Complex quantum systems, such as a quantum computer, will always be coupled in some way to the environment. This can cause what's called decoherence, a destructive process by which information is lost from the system into the environment. In this bachelor thesis paper, we discuss decoherence-free subspaces within networks of coupled quantum harmonic oscillators (or QHOs). We investigate where such noiseless subspaces (or NSs) occur most frequently in an ensemble of Erdos-Renyi networks, for which we do not yet consider the influence of the bath. We then proceed by adding the bath into the equation, using some of the theory of open quantum systems. Specifically, we derive the Lindblad master equation and show its form for the case of our networks. Consequently, we simulate the behavior of the moments of the position operators for the graphs with 3 nodes, both by means of the full Lindblad equation, and by first tracing out those moments to obtain their differential equations. We compare those two results to each other, and also look back to the situation before adding the bath to see if the NSs are still present. From the results of the simulations, we can conclude several things. Firstly, we see that for ensembles with probability of connection p very close to either 0 or 1, both to number of noiseless modes and the probability of finding at least one is largest. This is credited to their relatively high degrees of symmetry. Secondly, in the results of the density matrix and moment simulations, we see that, indeed, the noiseless modes are preserved when considering the influence of the bath. Furthermore, we can conclude that simulation of the density matrix for the case of coupled QHOs in a network is in many cases not stable; the cutoff at a finite level s needed to simulate an otherwise infinite- dimensional operator leads to non-positivity of the density matrix. Therefore, it is best to simulate the moments from their respective differential equations, as opposed to the full Lindblad master equation. Finally, the differential equations for the moments and their solutions show that there are indeed noiseless clusters for eigenmodes perpendicular to the center of mass, as predicted.
Complex quantum systems, such as a quantum computer, will always be coupled in some way to the environment. This can cause what's called decoherence, a destructive process by which information is lost from the system into the environment. In this bachelor thesis paper, we discuss decoherence-free subspaces within networks of coupled quantum harmonic oscillators (or QHOs). We investigate where such noiseless subspaces (or NSs) occur most frequently in an ensemble of Erdos-Renyi networks, for which we do not yet consider the influence of the bath. We then proceed by adding the bath into the equation, using some of the theory of open quantum systems. Specifically, we derive the Lindblad master equation and show its form for the case of our networks. Consequently, we simulate the behavior of the moments of the position operators for the graphs with 3 nodes, both by means of the full Lindblad equation, and by first tracing out those moments to obtain their differential equations. We compare those two results to each other, and also look back to the situation before adding the bath to see if the NSs are still present. From the results of the simulations, we can conclude several things. Firstly, we see that for ensembles with probability of connection p very close to either 0 or 1, both to number of noiseless modes and the probability of finding at least one is largest. This is credited to their relatively high degrees of symmetry. Secondly, in the results of the density matrix and moment simulations, we see that, indeed, the noiseless modes are preserved when considering the influence of the bath. Furthermore, we can conclude that simulation of the density matrix for the case of coupled QHOs in a network is in many cases not stable; the cutoff at a finite level s needed to simulate an otherwise infinite- dimensional operator leads to non-positivity of the density matrix. Therefore, it is best to simulate the moments from their respective differential equations, as opposed to the full Lindblad master equation. Finally, the differential equations for the moments and their solutions show that there are indeed noiseless clusters for eigenmodes perpendicular to the center of mass, as predicted.
In this thesis, networks of coupled quantum harmonic oscillators are studied. The dynamics of these networks are determined by single-frequency vibrations of the entire network called normal modes. We study the behavior of the nor- mal modes when the network is coupled to a thermodynamical heat bath by looking at the Lindblad Master Equation of the system. From this equation, we determine the rate at which the normal modes decay. Certain normal modes decay very slowly, and some do not decay at all. These normal modes are called quasi-noiseless and noiseless clusters respectively. We determine what happens to the noiseless clusters when the network pa- rameters are very slightly perturbed. We have found that two distinct types of noiseless clusters can be identified. The first type disappears with even the slightest perturbation, making it useless in practice. The second type instead be- comes quasi-noiseless, making it a viable candidate for applications. We show how to determine the degree to which these noiseless clusters become quasi- noiseless by looking at the other normal modes of the network. We also explain how a network of oscillators, including an optional heat bath, can be simulated with an optical setup as described in [3]. We suggest this setup can be used to verify our findings.
Known as decoherence, the unavoidable interaction of a quantum system with its surrounding environment is usually considered to be detrimental for quantum information processing. In this thesis the coherent, open loop control of such open systems is studied. Concepts from quantum control theory and the theory of open quantum system are adopted in order to fight decoherence and implement quantum gates in a noiseless manner. In particular, Lie algebraic methods and numerical optimization tools are used to investigate the control properties of a single spin interacting with a spin environment. We show that, independent of the size of the environment, every unitary transformation can be implemented on the system spin through a single control field. We proceed by investigating dynamical decoupling, a method to suppress the interactions with the environment, for finite- and for in- finite dimensional systems. We prove that every finite dimensional system can be protected from decoherence, even if the environment is infinite dimensional, whereas for noise described by a Lindblad master equation dynamical decoupling will never succeed. This will lead to a new method to distinguish decoherence from intrinsic noise terms. We further prove that not every infinite dimensional system can be protected from decoherence through dynamical decoupling. Afterwards we investigate dynamical decoupling of systems that are described by quadratic Hamiltonians, showing that such interactions can always be suppressed with two simple operations. In the last part we investigate the coherent control of a Lindblad master equation. We show that a strong noise process exhibiting a decoherence free subspace can substantially increase the number of unitary operations that can be implemented, allowing us to fully control parts of the system. Afterwards we develop a scheme to make Hamiltonians and Lindbladians commutative by adding an auxiliary system. The old, possibly non-commutative dynamics, is recovered through a non-selective measurement.
