Small Matrix Path Integral for System-Bath Dynamics

An ensemble variational quantum algorithm for non-Markovian quantum dynamics

Many physical and chemical processes in a condensed phase environment exhibit non-Markovian quantum dynamics. As such simulations are challenging on classical computers, we developed a variational quantum algorithm that is capable of simulating non-Markovian dynamics on noisy intermediate-scale quantum (NISQ) devices. We used a quantum system linearly coupled to its harmonic bath as the model Hamiltonian. The non-Markovianity is captured by introducing auxiliary variables from the bath trajectories. With Monte Carlo sampling of the bath degrees of freedom, finite temperature dynamics is produced. We validated the algorithm on a simulator and demonstrated its performance on an IBM quantum device. The framework developed naturally adapts to any anharmonic bath with non-linear coupling to the system, and is also well suited for simulating spin chain dynamics in a dissipative environment.

Mean-Field Ring Polymer Rates Using a Population Dividing Surface

Mean-field ring polymer molecular dynamics offers a computationally efficient method for the simulation of reaction rates in multilevel systems. Previous work has established that, to model a nonadiabatic state-to-state reaction accurately, it is necessary to ensure reactive trajectories form kinked ring polymer configurations at the dividing surface. Building on this idea, we introduce a population difference coordinate and a reactive flux expression modified to only include contributions from kinked configurations. We test the accuracy of the resulting mean-field rate theory on a series of linear vibronic coupling model systems. We demonstrate that this new kMF-RP rate approach is efficient to implement and quantitatively accurate for models over a wide range of driving forces, coupling strengths, and temperatures.

Direct Numerical Solutions to Stochastic Differential Equations with Multiplicative Noise

Inspired by path integral solutions to the quantum relaxation problem, we develop a numerical method to solve classical stochastic differential equations with multiplicative noise that avoids averaging over trajectories. To test the method, we simulate the dynamics of a classical oscillator multiplicatively coupled to non-Markovian noise. When accelerated using tensor factorization techniques, it accurately estimates the transition into the bifurcation regime of the oscillator and outperforms trajectory-averaging simulations with a computational cost that is orders of magnitude lower.

Parsing the Influence Functional: Harmonic Bath Mapping and Anharmonic Small Matrix Path Integral

The influence functional (IF) encodes all of the information required for calculating the dynamical properties of a system in contact with its environment. A direct and simple procedure is introduced for extracting from a few numerical evaluations of the IF, without computing time correlation functions or evaluating integrals, the parameters required for path integral calculations, either within or beyond the harmonic mapping, and for assessing the accuracy of the harmonic bath approximation. In addition, the small matrix decomposition of the path integral (SMatPI) is extended to anharmonic environments and the required matrices are constructed directly from the IF.

Seeking a quantum advantage with trapped-ion quantum simulations of condensed-phase chemical dynamics

Simulating the quantum dynamics of molecules in the condensed phase represents a longstanding challenge in chemistry. Trapped-ion quantum systems may serve as a platform for the analog-quantum simulation of chemical dynamics that is beyond the reach of current classical-digital simulation. To identify a 'quantum advantage' for these simulations, performance analysis of both analog-quantum simulation on noisy hardware and classical-digital algorithms is needed. In this Review, we make a comparison between a noisy analog trapped-ion simulator and a few choice classical-digital methods on simulating the dynamics of a model molecular Hamiltonian with linear vibronic coupling. We describe several simple Hamiltonians that are commonly used to model molecular systems, which can be simulated with existing or emerging trapped-ion hardware. These Hamiltonians may serve as stepping stones towards the use of trapped-ion simulators for systems beyond the reach of classical-digital methods. Finally, we identify dynamical regimes in which classical-digital simulations seem to have the weakest performance with respect to analog-quantum simulations. These regimes may provide the lowest hanging fruit to make the most of potential quantum advantages.

