In this paper, we provide a complete description of the life cycle of entanglement during the real-time motion of open quantum systems. The quantum environment can have arbitrary (e.g., structured) spectral density. The entanglement can be seen constructively as a Lego: its bricks are the modes of the environment. These bricks are connected to each other via operator transforms. The central result is that during each infinitesimal time interval one new (incoming) mode of the environment gets coupled (entangled) to the open system, and one new (outgoing) mode gets irreversibly decoupled (disentangled from the future). Moreover, each moment of time, only a few relevant modes (three to four in the considered cases) are non-negligibly coupled to the future quantum motion. These relevant modes change (flow or renormalize) with time. As a result, the temporal entanglement has the structure of a matrix-product operator. This allows us to pose a number of questions and to answer them in this paper: What is the intrinsic quantum complexity of a real time motion? Does this complexity saturate with time or grow without bounds? How does one do the real-time renormalization group in a justified way? How do the classical Brownian stochastic trajectories emerge from the quantum evolution? How does one construct the few-mode representations of non-Markovian environments? We provide illustrative simulations of the spin-boson model for various spectral densities of the environment: semicircle, subohmic, ohmic, and superohmic.
In this paper, we provide a complete description of the life cycle of entanglement during the real-time motion of open quantum systems. The quantum environment can have arbitrary (e.g., structured) spectral density. The entanglement can be seen constructively as a Lego: its bricks are the modes of the environment. These bricks are connected to each other via operator transforms. The central result is that during each infinitesimal time interval one new (incoming) mode of the environment gets coupled (entangled) to the open system, and one new (outgoing) mode gets irreversibly decoupled (disentangled from the future). Moreover, each moment of time, only a few relevant modes (three to four in the considered cases) are non-negligibly coupled to the future quantum motion. These relevant modes change (flow or renormalize) with time. As a result, the temporal entanglement has the structure of a matrix-product operator. This allows us to pose a number of questions and to answer them in this paper: What is the intrinsic quantum complexity of a real time motion? Does this complexity saturate with time or grow without bounds? How does one do the real-time renormalization group in a justified way? How do the classical Brownian stochastic trajectories emerge from the quantum evolution? How does one construct the few-mode representations of non-Markovian environments? We provide illustrative simulations of the spin-boson model for various spectral densities of the environment: semicircle, subohmic, ohmic, and superohmic.
When an open quantum system is coupled to a noninteracting environment, the interaction disturbance starts to propagate and gradually involves more and more environment modes. This leads to a spread of entanglement and to a combinatorial growth of complexity, the so-called 'entanglement barrier'. However one expects this process to be balanced by the continuous emergence of modes which are effectively and irreversibly decoupled from the future motion of open quantum system. In this work it is demonstrated that this is indeed the case. Under controlled and exponentially converging approximations, we derive a quantum circuit model of evolution. It shows that each infinitesimal time interval one new (incoming) mode gets coupled to open system, and one new (outgoing) mode gets irreversibly decoupled. This indicates that the time plays the role of the renormalization group (RG) flow parameter: the emerging outgoing modes are not entangled to the future real-time flow, thus they are the 'irrelevant degrees of freedom'. In the conventional RG procedure the latter would be iteratively traced out during the evolution. However there is another interesting opportunity: to measure them. This results in a stochastic RG for pure states, in which the irrelevant degrees of freedom collapse to a classical stochastic measurement signal. We provide an illustrative simulations of the spin-boson model for various spectral densities of the environment.
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.
