The statistical model developed in this study allows us to calculate the shape of multiple quantum (MQ) NMR spectra (the dependence of the amplitudes of the corresponding MQ coherences on their orders) by decomposing the desired time-correlation functions (TCFs) over an infinite set of orthogonal operators and by using some well-known facts from the physics of traditional model systems. The resulting expression contains series of gradually increasing numbers of spins in clusters of correlated spins depending on time. The influence of the possible degradation of these clusters on the shape of the spectra is taken into account. Analytical and numerical calculations are performed for various parameter values included in the final expressions. The developed theory adequately describes the results of the numerical calculations of the MQ spectra performed by us and experiments: the transformation of the Gaussian profile into an exponential one, the asymptotics (wings) of the spectrum depending on the coherence order M, and the dependence of the relaxation rate of the MQ spectrum on M, as well as the narrowing and stabilization of the MQ spectrum under the influence of a perturbation.

В настоящей работе впервые прямым способом в мезоскопическом масштабе длин решена «флуктуационная теорема». Получено аналитическое выражение для парной корреляционной функции критического состояния флюида, параметром которой является дисперсия плотности, легко восстанавливаемая из оптических экспериментов. Сравнение нашего результата с парной корреляционной функцией Орнштейна—Цернике позволило определить поправку Фишера η = 0,25, повышающую сходимость. Показано, что полученные результаты дают надежную теоретическую основу для оптической диагностики статистического состояния критических флюидов. In this paper we have solved the «fluctuation theorem» on the mesoscopic length scale for the first time in a direct way. An analytical expression for the pair correlation function of the critical state of fluid was obtained, where the density variance is the parameter that could be easily defined by optical measurements. By comparison of our result with the Ornstein—Zernike pair correlation function we have got the Fisher index η= 0.25 to increase the convergence. It was shown that the obtained results provide a reliable theoretical basis for optical diagnostics of the critical state of fluids.

The relaxation of the components of the multiple-quantum NMR spectrum of a solid under the effect of the dipole–dipole interactions during the evolution period is considered. It is taken into account that clusters of dynamically correlated spins of different sizes are formed in the preparatory period, and their degradation depends on their size and coherence order. To calculate the size distribution function of clusters and their degradation function, a physical model including relaxation processes is developed. Using this model, an analytical result for a multiple-quantum spectrum is obtained. Agreement is obtained between the theoretical and experimental dependences of the coherence degradation rates in adamantane scaled by the square root of the average cluster size. The parameters of the above functions are found from the comparison of these dependences.

The relaxation of the components of the multiple-quantum NMR spectrum of a solid under the effect of the dipole–dipole interactions during the evolution period is considered. It is taken into account that clusters of dynamically correlated spins of different sizes are formed in the preparatory period, and their degradation depends on their size and coherence order. To calculate the size distribution function of clusters and their degradation function, a physical model including relaxation processes is developed. Using this model, an analytical result for a multiple-quantum spectrum is obtained. Agreement is obtained between the theoretical and experimental dependences of the coherence degradation rates in adamantane scaled by the square root of the average cluster size. The parameters of the above functions are found from the comparison of these dependences.

A fundamental problem of the optical diagnostics of the supercritical fluid (SF) structure and its statistical properties in the vicinity of the Widom line is considered. The solution of this problem requires approaches that allow us to bring the peculiarities of optical measurement data in line with the peculiarities of the state of the fluid. The results obtained in the past ten years on the problem of the Widom lines in nonpolar supercritical media are briefly reviewed. Particular attention is given to the optical measurement data in the Widom region, namely, measurements of the nonlinear contribution to the refractive index and measurements of the Rayleigh light scattering intensity. It is demonstrated that these data can serve as a base for mutually complementary methods for the optical diagnostics of an SF state. As an example, the data on small-angle light scattering by SF-CO2 were used to restore its pair correlation function, the temperature dependence of which fundamentally differs from that of the Ornstein–Zernike pair correlation function. It is noted that the method based on measuring the Rayleigh scattering intensity is general in nature and can be applied to any random unordered molecular media, including the atmosphere.

