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Concept# Open quantum system

Résumé

In physics, an open quantum system is a quantum-mechanical system that interacts with an external quantum system, which is known as the environment or a bath. In general, these interactions significantly change the dynamics of the system and result in quantum dissipation, such that the information contained in the system is lost to its environment. Because no quantum system is completely isolated from its surroundings, it is important to develop a theoretical framework for treating these interactions in order to obtain an accurate understanding of quantum systems.
Techniques developed in the context of open quantum systems have proven powerful in fields such as quantum optics, quantum measurement theory, quantum statistical mechanics, quantum information science, quantum thermodynamics, quantum cosmology, quantum biology, and semi-classical approximations.
A complete description of a quantum system requires the inclusion of the environment. Completely describing the resulting combined system then requires the inclusion of its environment, which results in a new system that can only be completely described if its environment is included and so on. The eventual outcome of this process of embedding is the state of the whole universe described by a wavefunction . The fact that every quantum system has some degree of openness also means that no quantum system can ever be in a pure state. A pure state is unitary equivalent to a zero-temperature ground state, forbidden by the third law of thermodynamics.
Even if the combined system is in a pure state and can be described by a wavefunction , a subsystem in general cannot be described by a wavefunction. This observation motivated the formalism of density matrices, or density operators, introduced by John von Neumann in 1927 and independently, but less systematically by Lev Landau in 1927 and Felix Bloch in 1946. In general, the state of a subsystem is described by the density operator and the expectation value of an observable by the scalar product .

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Open quantum system

In physics, an open quantum system is a quantum-mechanical system that interacts with an external quantum system, which is known as the environment or a bath. In general, these interactions significantly change the dynamics of the system and result in quantum dissipation, such that the information contained in the system is lost to its environment. Because no quantum system is completely isolated from its surroundings, it is important to develop a theoretical framework for treating these interactions in order to obtain an accurate understanding of quantum systems.

Lindbladian

In quantum mechanics, the Gorini–Kossakowski–Sudarshan–Lindblad equation (GKSL equation, named after Vittorio Gorini, Andrzej Kossakowski, George Sudarshan and Göran Lindblad), master equation in Lindblad form, quantum Liouvillian, or Lindbladian is one of the general forms of Markovian master equations describing open quantum systems. It generalizes the Schrödinger equation to open quantum systems; that is, systems in contacts with their surroundings.

Quantum operation

In quantum mechanics, a quantum operation (also known as quantum dynamical map or quantum process) is a mathematical formalism used to describe a broad class of transformations that a quantum mechanical system can undergo. This was first discussed as a general stochastic transformation for a density matrix by George Sudarshan. The quantum operation formalism describes not only unitary time evolution or symmetry transformations of isolated systems, but also the effects of measurement and transient interactions with an environment.

The aim of this course is to familiarize the student with the concepts, methods and consequences of quantum physics.

This course provides an in-depth treatment of the latest experimental and theoretical topics in quantum sciences and technologies, including for example quantum sensing, quantum optics, cold atoms, th

This first part of the course covers non-equilibrium statistical processes and the treatment of fluctuation dissipation relations by Einstein, Boltzmann and Kubo. Moreover, the fundamentals of Markov

Observabilité et gouvernabilitéME-273: Introduction to control of dynamical systems

Explore l'observabilité, la gouvernabilité, les régulateurs étatiques et les formes canoniques dans les systèmes dynamiques.

Théorie des systèmes quantiques ouvertsPHYS-744: Advanced Topics in Quantum Sciences and Technologies

Couvre les cartes préservant les traces, les équations maîtresses et les opérateurs Lindblad dans la théorie des systèmes quantiques ouverts.

Dynamique efficace pour les équations différentielles stochastiques

Explore la dynamique efficace pour les équations différentielles stochastiques non réversibles, couvrant la dynamique moléculaire, les goulets d'étranglement, la graine grossière et la connexion à l'énergie libre.

The dynamics of open quantum systems is often modeled using master equations, which describe the expected outcome of an experiment (i.e., the average over many realizations of the same dynamics). Quantum trajectories, instead, model the outcome of ideal single experiments-the "clicks" of a perfect detector due to, e.g., spontaneous emission. The correct description of quantum jumps, which are related to random events characterizing a sudden change in the wave function of an open quantum system, is pivotal to the definition of quantum trajectories. In this article, we extend the formalism of quantum trajectories to open quantum systems with ultrastrong coupling (USC) between light and matter by properly defining jump operators in this regime. In such systems, exotic higher-order quantum-state and energy transfer can take place without conserving the total number of excitations in the system. The emitted field of such USC systems bears signatures of these higher-order processes, and significantly differs from similar processes at lower coupling strengths. Notably, the emission statistics must be taken at a single quantum trajectory level, since the signatures of these processes are washed out by the "averaging" of a master equation. We analyze the impact of the chosen unraveling (i.e., how one collects the output field of the system) for the quantum trajectories and show that these effects of the higher-order USC processes can be revealed in experiments by constructing histograms of detected quantum jumps. We illustrate these ideas by analyzing the excitation of two atoms by a single photon [Garziano et al., Phys. Rev. Lett. 117, 043601 (2016)]. For example, quantum trajectories reveal that keeping track of the quantum jumps from the atoms allows one to reconstruct both the oscillations between one photon and two atoms as well as emerging Rabi oscillations between the two atoms.

