This chapter gives a self-contained review of the how standard open quantum system Hamiltonians can be mapped analytically onto representations in which the environments appear as one dimensional harmonic chains with nearest neighbour interactions. This mapping, carried out rigorously using orthogonal polynomial theory, then allows the full evolution of the system and environment to be simulated using time-adaptive density matrix renormalisation group methods. With the combination of these two techniques, numerically-exact results can be obtained for dissipative quantum systems in the presence of arbitrarily complex environmental spectral functions, and the correlations and processes in the environment which drive the effectively irreversible dynamics of the reduced state of the quantum system can be explored in real time. The chain representation also reveals a number of universal features of harmonic environments characterised by a spectral density which are discussed here.
This chapter gives a self-contained review of the how standard open quantum system Hamiltonians can be mapped analytically onto representations in which the environments appear as one dimensional harmonic chains with nearest neighbour interactions. This mapping, carried out rigorously using orthogonal polynomial theory, then allows the full evolution of the system and environment to be simulated using time-adaptive density matrix renormalisation group methods. With the combination of these two techniques, numerically-exact results can be obtained for dissipative quantum systems in the presence of arbitrarily complex environmental spectral functions, and the correlations and processes in the environment which drive the effectively irreversible dynamics of the reduced state of the quantum system can be explored in real time. The chain representation also reveals a number of universal features of harmonic environments characterised by a spectral density which are discussed here.
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.
The chain mapping of structured environments is a most powerful tool for the simulation of open quantum system dynamics. Once the environmental bosonic or fermionic degrees of freedom are unitarily rearranged into a one dimensional structure, the full power of Density Matrix Renormalization Group (DMRG) can be exploited. Beside resulting in efficient and numerically exact simulations of open quantum systems dynamics, chain mapping provides an unique perspective on the environment: the interaction between the system and the environment creates perturbations that travel along the one dimensional environment at a finite speed, thus providing a natural notion of light-, or causal-, cone. In this work we investigate the transport of excitations in a chain-mapped bosonic environment. In particular, we explore the relation between the environmental spectral density shape, parameters and temperature, and the dynamics of excitations along the corresponding linear chains of quantum harmonic oscillators. Our analysis unveils fundamental features of the environment evolution, such as localization, percolation and the onset of stationary currents.
This thesis is concerned with the study of quantum systems coupled linearly to a continuous bath of oscillators examples of which are the spin-boson model and the Nelson model. The main theme throughout is to develop a better understanding of the properties of the bath of oscillators such that more efficient representations of it can be made to facilitate the understanding of the system dynamics. The main difficulty in simulating the system dynamics is that the bath of oscillators composes an infinite number of degrees of freedom. In this thesis, we investigate the mathematical properties of an approach in which the bath modes are written as a semi-infinite chain of nearest neighbour interacting harmonic oscillators such that the efficient time dependent density-matrix renormalisation group (t-DMRG) methods can be applied for simulation. In the first section, we show how there are many different ways to represent the bath as semiinfinite chains and prove that seemingly unrelated methods can all be achieved using the same mathematical formalism. We show that in an iterative process the bath can be transformed into a chain of oscillators with nearest neighbour interactions. This is achieved using the formalism of orthogonal polynomials. This allows one to define a sequence of residual spectral densities at each site along the chain. We show that this sequence of residual spectral densities is provided by the so-called ”sequence of secondary measures”. We derive a systematic procedure to obtain the spectral density of the residual bath in each step. We find that these residual spectral densities are related to an old abstract problem in mathematics known as the ”secondary measures”. We solve this problem from the field of orthogonal polynomials to give an explicit expression for the residual spectral densities and go on to prove that these functions converge under very general conditions. That is, the asymptotic part of the chain is universal, translation invariant with universal spectral density. These results suggest efficient methods for handling the numerical treatment of the residual bath. In the second section, we take a different approach. Rather than studying the properties of residual baths, we look at how system observables are affected by truncating the semi-infinite chain of harmonic oscillators to a finite length chain. By developing locality bounds for the dynamics, we derive an upper bound to the error introduced by such a truncation and show that for all finite times it can be made arbitrarily small by including a sufficient number of harmonic oscillators 3 before truncating. Furthermore, it is shown that the speed at which the system communicates with different harmonic oscillators in the chain is proportional to the maximum frequency of the environment but that this speed also depends on the particular version of the chain. These bounds are given for when the dynamics are calculated in the Interaction picture and in the Schrodinger picture.
The chain mapping of structured environments is a most powerful tool for the simulation of open quantum system dynamics. Once the environmental bosonic or fermionic degrees of freedom are unitarily rearranged into a one dimensional structure, the full power of Density Matrix Renormalization Group (DMRG) can be exploited. Beside resulting in efficient and numerically exact simulations of open quantum systems dynamics, chain mapping provides an unique perspective on the environment: the interaction between the system and the environment creates perturbations that travel along the one dimensional environment at a finite speed, thus providing a natural notion of light-, or causal-, cone. In this work we investigate the transport of excitations in a chain-mapped bosonic environment. In particular, we explore the relation between the environmental spectral density shape, parameters and temperature, and the dynamics of excitations along the corresponding linear chains of quantum harmonic oscillators. Our analysis unveils fundamental features of the environment evolution, such as localization, percolation and the onset of stationary currents
By using the properties of orthogonal polynomials, we present an exact unitary transformation that maps the Hamiltonian of a quantum system coupled linearly to a continuum of bosonic or fermionic modes to a Hamiltonian that describes a one-dimensional chain with only nearest-neighbor interactions. This analytical transformation predicts a simple set of relations between the parameters of the chain and the recurrence coefficients of the orthogonal polynomials used in the transformation and allows the chain parameters to be computed using numerically stable algorithms that have been developed to compute recurrence coefficients. We then prove some general properties of this chain system for a wide range of spectral functions and give examples drawn from physical systems where exact analytic expressions for the chain properties can be obtained. Crucially, the short-range interactions of the effective chain system permit these open-quantum systems to be efficiently simulated by the density matrix renormalization group methods.
Classical non-perturbative simulations of open quantum systems' dynamics face several scalability problems, namely, exponential scaling of the computational effort as a function of either the time length of the simulation or the size of the open system. In this work, we propose the use of the Time Evolving Density operator with Orthogonal Polynomials Algorithm (TEDOPA) on a quantum computer, which we term as Quantum TEDOPA (Q-TEDOPA), to simulate non-perturbative dynamics of open quantum systems linearly coupled to a bosonic environment (continuous phonon bath). By performing a change of basis of the Hamiltonian, the TEDOPA yields a chain of harmonic oscillators with only local nearest-neighbour interactions, making this algorithm suitable for implementation on quantum devices with limited qubit connectivity such as superconducting quantum processors. We analyse in detail the implementation of the TEDOPA on a quantum device and show that exponential scalings of computational resources can potentially be avoided for time-evolution simulations of the systems considered in this work. We applied the proposed method to the simulation of the exciton transport between two light-harvesting molecules in the regime of moderate coupling strength to a non-Markovian harmonic oscillator environment on an IBMQ device. Applications of the Q-TEDOPA span problems which can not be solved by perturbation techniques belonging to different areas, such as the dynamics of quantum biological systems and strongly correlated condensed matter systems.