Multitime Correlation Functions Research Articles

Generalized quantum master equations can improve the accuracy of semiclassical predictions of multitime correlation functions

Multitime quantum correlation functions are central objects in physical science, offering a direct link between the experimental observables and the dynamics of an underlying model. While experiments such as 2D spectroscopy and quantum control can now measure such quantities, the accurate simulation of such responses remains computationally expensive and sometimes impossible, depending on the system’s complexity. A natural tool to employ is the generalized quantum master equation (GQME), which can offer computational savings by extending reference dynamics at a comparatively trivial cost. However, dynamical methods that can tackle chemical systems with atomistic resolution, such as those in the semiclassical hierarchy, often suffer from poor accuracy, limiting the credence one might lend to their results. By combining work on the accuracy-boosting formulation of semiclassical memory kernels with recent work on the multitime GQME, here we show for the first time that one can exploit a multitime semiclassical GQME to dramatically improve both the accuracy of coarse mean-field Ehrenfest dynamics and obtain orders of magnitude efficiency gains.

Memory Tensor for Non-Markovian Dynamics with Random Hamiltonian

In the theory of open quantum systems, the Markovian approximation is very widespread. Usually, it assumes the Gorini–Kossakowski–Sudarshan–Lindblad (GKSL) equation for density matrix dynamics and quantum regression formulae for multi-time correlation functions. Nevertheless, now, quantum non-Markovianity is being actively studied, especially the non-Markovianity of multi-time correlations. In this work, we consider dynamics with a random Hamiltonian, which can lead to GKSL dynamics of the density matrix for some special cases, but correlation functions generally do not satisfy the quantum regression formulae. Despite the fact that random Hamiltonians have been actively studied, dynamics with such Hamiltonians has been little discussed from the viewpoint of multi-time correlations. For specific models with a random Hamiltonian, we provide the formulae for multi-time correlations which occur instead of the usual regression formulae. Moreover, we introduce and calculate the memory tensor, which characterizes multi-time correlations against the Markovian ones. We think that, despite being applied to specific models, the methods developed in this work can be used in a much broader setup.

Matsubara dynamics approximation for generalized multi-time correlation functions.

We introduce a semi-classical approximation for calculating generalized multi-time correlation functions based on Matsubara dynamics, a classical dynamics approach that conserves the quantum Boltzmann distribution. This method is exact for the zero time and harmonic limits and reduces to classical dynamics when only one Matsubara mode is considered (i.e., the centroid). Generalized multi-time correlation functions can be expressed as canonical phase-space integrals, involving classically evolved observables coupled through Poisson brackets in a smooth Matsubara space. Numerical tests on a simple potential show that the Matsubara approximation exhibits better agreement with exact results than classical dynamics, providing a bridge between the purely quantum and classical descriptions of multi-time correlation functions. Despite the phase problem that prevents practical applications of Matsubara dynamics, the reported work provides a benchmark theory for the future development of quantum-Boltzmann-preserving semi-classical approximations for studies of chemical dynamics in condensed phase systems.

An exact imaginary-time path-integral phase-space formulation of multi-time correlation functions.

An exact representation of quantum mechanics using the language of phase-space variables provides a natural starting point to introduce and develop semiclassical approximations for the calculation of time correlation functions. Here, we introduce an exact path-integral formalism for calculations of multi-time quantum correlation functions as canonical averages over ring-polymer dynamics in imaginary time. The formulation provides a general formalism that exploits the symmetry of path integrals with respect to permutations in imaginary time, expressing correlations as products of imaginary-time-translation-invariant phase-space functions coupled through Poisson bracket operators. The method naturally recovers the classical limit of multi-time correlation functions and provides an interpretation of quantum dynamics in terms of "interfering trajectories" of the ring-polymer in phase space. The introduced phase-space formulation provides a rigorous framework for the future development of quantum dynamics methods that exploit the invariance of imaginary time path integrals to cyclic permutations.

An accurate and efficient Ehrenfest dynamics approach for calculating linear and nonlinear electronic spectra.

Linear and nonlinear electronic spectra provide an important tool to probe the absorption and transfer of electronic energy. Here, we introduce a pure state Ehrenfest approach to obtain accurate linear and nonlinear spectra that is applicable to systems with large numbers of excited states and complex chemical environments. We achieve this by representing the initial conditions as sums of pure states and unfolding multi-time correlation functions into the Schrödinger picture. By doing this, we show that one can obtain significant improvements in accuracy over the previously used projected Ehrenfest approach and that these benefits are particularly pronounced in cases where the initial condition is a coherence between excited states. While such initial conditions do not arise when calculating linear electronic spectra, they play a vital role in capturing multidimensional spectroscopies. We demonstrate the performance of our method by showing that it is able to quantitatively capture the exact linear, 2D electronic spectroscopy, and pump-probe spectra for a Frenkel exciton model in slow bath regimes and is even able to reproduce the main spectral features in fast bath regimes.

