Stochastic Liouville Equation Research Articles

Quantum decoherence dynamics in stochastically fluctuating environments.

We theoretically study the decoherence of a two-level quantum system coupled to noisy environments exhibiting linear and quadratic fluctuations within the framework of a stochastic Liouville equation. It is shown that the intrinsic energy levels of the quantum system renormalize under either the linear or quadratic influence of the environmental noise. In the case of quadratic dependence, the renormalization of the energy levels of the system emerges even if the environmental noise exhibits stationary statistical properties. This is in contrast to the case under linear influence, where the intrinsic energy levels of the system renormalize only if the environmental noise displays nonstationary statistics. We derive the analytical expressions of the decoherence function in the cases where the fluctuation of the frequency difference depends linearly and quadratically on the nonstationary Ornstein-Uhlenbeck noise (OUN) and random telegraph noise (RTN) processes, respectively. In the case of the linear dependence of the OUN, the environmental nonstationary statistical property can enhance the dynamical decoherence. However, the nonstationary statistics of the environmental noise can suppress the quantum decoherence in this case under the quadratic influence of the OUN. In the presence of the RTN, the quadratic influence of the environmental noise does not give rise to decoherence but only causes a determinate frequency renormalization in dynamical evolution. The environmental nonstationary statistical property can suppress the quantum decoherence of the case under the linear influence of the RTN.

Using the Haken-Strobl-Reineker Model to Determine the Temperature Dependence of the Diffusion Coefficient.

Stochastic quantum Liouville equations (SQLE) are widely used to model energy and charge dynamics in molecular systems. The Haken-Strobl-Reineker (HSR) SQLE is a particular paradigm in which the dynamical noise that destroys quantum coherences arises from a white noise (i.e., constant-frequency) spectrum. A system subject to the HSR SQLE thus evolves to its "high-temperature" limit, whereby all the eigenstates are equally populated. This result would seem to imply that the predictions of the HSR model, e.g., the temperature dependence of the diffusion coefficient, have no validity for temperatures lower than the particle bandwidth. The purpose of this paper is to show that this assumption is incorrect for translationally invariant systems. In particular, provided that the diffusion coefficient is determined via the mean-squared-displacement, considerations about detailed-balance are irrelevant. Consequently, the high-temperature prediction for the temperature dependence of the diffusion coefficient may be extrapolated to lower temperatures, provided that the bath remains classical. Thus, for diagonal dynamical disorder the long-time diffusion coefficient, D∞(T) = c1/T, while for both diagonal and off-diagonal disorder, D∞(T) = c1/T + c2T, where c2 ≪ c1. An appendix discusses an alternative interpretation from the HSR model of the "quantum to classical" dynamics transition, whereby the dynamics is described as stochastically punctuated coherent motion.

Memory effects in the efficiency control of energy transfer under incoherent light excitation in noisy environments.

Fluctuations in energy gap and coupling constants between chromophores can play an important role in absorption and energy transfer across a collection of two-level systems. In photosynthesis, light-induced quantum coherence can affect the efficiency of energy transfer to the designated "trap" state. Theoretically, the interplay between fluctuations and coherence has been studied often, employing either a Markovian or a perturbative approximation. In this study, we depart from these approaches to incorporate memory effects by using Kubo's quantum stochastic Liouville equation. We introduce the effects of decay of the created excitation (to the ground state) on the desired propagation and trapping that provides a direction of flow of the excitation. In the presence of light-induced pumping, we establish a relation between the efficiency, the mean survival time, and the correlation decay time of the bath-induced fluctuations. A decrease in the steady-state coherence during the transition from the non-Markovian regime to the Markovian limit results in a decrease in efficiency. As in the well-known Haken-Strobl model, the ratio of the square of fluctuation strength to the rate plays a critical role in determining the mechanism of energy transfer and in shaping the characteristics of the efficiency profile. We recover a connection between the transfer flux and the imaginary part of coherences in both equilibrium and excited bath states, in both correlated and uncorrelated bath models. We uncover a non-monotonic dependence of efficiency on site energy heterogeneity for both correlated and uncorrelated bath models.