Incorporation of Empirical Gain and Loss Mechanisms in Open Quantum Systems through Path Integral Lindblad Dynamics

Path integrals offer a robust approach for simulating open quantum dynamics with advancements transcending initial system size limitations. However, accurately modeling systems governed by mechanisms that do not conserve the number of quantum particles, such as lossy cavity modes, remains a challenge. We present a method to incorporate such empirical source and drain mechanisms within a path integral framework using quantum master equations. This technique facilitates rigorous inclusion of bath degrees of freedom while accommodating empirical time scales via Lindbladian dynamics. Computational costs are primarily driven by the path integral method with minimal overhead from Lindbladian terms. We use it to study exciton transport in a four-site Fenna-Matthews-Olson model, examining the potential loss of the exciton to the reaction center. This path integral Lindblad method promises an enhanced ability to simulate dynamics and will be fundamental to simulation of spectra in diverse quantum processes in open systems.

Kink Sum for Long-Memory Small Matrix Path Integral Dynamics

The small matrix decomposition of the real-time path integral (SMatPI) allows for numerically exact and efficient propagation of the reduced density matrix (RDM) for system-bath Hamiltonians. Its high efficiency lies in the small size of the SMatPI matrices employed in the iterative algorithm, whose size is equal to that of the full RDM. By avoiding the storage and multiplication of large tensors, the SMatPI algorithm is applicable in multistate systems under a variety of conditions. The main computational effort is the evaluation of path sums within the entangled memory length to construct the SMatPI matrices. A number of methods are available for this task, each with its own favorable parameter regime, but calculations with strong system-bath coupling and long memory at low temperatures remain out of reach. The present paper evaluates the path sums by binning the paths (in forward time only) based on their amplitudes, which depend on the number and type of kinks they contain. The algorithm is very efficient, leading to a dramatic acceleration of path sums and significantly extending the accessible memory length in the most challenging regimes.

Exact Non-Markovian Quantum Dynamics on the NISQ Device Using Kraus Operators

The theory of open quantum systems has many applications ranging from simulating quantum dynamics in condensed phases to better understanding quantum-enabled technologies. At the center of theoretical chemistry are the developments of methodologies and computational tools for simulating charge and excitation energy transfer in solutions, biomolecules, and molecular aggregates. As a variety of these processes display non-Markovian behavior, classical computer simulation can be challenging due to exponential scaling with existing methods. With quantum computers holding the promise of efficient quantum simulations, in this paper, we present a new quantum algorithm based on Kraus operators that capture the exact non-Markovian effect at a finite temperature. The implementation of the Kraus operators on the quantum machine uses a combination of singular value decomposition (SVD) and optimal Walsh operators that result in shallow circuits. We demonstrate the feasibility of the algorithm by simulating the spin-boson dynamics and the exciton transfer in the Fenna-Matthews-Olson (FMO) complex. The NISQ results show very good agreement with the exact ones.

Path Integral over Equivalence Classes for Quantum Dynamics with Static Disorder

An efficient, fully quantum mechanical, real-time path integral method for including the effects of static disorder in the dynamics of systems coupled to common or local harmonic baths is presented. Rather than performing a large number of demanding calculations for different realizations of the system Hamiltonian, the influence of the bath is captured through a single evaluation of the path sum by grouping the system paths into equivalence classes of fixed system amplitudes. The method is illustrated with several analytical and numerical examples that show a variety of nontrivial effects arising from the interplay among coherence, dissipation, thermal fluctuations, and geometric phases.

Quantum correlation functions through tensor network path integral

Tensor networks have historically proven to be of great utility in providing compressed representations of wave functions that can be used for the calculation of eigenstates. Recently, it has been shown that a variety of these networks can be leveraged to make real time non-equilibrium simulations of dynamics involving the Feynman-Vernon influence functional more efficient. In this work, a tensor network is developed for non-perturbatively calculating the equilibrium correlation function for open quantum systems using the path integral methodology. These correlation functions are of fundamental importance in calculations of rates of reactions, simulations of response functions and susceptibilities, spectra of systems, etc. The influence of the solvent on the quantum system is incorporated through an influence functional, whose unconventional structure motivates the design of a new optimal matrix product-like operator that can be applied to the so-called path amplitude matrix product state. This complex-time tensor network path integral approach provides an exceptionally efficient representation of the path integral, enabling simulations for larger systems strongly interacting with baths and at lower temperatures out to longer time. The derivation, design, and implementation of this method are discussed along with a wide range of illustrations ranging from rate theory and symmetrized spin correlation functions to simulation of response of the Fenna-Matthews-Olson complex to light.

