Recent tensor network techniques for simulating system-environment wave functions have provided profound insights into non-Markovian dissipation and decoherence in open quantum systems. Here, we propose a dynamically adaptive one-site time-dependent-variational-principle (A1TDVP) method for matrix product states in which local bond dimensions grow to capture developing system-bath entanglement. This avoids the need for multiple convergence runs with respect to bond dimensions and the unfavorable local Hilbert space scaling of two-site methods. A1TDVP is thus ideally suited for open quantum dynamics in finite-temperature bosonic environments, as the initial states typically have low bond dimension but require very large local physical dimensions. We demonstrate this with simulations of nonequilibrium heat flows through a qubit spin, finding a $30\times$ and $10\times$ speed-up over 2TDVP and 1TDVP, respectively.

Modeling linear absorption spectra of solvated chromophores is highly challenging as contributions are present both from coupling of the electronic states to nuclear vibrations and from solute–solvent interactions. In systems where excited states intersect in the Condon region, significant non-adiabatic contributions to absorption line shapes can also be observed. Here, we introduce a robust approach to model linear absorption spectra accounting for both environmental and non-adiabatic effects from first principles. This model parameterizes a linear vibronic coupling (LVC) Hamiltonian directly from energy gap fluctuations calculated along molecular dynamics (MD) trajectories of the chromophore in solution, accounting for both anharmonicity in the potential and direct solute–solvent interactions. The resulting system dynamics described by the LVC Hamiltonian are solved exactly using the thermalized time-evolving density operator with orthogonal polynomials algorithm (T-TEDOPA). The approach is applied to the linear absorption spectrum of methylene blue in water. We show that the strong shoulder in the experimental spectrum is caused by vibrationally driven population transfer between the bright S₁ and the dark S₂ states. The treatment of the solvent environment is one of many factors that strongly influence the population transfer and line shape; accurate modeling can only be achieved through the use of explicit quantum mechanical solvation. The efficiency of T-TEDOPA, combined with LVC Hamiltonian parameterizations from MD, leads to an attractive method for describing a large variety of systems in complex environments from first principles.

The MPSDynamics.jl package provides an easy-to-use interface for performing open quantum systems simulations at zero and finite temperatures. The package has been developed with the aim of studying non-Markovian open system dynamics using the state-of-the-art numerically exact Thermalized-Time Evolving Density operator with Orthonormal Polynomials Algorithm based on environment chain mapping. The simulations rely on a tensor network representation of the quantum states as matrix product states (MPS) and tree tensor network states. Written in the Julia programming language, MPSDynamics.jl is a versatile open-source package providing a choice of several variants of the Time-Dependent Variational Principle method for time evolution (including novel bond-adaptive one-site algorithms). The package also provides strong support for the measurement of single and multi-site observables, as well as the storing and logging of data, which makes it a useful tool for the study of many-body physics. It currently handles long-range interactions, time-dependent Hamiltonians, multiple environments, bosonic and fermionic environments, and joint system–environment observables.

For complex molecules, nuclear degrees of freedom can act as an environment for the electronic “system” variables, allowing the theory and concepts of open quantum systems to be applied. However, when molecular system-environment interactions are non-perturbative and non-Markovian, numerical simulations of the complete system-environment wave function become necessary. These many body dynamics can be very expensive to simulate, and extracting finite-temperature results—which require running and averaging over many such simulations—becomes especially challenging. Here, we present numerical simulations that exploit a recent theoretical result that allows dissipative environmental effects at finite temperature to be extracted efficiently from a single, zero-temperature wave function simulation. Using numerically exact time-dependent variational matrix product states, we verify that this approach can be applied to vibronic tunneling systems and provide insight into the practical problems lurking behind the elegance of the theory, such as the rapidly growing numerical demands that can appear for high temperatures over the length of computations.

Nanoscale devices, either biological or artificial, operate in a regime where the usual assumptions of a structureless Markovian bath do not hold. Being able to predict and study the dynamics of such systems is crucial and is usually done by tracing out the bath degrees of freedom, which implies losing information about the environment. To go beyond these approaches we use a numerically exact method relying on a matrix product state representation of the quantum state of a system and its environment to keep track of the bath explicitly. This method is applied to a specific example of interaction that depends on the spatial structure of a system made of two sites. The result is that we predict a non-Markovian dynamics where long-range couplings induce correlations into the environment. The environment dynamics can be naturally extracted from our method and shine a light on long-time feedback effects that are responsible for the observed non-Markovian recurrences in the eigenpopulations of the system.

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.