Two-dimensional spectroscopy of two-dimensional materials: A Mahan-Nozières-De Dominicis model of electron-exciton scattering

In this paper, we provide a systematically convergable and efficient numerical approach to simulate multitime correlation functions in the Mahan-Nozi\`eres-De Dominicis model, which crudely mimics the spectral properties of doped two-dimensional (2D) semiconductors such as monolayer transition metal dichalcogenides. We apply this approach to study the coherent 2D electronic spectra of the model. We show that several experimentally observed phenomena, such as peak asymmetry and coherent oscillations in the waiting-time dependence of the trion-exciton cross peaks of the 2D rephasing spectrum, emerge naturally in our approach. Additional features are also present which find no correspondence with experimentally expected behavior. We trace these features to the infinite hole mass property of the model. We use this understanding to construct an efficient approach which filters out configurations associated with the lack of exciton recoil, enabling the connection to previous work and providing a route to the construction of realistic 2D spectra over a broad doping range in 2D semiconductors.

Functional renormalisation group for turbulence

Turbulence is a complex nonlinear and multi-scale phenomenon. Although the fundamental underlying equations, the Navier–Stokes equations, have been known for two centuries, it remains extremely challenging to extract from them the statistical properties of turbulence. Therefore, for practical purposes, a sustained effort has been devoted to obtaining an effective description of turbulence, that we may call turbulence modelling, or statistical theory of turbulence. In this respect, the renormalisation group (RG) appears as a tool of choice, since it is precisely designed to provide effective theories from fundamental equations by performing in a systematic way the average over fluctuations. However, for Navier–Stokes turbulence, a suitable framework for the RG, allowing in particular for non-perturbative approximations, has been missing, which has thwarted RG applications for a long time. This framework is provided by the modern formulation of the RG called the functional renormalisation group (FRG). The use of the FRG has enabled important progress in the theoretical understanding of homogeneous and isotropic turbulence. The major one is the rigorous derivation, from the Navier–Stokes equations, of an analytical expression for any Eulerian multi-point multi-time correlation function, which is exact in the limit of large wavenumbers. We propose in thisJFM Perspectivesarticle a survey of the FRG method for turbulence. We provide a basic introduction to the FRG and emphasise how the field-theoretical framework allows one to systematically and profoundly exploit the symmetries. We stress that the FRG enables one to describe fully developed turbulence forced at large scales, which was not accessible by perturbative means. We show that it yields the energy spectrum and second-order structure function with accurate estimates of the related constants, and also the behaviour of the spectrum in the near-dissipative range. Finally, we expound the derivation of the spatio-temporal behaviour of$n$-point correlation functions, and largely illustrate these results through the analysis of data from experiments and direct numerical simulations.

2D spectroscopies from condensed phase dynamics: Accessing third-order response properties from equilibrium multi-time correlation functions

The third-order response lies at the heart of simulating and interpreting nonlinear spectroscopies ranging from two-dimensional infrared (2D-IR) to 2D electronic (2D-ES), and 2D sum frequency generation (2D-SFG). The extra time and frequency dimensions in these spectroscopic techniques provide access to rich information on the electronic and vibrational states present, the coupling between them, and the resulting rates at which they exchange energy that are obscured in linear spectroscopy, particularly for condensed phase systems that usually contain many overlapping features. While the exact quantum expression for the third-order response is well established, it is incompatible with the methods that are practical for calculating the atomistic dynamics of large condensed phase systems. These methods, which include both classical mechanics and quantum dynamics methods that retain quantum statistical properties while obeying the symmetries of classical dynamics, such as LSC-IVR, centroid molecular dynamics, and Ring Polymer Molecular Dynamics (RPMD), naturally provide short-time approximations to the multi-time symmetrized Kubo transformed correlation function. Here, we show how the third-order response can be formulated in terms of equilibrium symmetrized Kubo transformed correlation functions. We demonstrate the utility and accuracy of our approach by showing how it can be used to obtain the third-order response of a series of model systems using both classical dynamics and RPMD. In particular, we show that this approach captures features such as anharmonically induced vertical splittings and peak shifts while providing a physically transparent framework for understanding multidimensional spectroscopies.

Quantum regression theorem for multi-time correlators: A detailed analysis in the Heisenberg picture

The quantum regression theorem is one of the central results in open quantum systems and is extensively used for computing multi-point correlation functions. Traditionally it is derived for two-time correlators in the Markovian limit employing the Schr\"odinger picture. In this paper we make use of the Heisenberg picture to derive the quantum regression theorems for multi-time correlation functions, which in the special limit reduce to the well-known two-time regression theorem. For the multi-time correlation function we find that the regression theorem takes the same form as it takes for the two-time correlation function with a mild restriction that one of the times should be greater than all other time variables. Interestingly, the Heisenberg picture also allows us to derive an analog of regression theorem for out-of-time-ordered correlators. We further extend our study for the case of non-Markovian dynamics and report the modifications to the standard quantum regression theorem. We illustrate all of the above results using the paradigmatic dissipative spin-boson model.

