Open quantum systems, i.e., quantum systems that exchange energy or particles with their environment, are ubiquitous in a wide variety of fields including chemical physics, condensed matter physics, quantum information, quantum optics, and quantum thermodynamics.1–3 Examples in the context of chemical physics range from molecules in solution and at surfaces to molecular junctions, where single molecules are coupled to electrodes at different chemical potentials or temperatures. The coupling to the environment gives rise to dynamical processes such as fluctuations, dephasing, relaxation, thermalization, nonequilibrium excitation, and transport. The understanding of these processes is a major goal in the field of condensed-phase chemical dynamics.4
The theory and simulation of dynamical processes in open quantum systems have a long tradition in chemical physics. Recent years have seen a wealth of activities in the field, in the development of advanced theories and sophisticated numerical techniques, and in their application to a variety of challenging and interesting systems. The 55 research articles in this special topic report recent advances in the field and showcase the diversity of this rich class of problems.
A variety of strategies are used to describe the dynamics of open quantum systems. They differ in the way the overall system is represented (directly as an open system or mimicked by a large but formally closed system), which quantity is simulated (wave function, density matrix, or correlation function), and how the resulting dynamical problem is solved. Naturally, most approximate techniques have a domain of validity with respect to parameters of the problem—such as the subsystem Hamiltonian, the environment Hamiltonian, the strength of the coupling between the two, or the temperature—and more accurate or more general techniques have an increased computational cost consistent with the exponential scaling of exact many-body quantum mechanics.
One of the most widely used approaches to the dynamics of open quantum systems is based on reduced density matrix theory. Within this framework, integrating out the environment degrees of freedom results in equations of motion for the reduced density matrix of the system. Such formally exact generalized quantum master equations can be written in a time-nonlocal form5,6 or a time-local form,7,8 which form the basis for a variety of approximate theories. Employing basic perturbation theory and time scale separation, Born-Markov master equations are obtained, such as Redfield’s theory9 or the Lindblad formalism. Several articles in this volume employ this framework to study conceptual questions of open quantum system dynamics and interesting applications.10–15 These methods can also be extended to include non-Markovian effects, higher order contributions, and nonequilibrium fluctuations.16–24 In certain forms, deterministic quantum master equations can be simulated as an ensemble of stochastic wave function trajectories, which enables a reduced computational scaling and potential physical insights at the level of individual trajectories.25–27 Neglecting quantum coherence in quantum master equations leads to quantum mechanical rate theories and kinetic schemes.28–31
The dynamics of the reduced density matrix can also be simulated in a way that allows systematic convergence, i.e., as a numerically exact method. Such formulations often start from the path-integral approach and make use of the Feynman-Vernon influence functional.32 Modern formulations of this basic idea use either Monte Carlo sampling or diagrammatic resummation.19,33–35 A complementary approach is the hierarchical equation of motion method36–43 and a reformulation thereof using tensor networks.44
As an alternative to the reduced density matrix, various time correlation functions can be directly calculated, either in equilibrium or out of equilibrium. For fermionic systems, the one-particle Green’s function is a natural object of interest and uses self-energy to capture the interaction with the environment. The Green’s function is commonly used for dynamical embedding techniques in electronic structure theory45,46 or, in its nonequilibrium form, for transport problems.46–48
A powerful way to describe the dynamics of open quantum systems is based on the full many-body wavefunction of a large but finite total system. The explicit treatment of the combined system and a large environment requires methods capable of simulating quantum systems with many degrees of freedom. Examples of wave function-based approaches in this context include multilayer and Gaussian-based extensions of the multiconfiguration time-dependent Hartree theory,49–51 truncated Hilbert space techniques,52,53 the surrogate Hamiltonian approach,13 the Davydov ansatz,26,54,55 and coupled coherent state approaches.56 Importantly, many of these methods enable the study of generic environments beyond the paradigmatic model of noninteracting baths and may include coupling between the bath degrees of freedom or the use of anharmonic potentials.
This latter generality is also present in semiclassical and quasiclassical methods, which use classical trajectory-based concepts for an approximate description of quantum dynamics. Importantly, this feature enables the straightforward combination with atomistic molecular dynamics. Approaches along this line represented in this special topic involve extensions of the surface hopping approach, the Ehrenfest method, and the ring polymer molecular dynamics method.57–59 Combined with mapping approaches, which allow a representation of discrete electronic states by continuous degrees of freedom, semiclassical and quasiclassical methods can also be applied to electronically nonadiabatic dynamics as well as electron transport in impurity models and molecular junctions.59–62
Finally, we highlight that promising methods can be obtained that balance accuracy with efficiency by combining the above techniques with one another. A popular strategy in this direction is the use of approximate semiclassical or quantum mechanical approaches to obtain the kernel in formally exact generalized quantum master equations.63–66
As can be seen in this special topic, a broad set of modern applications demands treatment with dynamical open quantum systems. As examples, we highlight transport in molecular junctions,12,21,23,47,48,62,66 strong light-matter interactions in cavities,10,29,39,53,67 coupling to molecular vibrations or solid-state phonons,11,18,25,27,37,65 and the counting statistics of quantum events.16,21,35,47
In summary, this special topic showcases the great diversity of theoretical and computational tools being pioneered and applied to the dynamics of open quantum systems in chemical physics. Despite its long history and enormous recent progress, this area is still ripe for advances. In particular, state-of-the-art methods still struggle with the accurate treatment of open-system quantum dynamics at low temperatures, for nonperturbatively strong system-bath interactions or for realistic environments beyond those that are harmonic or noninteracting. We hope that this special topic, as a survey of the present-day field, will help identify current limitations and promising avenues for future breakthroughs.
