Three-time Correlation Function Research Articles

Persistent oscillations in correlation function: from the perspective of past quantum states

Abstract The recent past quantum state formalism (2013 Phys. Rev. Lett. 111, 160401) indicates that the outcome probability of the measurement performed on a quantum system at time t shows dependence on the measurement events recorded until that time, and is also revised by the outcomes acquired after time t. We apply the past quantum state formalism to calculate the three-time correlation function of a radiation field and to analyze the conditional dynamics on the pre- and postselection by the first and last detection events. The persistent oscillations in the three-time field correlation function, which cannot be well understood by the usual quantum regression theorem, are straightforwardly explained by the probabilistic knowledge of the past and the future through the past quantum state reasoning.

Snapshotting quantum dynamics at multiple time points

Measurement-induced state disturbance is a major challenge in obtaining quantum statistics at multiple time points. We propose a method to extract dynamic information from a quantum system at intermediate time points, namely snapshotting quantum dynamics. To this end, we apply classical post-processing after performing the ancilla-assisted measurements to cancel out the impact of the measurements at each time point. Based on this, we reconstruct a multi-time quasi-probability distribution (QPD) that correctly recovers the probability distributions at the respective time points. Our approach can also be applied to simultaneously extract exponentially many correlation functions with various time-orderings. We provide a proof-of-principle experimental demonstration of the proposed protocol using a dual-species trapped-ion system by employing 171Yb+ and 138Ba+ ions as the system and the ancilla, respectively. Multi-time measurements are performed by repeated initialization and detection of the ancilla state without directly measuring the system state. The two- and three-time QPDs and correlation functions are reconstructed reliably from the experiment, negativity and complex values in the QPDs clearly indicate a contribution of the quantum coherence throughout dynamics.

Measurement of linear response functions in Nuclear Magnetic Resonance

We measure multi-time correlation functions of a set of Pauli operators on a two-level system, which can be used to retrieve its associated linear response functions. The two-level system is an effective spin constructed from the nuclear spins of 1H atoms in a solution of 13C-labeled chloroform. Response functions characterize the linear response of the system to a family of perturbations, allowing us to compute physical quantities such as the magnetic susceptibility of the effective spin. We use techniques exported from quantum information to measure time correlations on the two-level system. This approach requires the use of an ancillary qubit encoded in the nuclear spins of the 13C atoms and a sequence of controlled operations. Moreover, we demonstrate the ability of such a quantum platform to compute time-correlation functions of arbitrary order, which relate to higher-order corrections of perturbative methods. Particularly, we show three-time correlation functions for arbitrary times, and we also measure time correlation functions at fixed times up to tenth order.

Couplings between hierarchical conformational dynamics from multi-time correlation functions and two-dimensional lifetime spectra: Application to adenylate kinase.

An analytical method based on a three-time correlation function and the corresponding two-dimensional (2D) lifetime spectrum is developed to elucidate the time-dependent couplings between the multi-timescale (i.e., hierarchical) conformational dynamics in heterogeneous systems such as proteins. In analogy with 2D NMR, IR, electronic, and fluorescence spectroscopies, the waiting-time dependence of the off-diagonal peaks in the 2D lifetime spectra can provide a quantitative description of the dynamical correlations between the conformational motions with different lifetimes. The present method is applied to intrinsic conformational changes of substrate-free adenylate kinase (AKE) using long-time coarse-grained molecular dynamics simulations. It is found that the hierarchical conformational dynamics arise from the intra-domain structural transitions among conformational substates of AKE by analyzing the one-time correlation functions and one-dimensional lifetime spectra for the donor-acceptor distances corresponding to single-molecule Förster resonance energy transfer experiments with the use of the principal component analysis. In addition, the complicated waiting-time dependence of the off-diagonal peaks in the 2D lifetime spectra for the donor-acceptor distances is attributed to the fact that the time evolution of the couplings between the conformational dynamics depends upon both the spatial and temporal characters of the system. The present method is expected to shed light on the biological relationship among the structure, dynamics, and function.