Multiple-quantum (MQ) solid-state NMR spectroscopy allows the growth of multiple-spin correlations and, thus, the spreading of quantum information in the object under study to be observed. Recently, in [11] it was proposed to control this process through a controlled perturbation added to the effective Hamiltonian that causes degradation of correlated spin clusters with a rate determined by the number of spins K in a cluster. However, this perturbation can also lead to degradation whose rate is determined by the coherence order M . In this paper, to investigate the influence of a small added perturbation, we used an expansion into orthogonal operators that allowed the cluster size distribution to be taken into account. In our calculations we realized a simple model with known amplitudes of the expansion into a complete set of orthogonal operators in the absence of a perturbation. We performed numerical calculations of the “preparation time” dependences of the MQ spectra, their second moments, and the coherence orders at which the MQ spectra decrease by a factor of e as well as the average correlated spin cluster sizes K̅ . The coherence-order-dependent contribution to the degradation is shown to change the shape of the MQ spectrum. In particular, as the preparation time increases, the MQ spectrum can be stabilized, while the growth of K̅ is retained. Due to the change in the shape of the MQ spectrum, the relations of its characteristics to the number K̅ change compared to those for the Gaussian function (traditionally used to process the experiments). These changes should be taken into account when studying the spreading of quantum information through MQ spectroscopy.

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).

Multiple quantum (MQ) NMR spectroscopy of solids allows one to observe the growth and decay of multispin correlations. As a rule, the average size of the cluster of correlated spins is extracted from the width of the MQ spectrum. In the present article, the size distribution of such clusters is explored. To obtain the above distribution, the solutions for the amplitudes of the decomposition over complete sets of orthogonal operators for the two different models were used. By means of these models, we have taken into account the dependence of cluster degradation (the degradation of a cluster means, e.g., destruction of correlations in cluster or loss of particles in it) through two positions. The first one defines by the cluster size while the second one depends on the MQ coherence order of the cluster. It is shown that in dependence of the relation the rates of these degradation processes, the width of the MQ spectrum carries different information. If the first process is faster that the second one, then the width of the MQ spectrum is still determined by the average cluster size. When the velocity ratio becomes inverse, the width of the MQ spectrum takes on a smaller value, which is a consequence of the faster degradation of the MQ spectrum components with large orders of coherence.

On the basis of the earlier developed statistical theory of the growth of the effective size of correlated clusters (the number of correlated spins), an expression for the shape of the multiple-quantum (MQ) NMR spectrum is obtained that takes into account the loss of coherence in a spin system due, for example, to the controlled intervention of the experimenter. It is shown that the scrambling and decoherence processes in the MQ spectrum of the multiparticle system of a solid are not separated, unlike the corresponding spectra of some large isolated molecules [27] in a solution. The relations obtained allow one to extract the necessary information about the above processes (scrambling and decoherence) from the dependence of the MQ spectra on experimental parameters.

Based on the proposed theory, we have investigated the shape of the NMR absorption spectra for 13C and 29Si nuclei in diamond and silicon crystals attributable to the internuclear dipole–dipole interaction. In accordance with the available experimental data, we have considered both crystals with a 100% content of magnetoactive isotopes and crystals with a comparatively low dilution by nonmagnetic nuclei. The time correlation functions (the first of which is the Fourier transform of the NMR spectrum) arising in an infinite chain of coupled differential equations are shown to be mutually similar with a slight time delay. The proposed theory allows the spectrum to be calculated analytically. The results obtained agree satisfactorily with the experimental ones. It is noted that the mutual similarity of the time correlation functions is probably a corollary of the development of dynamical chaos in the system

A change in the time dependence of the second moment of the distribution of intensities of coherences with various orders in the spectrum of multiple-quantum NMR in a solid at the inclusion of an inhomogeneous magnetic field in the effective interaction is studied. Both the secular dipole–dipole and nonspecular twoquantum interactions are considered as nucleus–nucleus interactions, which correspond to traditional experimental realizations. It is shown that, with an increase in the magnitude of the inhomogeneous field, an exponential increase in the second moment of multiple-quantum NMR with time changes to a power-law increase. The results obtained in this work indicate that this second moment, which determines the average number of dynamically correlated spins, can be used as a convenient characteristic for studying a transition to a many-body localized state.

