Open Quantum System

An open quantum system is a quantum mechanical system that interacts with an external environment, resulting in the exchange of energy, information, or matter, which leads to irreversible dynamics, dissipation, and decoherence that fundamentally alter its evolution [8]. This interaction distinguishes it from a closed quantum system, which is perfectly isolated [1]. The study of open quantum systems is a fundamental framework in quantum physics, providing the theoretical tools to describe how real-world quantum systems behave when exposed to their surroundings, a condition that is ubiquitous in nature and essential for practical quantum technologies [2]. The central physical concepts and mathematical techniques of this theory are crucial for understanding a vast range of phenomena across physics [4]. The key characteristics of open quantum systems arise from their coupling to an environment, which introduces effects such as decoherence—the loss of quantum coherence and superposition—and dissipation, the irreversible loss of energy [8]. These processes are mathematically described using advanced techniques including quantum master equations and the theory of quantum dynamical semigroups [4][5]. While the interaction is often modeled as weak for simplification [7], more general and numerically exact approaches, such as the hierarchical equations of motion (HEOM) method, have been developed to handle strong coupling and non-Markovian effects [6]. The formalism allows for the systematic derivation of effective equations of motion for the system alone, tracing out the complex degrees of freedom of the environment. The significance and applications of open quantum system theory are extraordinarily broad, spanning from foundational physics to cutting-edge engineering. It is indispensable for the development of practical quantum computing, where understanding and mitigating decoherence is the primary challenge for building functional qubits [2]. Beyond quantum information science, the framework probes fundamental phenomena such as gravitational decoherence and is applied to diverse areas including particle physics, as in the study of fast neutrino flavor conversions, and cosmology [3]. The approach also provides insights into thermodynamic processes at the quantum level [7]. Consequently, the theory of open quantum systems forms a critical bridge between abstract quantum theory and the description of real, observable physical systems across microscopic and macroscopic scales.

This framework stands in contrast to a closed quantum system, which is isolated from its surroundings and evolves unitarily according to the Schrödinger equation. The theory of open quantum systems provides the essential mathematical and conceptual tools for describing realistic quantum systems that cannot be perfectly isolated, making it a cornerstone of modern quantum physics with applications across quantum information science, condensed matter physics, quantum optics, and chemical physics.

Fundamental Distinction from Closed Systems

The evolution of a closed quantum system is governed by the time-dependent Schrödinger equation, $i\hbar \frac{d}{dt}|\psi(t)\rangle = \hat{H} |\psi(t)\rangle$, where $\hat{H}$ is the system's Hamiltonian. This leads to a unitary evolution operator $\hat{U}(t) = \exp(-i\hat{H}t/\hbar)$ that preserves the purity of quantum states. In stark contrast, an open quantum system is coupled to an environment, often modeled as a large quantum system with many degrees of freedom (e.g., a thermal bath of photons, phonons, or other particles). The composite system (system + environment) may be treated as closed, but the subsystem of interest—the open system—undergoes non-unitary evolution. When the environment's degrees of freedom are traced out, the open system's state is described by a reduced density matrix $\rho_S(t) = \text{Tr}_E [\rho_{SE}(t)]$, where $\rho_{SE}$ is the density matrix of the total system and $\text{Tr}_E$ denotes the partial trace over the environmental degrees of freedom. This reduced description loses information and manifests key phenomena like decoherence and dissipation [12].

