Open quantum systems and their dynamics are usually studied in terms of reduced density matrices. This approach allows a nonsymmetric description, which is useful when one of the two subsystems is to be considered an environment, at the expense of a loss of information, as tracing out the environmental degrees of freedom is an irreversible process. This has a series of consequences which can be severe. In this work we present an alternative description, which is still nonsymmetric but yet exact: It is based on a parametric representation of composite systems, as obtained by introducing environmental coherent states, such that the principal system get to be described by a set of pure states parametrically dependent on environmental variables. The representation allows one to relate properties which typically arise in studying systems with parametrically dependent Hamiltonians, such as the emergence of geometrical phases, with features which specifically characterize open quantum systems, such as decoherence and entanglement generation.
Open quantum systems and their dynamics are usually studied in terms of reduced density matrices. This approach allows a nonsymmetric description, which is useful when one of the two subsystems is to be considered an environment, at the expense of a loss of information, as tracing out the environmental degrees of freedom is an irreversible process. This has a series of consequences which can be severe. In this work we present an alternative description, which is still nonsymmetric but yet exact: It is based on a parametric representation of composite systems, as obtained by introducing environmental coherent states, such that the principal system get to be described by a set of pure states parametrically dependent on environmental variables. The representation allows one to relate properties which typically arise in studying systems with parametrically dependent Hamiltonians, such as the emergence of geometrical phases, with features which specifically characterize open quantum systems, such as decoherence and entanglement generation.
The behavior of most physical systems is affected by their natural surroundings. A quantum system with an environment is referred to as open, and its study varies according to the classical or quantum description adopted for the environment. We propose an approach to open quantum systems that allows us to follow the cross-over from quantum to classical environments; to achieve this, we devise an exact parametric representation of the principal system, based on generalized coherent states for the environment. The method is applied to the s = 1/2 Heisenberg star with frustration, where the quantum character of the environment varies with the couplings entering the Hamiltonian H. We find that when the star is in an eigenstate of H, the central spin behaves as if it were in an effective magnetic field, pointing in the direction set by the environmental coherent-state angle variables (θ, ϕ), and broadened according to their quantum probability distribution. Such distribution is independent of ϕ, whereas as a function of θ is seen to get narrower as the quantum character of the environment is reduced, collapsing into a Dirac-δ function in the classical limit. In such limit, because ϕ is left undetermined, the Von Neumann entropy of the central spin remains finite; in fact, it is equal to the entanglement of the original fully quantum model, a result that establishes a relation between this latter quantity and the Berry phase characterizing the dynamics of the central spin in the effective magnetic field.
Based on the superposition principle, any two states of a quantum system may be coherently superposed to yield a novel state. Such a simple construction is at the heart of genuinely quantum phenomena such as interference of massive particles or quantum entanglement. Yet, these superpositions are susceptible to environmental influences, eventually leading to a complete disappearance of the system's quantum character. In principle, two distinct mechanisms responsible for this process of decoherence may be identified. In a classical decoherence setting, on the one hand, stochastic fluctuations of classical, ambient fields are the relevant source. This approach leads to a formulation in terms of stochastic Hamiltonians; the dynamics is unitary, yet stochastic. In a quantum decoherence scenario, on the other hand, the system is described in the language of open quantum systems. Here, the environmental degrees of freedom are to be treated quantum mechanically, too. The loss of coherence is then a direct consequence of growing correlations between system and environment. The purpose of the present thesis is to clarify the distinction between classical and quantum decoherence. It is known that there exist decoherence processes that are not reconcilable with the classical approach. We deem it desirable to have a simple, feasible model at hand of which it is known that it cannot be understood in terms of fluctuating fields. Indeed, we find such an example of true quantum decoherence. The calculation of the norm distance to the convex set of classical dynamics allows for a quantitative assessment of the results. In order to incorporate genuine irreversibility, we extend the original toy model by an additional bath. Here, the fragility of the true quantum nature of the dynamics under increasing coupling strength is evident. The geometric character of our findings offers remarkable insights into the geometry of the set of non-classical decoherence maps. We give a very intuitive geometrical measure---a volume---for the quantumness of dynamics. This enables us to identify the decoherence process of maximum quantumness, that is, having maximal distance to the convex set of dynamics consistent with the stochastic, classical approach. In addition, we observe a distinct correlation between the decoherence potential of a given dynamics and its achievable quantumness. In a last step, we study the notion of quantum decoherence in the context of a bipartite system which couples locally to the subsystems' respective environments. A simple argument shows that in the case of a separable environment the resulting dynamics is of classical nature. Based on a realistic experiment, we analyze the impact of entanglement between the local environments on the nature of the dynamics. Interestingly, despite the variety of entangled environmental states scrutinized, no single instance of true quantum decoherence is encountered. In part, the identification of the classical nature relies on numerical schemes. However, for a large class of dynamics, we are able to exclude analytically the true quantum nature.
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.
A quantum system is always in interaction with its environment. This interaction inevitably alters the state of the quantum system, which causes loss of information encoded in the system. The system under consideration is not a closed system any more when interaction with the outside world is in action. We formulate the theory of an open quantum system in this chapter by regarding the combined system of the quantum system and its environment as a closed system and subsequently trace out the environmental degrees of freedom. Let ρS and ρE be the density matrices of the system and the environment, respectively. Even when the initial state is an uncorrelated state ρS ⊗ ρE , the system-environment interaction entangles the total system so that the total state develops to an inseparable entangled state in general. Decoherence is a process in which environment causes various changes in the quantum system, which manifests itself as undesirable noise.
By exploiting the peculiarities of a recently introduced formalism for describing open quantum systems (the parametric representation with environmental coherent states) we derive an equation of motion for the reduced density operator of an open quantum system that has the same structure of the celebrated Gorini–Kossakowski–Sudarshan–Lindblad equation, but holds regardless of Markovianity being assumed. The operators in our result have explicit expressions in terms of the Hamiltonian describing the interactions with the environment, and can be computed once a specific model is considered. We find that, instead of a single set of Lindblad operators, in the general (non-Markovian) case there one set of Lindblad-like operators for each and every point of a symplectic manifold associated to the environment. This intricacy disappears under some assumptions (which are related to Markovianity and the classical limit of the environment), under which it is possible to recover the usual master-equation formalism. Finally, we find such Lindblad-like operators for two different models of a qubit in a bosonic environment, and show that in the classical limit of the environment their renown master equations are recovered.
Open quantum systems describe quantum systems that interact unavoidably with their environments. These notes introduce the physical origins of decoherence and dissipation in realistic settings, developing the theoretical framework of reduced dynamics and quantum master equations within the connection to thermodynamics in the quantum regime. Additionally, a set of relevant exercises are proposed to reinforce the acquisition of knowledge. The text equips readers with the tools to model and analyze open quantum dynamics.