An understanding of strongly coupled open quantum systems is crucial for the realisation of many of today’s emerging quantum technologies. However, actually simulating a quantum system that strongly interacts, or couples, with its environment is no easy task. This thesis tackles a number of topics in the field of strongly coupled open quantum systems. Given the computational challenges of solving the full open system dynamics, we first discuss how and why a quasiclassical stochastic approach can be used to simulate the dynamics of a quantum harmonic oscillator that is strongly coupled to its environment. We build upon the established analytical agreement by implementing the quasiclassical stochastic case numerically in the SpiDy.jl Julia package. Motivated by the applications to magnetic modelling, we then solve the full open quantum systems model for a single spin coupled arbitrarily strongly to multiple heat baths. In doing so, we find that the presence of multiple baths can actually increase the system-environment entanglement in the steady state. We also compare this to a fully classical approach and find that, for isotropic environmental coupling, the environment-induced effects in the equilibrium state only appear quantum mechanically. Finally, we couple these two topics together by considering the quasiclassical stochastic simulation of a single quantum spin, helping to understand the regimes of validity beyond the linear harmonic oscillator. This is with the larger aim to eventually be able to encode quantum effects into more realistic systems, such as those that describe magnetic materials.
An understanding of strongly coupled open quantum systems is crucial for the realisation of many of today’s emerging quantum technologies. However, actually simulating a quantum system that strongly interacts, or couples, with its environment is no easy task. This thesis tackles a number of topics in the field of strongly coupled open quantum systems. Given the computational challenges of solving the full open system dynamics, we first discuss how and why a quasiclassical stochastic approach can be used to simulate the dynamics of a quantum harmonic oscillator that is strongly coupled to its environment. We build upon the established analytical agreement by implementing the quasiclassical stochastic case numerically in the SpiDy.jl Julia package. Motivated by the applications to magnetic modelling, we then solve the full open quantum systems model for a single spin coupled arbitrarily strongly to multiple heat baths. In doing so, we find that the presence of multiple baths can actually increase the system-environment entanglement in the steady state. We also compare this to a fully classical approach and find that, for isotropic environmental coupling, the environment-induced effects in the equilibrium state only appear quantum mechanically. Finally, we couple these two topics together by considering the quasiclassical stochastic simulation of a single quantum spin, helping to understand the regimes of validity beyond the linear harmonic oscillator. This is with the larger aim to eventually be able to encode quantum effects into more realistic systems, such as those that describe magnetic materials.
When a quantum system interacts with many other quantum mechanical objects, the behaviour of the system is strongly affected; this is referred to as an open quantum system (OQS). Since the inception of quantum theory the development of OQSs has been synonymous with realistic descriptions of quantum mechanical models. With recent activity in the advancement of quantum technologies, there has been vested interest in manipulating OQSs. Therefore understanding and controlling environmental effects, by structuring environments, has become an important field. The method of choice for tackling OQSs is the master equation approach, which requires approximations and doesn’t allow direct assessment of the environment. This thesis tackles the issues of OQSs with an unorthodox method; we employ a series of coupled quantum harmonic oscillators to simulate an OQS. This permits the use of the covariance matrix technique which allows us to avoid approximations and analyse the environment modes. We investigate the Markov approximation and Rotating-Wave approximation (RWA), which are commonly used in the field. By considering four OQS models, we study an entanglement-based non-Markovian behaviour (NMB) quantifier (ENMBQ). The relevance of detuning, coupling strength and bath structures in determining the amount of NMB is noted. A brief study of the factors that affect a fidelity-based NMB quantifier is also conducted. We also analyse the effect on the ENMBQ if the terms excluded by the RWA are included in the models. Finally, an examination of the applicability of the RWA in the presence of strong coupling is undertaken in a three oscillator model. The fidelity-based analysis utilised could allow one to ascertain when and if the RWA can be applied to a model of interest, including OQSs. The knowledge within, and the methodology used throughout this thesis, could arm researchers with insights to control the flow of quantum information in their systems.