Multiple length and time scales of dynamic heterogeneities in model glass-forming liquids: A systematic analysis of multi-point and multi-time correlations

We report an extensive and systematic investigation of the multi-point and multi-time correlation functions to reveal the spatio-temporal structures of dynamic heterogeneities in glass-forming liquids. Molecular dynamics simulations are carried out for the supercooled states of various prototype models of glass-forming liquids such as binary Kob-Andersen, Wahnström, soft-sphere, and network-forming liquids. While the first three models act as fragile liquids exhibiting super-Arrhenius temperature dependence in their relaxation times, the last is a strong glass-former exhibiting Arrhenius behavior. First, we quantify the length scale of the dynamic heterogeneities utilizing the four-point correlation function. The growth of the dynamic length scale with decreasing temperature is characterized by various scaling relations that are analogous to the critical phenomena. We also examine how the growth of the length scale depends upon the model employed. Second, the four-point correlation function is extended to a three-time correlation function to characterize the temporal structures of the dynamic heterogeneities based on our previous studies [K. Kim and S. Saito, Phys. Rev. E 79, 060501(R) (2009); and J. Chem. Phys. 133, 044511 (2010)]. We provide comprehensive numerical results obtained from the three-time correlation function for the above models. From these calculations, we examine the time scale of the dynamic heterogeneities and determine the associated lifetime in a consistent and systematic way. Our results indicate that the lifetime of the dynamical heterogeneities becomes much longer than the α-relaxation time determined from a two-point correlation function in fragile liquids. The decoupling between the two time scales is remarkable, particularly in supercooled states, and the time scales differ by more than an order of magnitude in a more fragile liquid. In contrast, the lifetime is shorter than the α-relaxation time in tetrahedral network-forming strong liquid, even at lower temperatures.

NMR Multi-Time Correlation Functions of Ion Dynamics in Solids

Abstract We show that NMR multi-time correlation functions provide interesting new insights into the nature of lithium and silver ion dynamics in solids. For solid ion conductors, they usually probe the elementary jumps of the long-range charge transport. NMR two-time correlation functions yield rates and activation energies of the ionic hopping motion. Moreover, they reveal that the ionic relaxation exhibits a strong nonexponentiality in the studied crystals and glasses. NMR three-time correlation functions enable quantitative determination of the origin of the nonexponentiality. For various solid-state electrolytes, the analysis shows that pronounced dynamical heterogeneities govern the ionic hopping motion. If dynamical heterogeneities exist, NMR four-time correlation functions allow one to measure the time scale of exchange processes within the rate distributions. For silver ion dynamics in a crystalline and a glassy material, the results indicate that initially slow ions exhibit random new rates from broad rate distributions after very few jump events.

Role of the Lifetime of Dynamical Heterogeneity in the Frequency-Dependent Stokes–Einstein Relation of Supercooled Liquids

We previously investigated the lifetime of dynamical heterogeneity in supercooled liquids in terms of a four-point, three-time density correlation function via molecular dynamics simulations [Phys. Rev. E 79 (2009) 060501(R); J. Chem. Phys. 133 (2010) 044511]. In this letter, we examine the physical role of the lifetime of dynamicsl heterogeneity in the violation of the Stokes–Einstein (SE) relation by calculating the frequency-dependent viscosity and self-diffusion constant. Our results demonstrate that the SE relation holds up to the time scale at which the non-Gaussian parameter is maximum. Large violations of the SE relation are characterized by time scales longer than the lifetime of the dynamical heterogeneity in highly supercooled liquids.

Hidden slow time scale of correlated motions in supercooled liquids: Multi-time correlation function approach

A three-time correlation function of particle displacements is introduced and numerically calculated by performing molecular dynamics simulations of binary soft-sphere supercooled systems. It is found that the two-dimensional representations of the three-time correlation function reveal couplings of particle motions that exist over a wide range of time scales. Furthermore, it is demonstrated that the systematic change in the second time interval in the three-time correlation function enables us to analyze how the correlations in mobility decay with time. From this analysis, the characteristic time scale of dynamical heterogeneity is quantified. We find that the dynamical heterogeneity time scale becomes much slower than the α-relaxation time as the temperature decreased.