Evaluating the motional timescales contributing to averaged anisotropic interactions in MAS solid-state NMR

Abstract. Dynamic processes in molecules can occur on a wide range of timescales, and it is important to understand which timescales of motion contribute to different parameters used in dynamics measurements. For spin relaxation, this can easily be understood from the sampling frequencies of the spectral-density function by different relaxation-rate constants. In addition to data from relaxation measurements, determining dynamically averaged anisotropic interactions in magic-angle spinning (MAS) solid-state NMR allows for better quantification of the amplitude of molecular motion. For partially averaged anisotropic interactions, the relevant timescales of motion are not so clearly defined. Whether the averaging depends on the experimental methods (e.g., pulse sequences) or conditions (e.g., MAS frequency, magnitude of anisotropic interaction, radio-frequency field amplitudes) is not fully understood. To investigate these questions, we performed numerical simulations of dynamic systems based on the stochastic Liouville equation using several experiments for recoupling the dipolar coupling, chemical-shift anisotropy or quadrupolar coupling. As described in the literature, the transition between slow motion, where parameters characterizing the anisotropic interaction are not averaged, and fast motion, where the tensors are averaged leading to a scaled anisotropic quantity, occurs over a window of motional rate constants that depends mainly on the strength of the interaction. This transition region can span 2 orders of magnitude in exchange-rate constants (typically in the microsecond range) but depends only marginally on the employed recoupling scheme or sample spinning frequency. The transition region often coincides with a fast relaxation of coherences, making precise quantitative measurements difficult. Residual couplings in off-magic-angle experiments, however, average over longer timescales of motion. While in principle one may gain information on the timescales of motion from the transition area, extracting such information is hampered by low signal-to-noise ratio in experimental spectra due to fast relaxation that occurs in the same region.

Tunable zero-field magnetoresistance responses in Si transistors: Origins and applications

The near-zero-field magnetoresistance (NZFMR) response has proven to be a useful tool for studying atomic-scale, paramagnetic defects that are relevant to the reliability of semiconductor devices. The measurement is simple to make and, in some cases, simple to interpret. In other cases, more sophisticated modeling based on the stochastic Liouville equation (SLE) is needed to access valuable information from NZFMR results. It has been shown that hyperfine and dipolar coupling interactions at atomic-scale defects affect the NZFMR line shape, but experimental parameters related to the detection method of NZFMR can also affect the nature of the response. Here, we demonstrate four distinct NZFMR detection methods in Si MOSFETs, which all access identical Si/SiO2 interface defects. In all four cases, we show that the line shape of the response is tunable based on experimental parameters alone. Using SLE-based modeling, we verify that time constants connected to physical carrier capture rates at the defect sites lead to these NZFMR line shape changes. The results demonstrate a method to extract some atomic-scale information for the purpose of defect identification. They also have broader applications to the continued development of ultra-sensitive magnetometers based on NZFMR in semiconductors. Additionally, the NZFMR effect in common Si-based devices may provide an inexpensive and accessible platform that mimics similar radical pair mechanisms that have become increasingly important in various biology fields.

Control of electronic and exchange coupling by bridge substituents in donor acceptor triads with triptycene bridges.

A series of triads, consisting of a triarylamine electron donor and a perylene diimide electron acceptor which were attached to two different wings of a triptycene bridging unit, was investigated concerning the dynamics of photoinduced charge separation and charge recombination processes with a particular focus on the involved spin-chemical aspects. Attaching electron-donating or electron-withdrawing substituents to the third wing of the triptycene bridge allowed tuning the electron transfer processes. These processes were investigated via fs-transient absorption spectroscopy and ns-transient absorption spectroscopy in an external magnetic field. The resulting magnetic field-dependent decay dynamics were analysed and modelled using the stochastic Liouville equation which yielded rate constants for the charge recombination and the exchange energy. In combination with a diabatic rate theory and Anderson's perturbative treatment of the exchange energy, these data gave a complete set of rate constants for charge separation and charge recombination from which the diverse electronic couplings between the involved states were derived. These couplings depend linearly on the inverse energy of virtual triptycene bridge states which allows tuning the electron transfer dynamics by modifying the triptycene bridge.

