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Physical review. B, Condensed matter

Coherent aspects of the local dipolar field in echo dynamics: A spectral narrowing effect

R. Boscaino, F. M. Gelardi, M. Cannas

Published Jan 1, 1996• 6 citations
Publisher

Abstract

We have measured the spectral content of the echo radiation emitted by a two-level spin system ([${\mathrm{AlO}}_{4}$${]}^{0}$ centers in quartz), after a two-pulse excitation sequence, ${\mathrm{\ensuremath{\theta}}}_{1}$-\ensuremath{\tau}-${\mathrm{\ensuremath{\theta}}}_{2}$. In general, the experimental spectral shapes reproduce qualitatively those calculated for an inhomogeneous system of noninteracting spins. However, we found that in certain ranges of the sequence parameters (${\mathrm{\ensuremath{\theta}}}_{1}$, ${\mathrm{\ensuremath{\theta}}}_{2}$, and \ensuremath{\tau}) the widths of the experimental spectra are less than expected by a factor that amounts up to 0.5. The observed narrowing effect involves the spin packets located in a narrow spectral bandwidth (of the order of the Rabi frequency) around the excitation frequency and occurs whenever the excitation sequence induces a transverse oscillating moment of the strictly resonant packets. The spectral narrowing is a transient effect, in the sense that the spectral widths tend to recover their theoretical values for long interpulse distances \ensuremath{\tau}. We suggest a tentative interpretation which relates the narrowing to the coherence properties of the local field generated by precessional motion of the echo-active spins. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physrevb.53.302

