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Physical Review A

Clustering and decoherence of correlated spins under double quantum dynamics

Claudia M. Sánchez, Rodolfo Hector Acosta, Patricia R. Levstein, Horacio M. Pastawski, Ana Karina Chattah

Published Oct 30, 2014• 29 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.90.042122

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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.
In this work we present solid-state nuclear magnetic-resonance experiments to study decoherence in the dynamics of many-spin systems. We characterize the global Loschmidt echo and the distribution of multiple-quantum coherence orders during the evolution under dipolar and double-quantum Hamiltonians. To study an infinite 1H^{1}\text{H}1H system and a closed cluster of nuclear spins we use polycrystalline adamantane and the liquid crystal 5CB in the nematic mesophase, respectively, as model systems. The infinite or finite nature of the system is clearly manifested through spin counting measurements. Comparison between experimental and numerical results gives insights on the decoherence mechanisms in these systems. Contrastingly, the dominating mechanism in 5CB affects all coherence orders in the same way, while in adamantane higher orders of coherence decay faster than the lower ones.
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.
By varying the magnitude of the effective interaction between spins in relation to the perturbations, we study the decoherence behavior in a connected proton system.Making use of the Magnus expansion, we introduce a NMR pulse sequence that generates an average Hamiltonian with Double Quantum terms multiplied by a scaling factor, 𝛿, with the possibility to take positive and negative values.The performance of the pulse sequence for different values of the scaling factors was validated in polycrystalline adamantane, by observing the evolution of the polarization.A time reversal procedure, accessible through the change of sign in the controlled Hamiltonian, was necessary to observe multiple quantum coherences.The spin counting develops a characteristic growth in two species of clusters for the scaled time.The influence of the scaling factor on the reversibility was observed through the behavior of the Loschmidt echoes, which decayed faster as the scaling factor increases.From the analysis of dynamics and its reversibility, we extracted characteristic times for the spin diffusion, 𝑇 𝛿 2 and the intrinsic decoherence decay, 𝑇 𝛿 3 for each scaling factor 𝛿, and perturbation time scale, 𝑇 𝛴 .Observing the dependence of reversibility vs. perturbation rates, both normalized with the spin diffusion rate, we find that in the limit of low perturbations, 𝑇 𝛿 2 ∕𝑇 𝛿 3 deviates from the linear dependence on 𝑇 𝛿 2 ∕𝑇 𝛴 that corresponds to strong perturbation.The asymptotic value 𝑇 2 ∕𝑇 3 ≈ 0.15 as 𝑇 𝛿 2 ∕𝑇 𝛴 vanishes, gives evidence that the main source of irreversibility is the intrinsic decoherence associated to the chaotic many-body dynamics of the system.
We performed Loschmidt echo nuclear magnetic resonance experiments to study decoherence under a scaled dipolar Hamiltonian by means of a symmetrical time-reversal pulse sequence denominated Proportionally Refocused Loschmidt (PRL) echo. The many-spin system represented by the protons in polycrystalline adamantane evolves through two steps of evolution characterized by the secular part of the dipolar Hamiltonian, scaled down with a factor |k| and opposite signs. The scaling factor can be varied continuously from 0 to 1/2, giving access to a range of complexity in the dynamics. The experimental results for the Loschmidt echoes showed a spreading of the decay rates that correlate directly to the scaling factors |k|, giving evidence that the decoherence is partially governed by the coherent dynamics. The average Hamiltonian theory was applied to give an insight into the spin dynamics during the pulse sequence. The calculations were performed for every single radio frequency block in contrast to the most widely used form. The first order of the average Hamiltonian numerically computed for an 8-spin system showed decay rates that progressively decrease as the secular dipolar Hamiltonian becomes weaker. Notably, the first order Hamiltonian term neglected by conventional calculations yielded an explanation for the ordering of the experimental decoherence rates. However, there is a strong overall decoherence observed in the experiments which is not reflected by the theoretical results. The fact that the non-inverted terms do not account for this effect is a challenging topic. A number of experiments to further explore the relation of the complete Hamiltonian with this dominant decoherence rate are proposed.
The manipulation of quantum information in large systems requires precise control of quantum systems that are out of equilibrium. As the size of the system increases, its fragility in response to external perturbations and intrinsic decoherence processes also increases. The degradation of the system response makes accurate measurements a challenging and time-consuming task. However, quantum information lifetime enhancement can be achieved by dynamical decoupling techniques, where an external drive with a frequency much higher than the system's internal evolution renders signal acquisition with decay times greater than 1000-fold. In this study, we demonstrate that the system response during a prethermal period, subject to Floquet control, can be utilized to probe the multiple quantum evolution of dense and highly connected spin systems. This approach exhibits an enhanced sensitivity at a reduced experimental time. The enhanced signal-to-noise ratio achieved enabled the use of numerical inversion strategies to model the evolution of the excited multiple quantum coherences, which describe the number of correlated spins within a cluster. We observed that the increase in the number of correlated spins with multiple quantum evolution is accompanied by an increase in the distribution of spin cluster sizes, which follows a quadratic law.
Among the most important parameters for the usefulness of quantum computers are the size of the quantum register and the decoherence time for the quantum information. The decoherence time is expected to get shorter with the number of correlated qubits, but experimental data are only available for small numbers of qubits. Solid-state nuclear magnetic resonance allows one to correlate large numbers of qubits (several hundred) and measure their decoherence rates. We use a modified magnetic dipole-dipole interaction to correlate the proton spins in a solid sample and observe the decay of the resulting highly correlated states. By systematically varying the number of correlated spins, we measure the increase of the decoherence rate with the size of the quantum register.