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Physical Review A

Oscillatory free-induction decay and oscillatory spin echoes

Masakazu Kunitomo, T. Endo, Shunsuke Nakanishi, T. Hashi

Published Apr 1, 1982• 30 citations
Publisher

Abstract

Oscillatory free-induction decay (FID) theoretically predicted by Schenzle, Wong, and Brewer was observed by using NMR in an inhomogeneously broadened proton system. Oscillatory behaviors were also observed, in the same system, in two-pulse echoes generated by large-area pulses. These phenomena appear in a highly inhomogeneous-broadened atomic or spin two-level system driven by resonant laser or rf pulses with areas equal to or larger than $2\ensuremath{\pi}$. Detailed studies were made for spin-locked and notched echoes. Behaviors of the oscillatory FID and the oscillatory echoes were theoretically examined by solving the Bloch equations and numerically integrating the solution over the inhomogeneous broadening. Excellent agreement between experiment and theory was obtained. Analytic expressions for the shapes of the locked- and notched-type echoes are obtained, which interpret the main characteristics of these echoes, when the infinite linewidth of the inhomogeneous broadening and the $\ensuremath{\delta}$-function-type excitation for one of the pulses are assumed.

DOI
10.1103/physreva.25.2235

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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 $T$. 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 $T$ 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.

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Physical Review A•Dec 1, 1974•44 citations
Oscillatory free-induction decay (FID) theoretically predicted by Schenzle, Wong, and Brewer was observed by using NMR in an inhomogeneously broadened proton system. Oscillatory behaviors were also observed, in the same system, in two-pulse echoes generated by large-area pulses. These phenomena appear in a highly inhomogeneous-broadened atomic or spin two-level system driven by resonant laser or rf pulses with areas equal to or larger than 2\ensuremathπ2\ensuremath{\pi}2\ensuremathπ. Detailed studies were made for spin-locked and notched echoes. Behaviors of the oscillatory FID and the oscillatory echoes were theoretically examined by solving the Bloch equations and numerically integrating the solution over the inhomogeneous broadening. Excellent agreement between experiment and theory was obtained. Analytic expressions for the shapes of the locked- and notched-type echoes are obtained, which interpret the main characteristics of these echoes, when the infinite linewidth of the inhomogeneous broadening and the \ensuremathδ\ensuremath{\delta}\ensuremathδ-function-type excitation for one of the pulses are assumed.
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.
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.
A simple application of the Stark-pulse technique, developed by Brewer and Shoemaker, demonstrates optical free induction decay-the optical analog of free induction decay in NMR. A molecular sample which is coherently prepared by a cw laser beam exhibits such a decay when it is suddenly switched out of resonance by a Stark field. Observations are presented for a nondegenerate Doppler-broadened infrared transition of NH2{\mathrm{H}}_{2}H2​D, where the sample is optically thin, and the decay behavior can be compared quantitatively with a solution of the Bloch equations. When the molecular sample is prepared under steady-state conditions, the solutions are analytic; for pulse excitation, a numerical solution is required. The treatment invokes a hard-collision relaxation model. Such characteristics as the abrupt termination of the decay and the related edge echo, which result from Doppler dephasing, can be examined for Stark pulses of finite extent.
A fundamental theorem in nuclear magnetic resonance and quantum optics is derived for an atomic system subject to inhomogeneous broadening and coherent preparation by a pulsed electromagnetic field of arbitrary shape. We show that the coherent emission, free induction decay or echo, which follows is confined to an interval which precisely equals the duration of the applied pulse sequence. The theorem applies to two-level or multilevel transitions for which the inhomogeneous linewidth is infinitely broad and the sample is optically thin.
A new feature of optical free induction decay (FID) is observed in a coherently prepared sample of 13CH^{13}\mathrm{CH}13CH3F_{3}\mathrm{F}3​F by the method of Stark switching. The preparative phase is due to the resonant steady-state excitation by a cw laser beam, and is followed by FID upon sudden application of a dc Stark field that switches the molecular sample out of resonance. The emission is not observed as a simple decay but instead appears as a train of sharp pulses regularly spaced in time as a result of a repetitive interference. This situation arises because an entire set of infrared transitions within the Stark split manifold are initially prepared, in contrast to our previous study of a nondegenerate transition. The emission, which beats with the laser, produces a heterodyne beat spectrum, consisting of a set of regularly spaced frequencies, that is the Fourier transform of the slowly decaying pulse progression observed. We thus demonstrate what is the optical analog of the well-known NMR method of high-resolution pulse Fourier spectroscopy. The detailed behavior of the pulse train agrees well with an FID theory that assumes the transitions to be uncoupled. The subtle behavior of FID near the time origin is explored also by approximate analytic expressions that reveal either a near zero amplitude or sizable amplitude depending on the degree of saturation in the preparative stage. The experimental technique discussed offers an attractive way for obtaining high-resolution optical spectra without Doppler broadening, and for generating an optical pulse train whose time scale can be compressed by simply increasing the Stark field.