Key Physical Phenomena

The interaction with an environment gives rise to several defining physical processes:

• Decoherence: This is the process by which quantum superpositions (coherences) in the system are suppressed due to entanglement with the environment. Off-diagonal elements in the system's density matrix (in a preferred basis often determined by the system-environment coupling) decay over time. For a simple qubit superposition $|\psi\rangle = \alpha|0\rangle + \beta|1\rangle$, decoherence causes the decay of terms like $\alpha\beta^*$ towards zero, effectively transforming a pure quantum state into a statistical mixture. Decoherence timescales can vary from microseconds in superconducting qubits to milliseconds in trapped ions, setting fundamental limits on quantum coherence for computation and sensing [12].
• Dissipation: This refers to the exchange of energy between the system and the environment. The system can lose energy (relaxation) to the environment, typically approaching thermal equilibrium. The characteristic timescale for energy relaxation, often denoted $T_1$, is generally different from the decoherence time $T_2$. In the weak-coupling limit, the relation $T_2 \leq 2T_1$ typically holds.
• Dephasing: A specific form of decoherence where energy exchange is negligible, but the interaction with the environment randomizes the quantum phase of the system's state. This leads to the decay of coherences without a change in the diagonal (population) elements of the density matrix.

Mathematical Description and Approximations

The dynamics of the reduced density matrix $\rho_S(t)$ are not described by the von Neumann equation but by more complex master equations. Their derivation relies on specific approximations about the nature of the system, environment, and their interaction:

• Weak Coupling Approximation: Very often the interaction between an open system and the environment is assumed to be weak [13]. This allows the use of perturbation theory in the system-environment coupling strength. The standard approach involves the Born approximation, which assumes the environment is largely unaffected by the system and remains in a stationary state (e.g., a thermal state).
• Markov Approximation: This assumes the environment has no memory, meaning that correlations within the environment decay on a timescale much faster than the characteristic evolution time of the system. This leads to a master equation that is local in time.
• Lindblad Master Equation: Under the combined Born-Markov approximations and an additional secular approximation, the most general form of a Markovian, completely positive, trace-preserving dynamical map is given by the Gorini–Kossakowski–Sudarshan–Lindblad (GKSL) master equation: \frac{d\rho_S}{dt} = -\frac{i}{\hbar} [\hat{H}_S, \rho_S] + \sum_k \gamma_k \left( \hat{L}_k \rho_S \hat{L}_k^\dagger - \frac{1}{2} \{ \hat{L}_k^\dagger \hat{L}_k, \rho_S \} \right). \] Here, \( \hat{H}_S \) is the system Hamiltonian (possibly renormalized by Lamb shifts), \( \hat{L}_k \) are Lindblad operators representing different decay channels, and \( \gamma_k \geq 0 \) are the corresponding non-negative decay rates. This form guarantees the physicality of the density matrix evolution. For stronger system-environment couplings or structured environments (e.g., with a non-flat spectral density), non-Markovian master equations or path integral techniques like the Feynman-Vernon influence functional are required.

Role in Quantum Technologies

The theory of open quantum systems is not merely a description of unwanted noise but a fundamental framework for quantum engineering. Building on the indispensable role for quantum computing mentioned previously, its applications extend further:

• Quantum Reservoir Engineering: Deliberately tailoring system-environment interactions can be used to prepare desired quantum states or simulate specific dissipative dynamics.
• Quantum Thermodynamics: It provides the framework for analyzing heat, work, and entropy production at the quantum scale, studying quantum heat engines and the emergence of thermodynamic laws from quantum dynamics.
• Quantum Metrology and Sensing: Understanding environmental noise is crucial for designing robust quantum sensors that approach the Heisenberg limit, and certain non-Markovian features can even be harnessed for enhanced sensing protocols.
• Quantum Biology: Processes such as photosynthesis and magnetoreception are investigated using open quantum system models to understand the potential role of quantum coherence in biological environments. In summary, the study of open quantum systems transcends the idealizations of textbook quantum mechanics, providing the essential language for describing realistic quantum dynamics in contact with a complex world. Its mathematical formalism captures the emergence of irreversibility and classicality from quantum foundations while enabling the design and control of next-generation quantum technologies.

History

The theoretical framework for open quantum systems emerged from the need to reconcile the unitary, reversible evolution of isolated quantum systems with the irreversible, dissipative behavior observed in real physical systems interacting with their environments. This historical development spans from early 20th-century quantum foundations to contemporary research in quantum information science.

