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arXiv:2101.01098 · Frontiers in Chemistry

Simulating Quantum Vibronic Dynamics at Finite Temperatures With Many Body Wave Functions at 0 K

Angus Dunnett, Alex W. Chin

Published Jan 7, 2021• 24 citations
arXivHTMLPDF

Abstract

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.

DOI
10.3389/fchem.2020.600731

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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.
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.
Understanding decoherence and dissipation in quantum systems is essential for exploiting quantum effects in emerging technological applications. A proper understanding requires accurate modelling of the system-environment interaction which can only be achieved through treating the system and environment on an equal footing. Simulating the entire system+environment wave-function is a challenging task due to the exponential scaling which is characteristic of quantum mechanics. By using many-body methods based on the powerful concept of a tensor network, one can, nonetheless, obtain numerically exact quantum dynamics for systems with hundreds of degrees of freedom. In addition, a recent innovation has shown that this many-body approach can be extended to finite temperatures without a significant increase in computational cost or algorithmic complexity. This has allowed us to directly compare quantum dynamical simulations with experimental results. Specifically, simulations of the linear vibronic coupling Hamiltonian, parameterised using abinitio electronic structure methods, were used to compute the absorption spectrum of the dye Methylene blue in water. This study has elucidated certain unexplained features of the absorption and shown the how non-adiabatic dynamics resulting from a conical intersection can lead to intensity borrowing between bright and dark excited states. Several methodological advances are also made, including an improvement to the underlying time evolution method TDVP which brings with it an order of magnitude speed-up compared to existing methods.
We report the development of an efficient many-body algorithm for simulating open quantum system dynamics that utilizes a time-dependent variational principle for matrix product states to evolve large system-environment states. Capturing all system-environment correlations, we reproduce the nonperturbative, quantum-critical dynamics of the zero-temperature spin-boson model, and then exploit the many-body information to visualize the complete time-frequency spectrum of the environmental excitations. Our ``environmental spectra'' reveal correlated vibrational motion in polaronic modes which preserve their vibrational coherence during incoherent spin relaxation, demonstrating how environment information could yield valuable insights into complex quantum dissipative processes.
Experimental advances have made it possible to realize and control quantum many-body systems, allowing the experimental study of non-equilibrium phenomena. Due to the Hilbert space growing exponentially with the number of particles, advanced numerical techniques are usually required to understand many-body effects at the theoretical level. In this thesis, we address the question of how to efficiently simulate the non-equilibrium dynamics of many-body systems using matrix product states (MPS). In particular, we consider quantum systems in interaction with macroscopic environments, as they appear in Anderson impurity problems, quantum thermodynamics, or open quantum systems. We develop a low-entanglement representation of the environment with only short-range interactions, perfectly suitable for simulations with MPS. We further show that an interleaved ordering of tight-binding chains can significantly reduce the creation of entanglement, as opposed to an intuitive implementation of the Hamiltonian's geometry. Our approach allows long-time simulations and an analysis of the environmental dynamics, in which, after a sudden quench, we find clear signatures of many-body effects. We employ our techniques to compute spectral functions of impurity models, with possible applications to dynamical mean-field theory impurity solvers, and to calculate dissipation in the non-equilibrium Anderson model with explicit time-dependence, as relevant for ongoing experiments with oscillating tip atomic force microscopes.
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 $30\ifmmode\times\else\texttimes\fi{}$ and $10\ifmmode\times\else\texttimes\fi{}$ speed-up over 2TDVP and 1TDVP, respectively.
Dynamical electronic- and vibrational-structure theories have received a growing interest in the past few years due to their ability to simulate spectra recorded with ultrafast experimental techniques. The exact time evolution of a molecular system can, in principle, be obtained from the time-dependent version of full configuration interaction. Such an approach is, however, limited to few-atom systems due to the exponential increase of its cost with the system dimension. In the present work, we overcome this unfavorable scaling by employing the time-dependent density matrix renormalization group (TD-DMRG) which parametrizes the time-dependent wave function as a matrix product state. The time-dependent Schrödinger equation is then integrated with a sweep-based algorithm, as in standard time-independent DMRG. Unlike other TD-DMRG approaches, the one presented here leads to a set of coupled equations that can be integrated exactly. The resulting theory enables us to study real- and imaginary-time evolutions of Hamiltonians comprising more than 20 degrees of freedom that are challenging for current state-of-the-art quantum dynamics algorithms. We apply our algorithm to the simulation of quantum dynamics of models of increasing complexity, ranging from simple excitonic Hamiltonians to more complex ab initio vibronic ones.