Clustering and decoherence of correlated spins under double quantum dynamics

We present an improved approach for the study of the evolution of spin correlations and decoherence in multiple quantum nuclear magnetic resonance experiments. The infinite system, constituted by the protons of a polycrystalline adamantane sample, evolves under a double quantum Hamiltonian. The distribution of multiple quantum coherence orders is represented by a contribution of spin clusters with different sizes that exchange spins, increasing their size with the evolution time. A cluster with nearly exponential growth at all times is observed, in agreement with previous models. Remarkably, a small cluster that stabilizes in a size corresponding to 18 correlated spins is revealed. In addition, by performing a renormalization of the obtained data with the experimental Loschmidt echo, the contribution of the different clusters to the observable signal is determined. This procedure accounts for the effect of decoherence on the evolution of the system, and allows setting the range of confidence of the experimental data. Our analysis confirms the natural hint that, correlated states involving higher coherence orders are far more sensitive to the uncontrolled decoherent interactions, than those involving lower orders.

Figure 1

Schematic representation of the pulse sequence used to study the multiple quantum dynamics under a double quantum Hamiltonian. The reversed Hamiltonian $-H$ is obtained by changing the phases $x$ by $y$ and $-x$ by $-y$. The read-out pulse is cycled with ($\pm y$).

Figure 2

Allowed pathways through the Liouville space for a double quantum evolution.

Figure 3

Experimental MQC distribution $S_n$ at 960 μs, together with the fitting functions for $L=2$. Gauss 1 and Gauss 2 have been displayed separately. The inset displays the total fitting to the experimental curve, i.e., the sum of the two-Gaussian functions. The arrow in the inset marks the zero-order difference (ZOD).

Figure 4

Experimental MQC distributions at different evolution times (symbols) together with the corresponding total fittings to Eq. (5) (solid lines) in logarithmic scale. In all the cases, two-Gaussian functions ($L=2$) have been used. In all cases the coefficient of determination, $R^2$, is better than 99%.

Figure 5

ZOD as a function of evolution time.

Figure 6

Number of correlated spins (cluster size) for clusters $i=1$,2 as a function of time, representing the distribution of correlations in the system.

Figure 7

(Left) parameters ${\tilde{A}}_i$ for $i=1,2$, together with $\tilde{A}={\tilde{A}}_1+{\tilde{A}}_2$ and the LE as a function of the evolution time. (Right) parameters normalized to the LE, $A_i={\tilde{A}}_i/LE$ for $i=1,2$.

Figure 8

Experimental data for $\tau =1080\mu \mathrm{s}$. (Left) Fitting with a single-Gaussian function: $R^2=0.9937$. (Right) Fitting with two-Gaussian functions: $R^2=0.9993$.