The shape of NMR absorption line for typical ionic crystals, molecular crystals, and glasses is studied. The proposed theory and available experimental results suggest that the shape of NMR spectra of conventional dielectric crystals (and even molecular) is the convolution of a frequency-truncated nearly-rectangle-shaped function (characteristic oscillations in free precession signals) and a Gaussian-like function. A Gaussian-like shape of the spectra of glasses with a rigid structure (no oscillations in the free precession signal) is probably associated with a random scatter of interatomic distances. The results are interpreted within the framework of the proposed theory. It is demonstrated that, at least for solids the lattice of which contains no isolated groups of nuclei and are not quasi-one-dimensional, the structure only weakly affects the shape of the spectrum, which in turn is associated with the onset of dynamic chaos in the spin system.

As opposed to traditional methods of multiple-quantum NMR spectroscopy authors of the article [G.A. Alvarez, D. Suter, Phys. Rev. A 84, 012320 (2011)] generated the effective double-quantum Hamiltonian, with the slight adding usual secular dipole-dipole Hamiltonian to the first one at the stage of the correlations preparing (so the perturbation appeared). At this framework the width of MQ spectra reveals tending to any constant amount with the growing up of the time. Also there were shown that the spectral width decreases with increasing perturbation strength. The growing up of the cluster size as it was supposed by G.A. Alvarez and D. Suter was restricted by mentioned perturbation. We are assuming an alternative explanation. The growing up of cluster size still goes on but the width of MQ spectrum becomes stable because of different decay rate of MQ-spectral components in dependence on theirs location in spectrum. Here we are calculating the widths of MQ spectra in dependence on the time of "preparation" for different values of the perturbation strength and by this way we get the dependence of stabilized amount of MQ-spectrum width on mentioned strength. So, the excellent compliance obtains with the analogues experimental relation observed in Ref. [Phys. Rev. A 84, 012320 (2011)].

A modification of the widespread phenomenological model theory of multiple-quantum (MQ) nuclear magnetic resonance spectra of a single cluster of correlated spins has been developed. In contrast to the mentioned theory, the size distribution of such clusters has been consistently taken into account. To obtain the distribution, solutions for the amplitudes of the expansion in the complete set of orthogonal operators are used. Expressions specifying the dependence of the profile of the intensities of MQ coherences on their number n (order) have been obtained. The total form of the dependence has been evaluated by means of the numerical implementation of the resulting expressions. The asymptotic expressions for large n values (wings of the spectrum) have been obtained analytically by the saddle-point method. It has been shown that the dependence under study has a Gaussian central part and exponential wings. The results obtained are in agreement with the previous calculations for some model systems and existing experimental data.

We develop a theory that allows considering and describing the development of multiparticle correlations in paramagnetic spin systems. We show that in crystals with many equivalent nearest neighbors around a spin in a lattice, an infinite system (of size ∼ 1023) of coupled differential equations for time correlation functions describing multiparticle correlations is reducible to the diffusion equation with an imaginary diffusion coefficient. The equation can be solved analytically in the lowest-order approximation of the theory. The equation obtained in the next approximation must be solved numerically because a discontinuity of the diffusion coefficient appears. The obtained results agree well with experimental data. The observed mutual similarity of the calculated time correlation functions and several other characteristic features appearing in the spin system dynamics are consequences of the development of dynamical chaos.