Heisenberg Spin Exchange Between Nitroxide Probes Diffusing in a Percolation Network

AbstractHeisenberg spin exchange between nitroxide (Tempone) spin probes has been measured as a function of concentration in the aqueous phase of the hydrated ion exchange membrane Nafion 117. The observed fast-motional electron paramagnetic resonance spectra were analyzed in terms of the stochastic Liouville equation lineshape calculation of Freed and coworkers and the “new paradigm” for interpreting spin exchange effects proposed by Salikhov. Differences between the effective spin exchange measured from the spectrum by these methods are presented and compared, and indicate that dipolar interactions make a significant contribution to spin exchange in this system. In acidic Nafion membranes, the spin probes are deactivated over time, allowing simultaneous measurement of the decay kinetics and spin exchange as a function of paramagnetic probe concentration. Both these processes deviate from the behavior that would be expected from classical chemical kinetics in isotropic media. The results are discussed in terms of currently available models for diffusion and reaction in a percolation network.

The solid effect of dynamic nuclear polarization in liquids.

The solid-state effect of dynamic nuclear polarization (DNP) is operative also in viscous liquids where the dipolar interaction between the electronic and nuclear spins is partially averaged. The proper way to quantify the degree of averaging, and thus calculate the efficiency of the effect, should be based on the time-correlation function of the dipolar interaction. Here we use the stochastic Liouville equation formalism to develop a general theoretical description of the solid effect in liquids. The derived expressions can be used with different dipolar correlations functions depending on the assumed motional model. At high magnetic fields, the theory predicts DNP enhancements at small offsets, far from the classical solid-effect positions that are displaced by one nuclear Larmor frequency from the electronic resonance. The predictions are in quantitative agreement with such enhancement peaks observed at 9.4 T . These non-canonical peaks are not due to thermal mixing or the cross effect but exactly follow the dispersive component of the EPR line.

Effect of active layer thickness on the charge recombination and dissociation in bulk heterojunction polymer solar cells under open circuit conditions

Bulk heterojunction organic photovoltaic cells (OPVs), which are based on anthracene-containing poly (p-phenylene-ethynylene)-alt-poly (p-phenylene-vinylene) (PPE-PPV) polymer denoted (AnE-PVstat) and phenyl C60 butyric acid methyl ester (PCBM). The active layer thickness was varied from 100 nm to 300 nm. The solar cell parameters and magnetoconductance (MC) effect were analyzed, and the results showed a positive MC effect (+MC) for active layer thicknesses of 100 nm, 150 nm, and 200 nm, while a negative MC effect (-MC) was observed for thicknesses of 250 nm and 300 nm. The Power conversion efficiency (PCE) of 4 ℅ was achieved for an active layer thickness of 200 nm. The triplet-doublet quenching (TQD) model was used to describe the MC effect, which is influenced by triplet-charge reactions on the current based on density matrix formalism using the Stochastic Liouville Equation. The dissociation and recombination rates of the T-D pairs increased with the thickness for thinner layers (100, 150, 200 nm) while decreased for thicker layers (250 nm and 300 nm).

Quantitative Multistate Binding Model of Silica Nanoparticle-Protein Interactions Obtained from Multinuclear Spin Relaxation.

Nanoparticle-assisted NMR spin relaxation (NASR), which makes internal protein dynamics in solution directly observable on nanosecond to microsecond time scales, has been applied to different nuclei and relaxation processes of the same protein system. A model is presented describing the transient interaction between ubiquitin and anionic silica nanoparticles for the unified interpretation of a wealth of experimental data including 2H, 13C, and 15N relaxation of methyl side chain and backbone moieties. The best model, implemented using a stochastic Liouville equation, describes the exchange process via an intermediary encounter state between free and fully nanoparticle-bound protein. The implication of the three-state binding model on the interpretation of NASR data is discussed.

Piecewise ensemble averaging stochastic Liouville equations for simulating non-Markovian quantum dynamics

Here we present a novel stochastic Liouville equation with piecewisely correlated noises, in which the inter-piece correlation is rigorously incorporated by a convolution integral involving functional derivatives. Due to the feature of piecewise correlation, we can perform piecewise ensemble average and serve the average of the preceding interval as the initial condition of the subsequent propagation. This strategy avoids the long-time stochastic average and the statistical errors are saturated at long times. By doing so, we circumvent the intrinsic difficulty of the stochastic simulations caused by the fast increase in the variance of the quantum Brownian motion. Therefore, as demonstrated by the numerical examples, the proposed method enables us to simulate the long-time quantum dissipative dynamics with long memories in the non-perturbative regime.