Real-Time Simulation of Open Quantum Spin Chains with the Inchworm Method

We study the real-time simulation of open quantum systems, where the system is modeled by a spin chain with each spin associated with its own harmonic bath. Our method couples the inchworm method for the spin-boson model and the modular path integral methodology for spin systems. In particular, the introduction of the inchworm method can significantly suppress the numerical sign problem. Both methods are tweaked to make them work seamlessly with each other. We represent our approach in the language of diagrammatic methods and analyze the asymptotic behavior of the computational cost. Extensive numerical experiments are performed to validate our method.

Impact of Spatial Inhomogeneity on Excitation Energy Transport in the Fenna-Matthews-Olson Complex

The dynamics of the excitation energy transfer (EET) in photosynthetic complexes is an interesting question both from the perspective of fundamental understanding and the research in artificial photosynthesis. Over the past decade, very accurate spectral densities have been developed to capture spatial inhomogeneities in the Fenna-Matthews-Olson (FMO) complex. However, challenges persist in numerically simulating these systems, both in terms of parameterizing them and following their dynamics over long periods of time because of long non-Markovian memories. We investigate the dynamics of FMO with the exact treatment of various theoretical spectral densities using the new tensor network path integral-based methods, which are uniquely capable of addressing the difficulty of long memory length and incoherent Förster theory. It is also important to be able to analyze the pathway of EET flow, which can be difficult to identify given the non-trivial structure of connections between bacteriochlorophyll molecules in FMO. We use the recently introduced ideas of relating coherence to population derivatives to analyze the transport process and reveal some new routes of transport. The combination of exact and approximate methods sheds light on the role of coherences in affecting the fine details of the transport and promises to be a powerful toolbox for future exploration of other open systems with quantum transport.

Impact of Solvent on State-to-State Population Transport in Multistate Systems Using Coherences

Understanding the pathways taken by a quantum particle during a transport process is an enormous challenge. There are broadly two different aspects of the problem that affect the route taken. First is obviously the couplings between the various sites, which translates into the intrinsic "strength" of a state-to-state channel. Apart from these inter-state couplings, the relative coupling strengths and timescales of the solvent modes form the second factor. This impact of the dissipative environment is significantly more difficult to analyze. Building on the recently derived relations between coherences and population derivatives, we present an analysis of the transport that allows us to account for both the effects in a rigorous manner. We demonstrate the richness hidden behind the transport even for a relatively simple system, a 4-site coarse-grained model of the Fenna-Matthews-Olson complex. The effect of the local dissipative media is highly nontrivial. We show that while the impact on the total site population may be small, there are noticeable changes to the pathway taken by the transport process. We also demonstrate how an analysis in a similar spirit can be done using the Förster approximation. The ability to untangle the dynamics at a greater granularity opens up possibilities in terms of design of novel systems with an eye toward quantum control.

Coherence Maps and Flow of Excitation Energy in the Bacterial Light Harvesting Complex 2

We present and analyze coherence maps [ J. Phys. Chem. B 2022, 126, 9361-9375] to investigate the quantum coherences that are created, sustained, and damped by vibrational modes during the transfer of excitation energy from the B800 (outer) to the B850 (inner) ring of the light harvesting complex 2 (LH2) of purple bacteria with a variety of initial conditions. The reduced density matrix of the 24-pigment complex, where the ground and excited electronic states of each bacteriochlorophyll are explicitly coupled to 50 intramolecular vibrations at room temperature, is obtained from fully quantum-mechanical small matrix path integral (SMatPI) calculations. The coherence maps show a very rapid localization within the outer ring, accompanied by the formation of inter-ring quantum superpositions that evolve to a partial quantum delocalization at equilibrium, and quantify in state-to-state detail the flow of energy within the complex.