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We have measured the spectral content of the echo radiation emitted by a two-level spin system ([{\mathrm{AlO}}_{4}$${]}^{0} centers in quartz), after a two-pulse excitation sequence, \ensuremathθ1{\mathrm{\ensuremath{\theta}}}_{1}\ensuremathθ1​-\ensuremath{\tau}-\ensuremathθ2{\mathrm{\ensuremath{\theta}}}_{2}\ensuremathθ2​. In general, the experimental spectral shapes reproduce qualitatively those calculated for an inhomogeneous system of noninteracting spins. However, we found that in certain ranges of the sequence parameters (\ensuremathθ1{\mathrm{\ensuremath{\theta}}}_{1}\ensuremathθ1​, \ensuremathθ2{\mathrm{\ensuremath{\theta}}}_{2}\ensuremathθ2​, and \ensuremath{\tau}) the widths of the experimental spectra are less than expected by a factor that amounts up to 0.5. The observed narrowing effect involves the spin packets located in a narrow spectral bandwidth (of the order of the Rabi frequency) around the excitation frequency and occurs whenever the excitation sequence induces a transverse oscillating moment of the strictly resonant packets. The spectral narrowing is a transient effect, in the sense that the spectral widths tend to recover their theoretical values for long interpulse distances \ensuremath{\tau}. We suggest a tentative interpretation which relates the narrowing to the coherence properties of the local field generated by precessional motion of the echo-active spins. \textcopyright{} 1996 The American Physical Society.
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.
Large-amplitude oscillations are predicted in the free-induction decay (FID) of atomic or spin two-level quantum systems that are coherently prepared by a resonant electromagnetic field in the form of a square amplitude pulse of duration TTT. The effect should be detectable in nuclear magnetic resonance (NMR) or in optical resonance experiments when the transition line shape is inhomogeneously broadened (${T}_{2}^{*}\ensuremath{\ll}{T}_{2}$) and the applied field is intense enough that the pulse area $\ensuremath{\chi}T\ensuremath{\ge}2\ensuremath{\pi}$, where $\ensuremath{\chi}$ is the Rabi frequency. Physically, the effect is due to Rabi oscillations which are generated in the preparative stage and are then reproduced because of atomic memory in the radiative period which follows. The atomic polarization derived from the Bloch equations assumes the form of an integral over the inhomogeneous line shape, and can be evaluated numerically for arbitrary linewidth $\ensuremath{\sigma}$ or analytically when $\ensuremath{\sigma}\ensuremath{\rightarrow}\ensuremath{\infty}$. The characteristics of the oscillation are unusual since the oscillation frequency, which is of order $\ensuremath{\chi}$, increases with time while the oscillation envelope vanishes identically for times $t\ensuremath{\ge}2T$. Coherent emission is therefore confined to one pulse width TTT immediately following the pulse. Previous NMR and infrared FID experiments with small-area pulses ($\ensuremath{\chi}T\ensuremath{\le}4\ensuremath{\pi}$), interpreted erroneously in terms of an edge echo effect, support the oscillatory FID behavior described here. For the case $\ensuremath{\chi}T>4\ensuremath{\pi}$, an experimental test would be even more decisive, as the oscillations are more pronounced.
The two-pulse echo sequence is examined for the case in which the frequency modulation time τm of the transition is intermediate between the well known limiting cases of very fast modulation (motional narrowing) and very slow or static modulation (inhomogeneous broadening). Within this spectral diffusion regime, the interpretation of the echo decay differs markedly from standard treatments. If the frequency-frequency correlation function initially decays as 1−tβ, the echo decay time TE is proportional to τmβ/(β+2). These results reduce to those of Yan and Mukamel [J. Chem. Phys. 94, 179 (1991)] for β=1. Drawing on a viscoelastic model, the theoretical results are compared to viscosity and temperature dependent vibrational echo experiments on myoglobin–CO. A τm1/3 dependence is observed, as is predicted for an exponential decay of the frequency–frequency correlation function.
Rabi oscillations of quantum population are known to occur in two-level systems driven by spectrally narrow laser fields. In this work we study Rabi oscillations induced by shaped broadband femtosecond laser pulses. Due to the broad spectral width of the driving field, the oscillations are initiated between a ground state and a coherent superposition of excited states, or a ``wave packet,'' rather than a single excited state. Our experiments reveal an intricate dependence of the wave-packet phase on the intensity of the laser field. We confirm numerically that the effect is associated with the strong-field nature of the interaction and provide a qualitative picture by invoking a simple theoretical model.
We study experimentally the dynamical and decay properties of the stimulated nutation echo (SNE) in a two-level spin system, the signal of which allows the observation timescale of the driven coherence relaxation to be extended. This signal appears in the transient response of the system to the second pulse at time τ 1 from its start and coinciding with the duration of the first pulse. The information about the first pulse duration is imprinted into the population difference of the inhomogeneously broadened ensemble of the two-level absorbers. The decay of the SNE signal has two contributions. One originates from the population decay during the time τ between the two pulses. Another is caused by the coherence loss during the excitation by the first pulse and the reading time of the second pulse. Experimental results on the decay properties induced by these mechanisms are presented for the first time. We investigate the dependence of these decay rates on the pulse intensity and we examine its relationship with the anomalous (non-Bloch) decay of other coherent transients in solids.
The physical reasons for observing the splitting of optical lines several orders of magnitude smaller than the spectral width of a laser pulse are investigated. A theory of coherent and incoherent photon echo (PE) in an external static magnetic field and in the presence of a pulsed magnetic field, which causes oscillations of the PE intensity, is elaborated. It is shown that the periods of oscillations in the echo intensity, the echo duration, and the dimensions of the regions in the inhomogeneous line, where the excited ions are coherent, do not depend on the degree of coherence of the laser pulse and on the external static magnetic field. As follows from the theory, in the case of the coherent excitation of the echo, the amplitude of the intensity oscillations is independent of the external static magnetic field if the inhomogeneous line is symmetric. It is shown that the amplitude of the oscillations at the incoherent excitation of the echo is equal to the autocorrelation function of the distribution function of the transition frequency along the inhomogeneous line with the argument equal to the Zeeman splitting of the optical line in the external magnetic field. In this case, the experimental values of the oscillation amplitude are in good agreement with the calculated values of the autocorrelation function for the total inhomogeneous line in LuLiF 4 :Er 3+ ( 4 I 15/2 ⇒F 9/2 transition). In the same way, the autocorrelation function has been obtained for YLiF 4 :Er 3+ on the same transition.