Early Foundations and the Problem of Irreversibility (1920s–1950s)

The origins of open quantum system theory are deeply intertwined with the foundational debates of quantum mechanics itself. Following the formulation of the Schrödinger equation in 1926, which described the deterministic evolution of a closed system's wave function, a central paradox arose: how could this reversible dynamics produce the irreversible processes—such as energy dissipation and the approach to thermal equilibrium—that are ubiquitously observed in nature? [1] This was a manifestation of the broader quantum measurement problem. Early attempts to address dissipation were phenomenological. In 1928, Paul Dirac introduced one of the first models of damping in quantum theory while treating the interaction of atoms with the electromagnetic field, laying conceptual groundwork for system-environment interactions [2]. A pivotal step came with the development of quantum statistical mechanics. In 1928, John von Neumann established the density matrix formalism in his seminal work Mathematische Grundlagen der Quantenmechanik, providing the essential mathematical language (states as operators in $\Mn$) to describe statistical ensembles and, later, reduced states of subsystems [3]. The full recognition of an "open system" as a distinct conceptual entity, however, is often credited to the work of physicists like Herbert Fröhlich and Nikolay Bogoliubov in the 1940s and 1950s, who studied the interaction of a system of interest (e.g., an electron or a collective mode) with a heat bath or reservoir to explain phenomena like superconductivity and superfluidity [4].

Birth of the Master Equation and the Markovian Paradigm (1950s–1970s)

The modern theory of open quantum systems began to crystallize in the 1950s with the derivation of dynamical equations for the reduced density matrix of the system alone. A landmark achievement was the work of physicist and mathematician Léon Van Hove. In 1957, using perturbation theory and a crucial "weak-coupling" assumption, he derived a master equation for a system interacting with a large reservoir, demonstrating how irreversible behavior emerges from reversible microscopic laws in the thermodynamic limit [5]. This approach was rigorously generalized by Felix Bloch in 1957 for magnetic resonance phenomena, leading to the celebrated Bloch equations that describe the non-unitary relaxation of spin populations and coherences [6]. The most influential formulation for quantum optics and quantum information was developed by Swiss physicist Walter H. Louisell in the 1960s and, in its most widely used form, by Vittorio Gorini, Andrzej Kossakowski, George Sudarshan, and Göran Lindblad in the 1970s. In 1976, Gorini, Kossakowski, and Sudarshan, and independently Lindblad, established the most general form of a Markovian, completely positive, trace-preserving dynamical semigroup for a density matrix $\rho$ in $\Mn$ [7][8]. The Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) master equation, or simply the Lindblad equation, is expressed as:

$$\dot{\rho} = -\frac{i}{\hbar}[H, \rho] + \sum_k \left( L_k \rho L_k^\dagger - \frac{1}{2} \{ L_k^\dagger L_k, \rho \} \right)$$

where $H$ is the system Hamiltonian, and the Lindblad operators $L_k$ ($\in \Mnm$) model the incoherent effects of the environment. This equation became the standard workhorse for modeling quantum dissipation and decoherence under the Born-Markov approximation, which assumes weak, memoryless coupling to a large, featureless bath [9].

The Rise of Decoherence Theory and Quantum Information (1980s–1990s)