Multi-time density correlation functions in glass-forming liquids: Probing dynamical heterogeneity and its lifetime

A multi-time extension of a density correlation function is introduced to reveal temporal information about dynamical heterogeneity in glass-forming liquids. We utilize a multi-time correlation function that is analogous to the higher-order response function analyzed in multidimensional nonlinear spectroscopy. Here, we provide comprehensive numerical results of the four-point, three-time density correlation function from longtime trajectories generated by molecular dynamics simulations of glass-forming binary soft-sphere mixtures. We confirm that the two-dimensional representations in both time and frequency domains are sensitive to the dynamical heterogeneity and that these reveal the couplings of correlated motions, which exist over a wide range of time scales. The correlated motions detected by the three-time correlation function are divided into mobile and immobile contributions that are determined from the particle displacement during the first time interval. We show that the peak positions of the correlations are in accord with the information on the non-Gaussian parameters of the van Hove self-correlation function. Furthermore, it is demonstrated that the progressive changes in the second time interval in the three-time correlation function enable us to analyze how correlations in dynamics evolve in time. From this analysis, we evaluated the lifetime of the dynamical heterogeneity and its temperature dependence systematically. Our results show that the lifetime of the dynamical heterogeneity becomes much slower than the alpha-relaxation time that is determined from the two-point density correlation function when the system is highly supercooled.

Multiple time scales hidden in heterogeneous dynamics of glass-forming liquids

A multitime probing of density fluctuations is introduced to investigate hidden time scales of heterogeneous dynamics in glass-forming liquids. Molecular-dynamics simulations for simple glass-forming liquids are performed and a three-time correlation function is numerically calculated for general time intervals. It is demonstrated that the three-time correlation function is sensitive to the heterogeneous dynamics and that it reveals couplings of correlated motions over a wide range of time scales. Furthermore, the time scale of the heterogeneous dynamics tauhetero is determined by the change in the second time interval in the three-time correlation function. The present results show that the time scale of the heterogeneous dynamics tauhetero becomes larger than the alpha-relaxation time at low temperatures and large wavelengths. We also find a dynamical scaling relation between the time scale tauhetero and the length scale xi of dynamical heterogeneity as tauhetero approximately xiz with z=3.

Dynamics of a Polyphosphazene Melt Studied by Solid-State 2H NMR

Poly[bis(methoxy)phosphazene] (PBMP) is used as a model to investigate the backbone dynamics of polyphosphazenes approaching their glass transitions. Specifically, we study PBMP featuring deuterated methyl groups so that, as a consequence of fast rotation of the methyl groups, 2H NMR probes the reorientation of their 3-fold symmetry axes and, thus, of the inorganic backbone. Combining 2H NMR spin−lattice relaxation, line-shape, and stimulated-echo analyses, we follow the slowdown of the segmental motion upon cooling over a broad temperature/time range. Comparison of present and previous results provides no evidence that polymers featuring inorganic and organic backbones, respectively, show fundamentally different dynamical behaviors during vitrification. In particular, typical of glass-forming polymer melts, we find for the α-process of PBMP that its temperature dependence deviates from an Arrhenius law and its time dependence differs from a single-exponential function. 2H NMR three-time correlation functions indicate that both homogeneous and heterogeneous dynamics contribute to the nonexponential relaxation. In addition, 2H NMR spin−lattice relaxation and line-shape analyses reveal the existence of some large-angle anisotropic precursor motion in the moderately viscous melt, which may be a peculiarity of polyphosphazene.

Nonexponential polymer segmental motion in the presence and absence of ions: H2 NMR multitime correlation functions for polymer electrolytes poly(propylene glycol)-LiClO4

The authors measure 2H NMR multitime correlation functions to investigate the segmental motion of poly(propylene glycol) containing various amounts of the salt LiClO4. 2H NMR two-time correlation functions indicate that addition of salt affects not only the time scale of the segmental motion, but also the degree of the nonexponential relaxation behavior. To quantify the origin of the nonexponential segmental motion, the authors analyze 2H NMR three-time correlation functions. In general, nonexponential relaxation can result from homogeneous dynamics, i.e., intrinsic nonexponentiality, and from heterogeneous dynamics, i.e., existence of a distribution of correlation times G(ln tau). For the studied high and low salt concentrations, including neat poly(propylene glycol), the analysis shows that both homogeneous and heterogeneous contributions are important. 2H NMR four-time correlation functions allow the authors to measure the lifetime of the dynamical heterogeneities. For the studied salt concentrations, the rate exchange occurs on the same time scale as the segmental motion, indicating short-lived dynamical heterogeneities. To arrive at these results, the authors reconsider the interpretation of (2)H NMR three-time correlation functions. Results of analytical calculations and computer simulations show that it is necessary to extend the previous way of analysis so as to include effects due to correlated back-and-forth jumps.

