Matrix Product State Simulations of Non-Equilibrium Steady States and Transient Heat Flows in the Two-Bath Spin-Boson Model at Finite Temperatures

Abstract

Simulating the non-perturbative and non-Markovian dynamics of open quantum systems is a very challenging many body problem, due to the need to evolve both the system and its environments on an equal footing. Tensor network and matrix product states (MPS) have emerged as powerful tools for open system models, but the numerical resources required to treat finite-temperature environments grow extremely rapidly and limit their applications. In this study we use time-dependent variational evolution of MPS to explore the striking theory of Tamascelli et al. (Phys. Rev. Lett. 2019, 123, 090402.) that shows how finite-temperature open dynamics can be obtained from zero temperature, i.e., pure wave function, simulations. Using this approach, we produce a benchmark dataset for the dynamics of the Ohmic spin-boson model across a wide range of coupling strengths and temperatures, and also present a detailed analysis of the numerical costs of simulating non-equilibrium steady states, such as those emerging from the non-perturbative coupling of a qubit to baths at different temperatures. Despite ever-growing resource requirements, we find that converged non-perturbative results can be obtained, and we discuss a number of recent ideas and numerical techniques that should allow wide application of MPS to complex open quantum systems.

Influence of non-adiabatic effects on linear absorption spectra in the condensed phase: Methylene blue

“…The zero-temperature equilibrium correlation function of this spectrally extended environment is identical to that of the original finite temperature environment, and as the reduced system dynamics defined by Tr E {ρ(t)} can be shown to be uniquely determined by the bath correlation function, the proxy, extended zero temperature environment can be used to obtain finite temperature results [47]. Moreover, the mapping can also be inverted on the bath modes to provide thermal expectations for the nuclei in the original basis [48,49].…”

“…However, by applying Tamascelli et al's T-TEDOPA mapping to each bath, we transform these density matrices into vacuum states ρ {α} → |0 α -single pure state wave functions -while the bath spectral density picks up a temperature dependence J(ω) → J β (ω). [47,48] This new thermal spectral density, which encodes the detailed balance of thermal emission and absorption in the mode coupling strengths, is defined on the domain ω ∈ [−∞, ∞] and thus introduces effective negative energy modes into the bath. The zero-temperature equilibrium correlation function of this spectrally extended environment is identical to that of the original finite temperature environment, and as the reduced system dynamics defined by Tr E {ρ(t)} can be shown to be uniquely determined by the bath correlation function, the proxy, extended zero temperature environment can be used to obtain finite temperature results [47].…”

“…This limitation was overcome, first with the introduction of the thermofield approach, in which a bath of negative energy modes that act like a source of thermal energy are introduced, and later with the mapping of Tamascelli et al to treat a thermal en-vironment by defining a temperature dependent spectral density, leading to so called T-TEDOPA. [47,48] These advances now allow finite temperature OQS dynamics to be extracted from many-body pure state wave function simulations at 0K -offering substantial advantages over other methods in terms of efficiency [49]. In this article we demonstrate how this new capability allows us to predict optical spectra of real-world molecules in realistic environments that can be directly compared to experiments in solution.…”

Simulations of photoinduced processes with the exact factorization: State of the art and perspectives

“…It is worth mentioning here that other classes of methodologies exist to simulate nonadiabatic dynamics, those based for instance on the density-matrix formalism 10,[84][85][86][87][88][89][90][91][92] or on quantum wavepackets propagation [93][94][95][96][97][98] , but will not be covered in this Perspective.…”

New Density Matrix Renormalization Group Approaches for Strongly Correlated Systems Coupled with Large Environments

“…In recent years many algorithms have been proposed to accelerate TD-DMRG, such as increasing the basis number adaptively in the single-site algorithm − and utilizing the purification-projection (PP) to restore the U (1)-particle number conservation symmetry in the phononic system , as well as using the graphical processing units (GPUs) to accelerate the heavy tensor contractions, but nowadays it is still impossible to include all environmental phononic modes into TD-DMRG. Rather than including all environmental phononic modes into the embedded active subsystem explicitly, latest developments in TD-DMRG incline for methods based on open quantum systems or developing new algorithms to identify a small number of environmental modes which have strongest couplings with the interested subsystem. − For this purpose, we recently proposed a hierarchical mapping (HM) method which uses QIT to identify a small number of renormalized environmental phononic modes directly coupled to the quantum subsystem followed by successively transforming the vibronic Hamiltonian matrix to a nearly block-tridiagonal form by the block Lanczos algorithm. Numerical tests on model spin-boson systems and realistic singlet fission models in rubrene crystal environment with up to 7000 phononic modes and strong system–environment interactions indicate the HM can reduce the size of full quantum subsystem by 1–2 orders of magnitude and accelerate the calculation by ∼80% with negligible accuracy loss.…”