While master equations described dissipation, a deeper understanding of the quantum-to-classical transition was propelled by the theory of decoherence, pioneered by H. Dieter Zeh in 1970 and extensively developed by Wojciech Zurek and others in the 1980s [10]. Decoherence theory framed the environment as a mechanism for the selective suppression of quantum superpositions through the rapid dispersal of phase information into environmental degrees of freedom. This provided a dynamical explanation for the apparent collapse of the wave function and the emergence of "pointer states" robust to interaction. The study of open systems was thus no longer just about energy relaxation but about the loss of quantum coherence, a fundamental obstacle for quantum technologies [11]. This period also saw the development of powerful non-perturbative techniques for specific system-bath models. The spin-boson model, describing a two-level system ($\Mq$) coupled to a continuum of harmonic oscillators, was solved using path-integral methods by Anthony J. Leggett and others in the 1980s, revealing rich non-Markovian dynamics and quantum phase transitions in dissipation strength [12]. Concurrently, the quantum trajectories (or Monte Carlo wave function) method was invented by Jean Dalibard, Yvan Castin, and Klaus Mølmer in 1992, providing a stochastic unraveling of the master equation into individual quantum jump trajectories, offering both computational advantages and a compelling picture of continuous measurement [13].

Non-Markovian Dynamics and Quantum Control (2000s–Present)

The 21st century has been characterized by a move beyond the Markovian approximation, driven by advances in experimental control of nanoscale and mesoscopic systems where environmental correlations are significant. The study of non-Markovian effects, where the system's evolution depends on its historical states due to strong or structured environmental correlations, has become a major subfield [14]. Formal measures for non-Markovianity based on the backflow of information were established, such as those by Heinz-Peter Breuer, Francesco Petruccione, and others around 2009 [15]. This framework is crucial for accurately modeling short-time-scale processes, quantum biological phenomena, and systems coupled to engineered or finite baths. The historical trajectory culminated in the intimate connection with quantum information science. As noted earlier, understanding open systems is indispensable for the development of practical quantum computing. This has driven research into quantum error correction (first proposed by Peter Shor in 1995) and dynamical decoupling (conceptualized by Lev Viola and Seth Lloyd in 1998), which are explicit control strategies to mitigate decoherence by treating the system-environment interaction as a form of noise to be suppressed or corrected [16][17]. Furthermore, the theory has been inverted to exploit system-environment interactions for beneficial purposes, such as dissipative state engineering and reservoir engineering, where a tailored bath is used to drive a system into a desired quantum state, a key concept in approaches to quantum computation proposed by Ignacio Cirac and Peter Zoller in 2009 [18]. Today, the history of open quantum systems continues to be written at the intersection of fundamental quantum theory, condensed matter physics, quantum optics, and quantum information engineering, providing the essential toolkit for describing and manipulating the non-isolated quantum reality of our world.

Significance

The theoretical framework of open quantum systems provides the essential mathematical and conceptual foundation for understanding, predicting, and controlling quantum behavior in real-world settings where perfect isolation is impossible. Beyond this critical application, the framework underpins diverse fields across quantum technologies and fundamental physics, enabling the modeling of irreversible dynamics, dissipation, and decoherence that fundamentally alter a system's evolution compared to idealized closed-system predictions [12]. Its significance extends from enabling precise quantum control protocols to offering a testing ground for foundational questions in quantum mechanics and statistical physics.

Foundational Role in Quantum Technologies

The open quantum systems framework is the workhorse for modeling and engineering real quantum devices. In quantum information processing, the coherence time of qubits—the duration over which quantum superpositions are maintained—is directly limited by their coupling to environmental degrees of freedom [14]. Quantitative modeling of this decoherence via master equations is a prerequisite for designing error correction codes and assessing quantum advantage thresholds. In quantum optics, the formalism describes laser cooling of atoms, where spontaneous emission provides the dissipative mechanism to remove entropy, and cavity quantum electrodynamics (QED), where a two-level atom interacts with the confined modes of an optical cavity, exchanging energy and information in a controlled manner [14]. For condensed matter systems, particularly mesoscopic structures, the framework is used to study transport phenomena where quantum coherence of electrons persists across ballistic microstructures, linking to concepts in quantum chaos [14]. The design of control pulses for quantum technologies explicitly relies on open systems theory. Optimal control frameworks, such as those based on the Pontryagin Maximum Principle, are applied to design energy- and time-efficient control pulses that achieve target quantum operations while accounting for dissipative environmental effects [9]. This is critical in architectures like circuit quantum electrodynamics, where superconducting qubits are inherently open systems. Similarly, system identification techniques are necessary to characterize and subsequently control specific open system parameters, as demonstrated in studies of two-level systems [10].

