Multiple-time Correlation Functions Research Articles

Dynamical properties of single-file diffusion

We study the statistics of a tagged particle in single-file diffusion, a one-dimensional interacting infinite-particle system in which the order of particles never changes. We compute the two-time correlation function for the displacement of the tagged particle for an arbitrary single-file system. We also discuss single-file analogs of the arcsine law and the law of the iterated logarithm characterizing the behavior of Brownian motion. Using a macroscopic fluctuation theory we devise a formalism giving the cumulant generating functional. In principle, this functional contains the full statistics of the tagged particle trajectory—the full single-time statistics, all multiple-time correlation functions, etc are merely special cases.

Equivalence of two Bochkov-Kuzovlev equalities in quantum two-level systems.

We present two kinds of Bochkov-Kuzovlev work equalities in a two-level system that is described by a quantum Markovian master equation. One is based on multiple time correlation functions and the other is based on the quantum trajectory viewpoint. We show that these two equalities are indeed equivalent. Importantly, this equivalence provides us a way to calculate the probability density function of the quantum work by solving the evolution equation for its characteristic function. We use a numerical model to verify these results.

Hilbert-space treatment of incoherent, time-resolved spectroscopy. I. Formalism, a tensorial classification of high-order orientational gratings and generalized MUPPETS “echoes”

Time-resolved spectroscopy that uses more than one incoherent excitation, and thus has multiple periods of time evolution, is becoming more common. A recent example is multiple population-period transient spectroscopy (MUPPETS), which is implemented as a high-order transient grating. In this paper, a formalism is developed to treat these types of incoherent spectroscopy in a manner that parallels the Liouville-pathway formalism used to treat multidimensional coherent spectroscopy. A Hilbert space of incoherent (population) states is defined and general expressions for transition and time-evolution operators acting on these states are derived from the corresponding quantum operators. This formalism describes incoherent experiments that involve an arbitrary number of temporal dimensions in terms of pathways through the Hilbert space. Each pathway is associated with a multiple-time rate-correlation function. Previous work has shown that these multiple-time correlation functions can measure heterogeneity in electronic-relaxation rates. Thus, they are an analog of coherent "echo" experiments, which measure heterogeneity in frequencies. We show that similar "MUPPETS echo" experiments can be done on any incoherent variable. For a dilute molecular solute, the Hilbert-space method leads to a systematic treatment of multidimensional transient gratings. The extension of irreducible-tensor methods to the incoherent Hilbert space results in a classification of orientational gratings of arbitrary order. The general methods developed in this paper are applied more specifically to single-photon, dipole transitions in the following paper.

Dynamics of fluctuations in non-Markovian systems

The dynamics of multiple time correlation functions (MTCFs) of a system in contact with a non-Markovian environment is relevant in many different contexts. In this article we study the dynamical equations that satisfy MTCFs of a system and how they are used to characterize its fluctuations. The non-Markovian character of the interaction between the system and its environment leads to dynamical equations different from those derived from the Quantum Regression Theorem. We further discuss an stochastic method to compute MTCFs. To illustrate the theory we apply it to the evaluation of the emission spectrum of an atom in contact with a non-Markovian environment. The emission spectrum strongly depends on the way the environment reacts to the absorption of a photon emitted by the atom. To cite this article: D. Alonso et al., C. R. Physique 8 (2007).

Hierarchy of equations of multiple-time correlation functions

In this paper we derive the evolution equations for non-Markovian multiple-time correlation functions of an open quantum system without using any approximation. We find that these equations conform an open hierarchy in which $N$-time correlation functions are dependent on $(N+1)$-time correlations. This hierarchy of equations is consistently obtained with two different methods: A first one based on Heisenberg equations of system operators, and a second one based on system propagators. The dependency on higher order correlations, and therefore the open hierarchy structure, only disappears in certain particular cases and when some hypothesis or approximations are considered in the equations. In this paper we consider a perturbative approximation and derive the general evolution equation for $N$-time correlations. This equation turns to depend only on $N$-time and lower order correlation functions, conforming a closed hierarchy structure that is useful for computational purposes.

Non-Markovian reduced propagator, multiple-time correlation functions, and master equations with general initial conditions in the weak-coupling limit

In this paper we derive the evolution equation for the reduced propagator, an object that evolves vectors of the Hilbert space of a system $\mathcal{S}$ interacting with an environment $\mathcal{B}$ in a non-Markovian way. This evolution is conditioned to certain initial and final states of the environment. Once an average over these environmental states is made, reduced propagators permit the evaluation of multiple-time correlation functions of system observables. When this average is done stochastically the reduced propagator evolves according to a stochastic Schr\"odinger equation. In addition, it is possible to obtain the evolution equations of the multiple-time correlation functions which generalize the well-known quantum regression theorem to the non-Markovian case. Here, both methods, stochastic and evolution equations, are described by assuming a weak coupling between system and environment. Finally, we show that reduced propagators can be used to obtain a master equation with general initial conditions, and not necessarily an initial vacuum state for the environment. We illustrate the theory with several examples.