The exploration of open quantum many-body systems -systems of microscopic size exhibiting quantum coherence and interacting with their surrounding- has emerged as a key research area over the last years. The recent advances in controlling and preserving quantum coherence at the level of a single particle, developed in a wide variety of physical platforms, have been a major driving force in this field. The driven dissipative nature is a common characteristic of a wide class of modern experimental platforms in quantum science and technology, such as photonic systems, ultracold atoms, optomechanical systems, or superconducting circuits. The interplay between the coherent quantum dynamics and dissipation in open quantum systems leads to a wide range of novel out-of-equilibrium behaviours. Among them, the emergence in these systems of dynamical phases with novel broken symmetries, topological phases and the occurrence of dissipative phase transitions are of particular interest. This thesis aims at establishing a theoretical framework to engineer, characterize and control nonclassical states of light in photonic quantum optical networks in different regimes. The emphasis is put on its implementation, in particular with respect to integration and scalability in photonic platforms. In this thesis, we tackle some interesting aspects arising in the study of the dynamics of driven dissipative coupled nonlinear optical resonators. In that context, we consider the dynamics of two coupled nonlinear photonic cavities in the presence of inhomogeneous coherent driving and local dissipations using the Lindblad master equation formalism.We show that this simple open quantum many-body system can be subject to dynamical instabilities. In particular, our analysis shows that this system presents highly nonclassical properties and its dynamics exhibits dissipative Kerr solitons (DKSs), characterized by the robustness of its specific temporal or spatial waveform during propagation.In a second step, our intuition gained from this system composed of only few degrees of freedom is expanded to the study of systems of bigger size. In particular, we study DKSs originating from the parametric gain in Kerr microresonators. While DKSs are usually described using a classical mean-field approach, our work proposes a quantum-mechanical model formulated in terms of the truncated Wigner formalism. This analysis is motivated by the fact that technological implementations push towards the realization of DKSs in miniaturized integrated systems. These are operating at low power, a regime where quantum effects are expected to be relevant. Using the tools provided by the theory of open quantum systems, we propose a detailed investigation of the impact of quantum fluctuations on the spectral and dynamical properties of DKSs. We show that the quantum fluctuations arising from losses engender a finite lifetime to the soliton, and demonstrate that DKSs correspond to a specific class of dissipative time crystals.

The characterization of open quantum systems is a central and recurring problem for the development of quantum technologies. For time-independent systems, an (often unique) steady state describes the average physics once all the transient processes have faded out, but interesting quantum properties can emerge at intermediate timescales. Given a Lindblad master equation, these properties are encoded in the spectrum of the Liouvillian whose diagonalization, however, is a challenge even for small-size quantum systems. Here, we propose a new method to efficiently provide the Liouvillian spectral decomposition. We call this method an Arnoldi-Lindblad time evolution, because it exploits the algebraic properties of the Liouvillian superoperator to efficiently construct a basis for the Arnoldi iteration problem. The advantage of our method is double: (i) It provides a faster-than-the-clock method to efficiently obtain the steady state, meaning that it produces the steady state through time evolution shorter than needed for the system to reach stationarity. (ii) It retrieves the low-lying spectral properties of the Liouvillian with a minimal overhead, allowing to determine both which quantum properties emerge and for how long they can be observed in a system. This method is general and model-independent, and lends itself to the study of large systems where the determination of the Liouvillian spectrum can be numerically demanding but the time evolution of the density matrix is still doable. Our results can be extended to time evolution with a time-dependent Liouvillian. In particular, our method works for Floquet (i.e., periodically driven) systems, where it allows not only to construct the Floquet map for the slow-decaying processes, but also to re- trieve the stroboscopic steady state and the eigenspectrum of the Floquet map. Although the method can be applied to any Lindbla- dian evolution (spin, fermions, bosons, ... ), for the sake of simplicity we demonstrate the efficiency of our method on several examples of coupled bosonic resonators (as a particular example). Our method outperforms other di- agonalization techniques and retrieves the Li- ouvillian low-lying spectrum even for system sizes for which it would be impossible to per- form exact diagonalization.