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arXiv:1909.09589 · Physical review. B./Physical review. B

Efficient simulation of open quantum systems coupled to a fermionic bath

Alexander Nüßeler, Ish Dhand, Susana F. Huelga, Martin Bodo Plenio

Published Apr 27, 2020• 49 citations
arXivHTMLPDF

Abstract

The problem of open quantum systems coupled to fermionic environments arises in a variety of fields, ranging from quantum dots to quantum transport at zero and finite temperature. This work presents and analyzes a tensor network based time integration scheme for this problem, which enables simulations of such systems with controllable and certified error. Furthermore, it is shown that performing simulations in the Heisenberg picture can lead to significant efficiency gains.

DOI
10.1103/physrevb.101.155134

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The problem of open quantum systems coupled to fermionic environments arises in a variety of fields, ranging from quantum dots to quantum transport at zero and finite temperature. This work presents and analyzes a tensor network based time integration scheme for this problem, which enables simulations of such systems with controllable and certified error. Furthermore, it is shown that performing simulations in the Heisenberg picture can lead to significant efficiency gains.
We introduce a general numerical method to compute the dynamics and multitime correlations of chains of quantum systems, where each system may couple strongly to a structured environment. The method combines the process tensor formalism for general (possibly non-Markovian) open quantum systems with time-evolving block decimation for one-dimensional chains. It systematically reduces the numerical complexity originating from system-environment correlations before integrating them into the full many-body problem, making a wide range of applications numerically feasible. We illustrate the power of this method by studying two examples. First, we study the thermalization of individual spins of a short XYZ Heisenberg chain with strongly coupled thermal leads. Our results confirm the complete thermalization of the chain when coupled to a single bath, and they reveal distinct effective temperatures in low-, mid-, and high-frequency regimes when the chain is placed between a hot and a cold bath. Second, we study the dynamics of diffusion in a longer XY chain, when each site couples to its own bath.
Open quantum many-body systems play an important role in quantum optics and condensed matter physics, and capture phenomena like transport, the interplay between Hamiltonian and incoherent dynamics, and topological order generated by dissipation. We introduce a versatile and practical method to numerically simulate one-dimensional open quantum many-body dynamics using tensor networks. It is based on representing mixed quantum states in a locally purified form, which guarantees that positivity is preserved at all times. Moreover, the approximation error is controlled with respect to the trace norm. Hence, this scheme overcomes various obstacles of the known numerical open-system evolution schemes. To exemplify the functioning of the approach, we study both stationary states and transient dissipative behavior, for various open quantum systems ranging from few to many bodies.
Transport through correlated nanoscale systems underpins the operation of quantum-dot and molecular-scale devices, yet accurate simulations of large open quantum systems remain computationally challenging as system size increases. Tensor-network methods offer a promising route past this scaling barrier by efficiently compressing quantum states. Here we extend a tensor-based solver with a jump-counting estimator that enables direct computation of steady-state electron currents from lead-induced tunneling events. We benchmark the resulting currents against the state-of-the-art master-equation solver QmeQ across a range of lead-dot and inter-dot coupling parameters and find quantitative agreement in the tractable regime. Compared with classical approaches, TJM reduces memory requirements and wall-clock time by orders of magnitude, enabling simulations of interacting quantum-dot arrays far beyond the range accessible to density-matrix-based transport solvers and systematic studies of size-dependent nonequilibrium transport in larger arrays. Our approach allow us to model quantum transport in an array of up to fifty (50) quantum dots.
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 consider a thermofield approach to analyze the evolution of an open quantum system coupled to an environment at finite temperature. In this approach, the finite-temperature environment is exactly mapped onto two virtual environments at zero temperature. These two environments are then unitarily transformed into two different chains of oscillators, leading to a one-dimensional structure that can be numerically studied using tensor network techniques. Compared to previous approaches using a single chain mapping, our strategy offers the advantage of an exact description of the initial state at arbitrary temperatures, which results in a gain in computational efficiency and a reduced truncation error.
The nonequilibrium effects of dissipation and drive play a key role in an immense variety of nanoscale and mesoscale quantum systems. To understand the behaviour of open quantum systems, we need accurate methods that capture the influence of the environment on the system, while managing the exponentially large Hilbert space required to describe the system. Tensor network algorithms offer an efficient way to approach this challenge. In this thesis, we develop and apply tensor network techniques to study the dynamics and steady states of various open quantum systems. The first part of the thesis focuses on the driven dissipative many body physics in coupled cavity arrays described by Born-Markov master equations. We extend transfer matrix product operator methods to Liouvillian dynamics, and utilize them to compute dynamical correlation functions and fluorescence spectrum of an infinite coupled cavity array in 1D. We also investigate thermalization, and observe the emergence of a quasi-thermal steady state with a negative effective temperature. In another study, we use infinite projected entangled pair state (iPEPS) methods to compute steady states of coupled cavity lattices in 2D. We find that a straightforward adaptation of iPEPS to Liouvillian dynamics is unstable, contradicting a recent publication in the field. The second part investigates more general systems involving strong couplings and structured environments that induce non-Markovian dynamics. We develop a powerful time-evolving matrix product operator (TEMPO) algorithm that builds on Feynman-Vernon influence functional formalism, and uses matrix product states (MPS) to represent the temporal non-Markovian correlations efficiently. We apply TEMPO to study the localization phase transition of the spin-boson model and the dynamics of two spatially separated two-level systems coupled to a common environment. Finally, we propose the Toblerone TEMPO algorithm, which extends TEMPO to many-body systems interacting with general bosonic environments.