Topological aspects of system-bath Hamiltonians and a vector model for multisite systems coupled to local, correlated, or common baths

Some topological features of multisite Hamiltonians consisting of harmonic potential surfaces with constant site-to-site couplings are discussed. Even in the absence of Duschinsky rotation, such a Hamiltonian assumes the system-bath form only if severe constraints exist. The simplest case of a common bath that couples to all sites is realized when the potential minima are collinear. The bath reorganization energy increases quadratically with site distance in this case. Another frequently encountered situation involves exciton-vibration coupling in molecular aggregates, where the intramolecular normal modes of the monomers give rise to local harmonic potentials. In this case, the reorganization energy accompanying excitation transfer is independent of site-to-site separation, thus this situation cannot be described by the usual system-bath Hamiltonian. A vector system-bath representation is introduced, which brings the exciton-vibration Hamiltonian in system-bath form. In this, the system vectors specify the locations of the potential minima, which in the case of identical monomers lie on the vertices of a regular polyhedron. By properly choosing the system vectors, it is possible to couple each bath to one or more sites and to specify the desired initial density. With a collinear choice of system vectors, the coupling reverts to the simple form of a common bath. The compact form of the vector system-bath coupling generalizes the dissipative tight-binding model to account for local, correlated, and common baths. The influence functional for the vector system-bath Hamiltonian is obtained in a compact and simple form.

Time-Evolving Quantum Superpositions in Open Systems and the Rich Content of Coherence Maps

We discuss the general features of the time-evolving reduced density matrix (RDM) of multistate systems coupled to dissipative environments and show that many important aspects of the dynamics are visualized effectively and transparently through coherence maps, defined as snapshots of the real and imaginary components of the RDM on the square grid of system sites. In particular, the spread, signs, and shapes of the coherence maps collectively characterize the state of the system and the nature of the dynamics, as well as the equilibrium state. The topology of the system is readily reflected in its coherence map. Rows and columns show the composition of quantum superpositions, and their filling indicates the extent of the surviving coherence. Linear combinations of imaginary RDM elements specify instantaneous population derivatives. The main diagonal comprises the incoherent component of the dynamics, while the upper/lower triangular areas give rise to coherent contributions that increase the purity of the RDM. In open systems, the coherence map evolves to a band surrounding the principal diagonal whose width decreases with increasing temperature and dissipation strength. We illustrate these behaviors with examples of 10-site model molecular aggregates with Frenkel exciton couplings, where the electronic states of each monomer are coupled to harmonic vibrational baths.

Tight inner ring architecture and quantum motion of nuclei enable efficient energy transfer in bacterial light harvesting

The efficient, directional transfer of absorbed solar energy between photosynthetic light-harvesting complexes continues to pose intriguing questions. In this work, we identify the pathways of energy flow between the B800 and B850 rings in the LH2 complex of Rhodopseudomonas molischianum using fully quantum mechanical path integral methods to simulate the excited-state dynamics of the 24 bacteriochlorophyll molecules and their coupling to 50 normal mode vibrations in each chromophore. While all pigments are identical, the tighter packing of the inner B850 ring is responsible for the thermodynamic stabilization of the inner ring. Molecular vibrations enable the 1-ps flow of energy to the B850 states, which would otherwise be kinetically inaccessible. A classical treatment of the vibrations leads to uniform equilibrium distribution of the excitation, with only 67% transferred to the inner ring. However, spontaneous fluctuations associated with the quantum motion of the nuclei increase the transfer efficiency to 90%.

Effect of temperature gradient on quantum transport

The recently introduced multisite tensor network path integral (MS-TNPI) method [Bose and Walters, J. Chem. Phys., 2022, 156, 24101] for simulating quantum dynamics of extended systems has been shown to be effective in studying one-dimensional systems coupled with local baths. Quantum transport in these systems is typically studied at a constant temperature. However, temperature seems to be a very obvious parameter that can be spatially changed to control this transport. Here, MS-TNPI is used to study the "non-equilibrium" effects of an externally imposed temperature profile on the excitonic transport in one-dimensional Frenkel chains coupled with local vibrations. We show that in addition to being important for incorporating heating effects of excitation by lasers, temperature can also be an interesting parameter for quantum control.