Dynamic heterogeneities in a supercooled diatomic molecular system

Dynamic heterogeneities in a supercooled system of diatomic molecules with an associated dipole moment have been investigated. To this end, three-time correlation functions have been evaluated. Correlations between molecular displacements performed during consecutive time intervals are apparent at low temperatures in the beta -relaxation regime, whereas they tend to disappear during the alpha -relaxation regime. These correlations maximize when the deviation from Gaussian dynamics takes a maximum, and they reveal the existence of different dynamic domains. Directionality of translational motions has also been studied. At low temperatures, and in the beta -relaxation zone, the molecular vector displacement in a given time interval has an important component in the opposite direction of the vector displacement corresponding to the initial time interval. The amplitudes associated with this quasi-oscillatory behavior become larger as the system is cooled. Dynamic heterogeneities in reorientation have been observed in the beta -relaxation regime, and it has been obtained that molecules that perform faster translational motions experience faster reorientational motions too. This effect increases as temperature decreases.

Silver ion dynamics in the Ag5Te2Cl-polymorphs revealed by solid state NMR lineshape and two- and three-time correlation spectroscopies

The relation between structure and ion dynamics in the three polymorphs of Ag(5)Te(2)Cl has been investigated using (109)Ag, (125)Te, and (35)Cl NMR spectroscopies. Specifically, the influence of the structural phase transitions observed near 240 K (P2(1)/c<-->P2(1)/n) and near 332 K (P2(1)/n<-->I4/mcm) upon silver ion mobilities has been studied by temperature dependent (109)Ag NMR lineshapes and spin-lattice relaxation times. While the superionic high temperature phase alpha-Ag(5)Te(2)Cl is characterized by a molten cationic sublattice, fast ion dynamics in the medium-temperature phase beta-Ag(5)Te(2)Cl occurs in spatially restricted regions comprising all the crystallographically distinct silver sites. Temperature dependent magic-angle-spinning linewidths yield an activation energy of 0.38 eV, consistent with 0.44 eV measured from dc electric conductivities. For the low-temperature gamma-modification, results of two- and three-time (109)Ag correlation spectroscopies provide a detailed insight into the nature of the silver ionic hopping motion. Temperature dependent jump rates measured by two-time correlation functions yield an activation energy E(a)= 0.48 eV. (109)Ag NMR three-time correlation functions indicate that the non-exponential relaxation behavior of the silver ions can be attributed to a broad distribution of jump rates rather than correlated forward-backward jumps. Nevertheless, all the silver ions are mobile down to temperatures of about 185 K.

On the spectral relations for multitime correlation functions

A general approach for derivation of the spectral relations for the multitime correlation functions is presented. A special attention is paid to the consideration of the non-ergodic (conserving) contributions and it is shown that such contributions can be treated in a rigorous way using multitime temperature Green functions. Representation of the multitime Green functions in terms of the spectral densities and solution of the reverse problem -- finding of the spectral densities from the known Green functions are given for the case of the three-time correlation functions.

Complex lithium ion dynamics in simulatedLiPO3glass studied by means of multitime correlation functions

