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arXiv:cond-mat/0411345 · Physical Review Letters

Long-Time Behavior of Spin Echo

Boris V. Fine

Published Jun 20, 2005• 31 citations
arXivHTMLPDF

Abstract

It is predicted that (i) spin echoes have two kinds of generic long-time decays: either simple exponential, or a superposition of a monotonic and an oscillatory exponential decays; and (ii) the long-time behavior of spin echo and the long-time behavior of the corresponding homogeneous free induction decay are characterized by the same time constants. This prediction extends to various echo problems both within and beyond nuclear magnetic resonance. Experimental confirmation of this prediction would also support the notion of the eigenvalues of time evolution operators in large quantum systems.

DOI
10.1103/physrevlett.94.247601

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It is predicted that (i) spin echoes have two kinds of generic long-time decays: either simple exponential, or a superposition of a monotonic and an oscillatory exponential decays; and (ii) the long-time behavior of spin echo and the long-time behavior of the corresponding homogeneous free induction decay are characterized by the same time constants. This prediction extends to various echo problems both within and beyond nuclear magnetic resonance. Experimental confirmation of this prediction would also support the notion of the eigenvalues of time evolution operators in large quantum systems.
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 long-time behavior of the infinite temperature spin correlation functions describing the free induction decay in nuclear magnetic resonance and intermediate structure factors in inelastic neutron scattering is considered. These correlation functions are defined for one-, two- and three-dimensional infinite lattices of interacting spins, both classical and quantum. It is shown that, even though the characteristic time-scale of the long-time decay of the correlation functions considered is non-Markovian, the generic functional form of this decay is either simple exponential or exponential multiplied by cosine. This work contains (i) the summary of the existing experimental and numerical evidence of the above asymptotic behavior; (ii) theoretical explanation of this behavior; and (iii) semi-empirical analysis of various factors discriminating between the monotonic and the oscillatory long-time decays. The theory is based on a fairly strong conjecture that, as a result of chaos generated by spin dynamics, a Brownian-like Markovian description can be applied to the long-time properties of ensemble average quantities on a non-Markovian time-scale. The formalism resulting from that conjecture can be described as "correlated diffusion in finite volumes."
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.
The quantum properties of dynamic correlations in a system of an electron spin surrounded by nuclear spins under the conditions of free induction decay and electron spin echo have been studied. Analytical results for the time evolution of mutual information, classical part of correlations, and quantum part characterized by quantum discord have been obtained within the central-spin model in the high-temperature approximation. The same formulas describe discord in both free induction decay and spin echo although the time and magnetic field dependences are different because of difference in the parameters entering into the formulas. Changes in discord in the presence of the nuclear polarization β I in addition to the electron polarization β S have been calculated. It has been shown that the method of reduction of the density matrix to a two-spin electron-nuclear system provides a qualitatively correct description of pair correlations playing the main role at β S ≈ β I and small times. At large times, such correlations decay and multispin correlations ensuring nonzero mutual information and zero quantum discord become dominant.
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.
The amplitude of the electron-spin or photon echo from an ensemble of isolated atoms (echo atoms) interacting with magnetic-nuclear neighbors is calculated as a function of the magnitude and the orientation of an applied magnetic field $\stackrel{\ensuremath{\rightarrow}}{\mathrm{H}}$ and the separation between the two excitation pulses, in the limit that the magnitude of $\stackrel{\ensuremath{\rightarrow}}{\mathrm{H}}$ is large compared to the effective local fields at the echo-atom sites due to the nuclear neighbors. The general problem is reduced to the equivalent problem of two-level echo atoms interacting with nuclear neighbors. The latter problem is readily solved and applied to both spin and photon echoes. Experimentally observed spin-echo behavior in ruby is compared to calculated spin-echo behavior using no adjustable parameters, and is found to be in good agreement even when the magnetic field is tilted away from the optic axis thereby causing the echoes to disappear. Calculated photon-echo behavior, in which it is assumed that the interaction constants in the excited state are simply related to those in the ground state, shows the same general features as those that have been observed in experiment. Several results are presented to show the wide range of photon-echo behavior for different values of the interaction constants. A simple theory is presented which explains most observed echo behavior in a straightforward way.