Hidden Phase of the Spin-Boson Model

A quantum two-level system immersed in a sub-Ohmic bath experiences enhanced low-frequency quantum statistical fluctuations which render the nonequilibrium quantum dynamics highly non-Markovian. Upon using the numerically exact time-evolving matrix product operator approach, we investigate the phase diagram of the polarization dynamics. In addition to the known phases of damped coherent oscillatory dynamics and overdamped decay, we identify a new third region in the phase diagram for strong coupling showing an aperiodic behavior. We determine the corresponding phase boundaries. The dynamics of the quantum two-state system herein is not coherent by itself but slaved to the oscillatory bath dynamics.

Quantum State-to-State Rates for Multistate Processes from Coherences

For a multistate system coupled to a general environment through terms local in the system basis, we show that the time derivatives of populations are given in terms of imaginary components of coherences, i.e., off-diagonal elements of the reduced density matrix. When the process exhibits rate dynamics, we show that all state-to-state rates can be obtained from the early "plateau" values of these imaginary components. The evolution of the state populations is then obtained from the short-time simulation results and the solution of the kinetic equations with the computed rate matrix. These expressions generalize the reactive flux method and its nonequilibrium version to multistate processes and show that even in the completely incoherent limit of rate kinetics, the time evolution of populations is governed by coherences. Further, we show that by virtue of detailed balance, the short-time values of the imaginary components of coherences fully determine the equilibrium populations.

B800-to-B850 relaxation of excitation energy in bacterial light harvesting: All-state, all-mode path integral simulations

We report fully quantum mechanical simulations of excitation energy transfer within the peripheral light harvesting complex (LH2) of Rhodopseudomonas molischianum at room temperature. The exciton–vibration Hamiltonian comprises the 16 singly excited bacteriochlorophyll states of the B850 (inner) ring and the 8 states of the B800 (outer) ring with all available electronic couplings. The electronic states of each chromophore couple to 50 intramolecular vibrational modes with spectroscopically determined Huang–Rhys factors and to a weakly dissipative bath that models the biomolecular environment. Simulations of the excitation energy transfer following photoexcitation of various electronic eigenstates are performed using the numerically exact small matrix decomposition of the quasiadiabatic propagator path integral. We find that the energy relaxation process in the 24-state system is highly nontrivial. When the photoexcited state comprises primarily B800 pigments, a rapid intra-band redistribution of the energy sharply transitions to a significantly slower relaxation component that transfers 90% of the excitation energy to the B850 ring. The mixed character B850* state lacks the slow component and equilibrates very rapidly, providing an alternative energy transfer channel. This (and also another partially mixed) state has an anomalously large equilibrium population, suggesting a shift to lower energy by virtue of exciton–vibration coupling. The spread of the vibrationally dressed states is smaller than that of the eigenstates of the bare electronic Hamiltonian. The total population of the B800 band is found to decay exponentially with a 1/ e time of 0.5 ps, which is in good agreement with experimental results.

Small Matrix Quantum-Classical Path Integral

The quantum-classical path integral (QCPI) is a rigorous formulation of nonadiabatic dynamics, where the dynamical interaction between a quantum system and its environment is captured consistently through classical trajectories driven by forces along quantum paths of the system. In this Letter, we develop a small matrix decomposition (SMatQCPI) that eliminates the tensor storage requirements of the iterative QCPI algorithm. In the case of a system coupled to a harmonic bath, SMatQCPI provides fully quantum mechanical propagation, which also reduces the computational cost to that of a single QCPI step. Further, the SMatQCPI matrices only need to account for quantum contributions to decoherence, allowing high efficiency in challenging regimes of incoherent dynamics. Overall, this new composite algorithm combines the best features of two powerful path integral formulations and offers a versatile tool for simulating condensed phase quantum dynamics.