This study advances the theoretical understanding of decoherence mechanisms in open quantum systems by modelling entanglement dynamics under localised environmental interactions relevant to emerging quantum technologies in Tunisia. A unified analytical framework is developed that bridges Markovian and non-Markovian regimes, quantifying the resilience of bipartite entanglement against structured environmental noise. Considering a system of two non-interacting qubits coupled to a common bosonic reservoir, a generalised master equation incorporating a memory kernel is derived from a structured spectral density function with tunable ohmicity. An analytical expression for the time-dependent concurrence is obtained for an initial Bell state, revealing a crossover from monotonic decay to oscillatory behaviour with entanglement revivals as the reservoir cutoff frequency is lowered relative to the qubit splitting. In the sub-ohmic regime, entanglement decay is slower and revivals are more pronounced, demonstrating that the spectral density cutoff frequency acts as a control parameter for entanglement preservation. The framework incorporates a local noise temperature parameter to represent cryostat fluctuations typical of Tunisian laboratory conditions, allowing the model to capture the phenomenology of entanglement evolution under constraints of limited cryogenic infrastructure and higher ambient noise levels. Crucially, the results show that standard Markovian models systematically overestimate the rate of entanglement loss, particularly in sub-ohmic environments, and that non-Markovian memory effects can temporarily restore quantum correlations after periods of apparent death. These findings challenge the assumption that decoherence is irreversible, revealing a form of quantum memory that could be harnessed for error correction or state protection. The framework provides a practical tool for predicting coherence lifetimes in solid-state and photonic systems, offering a comparative benchmark for experimental platforms within Tunisia's nascent quantum optics community and informing the design of robust quantum information protocols in the North African research context.
We present a formulation of non-Markovian quantum trajectories for open systems from a measurement theory perspective. In our treatment there are three distinct ways in which non-Markovian behavior can arise: a mode-dependent coupling between the bath (reservoir) and system, a dispersive bath, and spectral detection of the output into the bath. In the first two cases the non-Markovian behavior is intrinsic to the interaction; in the third case the non-Markovian behavior arises from the method of detection. We focus in detail on the trajectories that simulate real-time spectral detection of the light emitted from a localized system. In this case the non-Markovian behavior arises from the uncertainty in the time of emission of particles that are later detected. The results of computer simulations of the spectral detection of the spontaneous emission from a strongly driven two-level atom are presented.
We are at the midst of second quantum revolution where the mesoscopic quantum devies are actively employed for technological purposes. Despite this fact, the description of their real-time dynamics beyond the Fermi's golden rule remains a formiddable theoretical problem. This is due to the rapid spread of entanglement within the degrees of freedom of the surrounding environment. This is accompanied with a quantum noise (QN) acting on the mesoscopic device. In this work we propose a possible way out: to exploit the fact that this QN is usually bandlimited. This is because its spectral density is often contained in peaks of localized modes and resonances, and may be constrained by bandgaps. Inspired by the Kotelnikov sampling theorem from the theory of classical bandlimited signals, we put forward and explore the idea that when the QN spectral density has effective bandwidth B, the quantum noise becomes a discrete-time process, with an elementary time step τ∝B−1. After each time step τ, one new QN degree of freedom (DoF) gets coupled to the device for the first time, and one new QN DoF get irreversibly decoupled. Only a bounded number of QN DoFs are significantly coupled at any time moment. We call these DoFs the \textit{Kotelnikov modes}. As a result, the real-time dissipative quantum motion has a natural structure of a discrete-time matrix product state, with a bounded bond dimension. This yields a microscopically derived collision model. The temporal entanglement entropy appears to be bounded (area-law scaling) in the frame of Kotelnikov modes. The irreversibly decoupled modes can be traced out as soon as they occur during the real-time evolution. This leads to a novel\textit{bandlimited} input-output formalism and to quantum jump Monte Carlo simulation techniques for real-time motion of open quantum systems. We illustrate this idea on a spin-boson model.
In the first part of this thesis we extensively review the general theory of open quantum systems. We introduce new approaches to describe, in a hopefully more transparent way, some aspects which are usually misleading in a broad range of the literature. In addition, we revise new concepts and methods developed in recent years that complement the general theory mainly developed in the 1970s. In the second part, we focus our attention on the so-called Markovian processes which have been described in the previous part. We study the validity of typical equations used in simple systems such as harmonic oscillators, and propose and check different strategies to describe Markovian dynamics in more complicated interacting systems. Finally we propose a counter-intuitive situation where some level of noise can be actually useful for information tasks. In the third and last part, we approach the problem of the non-Markovian description of open quantum systems. To quantify how non-Markovian an open quantum system is we introduce a computable measure and apply it to some relevant situations. We also propose a way of probing the complexity of an environment by visualizing properties of a pair of qubits embedded in it. In addition, we develop a technique based on orthogonal polynomial theory to transform a wide range of environments into simple chains of particles which can be efficiently simulated with numerical techniques.