Abstract We transform the system/reservoir coupling model into a one-dimensional semi-infinite discrete chain through unitary transformation to simulate the open quantum system numerically with the help of time evolving block decimation (TEBD) algorithm. We apply the method to study the dynamics of dissipative systems. We also generate the thermal state of a multimode bath using minimally entangled typical thermal state (METTS) algorithm, and investigate the impact of the thermal bath on an empty system. For both cases, we give an extensive analysis of the impact of the modeling and simulation parameters, and compare the numerics with the analytics.
Abstract We transform the system/reservoir coupling model into a one-dimensional semi-infinite discrete chain through unitary transformation to simulate the open quantum system numerically with the help of time evolving block decimation (TEBD) algorithm. We apply the method to study the dynamics of dissipative systems. We also generate the thermal state of a multimode bath using minimally entangled typical thermal state (METTS) algorithm, and investigate the impact of the thermal bath on an empty system. For both cases, we give an extensive analysis of the impact of the modeling and simulation parameters, and compare the numerics with the analytics.
Open quantum systems have drawn attention over decades due to its applicability in the foundation of theoretical physics, e.g. statistical mechanics, quantum mechanics and condensed matter physics. The dynamics of open quantum systems has been described as separate entities from their surrounding environment that consists of a very large number of modes, somehow coupled to the mode of the system. Even though the exact solution of the dynamical behavior of the system is impossible to calculate, we obtain a tentative solution using the crucial Markov approximation. The input-output formalism of the quantum Langevin equation (QLE) has been considered as a useful tool which provides a semi-classical description of the dynamics of the system, whereas the master equation provides a complete picture of the dynamics of the system expressed in terms of density matrix. While studying the dynamics of nonlinear system/environment coupling using QLE, we see, for a small value of external field, that the steady state system field does not change much from the steady state field obtained in the absence of nonlinear dissipation. However, in a case where a stronger external field is applied, we see that the deviation becomes substantial from the solution of linear system. We also see that the nonlinear coupling introduces significant difference in the cavity fluctuation spectrum. The description, therefore, provides a potential explanation of parametric effects in terms of nonlinear dissipation phenomena associated with the nonlinear coupling. Even though the theories developed in the context of open quantum systems have proven to be powerful tools, they do not provide a satisfactory platform to be implemented on non-linear Hamiltonians. We often approximate it by linearizing over nonlinear steady state field amplitude, and therefore, the interesting effects are often overlooked. The limitation of the analytics provokes us to simulate open quantum dynamics numerically. The numerical method consists of transformation of the environmental degrees of freedom to a one-dimensional many-body chain, and the computational technique includes numerical diagonalization and renormalization process. The time-adaptive density matrix renormalisation group (t-DMRG) is known as one of the most powerful techniques for the simulation of strongly-correlated many-body quantum systems. In this thesis, along with the theoretical modeling, we implement DMRG numerical scheme for the simulation of canonical S/B model by mapping it to one-dimensional harmonic chain with nearest neighbor interactions, and use the method to investigate the dynamics of the free dissipative system. The thermalization of open quantum systems is also studied by generating minimally entangled typical thermal states (METTS) through imaginary time evolution, and real-time evolving an empty system in the presence of the thermal bath. Further, we simulate coherently driven free dissipative Kerr nonlinear system numerically using Euler's method by solving Heisenberg equation of motion and t-DMRG algorithm, and demonstrate how the numerical results are analogous to classical bistability. By comparing with analytics, we see that the DMRG numerics is analogous to the quantum-mechanical exact solution obtained by mapping the equation of motion of the density matrix of the system to a Fokker-Plank equation. The comparison between two different numerical techniques shows that the semi-classical Euler's method determines the dynamics of the system field of one among two coherent branches, whereas DMRG numerics gives the superposition of both of them. Hence, DMRG-determined time dynamics undergoes generating non-classical states. Our approach of dealing with nonlinearity represents an important contribution in the developments of technique to study the dynamical and steady-state behavior of open quantum systems, which is a fundamental aspect of quantum physics.
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.
In this introductory review, we present and compare four algorithms for the numerical integration of the Lindblad equation for one-dimensional quantum lattice systems. All four methods are based on Matrix Product State representations and can be viewed as extensions of the Time-Evolving Block Decimation (TEBD) algorithm to open quantum systems. Two approaches directly integrate the vectorized Lindblad equation, one of them explicitly enforcing the positivity of the density matrix. The other two rely on stochastic unravelings of the Lindblad equation, namely the quantum trajectory and quantum state diffusion approaches. We discuss the principles, numerical implementation, accuracy, and computational efficiency of the different methods, and benchmark them against an exactly solvable free fermion model.
The infinite time-evolving block decimation algorithm [G. Vidal, Phys. Rev. Lett. 98, 070201 (2007)] allows to simulate unitary evolution and to compute the ground state of one-dimensional (1D) quantum lattice systems in the thermodynamic limit. Here we extend the algorithm to tackle a much broader class of problems, namely, the simulation of arbitrary one-dimensional evolution operators that can be expressed as a (translationally invariant) tensor network. Relatedly, we also address the problem of finding the dominant eigenvalue and eigenvector of a one-dimensional transfer matrix that can be expressed in the same way. New applications include the simulation, in the thermodynamic limit, of open (i.e., master equation) dynamics and thermal states in 1D quantum systems, as well as calculations with partition functions in two-dimensional (2D) classical systems, on which we elaborate. The present extension of the algorithm also plays a prominent role in the infinite projected entangled-pair states approach to infinite 2D quantum lattice systems.
Numerical methods for obtaining exact dynamics of non-Markovian open quantum systems are mostly limited to either small systems or to short-time evolution only. Here, we propose a new algorithm for computing process tensors--matrix product operator (MPO) representations that capture the environment influence--which achieves greatly enhanced computational scalings with system size, while maintaining linear scaling with simulation length. We build on recent developments in the field which allow for long-time evolutions through process tensors which have a time-translational invariance. These can be built for general Gaussian environments and generic coupling operators with the system using infinite time-evolving block decimation (iTEBD). We introduce a modified iTEBD algorithm using intermediate compression steps which bring down the computation time scaling with system size d from O(d8) to O(d4), as well as significantly lowering the required memory. To illustrate the power of this method, we apply it to the problem of dispersive qubit readout in circuit QED, which was previously out-of-reach numerically. The full treatment of the measurement resonator, which requires a large system space, combined with the long simulation times precipitated by the separation of timescales between the measurement drive and the environment dissipation, is now possible. The algorithm we introduce not only allows for capturing non-Markovian dynamics in large open quantum systems, but also further extends all the existing capabilities of process tensors, for example in quantum optimal control, or in computation of multi-time correlations or of steady states, to more complex systems with tens of levels.
We introduce a numerical approach to calculate the statistics of work done on 1D quantum lattice systems initially prepared in thermal equilibrium states. This approach is based on two tensor-network techniques: Time Evolving Block Decimation (TEBD) and Minimally Entangled Typical Thermal States (METTS). The former is an efficient algorithm used to simulate the dynamics of 1D quantum lattice systems, while the latter a finite-temperature algorithm for generating a set of typical states representing the Gibbs canonical ensemble. As an illustrative example, we apply this approach to the Ising chain in mixed transverse and longitudinal fields. Under an arbitrary protocol, the moment generating function of the work can be obtained, from which the work moments are numerically calculated and the quantum Jarzynski equality can be tested. Moreover, the numerical approach is also adapted to test the universal quantum work relation involving a functional of an arbitrary observable.