This thesis explores the dynamics of linearly coupled quantum and classical oscillators interacting with environments. In the quantum regime, such dynamics is described in terms of the Liouvillian superoperator, which leads to non-unitary time evolution of the system, specifically of the density matrix. In particular, when the correlation time of the environment is significantly shorter than the characteristic timescale of the system, the dynamics are classified as Markovian. Therefore, the equation of motion for the density matrix that fulfils this property is known as the Lindblad master equation. Systems of coupled cavities and waveguides in the presence of gain and loss can be described by this equation and, more importantly, by non-Hermitian Hamiltonians. In particular, when two coupled cavities exhibit balance between gain and loss, the system is PT -symmetric. However, for certain values in the parameter space, this symmetry is broken and the non-Hermitian Hamiltonian is no longer diagonalizable. Such values are denoted as exceptional points. In the case of large arrays of coupled quantum oscillators, the dynamics description becomes more challenging since the dimensions of the density matrix grow exponentially. To address this, we study the dynamics of these arrays via the quantum state diffusion approach. In such approach, the Lindblad master equation is unraveled by a stochastic Schrödinger equation. Then, we decouple this equation by diagonalizing the non-Hermitian Hamiltonian that describes the system. Such diagonalization leads to a transformation of a system of coupled oscillators into a decoupled one, offering a great reduction in the basis size. For systems of classical oscillators with dissipative nearest-neighbor coupling, the dynamics leads to synchronized motion. We study how an array of such oscillators reaches the synchronized state. This behavior is found when the system is dissipation free, which implies that all the eigenvectors of the dynamical matrix are damped, except one. To measure the degree of synchronization, we introduce the inverse of participation ratio, whose maximum value is associated with a delocalized state and, therefore, fully synchronized motion. For larger arrays of oscillators, we examine the case where the frequency fluctuations of each oscillator are obtained from a Gaussian distribution with zero mean and a given standard deviation. Then, we derive a scaling law for the largest value of the mentioned standard deviation that still allows the oscillators fully synchronize their motion.:1 Introduction 2 Open quantum systems 3 Dynamics of two coupled cavities in the presence of losses 4 Unraveling of Lindblad equation for system of oscillators 5 Synchronization of dissipatively coupled harmonic oscillators 6 Conclusions and outlook A Derivation of equations of motion for the amplitudes B Quantum state diffusion unraveling for N decoupled oscillators C Model for disspative coupling D Perturbation theory for disorder in the frequencies References
This thesis investigates two thematic lines of research, both underpinned by non-Markovian system-reservoir interactions in quantum optics. The overarching focus is on modelling the open system dynamics in a non-perturbative fashion, broadly on - though not restricted to - instances when the environment is structured. A theory is developed by means of enlarging the open system over environmental degrees of freedom to include memory effects in its dynamics. This is achieved using an established technique that involves mapping a bosonic environment onto a 1D chain of harmonic oscillators. Within this setting, we apply a Heisenberg equation-of-motion approach to derive an exact set coupled differential equations for the open system and a single auxiliary oscillator of the chain. The combined equations are shown to have their interpretation rooted in a quantum Markov stochastic process. Including the auxiliary chain oscillator as part of the original system then enables us to obtain an exact master equation for the enlarged system, avoiding any need for the Born-Markov approximations. Our method is valid for a dissipative two-state system, with cases of multiple excitations and added driving discussed. Separately, we apply the framework of quantum Darwinism to an atom-cavity system, and, subsequently, to a more general multiple-environment model. In both cases, the time-dependent spread of correlations between the open system and fractions of the environment is analysed during the course of the decoherence process. The degree to which information is redundant across different fractions is checked to infer the emergence of classicality. In the second case, we go further and present a decomposition of information in terms of its quantum and classical correlations. A quantitative measure of redundancy is also studied with regard to its ability to witness non-Markovian behaviour. Besides fundamental interest, our results have application to quantum information processing and quantum technologies, keeping in mind the potential beneficial use of non-Markovian effects in reservoir engineering.
The Lindblad theory of open quantum systems has been successfully applied in various fields ranging from quantum optics, condensed matter physics, and quantum thermodynamics to quantum chemistry. Nevertheless, there are situations where we can compare its predictions to those of exact methods or numerically exact results and find discrepancies between them. One example is the case where a single harmonic oscillator is coupled to a thermal bath of independent harmonic oscillators. In this case, for certain parameters, the exact master equation describes equilibration of the system to a thermal state with a temperature different from the one predicted by the Lindblad master equation. The Lindblad theory, based on a weak coupling between the system and environment as well as the Born-Markov approximation, predicts that the system will evolve to a thermal state whose temperature is equal to the initial temperature of the bath. This result is a consequence of the Born approximation, which supposes that throughout the evolution, the bath only fluctuates around its initial equilibrium state. In this master thesis we study the validity of the Born approximation as well as its consistency with other approximations. By starting from a Hamiltonian for the total system we show that, if the the Born approximation is not invoked, the dynamics of the system is determined by a hierarchy of equations for matrices which can be used to reconstruct the reduced density matrix of the system. This hierarchy approach is fundamentally different from the standard approach, which describes the system dynamics with a weak-coupling master equation, because the hierarchy does not restrict the amount of entanglement possible between the system and the environment. Furthermore, we show how invoking the Born approximation reduces to the usual result and how a generalised ansatz, which we refer to as a generalised Born approximation, reduces the hierarchy to a master equation where the bath temperature is an additional degree of freedom. Finally, we analyse the conservation of the energy of the total system and its relationship to the Markov and secular approximations, and discuss whether imposing it as a condition can fix the time-dependent temperature.
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.