Stochastic resets in the context of a tight-binding chain driven by an oscillating field

In this work, we study in the framework of the so-called driven tight-binding chain (TBC) the issue of quantum unitary dynamics interspersed at random times with stochastic resets mimicking non-unitary evolution due to interactions with the external environment, the driven TBC involves a quantum particle hopping between the nearest-neighbour sites of a one-dimensional lattice and subject to an external forcing field that is periodic in time. We consider the resets to be taking place at exponentially-distributed random times. Using the method of stochastic Liouville equation, we derive exact results for the probability at a given time for the particle to be found on different sites and averaged with respect to different realizations of the dynamics. We establish the remarkable effect of localization of the TBC particle on the sites of the underlying lattice at long times. The system in the absence of stochastic resets exhibits delocalization of the particle, whereby the particle does not have a time-independent probability distribution of being found on different sites even at long times, and, consequently, the mean-squared displacement of the particle about its initial location has an unbounded growth in time. One may induce localization in the bare model only through tuning the ratio of the strength to the frequency of the field to have a special value, namely, equal to one of the zeros of the zeroth order Bessel function of the first kind. We show here that localization may be induced by a far simpler procedure of subjecting the system to stochastic resets.

Triplet-Triplet Annihilation via the Triplet Channel in Crystalline 9,10-Diphenylanthracene.

Delayed fluorescence resulting from triplet-triplet annihilation in crystalline 9,10-diphenylanthracene was observed by means of steady-state fluorescence measurements under magnetic fields of ≤10 T. At five specific magnetic fields, four peaks and one dip in the magnetic field dependence of fluorescence intensity were observed, proving that exchange-coupled triplet pairs were generated in the course of triplet-triplet annihilation. The dip was in the opposite direction predicted for singlet channel triplet-triplet annihilation. Further analysis using the stochastic Liouville equation confirmed that the closest exchange-coupled triplet pair in crystalline 9,10-diphenylanthracene is quenched via both triplet channel and singlet channel triplet-triplet annihilation.

Nonlinear current response of Weyl semimetals to a strong dc-ac electric field in the ultraquantum regime

The linear charge transport properties of Weyl semimetals, such as negative magnetoresistance related to chiral anomaly, have been studied extensively. In this work, the nonlinear current response of Weyl semimetals to a strong dc-ac electric field in the ultraquantum regime with a strong magnetic field is explored by employing a nonperturbative treatment based on the stochastic Liouville equation. Our systematic studies of nonlinear charge transport for two types of ac fields (the cosinusoidal electric field and the periodic pulsed field) have revealed extraordinary modulation of nonlinear current response by the ac fields. For the case with the cosinusoidal electric field, we find the following: (i) in the high-frequency regime, dynamic localization (vanishing of current) and quasienergy band collapse occur under a suitable condition of ${J}_{0}(\frac{e{E}_{A}d}{\ensuremath{\hbar}{\ensuremath{\omega}}_{0}})=0$, where ${J}_{0}$ is the Bessel function with ${E}_{A}$ and ${\ensuremath{\omega}}_{0}$ being the strength and frequency of the ac field, and $d$ denoting the lattice constant of Weyl semimetals; and (ii) in the intermediate- and low-frequency regimes, the multiple-photon-assisted transport leads to extremal values of current responses whose patterns can be tuned by the magnetic field. As for the pulsed electric field, our results show that (i) the dynamic localization and quasienergy band collapse appear under a different condition of $cos(\frac{e{E}_{A}d}{\ensuremath{\hbar}{\ensuremath{\omega}}_{0}})=0$; and (ii) the influence of the ac field on the current response disappears when $\frac{e{E}_{A}d}{\ensuremath{\hbar}{\ensuremath{\omega}}_{0}}=m\ensuremath{\pi}$, with $m$ being an integer. The experimental conditions are also discussed and the predicted nonlinear transport effects could be observed in experiments.