Mathematical and Conceptual Framework

The significance of open quantum systems is deeply rooted in its rigorous mathematical structure, which generalizes the unitary evolution of closed systems. The state of an open system is described by a density matrix, $\rho$, which for a two-level system takes the Hermitian, positive semi-definite form $\rho=\begin{pmatrix}a & b\\ c & d\end{pmatrix}$ with $a,d\in\mathbb{R}^+$, $b,c\in\mathbb{C}$ where $b^*=c$, and a unit trace $a+d=1$ [17]. Its evolution is governed by dynamical equations derived from microscopic system-environment models. A central achievement is the derivation of the Lindblad master equation, a Markovian evolution equation of the form $\dot{\rho} = -\frac{i}{\hbar}[H, \rho] + \sum_k \gamma_k \left( L_k \rho L_k^\dagger - \frac{1}{2}\{L_k^\dagger L_k, \rho\} \right)$, which guarantees complete positivity of the density matrix [17]. This equation is the cornerstone for simulating quantum dynamics in the presence of memoryless dissipation and decoherence. Numerical methods to solve it include computing the propagator, solving the ordinary differential equation iteratively, or using quantum trajectory methods (Monte Carlo wavefunction approach) for sampling stochastic evolutions [17]. Building on the concept discussed above, the study of non-Markovian effects has become a major subfield. This involves deriving and solving integro-differential equations where the system's evolution depends on its history, which is crucial for systems with strong coupling to the environment or environments with structured spectral densities (e.g., photonic bandgap materials or biological complexes) [9]. This distinction between Markovian and non-Markovian regimes provides a comprehensive toolkit for modeling quantum dynamics across a vast range of physical timescales and coupling strengths.

Insights into Fundamental Physics

Open quantum systems offer a critical lens through which to examine foundational questions in physics. They provide the definitive framework for understanding the quantum-to-classical transition, where decoherence—the loss of quantum superposition into a classical mixture—explains the emergence of classical reality from quantum substrate without invoking wavefunction collapse axioms [12]. This process of environmental monitoring and information loss is the primary obstacle in quantum information processing but also the mechanism that selects quasi-classical pointer states in measurement theory. The framework also bridges quantum mechanics with statistical mechanics and thermodynamics. Dissipation involves the exchange of energy, leading to the relaxation of the system toward thermal equilibrium with its bath, thereby connecting quantum dynamics to concepts of entropy production and fluctuation theorems [12]. Furthermore, open systems are central to studies of quantum chaos in complex systems. Mesoscopic systems, where electron coherence combines with ballistic transport, serve as laboratories for testing quantum chaos ideas; signatures of chaos appear in the statistical properties of energy levels and in the sensitivity of time-reversal experiments, quantified by measures like the Loschmidt echo [14]. The Loschmidt echo measures the fidelity of revival after an imperfect time-reversal procedure and decays differently for chaotic versus regular systems, linking open dynamics to fundamental features of complex quantum evolution [15].

Enabling Practical Simulation and Design

The operational value of the open quantum systems framework is realized through its capacity for quantitative simulation, which guides experimental design and interprets results. The ability to simulate evolution using the Lindblad master equation or non-Markovian techniques allows researchers to:

• Predict qubit coherence times under specific noise models
• Model the efficiency of photon emission in quantum dots or NV centers
• Design pulse sequences for dynamical decoupling to mitigate environmental noise
• Calculate transport currents in nanoscale electronic devices
• Model the energy transfer efficiency in photosynthetic complexes

This predictive power transforms the framework from a theoretical construct into an engineering design tool. It enables the exploration of "what-if" scenarios in quantum device architecture, the optimization of material parameters to minimize unwanted environmental coupling, and the development of new protocols for quantum sensing that can leverage, rather than simply combat, environmental interactions.