Mode-Coupling Theory for Multiple-Time Correlation Functions of Tagged Particle Densities and Dynamical Filters Designed for Glassy Systems

The theoretical framework for higher-order correlation functions involving multiple times and multiple points in a classical, many-body system developed by Van Zon and Schofield [Phys. Rev. E 2002, 65, 011106] is extended here to include tagged particle densities. Such densities have found an intriguing application as proposed measures of dynamical heterogeneities in structural glasses. The theoretical formalism is based upon projection operator techniques which are used to isolate the slow time evolution of dynamical variables by expanding the slowly evolving component of arbitrary variables in an infinite basis composed of the products of slow variables of the system. The resulting formally exact mode-coupling expressions for multiple-point and multiple-time correlation functions are made tractable by applying the so-called N-ordering method. This theory is used to derive for moderate densities the leading mode coupling expressions for indicators of relaxation type and domain relaxation, which use dynamical filters that lead to multiple-time correlations of a tagged particle density. The mode coupling expressions for higher order correlation functions are also successfully tested against simulations of a hard sphere fluid at relatively low density.

Multiple-Time Correlation Functions for Non-Markovian Interaction: Beyond the Quantum Regression Theorem

We derive the dynamical equation of the reduced propagator, an object that evolves state vectors of the system conditioned to the dynamics of its environment, which is not necessarily in the vacuum state at the initial time. Such a reduced propagator is essential to obtain multiple-time correlation functions (MTCFs). We also study the evolution of MTCFs within the weak-coupling limit and show that the quantum regression theorem is, in general, not satisfied. We illustrate the theory in two different cases: first, solving an exact model, and, second, presenting the results of the numerical integration for a system coupled with a thermal environment through a nondiagonal interaction.

Silver ion dynamics in silver borate glasses: spectra and multiple-time correlation functions from 109Ag-NMR

The motion of silver ions in (AgI) x –(Ag 2O–B 2O 3) 1− x glasses with AgI concentrations of x = 0.5 and 0.7 was studied using 109Ag-NMR. The NMR spectra were analyzed in terms of a superposition of two different contributions. These are associated with Ag ions, which hop on vastly different time scales. The existence of dynamic heterogeneity, i.e. the existence of slow and of fast contributions to the hopping dynamics was directly demonstrated using a four-time stimulated-echo experiment. The results are compatible with an intrinsically exponential response. From measurements of the spectra and of two-time correlation functions a Gaussian distribution of energy barriers, hindering the ionic motion, could be mapped out.

Correlation experiments in nonlinear quantum mechanics

We show how one can compute multiple-time multi-particle correlation functions in nonlinear quantum mechanics in a way which guarantees locality of the formalism.

Multiple-point and multiple-time correlation functions in a hard-sphere fluid

A recent mode-coupling theory of higher-order correlation functions is tested on a simple hard-sphere fluid system at intermediate densities. Multiple-point and multiple-time correlation functions of the densities of conserved variables are calculated in the hydrodynamic limit and compared to results obtained from event-based molecular dynamics simulations. It is demonstrated that the mode-coupling theory results are in excellent agreement with the simulation results provided that dissipative couplings are included in the vertices appearing in the theory. In contrast, simplified mode-coupling theories in which the densities obey Gaussian statistics neglect important contributions to both the multiple point and multiple-time correlation functions on all time scales.

Mode-coupling theory for multiple-point and multiple-time correlation functions.

We present a theoretical framework for higher-order correlation functions involving multiple times and multiple points in a classical, many-body system. Such higher-order correlation functions have attracted much interest recently in the context of various forms of multidimensional spectroscopy, and have found an intriguing application as proposed measures of dynamical heterogeneities in structural glasses. The theoretical formalism is based upon projection operator techniques that are used to isolate the slow time evolution of dynamical variables by expanding the slowly evolving component of arbitrary variables in an infinite, "multilinear" basis composed of the products of slow variables of the system. Using the formalism, a formally exact mode coupling theory is derived for multiple-point and multiple-time correlation functions. The resulting expressions for higher-order correlation functions are made tractable by applying a rigorous perturbation scheme, called the N-ordering method, which is exact for systems with finite correlation lengths in the thermodynamic limit. The theory is contrasted with standard mode coupling theories in which the noise or fluctuating force appearing in the generalized Langevin equation is assumed to be Gaussian, and it is demonstrated that the non-Gaussian nature of the fluctuating forces leads to important contributions to higher-order correlation functions. Finally, the higher-order correlation functions are evaluated analytically for an ideal gas system for which it is shown that the mode coupling theory is exact.

Multiple-time correlation functions in spin-3/2 solid-state NMR spectroscopy

Stimulated echo spectroscopy of nonselectively excitable I = 3/2 nuclei offers new perspectives for the investigation of ultraslow motions predominantly in inorganic solids and solid-like materials. Conditions for the generation of pure, quadrupole modulated multipolar spin orders and for the detection of two- and four-time correlation functions are discussed. The case of spins I > 3/2 is also briefly considered. Copyright 2000 Academic Press.

Microscopic chaos from brownian motion?

We presented experimental evidence for the existence of microscopic chaos in the mesoscopic motion of a brownian particle in solution. We used standard techniques to analyse long trajectories of a brownian particle and inferred a dynamical entropy from this analysis. We showed that this dynamical entropy can be accessed experimentally by measuring multiple-time correlation functions for the particle. The dynamical entropy was then used to provide a positive lower bound for the sum of the Lyapunov exponents for the underlying deterministic dynamical system composed of the fluid and brownian particles. We concluded that the positive dynamical entropy, obtained experimen-tally, was evidence for the existence of positive Lyapunov exponents in the underlying dynamics, and hence for the existence of microscopic chaos generated by a dynamical instability.