Quantum unitary evolution interspersed with repeated non-unitary interactions at random times: the method of stochastic Liouville equation, and two examples of interactions in the context of a tight-binding chain

In the context of unitary evolution of a generic quantum system interrupted at random times with non-unitary evolution due to interactions with either the external environment or a measuring apparatus, we adduce a general theoretical framework to obtain the average density operator of the system at any time during the dynamical evolution. The average is with respect to the classical randomness associated with the random time intervals between successive interactions, which we consider to be independent and identically-distributed random variables. The formalism is very general in that it applies to any quantum system, to any form of non-unitary interaction, and to any probability distribution for the random times. We provide two explicit applications of the formalism in the context of the so-called tight-binding model relevant in various contexts in solid-state physics, e.g. in modelling nano wires. Considering the case of one dimension, the corresponding tight-binding chain models the motion of a charged particle between the sites of a lattice, wherein the particle is for most times localized on the sites, owing to spontaneous quantum fluctuations tunnels between the nearest-neighbour sites. We consider two representative forms of interactions, one that implements a stochastic reset of quantum dynamics in which the density operator is at random times reset to its initial form, and one in which projective measurements are performed on the system at random times. In the former case, we demonstrate with our exact results how the particle is localized on the sites at long times, leading to a time-independent mean-squared displacement (MSD) of the particle about its initial location. This stands in stark contrast to the behavior in the absence of interactions, when the particle has an unbounded growth of the MSD in time, with no signatures of localization. In the case of projective measurements at random times, we show that repeated projection to the initial state of the particle results in an effective suppression of the temporal decay in the probability of the particle to be found on the initial state. The amount of suppression is comparable to the one in conventional Zeno effect scenarios, but which it does not require us to perform measurements at exactly regular intervals that are hallmarks of such scenarios.

Extraction of dipolar coupling constants from low-frequency electrically detected magnetic resonance and near-zero field magnetoresistance spectra via least squares fitting to models developed from the stochastic quantum Liouville equation

We report low-frequency electrically detected magnetic resonance (EDMR) and near-zero field magnetoresistance (NZFMR) measurements observed through spin-dependent trap-assisted tunneling on unpassivated 28Si/28SiO2 metal–insulator–semiconductor (MIS) capacitors. This study both experimentally and theoretically explores the effects of the low-frequency EDMR response and the mechanisms responsible for spin-mixing, which leads to the NZFMR response in the absence of electron–nuclear hyperfine interactions. Previous reports that utilized high-frequency EDMR and NZFMR on these devices indicated that the observed trap-assisted tunneling spectra are dominated by silicon dangling bonds back bonded to silicon at the Si/SiO2 interface, Pb0 and Pb1 centers. These previous results also suggest that the rate limiting step in trap-assisted tunneling is the interface to an oxide trapping event. In this work, we extend the theory to show the explicit connection of the defects observed between the NZFMR response and the EDMR, which has not yet been demonstrated. We also extend a theoretical approach to the analysis of both the EDMR and NZFMR spectra and match the theory to experimental observations made in 28Si/28SiO2 MIS capacitors. The method utilizes a least squares fitting algorithm of models developed from the stochastic quantum Liouville equation. We find that we can extract a dipolar coupling constant by fitting both the NZFMR and EDMR spectra. Our experimental results and resulting fitted spectra from our quantitative model suggest the mechanism responsible for spin-mixing, which leads to the NZFMR response in the absence of electron–nuclear hyperfine interactions, is predominately magnetic dipolar interactions between Pb centers at the interface.

Autobiography of Yoshitaka Tanimura.

In this paper, two developments, the theory of hierarchical equations of motion (HEOM) for open quantum dynamics systems and the theory of ultrafast nonlinear two-dimensional (2D) spectroscopies, are described, following the history of their founder. The HEOM was discovered by attempting to derive an equation similar to Kuob's stochastic Liouville equation based on the system-bath Hamiltonian. The theory of 2D spectroscopy was developed in the process of investigating the effect of quantum coherence on nonlinear vibrational spectra based on the Feynman-Vernon influence functional. The paper also describes the development of these two theories for problems such as electron and exciton transfer in photosynthetic systems, quantum statistical thermodynamics, and solid-state physics.