Synthetic Control of Exciton Dynamics in Bioinspired Cofacial Porphyrin Dimers

Understanding how the complex interplay among excitonic interactions, vibronic couplings, and reorganization energy determines coherence-enabled transport mechanisms is a grand challenge with both foundational implications and potential payoffs for energy science. We use a combined experimental and theoretical approach to show how a modest change in structure may be used to modify the exciton delocalization, tune electronic and vibrational coherences, and alter the mechanism of exciton transfer in covalently linked cofacial Zn-porphyrin dimers (meso-beta linked ABm-β and meso-meso linked AAm-m). While both ABm-β and AAm-m feature zinc porphyrins linked by a 1,2-phenylene bridge, differences in the interporphyrin connectivity set the lateral shift between macrocycles, reducing electronic coupling in ABm-β and resulting in a localized exciton. Pump-probe experiments show that the exciton dynamics is faster by almost an order of magnitude in the strongly coupled AAm-m dimer, and two-dimensional electronic spectroscopy (2DES) identifies a vibronic coherence that is absent in ABm-β. Theoretical studies indicate how the interchromophore interactions in these structures, and their system-bath couplings, influence excitonic delocalization and vibronic coherence-enabled rapid exciton transport dynamics. Real-time path integral calculations reproduce the exciton transfer kinetics observed experimentally and find that the linking-modulated exciton delocalization strongly enhances the contribution of vibronic coherences to the exciton transfer mechanism, and that this coherence accelerates the exciton transfer dynamics. These benchmark molecular design, 2DES, and theoretical studies provide a foundation for directed explorations of nonclassical effects on exciton dynamics in multiporphyrin assemblies.

Intramolecular Vibrations in Excitation Energy Transfer: Insights from Real-Time Path Integral Calculations

Excitation energy transfer (EET) is fundamental to many processes in chemical and biological systems and carries significant implications for the design of materials suitable for efficient solar energy harvest and transport. This review discusses the role of intramolecular vibrations on the dynamics of EET in nonbonded molecular aggregates of bacteriochlorophyll, a perylene bisimide, and a model system, based on insights obtained from fully quantum mechanical real-time path integral results for a Frenkel exciton Hamiltonian that includes all vibrational modes of each molecular unit at finite temperature. Generic trends, as well as features specific to the vibrational characteristics of the molecules, are identified. Weak exciton-vibration (EV) interaction leads to compact, near-Gaussian densities on each electronic state, whose peak follows primarily a classical trajectory on a torus, while noncompact densities and nonlinear peak evolution are observed with strong EV coupling. Interaction with many intramolecular modes and increasing aggregate size smear, shift, and damp these dynamical features. Expected final online publication date for the Annual Review of Physical Chemistry, Volume 73 is April 2022. Please see http://www.annualreviews.org/page/journal/pubdates for revised estimates.

Quantum quench and coherent-incoherent dynamics of Ising chains interacting with dissipative baths

The modular path integral methodology is used to extend the well-known spin-boson dynamics to finite-length quantum Ising chains, where each spin is coupled to a dissipative harmonic bath. The chain is initially prepared in the ferromagnetic phase where all spins are aligned, and the magnetization is calculated with spin-spin coupling parameters corresponding to the paramagnetic phase, mimicking a quantum quench experiment. The observed dynamics is found to depend significantly on the location of the tagged spin. In the absence of a dissipative bath, the time evolution displays irregular patterns that arise from multiple frequencies associated with the eigenvalues of the chain Hamiltonian. Coupling of each spin to a harmonic bath leads to smoother dynamics, with damping effects that are stronger compared to those observed in the spin-boson model and more prominent in interior spins, a consequence of additional damping from the spin environment. Interior spins exhibit a transition from underdamped oscillatory to overdamped monotonic dynamics as the temperature, spin-bath, or spin-spin coupling is increased. In addition to these behaviors, a new dynamical pattern emerges in the evolution of edge spins with strong spin-spin coupling at low and intermediate temperatures, where the magnetization oscillates either above or below the equilibrium value.