Molecular dynamics simulations are performed to study the lithium jumps in ${\mathrm{LiPO}}_{3}$ glass. In particular, we calculate higher-order correlation functions that probe the positions of single lithium ions at several times. Three-time correlation functions show that the nonexponential relaxation of the lithium ions results from both correlated back-and-forth jumps and the existence of dynamical heterogeneities, i.e., the presence of a broad distribution of jump rates. A quantitative analysis yields that the contribution of the dynamical heterogeneities to the nonexponential depopulation of the lithium sites increases upon cooling. Further, correlated back-and-forth jumps between neighboring sites are observed for the fast ions of the distribution, but not for the slow ions and, hence, the back-jump probability depends on the dynamical state. Four-time correlation functions indicate that an exchange between fast and slow ions takes place on the time scale of the jumps themselves; i.e., the dynamical heterogeneities are short lived. Hence, sites featuring fast and slow lithium dynamics, respectively, are intimately mixed. In addition, a backward correlation beyond the first neighbor shell for highly mobile ions and the presence of long-range dynamical heterogeneities suggest that fast ion migration occurs along preferential pathways in the glassy matrix. In the melt, we find no evidence for correlated back-and-forth motions and dynamical heterogeneities on the length scale of the next-neighbor distance.

Off-resonant two-dimensional fifth-order Raman spectroscopy of liquid CS2: Detection of anharmonic dynamics

Off-resonant two-dimensional (2-D) fifth-order Raman signal of liquid CS2 is calculated by using a molecular dynamics (MD) simulation based on the fifth-order response function, which is characterized by a three-time correlation function of polarizability with the stability matrix of the phase space dynamics. The physical origin of the 2-D Raman signal pattern is investigated by analyzing anharmonic and nonlinear polarizability contributions. It is found that a rotation–rotation dynamical coupling arising from anharmonic dynamics is a dominant contribution to the fifth-order response function on the axis of the second time duration, t2. This coupling yields a novel feature of the signal, that is, a change in sign or a ridge in the vicinity along the t2 axis. The echo signal along t1=t2 obtained by the MD simulation accounting for the full dynamic effect, becomes blurred in comparison with that obtained by a harmonic description. The 2-D Raman signal at a very low temperature is also examined to find how the signal changes when structural transitions are suppressed. The 2-D Raman signal is also evaluated with a nonequilibrium MD simulation, which is found to be very similar to that with the equilibrium MD simulation based on the nonlinear response function formalism.

Backward correlations and dynamic heterogeneities: A computer study of ion dynamics

We analyse the correlated back and forth dynamics and dynamic heterogeneities, i.e. the presence of fast and slow ions, for a lithium metasilicate system via computer simulations. For this purpose we define, in analogy to previous work in the field of glass transition, appropriate three-time correlation functions. They contain information about the dynamics during two successive time intervals. First we apply them to simple model systems in order to clarify their information content. Afterwards we use this formalism to analyse the lithium trajectories. A strong back-dragging effect is observed, which also fulfills the time-temperature superposition principle. Furthermore, it turns out that the back-dragging effect is long-ranged and exceeds the nearest neighbor position. In contrast, the strength of the dynamic heterogeneities does not fulfill the time-temperature superposition principle. The lower the temperature, the stronger the mobility difference between fast and slow ions. The results are then compared with the simple model systems considered here as well as with some lattice models of ion dynamics.

Time-resolved x-ray diffraction: Statistical theory and its application to the photo-physics of molecular iodine

A theory is proposed to study time-resolved x-ray diffraction on the pico- and subpicosecond time scales. Electromagnetic fields are treated in the frame of Maxwellian electrodynamics, whereas the molecular system is treated by using quantum mechanics. An expression is given for the time-resolved x-ray signal; it involves a three-time correlation function of the Fourier transformed electronic density and of the electric dipole moment of the system. This theory is applied to the study of the recombination of photodissociated iodine molecules in solution. Both geminate and nongeminate recombination are considered. The feasibility of the real time visualization of atomic motions is discussed.

Iterative path integral formulation of equilibrium correlation functions for quantum dissipative systems

We present an iterative path integral algorithm for computing multitime correlation functions of a quantum system coupled to a dissipative bath of harmonic oscillators. By splitting the Boltzmann operator into two parts and reordering the propagators in the expression for canonical correlation functions, we are able to transform the evolution time contour into a symmetric one so that a forward propagation and a backward one are specified. Because the memory induced by the bath through the Feynman–Vernon influence functional decays rapidly in the complex time plane, long-time correlations are negligible. Taking advantage of this fact, we show that the correlation function can be obtained via an iterative procedure. The method is used to calculate three-time correlation functions of a dissipative two-level system.