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arXiv:1112.3626 · Physical Review Letters

Eigenmodes in the Long-Time Behavior of a Coupled Spin System Measured with Nuclear Magnetic Resonance

Benno Meier, Jonas Kohlrautz, Jürgen Haase

Published Apr 25, 2012• 26 citations
arXivHTMLPDF

Abstract

The many-body quantum dynamics of dipolar coupled nuclear spins I=1/2 on an otherwise isolated cubic lattice are studied with nuclear magnetic resonance. By increasing the signal-to-noise ratio by 2 orders of magnitude compared with previous reports for the free induction decay (FID) of (19)F in CaF(2) we obtain new insight into its long-time behavior. We confirm that the tail of the FID is an exponentially decaying cosine, but our measurements reveal a second decay mode with comparable frequency but twice the decay constant. This result is in agreement with a recent theoretical prediction for the FID in terms of eigenvalues for the time evolution of chaotic many-body quantum systems.

DOI
10.1103/physrevlett.108.177602

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The many-body quantum dynamics of dipolar coupled nuclear spins I=1/2 on an otherwise isolated cubic lattice are studied with nuclear magnetic resonance. By increasing the signal-to-noise ratio by 2 orders of magnitude compared with previous reports for the free induction decay (FID) of (19)F in CaF(2) we obtain new insight into its long-time behavior. We confirm that the tail of the FID is an exponentially decaying cosine, but our measurements reveal a second decay mode with comparable frequency but twice the decay constant. This result is in agreement with a recent theoretical prediction for the FID in terms of eigenvalues for the time evolution of chaotic many-body quantum systems.
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."
Abstract In the past decade, nuclear magnetic resonance (NMR) has been actively used to study the basic principles of quantum computers. It is assumed that quantum correlations play a significant role in their performance. They exist at both low and high temperatures. At the same time, the time correlation functions of nuclear spin systems of solids determine the observed signals in traditional NMR implementations. The separation of such signals into quantum and classical components has not previously been carried out and will be performed in this study for the most important of the correlational functions observed in magnetic resonance: the free induction decay (FID).
Quantum unitary evolution typically leads to thermalization of generic interacting many-body systems. There are very few known general methods for reversing this process, and we focus on the magic echo, a radio-frequency pulse sequence known to approximately "rewind" the time evolution of dipolar coupled homonuclear spin systems in a large magnetic field. By combining analytic, numerical, and experimental results we systematically investigate factors leading to the degradation of magic echoes, as observed in reduced revival of mean transverse magnetization. Going beyond the conventional analysis based on mean magnetization we use a phase encoding technique to measure the growth of spin correlations in the density matrix at different points in time following magic echoes of varied durations and compare the results to those obtained during a free induction decay (FID). While considerable differences are documented at short times, the long-time behavior of the density matrix appears to be remarkably universal among the types of initial states considered - simple low order multispin correlations are observed to decay exponentially at the same rate, seeding the onset of increasingly complex high order correlations. This manifestly athermal process is constrained by conservation of the second moment of the spectrum of the density matrix and proceeds indefinitely, assuming unitary dynamics.
The quantum properties of dynamic correlations in a system of an electron spin surrounded by nuclear spins under the conditions of free induction decay (FID) and 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 the quantum discord in both the FID and the spin echo although the forms of the dependences are different because of difference in the parameters entering into the formulas. Discord of the spin echo compared with the FID has a strong dependence on time at short times, and it tends to zero with decreasing of the magnetic field, whereas in the case of the FID it reaches a plateau.
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.
In this work, we overview time-reversal nuclear magnetic resonance (NMR) experiments in many-spin systems evolving under the dipolar Hamiltonian. The Loschmidt echo (LE) in NMR is the signal of excitations which, after evolving with a forward Hamiltonian, is recovered by means of a backward evolution. The presence of non-diagonal terms in the non-equilibrium density matrix of the many-body state is directly monitored experimentally by encoding the multiple quantum coherences. This enables a spin counting procedure, giving information on the spreading of an excitation through the Hilbert space and the formation of clusters of correlated spins. Two samples representing different spin systems with coupled networks were used in the experiments. Protons in polycrystalline ferrocene correspond to an 'infinite' network. By contrast, the liquid crystal N-(4-methoxybenzylidene)-4-butylaniline in the nematic mesophase represents a finite proton system with a hierarchical set of couplings. A close connection was established between the LE decay and the spin counting measurements, confirming the hypothesis that the complexity of the system is driven by the coherent dynamics.