Small Matrix Path Integral for Driven Dissipative Dynamics

The small matrix decomposition of the quasi-adiabatic propagator path integral (SMatPI) for a system coupled to a harmonic bath, which accounts for multitime memory correlations in the influence functional without the use of tensors, is extended to include a time-dependent term that drives the system. In the case of a periodic field, the algorithm requires the construction of SMatPI matrices initialized over a short time interval. The SMatPI algorithm circumvents the large array storage of tensor-based iterative path integral decompositions and, in the case of a periodic field, also eliminates the demanding tensor multiplication at each time step, leading to dramatic savings which allow the fully quantum mechanical treatment of multistate systems and long-memory environments.

Constructing tensor network influence functionals for general quantum dynamics

We describe an iterative formalism to compute influence functionals that describe the general quantum dynamics of a subsystem beyond the assumption of linear coupling to a quadratic bath. We use a space-time tensor network representation of the influence functional and investigate its approximability in terms of its bond dimension and time-like entanglement in the tensor network description. We study two numerical models, the spin-boson model and a model of interacting hard-core bosons in a 1D harmonic trap. We find that the influence functional and the intermediates involved in its construction can be efficiently approximated by low bond dimension tensor networks in certain dynamical regimes, which allows the quantum dynamics to be accurately computed for longer times than with direct time evolution methods. However, as one iteratively integrates out the bath, the correlations in the influence functional can first increase before decreasing, indicating that the final compressibility of the influence functional is achieved via non-trivial cancellation.

Time Evolution of Bath Properties in Spin-Boson Dynamics

The dynamical behaviors of a two-level system (TLS) coupled to a harmonic dissipative bath has been studied extensively using a variety of analytical and numerical methods. The focus of the vast majority of these studies has been on the properties of the TLS, averaged with respect to the bath degrees of freedom. In this work, we use real-time path integral methods to probe the behavior of select bath degrees of freedom during the dynamics of a symmetric two-level system (TLS) coupled to a dissipative bath by calculating system-bath densities (SBD) and coordinate expectation values. Overall, the SBD motion on each diabatic state is simpler than the motion of the total density. In the weak coupling regime, which characterizes the parameters of oscillators that comprise such a bath, the SBD on each TLS state remains primarily compact and Gaussian-like, such that its peak is well characterized by the mode expectation value. In the absence of a dissipative environment, nonadiabatic density depletion leads to spikes in coordinate expectation values. The evolution of the SBD peak trajectory for two discrete modes exhibits Lissajous patterns with frequency-dependent shapes that strongly resemble classical trajectory motion on a torus. These patterns become more complex when the coupling of the mode to the TLS is increased outside of this regime, leading to persistent small amplitude oscillations in the TLS populations characterized by a very slow decay and SBD trajectories that exhibit behaviors reminiscent of chaotic classical systems. Indirect coupling to a dissipative bath has a stabilizing effect on the dynamics, eliminating spikes, synchronizing the SBD motion on the two diabatic states and regularizing the SBD trajectory to simple rectangular Lissajous-like shapes with a slowly shrinking boundary, regardless of the mode frequencies.

Density matrix and purity evolution in dissipative two-level systems: I. Theory and path integral results for tunneling dynamics

The time evolution of the purity (the trace of the square of the reduced density matrix) and von Neumann entropy in a symmetric two-level system coupled to a dissipative harmonic bath is investigated through analytical arguments and accurate path integral calculations on simple models and the singly excited bacteriochlorophyll dimer. A simple theoretical analysis establishes bounds and limiting behaviors. The contributions to purity from a purely incoherent term obtained from the diagonal elements of the reduced density matrix, a term associated with the difference of the two eigenstate populations, and a third term related to the square of the time derivative of a site population, are discussed in various regimes. In the case of tunneling dynamics from a localized initial condition, the complex interplay among these contributions leads to the recovery of purity under low-temperature, weakly dissipative conditions. Memory effects from the bath are found to play a critical role to the dynamics of purity. It is shown that the strictly quantum mechanical decoherence process associated with spontaneous phonon emission is responsible for the long-time recovery of purity. These analytical and numerical results show clearly that the loss of quantum coherence during the evolution toward equilibrium does not necessarily imply the decay of purity, and that the time scales relevant to these two processes may be entirely different.