Quantum regression beyond the Born-Markov approximation for generalized spin-boson models

The quantum regression formula for an open quantum system consists in an infinite hierarchy of conditions for its multitime correlation functions, thus requiring full access to the total ``$\mathrm{system}+\mathrm{environment}$'' evolution, and providing a stronger requirement than completely positive (CP) divisibility. Here, we analyze CP divisibility and check the validity of quantum regression beyond the Born-Markov approximation (e.g., the weak-coupling limit) for a class of generalized spin-boson models giving rise to a multilevel amplitude-damping evolution; in all cases, it is possible to engineer the system-bath coupling in such a way that quantum regression is exactly satisfied.

Tracking Evaporative Cooling of a Mesoscopic Atomic Quantum Gas in Real Time

The fluctuations in thermodynamic and transport properties in many-body systems gain importance as the number of constituent particles is reduced. Ultracold atomic gases provide a clean setting for the study of mesoscopic systems; however, the detection of temporal fluctuations is hindered by the typically destructive detection, precluding repeated precise measurements on the same sample. Here, we overcome this hindrance by utilizing the enhanced light--matter coupling in an optical cavity to perform a minimally invasive continuous measurement and track the time evolution of the atom number in a quasi two-dimensional atomic gas during evaporation from a tilted trapping potential. We demonstrate sufficient measurement precision to detect atom number fluctuations well below the level set by Poissonian statistics. Furthermore, we characterize the non-linearity of the evaporation process and the inherent fluctuations of the transport of atoms out of the trapping volume through two-time correlations of the atom number. Our results establish coupled atom--cavity systems as a novel testbed for observing thermodynamics and transport phenomena in mesosopic cold atomic gases and, generally, pave the way for measuring multi-time correlation functions of ultracold quantum gases.

Accuracy of the Quantum Regression Theorem for Photon Emission from a Quantum Dot.

The quantum regression theorem (QRT) is the most widely used tool for calculating multitime correlation functions for the assessment of quantum emitters. It is an approximate method based on a Markov assumption for environmental coupling. In this Letter we quantify properties of photons emitted from a single quantum dot coupled to phonons. For the single-photon purity and the indistinguishability, we compare numerically exact path-integral results with those obtained from the QRT. It is demonstrated that the QRT systematically overestimates the influence of the environment for typical quantum dots used in quantum information technology.

Ring-polymer, centroid, and mean-field approximations to multi-time Matsubara dynamics.

Based on a recently developed generalization of Matsubara dynamics to the multi-time realm, we present a formal derivation of multi-time generalizations of ring-polymer molecular dynamics, thermostatted ring-polymer molecular dynamics (TRPMD), centroid molecular dynamics (CMD), and mean-field Matsubara dynamics. Additionally, we analyze the short-time accuracy of each methodology. We find that for multi-time correlation functions of linear operators, (T)RPMD is accurate up to order t3, while CMD is only correct up to t, indicating a degradation in the accuracy of these methodologies with respect to the single-time counterparts. The present work provides a firm justification for the use of path-integral-based approximations for the calculation of multi-time correlation functions.

Matrix product state of multi-time correlations

For an interacting spatio-temporal lattice system we introduce a formal way of expressing multi-time correlation functions of local observables located at the same spatial point with a time state, i.e. a statistical distribution of configurations observed along a time lattice. Such a time state is defined with respect to a particular equilibrium state that is invariant under space and time translations. The concept is developed within the rule 54 reversible cellular automaton, for which we explicitly construct a matrix product form of the time state, with matrices that act on the three-dimensional auxiliary space. We use the matrix-product state to express equal-space time-dependent density-density correlation function, which, for special maximum-entropy values of equilibrium parameters, agrees with the previous results. Additionally, we obtain an explicit expression for the probabilities of observing all multi-time configurations, which enables us to study distributions of times between consecutive excitations and prove the absence of decoupling of timescales in the rule 54 model.

Asymptotics of free fermions in a quadratic well at finite temperature and the Moshe–Neuberger–Shapiro random matrix model

Nous décrivons les statistiques locales de l’ensemble canonique de fermions libres dans un puits de potentiel quadratique à température finie, dans la limite où le nombre de particules tend vers l’infini. Ce modèle de fermions libres est équivalent à un modèle matriciel aléatoire proposé par Moshe, Neuberger et Shapiro. Les comportements à la limite précédemment obtenus pour l’ensemble grand-canonique sont observés dans l’ensemble canonique: Nous avons, au bord de l’ensemble, une transition de phase de la distribution de Tracy–Widom à la distribution de Gumbel via la distribution croisée de Kardar–Parisi–Zhang (KPZ), et dans l’ensemble, une transition de phase du processus ponctuel sinus au processus ponctuel de Poisson.

