The time-evolving matrix product operator (TEMPO) method is a powerful tool for simulating open system quantum dynamics. Typically, it is used in problems with diagonal system-bath coupling, where analytical expressions for discretized influence functional are available. In this work, we aim to address issues related to off-diagonal coupling by extending the TEMPO algorithm to accommodate arbitrary basis sets. The proposed approach is based on computing the derivative of the discretized path integral expression of a generalized influence functional when increasing one time step, which yields an equation of motion valid for non-diagonal basis set and arbitrary number of non-commuting baths. The generalized influence functional is then obtained by integrating the resulting differential equation. Applicability of the the new method is then tested by simulating one- and two- qubit systems coupled to both Z- and X-type baths.
The time-evolving matrix product operator (TEMPO) method is a powerful tool for simulating open system quantum dynamics. Typically, it is used in problems with diagonal system-bath coupling, where analytical expressions for discretized influence functional are available. In this work, we aim to address issues related to off-diagonal coupling by extending the TEMPO algorithm to accommodate arbitrary basis sets. The proposed approach is based on computing the derivative of the discretized path integral expression of a generalized influence functional when increasing one time step, which yields an equation of motion valid for non-diagonal basis set and arbitrary number of non-commuting baths. The generalized influence functional is then obtained by integrating the resulting differential equation. Applicability of the the new method is then tested by simulating one- and two- qubit systems coupled to both Z- and X-type baths.
Dynamics of open quantum systems with structured reservoirs can often be simulated efficiently with tensor network influence functionals. The standard variants of the time-evolving matrix product operator (TEMPO) method are applicable when the systems is coupled to Gaussian bosonic baths via hermitian coupling operators that mutually commute. In this work we introduce a generalization to cases where the system is coupled to a single reservoir through multiple non-commuting operators, representing the most general form of linear system-bath coupling. We construct a Gaussian influence functional that properly handles Trotter errors arising from a finite evolution time step, thus ensuring convergence for long evolution times. Based on this result, the uniform TEMPO scheme can be employed to obtain a matrix product operator form of the influence functional, enabling efficient simulations of the real-time dynamics of the open system. As a demonstration, we simulate the time evolution of driven two-level emitters coupled to a bosonic lattice at different lattice sites.
The time-evolving matrix product operator (TEMPO) method has proven to be a powerful method to study the long-time dynamics of bosonic impurity problems where a small system is linearly coupled to a noninteracting bosonic bath. However, current developments of TEMPO have mostly focused on the case of diagonal system-bath coupling, i.e., $\sum_k \Aop(V_k \bdop_k + \hc)$, with $\Aop$ a Hermitian operator of the system. Based on the process tensor framework, we extend TEMPO to the more general case of off-diagonal system-bath coupling in the form $\sum_k (V_k\Aop\bdop_k + \hc)$, where $\Aop$ could be non-Hermitian. As applications, we study the real-time dynamics of a spin that is coupled to a sub-ohmic bath via the Jaynes-Cummings-type system-bath coupling and compare it against the standard spin-boson model, where we show that the commonly used rotating-wave approximation could be very poor for this bath. We also study the imaginary-time evolution of a bosonic impurity with nonzero on-site interaction that is coupled to a sub-ohmic bath, to illustrate the flexibility of our method. Our method provides a unified framework to understand different variants of TEMPO, and is a promising building block for an impurity solver in the bosonic dynamical mean field theory for the normal phase with a scalar hybridization function.
The time-evolving matrix product operator (TEMPO) method efficiently simulates non-Markovian open quantum system dynamics by contracting a two-dimensional tensor network representing the time-discretized Feynman-Vernon influence functional. The standard form of this network has an unfavorable scaling with the size of the system. We show that, within the usual second-order time-step error, the dimensions of all internal network bonds can be reduced to (2m+1)(m+1), where m is the number of independent coupling channels. This dimension does not depend on the number of distinct eigenvalues of the coupling operators. The system size only enters in the final boundary layer of the network, making large system sizes accessible to TEMPO methods. This structure was discovered by Claude Opus 5.5 during guided algorithm development.
