This thesis examines dissipative forces in physical systems in both microscopic and quantum level. The system-environment interaction is modeled by assuming that the environment is composed of simple harmonic oscillators and that there is a linear interaction between these oscillators and the system. The classical case is solved by determining the retarded Green’s function, which yields the equation of motion of the system. For quantum dissipation, the Lindblad master equation is used to describe the time evolution of the reduced density matrix of the system, and the propagator is investigated to identify important parameters, including the term of Newton's friction force.
This thesis examines dissipative forces in physical systems in both microscopic and quantum level. The system-environment interaction is modeled by assuming that the environment is composed of simple harmonic oscillators and that there is a linear interaction between these oscillators and the system. The classical case is solved by determining the retarded Green’s function, which yields the equation of motion of the system. For quantum dissipation, the Lindblad master equation is used to describe the time evolution of the reduced density matrix of the system, and the propagator is investigated to identify important parameters, including the term of Newton's friction force.
Previous years researchers began to simulate open quantum system, taking into account the interaction between system and the environment. One approach to deal with this problem is to use the density matrix within the Liouville-von-Neumann formalism or the Markovian variant the Lindblad equations. Another way is to use a stochastic approach where a random force is added to the system. The benefit of the stochastic approach is to solve the dynamics of the system with less time and memory than the density matrix approaches. In this project we want to develop a stochastic approach that can deal with the stochastic wave functions approach. We did this on a 2-level system and found that it works well when comparing to a density matrix approach. Next, we tested a quantum particle connect to a bath of harmonic oscillators using the stochastic approach. We found that a friction term is necessary and applied it. Like in the classical Langevin equations the friction constant and the random force fluctuations are related by the fluctuation-dissipation constant. We showed that with friction the dynamics decays to an ensemble with energy of Egs+kBT. However, we also found here are problems. The system seems to absorb energy indefinitely if the temperature is higher than the zero point energy or if the system is a Morse oscillator. Thus more research is required to make this method work.
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
We develop an exact analytical framework for studying the nonequilibrium dynamics of interacting quantum harmonic oscillators coupled to a modelled thermal reservoir and driven by external classical fields. The total Hamiltonian is diagonalized via successive Bogoliubov transformations, enabling a nonperturbative treatment of dissipation and driving. For two coupled oscillators with an external source applied to the first, we derive explicit energy expressions and analyze their dependence on coupling and bath parameters. The phase-space structure is characterized through Husimi Q-functions for initial separable coherent and number states. We obtain the reduced density matrix elements in the number-state basis and demonstrate that, in the absence of driving and at zero temperature, the reduced density matrix of the main system satisfies the Lindblad master equation. A generating function for the Husimi function is introduced to facilitate computation of higher-order correlations. The analysis is then generalized to n interacting oscillators, with analytical energy expressions provided and the case n=3 examined in detail. Our results establish a versatile and exact framework for driven-dissipative quantum systems, with potential applications in quantum thermodynamics, open quantum systems, and many-body physics.
Within the framework of the quantum-statistical approach, utilizing both non-Hermitian Hamiltonian and Lindblad’s jump operators, one can derive various generalizations of the von Neumann equation for reduced density operators, also known as hybrid master equations. If one considers the evolution of pure states only, i.e., disregarding the coherence between states and spontaneous transitions from pure to mixed states, then one can resort to quantum-mechanical equations of the Schrödinger type. We derive them from the hybrid master equations and study their main properties, which indicate that our equations have a larger range of applicability compared to other generalized Schrödinger equations proposed hitherto. Among other features, they can describe not only systems which remain in the stationary eigenstates of the Hamiltonian as time passes, but also those which evolve from those eigenstates. As an example, we consider a simple but important model, a quantum harmonic oscillator driven by both Hamiltonian and non-Hamiltonian terms, and derive its classical limit, which turns out to be the damped harmonic oscillator. Using this model, we demonstrate that the effects of dissipative environments of different types can cancel each other, thus resulting in an effectively dissipation-free classical system. Another discussed phenomenon is whether a non-trivial quantum system can reduce to a classical system in free motion, i.e., without experiencing any classical Newtonian forces. This uncovers a large class of quantum-mechanical non-Hamiltonian systems whose dynamics are not determined by conventional mechanics’ potentials and forces, but rather come about through quantum statistical effects caused by the system’s environment.
The time evolution of an open quantum system is governed by the Gorini-Kossakowski-Sudarshan-Lindlad equation for the reduced density operator of the system. This operator is obtained from the full density operator of the composite system involving the system itself, the bath, and the interactions between them, by performing a partial trace over the bath degrees of freedom. The entanglement between the system and the bath leads to a generalized Liouville evolution that involves, amongst other things, dissipation and decoherence of the system. In a similar fashion, the time evolution of a physical observable in a classically constrained dynamical system is governed by a generalization of the Liouville equation, in which the usual Poisson bracket is replaced by the so-called Dirac bracket. The generalization takes into account the reduction in the phase space of the system because of constraints which arise either because they are introduced by hand, or because of singular Lagrangians. We derive an intriguing, but precise classical-quantum correspondence between the aforementioned situations which connects the Lindblad operators to the constraints. The correspondence is illustrated in the ubiquitous example of coupled simple harmonic oscillators studied earlier in various contexts like quantum optics, dissipative quantum systems, Brownian motion, and the area law for the entropy of black holes.
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.