We investigate the entanglement dynamics of two local two-state quantum systems coupling to a bosonic mode and a common structured bath. Here the bosonic mode is damped by a broadband reservoir and acts as a probe mode for the use of continuous quantum measurement feedback, and the bath has a structure characterized by the Ohmic spectral density. We first present the non-Markovian master equation for the two-state quantum systems and then the corresponding hierarchical equations of motion (HEOM). We illustrate the equivalence between the HEOM approach and the perturbation theory. HEOM-calculations show that the quantum measurement feedback plays a positive role in the entanglement generation, and the non-Markovian effect of the Ohmic bath could greatly enhance this action. We reveal in detail the dependence of the scheme on the spectral density parameters, the temperature of the bath, and the measurement feedback strength. The numerical results are obtained with the HEOM truncated at the third order, which are equal to those obtained with sixth-order perturbation theory.
We investigate the entanglement dynamics of two local two-state quantum systems coupling to a bosonic mode and a common structured bath. Here the bosonic mode is damped by a broadband reservoir and acts as a probe mode for the use of continuous quantum measurement feedback, and the bath has a structure characterized by the Ohmic spectral density. We first present the non-Markovian master equation for the two-state quantum systems and then the corresponding hierarchical equations of motion (HEOM). We illustrate the equivalence between the HEOM approach and the perturbation theory. HEOM-calculations show that the quantum measurement feedback plays a positive role in the entanglement generation, and the non-Markovian effect of the Ohmic bath could greatly enhance this action. We reveal in detail the dependence of the scheme on the spectral density parameters, the temperature of the bath, and the measurement feedback strength. The numerical results are obtained with the HEOM truncated at the third order, which are equal to those obtained with sixth-order perturbation theory.
For many open quantum systems, a master equation approach employing the Markov approximation cannot reliably describe the dynamical behaviour. This is the case, for example, in a number of solid state or biological systems, and it has motivated a line of research aimed at quantifying the amount of non-Markovian behaviour in a given model. Within this framework, we investigate the dynamics of a quantum harmonic oscillator linearly coupled to a bosonic bath. We focus on Gaussian states, which are suitably treated using a covariance matrix approach. Concentrating on an entanglement based non-Markovian behaviour quantifier (NMBQ) proposed by Rivas et. al. [1], we consider the role that near resonant and off-resonant modes play in affecting the NMBQ. By using a large but finite bath of oscillators for both Ohmic and super Ohmic spectral densities we find, by systematically increasing the coupling strength, initially the near resonant modes provide the most significant non-Markovian effects, while after a certain threshold of coupling strength the off-resonant modes play the dominant role. We also consider the NMBQ for two other models where we add a single strongly coupled oscillator to the model in extra bath mode and 'buffer' configurations, which affects the modes that determine non-Markovian behaviour.
Optimal control theory is implemented with fully converged hierarchical equations of motion (HEOM) describing the time evolution of an open system density matrix strongly coupled to the bath in a spin-boson model. The populations of the two-level sub-system are taken as control objectives; namely, their revivals or exchange when switching off the field. We, in parallel, analyze how the optimal electric field consequently modifies the information back flow from the environment through different non-Markovian witnesses. Although the control field has a dipole interaction with the central sub-system only, its indirect influence on the bath collective mode dynamics is probed through HEOM auxiliary matrices, revealing a strong correlation between control and dissipation during a non-Markovian process. A heterojunction is taken as an illustrative example for modeling in a realistic way the two-level sub-system parameters and its spectral density function leading to a non-perturbative strong coupling regime with the bath. Although, due to strong system-bath couplings, control performances remain rather modest, the most important result is a noticeable increase of the non-Markovian bath response induced by the optimally driven processes.
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 investigate the dynamics of a two-qubit open quantum system, in particular the two-qubit spin-boson model in the strong coupling regime, coupled to two thermal bosonic baths under non-Markovian and non-equilibrium conditions. Two complementary approaches, the Hierarchical Equations of Motion (HEOM) and Reaction Coordinate Mapping (RCM), are employed to examine various coupling regimes between the qubits and their respective baths. The dynamical features of the model and the impact of the tunneling amplitude on quantum coherence of the system are probed using the l1-norm of coherence. The model is further shown to have non-Markovian evolution. The nontrivial task of calculating entropy production in the strong-coupling regime is performed using auxiliary density operators in HEOM. Motivated by the realization of a quantum thermal device in the strong-coupling regime, the non-equilibrium steady-state behavior of the system is investigated. Furthermore, the relationship between the heat and spin currents and the tunneling amplitude is probed.
We consider a model of non-Markovian quantum Brownian motion that consists of a harmonic oscillator bilinearly coupled to a thermal bath, both via its position and via its momentum operators. We derive the master equation for such a model, and we solve the equations of motion for a generic Gaussian system state. We then investigate the resulting evolution of the first and second moments for both an Ohmic and a super-Ohmic spectral density. In particular, we show that, irrespective of the specific form of the spectral density, the coupling with the momentum enhances the dissipation experienced by the system, accelerating its relaxation to the equilibrium as well as modifying the asymptotic state of the dynamics. Eventually, we characterize explicitly the non-Markovianity of the evolution using a general criterion which relies on the positivity of the master equation coefficients.
An open quantum system refers to a system that is further coupled to a bath system consisting of surrounding radiation fields, atoms, molecules, or proteins. The bath system is typically modeled by an infinite number of harmonic oscillators. This system-bath model can describe the time-irreversible dynamics through which the system evolves toward a thermal equilibrium state at finite temperature. In nuclear magnetic resonance and atomic spectroscopy, dynamics can be studied easily by using simple quantum master equations under the assumption that the system-bath interaction is weak (perturbative approximation) and the bath fluctuations are very fast (Markovian approximation). However, such approximations cannot be applied in chemical physics and biochemical physics problems, where environmental materials are complex and strongly coupled with environments. The hierarchical equations of motion (HEOM) can describe the numerically "exact" dynamics of a reduced system under nonperturbative and non-Markovian system-bath interactions, which has been verified on the basis of exact analytical solutions (non-Markovian tests) with any desired numerical accuracy. The HEOM theory has been used to treat systems of practical interest, in particular, to account for various linear and nonlinear spectra in molecular and solid state materials, to evaluate charge and exciton transfer rates in biological systems, to simulate resonant tunneling and quantum ratchet processes in nanodevices, and to explore quantum entanglement states in quantum information theories. This article presents an overview of the HEOM theory, focusing on its theoretical background and applications, to help further the development of the study of open quantum dynamics.