Multitime system correlation functions are relevant in various areas of physics and science, dealing with system-bath interaction including spectroscopy and quantum optics, where many of these schemes include an off-diagonal system bath interaction. Here we extend the enhanced time-evolving matrix product operator (eTEMPO) algorithm for quantum path integrals using tensor networks [Phys. Rev. Lett. 123, 240602 (2019)PRLTAO0031-900710.1103/PhysRevLett.123.240602 to open quantum systems with off-diagonal coupling beyond a single two level system. We exemplify the approach on a coupled cavity waveguide system with spatially separated quantum two-state emitters, though many other applications in material science are possible, including entangled photon propagation, photosynthesis spectroscopy, and on-chip quantum optics with realistic dissipation.
Developing numerical exact solvers for open quantum systems is a challenging task due to the non-perturbative and non-Markovian nature when coupling to structured environments. The Feynman-Vernon influence functional approach is a powerful analytical tool to study the dynamics of open quantum systems. Numerical treatments of the influence functional including the quasi-adiabatic propagator technique and the tensor-network-based time-evolving matrix product operator method have proven to be efficient in studying open quantum systems with bosonic environments. However, the numerical implementation of the fermionic path integral suffers from the Grassmann algebra involved. In this work, we present a detailed introduction to the Grassmann time-evolving matrix product operator method for fermionic open quantum systems. In particular, we introduce the concepts of Grassmann tensor, signed matrix product operator, and Grassmann matrix product state to handle the Grassmann path integral. Using the single-orbital Anderson impurity model as an example, we review the numerical benchmarks for structured fermionic environments for real-time nonequilibrium dynamics, real-time and imaginary-time equilibration dynamics, and its application as an impurity solver. These benchmarks show that our method is a robust and promising numerical approach to study strong coupling physics and non-Markovian dynamics. It can also serve as an alternative impurity solver to study strongly correlated quantum matter with dynamical mean-field theory.
Open quantum systems play an important role in developing quantum sciences, and therefore the study of corresponding numerical method is of great significance. For the open quantum systems, the quasi-adiabatic propagator path integral invented in 1990s is one of the few numerically exact methods. However, its computational complexity scales exponentially with system size and correlation length, and therefore its application is limited in practical calculation. In recent years, the study and application of tensor network have made rapid progress. Representing the path integral by tensor network makes the computational complexity increase polynomially, thus greatly improving the computational efficiency. Such a new method is called time-evolving matrix product operator. At the very beginning, the reduced density matrix is represented as a matrix product state. Then the time evolution of the system can be achieved by iteratively applying matrix product operators to the matrix product state. The iterative process is amenable to the standard matrix product states compression algorithm, which keeps the computational cost on a polynomial scale. The time-evolving matrix product operator is an efficient, numerically exact and fully non-Markovian method, which has a broad application prospect in the study of quantum open systems. For instance, it is already used in the study of the thermalization, heat statistic, heat transfer and optimal control of the quantum open systems, and conversely it can be also used to investigate the effect of the system on the environment. In addition, the TEMPO method is naturally related to the process tensor, and can be used to calculate the correlation function of the system efficiently. In this article we review this method and its applications. We give a brief introduction of the path integral formalism of Caldeira-Leggett model. According to the path integral formalism, we demonstrate the usage of quasi-adiabatic propagator path integral method. we give the basic idea of matrix product states, and we show how to recast quasi-adiabatic propagator path integral method into time-evolving matrix product operators method by employing the concept of matrix product states and matrix product operators, and give a review of its applications. In addition, we use the calculation results of physical quantities, correlation functions and heat currents in the spin-boson model to illustrate the applications of the time-evolving matrix product operator method.