Applications and Uses

The theory of open quantum systems provides the fundamental framework for understanding and manipulating quantum systems that interact with their environment, underpinning diverse applications across quantum technologies and fundamental physics [18]. Unlike isolated systems, which evolve unitarily, open systems exhibit non-unitary reduced dynamics when environmental degrees of freedom are traced out, requiring density operators for their description [12]. This framework is essential for modeling realistic quantum devices and phenomena where perfect isolation is impossible.

Quantum Information Processing and Computation

A primary application lies in quantum information science. As noted earlier, decoherence is the primary obstacle for building functional qubits [8][12]. The theory enables the modeling of qubit-environment interactions to predict coherence times (T₁ for energy relaxation and T₂ for dephasing) and design error correction codes. Beyond this, a critical task is quantum control, which involves designing external pulses and sequences to suppress decoherence, perform high-fidelity gate operations, and maintain entanglement in noisy environments [7]. This includes dynamical decoupling techniques, where sequences of control pulses average out environmental noise, and optimal control theory for state preparation and steering. Furthermore, the capacity for decoherence suppression is a benchmark for control methodologies, directly impacting the feasibility of scalable quantum computers and quantum communication networks [7].

Quantum Optics and Atomic Physics

Open system methods are indispensable in quantum optics for modeling the interaction of light with matter. Key applications include:

• Laser cooling and trapping of atoms and ions, where spontaneous emission (a dissipative process) is harnessed to remove kinetic energy, allowing particles to reach ultracold temperatures in the microkelvin to nanokelvin range [18].
• Cavity quantum electrodynamics (QED), which studies atoms coupled to the electromagnetic field of a high-finesse optical or microwave cavity. The cavity acts as a structured environment, and the system's dynamics—whether atoms, artificial atoms like quantum dots, or superconducting circuits—are governed by open quantum system models. This enables the generation of non-classical light states and the study of strong coupling regimes.
• Quantum optics experiments involving single-photon sources, detectors, and linear optical networks for quantum computing, where losses and mode mismatches are modeled as environmental interactions.

Condensed Matter Physics

In condensed matter, open quantum system theory is used to study transport phenomena and many-body systems coupled to baths. Applications include:

• Modeling quantum dots and molecular junctions connected to electronic leads, where electron transport is described by non-equilibrium dynamics. - Understanding energy transfer in photosynthetic complexes and other biological systems, where non-Markovian effects and environmental noise can potentially enhance efficiency through quantum coherence. - Studying ultracold atoms in optical lattices with controlled dissipation, used to engineer novel quantum phases and probe many-body localization.

Foundational Physics and Dynamics

The theory provides essential tools for deriving and solving dynamical equations for system evolution. A central task is the microscopic derivation of dynamical equations from the total system-environment Hamiltonian, encompassing both Markovian and non-Markovian evolutions [8]. Markovian approximations, valid when environmental correlation times are short, lead to master equations of Lindblad form (Gorini–Kossakowski–Sudarshan–Lindblad equation), which guarantee complete positivity. These are widely used in quantum optics and quantum information. Non-Markovian evolution, relevant for strong coupling or structured environments (e.g., photonic bandgap materials), requires more advanced techniques like path integrals, hierarchical equations of motion (HEOM), or time-convolutionless master equations [8][19]. Building on the concept discussed above, the study of these effects is crucial for accurate modeling in many physical scenarios.

Quantum Thermodynamics and Metrology

Emerging fields leverage open system concepts to extend thermodynamics to the quantum regime and enhance measurement precision.

• Quantum thermodynamics investigates heat, work, and entropy production in quantum engines and refrigerators. Open system models describe the exchange of energy and information with thermal baths, exploring fluctuations and the validity of classical laws at the nanoscale.
• Quantum metrology and sensing uses open systems to model the limits of measurement precision imposed by environmental noise. Strategies are developed to protect quantum states used as sensors (e.g., entangled spin ensembles or nitrogen-vacancy centers in diamond) from decoherence, thereby surpassing classical sensing limits for magnetic fields, temperature, or time.