Density matrix and purity evolution in dissipative two-level systems: II. Relaxation

We investigate the time evolution of the reduced density matrix (RDM) and its purity in the dynamics of a two-level system coupled to a dissipative harmonic bath, when the system is initially placed in one of its eigenstates. We point out that the symmetry of the initial condition confines the motion of the RDM elements to a one-dimensional subspace and show that the purity always goes through its maximally mixed value at some time during relaxation, but subsequently recovers and (under low-temperature, weakly dissipative conditions) can rise to values that approach unity. These behaviors are quantified through accurate path integral calculations. Under low-temperature, weakly dissipative conditions, we observe unusual, nonmonotonic population dynamics when the two-level system is initially placed in its ground state. We also analyze the origin of the system-bath interactions responsible for the nonmonotonic behavior of purity during relaxation. Our results show that classical dephasing processes arising from site level fluctuations lead to a monotonic decay of purity, and that the quantum mechanical decoherence events associated with spontaneous phonon emission are responsible for the subsequent recovery of purity. Last, we show that coupling with a low-temperature bath can purify a mixed two-level system. In the case of the maximally mixed initial RDM, the purity increases monotonically even during short time.

Electronic-vibrational density evolution in a perylene bisimide dimer: mechanistic insights into excitation energy transfer

The process of excitation energy transfer (EET) in molecular aggregates is etched with the signatures of a multitude of electronic and vibrational time scales that often are extremely difficult to resolve. The effect of the motion associated with one molecular vibration on that of another is fundamental to the dynamics of EET. In this paper we present simple theoretical ideas along with fully quantum mechanical calculations to develop a comprehensive mechanistic picture of EET in terms of the time evolution of electronic-vibrational densities (EVD) in a perylene bisimide (PBI) dimer, where 28 intramolecular normal modes couple to the ground and excited electronic states of each molecule. The EVD motion exhibits a plethora of dynamical features, which impart physical justification for the composite effects observed in the EET dynamics. Weakly coupled vibrations lead to classical-like motion of the EVD center on each electronic state, while highly nontrivial EVD characteristics develop under moderate or strong exciton-vibration interaction, leading to the formation of split or crescent-shaped densities, as well as density retention that slows down energy transfer and creates new peaks in the electronic populations. Pronounced correlation effects are observed in two-mode projections of the EVD, as a consequence of indirect vibrational coupling between uncoupled normal modes induced by the electronic coupling. Such indirect coupling depends on the strength of exciton-vibration interactions as well as the frequency mismatch between the two modes and leaves nontrivial signatures in the electronic population dynamics. The collective effects of many vibrational modes cause a partial smearing of these features through dephasing.

Small matrix modular path integral: iterative quantum dynamics in space and time

The modular path integral (MPI) formulation for one-dimensional extended systems, such as spin arrays or molecular aggregates, allows evaluation of spin- or exciton-vibration dynamics with effort that scales linearly with the number of units. This work presents a small matrix decomposition of the modular path integral (SMatMPI), which eliminates tensor storage and enables iterative long-time propagation.

Recovery of Purity in Dissipative Tunneling Dynamics

The time evolution of purity for an initially localized state of a symmetric two-level system coupled to a dissipative bath is investigated using numerically exact real-time path integral methods. With strong system-bath coupling and high temperature, the purity decays monotonically to its fully mixed value, with a short-time Gaussian behavior, which is subsequently followed by exponential evolution. However, under low-temperature and weak coupling conditions, a substantial recovery of purity is observed. A simple theoretical analysis reveals three contributions that correspond to a completely incoherent, eigenstate population difference and rate terms. The last two of these terms can counter the early drop of purity and are responsible for its rebound. These findings caution against using purity as a measure of decoherence in the dynamics of quantum dissipative systems.