependence on Δτ, where Δ is the angular frequency of the radiator relative to the central angular frequency of the distribution and τ the interpulse time. In the new formulation, the customary expression for the field radiated by an atomic dipole following a two-pulse excitation sequence is multiplied by a factor periodic in Δτ. It is shown that a sequence of echo pulses is predicted when the average over the radiators is calculated. One such mechanism is proposed: the selective absorption or amplification of the dephased free-induction decay field as they propagate through the sample.
A photon echo in a multilevel quantum medium excited by two extremely short pulses with durations of less than one period of oscillation of the light is investigated theoretically. It is shown thatQ echo signals (Q is the number of allowed transitions) can form at each frequency of the allowed transitions, and the number of echo responses for all the allowed transitions equalsQ 2. Of these,Q(Q−1) signals are separated in both time and space. The otherQ echo signals of all the allowed frequencies all arise at the time 2τ 21 (τ 21 is the time interval between application of the first and second pulses to the medium) and are collinear with one another.