Markovian treatment of non-Markovian dynamics of open Fermionic systems

We show that an open fermionic system coupled to a continuous environment with unitary system-environment evolution can be exactly mapped onto an auxiliary system consisting of the physical fermion system and a set of discrete fermionic modes subject to non-unitary Lindblad-type system-modes evolution in such a way that reduced dynamics of the fermionic system in the two cases are the same. Conditions for equivalence of reduced dynamics in the two systems are identified and a proof is presented. Our study extends recent work on Bose systems (Tamascelli et al 2018 Phys. Rev. Lett. 120 030402) to the case of open quantum Fermi systems and to multi-time correlation functions. Numerical simulations within a generic junction model are presented for illustration.

Multi-time formulation of Matsubara dynamics.

Matsubara dynamics has recently emerged as the most general form of a quantum-Boltzmann-conserving classical dynamics theory for the calculation of single-time correlation functions. Here, we present a generalization of Matsubara dynamics for the evaluation of multitime correlation functions. We first show that the Matsubara approximation can also be used to approximate the two-time symmetrized double Kubo transformed correlation function. By a straightforward extension of these ideas to the multitime realm, a multitime Matsubara dynamics approximation can be obtained for the multitime fully symmetrized Kubo transformed correlation function. Although not a practical method, due to the presence of a phase-term, this multitime formulation of Matsubara dynamics represents a benchmark theory for future development of Boltzmann preserving semiclassical approximations to general higher order multitime correlation functions. It also reveals a connection between imaginary time-ordering in the path integral and the classical dynamics of multitime correlation functions.

Nonlinear transport coefficients from large deviation functions.

Nonlinear response occurs naturally when a strong perturbation takes a system far from equilibrium. Despite its omnipresence in nanoscale systems, it is difficult to predict in a general and efficient way. Here, we introduce a way to compute arbitrarily high order transport coefficients of stochastic systems, using the framework of large deviation theory. Leveraging time reversibility in the microscopic dynamics, we relate nonlinear response to equilibrium multitime correlation functions among both time reversal symmetric and asymmetric observables, which can be evaluated from derivatives of large deviation functions. This connection establishes a thermodynamiclike relation for nonequilibrium response and provides a practical route to its evaluation, as large deviation functions are amenable to importance sampling. We demonstrate the generality and efficiency of this method in predicting transport coefficients in single particle systems and an interacting system exhibiting thermal rectification.

Path-integral approach for nonequilibrium multitime correlation functions of open quantum systems coupled to Markovian and non-Markovian environments

Using a real-time path integral approach we develop an algorithm to calculate multi-time correlation functions of open few-level quantum systems that is applicable to highly nonequilibrium dynamics. The calculational scheme fully keeps the non-Markovian memory introduced by the pure-dephasing type coupling to a continuum of oscillators. Furthermore, we discuss how to deal consistently with the simultaneous presence of non-Markovian and Markovian system reservoir interactions. We apply the method to a crucial test case, namely the evaluation of emission spectra of a laser-driven two-level quantum dot coupled to a continuum of longitudinal acoustic phonons, which give rise to non-Markovian dynamics. Here, we also account for the coupling to a photonic environment, which models radiative decay and can be treated as a Markovian bath. The phonon side bands are found on the correct side of the zero phonon line in our calculation, in contrast to known results where the quantum regression theorem is applied naively to non-Markovian dynamics. Combining our algorithm with a recently improved iteration scheme for performing the required sum over paths we demonstrate the numerical feasibility of our approach to systems with more than two levels. Results are shown for the second-order photonic two-time correlation function of a quantum dot-cavity system with seven states on the Jaynes-Cummings ladder taken into account.

Inclusion of nuclear quantum effects for simulations of nonlinear spectroscopy.

The computation and interpretation of nonlinear vibrational spectroscopy is of vital importance for understanding a wide range of dynamical processes in molecular systems. Here, we introduce an approach to evaluate multi-time response functions in terms of multi-time double symmetrized Kubo transformed thermal correlation functions. Furthermore, we introduce a multi-time extension of ring polymer molecular dynamics to evaluate these Kubo transforms. Benchmark calculations show that the approximations are useful for short times even for nonlinear operators, providing a consistent improvement over classical simulations of multi-time correlation functions. The introduced methodology thus provides a practical way of including nuclear quantum effects in multi-time response functions of non-linear optical spectroscopy.