Chemical Physics and Spectroscopy

In chemistry, open quantum system approaches model:

• Electronic energy transfer in light-harvesting complexes and conjugated polymers.
• Vibrational relaxation and electron transfer in solutions, where the solvent acts as a dissipative environment.
• Multidimensional spectroscopic signals, which can reveal non-Markovian system-bath correlations and coherent dynamics in molecular aggregates.

Engineering of Artificial Quantum Systems

The principles are actively used to design and characterize engineered quantum platforms:

• Superconducting quantum circuits (qubits) are designed with specific coupling to electromagnetic environments to control relaxation and dephasing rates, implementing filters and Purcell shields to suppress unwanted decay channels.
• Quantum noise spectroscopy techniques characterize the spectral density of environmental noise, informing the development of tailored control pulses for decoherence suppression [7].
• Quantum reservoir engineering deliberately couples a system to a designed bath to drive it into a desired steady state, such as an entangled or squeezed state, useful for quantum memories and continuous-variable quantum information. In summary, the theory of open quantum systems transcends a mere description of decoherence; it provides the essential toolkit for modeling, controlling, and exploiting realistic quantum dynamics across virtually all modern quantum science and technology domains [8][18][12]. Its continued development, including the refinement of numerical techniques for non-Markovian dynamics and the integration with control theory, remains central to advancing practical quantum technologies and understanding fundamental quantum phenomena in complex environments.

References

1. Introduction - https://matteoacrossi.github.io/oqs-jupyterbook/introduction.html
2. What Are Open Quantum Systems? A Complete Guide - https://www.spinquanta.com/news-detail/what-are-open-quantum-systems
3. Open Quantum Systems: Dissipative Dynamics from Quarks to the Cosmos - https://www.int.washington.edu/programs-and-workshops/25-3b
4. Concepts and methods in the theory of open quantum systems - https://arxiv.org/abs/quant-ph/0302047
5. On the generators of quantum dynamical semigroups - https://link.springer.com/article/10.1007/BF01608499
6. Tree tensor network state approach for solving hierarchical equations of motion - https://arxiv.org/abs/2304.05151
7. Characterization and control of open quantum systems beyond quantum noise spectroscopy - https://www.nature.com/articles/s41534-020-00332-8
8. Open Quantum Systems. An Introduction - https://arxiv.org/abs/1104.5242
9. Optimal Control for Open Quantum System in Circuit Quantum Electrodynamics - https://arxiv.org/abs/2412.20149
10. Identification and control of a two-level open quantum system - https://ieeexplore.ieee.org/document/6160475/
11. Open quantum systems - https://arxiv.org/html/2407.16855v1
12. Open quantum system - https://grokipedia.com/page/Open_quantum_system
13. Dynamical and thermodynamical approaches to open quantum systems - https://www.nature.com/articles/s41598-020-59241-7
14. Mesoscopic transport and quantum chaos - http://www.scholarpedia.org/article/Mesoscopic_transport_and_quantum_chaos
15. Loschmidt echo - Scholarpedia - http://www.scholarpedia.org/article/Loschmidt_echo
16. “Open Quantum Systems” Everything you need to start learning this interesting but a little abstruse… - https://omid-hassasfar.medium.com/open-quantum-systems-c0dd99e8fddd
17. Open systems - Dynamiqs - https://www.dynamiqs.org/stable/documentation/basics/open-systems.html
18. 6: Evolution of Open Quantum Systems - https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Advanced_Quantum_Mechanics_(Kok)/06%253A_Evolution_of_Open_Quantum_Systems
19. Lecture Notes on the Theory of Open Quantum Systems - https://arxiv.org/abs/1902.00967