Excitation Energy Transfer Efficiency in Fluctuating Environments: Role of Quantum Coherence in the Presence of Memory Effects.

Several recent studies have interrogated the role of quantum coherence in affecting the transfer efficiency of an optical excitation to the designated "trap" state where the energy can be used for subsequent reactions, as in photosynthesis. However, these studies invoke a Markovian approximation for the time correlation function describing the environment-induced stochastic fluctuations. Here, we employ Kubo's quantum stochastic Liouville equation (QSLE) to include memory effects. We extend the existing QSLE scheme to introduce decay of a newly created excitation due to radiative and nonradiative channels and also by desired trapping toward the targeted chromophore. We show that the theoretical formalism based on the QSLE correctly reproduces the rate equation description in the Markovian limit, with the rate constants determined by an appropriate quantum limiting procedure. We find that under certain conditions, the efficiency of excitation transfer to the trap gains from the combined presence of quantum coherence and temporally correlated stochastic fluctuations. We work out different limiting situations in order to discover and quantify the optimum conditions for the energy transfer to the trapped state. We find that maximum energy transfer efficiency is achieved in the intermediate limit between coherent and incoherent transport.

Blend ratio and applied voltage effects on the charge recombination in bulk heterojunction polymer solar cells based on anthracene-containing poly(arylene-ethynylene)-alt-poly(arylene-vinylene) studied using magnetoconductance

We have investigated the magnetic field effect on the current in bulk heterojunction polymer solar cells made of blends of an anthracene-containing poly(arylene-ethynylene)-alt-poly (arylene-vinylene) (AnE-PVstat) and fullerene molecules prepared in various ratios (1:1, 1:2, 1:3 and 1:4) and different applied voltages ranging from 0.7 to 0.95 V in steps of 0.05 V. The efficiencies and fill factors of the resulting devices improved upon increasing the PCBM content. The highest efficiency obtained was 2.7% for the device prepared using 75% PCBM. The measured magnetoconductance (MC) is positive and monotonic over the full range of the blend ratios and applied voltages studied. We analyzed the MC effect using a magnetoconductance model controlled by interconversion of the singlet and triplet (MIST) electron-hole (e-h) pairs states based on density matrix formalism using the stochastic Liouville equation considering that the singlet and triplet e-h pairs have different recombination and dissociation rates to theoretically reproduce the line shape of MC. For all the applied voltages studied, the recombination and dissociation rates of the e-h pairs on the singlet and triplet states decreased upon increasing the fullerene content until it reached a maximum with a PCBM content of 75 wt%. After this PCBM content, the rates started to increase. However, for each blend ratio, the recombination and dissociation rates of the singlet and triplet e-h pairs increase when the voltage is increased.

NMR Spin Relaxation Theory of Biomolecules Undergoing Highly Asymmetric Exchange with Large Interaction Partners

The transient interactions of proteins and other molecules with much larger structures, such as synthetic or biological nanoparticles, lead to certain types of enhanced nuclear magnetic resonance (NMR) spin relaxation effects, which can be accurately measured by multidimensional solution NMR techniques. These relaxation effects provide new information about the nanostructures and the protein, their interactions, internal dynamics, and associated kinetic and thermodynamic parameters, such as exchange rates and populations. Although theoretical treatments exist that cover either the fast or slow exchange limits, a theoretical treatment that applies to all practically relevant exchange processes is still missing. A unified theoretical framework is presented for this purpose based on a stochastic Liouville equation (SLE). It covers nuclear spin dynamics, overall rotational diffusion of both the protein and the nanostructure, the exchange process between a free state and a bound state, and internal protein dynamics. Although the numerical implementation of the SLE typically involves large matrices, it is shown here that it is computationally still tractable for situations relevant in practice. Application of the theory demonstrates how transverse relaxation is substantially impacted by the kinetics of binding on a wide range of exchange timescales. It is further shown that when exchange occurs on the appropriate timescale, transverse relaxation is able to report on internal dynamics far slower than observable by traditional transverse relaxation experiments. The SLE will allow the realistic and quantitative interpretation of experimental NMR data reporting about transient protein-nanoparticle interactions, thereby providing a powerful tool for the characterization of protein dynamics modes on a vast range of timescales including motions that may be functionally relevant.