Understanding the dynamics of dissipative quantum systems, particularly beyond the weak coupling approximation, is central to various quantum applications. While numerically exact methods provide accurate solutions, they often lack the analytical insight provided by theoretical approaches. In this study, we employ the recently developed method dubbed the effective Hamiltonian theory to understand the dynamics of system-bath configurations without resorting to a perturbative description of the system-bath coupling energy. Through a combination of mapping steps and truncation, the effective Hamiltonian theory offers both analytical insights into signatures of strong couplings in open quantum systems and a straightforward path for numerical simulations. To validate the accuracy of the method, we apply it to two canonical models: a single spin immersed in a bosonic bath and two noninteracting spins in a common bath. In both cases, we study the transient regime and the steady state limit at nonzero temperature and spanning system-bath interactions from the weak to the strong regime. By comparing the results of the effective Hamiltonian theory with numerically exact simulations, we show that although the former overlooks non-Markovian features in the transient equilibration dynamics, it correctly captures non-perturbative bath-generated couplings between otherwise non-interacting spins, as observed in their synchronization dynamics and correlations. Altogether, the effective Hamiltonian theory offers a powerful approach for understanding strong coupling dynamics and thermodynamics, capturing the signatures of such interactions in both relaxation dynamics and in the steady state limit.
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.
Generic open quantum systems are notoriously difficult to simulate unless one looks at specific regimes. In contrast, classical dissipative systems can often be effectively described by stochastic processes, which are generally less computationally expensive. Here, we use the paradigmatic case of a dissipative quantum oscillator to give a pedagogic introduction into the modelling of open quantum systems using quasiclassical methods, i.e. classical stochastic methods that use a ‘quantum’ noise spectrum to capture the influence of the environment on the system. Such quasiclassical methods have the potential to offer insights into the impact of the quantum nature of the environment on the dynamics of the system of interest whilst still being computationally tractable.
Generic open quantum systems are notoriously difficult to simulate unless one looks at specific regimes. In contrast, classical dissipative systems can often be effectively described by stochastic processes, which are generally less computationally expensive. Here, we use the paradigmatic case of a dissipative quantum oscillator to give a pedagogic introduction to the modeling of open quantum systems using quasiclassical methods, i.e., classical stochastic methods that use a "quantum" noise spectrum to capture the influence of the environment on the system. Such quasiclassical methods have the potential to offer insights into the impact of the quantum nature of the environment on the dynamics of the system of interest while still being computationally tractable.
All systems are open to an essentially uncontrollable environment that acts as a source of decoherence and dissipation. In some cases the environment's only effect is to add a weak relaxation mechanism and thus can be ignored for short timescales. In others, however, the presence of the environment can fundamentally alter the behavior of the system. Such is the case in mesoscopic superconductors where the environment can stabilize superconductivity and in spin-boson systems where the environment induces a localization transition. Likewise, in technological applications we are often interested in systems operating far from equilibrium. Here the environment might act as a particle reservoir or strong driving force. In all these examples, we need accurate methods to describe the influence of the environment on the system and to solve for the resulting dynamics or equilibrium states. In this thesis, we develop computational and conceptual approaches to efficiently simulate quantum systems in contact with an environment. Our starting point is the use of numerical renormalization techniques. Thus, we restrict our attention to one-dimensional lattices or small quantum systems coupled to an environment. We have developed several complementary algorithms: a superoperator renormalization algorithm for simulating real-time Markovian dynamics and for calculating states in thermal equilibrium; a blocking algorithm for simulating integro-differential equations with long-time memory; and a tensor network algorithm for branched lattices, which can be used to simulate strongly dissipative systems. Further, we provide support for an idea that to generically and accurately simulate the real-time dynamics of strongly dissipative systems, one has to include all or part of the environment within the simulation. In addition, we discuss applications and open questions.